quantum field theory of the standard model - …thorn/homepage/qflecturesla.pdf · quantum field...

Quantum Field Theory of the Standard Model

Charles B. Thorn1

Institute for Fundamental TheoryDepartment of Physics, University of Florida, Gainesville FL 32611

1E-mail address: [email protected]

Contents

1 Introduction 91.1 Problems with relativistic quantum mechanics . . . . . . . . . . . . . . . . . 91.2 Lorentz Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 The Free Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.1 Space as a Discrete Lattice . . . . . . . . . . . . . . . . . . . . . . . . 141.3.2 Energy Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . . 171.3.3 Multi particle States . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Interacting Scalar Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . 191.5 The Free Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.5.1 Quantum Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . 191.5.2 Energy momentum tensor . . . . . . . . . . . . . . . . . . . . . . . . 211.5.3 Polarization and Helicity of Photons. . . . . . . . . . . . . . . . . . . 21

1.6 Particles and “Particles” of the Standard Model . . . . . . . . . . . . . . . . 24

2 Representations of the Poincare Group for General Spin 25

3 The Dirac Equation 333.1 Single particle interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1.1 The Matrices γµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1.2 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.3 Explicit Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.4 The Dirac sea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.1 General Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.2 Second Quantized Dirac Equation . . . . . . . . . . . . . . . . . . . . 43

4 The Discrete Symmetries of the Dirac Equation 514.1 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Charge Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3 Majorana Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.4 Weyl Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.5 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.6 Violation of the Discrete Symmetries and the CPT Theorem . . . . . . . . . 59

3

5 Time Dependent Perturbation Theory 615.1 Heisenberg and Schrodinger Pictures . . . . . . . . . . . . . . . . . . . . . . 615.2 Asymptotic States and Matrix Elements . . . . . . . . . . . . . . . . . . . . 625.3 General Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.4 Scattering in an External Field: Born Approximation . . . . . . . . . . . . . 66

5.4.1 Scattering of scalar particles . . . . . . . . . . . . . . . . . . . . . . . 665.4.2 Scattering of Dirac particle . . . . . . . . . . . . . . . . . . . . . . . . 67

5.5 Pair production in a time varying external field . . . . . . . . . . . . . . . . 695.6 Perturbation theory for Time Ordered Products . . . . . . . . . . . . . . . . 705.7 A Technical Comment on Time Derivatives in Time Dependent Perturbation

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.8 Propagators for Scalar and Dirac Fields . . . . . . . . . . . . . . . . . . . . . 735.9 Vacuum expectations from large time limits of general transition amplitudes. 77

5.9.1 Adiabatic switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.9.2 Long time evolution suppression of excited states. . . . . . . . . . . . 78

6 Cross Sections and Rates and Spin Sums 816.1 Cross section for 2 → N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2 Decay rate of a single metastable particle . . . . . . . . . . . . . . . . . . . . 826.3 Spin Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.3.1 Gamma matrix identities . . . . . . . . . . . . . . . . . . . . . . . . . 856.4 Tree diagrams in momentum space . . . . . . . . . . . . . . . . . . . . . . . 85

6.4.1 Scattering amplitudes from time ordered products . . . . . . . . . . . 86

7 Quantum Field Equations with External Fields 877.1 Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.2 Nonabelian Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907.3 External Gravitational Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 927.4 Asymptotic States and Matrix Elements . . . . . . . . . . . . . . . . . . . . 93

8 External Field Perturbations 958.1 Connected Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968.2 Furry’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

9 Scattering in External Fields 1039.1 Relation to Time Ordered Products. . . . . . . . . . . . . . . . . . . . . . . 109

10 Vacuum Polarization 11310.1 Retarded Commutators from Time Ordered Products . . . . . . . . . . . . . 11410.2 Calculation of Vacuum Polarization . . . . . . . . . . . . . . . . . . . . . . . 11510.3 The Physics of Vacuum Polarization . . . . . . . . . . . . . . . . . . . . . . 11910.4 Charge Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

10.4.1 Pair production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122


10.5 Superconductivity and the Higgs Mechanism . . . . . . . . . . . . . . . . . . 125

11 Perturbation Theory for φ3 Scalar Field Theory 127

11.1 The Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

11.2 One-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

11.3 Two Point Function and the Physical Mass . . . . . . . . . . . . . . . . . . . 128

11.3.1 Reduction Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

11.4 Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

12 Path History Quantization 133

12.1 The Classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

12.2 Imaginary Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

12.3 Matrix Elements of Time Ordered Products. . . . . . . . . . . . . . . . . . . 136

12.4 Coordinate Space Path Integral. . . . . . . . . . . . . . . . . . . . . . . . . . 137

12.5 Gaussian Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

13 Path Integrals for Anticommuting Quantum Fields 141

14 Operator Quantization of the Electromagnetic Field 143

14.1 Quantized Electromagnetic Field Interacting with a Conserved Current . . . 143

14.1.1 Polarization and Helicity of Photons. . . . . . . . . . . . . . . . . . . 147

14.2 Charged Fields Interacting with the Quantized Electromagnetic Field . . . . 149

15 Path Integrals for Gauge Fields 153

15.1 General Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

16 Fields, Charges, and Masses of the Standard Model 161

16.1 Particles and “Particles” of the Standard Model . . . . . . . . . . . . . . . . 161

17 Feynman Rules for QED 163

17.1 Lagrangian (density) for QED . . . . . . . . . . . . . . . . . . . . . . . . . . 163

17.2 Rules in Coordinate Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

17.2.1 Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

17.2.2 Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

17.2.3 Rules for Calculating⟨ψ(x1) · · ·ψ(xn)ψ(yn) · · · ψ(y1)A(z1) · · ·A(zm)

⟩. 164

17.2.4 Rules for Scattering Amplitudes . . . . . . . . . . . . . . . . . . . . . 164

17.2.5 Cross Sections and Decay Rates . . . . . . . . . . . . . . . . . . . . . 165

17.3 Rules in Momentum Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

17.3.1 Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

17.3.2 Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

17.4 Divergence structure of Feynman diagrams . . . . . . . . . . . . . . . . . . . 168


18 Scattering Amplitudes in Quantum Field Theory 169

18.1 Multiparticle States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

18.2 Reduction Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

18.3 Single Particle States are Handled Consistently . . . . . . . . . . . . . . . . 173

18.4 Two Particle Scattering Amplitudes . . . . . . . . . . . . . . . . . . . . . . . 174

19 One Loop Corrections in QED 179

19.1 Photon self energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

19.2 Proper Vertex Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

19.2.1 UV divergences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

19.2.2 UV finite remainder . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

19.2.3 Anomalous magnetic moment of the electron . . . . . . . . . . . . . . 182

19.3 Self energy of the electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

19.4 Ward Identities in QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

19.4.1 Coordinate space derivation . . . . . . . . . . . . . . . . . . . . . . . 184

19.4.2 Ward Identities from Diagrams in momentum space . . . . . . . . . . 184

19.5 Charge and mass renormalization to 1 Loop . . . . . . . . . . . . . . . . . . 186

19.6 Atomic bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

20 Soft Bremsstrahlung and Infrared divergences 189

20.1 One Soft Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

21 Nonabelian Gauge Theory 195

21.1 Some gauge groups and their representations . . . . . . . . . . . . . . . . . . 197

21.1.1 The Adjoint Representation . . . . . . . . . . . . . . . . . . . . . . . 197

21.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

21.2 Path integrals for gauge theories and gauge fixing . . . . . . . . . . . . . . . 199

21.2.1 The Fadeev-Popov Determinant . . . . . . . . . . . . . . . . . . . . . 199

21.3 Feynman Rules for a Nonabelian Gauge Theory . . . . . . . . . . . . . . . . 200

21.4 Gauge Invariant Regulation Procedures . . . . . . . . . . . . . . . . . . . . . 201

21.4.1 Minimal subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

21.5 UV divergences in nonabelian gauge theory . . . . . . . . . . . . . . . . . . . 205

21.5.1 Fermion Gluon Vertex . . . . . . . . . . . . . . . . . . . . . . . . . . 206

21.5.2 FP ghost loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

21.5.3 Gluon loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

21.6 BRST Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

21.7 Gauge theory of the Standard Model . . . . . . . . . . . . . . . . . . . . . . 212

21.7.1 Gauge-fixing with Higgs mechanism . . . . . . . . . . . . . . . . . . . 216

21.7.2 Massive vector boson couplings . . . . . . . . . . . . . . . . . . . . . 217

21.7.3 Fermion masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218


22 Systematics of Renormalization 22122.1 Renormalized perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . 221

22.1.1 Two loop example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22222.2 Renormalization group of Gell-Mann and Low . . . . . . . . . . . . . . . . . 222

23 Renormalization and Short Distance Properties of QCD 22723.1 Scaling properties of Green functions in massless gauge theory. . . . . . . . . 22723.2 Calculation of β, γg, γq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

23.2.1 Non universality of β, α . . . . . . . . . . . . . . . . . . . . . . . . . 22923.2.2 The Nature of the Callan-Symanzik Equation . . . . . . . . . . . . . 229

23.3 High momentum behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23023.4 Composite Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

23.4.1 Interpreting naive perturbation theory in QCD . . . . . . . . . . . . 23723.5 Operator product expansion (OPE) . . . . . . . . . . . . . . . . . . . . . . . 238

24 Spontaneous Global Symmetry Breaking: Chiral Dynamics 24124.1 Effective Action and Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 241

24.1.1 Effective potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24324.1.2 Feynman diagrams for the effective potential . . . . . . . . . . . . . . 24424.1.3 Symmetries of the effective action . . . . . . . . . . . . . . . . . . . . 24624.1.4 Spontaneous Symmetry breaking . . . . . . . . . . . . . . . . . . . . 246

24.2 Goldstone Bosons and Soft Pion Theorems . . . . . . . . . . . . . . . . . . . 24824.2.1 Goldstone’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 24824.2.2 SSB and matrix elements of operators. . . . . . . . . . . . . . . . . . 24824.2.3 Approximate symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 25024.2.4 (Approximate) Symmetries of QCD . . . . . . . . . . . . . . . . . . . 25024.2.5 Hypothesis of chiral SSB . . . . . . . . . . . . . . . . . . . . . . . . . 25124.2.6 Masses of Nambu-Goldstone bosons in QCD. . . . . . . . . . . . . . . 25224.2.7 Low energy pion scattering . . . . . . . . . . . . . . . . . . . . . . . . 254

24.3 Adler-Weisberger and Other Sum Rules . . . . . . . . . . . . . . . . . . . . . 25724.3.1 Soft pion scattering off nucleons . . . . . . . . . . . . . . . . . . . . . 25724.3.2 Dispersion relation for πN scattering . . . . . . . . . . . . . . . . . . 26124.3.3 Adler-Weisberger sum rule . . . . . . . . . . . . . . . . . . . . . . . . 266

24.4 Effective Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26724.4.1 Loop effects from chiral Lagrangian . . . . . . . . . . . . . . . . . . . 26724.4.2 Baryons and SU(3)× SU(3). . . . . . . . . . . . . . . . . . . . . . . 267

24.5 Review of Standard Model Lagrangian . . . . . . . . . . . . . . . . . . . . . 270

25 Ward Identities, Ultraviolet Divergences and Gauge Invariance Anomalies27325.1 Ward Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27325.2 Ultraviolet Divergences and Gauge Invariance . . . . . . . . . . . . . . . . . 275

25.2.1 Gauge Invariant Regulation Procedures . . . . . . . . . . . . . . . . . 28225.3 Lattice Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284


25.4 Chiral Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28525.4.1 Ambiguities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28725.4.2 Physical Consequences of the Chiral Anomaly . . . . . . . . . . . . . 28825.4.3 Consequences of Anomalies that Don’t Cancel . . . . . . . . . . . . . 29325.4.4 Mathematical Consequences of the Anomaly: Index Theorems . . . . 295

26 Higgs Mechanism and Custodial Symmetry 29926.1 New Look at the Higgs Sector . . . . . . . . . . . . . . . . . . . . . . . . . . 29926.2 Higgs Mechanism Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 30126.3 Gauge-fixing the Standard Model v2 . . . . . . . . . . . . . . . . . . . . . . 303

27 Electroweak Interactions of Leptons and Quarks 30927.1 Vector Boson decay to Leptons and Quarks . . . . . . . . . . . . . . . . . . . 30927.2 Lepton or Quark Beta Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 31727.3 Scattering Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

28 Electroweak Interactions of Hadrons 32128.1 Hadron Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32128.2 Vector and Axial Symmetries and Electroweak Processes . . . . . . . . . . . 33328.3 Semi-Leptonic Hadronic Decay Processes . . . . . . . . . . . . . . . . . . . . 33928.4 Electroweak Processes involving NGB’s . . . . . . . . . . . . . . . . . . . . . 34328.5 θ vacua and the Strong CP problem . . . . . . . . . . . . . . . . . . . . . . . 362

29 Quark Confinement 36929.1 String Model of Hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36929.2 Lattice Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

30 Physics at High energy and Low Momentum transfer 371

31 Beyond the Standard Model 373


Chapter 1

Introduction

1.1 Problems with relativistic quantum mechanics

One of the most basic facts about relativistic physics is the equivalence of mass and energy, asin Einstein’s famous relation E = mc2. This fact has profound consequences for relativisticquantum mechanics, because of Heisenberg’s uncertainty principle

∆x∆p ≥ ~. (1.1)

In ordinary nonrelativistic quantum mechanics this follows from the fact that the position andmomentum are represented by operators that do not commute: [x, p] = i~. According to (1.1)an accurate position measurement can be made only at the expense of a compensatingly largeinaccuracy in the momentum of the particle. This implies a correspondingly large expenseof energy in the accurate measurement of position. In nonrelativistic quantum mechanicsthe rest energy of a massive particle is effectively infinite so one can subject a particle toarbitrarily large energy changes without compromising its integrity as a particle, and accurateposition measurements are compatible with particle number conservation. The concept of anN particle wave function as a function of the N particle coordinates is sustainable. This isdrastically changed in relativistic quantum mechanics. As soon as one attempts to measurethe position of a particle to an accuracy much less than the particle’s Compton wavelength,~/mc, the associated momentum uncertainty is > mc, corresponding to an energy changelarger than mc2. Thus accurate position measurements require the supply of energy sufficientto produce additional particles. If the particle possesses a conserved charge, additional suchparticles can be produced but only in association with particles of opposite charge. Thus weshould expect any consistent version of relativistic quantum mechanics to require abandoningthe concept of a wave function for a system with a definite number of particles. Further, inretrospect, it is not surprising that when there are conserved charges, relativistic quantumphysics requires antiparticles.

The preceding is not meant to be a rigorous argument, but rather a simple explanationof why we must expect at least the complexities of quantum field theory when we attempt toextend quantum mechanics into the relativistic domain. The many-body aspects of relativis-tic quantum mechanics are universal, but the manner in which they make their appearance

9

varies. In quantum electrodynamics it is immediate once one tries to interpret classical radi-ation in terms of photons. For massive particles, like the electron, it arises from the existenceof negative energy solutions of the Dirac equation.

The first semester of quantum field theory will focus primarily on quantum electrody-namics, the theory of electrons and positrons interacting with the quantized electromagneticfield. We shall begin however with a much simpler quantum field theory– that of a selfinteracting scalar field. Then we shall show how Dirac’s theory of electrons and positronscan be converted to a quantum field theory through the device of “second quantization”. Weshall study this Dirac quantum field theory first in the presence of external (classical) elec-tromagnetic fields. Only then will we turn to the problem of quantizing the electromagneticfield and studying it in interaction with electrons and positrons.

1.2 Lorentz Invariance

In elementary discussions of special relativity we learn that frames of reference with a con-stant relative velocity V have their coordinates related by the Lorentz “boost”

x′ = γ(x+ V t), y′ = y, z′ = z, t′ = γ(t+ V x) (1.2)

where γ = 1/√

1− V 2. But more generally the boost could be in any direction specified

by a three vector ~V . in addition to boost invariance, we also require rotational invariance,a symmetry we are very familiar with in nonrelativistic mechanics. We define a generalLorentz transformation of space-time coordinates xµ = (t, x, y, z) ≡ (x0, x1, x2, x3) as alinear transformation

xµ → x′µ = Λµνx

ν (1.3)

where Λ preserves Minkowski scalar products v ·w = v1w1 + v2w2 + v3w3− v0w0 ≡ vµvνηµν .(η11 = η22 = η33 = −η00 = 1, ηµν = 0 for µ 6= ν. Note that we use the convention thatrepeated indices are summed. This requirement implies the following constraints on Λ:

ηρσΛρµΛσ

ν = ηµν . (1.4)

You should convince yourself that the special boost in the x direction does indeed preserveMinkowski scalar products. The Poincare group consists of Lorentz transformations togetherwith translations

xµ → Λµνx

ν + aµ. (1.5)

According to special relativity the laws of physics should look the same in all frames relatedby Lorentz transformations. The systematic way to implement this requirement is to identifyall physical quantities as the components of 4 tensors Aµν··· which transform as

A′µν··· = ΛµρΛ

νσ · · ·Aρσ··· (1.6)


For example energy and momentum are the components of a four vector pµ = (E, ~p) whichtransform just like the coordinates xµ.

pµ → p′µ = Λµνpν (1.7)

Tensors written with index superscripts are called contravariant tensors. It is also useful tointroduce covariant tensors Aµν···, written with index subscripts. A covariant index trans-forms like ∂/∂xµ:

∂

∂x′µ=∂xρ

∂x′µ∂

∂xρ=

∂

∂xρ(Λ−1)ρµ =

∂

∂xρηρκΛσ

κησµ ≡ Λ ρµ

∂

∂xρ(1.8)

A general tensor can have any number of upper and lower indices. the metric tensor is usedto raise and lower indices.

The Λ’s can be divided into 4 disjoint sets according to the signs of det Λ and Λ00. This

is because it is easy to show from the above property that (det Λ)2 = 1 and (Λ00)2 ≥ 1. Thus

a continuous variation of Λ always stays within one of these sets. In the following we restrictourselves to the proper Lorentz Group, i.e. with det Λ = +1 and Λ0

0 ≥ +1. The completeLorentz group is then obtained by adjoining parity and time reversal.

Lorentz transformations with Λ0k = Λk

0 = 0 are simply rotations and form a subgroup.We know from basic quantum mechanics all the unitary irreducible representations of theRotation group, namely those labeled by angular momentum j = 0, 1/2, 1, 3/2, . . .. This con-clusion is reached by considering infinitesimal rotations which are generated by the angularmomentum ~J , U(R) = e−iθu·

~J ≈ 1− iθu · ~J . The commutators of Jk are well-known:

[Jk, J l] = i~εklmJm (1.9)

and the |jm〉 basis of eigenstates of ~J2, J3 is constructed in the familiar way. The unitaryrepresentations of the Lorentz group must be extensions of these, U(Λ) = U(R) when Λ0

k =Λk

0 = 0. To find the generators of boosts, consider the boost in the x direction of themomentum of a particle of mass m:

p′1 = γ(p1 + V√~p2 +m2) ≈ p1 + V

√~p2 +m2 (1.10)

for infinitesimal V . This should be identified with p1 − iV [p1, K1], from which we infer

K1 = −(x1√~p2 +m2 +

√~p2 +m2x1)/2 + F (~p) (1.11)

. The symmetrized product in the first term is to keep K hermitian. To find F we considerthe transform of the coordinate

x′1(t′) ≈ x1(t) + V t ≈ x1(t′)− V (x1x1(t) + x1(t)x1)/2 + V t (1.12)

But

−iV [x1, K1] = −1

2

(x1 p1√

~p2 +m2+

p1√~p2 +m2

x1

)+∂F

∂p1= −1

2

(x1x1 + x1x1

)+∂F

∂p1(1.13)


Thus we are led to

K1 = −(x1√~p2 +m2 +

√~p2 +m2x1)/2 + p1t (1.14)

We easily see that K1 is constant in time. Boosts in all three directions are generated by

Kk = −(xk√~p2 +m2 +

√~p2 +m2xk)/2 + pkt (1.15)

It is now a straightforward exercise to complete the commutator algebra of Lorentz genera-tors:

[Jk, J l] = i~εklmJm, [Kk, J l] = i~εklmKm, [Kk, K l] = −i~εklmJm (1.16)

An alternative more covariant presentation of the Lorentz algebra is to define Mij = εijkJk,

M0i = Ki, and find

[Mµν ,Mρσ] = i (ηµρMνσ − ηνρMµσ − ηµσMνρ + ηνσMµρ) (1.17)

where now and henceforth we will choose units so that ~ = c = 1.In classical electromagnetic theory we learn that Maxwell’s equations are covariant under

Lorentz transformations of the electromagnetic field:

F ′µν(x′) = ΛµρΛ

νσF

ρσ(x) = ΛµρΛ

νσF

ρσ(Λ−1x′) (1.18)

Since F µν(x) is a field with a different value at each point, notice that two changes are goingon simultaneously: the components of F are mixed and the old and new fields are comparedat different space-time coordinates x′, x related by x′ = Λx. We are going to consider at thebeginning a scalar field φ(x) with only one component. The scalar field then has the muchsimpler Lorentz transformation

φ′(x′) = φ(Λ−1x′) (1.19)

The simplest Lorentz covariant scalar field equation is the Klein-Gordon equation

−ηµν ∂2

∂xµ∂xνφ+ µ2φ = 0 (1.20)

To show Lorentz covariance we want to prove

−ηµν ∂2

∂x′µ∂x′νφ(Λ−1x′) + µ2φ(Λ−1x′) = 0. (1.21)

this follows through the chain rule

ηµν∂2

∂x′µ∂x′ν= ηµν(Λ−1)ρµ(Λ−1)σν

∂2

∂xρ∂xσ= ηρσ

∂2

∂xρ∂xσ(1.22)


1.3 The Free Scalar Field

The field concept goes all the way back to Faraday in the first part of the nineteenth century.It was of course indispensable to a proper understanding of electromagnetism which hepioneered. This first field theory is actually quite complex involving three components foreach of the electric and magnetic fields. For our first look at quantum field theory we considerthe much simpler case of the single component scalar field.

A field is a dynamical variable φ(x, t) assigned to each point of space. One is thereforepositing from the beginning an infinite number of degrees of freedom. The simplest rela-tivistic equation of motion for such a field is a linear wave equation (henceforth we shall useunits in which the speed of light c = 1):

(∂2

∂t2−∇2 + µ2)φ ≡ (−∂2 + µ2)φ = 0. (1.23)

The parameter µ clearly has dimensions of 1/Length, and represents an inverse wavelength(or wave number) rather than a mass. After quantization, we shall see that the particlesassociated with the quantum field have mass ~µ. We are of course very familiar with thenature of the solutions to these equations: general superpositions of plane waves, called wavepackets:

φ(x, t) =

∫d3kf(k)eik·x−iω(k)t (1.24)

≈∫d3kf(k)eik·x−iω(k0)t−i(k−k0)·∇kω|k0 (1.25)

≈ e−iω(k0)t+ik0·vgtφ(x− vgt, 0) (1.26)

where ω(k) =√

k2 + µ2, moving with group velocity vg = k/√

k2 + µ2.

Notice that the wave equation is invariant in form under Lorentz transformations φ′(x′) =φ(x) where x′

µ= Λµ

νxν and Λ is a Lorentz transformation satisfying ηρσΛρ

µΛσν = ηµν . This

condition on Λ is just the requirement that scalar products of four vectors v ·w ≡ ηµνvµwν =

~v · ~w− v0w0 are invariant. One easily can see that ∂/∂xµ transforms as a four-vector so that∂2 = ∂µ∂νηµν is invariant.

To discuss the quantum mechanics of this field we must identify the canonical variables ofthe system. Since the equation of motion is of second order we can easily guess that the mo-mentum conjugate to φ(x) is just π(x) = φ. A more systematic approach is to find an actionprinciple which implies the field equation. Recall that for dynamical systems with discretelylabeled degrees of freedom qi(t) one forms the action from the Lagrangian L(q(t), q(t), t)via S =

∫ t2t1dtL. Hamilton’s principle states that the equations of motion follow from the

requirement that S is stationary under variations δqi(t) with δqi(t1) = δqi(t2) = 0. By defi-nition the momentum conjugate to qi is pi ≡ ∂L/∂qi. For our field the spatial coordinates xplay the role of the index i, and the Lagrangian will be an integral over spatial coordinates.


It is easy to see that Hamilton’s principle applied to the action

S =

∫ t2

t1

dt

∫d3x

1

2(φ2 − (∇φ)2 − µ2φ2) ≡

∫d4xL (1.27)

L = −1

2[ηµν∂µφ∂νφ+ µ2φ2] (1.28)

implies the Klein-Gordon equation. The second form shows that the Lagrange density L isa Lorentz covariant scalar field, so its integral is a Lorentz invariant.

From this action, it is evident that the above guess for π ≡ ∂L/∂φ = φ is correct. Thecanonical Hamiltonian is

H ≡∫d3x[φπ − L] =

∫d3x

1

2(π2 + (∇φ)2 + µ2φ2). (1.29)

One can easily check that Hamilton’s equations obtained from this Hamiltonian give thatsame old wave equation. To quantize canonically, we promote φ, π to operators with thecanonical commutation relations

[φ(x), π(y)] = i~δ(x− y). (1.30)

In the language of Hamiltonian mechanics with canonical variables qi, pi, φ is analogous to qand π to p, and the spatial coordinate to the index i that distinguishes independent degreesof freedom.

1.3.1 Space as a Discrete Lattice

To find the energy eigenstates, we note the close resemblance of our Hamiltonian to thatof a system of coupled harmonic oscillators. The only thing a little strange is that theoscillator coordinates are labeled by a continuous index. This can be remedied by replacingspace by a lattice an of spacing a which we send to zero after finding the eigenstates. Thuswe replace φ(an) with φn and π(an) by πn/a

3. We divide by a3 so that [φn, πm] = i~δn,mDoing this the Hamiltonian becomes

Hlattice =a3

2

∑n

1

a6π2n + µ2φ2

n +1

a2

∑i

(φn+i − φn)2

. (1.31)

On the lattice our system is a system of coupled oscillators. To solve it, we just have tofind the normal modes. This is done by a change of variables which renders diagonal thecoupling matrix

Vn,m = (µ2 +2d

a2)δn,m −

1

a2

∑i

(δn+i,m + δn,m+i), (1.32)


i.e. we want to find the eigenvectors and eigenvalues of V . It is easy to see that theeigenvectors are um = eiK·m, −π < Ki < π, belonging to eigenvalue

λ ≡ ω2 = µ2 +2

a2(d−

∑i

cos K · i). (1.33)

Thus the transformation to normal modes is given by

φn =

∫ π

−π

d3K

(2π)3/2eiK·nQ(K) (1.34)

πn =

∫ π

−π

d3K

(2π)3/2eiK·nP (K) (1.35)

or inversely by

Q(K) =∑n

1

(2π)3/2e−iK·nφn (1.36)

P (K) =∑n

1

(2π)3/2e−iK·nπn. (1.37)

The inverse equations directly imply the commutation relations

[Q(K), P (L)] = i~δ(K + L). (1.38)

Note also that if φ and π are hermitian, as we assume, then we have Q(K)† = Q(−K) andP (K)† = P (−K).

Expressed in terms of normal modes the lattice Hamiltonian becomes

Hlattice =

∫ π

−πd3K

(1

2a3P (K)P (−K) +

a3

2ω2(K)Q(K)Q(−K)

). (1.39)

In the standard fashion raising and lowering operators can be constructed in terms of which

Q(K) =

√~

2ω(K)a3(A(K) + A†(−K)) (1.40)

P (K) = −i√

~ω(K)a3

2(A(K)− A†(−K)) (1.41)

with [A(K), A†(L)] = δ(K− L), and the Hamiltonian becomes the familiar

Hlattice =

∫ π

−πd3K~ω(K)

1

2(A(K)A†(K) + A†(K)A(K)). (1.42)

A† and A are of course eigenoperators of HLattice with eigenvalues ±ω(K). Now we candescribe the exact energy eigenstates of the system. The ground state (the vacuum) |0〉 is


annihilated by all the A’s: A(K)|0〉 = 0, and its energy E0 = (~/2)δ(0)∫d3Kω(K). We

shall measure all energies relative to E0, i.e. from now on we take E0 = 0. This means ourenergy operator is redefined to be

H − E0 =

∫ π

−πd3K~ω(K)A†(K)A(K) (1.43)

The excited states are obtained by applying any number of A†’s to the vacuum. The simplestone is A†(K)|0〉 with energy

E(K) = ~√µ2 +

2

a2(d−

∑i

cos K · i). (1.44)

At this point we can consider the continuum limit a → 0. All excited states with finite Kwill have infinite energy and be dynamically irrelevant. However since the components of Kare continuous, we may consider states with K = ak. The energy of these states becomes~√µ2 + k2 independent of a in the continuum limit. We identify this energy as that of a

relativistic particle of mass ~µ and momentum ~k.We can pass to the continuum limit in all of our results by changing variables from K

to k whose components range from −π/a to π/a which become −∞ to ∞ as a → 0. Atthe same time we replace A(K) ≡ a(k)/a3/2 so that [a(k), a†(q)] = δ(k − q). After thesechanges we find as a→ 0:

φn(t)→ φ(x, t) =

∫d3k

(2π)3/2

√~

2ω(k)(a(k)eik·x−iω(k)t + a†(k)e−ik·x+iω(k)t) (1.45)

πna3→ π(x) = −i

∫d3k

(2π)3/2

√~ω(k)

2(a(k)eik·x−iω(k)t − a†(k)e−ik·x+iω(k)t)

= φ (1.46)

H − E0 → Hφ − E0 =

∫d3k~ω(k)a†(k)a(k) (1.47)

where now ω(k) =√µ2 + k2, and we have given the time dependence of the fields in

Heisenberg picture. It should of course be clear that we really never needed the lattice inobtaining these results: we simply had to substitute the above expansions for φ, π into Hφ.The lattice only served to make absolutely concrete our assertion that the field system wasa set of coupled oscillators.

We should also note that the momentum operator of the continuum field theory, identifiedas the generator of translations, is

P = −∫d3xπ(x)∇φ(x) =

∫d3k~ka†(k)a(k) (1.48)

confirming the interpretation of ~k as the momentum of the one particle state a†(k)|0〉.


1.3.2 Energy Momentum Tensor

As we have seen the energy and momentum of a quantum field system are integrals over allspace of corresponding densities. This is a little like expressing charge as the integral of acharge density. As we know the charge density is the time component of a locally conserved4-vector current density field Jµ(x), ∂µJ

µ = 0. Similarly the densities of energy momentumare time components of a locally conserved 4-tensor field T µν(x), ∂νT

µν = 0:

P µ =

∫d3xT µ0(x, t), T i0 = −∂0φ∇iφ, T 00 =

1

2

(φ2 + (∇φ)2 + µ2φ2

)(1.49)

You can confirm by inspection that these and the remaining components of T µν are given by

Tµν = ∂µφ∂νφ+ ηµνL (1.50)

where L is the Lagrange density

L = −1

2

((∂φ)2 + µ2φ2

)(1.51)

You can think of this as a covariant analogue of the Legendre transformation which definesthe Hamiltonian in terms of the Lagrangian. You can easily show that ∂νT

µν = 0, by virtueof the Klein-Gordon field equations.

From the energy momentum tensor we can construct a conserved third rank tensor,Mµνρ ≡ xµT νρ − xνT µρ:

∂ρMµνρ = T νµ − T µν + xµ∂ρTνρ − xν∂ρT µρ (1.52)

since Tµν = Tνµ and T µν is locally conserved. Then Mµν ≡∫d3xMµν0 are conserved

quantities which can be identified with the generators of the Lorentz group:

1.3.3 Multi particle States

If we apply two a†’s to the vacuum we get a two particle state

a†(k1)a†(k2)|0〉 (1.53)

with total momentum ~(k1 +k2) and total energy ~(ω(k1) +ω(k2)). Clearly this energeticsis that of noninteracting particles. It is highly significant that all multiparticle states arecompletely symmetric under interchange of the labels of any pair of particles: the scalar fieldtheory predicts Bose statistics for the associated particles!

The example of the scalar field illustrates the main physical aspects of quantum fieldtheory

1. It predicts multi-particle states together with their statistics (Bose or Fermi).

2. It incorporates the requirements of Special Relativity (Poincare invariance).


3. One can consider two classical limits (~→ 0):

With µ fixed the limit gives a classical field theory. In this case the Comptonwavelength 1/µ is finite in the classical limit.

With m = ~µ fixed it gives a classical theory of point particles (the Comptonwavelength vanishes).

Thus quantum field theory unites the particle and field concepts.

4. Quantum field theories with linear field equations predict noninteracting (free) parti-cles. Interactions will arise if the field equations have nonlinear terms. Such terms areassociated with terms in the action or Hamiltonian with three or more powers of fields.The presence of such terms implies that particle number is not conserved.

5. The scalar field describes spinless particles; more general fields e.g. spinor, vector,tensor describe particles of higher spin.

Finally, let us consider how an approximate quantum particle interpretation can be re-trieved from this quantum field theory. Remember the observables are quantum fields, notparticle coordinates! To discover properties of a single particle wave packet,

|f〉 ≡∫d3kf(k)a†(k)|0〉, (1.54)

we must consider what we can get from measurements of the quantum field

φ(x, t) =

∫d3k

(2π)3/2

√~

2ω(k)(a(k)eik·x−iωt + a†(k)e−ik·x+iωt). (1.55)

First it is easily shown that 〈f |φ(x, t)|f〉 = 0. That is, if we make many measurements ofφ, we find zero on average. This means nothing more than φ is negative as often as it ispositive. A more sophisticated measurement is to measure φ(x, t)φ(y, t), i.e. the product ofthe results of simultaneous measurement of the fields at x and at y. If no particle is present,the average of many repeats of this measurement is

〈0|φ(x, t)φ(y, t)|0〉 =1

2

∫d3k

(2π)3

~ω(k)

eik·(x−y). (1.56)

Note that due to the 1/ω(k), this is not 0 even when the fields are not measured at the samepoint. This implies a correlation between the measurements at separate points, but doesnot contradict causality: [φ(x, t), φ(y, t)] = 0! The vacuum is an energy eigenstate whichrequires an infinite time to set up. Thus there is plenty of time to set up correlations atdistant points.

Now, suppose a particle is present in the wave packet f , and the same quantity is mea-sured.

〈f |φ(x, t)φ(y, t)|f〉 = 〈0|φ(x, t)φ(y, t)|0〉+ ψ∗(x, t)ψ(y, t) + ψ∗(y, t)ψ(x, t). (1.57)


The presence of the particle causes the change in average results given by the last 2 termson the r.h.s. which are modulated by the function

ψ(x, t) ≡∫

d3k

(2π)3/2

√~

2ω(k)eik·x−iωtf(k) (1.58)

≈ ψ(x− vgt, 0) (1.59)

≈

√~

2ω(k0)ψS(x− vgt, 0). (1.60)

The last two forms hold if the packet function f is narrowly peaked about k0. ψS is theone particle Schrodinger wave function corresponding to momentum wave function f . vg ≡∂ω/∂k|k=k0 is the usual group velocity. The disturbance is nonzero only when the point isin the support of the wave function.

1.4 Interacting Scalar Field Theory

So far we have only considered linear field equations, whose associated Lagrangians arequadratic functions of the the field. As soon as the field equations become non-linear, theparticles associated with the quantum fields will interact with each other. In particular theycan scatter.

The simplest way to introduce non-linear terms in the field equations, while preservingLorentz invariance, is to add Lorentz scalar terms cubic and higher in the fields to theLagrangian density. for scalar fields this is easy to do. For example any term

−gnn!φn(x), for n > 2 (1.61)

is a scalar which produces a term gnφn−1/(n− 1)! in the field equation. Since such terms do

not involve time derivatives, they do not alter the definition of the conjugate momentum northe commutation relations. But clearly plane waves no longer solve the new field equationsso there will be interactions.

Nonlinear quantum field theory is too complicated to solve exactly, so we shall firstassume the nonlinear terms are small so they can be handled using perturbation theory. Inthe next chapter we review the formalism of time dependent perturbation theory.

1.5 The Free Electromagnetic Field

1.5.1 Quantum Electromagnetic Field

Treated classically the e.m. field Fµν = ∂µAν − ∂νAµ satisfies Maxwell’s equations

∂νFµν = Jµ, (1.62)


which imply current conservation ∂µJµ = 0 for consistency.

The source free (jµ = 0) Maxwell’s equations follow from stationarity of the action

S = −1

4

∫d4xFµνF

µν =1

2

∫d4x(E2 −B2), (1.63)

under Aµ → Aµ + δAµ. Then the momentum conjugate to Aµ is

Πµ = F µ0 = ∂0A

µ − ∂µA0. (1.64)

The spatial components of Π are just those of minus the electric field strength Π = −E, butΠ0 = 0. This last fact poses a difficulty for quantization since it is inconsistent with nonva-nishing canonical commutation relations. We know how to assign operator properties to Aand Π, but not to A0. Before facing this difficulty, we construct the canonical Hamiltonian

HCAN =

∫d3x[Π · A− L] (1.65)

=

∫d3x

[1

2Π2 +

1

2(∇×A)2 +∇A0 ·Π

]. (1.66)

Notice that the troublesome variable A0 appears only linearly and in the last term. After anintegration by parts the coefficient is just −∇·Π, which would vanish if we could use Gauss’Law ∇·E = 0. Dirac pioneered a way to quantize systems in the presence of such constraintson the phase space, which we shall return to later. However, here we use a more heuristicmethod, which is to use the constraints to eliminate physically redundant field components.This is called “fixing the gauge”, and is motivated by gauge invariance: the field strengthsare unchanged under the transformation Aµ → Aµ + ∂µΛ where Λ is an arbitrary functionof space and time.

To fix the gauge ambiguity one specifies a gauge condition, here we take Coulomb gauge∇ ·A = 0. Then Gauss’ law simplifies to

∇ ·E = −∇ ·Π = −∇2A0 = J0 (1.67)

which is solved for A0 by setting

A0(x, t) = − 1

∇2J0(x, t) =

∫d3y

J0(y, t)

4π|x− y|→ 0, for Jµ = 0. (1.68)

The longitudinal component of Π is also eliminated because ∇ ·Π = ∇2A0. One passes toquantum mechanics by promoting only A = AT and ΠT to operators. (If the currents areoperators, A0 is an operator by virtue of the constraint, but it is not independent.) Then thetransverse projector must appear on the right side. of the canonical commutation relations

[Ak(x),Πl(y)] = i

(δkl −

∇k∇l

∇2

)δ(x− y). (1.69)

The subscript T is to rmind us that Π and A have zero divergence.


1.5.2 Energy momentum tensor

From classical electrodynamics a gauge invariant and symmetric energy momentum tensorfor the free electromagnetic field can be taken to be

T µν = F µρF

νρ − ηµν 1

4FρσF

ρσ (1.70)

It is conserved by virtue of Maxwell’s equations

∂νTµν = (∂νF

µρ)F

νρ + F µρ∂νF

νρ − 1

2Fρσ∂

µF ρσ

= (∂µ(∂νAρ)− ∂ν∂ρAµ)F νρ − 1

2Fρσ∂

µF ρσ

= (∂µ∂νAρ)Fνρ − 1

2Fρσ∂

µF ρσ = 0 (1.71)

the second line uses Maxwell’s equations, and the third line exploits the antisymmetry ofF ρσ.

The energy and momentum densities are

T 00 = E2 +1

2(B2 −E2) =

1

2(B2 +E2) (1.72)

T i0 = F ikF

0k = εikjBjEk = (E ×B)i (1.73)

Since T µν is symmetric we can immediately write down the Lorentz generators

M0k =

∫d3x(x0T k0 − xkT 00), Mkl =

∫d3x(xkT l0 − xlT k0) (1.74)

We shall need this information in the next section.

1.5.3 Polarization and Helicity of Photons.

An explicit realization of the commutation relations can be given in terms of creation andannihilation operators as follows:

ATk(x, 0) =

∫d3k√

(2π)32|k|[ak(k)eik·x + a†k(k)e−ik·x] (1.75)

ΠTk(x, 0) = −i∫

d3k√|k|√

(2π)32[ak(k)eik·x − a†k(k)e−ik·x]. (1.76)

The Coulomb gauge condition ∇ ·A = Π = 0 implies that k · a = 0, so the commutationrelations for the a’s read:

[ak(k), a†m(q)] =

(δkm −

kkkmk2

)δ(k − q). (1.77)


Inserting these into Heff gives

Heff +H − E0 =

∫d3k|k|a†(k) · a(k), (1.78)

where E0 is the usual (infinite) zero point energy of the oscillators which will be droppedfrom now on. This formula shows us immediately that for Jµ = 0, the quantum e.m. fieldis interpretable as a system of massless bosons (photons). The vacuum |0〉 is defined byak(k)|0〉 = 0 and the n photon state is represented by

a†m1(q1)a†m2

(q2) · · · a†mn(qn)|0〉. (1.79)

Because of transversality there are two photon states for each momentum. These two polar-ization states will next be shown to correspond to the two helicities ±1 of the photon. Thehelicity acting on a single particle momentum eigenstate is defined by

h|k〉 =k · J|k||k〉. (1.80)

The orbital angular momentum part of J does not contribute to the helicity because p · (x×p) = 0. Thus the helicity of a spinless particle is zero.

First, for fixed k, let us introduce two (in general complex) basis vectors εa, a = 1, 2 forthe plane perpendicular to k, satisfying k · εa = 0 and the orthonormality and completenessrelations

εa · ε∗b = δab (1.81)∑a

εma · εn∗a = δmn −kmkn

k2. (1.82)

We can then introduce two independent sets of creation and annihilation operators via

a(k) =∑a

εaaa(k). (1.83)

We shall relate the multiplicity associated with the index a to the spin of the photon. Firstrecall the classical expression for the angular momentum carried by the e.m. field,

J =

∫d3xx× (E ×B) (1.84)

=

∫d3x

∑k

Ek(x×∇)Ak −∫d3xx× (E ·∇)A. (1.85)

We can recognize the first term in the last line as the “orbital” angular momentum, whichwill not contribute to the helicity of a one photon state. This is because acting on a one


photon state the ∇ is replaced by k and because of the cross product the term will beperpendicular to k. The second term, after an integration by parts becomes

S =

∫d3xE ×A = −

∫d3xA×A (1.86)

= −i∫d3ka†(k)× a(k). (1.87)

Applying S to a one photon state a†a(k)|0〉, yields

Sa†a(k)|0〉 = i∑b

(εa × ε∗b)a†b(k)|0〉. (1.88)

Thus we see that the 2× 2 matrix Sab = iεa × ε∗b acts as a spin matrix on the index of thecreation operator. Note that the transverse components of S are zero! To get the helicityinterpretation, consider the case of k = kz. Then the helicity matrix is

S3ab = i(ε1aε

2∗b − ε2aε1∗b ). (1.89)

This matrix is diag1,−1 with the choices

ε1 = (1, i, 0)/√

2 ε2 = (1,−i, 0)/√

2, (1.90)

so with this choice of polarization vectors, a†1 creates a photon with helicity +1 and a†2 createsa photon with helicity −1. This establishes that the photon is a spin one particle. There isno zero helicity state for the photon: this is consistent with Poincare invariance because thephoton is massless, and the helicity of a massless particle is Poincare invariant.

Notice that the other components of photon spin vanish S1 = S2 = 0 for this momentum.This means that the three matrices S1, S2, S3 do not satisfy the Lie algebra of O(3). Instead[Sk, Sl] = 0. The latter algebra holds for general k. Indeed we can write S = kh. Thesepeculiar properties of photon spin are due to the photon’s masslessness, the photon cannotbe brought to rest. The spin of a massive particle, which can be brought to rest, mustgenerate O(3). In particular a massive spin 1 particle must have all three helicities ±1, 0.

The polarization vector enters scattering amplitudes multilinearly, with a factor of ε foreach incoming photon and a factor ε∗ for each outgoing photon. Its four-vector index forms aMinkowski scalar product with that of the vertex coupling the gauge potential to the chargedfields. According to gauge invariance, this vertex satisfies current conservation: its scalarproduct with the momentum entering it gives zero. Thus changing each polarization vectorby an amount proportional to its four-momentum leaves the scattering amplitude unaltered.In Coulomb gauge the polarization vector is of the form ε = (ε, 0) with k ·ε = 0, so kµε

µ = 0.But kµk

µ = 0 since the photon is massless. Thus we can characterize the polarizationvector completely by the covariant condition kµε

µ = 0. Any further specification, e.g. ε0 =0, is merely a gauge choice which can be made at will and exploited to simplify detailedcalculations. This is particularly advantageous in the calculation of Compton scattering forpolarized photons.


1.6 Particles and “Particles” of the Standard Model

LeptonsParticle Charge Spin Masse± ±e 1/2 0.511 MeVµ± ±e 1/2 106 MeVτ± ±e 1/2 1.78 GeVν1 0 1/2 m1

ν2 0 1/2 [m21 + 8× 10−5]1/2eV

ν3 0 1/2 [m21 ± 2× 10−3]1/2eV

Electro-weak Bosonsγ 0 1 0W± ±e 1 80 GeVZ 0 1 91 GeVh 0 0 125 GeV

Quarks“Particle” Charge Spin “Mass”d (×3) −e/3 1/2 “4.7” MeVs (×3) −e/3 1/2 “94” MeVb (×3) −e/3 1/2 “4.2” GeVu (×3) 2e/3 1/2 “2.2” MeVc (×3) 2e/3 1/2 “1.3” GeVt (×3) 2e/3 1/2 “173” GeV

Strong BosonsG (×8) 0 1 “0”

The preceding table lists all of the particles in the standard model for which fundamentalfields are introduced. We have listed quarks and gluons as “particles” because none of themcan exist in isolation: they are trapped (confined) inside of hadrons. Quarks always comein groups of three or quark antiquark pairs. The gluons come in groups of two or more.Because of confinement the notion of mass is not precise and hence we put the mass valuesin quotes. They are theoretical parameters that roughly correspond to our heuristic notionsof mass.


Chapter 2

Representations of the PoincareGroup for General Spin

We have so far encountered via simple quantum field theories the relativistic quantum de-scription of free particles of spin 0 or spin 1/2. It is useful at this point to realize that muchof what we have found is not really tied to field theory but rather to simple requirements ofPoincare invariance. In quantum mechanics it is a general fact that a symmetry group mustbe realized by a unitary or antiunitary representation. The latter possibility only occursfor some discrete symmetries (time reversal being the physical example). Our goal in thischapter is to obtain the unitary realization of the Poincare group for multiparticle states.

The Poincare group consists of Lorentz transformations together with translations

xµ → Λµνx

ν + aµ. (2.1)

where Λ preserves Minkowski scalar products

ηρσΛρµΛσ

ν = ηµν . (2.2)

The Λ’s can be divided into 4 disjoint sets according to the signs of det Λ and Λ00. This is

because it is easy to show from the above property that (det Λ)2 = 1 and (Λ00)2 ≥ 1. Thus a

continuous variation of Λ always stays within one of these sets. In the following we restrictourselves to the proper Lorentz Group, i.e. with det Λ = +1 and Λ0

0 ≥ +1. The completeLorentz group is then obtained by adjoining parity and time reversal.

Lorentz transformations with Λ0k = Λk

0 = 0 are simply rotations and form a subgroup.We know from basic quantum mechanics all the unitary irreducible representations of theRotation group, namely those labeled by angular momentum j = 0, 1/2, 1, 3/2, . . .. Theunitary representations of the Lorentz group must be extensions of these. Let us ask thenhow to construct this extension for a free massive particle of spin s. Such a particle mustbe described by a set of at least 2s + 1 momentum space wave functions fa(p). This muchfollows just from the Rotation group. We shall find a representation of the Poincare groupwith this minimal number of components.

25

The basic idea, due to Wigner, is to exploit the fact that one can always bring a massiveparticle to rest by a Lorentz transformation. Define a “standard boost” Bp which boosts aparticle at rest to one with momentum p. Let us introduce momentum eigenstates via

|f〉 ≡∫d3p∑a

|p, a〉fa(p). (2.3)

Then a momentum eigenstate of the particle can be related to the state at rest by

|p, a〉 ≡√

m

ω(p)U(Bp)|0, a〉. (2.4)

The multiplicative constant is necessary because we want U to be unitary. To understandthis point, notice that the relation of the three momentum p′ of a boosted particle to itsinitial momentum is nonlinear:

p′k = Λklpl + Λk

0ω(p). (2.5)

The Jacobian of this nonlinear transformation of variables is ∂(p′)/∂(p) = ω(p′)/ω(p). Theeasiest way to see this is to observe that

∫d4pδ(p2 +m2) is a Lorentz invariant; integrating

over p0 then shows that∫d3p/ω(p) is an invariant, which implies the above value for the

Jacobian. A general Lorentz transformation on a particle state of momentum p, which boostsit to p′, can be expressed

Λ = Bp′(B−1p′ ΛBp)B

−1p . (2.6)

The transformation in parentheses leaves a particle at rest at rest, and is therefore simply arotation. Applying U(Λ) to (2.4) we have

U(Λ)|p, a〉 =√

mω(p)

U(Bp′)U(B−1p′ ΛBp)|0, a〉

=√

mω(p)

U(Bp′)|0, b〉Dsba(B

−1p′ ΛBp)

=√

ω(p′)ω(p)|p′, b〉Ds

ba(B−1p′ ΛBp). (2.7)

Here Dsba(R) is just the standard representation matrix of the rotation group with spin s. It

is now easily checked that this defines a unitary representation of the Lorentz group on singleparticle states of spin s. Of course, on momentum eigenstates of a free particle, space-timetranslations are trivially realized by multiplication of the state by the phase e−ia·p for thespace-time translation by amount aµ.

Let us return to the “standard boost” Bp. It is clearly not uniquely determined since itcan be preceded by an arbitrary rotation and followed by a rotation about the axis parallelto p. There are two widely used choices for this boost. The simplest choice is the pure boostparallel to p which we will call B0

p. The second choice is dictated by choosing helicity states.It is described as follows. First agree that the spin states of the particle at rest be labeled


by λ the eigenvalue of J3. Then first boost the particle along the z axis to momentum |p|z.Then apply a rotation that carries the z axis to the direction of p. This latter rotation canbe taken to be

R0(p) ≡ e−iφJ3e−iθJ2e+iφJ3 (2.8)

where (θ, φ) are the polar angles of p. Then the helicity preserving standard boost is givenby

Bhp = R0(p)B0(|p|z). (2.9)

Then, clearly, |p, λ〉 ≡√m/ωBh

p|0, λ〉 is an eigenstate of momentum p with helicity λ,since rotations do not change helicity. This is easy to see from the general transformationlaw. For a massive particle, helicity is changed by a general Lorentz transformation, i.e.B−1p′ ΛBp can be any rotation. However, if one considers the massless limit of this rotation

for any fixed Λ, he discovers that it always approaches a rotation about the z axis. Thusfor massless particles helicity is actually a Lorentz invariant. Thus it can be consistent formassless particles with spin to exist in only one helicity state.

Having understood how a single particle state transforms under Lorentz transformationsit is straightforward to find the transformation law for states with any number of free particlesof varying mass and spin, which can be viewed as tensor products of single particle states.We can incorporate bose or fermi statistics by introducing a vacuum state |0〉 and creationand annihilation operators for each species of particle,e.g. :

|p, λ〉 ≡ bi†λ (p)|0〉, (2.10)

where i labels the species. When we are dealing with only a limited number of species, wetypically choose different letters for different types of particles, e.g. a, a† for neutral scalarparticles, b, b† (d, d†) for particles (antiparticles), etc. Then the transformation law for ageneral multiparticle state is completely defined by

U(Λ)|0〉 = |0〉U(Λ)bi†λ (p)U−1(Λ) = bi†λ′(Λp)Ds

λ′λ(B−1ΛpΛBp)

U(Λ)biλ(p)U−1(Λ) = biλ′(Λp)Ds∗λ′λ(B

−1ΛpΛBp). (2.11)

Of course the last transformation law is just the Hermitian conjugate of the second one sinceU is supposed to be unitary. We complete the description by writing down the energy andmomentum operators

H =∫d3p∑

i

√p2 +m2

i

∑λ b

i†λ (p)biλ(p) (2.12)

P =∫d3pp

∑i

∑λ b

i†λ (p)biλ(p). (2.13)

The above discussion might mislead one into thinking that the problem of free quantum fieldtheory for any spin is completely solved. Indeed, we have solved the problem of constructing


a relativistic quantum description of any number of free particles with any spins. However,the equivalence of this description to local field theory is not yet transparent. We haveexplicitly seen how this works for spin 0 and 1/2. which are described by scalar and Diracfields respectively. The scalar field is supposed to have the Lorentz transformation propertiesφ′(x′) = φ(x). We relate this to the general discussion by first identifying the momentumeigenstates with a creation operator applied to the vacuum |p〉 = a†(p)|0〉. Assuming thevacuum is Lorentz invariant, we conclude from the general discussion that

U(Λ)a†(p)U−1(Λ) =

√ω(p′)

ω(p)a†(p′). (2.14)

With this result we can then evaluate how the scalar field transforms

U †(Λ)φ(x)U(Λ) =∫

d3p

(2π)3/2√

2ω(p)

√ω(p′)ω(p)

(a(p′)eix·p + a†(p′)e−ix·p)

=∫

d3p′

(2π)3/2√

2ω(p′)(a(p′)eix·Λp

′+ a†(p′)e−ix·Λp

′)

= φ(Λ−1x) (2.15)

as desired.Notice that Lorentz covariance alone is achieved by the positive frequency part of the

field

φ+(x) =

∫d3p

(2π)3/2√

2ω(p)a(p)eix·p. (2.16)

But such a field would not commute with its adjoint at space-like separations as a local fieldmust.

[φ(+)(x), φ(−)(y)] =

∫d3p

(2π)3

1

2ωei(x−y)·p. (2.17)

Compare this to the result for the total field

[φ(x), φ(y)] =

∫d3p

(2π)3

1

2ω(ei(x−y)·p − e−i(x−y)·p). (2.18)

If (x − y)2 > 0 there is a Lorentz frame for which x0 = y0. In that frame the r.h.s. ismanifestly zero. It must be zero in all frames by Lorentz covariance. Notice also that the allimportant minus sign on the r.h.s came because we chose commutation relations for a, a†.Had we tried to make the scalar particles fermions by imposing anticommutation relations,the two terms would add and locality would be lost. This is the famous spin-statisticsconnection for the scalar field. For the Dirac field fermi statistics was necessary for stabilityrather than locality, but the spin-statistics connection is nonetheless fixed.

The Dirac field shows us how we must generalize these considerations to develop a fieldtheory for particles with spin. As shown in an exercise the Dirac field transforms underLorentz transformations as

ψ′(x′) = e−iλµνσµν/2ψ(x), (2.19)


where (e−λ)µν = Λµν . The 4 × 4 matrices σµν provide a finite dimensional representation of

the Lorentz group which is necessarily not unitary. This nonunitarity is associated with thenoncompactness of the Lorentz group. The nonunitarity of these matrices does not conflictwith the unitarity of the action of the Lorentz group on the state space which is just that onmulti-particle states we have just discussed. By expressing the field in terms of creation andannihilation operators we can see how the unitary representation on particle states inducesthe desired field transformation.

From the transformation properties of a spin 1/2 particle it follows that

U †(Λ)d†λ(p)U(Λ) =√

ω(p′)ω(p)

d†λ′(p′)D

1/2λ′λ(B−1

p′ Λ−1Bp)

U †(Λ)bλ(p)U(Λ) =√

ω(p′)ω(p)

bλ′(p′)D

1/2∗λ′λ (B−1

p′ Λ−1Bp). (2.20)

where p′ = Λ−1p. Focus on the way b enters the Dirac field:

U †(Λ)∫

d3p

(2π)3/2√

2ω(p)

∑λ bλ(p)uλ(p)eix·pU(Λ)

=∫

d3p′

(2π)3/2√

2ω(p′)bλ′(p

′)uλ(Λp′)eiΛ

−1x·p′D1/2∗λ′λ (B−1

p′ Λ−1Bp)

=∫

d3p′

(2π)3/2√

2ω(p′)bλ′(p

′)uλ(Λp′)eiΛ

−1x·p′D1/2λλ′ (B

−1p ΛBp′)

= e−iλµνσµν/2

∫d3p′

(2π)3/2√

2ω(p′)

∑λ′ bλ′(p

′)uλ′(p′)eiΛ

−1x·p′ , (2.21)

which is exactly the desired field transformation. The term involving d† works in an exactlysimilar way. In obtaining the last line we used the identity

e−iλµνσµν/2uλ(p) =

∑λ′

uλ′(Λp)D1/2λ′λ(B−1

ΛpΛBp), (2.22)

which is a simple consequence of the way Lorentz covariance works in the first quantizedinterpretation of the Dirac wave function.

Clearly the first step in generalizing to higher spin fields is to classify all of the finitedimensional representations of the Lorentz group. We would like to find the possible rep-resentation matrices so that a multi-component field ψα(x) will have the transformationlaw

ψ′α(x) = Dαβ(Λ)ψβ(Λ−1x). (2.23)

Let us first cast the algebra of Lorentz generators

[Mµν ,Mρσ] = i(ηµρMνσ − ηνρMµσ − ηµσMνρ + ηνσMµρ) (2.24)

in terms of the generators for rotations Jk = εklmMlm/2 and for boosts Kk = M0k:

[Jk, Jl] = iεklmJm (2.25)

[Jk, Kl] = iεklmKm (2.26)

[Kk, Kl] = −iεklmJm. (2.27)


Now notice that the linear combinations J± ≡ (J±iK)/2 satisfy the algebra of two mutuallycommuting angular momentum algebras

[J±k , J±l ] = iεklmJ

±m (2.28)

[J+k , J

−l ] = 0. (2.29)

We know from elementary quantum mechanics what all of the finite dimensional represen-tations of the rotation group are: they are labeled by the eigenvalues j(j+ 1) of the Casimiroperator

∑k J

k2 with j = 0, 1/2, 1, 2, . . . all nonnegative integers and half integers. The rep-resentation j has dimension 2j + 1. It follows that all the finite dimensional representationsof the Lorentz group are labeled by the pair of eigenvalues j+(j+ + 1), j−(j− + 1) of the pairof Casimir operators

∑k J

k2± where 2j+, 2j− are any pair of nonnegative integers. All of these

representations are equivalent to a unitary representation, i.e. J± are both represented byhermitian matrices. This means that the rotation generators J = J+ + J− are representedby hermitian matrices, but the boost generators K = −i(J+ − J−) are represented by an-tihermitian matrices. Thus the finite dimensional representations of the Lorentz group arenot equivalent to unitary ones. This is associated with the fact that the Lorentz group isnoncompact. We already encountered this nonunitarity in the representation of the Lorentzgroup by gamma matrices.

We denote the representation matrices by D(j+, j−). The simplest nontrivial representa-tions are D(1/2, 0) and D(0, 1/2). In the first the generators J− are represented by 0 and thegenerators J+ by σ/2. This means that the angular momentum is represented by J = σ/2and the boost generators by K = −iσ/2. The other two dimensional representation hasthe same representative for J but the boost is represented by K = +iσ/2. It is clear thatthese representations are not equivalent to each other, since any similarity transformationwhich could reverse the sign of K would do the same to J. However these two inequivalentrepresentations can be related by complex conjugation. In fact it is easy to see from theproperties of the Pauli matrices that D(1/2, 0)∗ = σ2D(0, 1/2)σ2. In general, the represen-tation D(k,m)∗ is equivalent to D(m, k)1, so the only real irreducible representations havej+ = j−. Of course D(k,m)⊕D(m, k) is real but it is also reducible.

Notice that parity reverses the sign of the boost generators but not the sign of the angularmomentum. Thus the representations D(k,m) and D(m, k) are also related by parity. Weencountered this fact with the Dirac field which admits the parity symmetry. It exploits thereducible representation D(1/2, 0) ⊕ D(0, 1/2) to achieve this. To make this quite explicitwe note that the representatives of the Lorentz generators are the 4× 4 matrices

J = 12

(σ 00 σ

)(2.30)

K = −i2

(σ 00 −σ

)(2.31)

1In D(k,m)∗ the generators are −J∗,−K∗ so e.g. J+ is represented by −J∗ − iK∗ = −(J−)∗ and viceversa.


which are just the components of σµν/2 constructed out of gamma matrices in the so-callednatural representation

γ0 =

(0 II 0

)(2.32)

γ =

(0 σ−σ 0

). (2.33)

Defining σµ = (I,σ), and σµ = (I,−σ) we can unify these two in the single equation

γµ =

(0 σµ

σµ 0

)(2.34)

The potential spin content of a field in a given representation is typically richer than onemight desire. For example, since J = J+ + J− the representations of the rotation groupcontained in D(k,m) include all spins that arise from adding spin k to spin m: |k−m|, |k−m|+ 1, . . . , k +m. Thus if our desire is to describe a given spin, depending on our choice ofrepresentation, we might bring in several other spins as well. The choice of field content is notunambiguous. Even for spin 1/2 we have noted various possibilities, e.g. Dirac, Majorana,and Weyl. Weyl fermions make use of D(1/2, 0), but since this is not a real representation,the hermitian conjugate field, which transforms under D(0, 1/2), must also be introducedand represents the anti-particle.

When we come to spin 1, two possibilities come to mind. D(1, 0)⊕D(0, 1) or D(1/2, 1/2).The latter contains potentially both spin 1 and spin 0 and is in fact the representation of afour-vector field. The former seems to contain spin 1 twice. It is easy to see that it is theantisymmetric tensor product of D(1/2, 1/2) with itself and thus represents an antisymmetricsecond rank tensor. The field strengths Fµν of electromagnetism spring to mind, so we mightdecide that the first choice is best. However, we know that in quantum mechanics it isnecessary to use the potentials Aµ which transform under the second choice. Then gaugeinvariance is essential to eliminate unwanted spin states.

This general discussion of higher spin serves to indicate some of the subtleties and com-plexities that must be confronted. In fact, consistent fully interacting quantum field theorieshave never been constructed for spins higher than 2 (the graviton). Furthermore, ultravio-let divergences have so far caused incurable difficulties for theories with spin higher than 1including quantum gravity. Since gravity is very much present in the real world, it is clearthat there is much to do before we can claim that quantum field theory can describe all ofphysics.


Chapter 3

The Dirac Equation

In our development of relativistic quantum mechanics we shall study intensively the caseof spin 1/2 particles solved by Dirac. After this we shall return to the problem of higherspin. As a practical matter, the Dirac equation contains a tremendous amount of the presentunderstanding of elementary particles. In fact, before 2012 all of the known fundamentalconstituents of matter are either spin 1/2 (quarks and leptons), or spin 1 gauge particles(photon, gluons, weak vector bosons). To these we have to add a fundamental scalar (Higgsparticle, discovered in 2012). Of course there is always the spin 2 graviton that must beunderstood eventually. But we can go a tremendous distance to understanding the physicsof the standard model through study of the Dirac equation interacting with a gauge field.Indeed, for many applications the gauge field doesn’t even need to be quantized!

Dirac’s original motivation for his equation is still useful and inspiring to recall. The moststraightforward attempt to write down a relativistic version of the Schrodinger equation

i~∂ψ(x, t)

∂t=

√m2c4 − (~c)2∇2ψ(x, t) (3.1)

≡ m∞∑k=0

(1/2

n

)(−~2∇2

m2

)nψ(x, t) (3.2)

is horribly nonlocal, involving an arbitrarily high number of spatial derivatives. Dirac pro-posed to get around this by making ψ a multicomponent wave function and defining thesquare root in a local way by using matrices, in analogy with the properties of the Pauli spinmatrices

(σ · v)2 = v2. (3.3)

To achieve this he required the introduction of four anticommuting matrices γµ with µ =0, 1, 2, 3:

γµ, γν = −2ηµν . (3.4)

Using (3.4) it is simple to show that

(γµ∂µ)2 =1

2γµ, γν∂µ∂ν = −∂µ∂µ =

∂2

∂t2−∇2, (3.5)

33

where we have chosen units for which c = 1. We shall also choose units so that ~ = 1. Thenthe Dirac equation is

1

iγµ∂µψ +mψ = 0. (3.6)

To cast this equation as a relativistic Schrodinger equation we rewrite it as

i∂ψ(x, t)

∂t=

(1

iα · ∇+ βm

)ψ(x, t) (3.7)

where we have multiplied through by γ0 ≡ β using β2 = I and have defined α ≡ γ0γ.

3.1 Single particle interpretation

3.1.1 The Matrices γµ

So far we have not specified the gamma matrices. It is simple to show that they must be atleast 4× 4 (In D space-time dimensions the minimum size is 2D/2 × 2D/2 for even D.)

Let σk, k = 1, 2, 3 be the 2× 2 Pauli matrices

σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

). (3.8)

Then we have two popular representations:Standard Representation of γµ Chiral (Natural) Representation of γµ

γ0 =

(I 00 −I

)γ0 =

(0 II 0

)(3.9)

γ =

(0 σ−σ 0

)γ =

(0 σ−σ 0

)(3.10)

γ5 =

(0 II 0

)γ5 =

(−I 00 I

)(3.11)

σkm = εkmn

(σn 00 σn

)σkm = εkmn

(σn 00 σn

)(3.12)

where εkmn is the completely antisymmetric three tensor with ε123 = +1. The standardrepresentation is more convenient for slowly moving particles, whereas the chiral one is moreconvenient for massless fermions that move at the speed of light.

Clearly β = γ0 is hermitian, γ is anti-hermitian, and α is hermitian. Thus the Hamilto-nian

H =1

iα · ∇+ βm (3.13)

is a hermitian operator as it should be.


3.1.2 Spin

To justify the interpretation of the Dirac particle as spin 1/2, we need to construct an angularmomentum operator for which α transforms as a vector. This is clearly accomplished bychoosing the spin operator to be Σ/2 with

Σ =

(σ 00 σ

). (3.14)

Then the Dirac Hamiltonian obviously commutes with the total angular momentum

J = r×1

i∇+

Σ

2. (3.15)

The Dirac wave function transforms under the 12⊕ 1

2representation of the rotation group.

In particular it describes a spin 1/2 particle.To get the energy spectrum of the Dirac particle we must find all the eigenstates of the

Hamiltonian (3.13). Clearly, it is best to work with momentum eigenstates

ψp = u(p)eip·x, (3.16)

since the momentum operator commutes with H. Then the coefficient spinor must satisfy

(α · p + βm)u(p) = Eu(p). (3.17)

Since the square of the matrix on the l.h.s. is just m2 + p2, we immediately learn that

E(p) = ±√m2 + p2. (3.18)

3.1.3 Explicit Solutions

It is also fairly simple to find the eigenfunctions in the Standard Representation. It issufficient to find the positive energy eigenfunctions, because the negative energy ones canbe obtained from them by a simple operation: Complex conjugating (3.17) and multiplyingboth sides by iγ2, we learn that iγ2u∗(−p) is an eigenstate of H with eigenvalue −E if u(p)is an eigenstate with eigenvalue +E. An explicit solution for u, with E = +ω(p) > 0, iseasily shown to be

u(p) =

(√m+ ω φσ·p√m+ω

φ

), E = ω > 0 (3.19)

where ω =√m2 + p2 and φ is any two component spinor. The two independent components

just represent the two spin states of a spin 1/2 particle. As we have explained the negativeenergy eigenfunctions are then

u−(p) = iγ2u∗(−p), E = −ω < 0. (3.20)


In the Natural or Chiral representation, the positive energy eigenfunctions can be written

u(p) =

(√ω(p)− p · σ φ√ω(p) + p · σ φ

)(3.21)

and again the negative energy spinor is obtained from the positive energy one by (3.20). Inboth standard and natural representations, we have u†u = 2ω(p)φ†φ. the matrices underthe square roots in (3.21) have positive eigenvalues, and we always take the positive squareroot.

3.1.4 The Dirac sea

The positive energy states of the Dirac particle give the desired relativistic description ofa relativistic spin 1/2 particle. The negative energy states are a disaster for the singleparticle interpretation of the Dirac wave function. As soon as the Dirac particle in a positiveenergy state is subjected to external forces transitions to negative energy states will beinduced and the system is unstable. Dirac himself proposed the remedy for this problem,which exploits the Pauli Exclusion Principle for fermions: If one postulates that the Diracparticle is a fermion, then one can consider the “vacuum” to be the state in which all ofthe negative energy levels are filled, then any further particles added to this state must bythe exclusion principle occupy only positive energy states! This vacuum state is sometimescalled the “negative energy sea.” Although we call this state the vacuum, it is clear fromits construction that it is far from empty, since it contains an infinite number of particles.The presence of the sea will make itself felt as soon as one considers interactions of theDirac particles with themselves or with independent force fields. Strictly speaking, thereis an enormous (infinitely) negative energy associated with the sea. However in quantummechanics, only energy differences are measurable (in the absence of gravity) and we mightas well measure the energy of all states relative to that of the sea, i.e. we take Esea = 0.The total momentum contained in the sea is automatically zero because momentum statesin all directions are occupied.

In addition if the Dirac particle is charged the sea possesses an infinite uniform positivecharge density. It is convenient in this case to postulate a compensating negative backgroundcharge density, so our vacuum will be neutral

To understand the dynamical significance of the presence of the sea, it is instructive toconsider the excitation spectrum of the whole system. We can first of all add N particleswith momenta p1, p2, . . . , pN to the sea. The resulting state will have energy

EN particles =∑k

√m2 + p2

k (3.22)

and N units of charge.Next we can remove N particles with momenta q1, q2, . . . , qN from the sea. This state,

which we describe as a state of N holes, will also have a positive energy

EN holes =∑k

√m2 + q2

k (3.23)


since we have subtracted negative energy. However the momentum of this state is

PN holes = −∑k

qk (3.24)

for the same reason, and the charge is −N units. Thus this state appears to be a stateof N particles of opposite charge to the ordinary (positive energy) Dirac particle and withmomenta −q1,−q2, . . . ,−qN . This is how antiparticles appear in the theory.

Finally, one can simply excite N particles from the sea to positive energy states. Thisexcitation does not change the number of particles so it is neutral. The energy of this stateis the sum of the positive energy eigenvalues occupied minus the sum of the negative energylevels vacated. It is a state with N particles and N antiparticles (holes), with correspondingenergy. In principle, this state can be prepared from the sea by delivering sufficient energy,at least 2Nm, to the system.

If the original Dirac particle is the electron with charge −e, then the theory predictsthe existence of its antiparticle, the positron, with charge +e. The positron was unknownwhen Dirac discovered his equation, and the idea of identifying the positron with the protonfailed because the latter does not have the electron’s mass nor could it annihilate with anelectron into photons. Thus when the positron was eventually discovered, it was a spectacularconfirmation of Dirac’s theory.

3.2 Second Quantization

We have seen that the consistent interpretation of the Dirac theory requires the presence ofthe sea of an infinite number of negative energy particles. In practice, however, at least ifinteractions are weak, all but a finite number of particles in the sea are spectators in anycalculation. Thus we need a formalism that allows us to concentrate only on the part of thesea that is active in a given process. The formalism which achieves this has been developed,and for historical reasons goes under the name of second quantization. It should be stressedthat it is completely equivalent to a description in terms of the many body Schrodingerwave function for the sea, but it is much less cumbersome, and almost indispensable toefficient calculation. The formalism is applicable to any system of identical particles, bosonsor fermions. We shall present only the fermion case in lecture since we will only be applyingit to the Dirac theory. The changes needed for the boson case will be indicated in a seriesof exercises.

3.2.1 General Formalism

We begin with the concept of the occupation number basis. Consider a system of fermions,and let an orthonormal complete set of states for a single fermion be labeled by an index α:

ψα(x),

∫d3xψ∗α(x)ψβ(x) = δαβ. (3.25)


Then a multi-fermion antisymmetrized tensor product state is completely specified by theset of numbers nα which give the number of fermions (0 or 1) occupying each state α.

Now the crucial idea of second quantization is to define a creation operator b†α by

b†α|n1 . . . nα . . . 〉 = (−)∑γ<α nγ |n1 . . . nα + 1 . . . 〉. (3.26)

The state dependence of the prefactor is necessary for consistency with the antisymmetryof the state under fermion interchange. Since b†α adds a fermion in state α its square mustvanish. It is a direct consequence of this fact and its definition that

b†α, b†β = 0 (3.27)

By considering

〈n|b†α|n′〉∗ = 〈n′|bα|n〉 (3.28)

it is clear that bα removes a fermion from the state α and so is an annihilation operator. Itis then simple to check that

bα, b†β = δαβ. (3.29)

For our application, the state label will include a continuous momentum as well as a discretelabel. In that case the Kronecker delta will of course include a factor of a Dirac delta functionδ(p− p′).

Having introduced the creation and annihilation operators, we now have a very efficientnotation for the occupation number basis. Call the state with no states occupied |0′〉. Thenthe state with

nα1 = nα2 = · · ·nαN = 1 (3.30)

and all other occupation numbers zero is just

b†αN · · · b†α2b†α1|0′〉. (3.31)

We are now in a position to relate all this formalism to the standard many-body Schrodingerwave function description. The wave function describing the state in which the single particlestates α1 · · ·αN are occupied is just

Ψα1···αN (x1, · · · , x2) =1√N !

∑P

δPψα1(xP (1))ψα2(xP (2)) · · ·ψαN (xP (N)) (3.32)

where δP = −1 if P is an odd permutation and +1 if P is an even permutation. If we definethe field operator

ψ(x) ≡∑α

bαψα(x), (3.33)


with anticommutation relations

ψ(x), ψ(y) = ψ†(x), ψ†(y) = 0 (3.34)

ψ(x), ψ†(y) =∑

α ψα(x)ψ∗α(y) = δ(x− y), (3.35)

Then we have

Ψα1···αN (x1, · · · , xN) =1√N !〈0′|ψ(x1)ψ(x2) · · ·ψ(xN)b†αN · · · b

†α2b†α1|0′〉. (3.36)

A completely general N body wave function is of course a general superposition of suchstates

ΦN =∑

α1···αN

cα1···αNΨα1···αN =1√N !〈0′|ψ(x1)ψ(x2) · · ·ψ(xN)|Φ〉 (3.37)

where

|Φ〉 =∑

α1···αN

cα1···αN b†αN· · · b†α2

b†α1|0′〉 (3.38)

The state |Φ〉 thus can describe the most general many body Schrodinger wave function.Indeed, the second quantization formalism allows one to even superpose states of differentnumbers of particles, so it is the more general description!

We have shown how to describe general quantum states, it remains to show how linearoperators are related between the two formalisms. All possible observables in the Schrodingerwave function description are completely symmetric in the operators acting on each particle.Such operators can be classified according to the number of particles involved in each termof the operator. For example, a one body operator acting on N identical particles has theform

Ω(1) =∑k

ωk (3.39)

where ωk acts only on the variables of the kth particle. A two body operator would be asum of terms, each acting on a pair of particles:

Ω(2) =∑k<m

ωkm. (3.40)

A K body operator is obviously a sum of terms each acting on K particles. In a typicalnonrelativistic system with N identical particles, e.g. the electrons of a Z = N atom, thekinetic energy of the electrons is a 1 body operator, while the potential energy is a two bodyoperator (since the electrons interact in pairs). The interaction of the electrons with anexternal field would be described by another one body operator.


The second quantized description of these operators is straightforward. Starting with theone body case we first represent the single particle operator ω by its matrix elements in thesingle particle basis we have introduced

ωαβ = 〈α|ω|β〉. (3.41)

Then the one body operator Ω(1) is just

Ω(1) =∑αβ

b†αωαβbβ. (3.42)

To see that this definition has the correct action on multi-particle states, one simply appliesit to one of the basis states

Ω(1)b†αN · · · b†α2b†α1|0′〉 =

∑k

b†αN · · ·∑γk

(b†γk〈γk|ω|αk〉) · · · b†α1|0′〉 (3.43)

and we see that the r.h.s. is a sum of terms for the kth of which ω has the correct action onthe kth single particle state label.

Multi-body operators have the obvious analogous second quantized description. We limitthe discussion here to the two-body case. Let |β1〉|β2〉 denote the standard unsymmetrizedtwo particle tensor product state. It is convenient to reverse the order of factors in thecorresponding bra 〈α2|〈α1| so that

(〈α2|〈α1|)|β1〉|β2〉 = δα1β1δα2β2 . (3.44)

Then we define

ωα2α1,β1β2 = 〈α2|〈α1|ω|β1〉|β2〉 (3.45)

and the two-body operator Ω(2) is then just

Ω(2) =1

2

∑α1α2,β1β2

b†α2b†α1ωα2α1,β1β2bβ1bβ2 . (3.46)

A typical example which requires both one and two body operators is the hamiltonianfor a system of N nonrelativistic fermions with an interaction potential energy V (rk, rm)= V (rm, rk) between each pair (k,m). The Schrodinger picture Hamiltonian for this systemis just

H =∑k

−~2

2m∇2k +

∑k<m

V (rk, rm). (3.47)

According to the procedure just outlined, the second quantized version of the kinetic termis∑αβ

b†α〈α|(−~2

2m∇2k

)|β〉bβ =

∫d3rψ†(r)

(−~2

2m∇2

)ψ(r) =

∫d3r

~2

2m∇ψ†(r) · ∇ψ(r),(3.48)


where in the second form we have gone to the coordinate basis, using the definition of thesecond quantized field operator (3.33). The second quantized version of the potential termis the two-body operator

1

2

∑α1α2,β1β2

b†α2b†α1〈α2|〈α1|V |β1〉|β2〉bβ1bβ2 =

1

2

∫d3xd3yV (x,y)ψ†(y)ψ†(x)ψ(x)ψ(y). (3.49)

Two details to note about this expression are the overall factor of 1/2 and the order ofoperators. These are necessary to arrange that the potential energy has the correct sign andnormalization. The complete hamiltonian for this system can now be compactly written as

H =

∫d3r

~2

2m∇ψ†(r) · ∇ψ(r) +

1

2

∫d3xd3yV (x,y)ψ†(y)ψ†(x)ψ(x)ψ(y). (3.50)

This is the hamiltonian for a quantum field theory. The fundamental quantum operators arethe local fields ψ(x) or the corresponding creation and annihilation operators. The operatorsx, p etc. of the Schrodinger description have been demoted to c number labels and derivativeswith respect to them.

One final feature of the formalism to explain is the role of the field equations. In theSchroedinger picture the quantum dynamics is given by the Schroedinger equation

i~∂

∂t|Φ, t〉 = HS(t)|Φ, t〉 (3.51)

where we stress that we allow time varying external forces to be present (hence the timedependence of H). The field equations arise in the Heisenberg picture wherein the timedependence resides in the operators rather than in the system states which are constant intime. To pass to the Heisenberg picture we write

|Φ, t〉 = U(t)|Φ, 0〉 (3.52)

where

i~∂

∂tU = HS(t)U U(0) = I, (3.53)

and give the time independent Schroedinger picture operators Ω time dependence accordingto

Ω(t) ≡ U †ΩU. (3.54)

The Heisenberg picture Hamiltonian is similarly related to the Schrodinger picture one by

H(t) ≡ U †HS(t)U. (3.55)

Then the Heisenberg picture operators corresponding to constant Schrodinger picture oper-ators satisfy the Heisenberg equations

i~Ω(t) = [Ω(t), H(t)]. (3.56)


Returning to our system of nonrelativistic fermions we find that the Heisenberg equationfor the field operator ψ implies

i~∂

∂tψ(x, t) =

−~2

2m∇2ψ(x, t) +

∫d3zV (x, z)ψ†(z)ψ(z)ψ(x). (3.57)

This is the quantum field equation. It is a nonlinear differential equation for a quantumoperator, and because of its operator nature it has much more information packed in it thanis immediately apparent. The origin of the name “second quantization” for this formalismis that (3.57) looks like a nonlinear version of the Schroedinger equation. Indeed if theparticles did not mutually interact, it would be exactly the Schrodinger equation. If “firstquantization” produced the Schroedinger equation, we have now reinterpreted the latter asa classical field equation, which is then “second quantized”. Of course we know that all wehave really done is given a clever reformulation of ordinary many body quantum mechanics,and the name is really a misnomer, which has stuck.

We should mention that (3.57) is an elegant point of departure for the Hartree-Fockapproximation, in which one approximates the nonlinear term by a one body term∫

d3yK(x,y)ψ(y) (3.58)

in which K is chosen to make this term as close as possible to the original two body term.How much of the quantum dynamics is captured in the quantum field equation? The

time dependent many body wave function in Schrodinger picture, which summarizes thecomplete quantum dynamics of the system can be recast

1√N !〈0′|ψ(x1) · · ·ψ(xN)|Φ, t〉 =

1√N !〈0′|U(t)ψ(x1, t) · · ·ψ(xN , t)|Φ, 0〉. (3.59)

So in addition to the time dependence of ψ(x, t) which we could get from the field equation,we would also need to find the time dependence of the state

〈0′|U(t). (3.60)

In our example, the state 〈0′| is really dynamically inert i.e.

〈0′|U(t) = 〈0′|, (3.61)

because each term of HS(t) has a b† on the left so that 〈0′|HS(t) = 0. Thus knowledgeof ψ(x, t) at all times allows us to reconstruct the time dependence of the many bodySchrodinger wave function completely. If we were transforming a different matrix element, inwhich 〈0′| were replaced by a nonempty state, to the Heisenberg picture, the time dependenceof U acting on this state would have to be found in addition to the time dependence ofthe Heisenberg operators. The time dependence of 〈A|U(t) has a simple interpretation if〈A| is characterized as an eigenstate, say, of some definite Schrodinger picture observableΩ. Then 〈A|U(t) is the corresponding eigenstate of the corresponding Heisenberg picture


operator Ω(t) = U †ΩU . In particular if the time dependent terms in the Schrodinger pictureHamiltonian HS(t) = H0 +H ′(t) vanish initially and finally,

H ′(T ) = H ′(0) = 0, (3.62)

and |A〉 is an eigenstate of H0 = HS(0) = H(0), then 〈A|U(T ) is an eigenstate of theHeisenberg picture operator H0(T ) = H(T ).

Let us consider what is gained and lost in the alternative formulations of many bodyquantum physics. The second quantization formalism contains all the information containedin the (anti)symmetrized wave functions. The (unphysical) non-symmetrized wave functionsare of course lost, but that is desirable. The second quantization machinery allows a broaderrange of dynamical options. For example, particle number conservation is built into thewave function description. This conservation law is reflected in the existence of the numberoperator

N =∑α

b†αbα =

∫d3xψ†(x)ψ(x) (3.63)

which counts the number of particles in a given state. The conservation law is the statementthat N commutes with the Hamiltonian: [N,H] = 0. We can also identify ψ†(x)ψ(x) asthe number density operator. There is, however, no principle which excludes considering aHamiltonian with terms that don’t commute with N . For example, a term∫

d3xd3x′A(x,x′)(ψ(x)ψ(x′)− ψ†(x)ψ†(x′)) (3.64)

would not commute with N . In that case energy eigenstates would not have a definitenumber of particles!

3.2.2 Second Quantized Dirac Equation

The formalism developed in the last chapter can now be fruitfully applied to the Diracequation, cast as a Schrodinger equation,

i∂ψ(x, t)

∂t=

(1

iα · ∇+ βm

)ψ(x, t). (3.65)

Let us regard the component label a as an additional coordinate: ψa(x) = ψ(x, a). Fordefiniteness let us choose our single particle basis to be momentum eigenstates, with anadditional label λ for spin and ± to distinguish positive and negative energy states. Thusthe role of ψα(x) of the previous chapter will be played by

ψ(±)λ,p (x, a) =

1

(2π)3/2√

2ω(p)uaλ±(p)eip·x. (3.66)


The prefactors are conventional and with them in place the condition of orthonormalityimplies that

u†λ′±′uλ± ≡∑a

ua∗λ′±′uaλ± = 2ω(p)δλ′λδ±′±. (3.67)

We therefore write the explicit solution (3.19) for u = u+ in the rescaled form

uλ(p) =√ω(p) +m

(φλ

σ·pm+ω(p)

φλ

)(3.68)

so that the normalization condition on the two spinor φ is

φ†λ′φλ = δλ′λ. (3.69)

There are two widely used choices for φλ. One is to simply choose the two orthogonalspinors (

1

0

) (0

1

). (3.70)

In the rest frame p = 0 these are just eigenstates of Σ3 with eigenvalues +1,−1 respectively.The other choice is to pick them to be eigenstates of helicity h = p · σ/2|p| denoted χλ(p):

hχλ(p) = λχλ(p) λ = ±1

2. (3.71)

Explicit forms for χλ are developed in the exercises. Notice that for the helicity basis theexpression for u simplifies to

uλ(p) =√ω(p) +m

(χλ

2λ|p|m+ω(p)

χλ

), (3.72)

and furthermore uλ is itself an eigenstate of helicity p ·Σ/2|p| with eigenvalue λ.In all cases we maintain our choice (3.20)

u−(p) = iγ2u∗(−p) (3.73)

for the negative energy basis functions. For the helicity basis choice for uλ, this constructiongives a negative energy spinor with the same helicity, as can easily be shown by applying hto both sides. For the rest frame Σ3 basis this construction reverses the sign of Σ3 in therest frame.

The properties of ψ(±)λ,p (x, a) needed for the second quantization formalism are

Orthonormality:

∫d3x

∑a

ψ(±)∗λ,p (x, a)ψ

(±)′

λ′,p′(x, a) = δλ′λδ(±)′±δ(p′ − p) (3.74)


and

Completeness:

∫d3p∑λ±

ψ(±)λ,p (x, a)ψ

(±)∗λ,p (x′, a′) = δaa′δ(x

′ − x). (3.75)

To pass to the second quantized formalism, we simply introduce creation and annihilationoperators b†λ±(p), bλ±(p) with anticommutation relations

bλ±(p), b†λ′±′(p′) = δλ′λδ(±)′±δ(p

′ − p), (3.76)

and define the Dirac quantum field operator

ψa(x) =

∫d3p∑λ±

bλ±(p)ψ(±)λ,p (x, a). (3.77)

By virtue of (3.75) the field operators satisfy

ψa(x), ψa′†(x′) = δaa′δ(x− x′). (3.78)

A principal virtue of second quantization is the efficiency with which we can constructthe state describing the negative energy sea. We simply apply to the empty state all of thecreation operators for negative energy states:

|sea〉 ≡ |0〉 = N∏p,λ

b†λ−(p)|0′〉. (3.79)

This looks terribly complicated, but we can uniquely characterize this state very simply: Itis annihilated by all of the positive energy annihilation operators and by all the negativeenergy creation operators

bλ+(p)|0〉 = b†λ−(p)|0〉 = 0. (3.80)

These conditions tell us everything we need to know about the sea. The annihilation operatorfor a negative energy electron creates a hole in the sea. Thus the construction of the sea iscompletely equivalent to interchanging the role of the creation and annihilation operatorsfor the negative energy Dirac particles.

To see the consequences of this interchange of roles, let us consider a few of the observablesof the theory. The Hamiltonian is just

H =

∫d3x

∑a

ψa†(x)(1

iα ·∇+ βm)ψa(x) (3.81)

=

∫d3pω(p)

∑λ

(b†λ+(p)bλ+(p)− b†λ−(p)bλ−(p)) (3.82)

=

∫d3pω(p)

∑λ

(b†λ+(p)bλ+(p) + bλ−(p)b†λ−(p))− 2

∫d3pω(p)δ(0) (3.83)


where in the last form we have reordered the creation and annihilation operators of thenegative energy contributions, the nonzero anticommutator producing the negative infiniteconstant term. Notice that thanks to the Fermi statistics both contributions to the energy arepositive. This constant is just the energy of the sea. the factor of δ(0) can be identified withV olume/(2π)3 so the sea has an infinite negative energy density. As we have already stressedwe can and will choose to measure all energies relative to that of the sea which amountsto dropping this constant1, so henceforth we shall take the free Dirac second quantizedHamiltonian to be

HDirac =

∫d3p ω(p)

∑λ

(b†λ+(p)bλ+(p) + bλ−(p)b†λ−(p)). (3.84)

When we pass to the Heisenberg picture we find the field equation for ψ to be nothingother than the Dirac equation

(1

iγ · ∂ +m)ψ = 0. (3.85)

Since we have selected our single particle basis to be eigenstates of 1iα ·∇ + βm the time

dependence of ψ in Heisenberg picture is simply

ψa(x, t) =

∫d3p∑λ±

bλ±(p)ψ(±)λ,p (x, a)e∓iω(p)t (3.86)

The annihilation operators for the positive and negative energy Dirac particles are thus iden-tified with the positive and negative frequency components of the Dirac field in Heisenbergpicture. This is a useful observation because when we introduce time dependent externalfields which are switched off at early and late times, it will allow us to easily relate theoperators that characterize the sea at late times to the ones that characterize the sea atearly times.

Returning to our survey of observables, the momentum operator is just

P =

∫d3x

∑a

ψa†(x)1

i∇ψa(x) (3.87)

=

∫d3pp

∑λ

(b†λ+(p)bλ+(p) + b†λ−(p)bλ−(p)) (3.88)

=

∫d3pp

∑λ

(b†λ+(p)bλ+(p) + bλ−(−p)b†λ−(−p)) (3.89)

The term 2∫d3ppδ(0) arising from reordering the negative energy operators automatically

vanishes and need not be dropped. We see from the explicit form of the momentum operatorthat bλ−(−p) creates from the sea a particle of momentum +p.

1Alternatively we could introduce a bare cosmological constant to cancel it.


The charge operator is just Q = qN where q is the unit of charge carried by the Diracparticle and N is the number operator

Q = q

∫d3x

∑a

ψa†(x)ψa(x) (3.90)

= q

∫d3p∑λ

(b†λ+(p)bλ+(p)− bλ−(−p)b†λ−(−p)) + 2q

∫d3pδ(0) (3.91)

from which we see that bλ−(−p) creates a state of charge −q. We shall also in future dropthe constant term in Q so the charge of the sea is then zero2 There is a convenient way tomake this subtraction. Instead of taking the charge density to be ρ(x) = qψ†ψ, take it tobe the symmetrized form

ρ(x) =q

2

∑a

[ψ†a(x)ψa(x)− ψa(x)ψ†a(x)] (3.92)

Then when the operators in Q =∫d3xρ are suitably reordered the piece coming from the

positive energy term exactly cancels that from the negative energy term. At this point wecan also identify the current operator from local current conservation

∂ρ

∂t+∇ · j = 0. (3.93)

Inserting (3.92) into (3.93) and using the Dirac equation, we identify

j =q

2

∑a

[ψ†a(x)(αψ)a(x)− (αψ)a(x)ψ†a(x)]. (3.94)

We can assemble (ρ, j) in a four vector jµ:

jµ(x, t) =q

2

∑a

[ψ†a(x, t)(βγµψ)a(x, t)− (βγµψ)a(x, t)ψ

†a(x, t)]

=q

2

∑a

[ψ†a(x, t), (βγµψ)a(x, t)] (3.95)

=q

2

∑a

[ψa(x, t), (γµψ)a(x, t)] (3.96)

We have made use of the Dirac adjoint

ψa ≡∑b

ψ†bβba (3.97)

2If there is more than one species of fermion in the universe, the coefficient of this term is∑f qf . One

way of getting rid of the sea charge is to insist that this sum of charges vanishes. As it happens the standardmodel of strong weak and electromagnetic interactions has this property, which is required for cancellationof the axial anomaly. Thanks to Mr. Yan-Bo Xie for drawing my attention to this circumstance. In a similarvein supersymmetry is often proposed so that the zero point energy of bosons exactly cancels the sea energyof fermions.


in the last form, which we will also sometimes shorten even more by suppressing the spinorindices, q

2[ψ, γµψ]. Current conservation ∂µj

µ = 0 is an immediate consequence of the Diracequation and its Dirac adjoint

ψ(−iγ ·←−∂ −m) = 0. (3.98)

The final observable we mention is the angular momentum

J =

∫d3x

∑a,b

ψa†(x)

(1

i(x×∇) +

1

2Σ

)ab

ψb(x). (3.99)

The sea is of course rotationally invariant

J|0〉 = 0, (3.100)

as can be confirmed by direct calculation. Of particular interest is the action of the helicityon the single particle states. On the particle states we find

p · Jb†λ+(p)|0〉 =1

2ω

∑a,b,λ′

ua∗λ′+(p)1

2p ·Σabu

bλ+(p)b†λ′+(p)|0〉 = λ|p|b†λ+(p)|0〉, (3.101)

confirming that this state carries helicity λ. The helicity of the one hole state bλ−(−p)|0〉which possesses momentum +p is also λ but the reason is slightly subtle. First of all

p · Jbλ−(−p)|0〉 = − 1

2ω

∑a,b,λ′

ua∗λ−(−p)1

2p ·Σabu

bλ′−(−p)bλ′−(−p)|0〉 (3.102)

where the minus sign arises because b†λ− occurs in ψ† and must anticommute with ψ beforeit can contract against bλ−. But then

1

2(−p) ·Σuλ−(−p) = λ|p|uλ−(−p). (3.103)

Thus

p · J|p|

bλ−(−p)|0〉 = λbλ−(−p)|0〉 (3.104)

as we claimed.This survey of single particle observables has established:

1. The state b†λ+(p)|0〉 is a one particle state of momentum p, energy ω =√p2 +m2,

charge q, and helicity λ.

2. The state bλ−(−p)|0〉 is a one particle state of momentum p, energy ω =√p2 +m2,

charge −q, and helicity λ.


In particular if the first state is an electron of charge −e, then the second is a positron ofcharge +e. To emphasize these facts it is traditional to rename the creation and annihilationoperators for negative energy particles. So define

bλ−(−p) ≡ d†λ(p) b†λ+(p) ≡ b†λ(p). (3.105)

So b†λ(p) creates a particle and d†λ(p) creates an antiparticle. Similarly it is useful to definethe Dirac spinor

vλ(p) ≡ uλ−(−p) = iγ2u∗λ(p). (3.106)

Note that u and v satisfy

(γ · p+m)uλ(p) = 0 (h− λ)uλ(p) = 0 (3.107)

(γ · p−m)vλ(p) = 0 (h+ λ)vλ(p) = 0 (3.108)

with opposite signs in front of the mass and opposite helicities.With these definitions the free Dirac field operator in Heisenberg picture has the repre-

sentation

ψ(x) =

∫d3p

(2π)3/2√

2ω

∑λ

(bλ(p)uλ(p)eix·p + d†λ(p)vλ(p)e−ip·x

), (3.109)

where p ·x = p ·x−ω(p)t is the Minkowski scalar product. A point to bear in mind with thisnew interpretation is that one body operators will generally contain terms like b†d† whichcreate a particle antiparticle pair and terms like bd which destroy such a pair. Thus whenwe couple currents to the electromagnetic field we will have charge conservation, but notparticle number conservation.

The Dirac field is now expressed in a way parallel to the scalar and electromagnetic fields:

φ(x, t) =

∫d3p

(2π)3/2√

2ω

(a(p)eix·p + a†(p)e−ip·x

)Ak(x, t) =

∫d3p

(2π)3/2√

2ω

∑λ

(εkλ(p)aλ(p)eix·p + εk∗λ (p)a†λ(p)e−ip·x

). (3.110)


Chapter 4

The Discrete Symmetries of the DiracEquation

4.1 Parity

The parity transformation x → −x can be extended to a symmetry of the Hamiltonian.Consider the following transformation on the field:

ψ(x, t)→ P−1ψ(x, t)P = eiφβψ(−x, t), (4.1)

where we have allowed a multiplicative phase. Then the Hamiltonian transforms to

P−1HP =

∫d3xψ†(−x)β(

1

iα ·∇+ βm)βψ(−x) (4.2)

=

∫d3xψ†(−x)(−1

iα · ∇+ βm)ψ(−x) = H (4.3)

after changing integration variables, so it is parity invariant. From the parity transformation(4.1) we can infer how parity acts on the particle states:

eiφβψ(−x, t) = eiφ∫

d3p

(2π)3/2√

2ω

∑λ

(bλ(−p)βuλ(−p)eix·p + d†λ(−p)βvλ(−p)e−ip·x

), (4.4)

where we have reversed the sign of p by a change of variables. Next we note that dependingon the spin basis we choose,

Helicity Basis: βuλ(p) = −ieiλ(π+2φp)u−λ(−p) (4.5)

βvλ(p) = −ie−iλ(π+2φp)v−λ(−p) (4.6)

where we have used the formula

χλ(p) = −ieiλ(π+2φp)χ−λ(−p) (4.7)

51

obtained in the exercises, or

Rest Frame Σ3: βuµ(p) = uµ(−p) (4.8)

βvµ(p) = −vµ(−p). (4.9)

Using these spinor properties, we learn that

Helicity Basis: P−1bλ(p)P = eiφie−iλ(π+2φp)b−λ(−p) (4.10)

P−1d†λ(p)P = eiφieiλ(π+2φp)d†−λ(−p) (4.11)

P−1b†λ(p)P = e−iφ(−i)eiλ(π+2φp)b†−λ(−p) (4.12)

where the last equation is just the hermitian conjugate of the first. And

Rest Frame Σ3: P−1d†µ(p)P = −eiφd†µ(−p) (4.13)

P−1b†µ(p)P = e−iφb†µ(−p). (4.14)

It should be noted that the arbitrary phase we allowed in the definition of parity cancels outfor neutral states, i.e. those with an equal number of b†’s and d†’s acting on the sea, whichwe can take to be parity invariant. This means that whereas the intrinsic parity of a singleparticle is conventional, that of a particle antiparticle pair is not. For example the aboveformulae imply that the parity of the ground state s wave of positronium is odd, i.e. theground states are 0− and 1−.

4.2 Charge Conjugation

It is apparent that a Dirac particle and its antiparticle are closely related to each other.They have identical mass and spin, but exactly opposite charges. In fact this relationship is areflection of a symmetry in the dynamics under interchange of a particle with its antiparticle.To explore this symmetry, first define a unitary transformation C, CC† = I by the rules

C−1b†C = d† C−1d†C = b† C|0〉 = |0〉. (4.15)

(The last of these equations implies an extremely complicated transformation of the emptystate, but we shall never see those complications.) From the definition of C we can work outhow the field transforms

C−1ψ(x)C =

∫d3p

(2π)3/2√

2ω

∑λ

(dλ(p)uλ(p)eix·p + b†λ(p)vλ(p)e−ip·x

)(4.16)

which we can relate to ψ†:

ψ†a(x) =

∫d3p

(2π)3/2√

2ω

∑λ

(b†λ(p)u∗λa(p)e−ix·p + dλ(p)v∗λa(p)eip·x

). (4.17)


But v∗ = iγ2u and u∗ = iγ2v so we can infer

C−1ψa(x)C = i(γ2)abψ†b(x). (4.18)

When we suppress spinor indices, it is usually convenient to think of ψ† as a row vector andψ as a column vector. To write (4.18) with suppressed indices we want the r.h.s. to be acolumn vector which we could indicate by (ψ†)T meaning the transpose on spinor indices.In that case (4.18) can be written

C−1ψ(x)C = iγ2(ψ†)T (x). (4.19)

It should also be noted that the occurrence of γ2 in the charge conjugation transformationlaw is specific to the standard representation; with other representations a different matrixwould appear.

The invariance of H is obvious from its expression in terms of creation and annihilationoperators, but it is also instructive to see it using the local definition

C−1HC =

∫d3xψT iγ2(

1

iα ·∇+ βm)iγ2(ψ†)T

= −∫d3xψT (−1

iαT · ∇+ βTm)(ψ†)T (4.20)

= H (4.21)

where the last step involves an integration by parts, a transposition of Dirac indices, and areordering of the order of ψ and ψ† giving a minus sign which cancels the overall minus signin the second line.

The transformation of the charge and current densities under C should simply changetheir signs. This is not hard to see:

C−1jµC =q

2

(ψT iγ2βγµiγ2(ψ†)T − ψ†iγ2γµTβiγ2ψ

)(4.22)

= −jµ (4.23)

where use is made of

iγ2γµiγ2 = −γµ∗ βγµβ = γµ†. (4.24)

Note that because we have used the symmetrized definition of the current, there is noreordering of operators necessary in arriving at this result.

4.3 Majorana Fermions

In our discussion so far it has seemed inevitable that the Dirac particle carries charge. Moreprecisely, it carries a conserved fermion number N which could be identified with charge. If


there were several species f of Dirac particle the fermion number Nf of each species mightbe separately conserved. Including terms of the form

ψfΓψf ′ (4.25)

with f ′ 6= f in the Hamiltonian could violate the individual Nf but∑

f Nf would still beconserved. Majorana pointed out that even with only one species of fermion it is possible tomake it totally neutral, i.e. carry no conserved quantum number at all.

Starting with the Dirac theory we can see that this is possible, because we can considerredefining creation and annihilation operators to be eigenoperators under charge conjugation:

bλ±(p) =1√2

(bλ(p)± dλ(p)) with C−1bλ±(p)C = ±bλ±(p). (4.26)

Then the Hamiltonian is the sum of two commuting pieces

H = H+ +H− with H± =

∫d3p ω(p)

∑λ

b†λ±(p)bλ±(p). (4.27)

Clearly, it is perfectly consistent to consider the quantum system defined by H+ (or H−)alone. The number operator of the Dirac theory

N =

∫d3p∑λ

(b†λ−(p)bλ+(p) + b†λ+(p)bλ−(p)), (4.28)

clearly has no meaning in the truncated theory, but that is to be expected.One might worry that truncating the theory in this way might spoil locality, but this is

not the case. We can just as easily redefine the local fields to be eigenoperators of chargeconjugation:

ψ±(x) =1√2

(ψ(x)± iγ2(ψ†)T (x)) with C−1ψ±C = ±ψ±, (4.29)

Which satisfy anticommutation relations

ψa±(x), ψb±(x′) = ±(iγ2)abδ(x− x′). (4.30)

Clearly

ψ±(x) =

∫d3p

(2π)3/2√

2ω

∑λ

(bλ±(p)uλ(p)eix·p ± b†λ±(p)vλ(p)e−ip·x

), (4.31)

and in terms of these fields

H± =1

2

∫d3xψ†±

(1

iα ·∇+ βm

)ψ± (4.32)

=1

2

∫d3x(±ψT±)iγ2

(1

iα ·∇+ βm

)ψ±. (4.33)


In the second line we have used ψ†± = ±ψT±iγ2, a consequence of (4.29).The appearance of iγ2 in the above discussion is due to our choice of the standard

representation for the gamma matrices. The Majorana representation is characterized by thecondition that the gamma matrices be pure imaginary γµ∗ = −γµ. In that case the chargeconjugation transformation does not involve a matrix at all and all of the iγ2’s disappear.

4.4 Weyl Fermions

In the case of massless fermions m = 0, it is possible to describe relativistic spin 1/2 particleswith only one helicity. In the Dirac theory the easiest way to see this is to consider the matrix

γ5 ≡ iγ0γ1γ2γ3, (4.34)

which anticommutes with the γµ. It therefore commutes with the Lorentz matrices

σµν =i

2[γµ, γν ]. (4.35)

γ5 commutes with the α · ∇ term of the Dirac Hamiltonian but not with the mβ term. Sowhen m = 0, and only then, the energy eigenstates can be simultaneously eigenstates of γ5.Since γ2

5 = 1 the eigenvalues of γ5, called chirality, are ±1 and

I ± γ5

2(4.36)

are projectors onto orthogonal two dimensional subspaces with chirality ±1 respectively.Defining

R =I + γ5

2ψ L =

I − γ5

2ψ, ψ = R + L (4.37)

R for “right-handed” and L for “left-handed,” the Dirac hamiltonian for m = 0 decomposesinto two commuting terms

H =

∫d3x

(R†

1

iα · ∇R + L†

1

iα · ∇L

)(4.38)

either of which could define a consistent dynamics of the corresponding subsystem. Thesesubsystems are called Weyl fermions. The corresponding momentum space spinors of definitechirality are

I ± γ5

2uλ(p) =

1

2

√|p|(1± 2λ)

(χλ±χλ

)= δλ,±1/2

I ± γ5

2uλ(p), (4.39)

from which it is clear that helicity is identical to Chirality/2. In other words right-handedWeyl fermions have helicity +1/2 and left-handed ones have helicity −1/2. Since γ5 is realand γ2 anticommutes with γ5, the antiparticle spinors

I ± γ5

2vλ(p) = iγ2

(I ∓ γ5

2uλ(p)

)∗= δλ,∓1/2

I ± γ5

2vλ(p) (4.40)


have the opposite correlation between chirality and helicity. So if the Weyl particle is righthanded the particle has helicity +1/2 and the antiparticle has helicity −1/2. Since chargeconjugation interchanges the role of particle and antiparticle one can even choose by con-vention all Weyl particles to be left(right)-handed. One can do this, for example, by writingR† = L′T iγ2. Of course, such an L′ has charge opposite to R.

The Weyl particle with helicity ±1/2 comes along with its antiparticle with helicity ∓1/2.Thus the Weyl system has the same helicity content as the massless Majorana system. Infact one can describe the Majorana system using Weyl fields. First, separate the Majoranafield ψ† = ψT iγ2 into two fields of definite chirality

ψR,L =I ± γ5

2ψ. (4.41)

Then notice that

ψ†R = ψ†I + γ5

2(4.42)

= ψT iγ2 I + γ5

2(4.43)

= ψTL iγ2 (4.44)

so that the right-handed component of the Majorana field can be eliminated in favor ofthe hermitian conjugate of the left-handed component. When this is done the Majoranahamiltonian simplifies to

HMaj =

∫d3xψ†L

1

iα · ∇ψL +

m

2

∫d3x[ψTL iγ

2βψL︸︷︷︸∆F=−2

+ (ψTL iγ2βψL)†︸︷︷︸

∆F=+2

] (4.45)

which reduces to the Weyl hamiltonian for m = 0. Notice that the massless limit conservesfermion number but the mass term violates fermion number conservation by ±2 units. Thisis the so-called Majorana mass term for a Weyl fermion. If the system contains severalleft handed Weyl fermions, Lk then one can consider a general Majorana mass term whichincludes off diagonal couplings

mkl

2LTk iγ

2βLl +m∗kl2

(LTk iγ2βLl)

†. (4.46)

Since the matrix iγ2β is antisymmetric and the Lk obey anticommutation relatioins, thematrix mkl can be taken to be symmetric although it is allowed to be complex.

To construct the Dirac mass term which conserves fermion number, one must add aright-handed Weyl fermion to the theory. Writing the Dirac field ψ = L + R we find thatLL = RR = 0 because (I ∓ γ5)β(I ± γ5) = 0 and

mψψ = mRL+mLR. (4.47)


But we can relate R to a left handed field by R† = L′T iγ2 after which the Dirac mass termtakes the form

mψψ = mL′T iγ2βL−mL†iγ2βL′†T = mL′T iγ2βL+ (mL′T iγ2βL)† (4.48)

Written this way in terms of a pair of left handed Weyl fields, the Dirac mass is seens as aspecial case of the Majorana mass (4.46) with m11 = m22 = 0 and m12 = m21 = m.

The Weyl theory violates parity invariance, essentially because β fails to commute withγ5:

P−1L(x)P =I − γ5

2βψ(−x) = βR(−x). (4.49)

How then have we managed to show that it is equivalent to the parity conserving Majoranatheory? The answer is that although parity is violated in the Weyl theory, CP the productof parity times charge conjugation remains a symmetry. The Majorana field is inert undercharge conjugation, and has only parity as a nontrivial symmetry. It is the CP symmetryof the Weyl theory that corresponds to the parity of the Majorana theory.

Notice that it is impossible to have a fermion that is simultaneously Majorana and Weyl:even at zero mass one always must have both helicities. This is true in four space-timedimensions but in other dimensions it need not be so. For example in 10 dimensions onecan define Majorana-Weyl fermions. A Dirac fermion in D = 2k dimensions has 2k degreesof freedom, 2k−1 states for the particle and the same number for the antiparticle. In somedimensions (including 4 and 10) one can have Majorana fermions with only 2k−1 degrees offreedom. In the massless case one can define Weyl fermions in all even dimensions giving2k−1 degrees of freedom. In 2 + 8n dimensions one can have Majorana-Weyl fermions withonly 2k−2 degrees of freedom. For example in 10 dimensions a Dirac fermion has 32 states,a Majorana or Weyl fermion has 16 states, and a Majorana-Weyl fermion has only 8 states.This possibility is crucial for the consistency of superstring theory.

4.5 Time Reversal

The last discrete symmetry we discuss is time reversal T . It is well-known that T must bean antiunitary transformation, meaning that it is antilinear, and furthermore

〈TΦ|TΨ〉 = 〈Ψ|Φ〉 . (4.50)

With this in mind, we search for a transformation of the form

T−1ψ(x, t)T = T ψ(x,−t), (4.51)

with T an appropriate matrix. From antiunitarity we have⟨Φ|T−1ψTΨ

⟩= 〈ψTΨ|TΦ〉 =

⟨TΨ|ψ†TΦ

⟩=⟨T−1ψ†TΦ|Ψ

⟩, (4.52)


from which it follows that

T−1ψ†(t)T = (T−1ψT )† = ψ†(−t)T †. (4.53)

Thus

T−1HT =

∫d3xψ†(x,−t)T †

(−1

iα∗ · ∇+ βm

)T ψ(x,−t). (4.54)

If we choose T to be unitary, then invariance of H will be achieved (given conservation ofenergy dH/dt = 0, which follows from the field equation) if and only if T commutes withβ and α2 and anticommutes with α1 and α3. Clearly the most general solution of theseconditions is

T = eiτγ1γ3 = ieiτΣ2, (4.55)

so that the transformation law becomes

T−1ψ(x, t)T = ieiτΣ2ψ(x,−t). (4.56)

This transformation law on ψ implies that for b, d. It is easiest to do this for the rest frameΣ3 basis, because then the two-spinors φµ are real. Thus

u∗µ(−p) =√ω(p) +m

(φµ

−σ∗·pm+ω(p)

φµ

)(4.57)

= Σ2

√ω(p) +m

(σ2φµ

σ·pm+ω(p)

σ2φµ

)(4.58)

= i2µΣ2u−µ(p) (4.59)

Using v = iγ2u∗, it is just a few steps to show that

v∗µ(−p) = i2µΣ2v−µ(p). (4.60)

Because T is antilinear, the l.h.s. of (4.56) involves u∗e−ix·pT−1bT and v∗e+ix·pT−1d†T , so(4.59) and (4.60) allow us to infer from (4.56) that

T−1bµ(p)T = ieiτ i−2µb−µ(−p) T−1b†µ(p)T = −ie−iτ i+2µb†−µ(−p) (4.61)

T−1d†µ(p)T = ieiτ i−2µd†−µ(−p) T−1dµ(p)T = −ie−iτ i+2µd−µ(−p) (4.62)

The reversal of signs of momentum and spin label is intuitively correct since time reversinga motion reverses both momentum and angular momentum. The µ dependence of the phaseis perhaps less intuitive, but follows straightforwardly by using angular momentum raisingand lowering operators together with the action of time reversal on angular momentum. Ifwe had used the helicity basis, the helicity label would not be reversed by time reversal(remember it is J · p/|p|); unfortunately the phases that are induced are angle dependentand not very illuminating.


4.6 Violation of the Discrete Symmetries and the CPT

Theorem

Having shown that C, P , and T are symmetries of the Dirac equation it is instructive tocontemplate how things must be changed to violate these symmetries. For example, we haveseen that both parity and charge conjugation are violated with Weyl fermions, but in sucha way that CP remains a symmetry. More generally, one can consider adding noninvariantterms to the Hamiltonian. In exercises, it is shown how the bilinears ψAΓψB transform underthese symmetries for Γ = (I, iγ5, γ

µ, γ5γµ, σµν). Under parity they transform with a factor of

(+,−,+,−,+)(−)S times the bilinear evaluated with x→ −x, and where S is the numberof spatial indices in the tensor component. So, examples of parity odd Lorentz invariantswould be

ψiγ5ψ ψγµψψγ5γµψ. (4.63)

Adding such terms to the energy density would appear to violate parity. One must be carefulthat the violation is not an illusion. For example the term ψ†α · ∇ψ is invariant under thechiral symmetry

ψ → eiαγ5ψ (4.64)

under which

ψψ → cos 2αψψ + ψiγ5ψ sin 2α (4.65)

so the added term ψiγ5ψ can be rotated into ψψ and parity violation disappears.

With regard to charge conjugation, ψAΓψB transforms to (+,+,−,+,−) times ψBΓψA.For example the first of (4.63) is invariant under charge conjugation (if A = B) but thesecond is odd. Under CP the result is (+,−,−,−,−)(−)S. Thus it is the first of (4.63) thatwould violate CP . Since such a term can be rotated away by a chiral transformation we seethat CP is a bit tricky to violate. For example in the Standard Model one needs at leastthree generations of quarks and leptons to frustrate the ability to transform away apparentlyCP violating couplings! Fortunately there is solid evidence for this number of generations.1

Finally we come to time reversal, under which the bilinears can be shown to transforminto (+,−,+,+,−)(−)S times the bilinear with t → −t. Note that ψγµψ transforms asexpected for a current and ψσklψ as expected for an angular momentum. It is only the firstof (4.63) that violates time reversal: it is as tricky to violate as CP . In fact there is a deepconnection between T and CP known as the CPT theorem.

1The simplest version of QCD, the strong interaction sector of the standard model, violates the chiralsymmetry used to rotate away ψiγ5ψ. Then one could get CP violation with a smaller number of generations.To be compatible with the experimental size of CP violation the coefficient of such a term would have tobe so tiny that a modified form of QCD which restores this symmetry (and predicts axions) is usuallypostulated. Then one is back to the three generation requirement.


Composing the three discrete symmetries we find

(CPT )−1ψ(x)CPT = e−iτ (iγ2γ0γ1γ3)ψ(−x) = −e−iτγ5(ψ†)T (−x). (4.66)

Now applying this transformation to the bilinears, remembering the antilinear property ofCPT , we find

(CPT )−1ψA(x)ΓψB(x)CPT = ψTA(−x)γ5βΓ∗γ5(ψ†B)T (−x) (4.67)

= −ψB(−x)βγ5Γ†βγ5ψA(−x) (4.68)

= (−)nΓψB(−x)βΓ†βψA(−x) (4.69)

= (−)nΓ(ψA(−x)ΓψB(−x))† (4.70)

where nΓ is the number of Lorentz indices carried by Γ. The CPT theorem states theimpossibility of violating this symmetry in quantum field theory. We shall not go throughthe rigorous proof here, but from the transformation law of the bilinears it is clear what isbehind the theorem. Since each Lorentz index must be contracted with another in forming aLorentz scalar polynomial of the bilinears, all of the (−)nΓ ’s will cancel in the CPT transformof the polynomial. If we denote the Hamiltonian by some function H(ψA(x)ΓψB(x)) of thebilinears, we have

(CPT )−1H(ψA(x)ΓψB(x))CPT = H∗((CPT )−1ψA(x)ΓψB(x)CPT ) (4.71)

= H∗((ψA(−x)ΓψB(−x))†) (4.72)

where by H∗ we mean that all of the complex numbers appearing in the formation of H asa function of the bilinears are complex conjugated. Apart from ordering of operators, thelast line is just what we mean by the hermitian conjugate of H, if we set t = 0 (conservationof energy means H is constant) and integrate over x. So up to operator ordering questions(which can be sorted out for local interactions), a hermitian Hamiltonian must be CPTinvariant.


Chapter 5

Time Dependent Perturbation Theory

Although there are special quantum field theories, for example some in 2 space-time dimen-sions, that can be solved exactly, the exact solution of realistic interacting quantum fieldtheories in 4 space-time dimensions is beyond reach. There are important cases for whichlinear QFT’s in the presence of certain external fields can be solved. For example the Diracequation in a Coulomb potential admits an exact solution for which one can find all the en-ergy eigenvalues and eigenstates. The Coulomb potential is of special importance since theexact solution in that case is the starting point for the relativistic theory of atomic energylevels. Still the complete dynamics is never exactly given by these special cases and perturba-tion theory is the important tool for evaluating corrections to the exactly soluble (idealized)case, which can be free field theory with no external fields or one of the above cases. Whenwe quantize the electromagnetic field, perturbation theory is essentially our only tool forcomputing radiative corrections due to the quantum nature of the electromagnetic field.

5.1 Heisenberg and Schrodinger Pictures

In the Schrodinger picture the quantum dynamics is given by the Schrodinger equation

i~∂

∂t|Φ, t〉 = HS(t)|Φ, t〉 (5.1)

where we stress that we allow time varying external forces to be present (hence the timedependence of H). The field equations arise in the Heisenberg picture wherein the timedependence resides in the operators rather than in the system states which are constant intime. To pass to the Heisenberg picture we write

|Φ, t〉 = U(t)|Φ, 0〉 (5.2)

where

i~∂

∂tU = HS(t)U U(0) = I, (5.3)

61

and give the time independent Schrodinger picture operators Ω time dependence accordingto

Ω(t) ≡ U †ΩU. (5.4)

The Heisenberg picture Hamiltonian is similarly related to the Schrodinger picture one by

H(t) ≡ U †HS(t)U. (5.5)

Then the Heisenberg picture operators corresponding to constant Schrodinger picture oper-ators satisfy the Heisenberg equations

i~Ω(t) = [Ω(t), H(t)]. (5.6)

It is most natural to formulate the time dependence in quantum field theory using Heisen-berg picture, since the field operators will then satisfy equations of motion that are the directquantum analogue of the classical field equations. We shall therefore always understand H(t)without subscripts to be the Hamiltonian in Heisenberg picture. When we work with anyother picture we will attach a subscript to H, e.g. HS(t) is the Hamiltonian in Schrodingerpicture.

5.2 Asymptotic States and Matrix Elements

In discussing time dependent processes, it is convenient to introduce asymptotic states whichare eigenstates of H(±∞). We denote by |in〉 the ground state of H(−∞) and by |out〉 theground state of H(+∞). The normal situation will be one in which all external fields vanishat sufficiently early and late times. Thus |in〉 and |out〉 will typically be ground states ofH0(−∞) andH0(+∞) respectively. Although these operators are not the same (because theirtime evolution is governed by H not H0), the spectra of the two Hamiltonians are identical:H0(t) = U−1(t,−∞)H0SU(t,−∞). By convention we are identifying the Schrodinger andHeisenberg pictures at t = −∞. Thus, if |in〉 is the ground state of H0(−∞) = H0S, thestate 〈in|U(∞,−∞) is an eigenstate of H0(+∞) with the same eigenvalue and hence theground state. Thus we can and shall fix phases by defining

〈out| ≡ 〈in|U(∞,−∞). (5.7)

We stress that this is the true “out” state only when HS(∞) = HS(−∞) ≡ H0.If the time dependence of HS is adiabatic, i.e. very slow on the time scale set by the

level spacings, the Adiabatic Theorem assures us that an eigenstate of HS(−∞) evolves toan eigenstate of HS(t) for all t for which adiabatic conditions apply, even after a long enoughtime to change HS by a finite amount. For example, the state |in〉 will be an eigenstateof H(t) for all t for which adiabatic time variation applies. In particular, the ground stateeigenvalue EG(t) must not get close to the next higher eigenvalue as t varies. If this situationholds for all time, it follows that the state |in〉 is a phase times the state |out〉, or 〈out| is this


same phase times 〈in|. This phase is easily evaluated in terms of the time dependent groundstate energy EG(t) of HS(t) by applying the Schrodinger equation to 〈in|U(t,−∞)|in〉 andusing the adiabatic theorem HS(t)U(t,−∞)|in〉 = EG(t)U(t,−∞)|in〉:

〈out|in〉 = exp

−i∫ ∞−∞

dtEG(t)

Adiabatic Conditions. (5.8)

Note carefully that adiabatic conditions would not apply if the ground state energy got closeto an excited level as time evolved. In particular, it would not apply in processes with pairproduction when | 〈out|in〉 | < 1.

5.3 General Formalism

We shall keep the initial discussion completely general and consider the situation in whichthe Heisenberg picture Hamiltonian is the sum of two pieces,

H(t) = H0(t) +H ′(t) (5.9)

where H0 can be exactly dealt with and H ′ is “small” in an appropriate sense. Note thateven when H is independent of time, H0 and H ′ still depend on time through the timedependence of the Heisenberg operators that enter it. The Heisenberg equations for thedynamical variables have the form

i∂Ω

∂t= [Ω, H(t)]. (5.10)

The goal of time dependent perturbation theory is to expand the evolution operator U(t, t0)which carries the time dependence of the Heisenberg picture operators Ω = U−1ΩSU , oralternatively the time dependence of the Schrodinger picture system states |ψ(t)〉 = U |ψ(t0)〉,in a power series in H ′. A complication is that there is time dependence in U even whenthe perturbation vanishes. To systematically deal with this complication a new InteractionPicture (sometimes called the Dirac Picture) has been devised in which the operators carrythe (known) time dependence due to H0 and the perturbation only enters the modifiedevolution operator UI , which is constructed to be the identity in the absence of H ′.

Thus for each Heisenberg picture operator with no explicit time dependence, we definean interaction picture operator by

ΩI(t) ≡ UI(t)Ω(t)U−1I (t) (5.11)

and require that ΩI satisfies the Heisenberg equation with Hamiltonian H0I = UIH0(t)U−1I

1:

i∂ΩI

∂t= [ΩI , H0I(t)]. (5.12)

1Note that if H0 = H0(Ωk(t), t), then H0I = H0(ΩIk(t), t).


differentiating (5.11) we find the requirement

[ΩI , H0I(t)] = iUIΩU−1I + iUIΩU

−1I + UI [Ω, H(t)]U−1

I (5.13)

= iUIU−1I ΩI + iΩIUIU

−1I + [ΩI , H0I(t)] + [ΩI , H

′I(t)] (5.14)

= [ΩI , H0I(t)] + [ΩI , H′I(t)− iUIU−1

I ]. (5.15)

Thus the equation for UI is just

iUI(t) = H ′I(t)UI(t) = UI(t)H′(t). (5.16)

We choose the initial condition UI(t0) = I, in which case it is a good idea to display two timearguments UI(t, t0) as we did for U . Notice that since U relates Heisenberg and Schrodingerpictures, the equation (5.3) for U can be also written

iU = UH(t) (5.17)

from which it is clear that we can express U = U0UI where

iU0 = U0H0I(t) (5.18)

To expand UI in powers of H ′ it is convenient first to incorporate initial condition infor-mation by writing the integral equation

UI(t, t0) = I − i∫ t

t0

dt′H ′I(t′)UI(t

′, t0), (5.19)

and then to generate the perturbation series by iteration

UI(t, t0) = I − i∫ t

t0

dt′H ′I(t′)(I − i

∫ t′

t0

dt′′H ′I(t′′)UI(t

′′, t0) (5.20)

= I − i∫ t

t0

dt′H ′I(t′) + (−i)2

∫ t

t0

dt′∫ t′

t0

dt′′H ′I(t′)H ′I(t

′′) + · · · . (5.21)

There is a useful way to summarize the entire perturbation series, which employs theconcept of the time ordered product of operators. Consider a set of operators each associ-ated with a different time, A1(t1), A2(t2), . . . , AN(tN). The time ordered product of theseoperators is defined as the ordinary product with the operators ordered according to the timeargument: the operator Ak(tk) to the left of Al(tl) if tk > tl. If there are any anticommutingoperators in the set, there is also an overall −1 if one achieves the time ordering by an oddpermutation of fermionic operators. Thus, for example,

T [A(t1)B(t2)] =

A(t1)B(t2) t1 > t2

±B(t2)A(t1) t2 > t1(5.22)

with the − for both operators fermionic. Now the factors of H ′I in the series for UI are alltime ordered due to the limits of integration. If we make use of the time ordering symbol, we


can extend all integrations to the full range t0 < t′ < t provided we divide the nth term byn! to account for the overcounting due to the n! orderings of the t’s. Thus the entire seriesbecomes

UI(t, t0) =∞∑n=0

1

n!(−i)n

∫ t

t0

dt1dt2 · · · dtnT [H ′I(t1)H ′I(t2) · · ·H ′I(tn)]. (5.23)

If it weren’t for the time ordering symbol this would be just the exponential series. It istherefore a useful mnemonic to write

UI(t, t0) = Te−i

∫ tt0dt′H′I(t′)

(5.24)

where it is understood that t > t0. This equation is known as the Dyson Formula. Since theformula just reflects the equation UI satisfies, we can write a similar formula for the full U :

U(t, t0) = Te−i

∫ tt0dt′HS(t′)

= T e−i

∫ tt0dt′H(t′)

= T e−i

∫ tt0dt′H0,I(t′)

Te−i

∫ tt0dt′H′I(t′)

(5.25)

where T denotes anti-time ordering (later times to the right).It will be useful to extend the definition of UI(t, t0) to times earlier than t0. We shall do

this in a way to preserve the closure property

UI(t, t1)UI(t1, t0) = UI(t, t0), (5.26)

which follows from the differential equation and initial condition for t > t1 > t0. If we sett = t0 in (5.26), the r.h.s. is just I so we have to define

UI(t, t0) ≡ U−1I (t0, t) = U †I (t0, t) for t < t0. (5.27)

(Note that U †IUI = I is a simple consequence of the differential equation and the hermiticityof H ′I .) It is then simple to check that (5.26) holds for all time orderings.

Next let us show how to express various physical quantities in the Interaction picture.One interesting quantity is the so called vacuum persistence amplitude 〈out|in〉, given by

〈out|in〉 = 〈in|U(∞,−∞)|in〉 = 〈in|U0(∞,−∞)UI(∞,−∞)|in〉. (5.28)

We shall identify all pictures at t = −∞. In the usual situation where external fields vanishat early times the state |in〉 will be the ground state of H0(−∞) = H0I . Furthermore, withno external fields in H0I(t) the latter will be time independent for all time (since its timeevolution is governed by H0I itself). In this situation

U0(t, t0) = e−iH0I(t−t0) (5.29)

and 〈in| is an eigenstate of U0(t, t0) with eigenvalue e−iE0(t−t0), where E0 is the ground stateenergy of H0I . By convention we can choose our zero of energy so that E0 = 0, in whichcase we have

〈out|in〉 = 〈0, I|UI(∞,−∞)|0, I〉 E0 = 0. (5.30)


The persistence amplitude carries a lot of information, because it can be defined forany choice of external fields. Its dependence on these external fields can then be exploitedto obtain numerous matrix elements relevant to the zero field situation. We shall see manyapplications of this remark in the course of our studies. One can also get the energy spectrumfor static external fields from this amplitude by switching them on at some early time keepingthem constant for a long time 2T and then switching them off. The T dependence of 〈out|in〉will then display the dependence e−iEk2T from which the energy eigenvalues can be read off.The states that are probed by this device will depend on the manner of the switching onprocedure. For adiabatic switching on, only the ground state in the presence of the staticfield will contribute.

5.4 Scattering in an External Field: Born Approxima-

tion

5.4.1 Scattering of scalar particles

One can equally well choose initial and final states that contain particles. For example, theamplitude for a scalar particle with momentum p initially making a transition to p′ at verylate times is

〈0, I|a(p′)UI(∞,−∞)a†(p)|0, I〉 ≈

δ(p′ − p)− i∫ ∞−∞

dt〈0, I|a(p′)H ′I(t)a†(p)|0, I〉, (5.31)

where we kept only terms to first order. As a concrete example, consider the interaction∫dtH ′I(t) = −

∫d4xφ2

I(x)B(x)/2, where B(x) is a fixed external scalar field. Since the φIare free fields, they can be expressed in terms of creation and annihilation operators and thematrix element evaluated:∫ ∞

−∞dt〈0, I|a(p′)H ′I(t)a

†(p)|0, I〉 (5.32)

= −∫d4xB(x)ei(p−p

′)·x 1

(2π)3√

2ω(p)√

2ω(p′)(5.33)

= − 1

(2π)3√

2ω(p)√

2ω(p′)B(p′ − p), (5.34)

where B(q) =∫d4xB(x)e−iq·x is the Fourier transform of the external field. Note that in

the case B is static the time integral gives a factor of 2πδ(ω′ − ω).Recall from basic scattering theory that if the scattering matrix for a particle from a

static potential is written

〈q, out|p, in〉 = δ(q − p)− 2πiδ(ω(q)− ω(p))T (q,p) (5.35)


then the differential scattering cross section is given by

dσ

dΩ=

d3q

dΩδ(ω(q)− ω(p))

(2π)4

v|T (q,p)|2, (5.36)

= q2dqδ(ω(q)− ω(p))(2π)4

v|T (q,p)|2, (5.37)

= pω(p)(2π)4

v|T (q,p)|2, (5.38)

= ω(p)2(2π)4|T (q,p)|2, (5.39)

where v is the speed of the incident particle. Defining the spatial Fourier transform B(k) ≡∫d3xe−ik·xB(x), we then obtain

TBorn(q,p) = − 1

(2π)32ω(p)B(q − p), (5.40)

giving the cross section

dσ

dΩ

Born

=1

16π2|B(q − p)|2. (5.41)

5.4.2 Scattering of Dirac particle

Another scattering example, the amplitude for a Dirac particle with momentum and helicityp, λ initially making a transition to p′, λ′ at very late times is

〈0, I|bλ′(p′)UI(∞,−∞)b†λ(p)|0, I〉 ≈

δλ′λδ(p′ − p)− i

∫ ∞−∞

dt〈0, I|bλ′(p′)H ′I(t)b†λ(p)|0, I〉, (5.42)

where we kept only terms to first order. In the case of a weak external electromagnetic field,∫dtH ′I(t) = −

∫d4xjµI (x)Aµ(x), where jµI = q

2[ψI , γ

µψI ]. Since the ψI are free fields, theycan be expressed in terms of creation and annihilation operators and the matrix elementevaluated:∫ ∞

−∞dt〈0, I|bλ′(p′)H ′I(t)b

†λ(p)|0, I〉

= −q∫d4xAµ(x)ei(p−p

′)·x 1

(2π)3√

2ω(p)√

2ω(p′)uλ′(p

′)γµuλ(p) (5.43)

= −q 1

(2π)3√

2ω(p)√

2ω(p′)uλ′(p

′)γµAµ(p′ − p)uλ(p), (5.44)

where A(q) =∫d4xA(x)e−iq·x is the Fourier transform of the potential. Note that in the

case A is static the time integral gives a factor of 2πδ(ω′ − ω).


Recall from basic scattering theory that if the scattering matrix for a particle from astatic potential is written

〈q, out|p, in〉 = δλ′λδ(q − p)− 2πiδ(ω(q)− ω(p))Tλ′λ(q,p) (5.45)


dσ

dΩ=

d3q


(2π)4

v|Tλ′λ(q,p)|2, (5.46)

= q2dqδ(ω(q)− ω(p))(2π)4

v|Tλ′λ(q,p)|2, (5.47)

= pω(p)(2π)4

v|Tλ′λ(q,p)|2, (5.48)

= ω(p)2(2π)4|Tλ′λ(q,p)|2, (5.49)

where v is the speed of the incident particle.Defining A(k) ≡

∫d3xe−ik·xA(x), we then obtain

TBornλ′λ (q,p) = − 1

(2π)32ω(p)uλ′(q)γ · A(q − p)uλ(p), (5.50)

giving the cross section

dσ

dΩ

Born

=1

16π2|uλ′(q)γ · A(q − p)uλ(p)|2. (5.51)

Consider the example of the scattering of an electron with q = −e from the Coulombpotential of a nucleus of atomic number Z (Mott Scattering), A0 = Ze/4πr, A = 0. ThenA0(k) = Ze/k2 and

dσ

dΩ

Born

=e4Z2

16π2(q − p)4|uλ′(q)γ0uλ(p)|2 =

α2Z2

(q − p)4|uλ′(q)γ0uλ(p)|2. (5.52)

Here we have introduced the fine structure constant α = e2/4π ≈ 1/137. Evaluating thespinor matrix element in terms of two component helicity spinors leads to (using |q| = |p|)

uλ′(q)γ0uλ(p) =

[ω(p) +m+

4λλ′p2

ω +m

]χ†λ′(q)χλ(p). (5.53)

The absolute square of χ′†χ can be evaluated by noting that the 2 × 2 matrix χχ† is aprojector onto the spin state of definite helicity:

χλ(p)χ†λ(p) =1 + 2λp · σ

2. (5.54)

Thus we have

|χ†λ′(q)χλ(p)|2 =1

4tr[(1 + 2λp · σ)(1 + 2λ′q · σ)] (5.55)

=1

2(1 + 4λλ′p · q) (5.56)


Inserting all this into the formula for the differential cross section, we obtain after simplifying

dσ

dΩ

Born

=α2Z2

(q − p)4[ω2 +m2 + q · p+ 4λλ′(p2 + (ω2 +m2)p · q)] (5.57)

To compare all of the details of this formula with experiment we would have to preparea polarized beam of electrons with definite helicity and also measure the spin of the finalelectron. A noteworthy feature of such a complete experiment is that at high energies thereis an overall factor of (1 + 4λλ′) = 2δλλ′ , which means that helicity is conserved at highenergy. If we don’t measure the final spin we should sum over λ′ = ±1/2 to obtain

dσ

dΩ

Born

Unobserved spin= 2

α2Z2

(q − p)4[ω2 +m2 + q · p]. (5.58)

Similarly, if we have a completely unpolarized beam, we need to average over λ to obtain

dσ

dΩ

Born

unpol=

α2Z2

(q − p)4[ω2 +m2 + q · p], (5.59)

independent of the final spin.Two simplifying limits can be considered. The nonrelativistic or low energy limit p2 <<

m2 is

dσ

dΩ

NR

∼ 2m2 α2Z2

(q − p)4[1 + 4λλ′p · q] (5.60)

Apart from the helicity dependence due to the spin of the electrons this is just the Rutherfordformula. The opposite limit, the ultrarelativistic or high energy limit p2 >> m2 is (assumep · q 6= −1)

dσ

dΩ

UR

∼ α2Z2

(q − p)4[p2 + q · p][1 + 4λλ′]. (5.61)

where the high energy helicity conservation is transparent.

5.5 Pair production in a time varying external field

Let us return to the scalar field external field perturbation H ′I(t) = −∫d3xφ2

IB(x)/2 in thecase where the initial state is the ground state of H0,I , |0〉I , and the final state contains twoparticles: 〈0, I|a(~p1)a(~p2). The transition amplitude to first order is

Tfi =i

2

∫d4x〈0, I|a(~k1)a(~k2)φ2

I(x)B(x)|0, I〉 (5.62)

=i

(2π)32√ω1ω2

∫d4xe−i(k1+k2)·xB(x) ≡ iB(k1 + k2)

(2π)32√ω1ω2

(5.63)


Squaring and integrating over final momenta we get the total pair production probability

Ppair =1

2

∫d3k1d

3k2

(2π)64ω1ω2

|B(k1 + k2)|2

=1

4(2π)5

∫d4Kθ(−K2 − 4m2)

√1 +

4m2

K2|B(K)|2 (5.64)

where the 1/2 out front corrects for double counting the same final state in doing the k1, k2

integration. Since ω1 + ω2 > 2m, Ppair 6= 0 only when B oscillates with frequencies greaterthan 2m. In particular a static external field will not produce pairs in perturbation theory.

5.6 Perturbation theory for Time Ordered Products

Another class of quantities that will be very useful to us is the matrix element of the timeordered product of a finite number of Heisenberg picture fields between asymptotic states:

〈out|T [A1(t1)A2(t2) · · ·AN(tN)]|in〉. (5.65)

The simplest way to transcribe this matrix element to interaction picture is to first assumethe ordering t1 > t2 · · · > tN so that the T symbol can be removed. Then

〈out|A1(t1)A2(t2) · · ·AN(tN)|in〉 (5.66)

= 〈0, I|UI(∞,−∞)U−1I (t1,−∞)AI1(t1)UI(t1,−∞)U−1

I (t2,−∞)AI2(t2)

UI(t2,−∞) · · ·U−1I (tN ,−∞)AIN(tN)U(tN ,−∞)|0, I〉 (5.67)

= 〈0, I|UI(∞, t1)AI1(t1)UI(t1, t2)AI2(t2)UI(t2, t3) · · ·UI(tN−1, tN)AIN(tN)

UI(tN ,−∞)|0, I〉, (5.68)

where use has been made of the closure property of UI . Now we notice that all of theinteraction picture operators that appear in the final matrix element, including those in theDyson formula for each UI are time ordered. Thus if we insert the time ordering symbol infront of all the operators we can combine all of the UI ’s into a single UI(∞,−∞) arriving at

〈out|T [A1(t1)A2(t2) · · ·AN(tN)]|in〉= 〈0, I|T [UI(∞,−∞)AI1(t1)AI2(t2) · · ·AIN(tN)]|0, I〉. (5.69)

Finally, we simply note that had the time ordering been any other, the same steps wouldhave led to the same final result.

In the usual situation where H0I is the Hamiltonian for free fields, all of the interactionpicture operators are free fields, and to evaluate each finite order in perturbation theoryone only needs to master the computation of the vacuum expectation values of the timeordered product of a finite number of free fields. Free fields can always be expressed as alinear functional of creation and annihilation operators. Thus if φk(x) is a free field, it canbe written

φk(x) = φ+k (x) + φ−k (x) (5.70)


where φ+k (x) annihilates |0〉 and φ−k (x) annihilates 〈0|. Thus we have

〈0|T [φk(x)φl(x′)]|0〉

= θ(t− t′)〈0|φ+k (x)φ−l (x′)|0〉 ± θ(t′ − t)〈0|φ+

l (x′)φ−k (x)|0〉 (5.71)

≡ θ(t− t′)Ckl(x− x′)± θ(t′ − t)Clk(x′ − x) (5.72)

where Ckl(x− x′) = [φ+k (x), φ−l (x′)]± is a c number since the fields are free.

Now consider a general time ordered product of N free fields

〈0|T [φ1(x1) · · ·φN(xN)]|0〉 (5.73)

and first assume t1 > t2 > · · · > tN . Then the leftmost field is φ1 and it can be replaced byits annihilation part φ+

1 , which is then moved via the commutation relations all the way tothe right where it kills the vacuum. The (anti)commutators

[φ+1 (x1), φk(xk)]± = [φ+

1 (x1), φ−k (xk)]± = 〈0|T [φ1(x1)φk(xk)]|0〉 (5.74)

since the (anti)commutators are c numbers and t1 > tk by assumption. Thus

〈0|T [φ1(x1) · · ·φN(xN)]|0〉= 〈0|T [φ1(x1)φ2(x2)]|0〉〈0|T [φ3(x3) · · ·φN(xN)]|0〉±〈0|T [φ1(x1)φ3(x3)]|0〉〈0|T [φ2(x2)φ4(x4) · · ·φN(xN)]|0〉 ± · · · (5.75)

where the sign in front of each term is dictated by the number of times the order of fermionicoperators is switched. The time ordering symbol is not needed with our assumed orderingof times. But now we notice that if the time ordering had been any other the same stepswould have led to the same result provided we keep the T symbol in place. Thus we haverelated the vacuum expectation value of the time ordered product of N free fields to thoseof 2 free fields and N − 2 free fields. By induction we can therefore express the vacuumexpectation value of the time ordered product of N free fields as sums of products of thevacuum expectation values of the time ordered product of pairs of free fields.

The result, known as Wick’s Theorem, can be expressed as follows. First note that theanswer is 0 unless N is even. Then the vacuum expectation value of the time ordered productof N free fields is the sum of terms, one for each distinct pairing off of all the N fields. Theterm for each such pairing off is simply ± the product of the vacuum expectation values ofthe time ordered product of each pair of fields in the given pairing off. The sign is determinedby comparing the ordering of the N operators in the original time ordered product with theorder they appear in the given term after being paired off. If the latter ordering is achievedby an odd permutation of fermionic operators the sign is −; otherwise it is +. It doesn’tmatter what order we display the factors within a given term, since switching their orderwould always be an even permutation: a pairing of a boson field with a fermion field wouldalways contribute zero!


5.7 A Technical Comment on Time Derivatives in Time

Dependent Perturbation Theory

It is important to appreciate some subtle differences between time derivatives of operatorsin different pictures. For example, interaction picture depends on a specific breakup of theHeisenberg picture Hamiltonian H = H0 +H ′, so in Heisenberg picture

Ω =1

i[Ω, H0] +

1

i[Ω, H ′]. (5.76)

The transformation to interaction picture, being a purely algebraic similarity transformationshows that

(Ω)I = 1i[ΩI , H0I ] + 1

i[ΩI , H

′I ] (5.77)

= ΩI + 1i[ΩI , H

′I ], (5.78)

so there is in general a discrepancy between the interaction picture operator correspondingto Ω and the time derivative of the operator ΩI . When we use the Dyson formula for timedependent perturbation theory to calculate a matrix element involving Ω, care must be takenabout this difference. However there is a very simple prescription to keep things straight.This is to always think of time derivatives of operators in the Dyson formula as acting outsidethe time ordering symbol. Note the following identity

∂

∂tT [e−i

∫∞−∞ dt′H′I(t′)ΩI(t)] = T [e−i

∫∞t dt′H′I(t′)](ΩI(t)

+1

i[ΩI(t), H

′I(t))])T [e−i

∫ t−∞ dt′H′I(t′)]

= T [e−i∫∞−∞ dt′H′I(t′)(Ω(t))I ]. (5.79)

This comment becomes particularly useful in cases such as scalar electrodynamics where therelation between π†(x) and φ(x) = π†(x) + iQA0φ(x) involves the interaction. Since theinteraction picture fields are free, the relationship in that picture is φI(x) = π†I(x). Thus theexponent in the Dyson formula shows a disquieting asymmetry between space and time:∫

dtH ′I(t) =

∫d4x

(iQA · (φ†I∇φI − (∇φ†I)φI) +Q2A2φ†IφI

−iQA0(φ†I φI − φ†φI)

). (5.80)

However, it is possible to prove a “reshuffling theorem” that if all time derivatives in theDyson formula are understood to be taken outside the time ordering symbol, covariance isrestored. In other words there are two sources of apparent non -covariance: the form of H ′Iand the time ordering operation itself. To present the results of the reshuffling theorem,we introduce the symbol T ∗ to signify time ordering in which all time derivatives are taken


outside the time ordering symbol. Then the reshuffling theorem for scalar electrodynamicscan be stated

T [e−i∫∞−∞ dt′H′I(t′)] = T ∗

[e−i

∫d4x(iQAµ(φ†I∂µφI−(∂µφ

†I)φI)+Q2AµAµφ

†IφI)]. (5.81)

When employing the Wick expansion to the r.h.s. one simply needs to remember that onenever uses a quantity such as 〈0|T [∂µφ(x)φ†(y)]|0〉, namely all derivatives occur outside notinside the time ordering symbols.

As an illuminating example of these ideas we quote the improved Dyson formula for theoutin matrix element of the current operator:

〈out|jµ(x)|in〉 = 〈0, I|T ∗[e−i

∫d4x(iQAµ·(φ†I∂µφI−(∂µφ

†I)φI)+Q2AµAµφ

†IφI) (5.82)

(−iQ(φ†I∂µφI − (∂µφ†I)φI)− 2Q2Aµφ

†IφI)

]|0, I〉. (5.83)

Take particular note of the manifest covariance of the r.h.s. of this formula.

5.8 Propagators for Scalar and Dirac Fields

Wick’s Theorem assures us that to obtain a general time ordered product of free fields, weonly need to know the two field case, 〈0|T [φ1(x1)φ2(x2)]|0〉, which is also called the two pointfunction and sometimes the propagator.

Let us first work out the propagator for a free scalar field which has the representation

φ(x) =

∫d3p

(2π)3/2√

2ω(p)

(a(p)eip·x + b†(p)e−ip·x

), (5.84)

where a annihilates a particle and b† creates an antiparticle. These operators satisfy thecommutation relations

[a(p), a†(p′)] = [b(p), b†(p′)] = δ(p′ − p), (5.85)

with all other commutators vanishing2. The Hamiltonian for the free scalar field is easy to

2Notice that the commutator

[φ(x1), φ†(x2)] =

∫d3p

(2π)32ω(p)(eip·(x1−x2) − eip·(x2−x1) (5.86)

vanishes for space-like separations (x2 − x1)2 > 0. To see this go to a Lorentz frame where t2 = t1 (alwayspossible for space-like separations). Then the second term cancels the first after the variable change p→ −p.If a, b satisfied anticommutation relations, the anticommutator would not have this locality property sincethe two terms would then add. This is the spin-statistics connection for scalar fields. Also notice that if theω(p) were absent anticommutation relations would be local.


write down

H =

∫d3pω(p)(a†(p)a(p) + b†(p)b(p)) (5.87)

=

∫d3x : (φ†φ+∇φ† · ∇φ+m2φ†φ) : (5.88)

where the double colons : (· · · ) : denotes normal ordering, i.e. all creation operators to theleft of all annihilation operators.

Clearly the vacuum expectation of the time ordered product of two φ’s or two φ†’svanishes, and

〈0|T [φ(x1)φ†(x2)]|0〉 = θ(t1 − t2)

∫d3p

(2π)32ω(p)eip·(x1−x2) (5.89)

+θ(t2 − t1)

∫d3p

(2π)32ω(p)eip·(x2−x1), (5.90)

where we recall that p · x = p · x − ωt. To make this expression less unwieldy, it is helpfulto use the following integral representation for the step function

θ(t) =

∫ ∞−∞

dp0

2πie−ip

0t 1

−p0 − iεε→ 0+. (5.91)

Including the factor e−iωt gives

θ(t)e−iωt =

∫ ∞−∞

dp0

2πie−ip

0t 1

ω − p0 − iε, (5.92)

after a shift of p0. Inserting this representation into (5.90), gives

〈0|T [φ(x1)φ†(x2)]|0〉 =

∫d4p

(2π)42ω(p)

(eip·(x1−x2) −i

ω − p0 − iε+ eip·(x2−x1) −i

ω − p0 − iε

),

where now p · x = p · x − p0t. Thus we can change p → −p in the second term and thencombine it with the first to obtain finally

〈0|T [φ(x1)φ†(x2)]|0〉 =

∫d4p

(2π)4eip·(x1−x2) −i

p2 +m2 − iε≡ ∆F (x1 − x2). (5.93)

From its definition the propagator should have the property that only positive frequencycomponents should be present as tk → +∞ and negative frequency components as tk → −∞.This property is assured in (5.93) by the −iε in the denominator. The propagator is a Greenfunction for the Klein-Gordon differential −∂2 +m2:

(−∂2 +m2)∆F (x, y) = −iδ(x− y). (5.94)


p0

p0

0x

0x

-ix

4-ip

4

Figure 5.1: Wick Rotations.

The iε prescription tells us which boundary conditions to impose. This prescription alsofollows if we define ∆F by analytically continuing the Euclidean space Green function toMinkowski space. To continue from Minkowski space to Euclidean space, one rotates thep0 integration contour to the imaginary axis in the counterclockwise direction (to avoid thepoles at ±(ω − iε)). In order to preserve convergence at infinity, x0 must be simultaneouslyrotated in the opposite (clockwise) direction. Changing variables p0 = −ip4 and callingx0 = −ix4 (positive p0 rotates to negative p4 but positive x0 rotates to positive x4) gives theEuclidean Green function

∆F → ∆E ≡∫

d4p

(2π)4ei(x·p+x4p4) 1

p2 + (p4)2 +m2. (5.95)

Clearly ∆E satisfies

(−∂24 −∇2 +m2)∆E(x− y) = δ4(x− y). (5.96)

Next we turn to the evaluation of the propagator for the Dirac field. Remembering thatψ is fermionic we have

SF (x1 − x2)ab ≡ 〈0|T [ψa(x1)ψb(x2)]|0〉= θ(t1 − t2)〈0|ψa(x1)ψb(x2)|0〉 − θ(t2 − t1)〈0|ψb(x2)ψa(x1)|0〉

=

∫d3p

(2π)32ω(p)[θ(t1 − t2)eip·(x1−x2)

∑λ

uaλ(p)ubλ(p)

−θ(t2 − t1)eip·(x2−x1)∑λ

vaλ(p)vbλ(p)], (5.97)

where in the integrand we have p0 = ω(p).To simplify the expression for SF we need to evaluate

∑λ u

aλ(p)ubλ(p) and

∑λ v

aλ(p)vbλ(p).

If we regard them as matrices with indices a, b, we know we can write each as a linear combi-nation of the 16 matrices Γ we used in constructing the bilinears. By virtue of the sum over


helicity, conjugation by the Lorentz transformation matrices simply does the correspondingLorentz transformation on pµ = (p, ω). Thus they must be scalars formed from p and thematrices Γ. The only possibilities are I and p · γ, so∑

λ

uaλ(p)ubλ(p) = Aδab +Bp · γab, (5.98)

and we only need to determine A,B. First notice that multiplying by the matrix m + p · γmust give 0. which determines A = −mB. Then, multiplying both sides by γ0 gives2ω∑

λB = −4ω or B = −1. Thus∑λ

uaλ(p)ubλ(p) = (m− p · γ)ab. (5.99)

The definition v = iγ2u∗ then determines∑λ

vaλ(p)vbλ(p) = [iγ2(m− p · γ∗)(−iγ2)]ab = −(m+ p · γ)ab. (5.100)

Inserting these relations into (5.97) yields

SF (x1 − x2)ab =∫

d3p(2π)32ω(p)

[θ(t1 − t2)eip·(x1−x2)(m− p · γ)ab (5.101)

+θ(t2 − t1)eip·(x2−x1)(m+ p · γ)ab]. (5.102)

The final step is to employ the integral representation for the step functions as we did forthe scalar propagator. This process results in a four dimensional momentum integral withp0 substituted for ω in the exponents but not in front of γ0.Then the two terms involvingm and those involving the spatial γk combine, after the change of variable p → −p in thesecond term, exactly as in the scalar case. The two terms involving γ0 have a factor of ωwhich cancels that in the denominator, but then they combine with the opposite relativesign to produce a 2p0 in the numerator. Thus the net result is simply

SF (x1 − x2)ab = −i∫

d4p

(2π)4eip·(x1−x2)

(m− p · γ

m2 + p2 − iε

)ab

. (5.103)

Just as with the scalar propagator SF may be recognized as a Green function for thedifferential dirac operator 1

iγ · ∂ +m:(

1

iγ · ∂ +m

)SF (x− y) = −iδ4(x− y), (5.104)

with boundary condition dictated by the iε prescription. As before this boundary conditioncan be enforced by defining SF as the continuation of the Euclidean Green function SE. Thecontinuation from SF to SE proceeds by rotating the p0 integration contour to the imagi-nary axis in the counterclockwise direction (of course rotating x0 in the opposite direction),changing variables p0 = −ip4, and defining x0 = −ix4, γ0 = −iγ4

E:

SF (x− y)→ SE(x− y) =

∫d4p

(2π)4eip·(x1−x2)

(m− p · γEm2 + p2

)ab

. (5.105)


SE satisfies the equation (1

iγ · ∂ +m

)SE(x− y) = δ4(x− y). (5.106)

5.9 Vacuum expectations from large time limits of gen-

eral transition amplitudes.

One obstacle to formulating an efficient perturbation theory for systems with interactingquantum fields is that one can’t “turn off” the interactions at early and late times as ispossible with externally applied fields. Thus out and in states are eigenstates of complicatedinteracting Hamiltonians. There are two approaches to this difficulty.

5.9.1 Adiabatic switching

The simplest is to temporarily make the coupling constants time dependent and force them tovanish at early and late times, adiabatically if at all possible, and let them be constant for alltimes−T < t < T .We take T large enough so that all of the times in the time ordered productare later than−T and earlier than T . Then the coupling constants are a kind of external field,for which the passage to interaction picture proceeds as we have discussed, with H0 just thefree field Hamiltonian. Of course, such a procedure introduces unwanted dependence of theout-in matrix element of time ordered products on the switching procedure. To isolate andremove this dependence, recall that the adiabatic theorem assures us that U(−T,−∞)|in〉 isthe ground state of HS(−T ). As for 〈in|U(∞, T ) we can say that the adiabatic evolution anyeigenstate of HS(T ), U(∞, T )|Er〉 will be an eigenstate of HS(+∞), and further since levelcrossings are forbidden in adiabatic evolution, 〈in|U(∞, T )|Er〉 will be zero unless r = 0 theground state. Thus we can write

〈out|T [Ω(t1) · · ·Ω(tN)]|in〉 =

〈in|U(∞, T )|0, T 〉〈0, T |U(T, t1)Ω(t1)U(t2, t1)Ω(t2) · · ·Ω(tN)U(tN ,−T )|0,−T 〉〈0,−T |U(−T,−∞)|in〉 (5.107)

The switch dependence is entirely contained in the first and last factors. But the samefactors appear in

〈out|in〉Ext=0 = 〈in|U(∞, T )|0, T 〉 e−2iTEG 〈0,−T |U(−T,−∞)|in〉 (5.108)

Si we can remove the dependence by division:

〈out|T [Ω(t1) · · ·Ω(tN)]|in〉〈out|in〉Ext=0

=

〈0, T |U(T, t1)Ω(t1)U(t2, t1)Ω(t2) · · ·Ω(tN)U(tN ,−T )|0,−T 〉 e2iEGT (5.109)


5.9.2 Long time evolution suppression of excited states.

A more general approach, which we shall favor, is to relax the requirement that the initialand final states be eigenstates of the Hamiltonian with vanishing external fields. Then onecalculates in first instance a quantity that is not of immediate interest, but which can besimply related to such quantities.

A quantity of more or less direct physical interest is the vacuum expectation value of thetime ordered product of several quantum fields. More generally the outin matrix element ofsuch a time ordered product is relevant if time varying external fields are present. So let usconsider how to obtain this quantity in perturbation theory by first calculating with generalinitial and final states. Using the evolution operator and assuming t1 > t2 > · · · > tn, wetherefore consider

〈f |U(∞,−∞)T [Ω1(t1) · · ·Ωn(tn)]|i〉 =

〈f |U(∞, t1)ΩS1U(t1, t2) · · ·U(tn−1, tn)ΩSnU(tn,−∞)|i〉. (5.110)

Choose the time T so that all external fields vanish for times earlier than −T and later thanT . Then

U(tn,−∞)|i〉 = U(tn,−T )e−i(∞−T )HS |i〉 (5.111)

= U(tn,−T )e−i(∞−T )EG∑r

e−i(∞−T )(Er−EG)|r〉〈r|i〉 .

We would now like to argue that the infinite oscillations wash out all contributions butthe (assumed nondegenerate3) ground state. In a field theory this is quite plausible since theexcited states correspond to particles so the sum over r is really an integral over a range ofcontinuous energies. But even without this smearing, we can make the washing out rigorousby calculating with imaginary time: it = β > 0. Then i∞ is really +∞ and all excitedstates are damped exponentially. Massless particle states could introduce a subtlety here,but the part of phase space that is not exponentially damped is infinitesimal: this has theeffect of changing exponential damping to a power law damping. If we buy this argument,then we can assert quite generally that U(tn,−∞)|i〉 = U(tn,−∞)|0〉〈0|i〉 and similarly〈f |U(∞, t1) = 〈f |0〉〈0|U(∞, t1).

Since we take (as usual) the Heisenberg and Schrodinger pictures to coincide at t = −∞,then |in〉 = |0〉 and 〈out| = 〈0|U(∞,−∞). Thus we have obtained the relation

〈f |U(∞,−∞)T [Ω1(t1) · · ·Ωn(tn)]|i〉 = (5.112)

〈f |0〉〈0|i〉〈out|T [Ω1(t1) · · ·Ωn(tn)]|in〉. (5.113)

In other words calculating with any initial and final states that have finite overlap4 with

3There are interesting cases of degenerate vacua, when there is “spontaneous symmetry breakdown.” Insuch cases the choice of initial and final states determines which of the degenerate vacua is picked out.

4The infinite number of degrees of freedom in quantum field theory requires care here: the overlap betweendifferent states in a theory with n degrees of freedom is typically fn with f < 1. Since n = ∞, we shouldexpect 〈f |0〉〈0|i〉 ∼ e−∞. In field theory n =∞ because the volume of space is infinite and because space iscontinuous. Thus strict application of the above relation should be done in the presence of both an infraredand ultraviolet cutoff, which can then be removed after extracting the desired amplitude.


the true ground state gives us a constant times the desired matrix element. We can easilyevaluate the multiplicative constant by considering by the same reasoning

〈f |U(∞,−∞)|i〉 = 〈f |0〉〈0|i〉〈out|in〉 (5.114)

→ e−2i∞EG 〈f |0〉〈0|i〉 , External Fields = 0. (5.115)

Putting this into our relation we obtain

〈out|T [Ω1(t1) · · ·Ωn(tn)]|in〉 = (5.116)

e−2i∞EG 〈f |U(∞,−∞)T [Ω1(t1) · · ·Ωn(tn)]|i〉〈f |U(∞,−∞)Ext=0|i〉

, (5.117)

where the subscript on U in the denominator denotes vanishing external fields. In fieldtheory applications EG is the energy of the vacuum, which is zero if we measure all energiesrelative to that of the vacuum. In the absence of gravity all physical quantities dependonly on energy differences, so we lose nothing by doing this. Gravity couples directly to theenergy density and therefore is sensitive to the energy as opposed to energy differences, butthen EG only appears in the combination Λ ≡ EG + Λ0, with Λ0 the “bare” cosmologicalconstant. Replacing Λ0 by Λ in effect sets EG = 0.

The formula (5.117) is a convenient starting point for developing perturbation theory.Any breakup

HS(t) = H0(t) +H ′(t) (5.118)

determines an interaction picture defined by

ΩI(t) = U−10 (t,−∞)ΩSU0(t,−∞) = UI(t,−∞)Ω(t)U−1

I (t,−∞), (5.119)

where

iU = HS(t)U = UH(t) (5.120)

iU0 = U0H0I(t) (5.121)

iUI = HI(t)UI (5.122)

and all U ’s are the identity at t = −∞. Then the evolution operator satisfies

U(t1, t2) = U(t1,−∞)U−1(t2,−∞) = U0(t1,−∞)UI(t1, t2)U−10 (t2,−∞). (5.123)

Plugging these relations into (5.117) then gives

〈out|T [Ω1(t1) · · ·Ωn(tn)]|in〉 = (5.124)

e−2i∞EG 〈f |U0(∞,−∞)T [UI(∞,−∞)Ω1I(t1) · · ·ΩnI(tn)]|i〉〈f |U0(∞,−∞)Ext=0UI(∞,−∞)Ext=0|i〉

. (5.125)

This formula is completely general: we have even allowed H0 to contain time varying externalfields, which is hardly ever done in practice. Since all operators in this formula are in inter-action picture, it is most convenient to choose |i〉, |f〉 to have simple properties with respect


to H0I(−∞). Let us call the ground state of this operator |in, 0〉. Then 〈in, 0|U0(∞,−∞)is the ground state of H0I(+∞) and therefore deserves the name 〈out, 0|. When all exter-nal fields vanish, H0I is time independent and we call its ground state |0, I〉 ≡ |in, 0〉 andits ground state energy E0. Then 〈in, 0|U0(∞,−∞)Ext=0 = e−2i∞E0〈0, I|. Thus choosing|i〉 = |f〉 = |0, I〉 = |in, 0〉 we obtain the useful formula

〈out|T [Ω1(t1) · · ·Ωn(tn)]|in〉 =

e−2i∞(EG−E0) 〈out, 0|T [UI(∞,−∞)Ω1I(t1) · · ·ΩnI(tn)]|in, 0〉〈0, I|UI(∞,−∞)Ext=0|0, I〉

. (5.126)

In the usual case where we do not include external fields in H0, the formula simplifies further

〈out|T [Ω1(t1) · · ·Ωn(tn)]|in〉 = e−2i∞EG〈G|T [Ω1(t1) · · ·Ωn(tn)]|G〉 = (5.127)

e−2i∞EG 〈0, I|T [UI(∞,−∞)Ω1I(t1) · · ·ΩnI(tn)]|0, I〉〈0, I|UI(∞,−∞)Ext=0|0, I〉

. (5.128)

Using the Wick expansion one can describe the perturbation series for the numeratorsand denominators of these formulas using Feynman diagrams. The diagrams contributing tothe denominator are all those completely disconnected from either external fields or from thepoints assigned to the operators in the numerator. The numerator contains this same sumof diagrams as a multiplicative factor. Thus the division by the denominator is achievedby simply deleting all such disconnected “vacuum” diagrams from the expansion of thenumerator.


Chapter 6

Cross Sections and Rates and SpinSums

6.1 Cross section for 2 → N

The scattering matrix is written as

〈out, f |in, i〉 = δfi − i(2π)4δ(Pf − p1 − p2)T (6.1)

And we define the Feynman ampltude M by

T ≡ M(2π)3

√4ω1ω2

∏f [(2π)3/2

√2ωf ]

(6.2)

For a scattering process the initial state is two separated wave packets aimed at each otherso they will meet and scatter. This situation can be arranged by a momentum space wavefunction f(~p1, ~p2), narrowly peaked about ~pi = zpi. For the lab frame p2 = 0, and for theCM frame p2 = −p1. Then

Amplitude =

∫d3p1d

3p2(2π)4δ(Pf − p1 − p2)T f(~p1, ~p2)

≈ (2π)4T∫d3p1δ(Ef − ω1 − ω2)f(~p1, ~Pf − ~p1) (6.3)

To deal with the last delta function write d3p1 = d2p⊥1 dpz1 = d2p⊥1 d(ω1 + ω2)/v12 where

v12 = |∂ω1/∂pz1 +∂ω2/∂p

z1| = |v1−v2| is the relative velocity of the two initial packets. Then

Amplitude ≈ (2π)4

v12

T∫d2p1f(~p1, ~Pf − ~p1) (6.4)

This function is very sharply peaked about Pf = p01 + p0

2 with width given by the initialpacket. Its square will also be sharply peaked, and safely approximated by a delta function:

|Amp|2 ≈ (2π)8

v212

|T |2δ(Pf − p01 − p0

2)

∫d4P

∣∣∣∣ ∫ d2p1f(~p1, ~P − ~p1)

∣∣∣∣2 (6.5)

81

Now consider

ψ(x, y, pz1, ~P ) =

∫d2p

2πeixp

x+iypyf(~p1, ~P − ~p1) (6.6)

It’s square is the probability per dpz1d3P that the relative transverse displacement of the

incident particles is in dxdy of the point (x, y). This quantity appears in our formula for the

case x = y = 0 and integrated over d3PdEf ≈ v12d3Pdpz1.

∫d4P

∣∣∣∣ ∫ d2p1f(~p1, ~P − ~p1)

∣∣∣∣2 ≈(2π)2v12× the total probability the incident particles are within dxdy of x = y = 0. Thecoefficient of this probability is just what we mean by the differential cross section

dσ =∏f

d3pf(2π)10

v12

δ(Pf − p1 − p2)|T |2

=∏f

d3pf(2π)32ωf

1

4ω1ω2v12

(2π)4δ(Pf − p1 − p2)|M|2 (6.7)

The last form is the one that is most convenient to remember. In itM has all external wavefunction factors 1/(2π)3/2/

√2ω removed. It is the Fourier transform of the appropriate time

ordered product, with the propagator on each external line amputated and replaced withthe appropriate

√Z factor.

6.2 Decay rate of a single metastable particle

An apparently simpler process than scattering is the decay of a single particle into any finalstate. In this case we write the decay probability amplitude as

(2π)4δ(Pf − p1)T ≡ (2π)4δ(Pf − p1)M

(2π)3/2√

2ω1

∏f [(2π)3/2

√2ωf ]

(6.8)

However in this case putting the initial particle in a wave packet f(~p) only takes care ofthe spatial delta function, and after this we still can’t meaningfully square the amplitudeto get a probability. The problem is that the formula assumes that the initial particle hasexisted for an infinite time, which is not consistent with the fact that it can decay. It hasa finite lifetimes τ , and it can only approximately be described as a particle with definitemass for a time 2T τ . The correct way to deal with unstable systems involves includingtheir production as well as decay. To deal with the application of the above formula, we canreplace the energy conserving delta function with a finite time version

2πδ(Ef − ω1)→∫ T

−Tdte−it(Ef−ω1) =

2 sin(Ef − ω1)T

Ef − ω1

(6.9)

where we understand that 2T τ . Application of the formula implicitly assumes thatT 1/∆E where ∆E characterizes the scale over which T varies. Thus we should only try


to apply this formula for the decay of a particle with lifetime τ 1/∆E. In this situationwe can approximate the square of the finite time delta function with a sharp delta function:∣∣∣∣2 sin(Ef − ω1)T

Ef − ω1

∣∣∣∣2 ≈ δ(Ef − ω1)4

∫dE

sin2ET

E2= 2πδ(Ef − ω1)(2T ) (6.10)

With this interpretation we arrive at the following formula for the differential decay proba-bility

dP =∏f

d3pf (2π)7|T |2δ(Ef − ω1)(2T )|f(P f )|2

≈∏f

d3pf (2π)7|T |2δ(Pf − p1)(2T )

∫d3p|f(p)|2

≈∏f

d3pf (2π)7|T |2δ(Pf − p1)(2T )

≈ 2T∏f

d3pf(2π)32ωf

1

2ω1

(2π)4δ(Pf − p1)|M|2 (6.11)

We see that in this approximate description the decay probability is proportional to theduration of the process 2T . This linear dependence is clearly only meaningful for 2T τ .However the probability per unit time, the decay rate, is independent of T :

dΓ ≡ dP

2T=∏f

d3pf(2π)32ωf

1

2ω1

(2π)4δ(Pf − p1)|M|2. (6.12)

Summing over all allowed final states, including integrating over phase space gives the totaldecay rate Γ. We can use it to determine the long time behavior of the decay process.Let P (t) be the probability that the particle has not decayed in a time t after its creationP (0) = 1. Then

dP

dt= −ΓP (t), → P (t) = e−Γt (6.13)

This provides a precise definition for the lifetime of the particle, namely, τ = 1/Γ. Also theprobability that the particle has decayed in time t is 1−P (t) = 1−e−Γt = 1−e−t/τ ≈ t/τ = tΓ,when t τ , showing the consistency of our rate formula.

As a very useful example, we specialize these formulas to the case of two particle finalstates. Four of the six phase space integrals can be evaluated using the delta functions.

d3pf1d3pf2δ(p

f1 + pf2 − Pi) = d3pf1δ(ω

f1 (pf1) + ωf2 (Pi − pf1)− Ei) = dΩ

pf21 ω

f1ω

f2

pf1Ei − Piωf1 cos θ

(6.14)

then we have

dσ

dΩ=

|M|2

64π2ω1ω2v12

pf21

pf1Ei − Piωf1 cos θ

(6.15)

dΓ

dΩ=

|M|2

32π2ω1

pf21

pf1ω1 − p1ωf1 cos θ

(6.16)


These formulas dramatically simplify in the center of mass system when ~Pi = 0. Thenω1ω2v12 = p1(ω1 + ω2) and

dσ

dΩCM=

pf1p1

|M|2

64π2(ω1 + ω2)2(6.17)

dΓ

dΩCM=

|M|2

32π2m21

pf1 (6.18)

We finally note that when calculating total rates and cross sections with identical particlesin the final state, the result of integrating over phase space must be divided by

∏c nc! when

there are nc identical particles of type c.

6.3 Spin Sums

Cross section and rate formulas simplify when we either choose not to measure final particlspins or use an unpolarized beam. To accomplish this for spin 1/2 particles the followingprojectors onto definite spin states are useful

uu =1

2(m− γ · p)(1− γ5γ · s) (6.19)

vv =1

2(−m− γ · p)(1− γ5γ · s) (6.20)

where the spin state information is carried by the four vector sµ, given by either

s0 =p · sm

(6.21)

s = s + pp · s

m(m+ ω)(6.22)

and s is a unit vector in the direction of the polarization in the rest frame, i.e.

s · σus(0) = us(0),

or, if spins are labelled by helicity,

s0 = 2h|p|m

(6.23)

s = 2hωp

m|p|. (6.24)

Note: in the case of zero mass the projectors have a smooth limit only for the helicity basis:

uhuh = −1

2(1 + 2hγ5)γ · p (6.25)

vhvh = −1

2(1− 2hγ5)γ · p. (6.26)


Then summation over spin states leads to∑λ

uλuλ = m− γ · p, vv = −m− γ · p (6.27)

For example, ∑λ′,λ

|uλ′(p′)Γuλ(p)|2 = Tr[(m− γ · p)γ0Γ†γ0(m− γ · p′)Γ

](6.28)

6.3.1 Gamma matrix identities

In evaluating traces of products of gamma matrices the following identities are extremelyuseful:

Tr[odd number of γ′s] = 0 (6.29)

Tr[γµγν ] = −4ηµν (6.30)

Tr[γκγλγµγν ] = 4[ηκληµν − ηκµηλν + ηκνηλµ] (6.31)

Tr[γκγλγµγνγ5] = −4iεκλµν

where εκλµν is completely antisymmetric and ε0123 = +1.

γµγνγρ = −iεµνρσγσγ5 − ηµνγρ + ηµργν − ηνργµ (6.32)

γµγλγµ = 2γλ (6.33)

γµγκγλγµ = 4ηκλ (6.34)

γµγκγλγργµ = 2γργλγκ (6.35)

εµνρσσρσ = −2iγ5σµν (6.36)

For example, we can evaluate

Tr(m− γ · p)γµ(m− γ · p′)γν = −4m2ηµν + 4(pµp′ν − ηµνp · p′ + pνp′µ) (6.37)

6.4 Tree diagrams in momentum space

At any vertex the∫d4x delivers a 4 momentum conserving delta function. For example∫

d4x∆F (x1 − x) · · ·∆F (xN − x)

∫d4q1

(2π)4· · · d

4qN(2π)4

(2π)4δ(∑

qk)ei∑qk·xk −i

q21 +m2

· · · −iq2N +m2

(6.38)

In momentum space external propagators have fixed momentum, but internal momentaare potentially integrated. Connected tree diagrams have the property that the numberof vertices is 1 more than the number of internal propagators. Thus all internal momentaare determined by the external momenta. Diagrams with L loops have L undeterminedmomenta which are each integrated with measure d4p/(2π)4. It is these momentum integalsthat produce the notorious divergences of QFT.


6.4.1 Scattering amplitudes from time ordered products

Instead of the S-matrix for two to two scattering suppose we calculate the Fourier transformof 〈0|Tφ(x1)φ(x2)φ(x3)φ(x4)|0〉. Then in gφ3 theory one lowest order tree diagram is

A = (ig)2

∫d4xd4y∆F (x1 − x)∆F (x2 − x)∆F (x3 − y)∆F (x4 − y)∆F (x− y) (6.39)

Then∫ ∏k

(d4xke

iqk·xk)A = (ig)2

∏k

(−i

q2k +m2

)∫d4xd4yei(q1+q2)·x+i(q3+q4)·y∆F (x− y)

= (ig)2∏k

(−i

q2k +m2

)∫d4xei(q1+q2)·x∆F (x)(2π)4δ(

∑k

qk)(6.40)

We see that we have for each external propagator a factor −i/(q2k +m2) which contains poles

at q0k = ±

√q2k +m2. The residue of either of these poles is proportional to the desired

scattering amplitude. The F.T. of a suitable time ordred product thus contains informationabout scattering. The prescription is to “amputate” the external propagators and sendq0k → ±ω(qk). The + choice gives an incoming particle with momentum qk and the − choice

to an outgoing particle of momentum −qk.The analogous procedure for Dirac particles leads to an external propagator SF (x − y)

where the external coordinate could be x 0r y. If it is x the F.T. would be∫d4xeiq·xSF (x− y) =

−i(m+ q · γ)

m2 + q2eiq·y (6.41)

If q0 > 0 the numerator on shell goes to −i(m+ q · γ) = +i∑

λ vλ(q)vλ(q) and correspondsto an incoming antiparticle of momentum q. Note the extra − in this case. If q0 < 0 it isoutgoing with momentum p = −q so the numerator is −i(m− p · γ) = −i

∑Uu, showing it

is an outgoing particle.On the other hand if y is the external coordinate the F.T. would be∫

d4yeiq·ySF (x− y) =−i(m− q · γ)

m2 + q2eiq·x (6.42)

If q0 > 0 the numerator on shell goes to −i(m−p ·γ) and corresponds to an incoming particlewith momentum q. If q0 < 0 one sees that it is an outgoing antiparticle with momentump = −q since the numerator is i

∑vv.


Chapter 7

Quantum Field Equations withExternal Fields

7.1 Electromagnetic Fields

The coupling of a quantum field to an external electromagnetic field is dictated by theprinciple of gauge invariance. In classical electrodynamics, it is possible to avoid potentialsand formulate all equations of motion in terms of the electric and magnetic fields Fµν .However, the potential Aµ(x) is indispensable to an economical description of the coupling ofa quantum particle to electromagnetism. Fundamentally, this is because the Hamiltonian andLagrangian play a much more central role in quantum dynamics than in classical dynamics,and the potential appears explicitly in the Hamiltonian and Lagrangian. (Recall that theSchrodinger equation involves the Hamiltonian explicitly.) The field strengths are related tothe potential via

Fµν(x) = ∂µAν(x)− ∂νAµ(x), (7.1)

but it is clear that the potential is not given uniquely in terms of the field strength: If thepotential is changed by a gauge transformation

Aµ → Aµ + ∂µΛ, (7.2)

the field strength Fµν is unchanged. It is therefore important to introduce the potential intothe Schrodinger equation in a way which preserves gauge invariance.

Let us first ask how gauge invariance is realized in classical particle electrodynamics. Inorder that the Euler-Lagrange equations reproduce the Lorentz force law, the scalar andvector potentials (A0,A) must enter the Lagrangian through the terms

−qA0(x) + qx ·A(x). (7.3)

Because of the term linear in velocity, the momentum conjugate to x becomes

p = pA=0 + qA, (7.4)

87

where pA=0 is the conjugate momentum with vanishing vector potential. Furthermore, whenwe form the Hamiltonian H = x · p− L, the term linear in velocity cancels, so

H = H0(pA=0,x) + qA0 = H0(p− qA,x) + qA0. (7.5)

If we subject A to a gauge transformation, the Lagrangian changes by the amount

q(∂0Λ + x · ∇Λ) = qdΛ

dt, (7.6)

a total time derivative, so that the action∫ t2t1L changes by the amount qΛ(x(t2), t2) −

qΛ(x(t1), t1) and the added terms have no effect on the Euler-Lagrange equations. At firstsight the Hamiltonian doesn’t look invariant, but notice that the transformation

p′ = p− q∇Λ x′ = x (7.7)

is a canonical transformation with generating function W2(x,p′, t) = x · p′ + qΛ(x, t). Fur-thermore if the generating function is time dependent the canonical transformation includeschanging the Hamiltonian by ∂tW2 = q∂tΛ, so that after the canonical transformation thegauge transformed Hamiltonian is identical to the old one with the substitutions x → x′,p→ p′.

The way Aµ enters the Schrodinger equation for a charged particle is now clear. Thesubstitution p→ p− qA corresponds in the Schrodinger equation to

1

i∇ → 1

i∇− qA. (7.8)

Furthermore the addition of qA0 = −qA0 to the Hamiltonian is prescribed by the substitutionrule

1

i

∂

∂t→ 1

i

∂

∂t− qA0, (7.9)

so both substitutions can be given the compact expression

∂µ → ∂µ − iqAµ (7.10)

which is known as the minimal substitution rule. The gauge invariance of the Schrodingerequation is achieved by postulating in addition to (7.2) the change

ψ → eiqΛψ. (7.11)

If we recall that in the semi-classical approximation ψ ∼ eiS, where S is the classical ac-tion, we recognize that this change of the wave function under gauge transformation is thequantum analogue of the classical change in the action.

The gauge transformation makes an arbitrary, local, redefinition of the phase of the wavefunction. In fact, one could take the attitude that invariance under such local phase changes


is desirable from a physical point of view. (Since global phase changes are unobservableperhaps local ones should also be.) In that case one would be forced to introduce theelectromagnetic field to realize the invariance! It is obvious that the Schrodinger equation isinvariant under the combined changes (7.2) and (7.11). As a special case, the Dirac equationis invariant under the same gauge transformations. When we interpret the Dirac equationas a field equation, (7.11) is a transformation on fields as is (7.2), so the two are really onsimilar footing.

To sum up the above discussion, we display the Dirac equation in a potential Aµ:

iγ · (∂ − iqA)ψ = mψ. (7.12)

Also the corresponding second quantized Hamiltonian is given by

HA(t) =

∫d3x

(ψ†(

1

iα ·∇+ βm

)ψ − jµAµ

). (7.13)

Notice that a simple consequence of the Dirac equation is current conservation

∂µjµ = 0, (7.14)

even when A 6= 0. This can be understood as a consequence of the gauge invariance ofquantum evolution. To see this we have to consider the unitary evolution operator U(t, t0)1

defined by

i∂tU(t, t0) = HS(t)U(t, t0) U(t0, t0) = I, (7.16)

where HS is the Schrodinger picture Hamiltonian. If we make a small change δAµ inAµ, keeping the Schrodinger picture dynamical variables fixed, U changes by the

1U gives the unitary transformation between Heisenberg and Schrodinger pictures. Its analogue in classicalmechanics is the generator S(q, P, t) of the canonical transformation mapping the initial phase space variablesQ,P to those at time t, q, p. The analogue of the following equation for U is the Hamilton-Jacobi equationfor S,

∂S

∂t= −H(q,

∂S

∂q, t). (7.15)


amount2

δU(t, t0) = i

∫ t

t0

dt′d3xU(t, t′)δAµ(x′)jµ(x)U(t′, t0) (7.19)

= U(t, t0)i

∫ t

t0

dt′d3xδAµ(x′)jµ(x, t′) (7.20)

with jµ(x, t′) = U †(t′, t0)jµ(x)U(t′, t0) the Heisenberg picture current operator3. An easyway to see this is to write U = Te−i

∫dt′HS(t′ . Alternatively, simply differentiate (7.20) with

respect to time and show that it satisfies (5.3) to first order in δA. Use is also made of theclosure relation

U(t, t′)U(t′, t0) = U(t, t0) (7.21)

which is a simple consequence of the differential equation satisfied by U . Under a gaugetransformation δA = ∂Λ so we see that

δU = Ui

∫ t

t0

d4x∂µΛjµ(x). (7.22)

The requirement that U be invariant under gauge transformations that vanish at t0, t is thenthat ∂µj

µ = 0.

7.2 Nonabelian Gauge Fields

The local phase transformation (7.11) on charged fields can be generalized (Yang-Mills).Suppose that ψ carries an internal index k. Then in place of (7.11) we can consider

ψk(x)→∑l

Ωkl(x)ψl(x) (7.23)

2In the classical theory a change in the parameters of the Hamiltonian would give rise to a change in Ssatisfying

∂δS

∂t= −

∑k

∂δS

∂qk

∂H

∂pk− δH(q,

∂S

∂q, t). (7.17)

Now if we put q = q(t) and ∂S/∂q = p(t) the Hamilton-Jacobi equation just says that q(t), p(t) satisfyHamilton’s equations so ∂H/∂p = q and we have

dδS

dt= −δH(q(t), p(t), t) (7.18)

or δS = −∫ tt0dt′δH(q(t′), p(t′), t′), the classical analogue of the following equation.

3Equation (7.20) has a generalization to an arbitrary physical system: If one makes any small changeδHS in the Schrodinger picture hamiltonian, with the Scrodinger picture operators fixed, the correspondingchange in the evolution operator U(t, t0) is given by δU = −iU

∫ tt0dt′δH(t′), where δH(t) ≡ U†δHSU is

the change in the Schrodinger hamiltonian transformed to the Heisenberg picture, which is the same as thechange in the Heisenberg Hamiltonian H(t), keeping the Heisenberg dynamical variables fixed. The proof ofthis is exactly the same as that of Eq.(7.20.


or, with suppressed indices

ψ(x)→ Ω(x)ψ(x) (7.24)

with Ω(x) in a unitary matrix representation of some continuous group G. In this language(7.11) corresponds to the choice G = U(1), multiplication by a phase, an abelian group. Ifwe require the dynamics to be invariant under the nonabelian local gauge transformations,we must introduce a nonabelian gauge field to absorb the noncovariance of

∂µψ → Ω(∂µ + Ω−1∂µΩ)ψ. (7.25)

In analogy with the electromagnetic case, we need to introduce a matrix valued potentialAµ(x) via the substitution

∂µ → ∂µ − igAµ(x) ≡ Dµ. (7.26)

Then the requirement Dµψ → ΩDµψ translates to

Aµ → ΩAµΩ−1 − i

g∂µΩΩ−1.. (7.27)

Clearly A takes values in the Lie algebra of the group G. If ψ is a Dirac field, a gaugeinvariant dynamics is given by the field equation

iγ ·Dψ = mψ, (7.28)

or the corresponding second quantized Hamiltonian

HA(t) =

∫d3x(ψ†(

1

iα ·∇+ βm)ψ − gψAµγµψ). (7.29)

Just as in the electromagnetic case we can consider how the quantum evolution operatorchanges under a small change of A, and identical steps lead to:

δU(t, t0) = U(t, t0)i

∫ t

t0

dt′d3xψgδAµγµψ. (7.30)

An infinitesimal gauge transformation Ω = I + igεG corresponds to

δAµ = ε(∂µG− ig[Aµ, G]). (7.31)

For G which vanish initially and finally the corresponding change in U is

δU(t, t0) = U(t, t0)igε

∫ t

t0

dt′d3x (7.32)

Gab(−∂µ(ψaγµψb)− ig(Aµ)caψcγ

µψb + ig(Aµ)bcψaγµψc) (7.33)

so gauge invariance implies the following generalization of current conservation (“covariantconservation”)

Dµjµ ≡ ∂µj

µ − ig[Aµ, jµ] = 0, (7.34)

where the current is a matrix operator

jµ(x)ba ≡ ψa(x)γµψb(x). (7.35)


7.3 External Gravitational Fields

According to the Principle of Equivalence, an external gravitational field is described byintroducing a space-time dependent metric ηµν → gµν(x) which then must enter the fieldequations in a generally covariant way. This prescription suffices for bosonic fields but newconcepts must be brought in for fermionic fields. At this stage we shall confine our discussionto a real scalar field. The minimal generally covariant classical action is given by

S = −1

2

∫d4x√−g(gµν(x)∂µφ∂νφ+m2φ2). (7.36)

To construct the Hamiltonian, we first define the conjugate momentum

π(x) ≡ δS

δφ(x)= −√−gg0ν(x)∂νφ. (7.37)

Then the Hamiltonian is

H(t) =

∫d3x(π(x)φ(x) +

1

2

√−g(gµν(x)∂µφ∂νφ+m2φ2)) (7.38)

=

∫d3x

(− 1

2

π2

g00√−g

+g0k

g00π∂kφ+

1

2

√−g(gkl − g0kg0l

g00

)∂kφ∂lφ

+m2

2

√−gφ2

). (7.39)

Just as with gauge fields we can ask how the evolution operator changes under a small changeδgµν in the metric, δU = −iU

∫ tt0dt′δH(t′), where δH is computed holding φ, π fixed. The

easiest way to do this is to evaluate the change in the Lagrangian L at fixed φ, φ. Since L isrelated to H by a Legendre transform, δH = −δL.

δL = −1

2

∫d3x√−gδgµν(x)Tµν(x) (7.40)

with Tµν the energy momentum tensor

Tµν = ∂µφ∂νφ−1

2gµν(g

ρσ∂ρφ∂σφ+m2φ2). (7.41)

Thus we have finally4

δU = iU1

2

∫ t

t0

d4x′√−gδgµν(x′)T µν(x′). (7.42)

4Note that since gµν is the inverse matrix to gµν , δgµν = −gµρδgρσgσν .


Under the infinitesimal general coordinate transformation xµ = x′µ + ξµ(x′), the metricchanges according to

gµν(x) = g′µν(x′)− ∂ξρ

∂xµgρν −

∂ξρ

∂xνgρµ (7.43)

= g′µν(x′)−Dµξν −Dνξµ + Γρµσξ

σgρν + Γρνσξσgρµ (7.44)

= g′µν(x′)−Dµξν −Dνξµ + ξρ∂ρgµν (7.45)

= g′µν(x)−Dµξν −Dνξµ (7.46)

An infinitesimal change of integration variables is just a surface term:∫d4x′L′(x′) =

∫d4x(1− ∂ρξρ)(1− ξρ∂ρ)L′(x) =

∫d4xL′(x)−

∫d4x∂ρ(ξ

ρL′(x)) (7.47)

so choosing δgµν = −Dµξν−Dνξµ should give δU = 0 for all ξ vanishing sufficiently rapidly atinfinity, if the quantum field dynamics is invariant under general coordinate transformations.Thus general coordinate invariance implies that the energy momentum tensor is covariantlyconserved:

DµTµν = 0. (7.48)

In the limit of flat space (no gravity) this condition reduces to ordinary energy-momentumconservation.

7.4 Asymptotic States and Matrix Elements

In discussing time dependent processes, it is convenient to introduce asymptotic states whichare eigenstates of H(±∞). We denote by |in〉 the ground state of H(−∞) and by |out〉 theground state of H(+∞). The normal situation will be one in which all external fields vanishat sufficiently early and late times. Thus |in〉 and |out〉 will typically be ground states ofH0(−∞) andH0(+∞) respectively. Although these operators are not the same (because theirtime evolution is governed by H not H0), the spectra of the two Hamiltonians are identical:H0(t) = U−1(t,−∞)H0SU(t,−∞). By convention we are identifying the Schrodinger andHeisenberg pictures at t = −∞. Thus, if |in〉 is the ground state of H0(−∞) = H0S, thestate 〈in|U(∞,−∞) is an eigenstate of H0(+∞) with the same eigenvalue and hence theground state. Thus we can and shall fix phases by defining

〈out| ≡ 〈in|U(∞,−∞). (7.49)

We stress that this is the true “out” state only when HS(∞) = HS(−∞) ≡ H0.If the time dependence of HS is adiabatic, i.e. very slow on the time scale set by the

level spacings, the Adiabatic Theorem assures us that an eigenstate of HS(−∞) evolves toan eigenstate of HS(t) for all t for which adiabatic conditions apply, even after a long enoughtime to change HS by a finite amount. For example, the state |in〉 will be an eigenstate


of H(t) for all t for which adiabatic time variation applies. In particular, the ground stateeigenvalue EG(t) must not get close to the next higher eigenvalue as t varies. If this situationholds for all time, it follows that the state |in〉 is a phase times the state |out〉, or 〈out| is thissame phase times 〈in|. This phase is easily evaluated in terms of the time dependent groundstate energy EG(t) of HS(t) by applying the Schrodinger equation to 〈in|U(t,−∞)|in〉 andusing the adiabatic theorem HS(t)U(t,−∞)|in〉 = EG(t)U(t,−∞)|in〉:

〈out|in〉 = exp

−i∫ ∞−∞

dtEG(t)

Adiabatic Conditions. (7.50)

Note carefully that adiabatic conditions would not apply if the ground state energy got closeto an excited level as time evolved. In particular, it would not apply in processes with pairproduction when | 〈out|in〉 | < 1.

It is only for very particular external fields that one can solve the Dirac equation or anyfield equation exactly. Important examples include the static Coulomb potential for whichone can find all the energy eigenvalues and eigenstates, arbitrary constant field strengths,and plane waves. The Coulomb potential is of special importance since the exact solutionin that case is the starting point for the relativistic theory of atomic energy levels. Still thecomplete dynamics is never exactly given by these special cases and perturbation theory isthe important tool for evaluating corrections to the exactly soluble (idealized) case, whichcan be zero external fields or one of the above cases. When we quantize the electromagneticfield perturbation theory is essentially our only tool for computing radiative corrections dueto the quantum nature of the electromagnetic field.


Chapter 8

External Field Perturbations

As an important example, let us apply perturbation theory to the Dirac field in an externalelectromagnetic field. Since we shall work in interaction picture throughout, we shall notappend the I subscript to indicate interaction picture: that will be understood throughoutthis section. Then the persistence amplitude in the presence of A is

〈out|in〉A = 〈0|Tei∫d4xjµ(x)Aµ(x)|0〉 (8.1)

=∞∑n=0

in

n!(q

2)n∫d4x1 · · · d4xnAµ1 · · ·Aµn (8.2)

〈0|T [[ψ(x1), γµ1ψ(x1)] · · · [ψ(xn), γµnψ(xn)]]|0〉. (8.3)

It is now a matter of applying Wick’s theorem to evaluate the nth term of the series.We shall organize the calculation with Feynman diagrams. The Wick expansion ex-

presses the time ordered product in terms of propagators completely characterized by twopoints. Each propagator is represented by a line connecting the two points, directed fromthe argument of ψ to that of ψ:

x,a

y,b

SF (x− y)ab.

The lines terminate on vertices associated with the field A:

A

b

a

iq

∫d4xAµ(x)γµab.

For each closed loop there is a trace over Dirac indices and a multiplicative factor of −1.This last factor is due to the anticommuting property of the Dirac field and arises because

95

the order of fields in the product of currents that is contracted to form the loop differs fromthat of the contributing propagators by an odd number of interchanges.

−

ψ

+ + + +

ψψ ψ ψ ψ ψ ψ

Finally there is a combinatoric factor arising from a sometimes partial cancellation of the1/n! multiplying the nth order term.

8.1 Connected Diagrams

Terms in the perturbation series described by disconnected diagrams factorize into a productof the values of each connected subdiagram. Thus all the useful information is contained inthe subset of connected diagrams, and it is useful to know how the final answer is expressedin terms of connected diagrams only. Roughly speaking, the sum of all diagrams is simply theexponential of the sum of all connected diagrams. This statement applies to the expansionof

〈0|Te−i∫dtH′I(t)|0〉 (8.4)

for a completely general Hamiltonian. The reason is purely combinatoric. Set

Gck =

⟨0|T ((−i

∫dtH ′I)

n|0⟩c

(8.5)

where by the subscript c we mean drop all contractions which produce any disconnectedparts. Then the value of the sum of connected graphs at order k, Gc

k/k!. Now considerthe terms at order n =

∑∞k=1 krk in the perturbation series described by rk appearances of

the connected graphs of order k. (Note that all but a finite number of the rk are 0.) Thecontractions leading to these terms can occur in n!/

∏k[(k!)rkrk!] ways. This is because there

are n! ways to assign the n Hamiltonians to the n vertices of all the connected subgraphs,but this overcounts the number of contractions by a factor of k! for each of the connectedsubgraphs of order k, because different orders of the assignment to each subgraph do not givedistinct contractions, and overcounts by a factor of rk! for each group of identical subgraphsfor the same reason. Thus the value of the order n terms is

1

n!

∑∑krk=n

n!∏k[(k!)rkrk!]

∏k

(Gck)rk =

∑∑krk=n

∏k

[1

rk!(Gck

k!)rk]. (8.6)

Note that the factors in square brackets are 1 if all rk = 0. Summing over all n simplyrelaxes the constraint on the summation over the rk, so we have finally

〈0|Te−i∫dtH′I(t)|0〉 =

∞∏k=1

[∞∑rk=0

1

rk!(Gck

k!)rk

]=∞∏k=1

eGck/k! = e

∑k G

ck/k!, (8.7)


which is the desired result.We now turn to the connected diagrams for an external electromagnetic field. At order

n in the fields, the diagrams contributing to ln 〈out|in〉 are

An-1

An

A1

A2

= −(iq)n∫d4x1 · · · d4xn

Tr[γ · A(x1)SF (x1 − x2) · · · γ · A(xn)SF (xn − x1)].

(8.8)

Since there are (n − 1)! distinct diagrams with the same value (after the coordinate inte-grations) the net combinatoric factor is (n − 1)!/n! = 1/n. This factor can be interpretedas 1/SΓ where SΓ is the symmetry number of the graph γ. It is also worth noting that thesame formula applies to the case of nonabelian gauge field Aµab with the understanding thatthe trace includes the trace over the internal indices as well as the spinor indices.

In fact, it is instructive to regard the coordinates as (continuous) indices, so SFx1a,x2b is amatrix and (γ ·A)x1x2 ≡ δ(x1− x2)γ ·A is a matrix, so the term as a whole can be regardedas a grand trace

− 1

nTr[iqγ · ASF ]n (8.9)

and the sum over all n is then recognized as the Taylor expansion for a logarithm:

ln 〈out|in〉A = Tr[ln(I − iqγ · ASF )]. (8.10)

Making use of the identities detA = expTr lnA and detAB = detA detB, which arefundamental properties of the determinant, and noting that in this matrix notation SF =−i(m+ (1/i)γ · ∂)−1, we see that

〈out|in〉A = det(I − qγ · A(m+1

iγ · ∂)−1) (8.11)

=det(m+ 1

iγ · ∂ − qγ · A)

det(m+ 1iγ · ∂)

(8.12)

≡ det(m− iγ ·D)

det(m− iγ · ∂)(8.13)

where we have defined the covariant derivative operator as D = ∂ − iqA. The denominatorin (8.13) serves to normalize 〈out|in〉 to 1 at A = 0, which amounts to choosing the zero ofenergy to be the ground state energy of H0. In practice these “explicit formulae” for 〈out|in〉can not be evaluated exactly for general A, although for special external potentials such asthose corresponding to constant field strength it is possible. Nonetheless they give the mostefficient derivation of the perturbation series in powers of the external fields, and as we shallsee give some general insight into the meaning of the amplitudes we are calculating.


8.2 Furry’s Theorem.

There appears to be a connected diagram (8.8) for every n. But for the electromagneticcase, only those with even n are nonvanishing. To see why this is true use the trace propertyTrAT = TrA to show that

Tr[γ · A(x1)SF (x1 − x2) · · · γ · A(xn)SF (xn − x1)] = (8.14)

Tr[STF (xn − x1)γT · A(xn) · · ·STF (x1 − x2)γT · A(xn)]. (8.15)

But γµT = −(γ0γ2)−1γµγ0γ2. From this it follows that

STF (x) = −i(γ0γ2)−1

∫d4p

(2π)4

m+ γ · pm2 + p2

eip·xγ0γ2 = (γ0γ2)−1SF (−x)γ0γ2 (8.16)

So we have

Tr[γ · A(x1)SF (x1 − x2) · · · γ · A(xn)SF (xn − x1)] = (8.17)

(−)nTr[SF (x1 − xn)γ · A(xn) · · ·SF (x2 − x1)γ · A(x1)]. (8.18)

After integrating over the x’s the only difference between the left and right sides is thelabeling of dummy integration variables and the factor (−)n on the right. Thus for odd nboth sides must vanish, i.e. all connected diagrams for 〈out|in〉 with an odd number of A’svanish. This is Furry’s theorem. A more basic way to understand the result is to note thatthe substitution A → −A can be undone by the charge conjugation transformation underwhich j → −j. Thus 〈out|in〉 must be an even function of A.

The perturbation series can be similarly “summed” for the propagator in the presence ofexternal gauge fields

〈out|Tψ(x)ψ(y)|in〉A =∞∑n=0

in

n!

∫d4x1 · · · d4xn (8.19)

〈0|T [A · j(x1) · · ·A · j(xn)ψ(x)ψ(y)]|0〉. (8.20)

The connected subdiagrams are of two types: (1) In a given term there are any number ofclosed loop diagrams of the sort contributing to 〈out|in〉A; (2) In each term there is exactlyone subdiagram with a continuous line running from y to x

y


Summing up all of the first type of diagram gives simply an overall factor of 〈out|in〉Amultiplying the value of each diagram of the second type. Apart from the net combinatoricfactor the latter diagram has at order n the value

x

y

= −(iq)n∫d4x1 · · · d4xn

SF (x− x1)γ · A(x1)SF (x1 − x2) · · · γ · A(xn)SF (xn − y).

(8.21)

In fact the combinatoric factor is just 1 since there are n! distinct contractions leadingto this diagram (giving identical values since they only differ in the labeling of dummyintegration variables), and that precisely cancels the 1/n! coming from the Dyson formula.In the matrix notation introduced above, we recognize the sum of all diagrams, includingthe disconnected closed loops, as

〈out|in〉A∞∑n=0

SF (iqγ · ASF )n = 〈out|in〉A SF1

I − iqγ · ASF(8.22)

= 〈out|in〉A−i

m− iε− iγ ·D. (8.23)

In other words the right hand side is just proportional to a Green function for the differentialoperator m− iγ ·D:

〈out|Tψ(x)ψ(y)|in〉A = 〈out|in〉A S(x, y;A) (8.24)

where

(m− iγ ·D)S(x, y;A) = −iδ4(x− y), (8.25)

and the boundary condition is specified by giving an infinitesimal negative imaginary partto m. Since A = 0 in the distant past and future, this prescription corresponds to therequirement of only positive frequencies as t → +∞ and negative frequencies as t → −∞,which are manifest properties of 〈out|Tψ(x)ψ(y)|in〉A. The fact that this latter quantityis this Green function is also immediately seen by applying the differential operator andremembering the contribution from differentiating the step functions implicit in the timeordering symbol.


By making use of our result for 〈out|Tψ(x)ψ(y)|in〉A, we can give a much quicker deriva-tion of the determinant formula for 〈out|in〉A. Looking back to (7.20) we see that if we makea change δA in A, the change in 〈out|in〉 is

δ 〈out|in〉A = 〈in|δU(∞,−∞)|in〉 = i

∫d4xδAµ(x)〈out|jµ(x)|in〉A. (8.26)

On the other hand the current matrix element can be related to the Green function of theprevious paragraph

〈out|jµ(x)|in〉A = 〈out|q2

[ψ(x), γµψ(x)]|in〉A (8.27)

= −q2

Trγµ〈out|[ψ(x), ψ(x)]|in〉A (8.28)

= −qTrγµ〈out|T [ψ(x)ψ(x)]|in〉A (8.29)

= −qTrγµS(x, x;A) 〈out|in〉A . (8.30)

where we are interpreting θ(0) = 1/2. Using δ 〈out|in〉 / 〈out|in〉 = δ ln 〈out|in〉 we obtain

δ ln 〈out|in〉A = −iq∫d4xTr[γ · δAS(x, x;A)] (8.31)

= Tr[−qγ · δA(m− iγ ·D)−1] (8.32)

= δTr ln[m− iγ ·D] (8.33)

which implies our previous formula up to a multiplicative A independent constant which isfixed by requiring 〈out|in〉A=0 = 1. Actually this initial condition is somewhat artificial; itwould be more natural to simply take 〈out|in〉A = det(m− iγ ·D). This corresponds to notremoving the sea contribution to the energy in the case A = 0.

It is instructive to see how the formal expression for the sea energy comes out of thisevaluation. When A = 0 the matrix element (in momentum basis) (p′| ln(m − γ · ∂)|p) =δ(p′− p) ln(m+ γ · p), so when we take the trace over the continuous momentum indices, weset p′ = p and get an overall factor of δ4(0). To interpret this singular factor, think of theintegral representation for δ(p) =

∫eip·xd4x/(2π)4. As p→ 0 this is just V T/(2π)4 where V

is the volume of space and T the duration of time. With this interpretation, we have

Tr ln(m− iγ · ∂) =V T

(2π)4

∫d4pTr ln(m+ γ · p) (8.34)

=V T

(2π)4

∫d4p 2 ln(m2 + p2) (8.35)

where we have used the fact that the eigenvalues of γ ·p are +√−p2 twice and −

√−p2 twice.

To compare this to the sea energy we obtained earlier, we need to interpret the integral overp0. This is of course a divergent integral, but if we differentiate once with respect to m2 toget the mass dependence, we get a convergent integral which can be evaluated by closing thecontour in the upper half complex p0 plane∫

dp0

2π

1

ω(p)2 − p02 − iε=

i

2ω= i

d

dm2ω(p). (8.36)


Thus

ln 〈out|in〉 = 2iTV

(2π)3

∫d3p[ω(p) + C] (8.37)

where the constant is at least independent of the mass. Since 〈out|in〉 ∼ e−iEseaT we see thatwe recover our previous result for Esea at least as far as the mass dependence is concerned.The constant C is itself infinite and complex: the Wick rotation to imaginary p0 gives afactor of i, but the contours at infinty are nonvanishing and complex.

The occurrence of complex energies in the Minkowski definition of 〈out|in〉 motivatesthe idea that these vacuum amplitudes should be defined fundamentally in Euclidean spaceand then continued back to Minkowski space as the application demands. In Euclideanspace-time we put T = −iTE and the vacuum amplitude would have the behavior 〈out|in〉 ∼exp(−EseaTE). If we repeat the calculation of Tr ln(m − iγ · ∂) in Euclidean space we getV TE

∫d4pE2 ln(m2 + p2)/(2π)4 which is manifestly real albeit divergent, and we get the

formula

Esea = −2V

(2π)4

∫d4pE ln(m2 + p2). (8.38)

This still has the mass dependence of our earlier result and the mass independent discrepancyis at least real. In effect, working in Euclidean space-time from the beginning discardsundesirable complex contours from Wick rotations that fail to vanish only because of the poorhigh momentum (ultra-violet) behavior of the integrands of vacuum diagrams. Consequently,among modern field theorists Euclidean space-time is widely accepted as the best way todefine vacuum amplitudes.

Finally, we consider the calculation of time ordered products of an arbitrary number ofDirac fields. Clearly the nonvanishing ones have an equal number of ψ’s and ψ’s:

〈out|T [ψ(x1) · · ·ψ(xn)ψ(yn) · · · ψ(y1)]|in〉. (8.39)

To each term there will correspond any number of connected closed loop diagrams and nconnected diagrams of exactly the type contributing to the n = 2 case. Each of these lattersubdiagrams consists of a line from one of the yk to one of the xPk . The closed loops sumto an overall factor of 〈out|in〉. For each distinct pairing of the x’s with the y’s the othersubdiagrams sum to a product of n factors S(xPk , yk;A). In other words, there is a Wickexpansion for time ordered products of Dirac fields in the presence of an external field exactlyas in the A = 0 case. All one does is substitute SF (x− y)→ S(x, y;A) and multiply by anoverall factor of 〈out|in〉A.


Chapter 9

Scattering in External Fields

Our time dependent formalism is ideally suited for defining transition amplitudes1. Sinceexternal fields are turned off in the far future and distant past, particle states can be definedin exactly the same manner as the in and out vacua. We can expand the Heisenberg fieldoperators at early(late) times in terms of annihilation and creation operators binλ (p), dinλ (p)(boutλ (p), doutλ (p)), since they are free fields there. These are of course eigenoperators forH0(−∞)(H0(+∞)) respectively. They are of course determined up to a phase by the labelsλ,p. We can fix the phases of the out operators in terms of the in operators by definingΩout ≡ U−1

I (∞,−∞)ΩinUI(∞,−∞)2. Then incoming particle states are obtained by ap-plying bin†, din† to |in〉 and outgoing particle states are obtained by applying bout, dout to〈out|.

The transition amplitudes between multiparticle states at early times and multiparticlestates at late times can be immediately transcribed to interaction picture:

〈out|bout1 · · · doutN din†M · · · bin†1 |in〉 = 〈0, I|bI1 · · · dINUI(∞,−∞)d†IM · · · b

†I1|0, I〉. (9.1)

The rules for expanding these amplitudes in perturbation theory are very similar to those forthe outin matrix elements of time ordered products of Dirac field operators. The disconnectedclosed loops sum up to an overall factor of 〈out|in〉A. The creation and annihilation operatorscan either contract against each other or against one of the Dirac fields in UI :

1This chapter brings together is a unified setting topics on scattering introduced in earlier chapters forthe purpose of giving concrete examples of general principles.

2Note that since U = U0UI and U−10 ΩinU0 is just a numerical phase times Ωin this definition does ensurethat Ωoutis an eigenoperator of H0(+∞).

103

〈0, I|bλ(p)ψ(x)|0, I〉 =1

(2π)3/2√

2ωuλ(p)e−ip·x

(9.2)

〈0, I|dλ(p)ψ(x)|0, I〉 =1

(2π)3/2√

2ωvλ(p)e−ip·x

(9.3)

〈0, I|ψ(x)b†λ(p)|0, I〉 =1

(2π)3/2√

2ωuλ(p)eip·x

(9.4)

〈0, I|ψ(x)d†λ(p)|0, I〉 =1

(2π)3/2√

2ωvλ(p)eip·x

(9.5)

To illustrate how to use these rules consider the process in which a single particle at earlytime is scattered by the external potential to a single particle at late times.

〈out|boutλ (q)bin†λ (p)|in〉 = 〈out|in〉A [δλ′λδ3(q − p) + 〈0, I|bλ(q)UIb

†λ(p)|0, I〉c] (9.6)

where the superscript c denotes the restriction to connected diagrams containing at least onevertex.

. . .+ + +

The first diagram has the value

1

(2π)3√

4ω(q)ω(p)

∫d4xei(p−q)·xuλ′(q)iqγ · A(x)uλ(p). (9.7)

The second diagram has two vertices and a factor of SF (y − x) between them. But noticethat the second plus all the higher diagrams just amounts to replacing this SF by SF (y, x;A),


so the sum of all the diagrams but the first has the value

1

(2π)3√

4ω(q)ω(p)

∫d4xd4yei(p·x−q·y)uλ′(q)iqγ · A(y)SF (y, x;A)iqγ · A(x)uλ(p). (9.8)

We put all this together in the form

〈out|boutλ (q)bin†λ (p)|in〉 = 〈out|in〉A [δλ′λδ3(q − p) +M(q,p;A)], (9.9)

where

M(q,p;A) =1

(2π)3√

4ω(q)ω(p)

∫d4xd4yei(p·x−q·y) (9.10)

uλ′(q)[iqγ · A(x)δ(x− y) + iqγ · A(y)SF (y, x;A)iqγ · A(x)]uλ(p).

We can put this last formula in a more suggestive form by defining the free Dirac plane wavefunctions

ψ0pλ(x) ≡ 1

(2π)3/2√

2ωuλ(p)eip·x (9.11)

and noting that

ψpλ(x) ≡ ψ0pλ(x) +

∫d4ySF (x, y;A)iqγ · A(y)ψ0

pλ(y) (9.12)

is a solution of the Dirac equation in the presence of A:

(m+1

iγ ·D)ψpλ = 0, (9.13)

with the boundary condition that at early times the only positive frequency components arecontained in the term ψ0

pλ(x). Thus we can write

M(q,p;A) =

∫d4xψ0

qλ′(x)iqγ · A(x)ψpλ(x), (9.14)

which is reminiscent of the corresponding formula in nonrelativistic quantum mechanics forthe scattering of a particle from an external potential.

In the case of a static potential, the time dependence of both ψ0pλ(x) and ψpλ(x) is given

by a multiplicative phase e−iω(p)t and the time integral then provides an energy conservingdelta function

M(q,p;A(x)) = 2πδ(ω(q)− ω(p))

∫d3xψ0

qλ′(x)iqγ · A(x)ψpλ(x). (9.15)

Recall from basic scattering theory that if the scattering matrix for a particle from a staticpotential is written

〈q, out|p, in〉 = δλ′λδ(q − p)− 2πiδ(ω(q)− ω(p))Tλ′λ(q,p) (9.16)



dσ

dΩ=

d3q


(2π)4

v|Tλ′λ(q,p)|2, (9.17)

= q2dqδ(ω(q)− ω(p))(2π)4

v|Tλ′λ(q,p)|2, (9.18)

= pω(p)(2π)4

v|Tλ′λ(q,p)|2, (9.19)

= ω(p)2(2π)4|Tλ′λ(q,p)|2, (9.20)

where v is the speed of the incident particle. Comparing with our expression we find thatfor electron scattering from a static potential

Tλ′λ(q,p) = i

∫d3xψ0

qλ′(x)iqγ · A(x)ψpλ(x). (9.21)

Note that in the absence of pair production, the factor 〈out|in〉, the vacuum persistenceamplitude, is a pure phase and doesn’t contribute in the absolute square of T . In lowestorder in A (the Born approximation) one simply replaces ψ by ψ0.

Let us note some tricks that are useful in calculating cross sections for processes withunobserved final spins and unpolarized beams. When we calculate the absolute square of aspinor matrix element we can make use of the identities∑

λ

uλuλ = m− γ · p (9.22)∑λ

vλvλ = −m− γ · p. (9.23)

For example these allow us to write∑λλ′

|uλ′γ · A(q − p)uλ(p)|2 = Tr[γ · A(m− γ · p)γ · A∗(m− γ · q)] (9.24)

= 4(−m2 − p · q)A · A∗ + 4(p · Aq · A∗ + q · Ap · A∗)(9.25)

= 2(p− q)2A · A∗ + 4(p · Aq · A∗ + q · Ap · A∗) (9.26)

Physical quantities should be gauge invariant and cross sections are no exception. We canreveal the gauge invariance of this last formula by introducing Fµν(q) ≡ i(qµAν−qνAµ) whichis the Fourier transform of the gauge invariant field strength. Then

FµνFµν∗ = 2(q2A · A∗ − q · Aq · A∗). (9.27)

Recalling that the argument of A is q−p we then find that the squared spinor matrix elementcan be rewritten∑

λλ′

|uλ′γ · A(q − p)uλ(p)|2 = FµνFµν∗ + 2(p+ q) · A (p+ q) · A∗. (9.28)


The first term is manifestly gauge invariant and the second is gauge invariant by virtue ofthe identity

(q + p) · (q − p) = q2 − p2 = −m2 +m2 = 0. (9.29)

The differential cross section for unpolarized electron scattering in a general static potentialwith final spin unobserved is then

∑λ′

dσ

dΩ

Born

Unpolarized=

1

16π2

1

2

∑λ′λ

|uλ′qγ · A(q − p)uλ(p)|2 (9.30)

=e2

16π2(1

2FµνF

µν∗ + (p+ q) · A (p+ q) · A∗) (9.31)

=α

4π(1

2FµνF

µν∗ + |(p+ q) · A|2). (9.32)

One can easily confirm that this agrees with our previous results for the special case of theCoulomb potential.

We expect that the transition amplitudes for antiparticles should be obtained from (9.10)by the substitution A→ −A. On the other hand the rules seem to give a different prescrip-tion:

〈out|doutλ (q)din†λ (p)|in〉 = 〈out|in〉A [δλ′λδ3(q − p) + M(q,p;A)], (9.33)

with

M(q,p;A) = − 1

(2π)3√

4ω(q)ω(p)

∫d4xd4yei(p·x−q·y) (9.34)

vλ(p)[iqγ · A(x)δ(x− y) + iqγ · A(x)SF (x, y;A)iqγ · A(y)]vλ′(q).

In fact it is not hard to show that M(q,p;A) = M(q,p;−A) by inserting v = iγ2u∗ into(9.34) and transposing the matrix element, using iγ2γ0γµTγ0iγ2 = −γµ and iγ2γ0ST (x, y;A)γ0iγ2 =−γµ = SF (y, x;−A). The latter fact can be seen either term by term in the expansion in Aor by examining the defining Green function equation for SF (x, y;A).

In addition to the scattering of electrons or positrons, an external field can induce electronpositron pair production and also pair annihilation. Taking the first case, for example, wehave

〈out|boutλ (q1)doutλ′ (q2)|in〉 = 〈out|in〉A [MPairCreate(q1, q2;A)], (9.35)

with

MPairCreate(q1, q2;A) = − 1

(2π)3√

4ω(q1)ω(q2)

∫d4xd4ye−i(q1·x+q2·y) (9.36)

uλ(q1)[iqγ · A(x)δ(x− y) + iqγ · A(x)SF (x, y;A)iqγ · A(y)]vλ′(q2).


In this case, of course, there is no delta function term representing only vacuum persistence.Similarly for pair annihilation, we have

〈out|bin†λ (q1)din†λ′ (q2)|in〉 = 〈out|in〉A [MPairAnnih(q1, q2;A)], (9.37)

with

MPairAnnih(q1, q2;A) =1

(2π)3√

4ω(q1)ω(q2)

∫d4xd4ye−i(q2·x+q1·y) (9.38)

vλ(q2)[iqγ · A(x)δ(x− y) + iqγ · A(x)SF (x, y;A)iqγ · A(y)]uλ′(q1).

Notice the prominent appearance of the Green function SF (x, y;A) for the Dirac Equationwith an external field in all of the four basic processes, electron scattering, positron scattering,pair production, and pair annihilation. Moreover, we have also seen how to express thevacuum persistence amplitude in terms of this same Green function. Thus we see that thesolutions of the first quantized Dirac equation are of direct utility in finding the physicalproperties of the second quantized theory.

As a final note we show how SF even contains information about the bound states of anelectron in a static external field. Actually it is better to deal not with an exactly staticfield, but with a field that is adiabatically switched on at some very early time −T , staticfor a very long time interval ≈ 2T and then adiabatically switched off at a very late timeT . Then according to the adiabatic theorem, |in〉 is proportional to the ground state of theHamiltonian H(t) at all times, and |out〉 = |in〉〈out|in〉. In particular, |in〉 is the groundstate |G〉 of HA the hamiltonian for the static potential under study, since H(t) = HA for−T << t << T . Now fix the times T >> x0 > y0 >> −T . Then we have

SF (x, y) 〈out|in〉 = 〈out|ψ(x)ψ(y)|in〉 (9.39)

= 〈out|in〉〈G|ψ(x)ψ(y)|G〉 (9.40)

= 〈out|in〉∑n

〈G|ψ(x)|n〉〈n|ψ(y)|G〉 (9.41)

= 〈out|in〉∑n

e−i(En−EG)(x0−y0)〈G|ψ(x, 0)|n〉〈n|ψ(y, 0)|G〉 (9.42)

From this last formula, we see that a harmonic analysis of SF in the time variable t = x0−y0

yields the possible energy eigenvalues En − EG of the energy eigenstates |n〉. These stateshave the quantum numbers of a one electron state because they are created from the vacuumby ψ. Note that the harmonic components for x0 < y0 have the interpretation as −(En−EG)where En are the energy levels of one positron states. Because SF satisfies the homogeneousDirac equation for t 6= 0, the possible energy eigenvalues are solutions of the time independentDirac equation

(m+1

iγ ·D + qA0γ0 − (En − EG))ψ = 0, t > 0 (9.43)

or

(m+1

iγ ·D + qA0γ0 + (En − EG))ψ = 0, t < 0. (9.44)


Thus, again, the solution of the energy eigenvalue problem for the first quantized theory isdirectly applicable to that for the second quantized theory.

9.1 Relation to Time Ordered Products.

(This subsection can be skipped in a first reading.) It is useful to establish how the scatteringamplitudes are related to matrix elements of time ordered products. First let us define

b†λ(p, t) =

∫d3x

1

(2π)3/2√

2ω(p)eip·xψ(x)γ0uλ(p) (9.45)

d†λ(p, t) =

∫d3x

1

(2π)3/2√

2ω(p)eip·xvλ(p)γ0ψ(x) (9.46)

bλ(p, t) =

∫d3x

1

(2π)3/2√

2ω(p)e−ip·xuλ(p)γ0ψ(x) (9.47)

dλ(p, t) =

∫d3x

1

(2π)3/2√

2ω(p)e−ip·xψ(x)γ0vλ(p). (9.48)

For free fields all these operators are constant in time and are just the creation and annihi-lation operators of the Dirac field. For a Dirac field in the presence of external fields whichvanish at early and late times they are not constant but approach the “out” creation andannihilation operators at t =∞ and the “in” operators at t = −∞.

Now we shall make use of the following “reduction” trick:

F (t = +∞)T [φ1(x1) · · ·φN(xN)]∓ T [φ1(x1) · · ·φN(xN)]F (t = −∞)

=

∫ ∞−∞

dt∂

∂tT [F (t)φ1(x1) · · ·φN(xN)] (9.49)

where F (t) is any function of Heisenberg operators at time t, and the φ’s are generic fieldoperators. The sign choice is + for F (t) and T [· · · ] fermionic operators, but − in all othercases. In particular, F can be any of the expressions (9.48) where the time appearing onthe r.h.s. is set to t. This leads to a series of formulae for the commutation of creation andannihilation operators with time ordered products. For instance, take F to be the r.h.s. ofthe expression for bλ(p).

boutλ (p)T [φ1(x1) · · ·φN(xN)]∓ T [φ1(x1) · · ·φN(xN)]binλ (p)

=uλ(p)

(2π)3/2√

2ω(p)

∫d4xγ0 ∂

∂t(e−ip·xT [ψ(x)φ1(x1) · · ·φN(xN)]) (9.50)

=uλ(p)

(2π)3/2√

2ω(p)

∫d4xe−ip·x

(γ0 ∂

∂t+ iγ0ω(p)

)T [ψ(x)φ1(x1) · · ·φN(xN)] (9.51)

=uλ(p)

(2π)3/2√

2ω(p)

∫d4xe−ip·xi (m− iγ · ∂)T [ψ(x)φ1(x1) · · ·φN(xN)] (9.52)


where in the last line we used

uγ0ωe−ip·x = u(m+ iγ · ∇)e−ip·x (9.53)

and then integrated by parts. Clearly this derivation can be repeated for each of the expres-sions in (9.48) leading to the reduction formulae:

boutλ (p)T [φ1(x1) · · ·φN(xN)]∓ T [φ1(x1) · · ·φN(xN)]binλ (p)

=uλ(p)

(2π)3/2√

2ω(p)

∫d4xe−ip·xi (m− iγ · ∂)T [ψ(x)φ1(x1) · · ·φN(xN)] (9.54)

bout†λ (p)T [φ1(x1) · · ·φN(xN)]∓ T [φ1(x1) · · ·φN(xN)]bin†λ (p)

=

∫d4xT [ψ(x)φ1(x1) · · ·φN(xN)]i

(−m− iγ ·

←−∂)eip·x

uλ(p)

(2π)3/2√

2ω(p)(9.55)

doutλ (p)T [φ1(x1) · · ·φN(xN)]∓ T [φ1(x1) · · ·φN(xN)]dinλ (p)

=

∫d4xT [ψ(x)φ1(x1) · · ·φN(xN)]i

(−m− iγ ·

←−∂)e−ip·x

vλ(p)

(2π)3/2√

2ω(p)(9.56)

dout†λ (p)T [φ1(x1) · · ·φN(xN)]∓ T [φ1(x1) · · ·φN(xN)]din†λ (p)

=vλ(p)

(2π)3/2√

2ω(p)

∫d4xeip·xi (m− iγ · ∂)T [ψ(x)φ1(x1) · · ·φN(xN)] (9.57)

The reduction formulae can be used to systematically reduce scattering amplitudes to ex-pressions directly involving time ordered products.

We shall illustrate the procedure for the case of a particle scattering in an external field.The application to antiparticle scattering and pair production and annihilation will be leftto the reader:

〈out|boutλ′ (q)bin†λ (p)|in〉 = 〈out|binλ′ (q)bin†λ (p)|in〉

+uλ′(q)

(2π)3/2√

2ω(q)

∫d4xe−iq·xi (m− iγ · ∂) 〈out|ψ(x)bin†λ (p)|in〉 (9.58)

= 〈out|in〉 δλλ′δ(q − p) +uλ′(q)

(2π)3/2√

2ω(q)

∫d4xd4y (9.59)

e−iq·xi (m− iγ · ∂) 〈out|T [ψ(y)ψ(x)]|in〉i(−m− iγ ·←−∂ y)e

ip·y uλ(p)

(2π)3/2√

2ω(p)

= 〈out|in〉[δλλ′δ(q − p) +

uλ′(q)

(2π)3/2√

2ω(q)

∫d4xd4y

e−iq·xi (m− iγ · ∂)SF (x, y;A)i(m+ iγ ·←−∂ y)e

ip·y uλ(p)

(2π)3/2√

2ω(p)

](9.60)

This is identical to our original expression as can be seen by writing m − iγ · ∂ = m − iγ ·D + qγ · A so that

i(m− iγ · ∂)SF (x, y;A)i(m+ iγ ·←−∂ y)

= δ(x− y)i(m+ iγ ·←−∂ y) + iqγ · ASF (x, y;A)i(m+ iγ ·

←−∂ y). (9.61)


The first term on the r.h.s contributes a term u(q)(m+ γ · q)u(p) = 0 by the Dirac equation.The second term can be simplified using the fact that S is also a Green function in its secondargument:

SF (m+ iγ ·←−∂ y − qγ · A) = −iδ(x− y). (9.62)

Thus we have

i(m− iγ · ∂)SF (x, y;A)i(m+ iγ ·←−∂ y) = iqγ · Aδ(x− y) + iqγ · ASF (x, y;A)iqγ · A (9.63)

as desired.

Now let us consider a little more closely the meaning of the reduction formula

〈out|boutλ′ (q)bin†λ (p)|in〉 = 〈out|in〉[δλλ′δ(q − p) +

uλ′(q)

(2π)3/2√

2ω(q)

∫d4xd4y (9.64)

e−iq·xi (m− iγ · ∂)SF (x, y;A)i(m+ iγ ·←−∂ y)e

ip·y uλ(p)

(2π)3/2√

2ω(p)

].

The factors (m − iγ · ∂) and i(m + iγ ·←−∂ y) look as though they should give zero because

after integrating by parts, they become (m + γ · q) and (m + γ · p) respectively and thesefactors give zero next to the Dirac spinors. The error in this reasoning is of course that thesurface terms at t = ±∞ are not zero (being in fact the scattering amplitudes themselves!).To get a clearer idea of what is happening, suppose we continue q0, p0 a little away fromtheir “on-shell” values of ω(q), ω(p). Then the surface terms oscillate at infinity and areeffectively zero (e.g. in any smooth wave packet). Then we can integrate by parts and thesecond term becomes

uλ′(q)

(2π)3/2√

2ω(q)i (m+ γ · q)

∫d4xd4ye−iq·x+ip·ySF (x, y;A)i(m+ γ · p) uλ(p)

(2π)3/2√

2ω(p).

≡ uλ′(q)

(2π)3/2√

2ω(q)i (m+ γ · q)T (q, p)i(m+ γ · p) uλ(p)

(2π)3/2√

2ω(p). (9.65)

where the r.h.s. defines T (q, p), the Fourier transform of the Green function. The only wayfor this to be nonzero as q0, p0 approach there on-shell values is for T to acquire poles inthis limit. The residues of these poles are then related to the scattering amplitudes. This isthe content of the reduction formula, but there is a rather direct way to see how these polescome about.

Consider first of all the region of integration x0 > T where ψ is a free field, with bout, dout†.Then this region contributes to T (q, p) the bit∫ ∞

T

dtei(q0−ω(q))t (2π)3/2√

2ω(q)〈out|

∑λ′

bout(q)uλ′(q)ψ(y)|in〉 (9.66)


For q0 = ω the integrand is time independent so the integral over an infinite range gives adivergence. To study it, give q0 a small positive imaginary part and do the integral:

iei(q0−ω(q))T

q0 − ω(q) + iε

(2π)3/2√2ω(q)

〈out|∑λ′

bout(q)uλ′(q)ψ(y)|in〉 (9.67)

For the range of integration with x0 < −T ψ contains bin, din† and stands on the right soonly the second operator contributes. The integral over this range is not singular for q0 → ω.

Similar considerations apply to the integral over y0. This time the region y0 < −Tcontributes a pole at p0 = ω(p) involving bin†. There is also a pole from the region y0 > Tinvolving bout†. This contribution doesn’t vanish because of the presence of bout from the firstreduction; it just gives delta functions. Going through all these steps leads to

T (q, p) ∼ i

q0 − ω(q)

i

p0 − ω(p)

[(2π)3√

4ω(q)ω(p)

]∑λ′,λ

uλ′(q)uλ(p) (9.68)

[〈out|boutλ′ (q)bin†λ (p)|in〉 − 〈out|in〉 δλ′λδ(q − p)], q0, p0 → ω(q), ω(p)

which is of course exactly the behavior required to satisfy the reduction formula.


Chapter 10

Vacuum Polarization

Before leaving external field problems, it is interesting to consider the effect an external fieldhas on the vacuum (negative energy sea in the Dirac case). We shall calculate the responseof the vacuum to the application of a weak electromagnetic field. For definiteness considerthe Dirac field described by

HA = H0 −∫d3xjµ(x, t)Aµ(x, t). (10.1)

Assume that A → 0 as t → −∞ and that the system starts out in the ground state ofHA(−∞) = H0(−∞).

We should expect the field to induce charge and current densities in the vacuum. Asimple measure of these induced currents is the expectation value of the Heisenberg picturecurrent operator in the system state 〈in|jµ(x)|in〉A. We may express this matrix element ininteraction picture and then develop it in an expansion in powers of A. In the limit of veryweak fields we can neglect all terms beyond those linear in A:

〈in|jµ(x)|in〉 = 〈in|U−1I (t,−∞)jµI (x)UI(t,−∞)|in〉A (10.2)

= 〈0, I|(Tei

∫ t−∞ jI ·A

)†jµI (x)Tei

∫ t−∞ jI ·A|0, I〉 (10.3)

≈ i

∫d4yθ(t− ty)〈0, I|[jµI (x, t), jνI (y)]|0, I〉Aν(y). (10.4)

There is no term independent of A because the vacuum expectation value of the currentvanishes in the absence of applied fields.

This is the linear response to an applied field and is characterized by the response function

Rµν(x) ≡ iθ(t)〈0|[jµ(x), jν(0)]|0〉 (10.5)

where here and in the following we drop the subscripts I and it is understood that thecurrents are those of free fields. An important physical property of the response function isEinstein causality: Rµν vanishes for spacelike argument x2 > 0 as follows from the fact thatlocal operators commute at space-like separations. Thus application of an external field atthe origin at t = 0 cannot evoke a response at x until enough time has elapsed for light totravel from the origin to x. This property is not shared by the time ordered product.

113

10.1 Retarded Commutators from Time Ordered Prod-

ucts

The Wick expansion we have developed works best for time ordered products, so it is helpfulthat we can work out a relationship between the response function and the expectation valueof the time-ordered product. This relationship is a general one that depends only on thetime variable, so we suppress spatial and internal labels and consider two hermitian operatorsO1(t), O2(t). We shall actually relate the Fourier transforms of the two quantities:

R(ω) ≡ i

∫dteiωtθ(t)〈G|[O1(t), O2(0)]|G〉 (10.6)

T (ω) ≡ i

∫dteiωt〈G|T [O1(t)O2(0)]|G〉 (10.7)

where |G〉 is the Ground State of the system, assumed to be nondegenerate. Now usingθ(t) = 1− θ(−t) we have

θ(t)[O1(t), O2(0)] = T [O1(t)O2(0)]−O2(0)O1(t) (10.8)

so the difference between R and T involves

−i∫dteiωt〈G|O2(0)O1(t)|G〉 (10.9)

= −i∫dteiωt

∑n

〈G|O2(0)|n〉〈n|O1(0)|G〉e−i(EG−En)t (10.10)

= −2πiδ(ω)〈G|O2(0)|G〉〈G|O1(0)|G〉 (10.11)

−i∫dteiωt

∑n6=G

〈G|O2(0)|n〉〈n|O1(0)|G〉e−i(EG−En)t (10.12)

The important feature of this result is that by virtue of the fact that EG is the lowest energyeigenvalue, the r.h.s. vanishes for positive frequency ω > 0. Thus in this case R(ω) = T (ω).Next we find a relation for negative frequency. For this case we relate R to the anti-time-ordered product:

θ(t)[O1(t), O2(0)] = O1(t)O2(0)− T [O1(t)O2(0)]. (10.13)

Now inserting a complete set of states allows us to conclude that the Fourier transform of thefirst term vanishes for negative frequency ω < 0. Thus in this case we have R(ω) = −T (ω)where

T (ω) ≡ i

∫dteiωt〈G|T [O1(t)O2(0)]|G〉 = −T ∗(−ω). (10.14)

where we used the assumption that O1, O2 are hermitian.


In summary we have found that

R(ω) =

T (ω) ω > 0

T ∗(−ω) ω < 0. (10.15)

For ω near zero we observe that the ground state contributes to T but not to R. Thecontribution to T is

TG(ω) = 2πiδ(ω)〈G|O2(0)|G〉〈G|O1(0)|G〉. (10.16)

If there is a gap separating EG from the rest of the spectrum this is the only zero frequencydiscrepancy between R and T .

Finally when we consider this relationship for field operators, it is natural to quote it forthe spatial and temporal Fourier transform:

R(p0,p) =

T (p0,p) p0 > 0

T ∗(−p0,−p) p0 < 0. (10.17)

Again the vacuum contributes to T (and not R) the amount

TG(p0) = (2π)4iδ4(p)〈0|O1|0〉〈0|O2|0〉. (10.18)

10.2 Calculation of Vacuum Polarization

If we Fourier transform the current induced by an external field, we obtain

Jµ(k) =

∫d4xe−ik·x〈in|jµ(x)|in〉 (10.19)

=

∫d4xe−ik·x

∫d4yRµν(x− y)Aν(y) +O(A2) (10.20)

= Rµν(k)Aν(k) +O(A2) (10.21)

and we have just obtained the relation of Rµν to T µν .To calculate T µν we first apply the Wick expansion to the time ordered product of four

Dirac fields contained in the two current amplitude. There are two distinct contractionscorresponding to the diagrams

x

0, ν

, µx

0, ν

, µ


The disconnected diagrams vanish because 〈0|jµ|0〉 = 0 in the absence of external fields.(This is a simple consequence of charge conjugation invariance.) The unique connecteddiagram has the value

−Q2Tr[γµSF (x)γνSF (−x)] (10.22)

where the minus sign comes from the single closed fermi loop. Inserting the known Fourierrepresentation for SF and carrying out the integration over x in the evaluation of T µν leadsto

T µν(k) = iQ2

∫d4p

(2π)4Tr

(γµ

m− p · γm2 + p2 − iε

γνm− (p− k) · γ

m2 + (p− k)2 − iε

). (10.23)

We immediately see from this expression that the integration over momentum is quadraticallydivergent at high momentum. The origin of this divergence is that SF (x) behaves like 1/x3

at small x which means that the two current amplitude behaves like 1/x6 which means thatits Fourier transform is ill-defined.

Before dealing with this divergence, let us simplify the integrand by first evaluating thetrace

Nµν(p, k) ≡ Tr(γµ(m− p · γ)γν(m− (p− k) · γ)) (10.24)

= 8pµpν − 4(pµkν + pνkµ)− 4ηµν(m2 + p · (p− k)) (10.25)

and secondly combining denominators using the Feynman trick

1

AB=

∫ 1

0

dx1

[Ax+B(1− x)]2(10.26)

which is trivial to derive. Then

T µν(k) = iQ2

∫d4p

(2π)4

∫ 1

0

dxNµν(p, k)

[m2 + (p− kx)2 − iε+ x(1− x)k2]2(10.27)

where we have completed the square in the denominator.Next we do a step which is not quite legitimate in view of the quadratic divergence,

which is to change integration variables p → p + kx. After this the denominator dependsonly on p2, so all terms in the numerator linear in pµ integrate to zero and can be dropped.Furthermore terms of the form pµpν can be replaced by p2ηµν/4, since

∫d4pf(p2)pµpν must be

proportional to ηµν and the proportionality constant is then determined to be∫d4pf(p2)p2/4

by comparing the trace of both sides. Thus we have the replacements

Nµν(p+ xk, k) → 8(p2

4ηµν + x2kµkν)− 8xkµkν − 4ηµν(m2 + p2 − x(1− x)k2)

→ −2ηµνp2 + 4x(1− x)(k2ηµν − 2kµkν)− 4m2ηµν (10.28)

After all these steps so far we have reduced the integrals to

T µν(k) = iQ2

∫d4p

(2π)4

∫ 1

0

dx−2ηµνp2 + 4x(1− x)(k2ηµν − 2kµkν)− 4m2ηµν

[m2 + p2 − iε+ x(1− x)k2]2. (10.29)


Next we wish to evaluate the p integral. It is easiest to think about this evaluation afterthe Wick rotation to Euclidean momenta, so the integral is over 4 dimensional Euclideanspace and the integrand is O(4) invariant. Then the angular integrals can be done and theintegral reduced to a one dimensional one. The rotation of the p0 contour to the imaginaryaxis must avoid the singularities due to the vanishing of the denominator which occurs at

p0 = ±√m2 + p2 + x(1− x)k2 − iε (10.30)

These poles remain in the fourth and second quadrant of the complex p0 plane for all valuesof k2. However they get infinitesimally close to the imaginary axis for x(1−x)k2 ≤ −m2−p2

which we shall see is responsible for singular behavior in the result as a function of k2. Aslong as we stick to k2 > −4m2, though, the poles stay well within their respective quadrants,and a counterclockwise contour rotation by 90 degrees encounters no singularities.

0p

4

p0

-ip

After the Wick rotation we change variables to p0 = ip4 so that d4p = id4pE and p2 =p2 + (p4)2. Going to polar coordinates, d4pE = p3dpdΩ, we wish to evaluate the angularintegrals dΩ. A useful trick to do this in any number of dimensions is to integrate a Gaussiane−~p

2in both Cartesian and polar coordinates. In Cartesian coordinates in D dimensions we

get πD/2. In polar coordinates it is

ΩD

∫ ∞0

pD−1dpe−p2

= ΩD1

2Γ(D/2) (10.31)

Comparing we arrive at

ΩD =2πD/2

Γ(D/2). (10.32)

Notice that for D = 1, 2, 3 this gives the well known results 2, 2π, 4π. For our case D = 4


and the result is 2π2. We also record here the useful formula∫dDp

(p2)m

(p2 + A2)n=

2πD/2

Γ(D/2)

∫ ∞0

dppD−1+2m

(p2 + A2)n

=AD+2m−2nπD/2Γ(m+D/2)Γ(n−m−D/2)

Γ(D/2)Γ(n). (10.33)

The fact that the r.h.s. is a perfectly defined analytic function of D,m, n allows for dimen-sional regularization as we shall see later.

Putting all this together and cutting off the p integral at Λ we have so far

T µνΛ (k) = − Q2

8π2

∫ Λ

0

p3dp

∫ 1

0

dx−2ηµν(p2 + 2m2) + 4x(1− x)(k2ηµν − 2kµkν)

[m2 + p2 − iε+ x(1− x)k2]2. (10.34)

The p integrals are now elementary:∫ Λ

0

p3dp

[p2 + C]2=

1

2

[ln

Λ2 + C

C− 1

]+

1

2

C

Λ2 + C(10.35)

=1

2

[ln

Λ2

C− 1

]+O(Λ−2) (10.36)∫ Λ

0

p5dp

[p2 + C]2= Λ2 − C ln

Λ2 + C

C− 1

2

Λ4

Λ2 + C(10.37)

=1

2Λ2 +

1

2C − C ln

Λ2

C+O(Λ−2), (10.38)

where for us C = m2 + x(1− x)k2− iε. Putting these results into the expression for T gives

T µνΛ (k) = − Q2

8π2

∫ 1

0

dx

[−ηµν(Λ2 − C − 2C(ln

Λ2

C− 1))

−(2ηµνm2 − 2x(1− x)(k2ηµν − 2kµkν)

[ln

Λ2

C− 1

]](10.39)

= − Q2

2π2

∫ 1

0

dxx(1− x)(k2ηµν − kµkν)(ln Λ2

m2 + x(1− x)k2 − iε− 1)

+Q2

8π2ηµν(Λ2 −m2 − k2

6). (10.40)

We could do the last integral over x, but it is actually easier to see the properties of Tdirectly from the integral representation (10.40).

We have been casual about the way we cutoff the momentum integral, and now we cansee a bad consequence of this: a violation of gauge invariance. In Fourier components agauge transformation on the external field has the form Aµ(k)→ Aµ + kµΛ(k). The inducedcurrent was given by Rµν(k)Aν(k), so gauge invariance would imply Rµν(k)kν = 0, and inview of the relation between R and T , T µν(k)kν = 0. Clearly, the last line of (10.40) fails to


satisfy this condition. The reason for this error can be traced to insufficient care with themanner in which we regularized the divergent integral. It is fortunate that the momentumdependence of the erroneous terms is a simple polynomial. This is in fact a characteristic ofall such errors induced by ultraviolet divergences: Differentiating the integral a finite numberof times with respect to the external momenta renders it convergent, so a finite number ofderivatives must kill the mistake. In this case it would require three derivatives to kill themistake. We shall later discuss gauge invariant regularization procedures that prevent suchmistakes from occurring, provided of course that the theory can be consistently quantized.But for now we shall be satisfied with simply adjusting the polynomial dependence of ourresults to be consistent with gauge invariance. The nonpolynomial part of T can of coursenot be removed by such an adjustment. Making this adjustment, and at the same timeabsorbing the −1 in a rescaling of the cutoff, we then obtain

T µνGI (k) = (kµkν − k2ηµν)T (k2) (10.41)

T (k2) =Q2

2π2

∫ 1

0

dxx(1− x) lnΛ2e−γ

m2 + x(1− x)k2 − iε. (10.42)

It is important to appreciate that gauge invariant regularization does not cure the problemof ultraviolet divergences, although it does reduce its severity. Our initial expression for Twas quadratically divergent, but we have seen that gauge invariance effectively reduces thedivergence to a logarithmic one. We shall see that this last divergence, although present inthe quantities we are calculating, disappears after expressing the answer in terms of physicallymeasurable parameters. Note also that our polynomial adjustment of T µν to make it gaugeinvariant allows an undetermined constant γ. However this ambiguity is linked to the cutoffdependence, and will disappear along with the latter in physical quantities.

10.3 The Physics of Vacuum Polarization

Our result for T µν can now be used to give us the response function

RµνGI(k) = (kµkν − k2ηµν)R(k2) (10.43)

where

R(k2) =

T (k2) k0 > 0

T ∗(k2) k0 < 0. (10.44)

Note that since T is real for k2 > −4m2, the two cases merge for that range of momentum.Since we have incorporated gauge invariance in our answer, we are free to fix a convenientgauge for discussing the physical interpretation of our result. Let us choose Lorentz gauge,kµAµ = 0. Then the Fourier transform of the current induced by the external field is simply

〈jµ(k)〉 = −k2R(k2)Aeµ(k). (10.45)


We must now recognize that the induced currents will produce induced fields via Maxwell’sequations. As long as Ae the external field is sufficiently weak the induced currents and theinduced field AIND will also be weak, and it will be consistent to assert that the total currentis given by the response function times the total field Ae + AIND.

〈jµ(k)〉TOTAL = −k2R(k2)(Aeµ(k) + AINDµ (k)). (10.46)

It is this total current that we should use in Maxwell’s equations to calculate AIND

k2AINDµ (k) = 〈jµ(k)〉TOTAL = −k2R(k2)(Aeµ(k) + AINDµ (k)). (10.47)

This gives a self-consistent equation for AIND in terms of the external field.

AINDµ (k) = − R(k2)

1 +R(k2)Aeµ(k). (10.48)

Finally, if we add the induced field to the external field, we obtain the total field

ATOTµ (k) =1

1 +R(k2)Aeµ(k). (10.49)

In summary we have calculated the total electromagnetic field that arises in the “medium”of the Dirac sea in the presence of an externally applied field. The externally applied fieldsare what are traditionally called the ~D and ~H fields. (Recall that the sources of thesefields are the external charge and current density respectively.) On the other hand the total

fields are traditionally given the name ~E and ~B. The dielectric “constant” of the mediumis defined by ~D = ε ~E and the magnetic permeability by ~B = µ ~H. Thus we can interpretour calculation by attributing a k dependent dielectricity and magnetic permeability to theDirac sea

ε(k2) = 1/µ(k2) = 1 +R(k2). (10.50)

Since R is positive for static fields (k0 = 0), the vacuum is a polarizable diamagnetic medium.The fact that ε = 1/µ means that the velocity of light is unaltered by the medium (i.e.the medium preserves Poincare invariance). The effectiveness of the medium in screeningexternal fields is reduced at shorter wavelength (R decreases as k2 increases).

10.4 Charge Renormalization

We now come to the resolution of the logarithmic divergence that remains in our expressionfor R(k2). We begin by asking how we measure charge. We seem to have particles andantiparticles of charge ±Q in the theory. However this is not the measured charge, the chargewe could define by e2

ph ≡ limR→∞ 4πR2F (R) where R is the spatial separation between twosuch charged particles and F is the force exerted by one on the other. In other words, Qrepresents the external or “bare” charge, which acts as source to the ~D field. The Fourier


component of the latter field is just −iQk/k2. The measured force is given by Q~E theFourier component of which is −iQ2k/[k2(1 + R(k2))]. The long distance part of the forceis controlled by the Fourier components with k ≈ 0 which are clearly those of a Coulombforce with effective charge squared of

e2 =Q2

1 +R(0). (10.51)

Putting k = 0 in our expression for R gives us

e2 =Q2

1 + (Q2/12π2) ln(Λ2/m2). (10.52)

It is e and not the parameter Q that we measure in experiments, all of which are performedwithin the “medium” represented by the vacuum. The fine structure constant is α = e2/4π ≈1/137.

If e is the measured charge, then the measured electric field should be related to measuredforce by

~F = e ~Emeas = Q~E =Q

1 +R(k2)~D (10.53)

=e(1 +R(0))

1 +R(0) + [R(k2)−R(0)]

~D√1 +R(0)

(10.54)

=e

1 + [R(k2)−R(0)]/(1 +R(0))

~D√1 +R(0)

(10.55)

Now since ~D is simply proportional to Q, ~Dmeas ≡ ~D/√

1 +R(0) is what we can call the

measured ~D field since it has Q replaced by e. Thus we have the following relationshipbetween measured fields

~Emeas =1

1 + [R(k2)−R(0)]/(1 +R(0))~Dmeas. (10.56)

The measured dielectric constant is accordingly

ε(k2) = 1 +R(k2)−R(0)

1 +R(0)≈ 1 +

e2

2π2

∫ 1

0

dxx(1− x) lnm2

m2 + x(1− x)k2(10.57)

and we see that the cutoff dependence has disappeared when we express measured quantitiesin terms of measured parameters. This is what is known as Renormalizability, and is a featureof a wide class of quantum field theories. Such theories encounter infinities in intermediatestages of a given calculation, but the measurable quantities always come out finite.

After renormalization, the dielectric constant is fixed to be 1 at k = 0, i.e. the medium iseffectively absent then. Before, we argued that the effects of the medium should be reducedat large k. By going to large k we should begin to see more and more of the bare charge.


We can phrase this by defining a k dependent coupling by α(k2) = e2/4πε(k2). Then ask2 increases from zero, corresponding to shorter distances, α increases until it blows up atsome finite value of k2. It’s clear that this will happen because the ln starts out at zerogoes negative and behaves monotonically without bound. The pole occurs at a value ofk2 ∼ m2 exp(12π2/e2). We have seen how poles in amplitudes are associated with particles.Unfortunately this pole occurs at space-like momentum, i.e. imaginary mass, a tachyon.This is the physically unacceptable Landau Ghost. If it were really present it would signifyan inconsistency of electrodynamics. Fortunately, the whole issue is completely open sincewe have made approximations in our calculation that amount to weak coupling perturbationtheory, and as we have seen the effective coupling gets strong at values of k2 much less thanthe ghost mass, thus invalidating perturbation theory.

One way to understand why this approximation has led to this problem is to return tothe bare expressions

e2 =1

1/Q2 + (1/12π2) ln(Λ2/m2). (10.58)

Now in the renormalization procedure, we attempt to take Λ → ∞ holding e fixed. Butthis is only possible if Q is imaginary, which would mean we started with a Hamiltonianwhich was not hermitian. In order to escape this conclusion, the relation between bare andmeasured coupling would have to be fundamentally altered by higher order corrections. Thisis a logical possibility, but many field theorists including Landau doubted that the problemwould go away. At this point it is appropriate to mention that for some quantum fieldtheories the sign in front of the logarithmic divergence is opposite to that in QED:

g2 =1

1/g20 − b ln(Λ2/m2)

, b > 0. (10.59)

In this case one can take Λ → ∞ with g fixed and g0 real. This phenomenon is known asasymptotic freedom since the bare coupling goes to 0 through real values as Λ→∞. Thesequantum field theories can be renormalized consistently in the weak coupling limit as long asone restricts one’s attention to very large momenta. The other side of the coin is that theybecome strong coupling theories at low momenta, and so weak coupling approximations areuseless for studying their particle spectrum.

10.4.1 Pair production

Next we turn to the interpretation of the singularity in T or R as k2 → −4m2. At this pointthe logarithm has a branch point, leading to a discontinuity depending on whether one goespast the branch point in the upper or lower half plane:

T±(k2) =Q2

2π2

∫ 1

0

dxx(1− x) lnΛ2

|m2 + x(1− x)k2|

∓iQ2

2π

∫ 1

0

dxx(1− x)θ(−x(1− x)k2 −m2). (10.60)


The integral in the second term contributes only if k2 < −4m2 when the range of x con-tributing is

1

2−√

1

4+m2

k2< x <

1

2+

√1

4+m2

k2. (10.61)

The integral over that range is elementary and yields

T±(k2) =Q2

2π2

∫ 1

0

dxx(1− x) lnΛ2

|m2 + x(1− x)k2|

∓iQ2

2π

1

3θ(−k2 − 4m2)

√1

4+m2

k2

(1− 2m2

k2

). (10.62)

The −iε prescription tells us to choose the lower half plane continuation, i.e. the lower (+)sign is to be taken.

T (k2) =Q2

2π2

∫ 1

0

dxx(1− x) lnΛ2

|m2 + x(1− x)k2|

+iα

3θ(−k2 − 4m2)

√1 +

4m2

k2

(1− 2m2

k2

). (10.63)

R(k2) =Q2

2π2

∫ 1

0

dxx(1− x) lnΛ2

|m2 + x(1− x)k2|

+iα0

3ε(k0)θ(−k2 − 4m2)

√1 +

4m2

k2

(1− 2m2

k2

), (10.64)

where we have used the definition of the bare fine structure constant α0 = Q2/4π and therelation between R and T .

We can associate the appearance of an imaginary part of T with pair production whichbecomes energetically possible for energies larger than 2m. To see the connection recall thevariational equation satisfied by the vacuum persistence amplitude

δ ln 〈out|in〉 = i

∫d4x〈out|jµ(x)|in〉〈out|in〉

δAµ(x) (10.65)

≈ −∫d4xd4y〈0, I|T [jµI (x)jνI (y)]|0, I〉Aν(y)δAµ(x) (10.66)

where we used first order perturbation theory in the external field to approximate the currentmatrix element. The results of our calculation gave

i〈0, I|T [jµI (x)jνI (y)]|0, I〉 =

∫d4k

(2π)4eik(x−y)(kµkν − k2ηµν)T (k2), (10.67)

so we find, in weak field approximation,

〈out|in〉 ≈ exp

i

2

∫d4k

(2π)4T (k2)Aµ(−k)(kµkν − k2ηµν)Aν(k)

. (10.68)


It is illuminating to express this in terms of the Fourier components of the field strengths

Fµν(k) = i(kµAν(k)− kνAµ(k)) (10.69)

Fµν(−k)F µν(k) = −2(kµkν − k2ηµν)Aµ(−k)Aν(k) (10.70)


− i

4

∫d4k

(2π)4T (k2)Fµν(−k)F µν(k)

. (10.71)

As long as T is real, this is a pure phase and | 〈out|in〉 | = 1. But when T acquires animaginary part as it does for k2 < −4m2 this is no longer true. Specifically,

| 〈out|in〉 |2 ≈ exp

1

2

∫d4k

(2π)4ImT (k2)Fµν(−k)F µν(k)

(10.72)

which is the probability that there is no pair creation. The probability of pair creation is1− | 〈out|in〉 |2 and is approximately given by

Ppair ≈ −1

2

∫d4k

(2π)4ImT (k2)Fµν(−k)F µν(k) (10.73)

≈ −∫

d4k

(2π)4

α

6θ(−k2 − 4m2)

√1 +

4m2

k2

(1− 2m2

k2

)Fµν(−k)F µν(k),

where we have replaced the bare fine structure constant α0 = α(1 + R(0)) → α which iscorrect to the order we are calculating1.

Note that one could also find this result by calculating |〈out|boutdout|in〉|2 directly in per-turbation theory and summing over all final states. It is of course important that thisprobability be positive (and | 〈out|in〉 | < 1). Indeed, Fµν(−k)F µν(k) is negative for ktimelike, because then there is a Lorentz frame where k = 0 which implies B = 0 soFµν(−k)F µν(k) = −2|E|2. Since it is an invariant, it must be negative in all frames.

When the fields have support only where T is real, the amplitude 〈out|in〉, even though itis a pure phase, gives information about the energy of the system in the presence of externalfields. To get this connection consider a static field with adiabatic switching off at early andlate times:

Fµν(x) = Fµν(x)ε(t). (10.74)

We take ε to be a symmetric function of t with central value 1 and gradual fall off to zeroat times roughly ±T . The exact shape is unimportant: we only need that ε(k0) peaked at 0with a width of order 1/T . Clearly∫

dk0ε(k0)2 = 2π

∫dtε(t)ε(−t) ≈ 4πT. (10.75)

1Strictly speaking, this calculation of 〈out|in〉 is for a fixed external field for which QA is fixed andfinite. Renormalization applies only when the induced fields are included. In that case the F appearing inthese formulas are the total fields, and for them QF is indeed held fixed as the cutoff is removed. That is,√

1 +R(0)F is the measured field strength.


Thus in the limit T → ∞, ε2 can be approximated by 4πTδ(k0). Thus in this limit ourapproximate formula for 〈out|in〉 reads


−2iT

∫d3k

(2π)3

1

4Fµν(−k)F µν(k)T (k2)

. (10.76)

The coefficient of −2iT is just the energy of the Dirac system in the presence of staticexternal fields. Since T is positive for spacelike k, and

1

4Fµν(−k)F µν(k) =

1

2[|B|2 − |E|2] (10.77)

we see that the energy increases under the addition of a magnetic field and decreases withan electric field in accord with our conclusion that the vacuum is a diamagnetic dielectricmedium.

10.5 Superconductivity and the Higgs Mechanism

There is one physical phenomenon which can occur in quantum gauge field theories, althoughnot for QED in four dimensions. It can happen in some theories that R(k2) possesses a poleat zero:

R(k2) ∼ K

k2, k2 → 0. (10.78)

Note that this infrared singularity does not occur in our previous calculation even for m = 0.But if it does occur, then

A(k)TOT =1

1 +R(k2)Ae ∼

k2

KAe ∼ 0, k2 → 0. (10.79)

Thus ATOT is screened at long wavelengths by the induced currents. In particular, for astatic Coulomb potential A0 ∼ δ(k0)/k2, the singularity at vanishing k disappears so incoordinate space the potential falls off faster than any power. Because of Lorentz covariancethis screening is effective for both electric and magnetic fields. The screening of the magneticfields means there is a Meissner effect, i.e. the vacuum in this situation is a relativisticsuperconductor. The vanishing of the total field at k2 = 0 means that the vacuum cannotsupport massless photons. On the other hand there most likely is a negative value of k2 call it−M2 for which 1+R(−M2) = 0. For such values of k, ATOT can be nonzero even for vanishingexternal field. These waves correspond to particles of mass M . Thus this phenomenon,sometimes called the Higgs mechanism, gives the photon a mass without violating gaugeinvariance. A nonrelativistic version of this effect was long known for superconductors. Itwas first discussed in the context of relativistic quantum field theory by Schwinger for QED intwo space-time dimensions. Four dimensional versions were first discussed by Higgs, Englertand Brout, and Guralnik, Hagen and Kibble. In spite of the long list of discoverers, it seemsthat Higgs’ name has stuck. The Higgs mechanism is at the heart of the electroweak unifiedgauge theory, because it is responsible for the masses of the W and Z bosons.


Chapter 11

Perturbation Theory for φ3 ScalarField Theory

Let us now apply what we have learned to the scalar field theory with Lagrangian density

L = −1

2(∂φ)2 − m2

0

2φ2 − g

3!φ3 (11.1)

One’s first thought is to take the perturbation in interaction picture to beH ′I(t) = (g/3!)∫d3xφ3(x, t).

We shall find that, because the interactions don’t turn off at early and late times, this istoo glib. But let’s see where it leads. We have no external fields, so the time dependentperturbation theory reads

〈G|T [φ(x1) · · ·φ(xn)]|G〉 =

〈0, I|T [exp− ig

3!

∫d4xφ3

I(x)φI(x1) · · ·φI(xn)]|0, I〉

〈0, I|T exp− ig

3!

∫d4xφ3

I(x)|0, I〉

. (11.2)

Her |G〉 is the exact ground state.

11.1 The Vacuum

The simplest application of the dyson formula is the case without any fields in the timeordered product:

〈G|I|G〉 = 〈G|G〉 =T [exp

− ig

3!

∫d4xφ3

I(x)

]|0, I〉〈0, I|T exp

− ig

3!

∫d4xφ3

I(x)|0, I〉

= 1. (11.3)

Which is consistent but pretty trivial. The numerator and denominator are separately com-plicated expressions but complications all cancel. Let’s look more closely at the structure ofthe numerator:

Num = 1 +1

2

(−ig)2

62

∫d4xd4y〈0, I|Tφ3

I(x)φ3I(y)|0, I〉. (11.4)

127

Applying the Wick expansion to the second term can be visualized as two Feynman diagrams.These diagrams are characterized by having no external lines and are sometimes calledvacuum bubbles. The bottom line for them is that they are always cancelled, so we neverhave to deal with them.

11.2 One-point function

The next simplest thing to consider is the VEV of the scalar field, 〈G|φ(x)|G〉 = 〈G|φ(0)|G〉by translation invariance. It is just a constant number, call it v. What if we start trying tocalculate it in perturbation theory. Applying the Dyson formula to first order gives

v = 〈G|φ(0)|G〉 ≈ −ig3!

∫d4x〈G|Tφ(0)φ3(x)|G〉+ · · · (11.5)

The Wick expansion shows this as a “tadpole” diagram. The actual value of the number vis not particularly significant. We chose H ′ so that at zeroth order in perturbation theory〈G|φ(0)|G〉0 = 0. But then the perturbation makes it non zero. As we continue perturbationtheory, it is much more convenient to write φ = v + φ where 〈G|φ(0)|G〉0 = 0 exactly to allorders in perturbation theory. In terms of φ the Lagrangian reads

L = −1

2(∂φ)2 − m2

0

2(v + φ)2 − g

3!(v + φ)3

= −1

2(∂φ)2 − m2

2φ2 −

[−m

2

2φ2 +

m20

2(v + φ)2 +

g

3!(v + φ)3

](11.6)

where m represents the true physical mass of the scalar particle, in general different fromthe “bare” input mass m0. The correct procedure is to take all the terms we have enclosedin square brackets to define the perturbation H ′.

H′ = −m2

2φ2 +

m20

2(v + φ)2 +

g

3!(v + φ)3

≡ g

3!φ3 +

δm2φ2 + δvφ+

m20

2v2 +

g

3!v3 (11.7)

The last two terms are constants which will cancel between numerator and denominator.The values of δm and δv are chosen at the end of the calculation so that 〈G|φ|G〉 = 0 and sothat m stays the physical mass. In practice we never need to find δv because its only effectis to set to zero all “tadpole” diagrams and subdiagrams. Thus all we have to do is deleteall tadpoles and then forget about δv.

11.3 Two Point Function and the Physical Mass

We have seen that the Fourier transform of the free field two point function has a pole atp2 = −m2. We now argue that this feature holds generally for two point functions even inthe interacting case.


Let’s first see how this happens in perturbation theory to second order∫d4xe−iq·x〈G|Tφ(x)φ(0)|G〉 =

−iq2 +m2

0

+1

2

(−ig)2

3!2

∫d4xe−iq·x〈0, I|TφI(x)φI(0)

∫d4yd4zφ3

I(y)φ3I(z)|0, I〉 (11.8)

In the second term, if both φ’s hit the same Hamiltonian we get a tadpole correction to thepropagator which is simply proportional to (q2+m2

0)−2. If each φ hits a different Hamiltonianwe are left with

−iq2 +m2

0

+(−ig)2

2

(−i

q2 +m20

)2 [∫d4ye−iq·(y−z)〈0, I|TφI(y − z)φI(0)|0, I〉

]2

≡ −iq2 +m2

0

[1 +

−iq2 +m2

0

(−i)Π(q2))

]≈ −iq2 +m2

0 + Π(q2)(11.9)

We see that the pole location has shifted to q2 = −m2 where m satisfies

−m2 +m20 + Π(−m2) = 0. (11.10)

We now see that it would be a good idea to use the true massm in the zeroth order propagatorand tune the δm counterterm so that corrections don’t change the pole location, i.e. so thatΠ(−m2) = 0 to all orders in perturbation theory.

Now, we give the general argument:∫d4xe−iq·x〈G|Tφ(x)φ(0)|G〉 (11.11)

We lose no generality in taking one point to be 0 because of translation invariance. Fix t > 0and insert a complete set of energy momentum eigenstates between the two fields:∫ ∞

0

dt

∫d3xe−iq·x

∑n

∫d3p〈G|φ(x)|n, ~p〉〈n, ~p|φ(0)|G〉

=

∫ ∞0

dt

∫d3xe−iq·x

∑n

∫d3peipn·x|〈n, ~p|φ(0)|G〉|2 (11.12)

=∑n

(2π)3

∫ ∞0

dtei(q0−En)t|〈n, ~q|φ(0)|G〉|2 (11.13)

=∑n

−(2π)3

i(q0 − En + iε)|〈n, ~q|φ(0)|G〉|2 (11.14)

If |n, ~q〉 is a single particle state, En =√~q2 +m2

n and the pole at q2 = −m2n is explicit. If it

is a multi-particle state the singularity is smeared and becomes a branch point. The residueof the pole depends on the matrix element 〈n, ~q|φ(0)|G〉 ≡

√Z/[(2π)3/2

√ω(~q)] for a single


particle state by Lorentz invariance. Z < 1 will not be unity as in the free particle case andthe single particle pole is

−ip2 +m2

0 − iε→ −iZ

p2 +m2 − iε(11.15)

This change in residue Z is called “wave-function” renormalization.

The argument we have just given shows that φ(x)|g〉 has a single particle component, aswell as multi-particle components. It would be nice to use φ(x) as a creation operator, butsince it also creates multi-particle states, we need somehow arrange these unwanted compo-nents to disappear. Taking the spatial Fourier transform

∫d3xei~q·~xφ(x) only guarantees that

it creates momentum ~q but not that it creates only a single particle state. If we could selectthe component with frequency eiω(~q)t, with ω(~q) =

√~q2 +m2. A temporal Fourier transform∫ T

−T dte−iq0t

∫d3xei~q·~xφ(x) would do this to accuracy 1/2T . Since the single particle energy

is separated by a gap O(m) from the vacuum and multi-particle energies this is good enoughas long as 2mT 1. In summary

1

2T

∫ t0+T

t0−Tdte−iω(~q)t

∫d3x

(2π)3ei~q·~xφ(x)|G〉 ≡

∫F (t− t0)dte−iω(~q)t

∫d3x

(2π)3ei~q·~xφ(x)|G〉

≡ A†(t0, ~q)|G〉 (11.16)

is a pure single particle state, with mass m and momentum ~q, of the exact interacting theory.Taking out the factors as shown gives A(t0) the normalization

〈G|A(t0, ~q′)A†(t0, ~q)|G〉 ≈

∫d3p〈G|A(t0, ~q

′)|~p〉〈~p|A†(t0, ~q)|G〉 (11.17)

≈ δ(~q′ − ~q)|〈~q|φ(0)|G〉|2 (11.18)

It is constructed from a Heisenberg picture field operator smeared in the neighborhood oftime t0.

We can make a two particle state by applying a second A†. However, it only has a cleantwo particle interpretation when the two particles are in wave packets space-like separatedfrom each other. In that case the two A†’s commute and represent two independent particles.However, if they are aimed to scatter, the packets will eventually overlap and scattering canoccur.


11.3.1 Reduction Formulae

Let −∞ < t1, t2, . . . , tn < ∞. Call f(~x, t) =∫d3qf(~q)ei~x·~q−iω(~q)t/(2π)3. Then it is easy to

show

T Aa (∞) Ω1(x1) . . .Ωn(xn) − T Ω1(x1) . . .Ωn(xn)Aa (−∞) (11.19)

=

∞∫−∞

dt0d

dt0TAa(t0)Ω1(x1) . . .Ωn(xn) (11.20)

=

∞∫−∞

dt0

∫d4xF (t− t0)

∂

∂tf ∗a (x, t)Tφ(x)Ω1(x1) . . .Ωn(xn (11.21)

=

∞∫−∞

dt0

∫d4xF (t− t0)f ∗a (x, t)

(iωa(−i∇) +

∂

∂t

)Tφ(x)Ω1(x1) . . .Ωn(xn)

TA†a (∞) Ω1(x1) . . .Ωn(xn)

− T

Ω1(x1) . . .Ωn(xn)A†a (−∞)

(11.22)

=

∞∫−∞

dt0

∫d4xF (t− t0)

∂

∂tfa(x, t)Tφ(x)Ω1(x1 . . .Ωn(xn) (11.23)

=

∞∫−∞

dt0

∫d4xF (t− t0)fa(x, t)

(−iωa(i∇) +

∂

∂t

)Tφ(x)Ω1(x1) . . .Ωn(xn) .

In these reduction formulae we have used the fact that by construction fa satisfies a Schrodingerequation with hamiltonian ωa(−i∇) =

√m2a −∇2. Spatial integration by parts then allows

the spatial derivatives to be transferred to φ(x).The reduction formulas can be used to relate scattering amplitudes represented as, e.g.

〈G|A(∞, ~q1)A(∞, ~q2)A†(−∞, ~p1)A†(−∞, ~p2)|G〉 (11.24)

to Fourier transforms of

〈G|T (φ(x1)φ(x2)φ(x3)φ(x4)|G〉 ≡∫ ∏

k

d4qk(2π)4

exp

i∑k

qk · xk

T (q1, q2, q3, q4)

The scattering amplitude is proportional to T in the on-shell limit q2k → −m2

k. Then applying∂/∂tk ± iω(∓i∇k) is equivalent to multiplying T by −iq0 ± iω(~q), which vanishes in one ofthe on-shell limits when q0 → ±ω(~q). There will be one such on-shell zero for each externalparticle. Thus there is only scattering if the F.T. has a mass-shell pole for each particle.The reduction formula gives the precise connection including all normalization factors. Butit is quicker to get all the right factors by just considering Feynman diagrams. The required


poles come from complete propagators attached to each external line. Lorentz invariancedetermines all factors up to overall Z factors. Therefore we can use lowest order perturbationtheory to determine everything. The rule is to “amputate” all external legs and replace eachpropagator with

√Ze±iq·x/(2π)3/2/

√2ω.

In summary, by choosing the perturbation carefully we set things up so that 〈G|φ|G〉 = 0to all orders in perturbation theory (this just amounts to dropping all tadpole corrections!),and so that Π(−m2) = 0 to all orders in perturbation theory. Then near mass shell Π(q2) ≈(q2 + m2)(Z−1 − 1), and we see how the residue shifts. Since we don’t know m2 and v inadvance we have to keep adjusting δm and δv as we proceed.

11.4 Feynman Diagrams

The Wick expansion is very efficiently analyzed via Feynman diagrams. These are a collec-tions of lines, one for each propagator in the Wick expansion, connected together at verticesdetermined by the interaction Lagrangian. In the case at hand, L′ = − g

3!φ3 − δm

2φ2 − δvφ.

1 List Feynman rules in x space for φ3.

2 Motivate and translate rules to p space.

3 Connectedness: sufficient to focus on connected diagram– general term is product ofconnected pieces.

4 Example: Sum of all vacuum graphs = exp of sum of all connected vacuum graphs.

5 Lowest order four point function.


Chapter 12

Path History Quantization

One of the principal drawbacks of the canonical operator formulation of quantum mechanicsis that it obscures symmetries that bring in time in an essential way. Lorentz boosts are ofthis type, so the operator approach inevitably hides the full symmetries of relativity. Thisis of course also true in the Hamilton equation form of classical mechanics. In classicalmechanics one can work with the Lagrangian and Action Principle which keep dynamicalsymmetries like Poincare transparent. The path integral approach to quantum mechanics isthe quantum analogue of this alternative and, as we shall see, is a much more convenientformulation of quantum field theory than the operator approach. Even in the operatorapproach we have seen the advantage of expressing results in terms of finite time evolutions,since it is these that reflect the true symmetries of the system. The central object in thepath integral approach then is not a state but an amplitude for the evolution of one stateinto another.

To keep notation simple, we shall suppress indices in dealing with a general quantummechanical system the coordinates of which are collectively denoted q and the conjugatemomenta of which are p. Then we seek an alternative scheme for calculating, for example,the amplitude

〈q′′|U(t1, t2)|q′〉. (12.1)

U is of course a very complicated operator for finite time differences, but as t1 → t2, it issimply related to the Hamiltonian.

Thus we are led to break up the time interval into infinitesimal pieces t1− t2 = (N + 1)aand employ the closure property of U to write it as a product of N+1 infinitesimal evolutions.

〈q′′|U(t1, t2)|q′〉 = 〈q′′|U(t1, t1 − a)U(t1 − a, t1 − 2a) · · ·U(t2 + a, t2)|q′〉 (12.2)

=

∫ N∏k=1

dqk〈q′′|U(t1, t1 − a)|qN〉〈qN |U(t1 − a, t1 − 2a)|qN−1〉

· · · 〈q1|U(t2 + a, t2)|q′〉. (12.3)

133

Next, we write, assuming HS is constant over a time interval a,

〈qk|U(t2 + ka, t2 + (k − 1)a)|qk−1〉

≈∫dpk−1〈qk|e−iaH(t2+(k−1/2)a)/2|pk−1〉〈pk−1|e−iaH(t2+(k−1/2)a)/2|qk−1〉

≡∫dpk−1 exp− i

2~a(Hk(qk, pk−1, ia) +Hk(qk−1, pk−1,−ia∗)∗)

〈qk|pk−1〉〈pk−1|qk−1〉

=

∫dpk−1

2π~exp i

~[pk−1(qk − qk−1) (12.4)

−a2

(Hk(qk, pk−1, ia) +Hk(qk−1, pk−1,−ia∗)∗)],

where for the moment Hk is defined by these equations. Putting everything together weobtain

〈q′′|U(t1, t2)|q′〉 =

∫ N∏k=1

dqkdpk2π~

dp0

2π~exp

i

~

N+1∑k=1

[pk−1(qk − qk−1) (12.5)

−a2

(Hk(qk, pk−1, ia) +Hk(qk−1, pk−1,−ia∗)∗)],

where qN+1 ≡ q′′ and q0 ≡ q′. Apart from the assumption that external fields are constantover the time interval a this formula is exact. But it is not useful until we get a simpleapproximation for Hk. For a→ 0 we should be able to approximate

〈qk|e−iaH(t2+(k−1/2)a)/2|pk−1〉 ≈ 〈qk|(1− ia

2H(t2 + (k − 1/2)a))|pk−1〉 (12.6)

≈ (1− ia2HWk (qk, pk−1)) 〈qk|pk−1〉 (12.7)

≈ exp−ia2HWk (qk, pk−1) 〈qk|pk−1〉 (12.8)

where HWk (q, p) is the operator H(t2 + (k − 1/2)a) rewritten through use of the canonical

commutation relations with all p’s on the right and all q’s on the left. After this is done q canthen be replaced with the eigenvalue qk and p with the eigenvalue pk−1. In the limit a → 0with t1 − t2 = (N + 1)a fixed it should be valid to replace Hk(qk, pk−1) with HW

k (qk, pk−1).We define the quantity

HQk (qk, qk−1, pk−1) =

1

2(HW

k (qk, pk−1) +HWk (qk−1, pk−1)∗) (12.9)

which appears in the path integral. Up to the reordering terms HWk is just the classical

Hamiltonian for the system. In the common case where the Hamiltonian is a function of p’splus a function of q’s it is nothing more nor less than the classical Hamiltonian. But noticethat the object appearing in the path integral is (12.9) which depends on the two coordinates


describing the initial and final states of the basic unit of propagation even in the case wherethere are no reordering terms, when it is simply the average of the classical Hamiltonianover the two coordinates. In this continuum limit we can think of the sum in the exponentin (12.5) as an integral

i

~

∫ t1

t2

dt[p(t)q(t)− 1

2(HW (q>(t), p(t), t) +HW (q<(t), p(t), t)∗)]. (12.10)

The coefficient of i~ is just the classical action

∫dtL plus terms of order ~2 expressed as a

Legendre transform of the Hamiltonian. In the continuum limit, the number of integrationvariables tends to infinity and the limit gives the definition of the path integral representationof 〈qf |U(t1, t2)|qi〉.

12.1 The Classical Limit

For a general Hamiltonian, this is as far as one can go without further approximations. Onesuch approximation one can always try is the limit ~→ 0, the classical limit. Such a limit isdominated by the functions q(t), p(t) for which the coefficient of i/~ is stationary. Since, inthis limit, this coefficient is just the classical action, the stationarity conditions are simply theclassical Hamilton equations: q = ∂H/∂p and p = −∂H/∂q. Thus the path history versionof the quantum principle is that for ~ 6= 0 transition amplitudes are computed by evaluatingei(Action)/~ for all possible histories and averaging this expression over all such histories. Theclassical limit is understood as the situation in which this average is dominated by solutionsof the classical equations of motion.

12.2 Imaginary Time.

In working with the path integral it is technically advantageous to work with actually dampedintegrands rather than the oscillating integrand occurring in the quantum path integral. Thiscan be achieved with the Wick rotation it = τ where real positive t is rotated to real positiveτ . Considering the basic unit of the path integral, the matrix element of the operator, e−iaH/~,we see that this rotation is mathematically justified when H is an operator bounded below,i.e. its eigenvalue spectrum is bounded below. It obviously should not be attempted if H haseigenvalues down to −∞. Fortunately, most reasonable physical systems have this property,and for these the Wick rotated path integral is the superior one to work with, especially forapplications outside of perturbation theory. For a constant Hamiltonian (no external fields)this path integral calculates 〈qf |e−βH |qi〉 where β = (τ1 − τ2)/~. If we identify qf = qi = qand integrate over q, it calculates Tre−βH , the statistical mechanical partition function fortemperature 1/β. In this way the Wick rotated path integral is related to a quantity ofdirect physical interest in another context. In the limit β → ∞ (low temperature) oneobtains information about the energy levels and degeneracies of the system. However, for


applications to quantum mechanics it is necessary to continue back to real time at the endof the calculation of physical transition amplitudes.

Technically the Wick rotation amounts to replacing ia by δ > 0 in (12.5). Thus oneobtains, after approximating Hk by HW

k ,

〈q′′|U(−iτ1,−iτ2)|q′〉 ≈∫ N∏

k=1

dqkdpk2π~

dp0

2π~exp1

~

N+1∑k=1

[ipk−1(qk − qk−1)

−δ2

(HWk (qk, pk−1) +HW

k (qk−1, pk−1)∗)]. (12.11)

The Wick rotated version of the quantum action (12.10) is of course

1

~

∫ τ1

τ2

dτ [ip(τ)q(τ)− 1

2(HW (q>(τ), p(τ), τ) +HW∗(q<(τ), p(τ), τ))]. (12.12)

Here we have identified q(t) ≡ q(τ) and the complex conjugation in HW∗ ignores the i’scoming from the Wick rotation.

12.3 Matrix Elements of Time Ordered Products.

For T > t1 > · · · > tn > −T

〈qf |U(T, t1)q1U(t1, t2)q2 · · · qnU(tn,−T )|qi〉 =

〈qf |U(T,−T )T [q1(t1)q2(t2) · · · qn(tn)]|qi〉 (12.13)

where the Heisenberg picture operators qk(tk) ≡ U−1(tk,−T )qkU(tk,−T ) are defined so thatHeisenberg and Schrodinger pictures agree at t = −T . Working with the l.h.s. of thisrelation we can insert a complete set of coordinate basis states between each pair of U ’s andthen replace each operator qk by its eigenvalue and each matrix element of U by its pathintegral representation. The integrals over the basis labels q′k then simply extend the sumover piecewise histories qi → qn · · · → q1 → qf to the sum over all histories qi → qf . Thuswe arrive at

〈qf |U(T,−T )T [q1(t1)q2(t2) · · · qn(tn)]|qi〉 (12.14)

=

∫DqDpq(τ1) · · · q(τn) exp1

~

∫ T−T

dτ [ip(τ)q(τ)−HQ(q(τ), p(τ), τ)].

where we have used the Wick rotated version with iT = T . Written in this way with thetime ordering symbol on the l.h.s. this formula is valid for any time ordering.

In field theory we are really interested in the vacuum (ground state) expectation value oftime ordered products. These can be obtained by taking the limit T → ∞. Then, inserting


energy eigenstates at the left and right, all states but the ground states are exponentiallysuppressed so we have

〈qf |U(∞,−∞)T [q1(t1)q2(t2) · · · qn(tn)]|qi〉= 〈qf |0〉〈0|U(∞,−∞)T [q1(t1)q2(t2) · · · qn(tn)]|0〉〈0|qi〉= 〈qf |0〉〈0|qi〉〈out|T [q1(t1)q2(t2) · · · qn(tn)]|in〉. (12.15)

The wave functions that multiply the desired result can be obtained from

〈qf |UExt=0(∞,−∞)|qi〉 = e−2∞EG 〈qf |0〉〈0|qi〉 , (12.16)

which of course has its own path integral representation. Thus by division we obtain, definingenergy so that EG = 0,

〈out|T [q1(t1)q2(t2) · · · qn(tn)]|in〉 (12.17)

=

∫DqDp q(τ1) · · · q(τn) exp1

~

∫∞−∞ dτ [ip(τ)q(τ)−HQ(q(τ), p(τ), τ)]∫

DqDp exp1~

∫∞−∞ dτ [ip(τ)q(τ)−HQ

Ext=0(q(τ), p(τ))].

12.4 Coordinate Space Path Integral.

A strong motivation for using the path integral formulation for quantum field theory is thatit makes possible a more symmetrical treatment of space and time. This symmetry is evenmore striking after the Wick rotation when the Lorentz group O(3, 1) becomes simply O(4)the group of rotations in four dimensions. To achieve the full force of this benefit though wewould like to be able to use the configuration space action

∫dtL(q, q, t) rather than the phase

space one. We can do this provided it is possible to “integrate out” the conjugate momentap. This is generally possible in quantum field theory because field theoretic Hamiltonianstypically only depend on the conjugate momenta Π through an additive term 1

2

∫d3xΠ2.

Thus the integral over the Π is gaussian and can be explicitly performed. In the languageof quantum mechanics, the field theoretic Hamiltonian is always of the form p2/2 + V (q) +[f(q)p + pf(q)]/2 = p2/2 + V (q) + f(q)p − i~f ′(q)/2. In this case we can integrate out thep’s even before the continuum limit which converts ordinary integrals to path integrals:∫

dpk−1√2π~

exp

1

~[ipk−1(qk − qk−1)− δ(1

2p2k−1 +

1

2(V (qk) + V (qk−1) (12.18)

+pk−1(f(qk) + f(qk−1)− i~4

(f ′(qk)− f ′(qk−1)))]

.

=1

δ1/2exp

− 1

~[(qk − qk−1 + iδ

2(f(qk) + f(qk−1)))2

2δ

+δ

2(V (qk) + V (qk−1)− i~

2(f ′(qk)− f ′(qk−1)))]

. (12.19)


So that the path integral expression (12.11) becomes

〈q′′|U |q′〉 ≈(

1

2π~δ

) (N+1)2∫ N∏

k=1

dqk exp

− 1

~

N+1∑k=1

[(qk − qk−1 + iδ

2(f(qk) + f(qk−1)))2

2δ

+δ

2(V (qk) + V (qk−1)− i~

2(f ′(qk)− f ′(qk−1))]

(12.20)

→∫Dq exp

− 1

~

∫ τ1

τ2

dτ [1

2

(dq

dτ+i

2(f(qk) + f(qk−1))

)2

+ V (q(τ))]

.

Note that the (divergent) prefactor is necessary to obtain the same result for the evolutionamplitude as with the usual operator formalism, and it naturally appears when we startthe path history formulation in phase space. However, notice also that the dependence onthe evolution time (N + 1)δ is exactly of the form that would come from adding a constant−(1/2δ) ln(2π~δ) to the overall energy of the system. Thus since only energy differencesare measurable, the physics is insensitive to the presence of this factor. In quantum fieldtheory we could lump this contribution into the zero point vacuum energy we are supposedto subtract in any case. We recognize the coefficient of 1/~ in the exponent as the (imagi-nary time) classical action for the system. It is this configuration space path integral thatgives quantum field theory its neatest expression. Then the classical action has the form∫d4xL(φ, ∂µφ) with L a Lorentz scalar field. The configuration variables of quantum field

theory are the fields φ(x, t) one for each point in space time. To define the sum over historiesof fields one therefore needs a lattice in space-time. In the Wick-rotated version this could betaken, for example, to be a hypercubic lattice in 4 dimensional Euclidean space. The pathintegral defined via such a lattice can be taken as the definition of quantum field theory.We shall find that perturbation theory can be developed directly from the continuum pathintegrals, essentially because one can avoid the actual evaluation of the integrals by varioustricks.

12.5 Gaussian Integrals

When evaluating ground state averages of physical quantities in perturbation theory, onecan manage to avoid ever having to do a functional integral. This is because the freefield functional integral will cancel between numerator and denominator after extracting thesource dependence. However there are cases where one needs to know the numerator (ordenominator) separately, for example, when one uses the path integral representation of thepartition function. Since the free field integral is simply gaussian, we can in fact calculateit.

We start by noting that the general multivariable gaussian ordinary integral is given by∫ ∞−∞

N∏i=1

(dxi√

2π

)e−

12

∑km xkMkmxm =

∏i

m−1/2i = det−1/2M, (12.21)


where mi are the eigenvalues of the real symmetric matrix M . If we always define Euclideanfunctional integrals in terms of a lattice, this result can be directly applied. Then aftertaking the continuum limit, we can write for the neutral scalar field∫

Dφ exp

−∫d4x

[1

2(∂φ)2 +

m2

2φ2

]≡ det−1/2[m2 − ∂2]. (12.22)

A charged scalar field can be decomposed φ = (φ1 + iφ2)/√

2 in terms of two real scalarfields so the corresponding formula is∫

DφDφ∗ exp

−∫d4xE

[(∂φ†∂φ) +m2φ†φ

]≡ det−1[m2 − ∂2]. (12.23)

We have already encountered determinants of differential operators in our study of ex-ternal field problems, for example, the outin matrix element for a charged scalar field in thepresence of an external gauge field is proportional to det−1[m2 − (∂ − iQA)2]. Thus we canimmediately write the path history representation for this matrix element:

〈out|in〉A =

∫DφDφ∗ exp

−∫d4xE

[(∂ + iQA)φ∗(∂ − iQA)φ+m2φ∗φ

]∫DφDφ∗ exp

−∫d4xE

[∂φ∗ · ∂φ+m2φ∗φ

] . (12.24)

Since this is what must be inserted into the gauge field path integral to couple gauge fields tocharged fields, this completes the process for converting to path integration language the qftof scalar fields interacting with gauge fields. To get outin matrix elements of time orderedproducts of fields we use the generating functional

〈out|T [ei∫d4x(J∗φ+φ†J)]|in〉 (12.25)

=

∫DφDφ∗ exp

−∫d4xE

[(∂ + iQA)φ∗(∂ − iQA)φ+m2φ∗φ− J∗φ− φ∗J

]∫DφDφ∗ exp

−∫d4xE

[∂φ∗ · ∂φ+m2φ∗φ

]=

det−1(m2 −D2)

det−1(m2 − ∂2)exp

∫d4xd4yiJ∗(x)∆F (x, y;A)iJ(y)

. (12.26)

We have written the functional integrals in Euclidean space, which we indicate with theE subscript. To express the results in terms of Minkowski space, just use d4xE = id4x.Remember that ∆F (x, y;A) is precisely the continuation back to Minkowski space of theEuclidean Green function ∆E(xE, yE;A). To confirm the last equality one simply completesthe square and changes variables in the by now familiar way. Doing this in Euclidean spaceleads to the exponent

∫d4xEd

4yEJ∗(x)∆E(xE, yE;A)J(y), which, when continued back to

Minkowski space, gives the result shown.We know from our experience with the operator formulation that the boundary conditions

that fix ∆F in Minkowski space are the requirements that it contains only positive frequencies


at very late times and only negative frequencies at very early times. In Euclidean space, theboundary condition on ∆E is simply that it vanish at infinity. Since ∆F ,∆E are analyticcontinuations of one another, either statement of the boundary conditions is satisfactory.From the path integral point of view it is more natural to choose the Euclidean version.


Chapter 13

Path Integrals for AnticommutingQuantum Fields

To convert the persistence amplitude for the Dirac field in the presence of external gauge fieldsto a path integral formalism, we must be able to produce a factor of detm− iγ ·D in thenumerator unlike the denominator as gaussian integrals tend to produce. This requires theintroduction of “anti-commuting” numbers or a numbers in contrast to ordinary c numbers.Any two a numbers e, f satisfy ef+fe = 0. In particular the square of an a number vanishes!Thus the most general function of a single a number a is the linear one c1 + c2a. The theoryof functions of a numbers is quite trivial. A function of N a numbers is at most linear ineach variable, but that involves terms with up to N factors. With N →∞ one can of coursehave any number of factors as long as each factor is a different a number.

How do we integrate over a numbers? To define this we define∫daf(a) to be a linear

operation that assigns a unique c number to each function f . We also require the fundamentaltranslation property

∫daf(a + e) =

∫daf(a). But there are only two linearly independent

functions of a namely 1 and a itself. So we only need to specify∫da1 and

∫daa. The

translation property for the second of these holds only if∫da1 = 0. Thus we only need to

specify∫daa to be some fixed c number, and then integration is completely defined! By

definition we take∫daa = 1. Then the integral of the arbitrary function c1 + c2a is simply

c2, the coefficient of the linear term in a.Now consider integration of a gaussian over 2M a numbers, ak, ak:∫

da1da1 · · · daMdaMeaTCa (13.1)

=1

M !

∫da1da1 · · ·

∫daMdaM(aTCa)M (13.2)

=

∫da1da1 · · ·

∫daMdaM a1C1k1ak1 a2C2k2ak2 · · · aMC1kMakM (13.3)

= C1k1C2k2 · · ·C1kM εk1k2···kM = detC. (13.4)

This shows that integrating gaussians over anticommuting numbers yields determinants withpositive powers. Next consider the case where the exponent of the gaussian is a bilinear in

141

M a numbers bk:12bTAb where A is an antisymmetric matrix. It can be reduced to the

previous case by considering its square:(∫db1 · · · dbMeb

TAb/2

)2

=

∫db1 · · · dbMdb′1 · · · db′Me(bTAb+b′TAb′)/2

= (−)M2

2

∫da1da1 · · · daMdaMea

TAa

= (−)M2

2 detA (13.5)

where the change of variables a = (b + ib′)/√

2, a = (b− ib′)/√

2 has been used. The phaseout front is never relevant, since when M is odd, both sides vanish. We conclude that∫

db1 · · · dbMebTAb/2 = det1/2A. (13.6)

Since the l.h.s. is a polynomial in the matrix elements of A we have proved an interestingcorollary that the determinant of an antisymmetric matrix is the square of a polynomial in thematrix elements. That polynomial is sometimes known as the Pfaffian, Pf(A) = det1/2[A]for antisymmetric matrices A.

Now we can repeat our discussion of bosonic functional integrals for the fermionic case:

〈out|in〉A =det(m− iγ ·D)

det(m− iγ · ∂)(13.7)

=

∫DψDψ exp

−i∫d4x(ψ(m− iγ ·D)ψ

∫DψDψ exp

−i∫d4x(ψ(m− iγ · ∂)ψ

. (13.8)

Introducing anticommuting sources, η, η in the combination i∫d4x[ηψ+ ψη] in the exponent

and competing the square, one can easily see that the source dependence is just a factor

ei∫η(m−iγ·D)−1η = e

∫d4xd4yiη(x)SF (x,y;A)iη(y). (13.9)

By differentiating with respect to the sources, one can show that this expression is just thegenerating function for the outin matrix element of time ordered products of fields:

〈out|Tei∫d4x[ηψ+ψη]|in〉

=

∫DψDψ exp

−i∫d4x

[(ψ(m− iγ ·D − ηψ − ψη)ψ

]∫DψDψ exp

−i∫d4x(ψ(m− iγ · ∂)ψ

=

det(m− iγ ·D)

det(m− iγ · ∂)exp

∫d4xd4yiη(x)SF (x, y;A)iη(y)

. (13.10)


Chapter 14

Operator Quantization of theElectromagnetic Field

[Note: This chapter describes the operator alternative to the path integraldescription ofQED. It may be skipped on a first reading.]

Treated classically the e.m. field Fµν = ∂µAν − ∂νAµ satisfies Maxwell’s equations

∂νFµν = Jµ, (14.1)

which imply current conservation ∂µJµ = 0 for consistency. So far we have developed the

theory of Dirac and scalar quantum fields interacting with fixed external e.m. fields. Forsuch systems the current is of course an operator, so Maxwell’s equations imply that thee.m. field must also be a quantum operator, which inserted into the Dirac equation producesa nonlinear quantum field equation.

14.1 Quantized Electromagnetic Field Interacting with

a Conserved Current

The first step in developing the quantum theory for Aµ is to understand the canonicalstructure of Maxwell’s equations. One first notes that the equations follow from stationarityof the action

S = −1

4

∫d4xFµνF

µν +

∫d4xAµJ

µ, (14.2)

which is gauge invariant provided ∂µJµ = 0. It is straightforward to find the momentum

conjugate to Aµ

Πµ = F µ0 = ∂0A

µ − ∂µA0. (14.3)

The spatial components of Π are just those of minus the electric field strength, but Π0 = 0.This last fact poses a difficulty for quantization since it is inconsistent with nonvanishing

143

canonical commutation relations. We know how to assign operator properties to ~A and ~Π,but not to A0. Before facing this difficulty, we construct the canonical Hamiltonian

HCAN =

∫d3x[~Π · ~A− L] (14.4)

=

∫d3x

[1

2~Π2 +

1

2(∇× ~A)2 − ~A · ~J +∇A0 · ~Π− A0J

0

]. (14.5)

Notice that the troublesome variable A0 appears only linearly and in the last two terms.After an integration by parts the coefficient is just −(∇ · ~Π + J0), which would vanish if wecould use Gauss’ Law.

Classically, we could certainly enforce Gauss’ Law and then the canonically uncertainA0 would disappear from the dynamics. Attempting to enforce Gauss’ Law as a quantumoperator equation would contradict the canonical commutation relations for ~A, ~Π

[Ak(~x, t),Πm(~y, t)] = iδmk δ(~x− ~y), (14.6)

so we postpone discussion of this point. If we leave the operator character of A0 unspecified,it is clearly important to know how quantum evolution will be affected if we make a change inA0. Let the evolution operator for a given A0 be UA0(t,−∞). Then by a familiar argument

δUA0(t,−∞) = iUA0(t,−∞)

∫d4xδA0(∇ · ~Π + J0 (14.7)

where the operators multiplying UA0 are in Heisenberg picture. The Heisenberg equationsfor ~A, ~Π imply

∂

∂t(∇ · ~Π + J0) =

∂J0

∂t+∇ · ~Ji

∫d3x[∇ · ~Π, A0](∇ · ~Π + J0) (14.8)

= i

∫d3x[∇ · ~Π, A0](∇ · ~Π + J0) (14.9)

Although we are not free to impose Gauss’ Law as an operator equation, we can restrictour incoming states to satisfy

(∇ · ~Π + J0)|t=−∞|in〉 = 0. (14.10)

It then follows from (14.9) that (∇ · ~Π + J0)|in〉 = 0 for all time. By making this restrictionwe therefore arrange that

δUA0|in〉 = 0 (14.11)

for arbitrary operator changes in A0, that is, the quantum evolution is independent of howwe treat A0. This is the quantum analog of what is true in the classical treatment, andsuffices to resolve the difficulties. By imposing Gauss’ Law on initial states, we are free tomake a choice for A0 which simplifies the dynamical problem.


For example, suppose we want to “solve” the constraint by “imposing Coulomb gauge.”We do this by writing

~Π = ~ΠT +1

∇2~∇∇ · ~Π (14.12)

= ~ΠT −1

∇2~∇J0 +

1

∇2~∇(∇ · ~Π + J0) (14.13)

so that ∇ · ~ΠT = 0. Inserting (14.13) into the term in the Hamiltonian containing Π andjudiciously integrating by parts, we obtain∫

d3x ~Π2 =

∫d3x ~Π2

T +

∫J0

(− 1

∇2

)J0 (14.14)

+

∫ (2

1

∇2J0 − 1

∇2(∇ · ~Π + J0)

)(∇ · ~Π + J0) +

∫[

1

∇2J0,∇ · ~Π + J0].

The last term, which arises because we reordered operators so that ∇ · ~Π + J0 stands onthe right wherever it appears, formally vanishes because the fields entering J are canonicallyindependent of Π and also J0 commutes with itself at equal times. The next to last termhas a factor of ∇ · ~Π + J0 on the right. By choosing

A0 =1

∇2J0 − 1

2∇2(∇ · ~Π + J0), (14.15)

this term is cancelled by the term linear in A0 and the Hamiltonian then simplifies to

HCOUL =

∫d3x ~Π2

T +

∫J0

(− 1

∇2

)J0 +

∫(1

2(∇× ~A)2 − ~A · ~J). (14.16)

Since only ΠT now enters the Hamiltonian, it is appropriate to make a similar decompositionof ~A = ~AT + ~∇(1/∇2)∇ · ~A. The gradient term does not contribute to the curl of A and anintegration by parts allows that term in HCOUL to be simplified as∫

(∇× ~A)2 =

∫(∇× ~AT )2 =

∫~AT · (∇× (∇× ~AT )) (14.17)

=

∫~AT · (−∇2) ~AT =

∫∇k

~AT · ∇k~AT . (14.18)

The longitudinal component of ~A appears in HCOUL through the coupling to ~J . To interpretthis term we have to consider a little more closely the inner product structure on the space ofstates. The states which satisfy the Gauss’ Law constraint have the character of momentumeigenstates with a fixed eigenvalue. Such states would of course have infinite norm. If werestrict our in states to satisfy Gauss’ Law, and we want our out states to have a finiteinner product with these in states, it follows that the out states should not be taken tosatisfy the constraint. In fact, the natural dual space to momentum eigenstates are positioneigenstates, with 〈q|p〉 ∝ eiqp. Thus we should choose our out states to be eigenstates of


∇· ~A, which is the conjugate variable to ∇· ~Π. Since ∇·A commutes with the Hamiltonian,as a Heisenberg picture operator it will be independent of time. Thus, if we choose our outstates to be eigenstates of ∇ · ~A(~x,+∞), with eigenvalue AL(~x) they will also be eigenstates

of ∇ · ~A(~x,−∞) with the same eigenvalue.We can always choose phases, as in the standard Schrodinger representation, so that

i〈A′, out|~Π = (δ/δA′)〈A′, out|. Then the dependence of persistence amplitudes on the choiceof AL is determined by

iδ 〈AL, out|in〉 =

∫d3x δAL(~x)〈AL, out|

(− 1

∇2

)∇ · Π(~x,+∞)|in〉 (14.19)

=

∫d3x δAL(~x)〈AL, out|

(+

1

∇2

)J0(~x,+∞)|in〉. (14.20)

Since J0(~x, t) is just the infinitesimal generator of local phase changes on the charged Heisen-berg fields at time t, we see that an infinitesimal change in AL can be compensated by alocal gauge transformation on the charged fields at t = +∞, which determine the definitionof the out states. The principle of gauge invariance includes the statement that state vec-tors differing by such gauge changes describe the same physical state, since they will giveidentical predictions for all gauge invariant observations. Thus we are free to fix AL to beany convenient function. Coulomb or radiation gauge corresponds to the choice AL = 0.

We have generally defined 〈out| by the t→∞ limit of 〈t| = 〈in|U(t,−∞). Our choice of〈AL, out| is implemented by the replacement 〈in| → 〈AL, in| where the latter is the eigenstateof ∇ · A(~x,−∞) with eigenvalue AL. Then the corresponding 〈AL, t| is the correspondingeigenstate of ∇ ·A(~x, t). Since 〈AL, t| satisfies i(∂/∂t)〈AL, t| = 〈AL, t|HCOUL(t), we see thatreplacing ∇ · A in HCOUL by AL will yield the same 〈AL, out|. In particular the radiationgauge choice AL = 0 leads to the effective Hamiltonian

Heff =

∫d3x ~Π2

T +

∫J0

(− 1

∇2

)J0 +

∫(1

2∂k ~AT · ∂k ~AT − ~AT · ~J). (14.21)

One may use Heff to compute any physical quantity. It only contains the transverse com-

ponents of ~A and ~Π. From the canonical commutation relations for these two quantities onecan easily evaluate those for the transverse components

[ATk(~x),ΠTm(~y)] = i

(δkm −

∇k∇m

∇2

)δ(~x− ~y). (14.22)

The operator acting on the delta function simply reflects the fact that the l.h.s. has vanishingdivergence because of the transversality of the operators.

We have obtained the effective Coulomb gauge Hamiltonian (14.21) by reduction from agauge independent quantization procedure. A much quicker route to the same answer is tofix the gauge before quantization by setting ∇ ·A = 0 from the beginning. Then the Gauss’law constraint can be “solved” by setting

A0(~x, t) = − 1

∇2J0(~x, t) =

∫d3y

J0(~y, t)

4π|~x− ~y|. (14.23)


The longitudinal component of ~Π is also eliminated because ∇ · ~Π = ∇2A0. One passes toquantum mechanics by promoting only ~A and ΠT to operators. (If the currents are operators,A0 is an operator by virtue of the constraint, but it is not independent.) Then the transverseprojector must appear on the r.h.s. of the canonical commutation relations.

14.1.1 Polarization and Helicity of Photons.

An explicit realization of the commutation relations can be given in terms of creation andannihilation operators as follows:

ATk(~x, 0) =

∫d3k√

(2π)32|k|[ak(~k)ei

~k·~x + a†k(~k)e−i

~k·~x] (14.24)

ΠTk(~x, 0) = −i∫

d3k√|k|√

(2π)32[ak(~k)ei

~k·~x − a†k(~k)e−i~k·~x] (14.25)

with

[ak(~k), a†m(~q)] = (δkm −kkkmk2

)δ(~k − ~q). (14.26)

Inserting these into Heff gives

Heff =

∫d3k|~k|~a†(~k) · ~a(~k)−

∫d3x ~AT · ~J + E0, (14.27)

where E0 is the usual (infinite) zero point energy of the oscillators which will be dropped

from now on. This formula shows us immediately that for ~J = 0, the quantum e.m. fieldis interpretable as a system of massless bosons (photons). The vacuum |0〉 is defined by

ak(~k)|0〉 = 0 and the n photon state is represented by

a†m1(~q1)a†m2

(~q2) · · · a†mn(~qn)|0〉. (14.28)

Because of transversality there are two photon states for each momentum. These two polar-ization states will next be shown to correspond to the two helicities ±1 of the photon.

First for fixed ~k let us introduce two (in general complex) basis vectors ~εa, a = 1, 2 for

the plane perpendicular to ~k, satisfying ~k ·~εa = 0 and the orthonormality and completenessrelations

~εa · ~ε∗b = δab (14.29)∑a

~εma · ~εn∗a = δmn −kmkn

k2. (14.30)

We can then introduce two independent sets of creation and annihilation operators via

~a(~k) =∑a

~εaaa(~k). (14.31)


We shall relate the multiplicity associated with the index a to the spin of the photon. Firstrecall the classical expression for the angular momentum carried by the e.m. field,

~J =

∫d3x~x× ( ~E × ~B) (14.32)

=

∫d3x

∑k

Ek(~x× ~∇)Ak −∫d3x~x× ( ~E · ~∇) ~A. (14.33)

We can recognize the first term in the last line as the “orbital” angular momentum, whichwill not contribute to the helicity of a one photon state. This is because acting on a onephoton state the ∇ is replaced by ~k and because of the cross product the term will beperpendicular to ~k. The second term, after an integration by parts becomes

~S =

∫d3x~E × ~A (14.34)

= −i∫d3k~a†(~k)× ~a(~k). (14.35)

Applying ~S to a one photon state a†a(~k)|0〉, yields

~Sa†a(~k)|0〉 = i

∑b

(~εa × ~ε∗b)a†b(~k)|0〉. (14.36)

Thus we see that the 2 × 2 matrix ~Sab = i~εa × ~ε∗b acts as a spin matrix on the index of the

creation operator. To get the helicity interpretation, consider the case of ~k = kz. Then thehelicity matrix is

S3ab = i(ε1aε

2∗b − ε2aε1∗b ). (14.37)

This matrix is diag1,−1 with the choices

~ε1 = (1, i, 0)/√

2 ~ε2 = (1,−i, 0)/√

2, (14.38)

so with this choice of polarization vectors, a†1 creates a photon with helicity +1 and a†2 createsa photon with helicity −1. This establishes that the photon is a spin one particle. There isno zero helicity state for the photon: this is consistent with Poincare invariance because thephoton is massless.

The polarization vector enters scattering amplitudes multilinearly, with a factor of ε foreach incoming photon and a factor ε∗ for each outgoing photon. Its four-vector index formsa Minkowski scalar product with that of the vertex coupling the gauge potential to thecharged fields. According to gauge invariance, this vertex satisfies current conservation: itsscalar product with the momentum entering it gives zero. Thus changing each polarizationvector by an amount proportional to its four-momentum leaves the scattering amplitudeunaltered. In Coulomb gauge the polarization vector is of the form ε = (ε, 0) with k ·epsilon = 0, so kµε

µ = 0. But kµkµ = 0 since the photon is massless. Thus we can

characterize the polarization vector completely by the covariant condition kµεµ = 0. Any

further specification, e.g. ε0 = 0, is merely a gauge choice which can be made at willand exploited to simplify detailed calculations. This is particularly advantageous in thecalculation of Compton scattering for polarized photons.


14.2 Charged Fields Interacting with the Quantized

Electromagnetic Field

We have seen that when the e.m. field interacts with a conserved current, the time componentof the current enters the Hamiltonian only through the coefficient in the term linear in A0

so that the coefficient of A0 is just the Gauss’ Law constraint with nonzero charge density.This feature is quite general, even when the e.m. field couples to dynamical charged fields.This is obvious in the case of the Dirac field because the Dirac field Hamiltonian is linearin Aµ. The A dependence of the scalar field Hamiltonian includes quadratic pieces in thespatial components of the potential, but nonetheless A0 still enters only linearly after theHamiltonian is expressed solely in terms of coordinates and momenta. This means thatthe elimination of A0 in the passage to Coulomb gauge proceeds exactly as in the previoussection.

Thus, the Hamiltonian for e.m. field plus charged fields in Coulomb gauge will quitegenerally be of the form

Heff =

∫d3x

(1

2ΠT

2 +1

2∂kAT · ∂kAT −AT · Je

)+

∫(J0e + j0)

(− 1

2∇2

)(J0e + j0) +Hfields|A0=0, (14.39)

where Jµe is an optional external current, jµ is the current operator for the fields, and thesubscript fields refers to the Hamiltonian operator for any dynamical fields in the system.For the Dirac field

j0 = Q[ψ, γ0ψ] (14.40)

Hfields =

∫d3xψ(

1

iγ · ∂ +m−Qγ ·A)ψ, (14.41)

and for the scalar field,

j0 = −iQ(πφ− φ†π†) (14.42)

Hfields =

∫d3x(ππ† +m2φ†φ+ (∇+ iQA)φ† · (∇− iQA)φ. (14.43)

One obstacle to formulating an efficient perturbation theory for systems with interactingquantum fields is that one can’t “turn off” the interactions at early and late times as ispossible with externally applied fields. Thus out and in states are eigenstates of complicatedinteracting Hamiltonians. One approach to this difficulty is to artificially make the couplingconstants time dependent and force them to vanish at early and late times. Another ap-proach, which we shall favor, is to relax the requirement that the initial and final states beeigenstates of the Hamiltonian with vanishing external fields. Then one calculates in firstinstance a quantity that is not of immediate interest, but which can be simply related tosuch quantities.


A quantity of more or less direct physical interest is the vacuum expectation value of thetime ordered product of several quantum fields. More generally the outin matrix element ofsuch a time ordered product is relevant if time varying external fields are present. So let usconsider how to obtain this quantity in perturbation theory by first calculating with generalinitial and final states. Using the evolution operator and assuming t1 > t2 > · · · > tn, wetherefore consider

〈f |U(∞,−∞)T [Ω1(t1) · · ·Ωn(tn)]|i〉 =

〈f |U(∞, t1)ΩS1U(t1, t2) · · ·U(tn−1, tn)ΩSnU(tn,−∞)|i〉. (14.44)

Choose the time T so that all external fields vanish for times earlier than −T and later thanT . Then

U(tn,−∞)|i〉 = U(tn,−T )e−i(∞−T )HS |i〉 (14.45)

= U(tn,−T )e−i(∞−T )EG∑r

e−i(∞−T )(Er−EG)|r〉〈r|i〉 . (14.46)

We would now like to argue that the infinite oscillations wash out all contributions butthe (assumed nondegenerate1) ground state. In a field theory this is quite plausible since theexcited states correspond to particles so the sum over r is really an integral over a range ofcontinuous energies. But even without this smearing, we can make the washing out rigorousby calculating with imaginary time: it = β > 0. Then i∞ is really +∞ and all excitedstates are damped exponentially. Massless particle states could introduce a subtlety here,but the part of phase space that is not exponentially damped is infinitesimal: this has theeffect of changing exponential damping to a power law damping. If we buy this argument,then we can assert quite generally that U(tn,−∞)|i〉 = U(tn,−∞)|0〉〈0|i〉 and similarly〈f |U(∞, t1) = 〈f |0〉〈0|U(∞, t1).

Since we take (as usual) the Heisenberg and Schrodinger pictures to coincide at t = −∞,then |in〉 = |0〉 and 〈out| = 〈0|U(∞,−∞). Thus we have obtained the relation

〈f |U(∞,−∞)T [Ω1(t1) · · ·Ωn(tn)]|i〉 =

〈f |0〉〈0|i〉〈out|T [Ω1(t1) · · ·Ωn(tn)]|in〉. (14.47)

In other words calculating with any initial and final states that have finite overlap2 withthe true ground state gives us a constant times the desired matrix element. We can easilyevaluate the multiplicative constant by considering by the same reasoning

〈f |U(∞,−∞)|i〉 = 〈f |0〉〈0|i〉〈out|in〉 (14.48)

→ e−2i∞EG 〈f |0〉〈0|i〉 External Fields = 0. (14.49)

1There are interesting cases of degenerate vacua, when there is “spontaneous symmetry breakdown.” Insuch cases the choice of initial and final states determines which of the degenerate vacua is picked out.

2The infinite number of degrees of freedom in quantum field theory requires care here: the overlap betweendifferent states in a theory with n degrees of freedom is typically fn with f < 1. Since n = ∞, we shouldexpect 〈f |0〉〈0|i〉 ∼ e−∞. In field theory n =∞ because the volume of space is infinite and because space iscontinuous. Thus strict application of the above relation should be done in the presence of both an infraredand ultraviolet cutoff, which can then be removed after extracting the desired amplitude.


Putting this into our relation we obtain

〈out|T [Ω1(t1) · · ·Ωn(tn)]|in〉 =

e−2i∞EG 〈f |U(∞,−∞)T [Ω1(t1) · · ·Ωn(tn)]|i〉〈f |U(∞,−∞)Ext=0|i〉

, (14.50)

where the subscript on U in the denominator denotes vanishing external fields. In fieldtheory applications EG is the energy of the vacuum, which is zero if we measure all energiesrelative to that of the vacuum. In the absence of gravity all physical quantities dependonly on energy differences, so we lose nothing by doing this. Gravity couples directly to theenergy density and therefore is sensitive to the energy as opposed to energy differences, butthen EG only appears in the combination Λ ≡ EG + Λ0, with Λ0 the “bare” cosmologicalconstant cosmological constant. Replacing Λ0 by Λ in effect sets EG = 0.

The formula (14.50) is a convenient starting point for developing perturbation theory.Any breakup

HS(t) = H0(t) +H ′(t) (14.51)

determines an interaction picture defined by

ΩI(t) = U−10 (t,−∞)ΩSU0(t,−∞) = U−1

I (t,−∞)Ω(t)UI(t,−∞), (14.52)

where

iU = HS(t)U = UH(t) (14.53)

iU0 = U0H0I(t) (14.54)

iUI = HI(t)UI (14.55)

and all U ’s are the identity at t = −∞. Then the evolution operator satisfies

U(t1, t2) = U0(t1,−∞)UI(t1, t2)U0(t2,−∞). (14.56)

Plugging these relations into (5.117) then gives

〈out|T [Ω1(t1) · · ·Ωn(tn)]|in〉 =

e−2i∞EG 〈f |U0(∞,−∞)T [UI(∞,−∞)Ω1I(t1) · · ·ΩnI(tn)]|i〉〈f |U0(∞,−∞)Ext=0UI(∞,−∞)Ext=0|i〉

. (14.57)

This formula is completely general: we have even allowed H0 to contain time varying externalfields, which is hardly ever done in practice. Since all operators in this formula are in inter-action picture, it is most convenient to choose |i〉, |f〉 to have simple properties with respectto H0I(−∞). Let us call the ground state of this operator |in, 0〉. Then 〈in, 0|U0(∞,−∞)is the ground state of H0I(+∞) and therefore deserves the name 〈out, 0|. When all exter-nal fields vanish, H0I is time independent and we call its ground state |0, I〉 ≡ |in, 0〉 and


its ground state energy E0. Then 〈in, 0|U0(∞,−∞)Ext=0 = e−2i∞E0〈0, I|. Thus choosing|i〉 = |f〉 = |0, I〉 = |in, 0〉 we obtain the useful formula

〈out|T [Ω1(t1) · · ·Ωn(tn)]|in〉 =

e−2i∞(EG−E0) 〈out, 0|T [UI(∞,−∞)Ω1I(t1) · · ·ΩnI(tn)]|in, 0〉〈0, I|UI(∞,−∞)Ext=0|0, I〉

. (14.58)

In the usual case where we do not include external fields in H0, the formula simplifies further

〈out|T [Ω1(t1) · · ·Ωn(tn)]|in〉 =

e−2i∞EG 〈0, I|T [UI(∞,−∞)Ω1I(t1) · · ·ΩnI(tn)]|0, I〉〈0, I|UI(∞,−∞)Ext=0|0, I〉

. (14.59)

Using the Wick expansion one can describe the perturbation series for the numeratorsand denominators of these formulas using Feynman diagrams. The diagrams contributing tothe denominator are all those completely disconnected from either external fields or from thepoints assigned to the operators in the numerator. The numerator contains this same sumof diagrams as a multiplicative factor. Thus the division by the denominator is achievedby simply deleting all such disconnected “vacuum” diagrams from the expansion of thenumerator.


Chapter 15

Path Integrals for Gauge Fields

The path integral for gauge theories presents special problems because of the gauge invarianceof the action. We shall establish the proper formulation for Quantum Electrodynamics byfirst obtaining the path integral in Coulomb gauge ∇·A = 0, following the general proceduresketched in this chapter. Once that has been done, we can discuss within the new pathintegral formalism more general gauges, including covariant ones.

We have already discussed the quantization of the free EM field in Coulomb gauge. Inthat case Gauss’ law ∇ · E = 0 implied that A0 = 0. In the presence of a current jµ, thisconclusion is modified:

∇ ·E = = −∇2A0 = j0, ∇ ·A = 0 (15.1)

A0(x, t) =

∫d3y

j0(y, t)

4π|x− y|(15.2)

Then the contribution of hte electric field to the Hamiltonian is

1

2

∫d3xE2 =

1

2

∫d3x

[E2

T + (∇A0)2]

=1

2

∫d3x

[E2

T − A0∇2A0]

=1

2

∫d3x

[E2

T − A0j0]

=1

2

∫d3xE2

T +1

2

∫d3x

j0(x, t)j0(y, t)

4π|x− y|(15.3)

The Hamiltonian operator for charged fields interacting with the quantized e.m. field Ain Coulomb gauge is

Heff =

∫d3x

(1

2ΠT

2 +1

2∂kAT · ∂kAT −AT · Je

)+

∫(J0e + j0)

(− 1

2∇2

)(J0e + j0) +Hfields(A)|A0=0, (15.4)

Here we have added an external current jµ → jµ+Jµe , and we understand all operators to bein Schrodinger picture. We also used the fact that A0 appears linearly in both the Dirac field

153

and charged scalar field in the form∫d3xA0j0. For the Dirac field j0 = Qψγ0ψ manifestly

has no A dependence, whereas for the scalar field j0 = iQ(πφ− π†φ†) has no A dependenceafter φ has been expressed in terms of π†.

We see that this is a case where the conjugate momentum appears only quadraticallyso we can immediately transcribe an infinitesimal evolution between eigenstates of A nearimaginary time τ

〈A′′|e−dτHeff |A′〉 (15.5)

≈ δ(∇ ·A′′) exp

−dτ

∫d3x

[1

2(A)2(τ) +

1

2(∂A)2(τ)−A(τ) · Je)

]−dτ

∫(J0e + j0)

(− 1

2∇2

)(J0e + j0)− dτHfields(A(τ)|A0=0

,

where the terms involving A(τ) with no time derivative are averaged over A(τ) = A′′,A′

and A(τ) symbolizes the continuum limit of (A′′ −A′)/dτ . The delta function multiplyingthe r.h.s. symbolizes the condition ∇·A = 0. It is necessary if we wish to formally integrateover all three components of A. Notice that for the moment we are only changing the e.m.field into path integral language, leaving the charged fields as operators.

Next we employ a widely used trick for simplifying the term describing the instantaneousCoulomb interaction. This involves introducing an auxiliary variable A4(x, t) and writingthe identity

exp

−dτ

∫(J0e + j0)

(− 1

2∇2

)(J0e + j0)

=∫

DA4(τ) exp−dτ

∫d3x[1

2(∇A4)2 − A4(J4

e + j4)]∫

DA4(τ) exp−dτ

∫d3x1

2(∇A4)2

(15.6)

where j4 = ij0 and J4e = iJ0

e as appropriate after the Wick rotation. This identity is provedby completing the square by a shift of integration variable A4 → A4 − (1/∇2)j4. Here weuse the fact that ultimately dτ → 0 to neglect any commutators that might arise due to thefact that j0 is really an operator. (This latter approximation is in exactly the same spirit asthe replacement of Hk by HW

k in our general derivation of the path integral.) The variableA4 enters now exactly as the imaginary time component of the vector potential would haveentered before it was eliminated by solving the Gauss’ law constraint. For example, using∇ ·A = 0 we can write∫

d3x[1

2A

2+

1

2(∇Ak)

2 −A · Je) +1

2(∇A4)2 − A4J4

e ] =∫d3x[

1

4FµνFµν − AµJµe ]. (15.7)

Furthermore Hfields|A4=0 −∫d3xA4(τ)j4 = Hfields with A4 6= 0 playing the role of the

imaginary time component of the vector potential.


Composing many infinitesimal evolutions, we arrive at the path integral representation

〈Af |U(∞,−∞)|Ai〉 = (15.8)∫DAµδ(∇ ·A)e−

∫d4x(FµνFµν/4−AµJµe )〈f |Te−

∫dτHfields(Aµ(τ))|i〉∫

DA4e−∫dτ(∇A4)2/2.

(15.9)

As discussed before, by taking the evolution over an infinite time interval we effectivelyproject onto the ground states (vacuum) at early and late times. We can therefore pickconvenient initial and final states, e.g. we can let them be eigenstates of A with vanishingeigenvalue and ground states of Hfields(Aµ(τ = ∓∞),∓∞) respectively. We then obtain thefollowing general formula for the outin matrix element of time ordered products of fields

〈out|T [ψ(y1) · · ·ψ(ym)ψ(zm) · · · ψ(z1)]|in〉Je,Ae = (15.10)∫DAµδ(∇ ·A)e−

∫d4x(FµνFµν/4−AµJµe )〈out|T [ψ(y1) · · · ψ(z1)]|in〉A(τ)+Ae(τ)∫

DAµδ(∇ ·A)e−∫d4xFµνFµν/4 〈out|in〉A(τ)

.

An important observation is that the set of operators in the time ordered product on thel.h.s. can be expanded to include any number of vector potentials by functionally differen-tiating w.r.t. the external current Jµe . This is clear from the pure exponential dependenceon Je on the r.h.s. Thus in addition to describing the presence of real external sources, theJe dependence provides a generating function for all correlation functions of any number ofvector potentials in the source free case: simply set Je = 0 after differentiating the appro-priate number of times w.r.t. Je. To illustrate this point, consider the path integral for thefree e.m. field (no charged fields).

〈out|in〉Je =

∫DAµδ(∇ ·A)e−

∫d4x(FµνFµν/4−AµJµe )∫

DAµδ(∇ ·A)e−∫d4xFµνFµν/4

. (15.11)

We can extract the dependence on Je by shifting the integration variable in the numeratorby Aµ → Aµ + Cµ, where C is chosen so that the linear term in A is cancelled. It must ofcourse also be restricted by the Coulomb gauge condition ∇ · C = 0. After the shift thecoefficient of the linear term in A becomes after an integration by parts

∂µ(∂µCν − ∂νCµ) + Jν = 0. (15.12)

This equation is only consistent if the external current is conserved ∂µJµ = 0 which we are

assuming. For ν = 4, remembering the Coulomb gauge condition, this equation determinesC4 = (−1/∇2)J4. After using current conservation to write ∂4C

4 = (1/∇2)∇·Je the spatialcomponents are determined to be C = (−1/∂2)(J−∇(1/∇2)∇·J). The question of boundaryconditions is settled in Euclidean space by requiring that C vanish in all four directions atinfinity. As we have discussed this prescription becomes the familiar iε one when continuedback to Minkowski space. Inserting these results into the path integral we find

〈out|in〉Je = e−∫d4x(FCµνF

Cµν/4−CµJ

µe ) (15.13)

= e∫d4xCµJ

µe /2 (15.14)

= e12

∫d4xd4yJµe (x)Dµν(x−y)Jνe (y). (15.15)


where

Djk(x) =

∫d4p

(2π)4eix·p

δjk − pjpk/p2

p2(15.16)

D44(x) =

∫d4p

(2π)4eix·p

1

p2= δ(t)

1

4π|x|(15.17)

D4k = Dk4 = 0. (15.18)

To get correlation functions of any number of vector potentials we differentiate 〈out|in〉Jewith respect to the current any number of times. Because of current conservation we mayonly differentiate w.r.t. three components of Jµ say the two transverse components JTand J4. But that is sufficient since the longitudinal component of A is zero by the gaugecondition. If we set Je = 0 after differentiating, it is easy to check that the Wick expansionfollows with the two point function

〈Aµ(x)Aν(y)〉 = Dµν(x− y). (15.19)

Note that with path integrals it is more precise to speak of correlation functions, whichare functional averages of some number of fields, than of the vacuum expectations of timeordered products of field operators. They are of course numerically equal to each other.

15.1 General Gauges

The path integral formulation of gauge theories is particularly suited to the discussion ofgeneral gauges. To keep the formalism covariant we would like to be able to replace the non-covariant Coulomb gauge condition with a covariant one such as ∂µAµ = 0. The procedurefor gauge fixing that follows is due to Fadeev and Popov, and is quite general. The methodstarts by selecting some general gauge fixing condition F (A, ∂A) = 0. This condition shouldhave the property that for any value of the gauge potential it is possible to find a gaugetransformation to a potential for which F = 0, and further that if F (A) = 0 then F (AΩ) 6= 0with Ω any nontrivial gauge transformation which vanishes at infinity. We don’t requirethis property for more general gauge transformations because that would rule out Coulombgauge which seems to be perfectly adequate. The path integrand in such a gauge shouldcontain a factor of δ(F (A)).

The F-P procedure is to define a functional ∆F (A) by the requirement

1 = ∆F (A)

∫DΩδ(F (AΩ)) (15.20)

where AΩ is the transformation of A by the gauge group element Ω(x), and the measureDΩ is gauge invariant. By this definition ∆F is clearly gauge invariant. Now insert thisrepresentation for 1 in the “unfixed” gauge field path integrand. Next change functionalintegration variables so that AΩ → A. Here we implicitly assume that the unfixed measure


DAµ is invariant under changes of variables which are gauge transformations1. Then theinfinite volume of the gauge group

∫DΩ comes out as a common factor in both the numerator

and denominator of the functional average and so cancels. We are left with the factors

∆F (A)δ(F (A)) (15.21)

in the functional integrand. The delta function fixes the gauge and the factor ∆A is ingeneral needed to guarantee that different choices for F yield the same answer for gaugeinvariant quantities. (It is only for functional averages of gauge invariant quantities thatthe rest of the integrand stays invariant under the variable change that removes Ω from thedelta function.)

Our Coulomb gauge path integral did not include a factor of ∆Coul(A) so we need toconfirm that it is inconsequential for the abelian case. For this we have to compute

1

∆Coul

=

∫DΛδ(∇ ·A+∇2Λ) =

1

det(−∇2). (15.22)

We see that ∆Coul is independent of A so that it will cancel between the numerator and de-nominator of functional averages. Thus our failure to put it in gives no differences in physicalquantities. Thus all gauge choices are formally equivalent to Coulomb gauge provided theFadeev-Popov factor ∆F (A) is included along with the gauge fixing delta functional.

In practice ∆F (A) always multiplies δ(F (A)) so the former is only needed for A satisfyingthe gauge condition. For such A, δ(F (AΩ)) only contributes for infinitesimal Ω:

F (A(x) + δA(x)) ≈∫d4y

δF (A(x))

δAµab(y)δAµab(y). (15.23)

Denoting the infinitesimal generators of the gauge group by G(z), δA is linear in the matrixelements of G: δAµab(y) =

∫d4zLµab,cd(y, z;A)Gcd(z) where L is a linear differential operator

depending on A in general. For example, the infinitesimal nonabelian gauge transformationreads δA = ∂G− ig[A,G]. Thus ∆F = det( δF

δAµLµ) where the determinant is that of a linear

differential operator which is also a matrix in the internal group space with matrix elementslabeled (ab, ef): ∫

d4yδFab(A(x))

δAµcd(y)Lµcd,ef (y, z;A). (15.24)

An important class of covariant gauges consists of the Lorentz gauges ∂ ·A = f where fis some fixed function. The F-P determinant for this gauge is det(−∂2) in the abelian caseof QED. Notice that it is independent of both f and A. In the nonabelian case, it woulddepend upon A of course, but is still independent of f . So for QED the F-P determinant

1In a completely general context this assumption might clash with the more basic translational invarianceof the measure. In such a case there would be a Jacobian accompanying the variable change. This is notneeded for the usual abelian and nonabelian gauge theories however.


cancels between numerator and denominator. and it is safe to ignore it2. Functional averagesof gauge invariant quantities will be independent of f . We can exploit this to get rid of thefunctional delta function in the path integrand by averaging over f with a gaussian weightfunction e−

∫d4xf2(x)/2α. This just multiplies the numerator and denominator by the same

constant so it won’t alter gauge invariant quantities. The net effect of this is to removethe delta function and instead add a non gauge invariant term

∫d4x(∂ · A)2(x)/2α to the

Euclidean action. For α = 1 (Feynman Gauge) the effect of this term is to simplify thekinetic term for A:∫

d4x

(1

4FµνFµν +

1

2α(∂ · A)2

)=

∫d4x

(1

2∂µAν∂µAν +

1− α2α

(∂ · A)2

)∼

∫d4x

1

2∂µAν∂µAν , α→ 1 (15.25)

Just as with the Coulomb gauge, the photon propagator for the “α” gauges is obtainedby evaluating the path integral for the gauge field in the presence of an external source. Thesource dependence is easily obtained by shifting A→ A+ C with C satisfying

∂2Cµ +1− αα

∂∂ · C = −Jµ (15.26)

which is solved first for ∂ · C = −α · J and then

Cµ = − 1

∂2(Jµ − (1− α)

∂µ∂ν∂2

Jν). (15.27)

Thus the propagator is

Dαµν(x) =

∫d4p

(2π)4eip·x

δµν − (1− α)pµpν/p2

p2. (15.28)

In the continuation to Minkowski space δµν is replaced by ηµν , p2 by p2 − iε, and there is

an additional factor of −i. We note the great simplification for Feynman gauge. The caseα = 0, known as Landau gauge is effectively the gauge ∂ ·A = 0 because the coefficient of thegauge breaking term in the action blows up damping out all contributions to the integral notsatisfying this condition. We see that in this case the divergence of the propagator vanishes,∂µD

0µν = 0. The fundamental reason that QED is independent of α is that A always couples

to a conserved current so that the terms involving α in the propagator decouple.To get a bit more insight into the role of current conservation consider a photon propa-

gator attached to two conserved vertices in momentum space

Amp ∼R1µR

2µ

p2=

R1 ·R2 −R10R

20

p2. (15.29)

2There are some applications where it is nonetheless important to keep it. For example, it contributesa constant to the zero point energy which just subtracts the spurious contribution of the time-like andlongitudinal components of the vector potential. Also when one computes the finite temperature partitionfunction in a covariant gauge, the F-P factor removes two photon degrees of freedom so the total number is4− 2 = 2. Without the F-P factor the Stefan-Boltzmann law would be off by a factor of 2!


It would seem that this coupling corresponds to the propagation of four photon states, 3“space” components and 1 time component, the last one coupling with the “wrong” sign.These wrong sign states are sometimes called “ghosts.” But by current conservation Ra

0 =p ·Ra/p0. In the limit of physical photon momentum,i.e. for which p2 = 0 the residue of thepole is just

R1 ·R2 − p ·R1p ·R2

p2(15.30)

which is to say only the transverse states (perpendicular to p) truly propagate.


Chapter 16

Fields, Charges, and Masses of theStandard Model

16.1 Particles and “Particles” of the Standard Model

LeptonsParticle Charge Spin Masse± ±e 1/2 0.511 MeVµ± ±e 1/2 106 MeVτ± ±e 1/2 1.78 GeVν1 0 1/2 m1

ν2 0 1/2 [m21 + 8× 10−5]1/2eV

ν3 0 1/2 [m21 ± 2× 10−3]1/2eV

Electro-weak Bosonsγ 0 1 0W± ±e 1 80 GeVZ 0 1 91 GeVh 0 0 125 GeV

Quarks“Particle” Charge Spin “Mass”d (×3) −e/3 1/2 “4.7” MeVs (×3) −e/3 1/2 “94” MeVb (×3) −e/3 1/2 “4.2” GeVu (×3) 2e/3 1/2 “2.2” MeVc (×3) 2e/3 1/2 “1.3” GeVt (×3) 2e/3 1/2 “173” GeV

Strong BosonsG (×8) 0 1 “0”

The preceding table lists all of the particles in the standard model for which fundamentalfields are introduced. We have listed quarks and gluons as “particles” because none of them

161

can exist in isolation: they are trapped (confined) inside of hadrons. Quarks always comein groups of three or quark antiquark pairs. The gluons come in groups of two or more.Because of confinement the notion of mass is not precise and hence we put the mass valuesin quotes. They are theoretical parameters that roughly correspond to our heuristic notionsof mass.


Chapter 17

Feynman Rules for QED

17.1 Lagrangian (density) for QED

Spinor QED

Fµν ≡ ∂µAν − ∂νAµ (17.1)

L1/2QED = −1

4FµνF

µν − ψ(m− iγµ(∂µ − iQ0Aµ)ψ (17.2)

Scalar QED

L0QED = −1

4FµνF

µν − (∂ + iQ0A)φ† · (∂ − iQ0A)φ− µ2φ†φ− λ

4(φ†φ)2 (17.3)

17.2 Rules in Coordinate Space

17.2.1 Propagators

x,a

y,b

SF (x− y)ab = −i∫

d4p

(2π)4

mδab − γab · pm2 + p2 − iε

ei(x−y)·p.

y,

µx,

ν

DFµν = −i∫

d4k

(2π)4

ηµν − (1− α)(kµkν/k2)

k2 − iεei(x−y)·k

163

17.2.2 Vertices

The vertices are determined from the nonquadratic terms of i times the Lagrange density.

iQ0Aµψγµψ, Spinor (17.4)

Q0Aµ(φ†∂µφ− ∂µφ†φ)− iQ20A · Aφ†φ (17.5)

µ

a

b

x,iQ0γ

µab

∫d4x.

17.2.3 Rules for Calculating⟨ψ(x1) · · ·ψ(xn)ψ(yn) · · · ψ(y1)A(z1) · · ·A(zm)

⟩1. Draw all possible graphs connecting together the points (xi, yi, zi). Drop all disconnected

vacuum bubbles. Associate with each graph the product of propagators and verticesaccording to the above table. Integrate over all internal points.

2. Each distinct graph has a weight ±1 as follows:

a) For each closed fermion loop include a factor (−1).

b) Two graphs of identical structure except for a permutation of the xi’s or of theyi’s have a relative minus (plus) sign if the total permutation is odd (even).

17.2.4 Rules for Scattering Amplitudes

1. Drop all corrections to all external lines.

2. Replace propagators associated with external lines by the following factors:

Outgoing Electron:

√Z2

(2π)3/2√

2ω(p)uλ(p)e−iy·p

Incoming Electron:

√Z2

(2π)3/2√

2ω(p)uλ(p)eiy·p


Outgoing Positron

√Z2

(2π)3/2√

2ω(p)vλ(p)e−iy·p

Incoming Positron:

√Z2

(2π)3/2√

2ω(p)vλ(p)eiy·p

Outgoing Photon:

y

k

√Z3

(2π)3/22|k|ε∗λ(k)e−iy·k

Incoming Photon:

k

y √Z3

(2π)3/22|k|ελ(k)eiy·k

3. Z2 and Z3 are obtained from examining the poles in the F.T. of the two point functions:∫d4xe−iq·x〈0|T [ψ(x)ψ(0)]|0〉 ∼ Z2

−i(m− γ · q)q2 +m2

, q2 → −m2 (17.6)∫d4xe−iq·x〈0|T [Aµ(x)Aν(0)]|0〉 ∼ Z3

−i(ηµν − (1− α′)qµqν/q2))

q2, q2 → 0(17.7)

Note that α′ 6= α in general. Also Z3 is gauge invariant whereas Z2 is not.

17.2.5 Cross Sections and Decay Rates

LetM be the amplitude obtained from the above rules by dropping the factors 1/[(2π)3/2√

2E]associated with the external lines and also dropping the overall momentum conserving deltafunction factor (2π)4δ(

∑f p′f −∑

i pi). Then the differential cross section for 2→ N particlescattering is:

dσ =d3p′1 · · · d3p′N

(2π)32E ′1 · · · (2π)32E ′N(2π)4δ(

∑f

p′f −∑i

pi)1

4E1E2|v2 − v1||M|2. (17.8)


Here vi are the velocities of the particles in the initial state. The initial state momenta arep1, p2 and the final state momenta all carry primes.

For a decay process 1 → N the differential decay rate in the rest frame of the decayingparticle is

dΓ =d3p′1 · · · d3p′N

(2π)32E ′1 · · · (2π)32E ′N(2π)4δ(

∑f

p′f − p)1

2M|M|2, (17.9)

where p = (M,0).To obtain total cross sections and total rates one must integrate these expressions over

all final state momenta and summing over all final spin states. In addition, if some of thefinal state particles are identical one must include the statistical factor 1/rk! for each subsetk of rk identical particles to compensate for multiple counting of identical states when thesefinal state sums are carried out independently.

17.3 Rules in Momentum Space

17.3.1 Propagators

pSF (p)ab = −imδab − γab · p

m2 + p2 − iε.

k DFµν(k) = −iηµν − (1− α)(kµkν/k2)

k2 − iε

17.3.2 Vertices

µ

a

b

iQ0γµab.

1. Each line carries a momentum obeying the constraint that energy-momentum is con-served at each vertex.

2. The rules for drawing graphs are identical to those in coordinate space.

3. Each unconstrained internal momentum is integrated with weight d4p/(2π)4.


4. The external line factors for M are simply

Outgoing Electron:

√Z2uλ(p)

Incoming Electron:

√Z2uλ(p)

Outgoing Positron

√Z2vλ(p)

Incoming Positron:

√Z2vλ(p)

Outgoing Photon:

y

k√Z3ε

∗λ(k)

Incoming Photon:

k

y √Z3ελ(k)


17.4 Divergence structure of Feynman diagrams

Since ultraviolet divergences are unavoidable in quantum field theory, it is useful to have acriterion for deciding the degree of divergence of a given (conneted) diagram. This is the(naive) degree of divergence D. let L be the number of loops; Vn be the number of verticeswith n powers of momentum, IB (EB) be the number of internal (external) boson lines IF(EF ) the number of internal (external) fermion lines. Then

D = DL− IF − 2IB +∑n

nVn (17.10)

where D is the spacetime dimension. The number of loops is IB + IF less the numberof vertices

∑N Vn plus 1, because there is a momentum conserving delta function at each

vertex, but one of them simply conserves overall momentum and doesn’t reduce the numberof momentum integrals. With this information we have

D = D(IB + IF −∑n

Vn + 1)− IF − 2IB +∑n

nVn

= D + IB(D − 2) + IF (D − 1) +∑n

(n−D)Vn (17.11)

To go further we need more detailed information about the vertices. So first consider spinorQED. Then the only vertex has n = 0 and two fermion lines and one boson line. Each vertexhas 1 end of a boson line. The number of such ends available is EB + 2IB so V = EB + 2IB.On the other hand each vertex has two ends of a fermion line so V + EF/2 + IF . Thus

IB =V − EB

2, IF = V − EF

2(17.12)

D = D +D − 4

2V − EB

D − 2

2− EF

D − 1

2(17.13)

For D < 4 adding vertices and/or external lines makes the degree of divergence less. ForD = 4 (our dimension) adding vertices doesn’t improve D. However adding external linesdoes. This is the criterion for a renormalizable theory. (The first case is superrenormalizable).

Coming back to spinor QED, we see that D ≥ 0 for the limited number of processesEF = 0, EB = 2, 3, 4, and EF = 2, EB = 0, 1. We shall find that the situation is even betterthan that: because of gauge invariance, the effective D is negative for EF = 0, EB = 4,and the process EB = 3, EF = 0 is zero by Furry’s theorem (Charge conjugation invariance.Furthermore the effective D for vacuum polarization is only 0, rather than 2.

1 Loop Examples: 2 photons, 3 photons, 4 photons, 2 electrons, 4 electrons, 2 photonsand 2 electrons, 2 electrons and 1 photon

Exercise: Find the formula for D for scalar electrodynamics, scalar φ3 theory, and scalar φ4

theory in arbitrary spacetime dimension D. In each case draw the one loop diagrams thatare divergent in 4 spacetime dimensions, and also, for the φ3 case in 6 spacetime dimensions.


Chapter 18

Scattering Amplitudes in QuantumField Theory

The observables of quantum field theory are local quantum fields. For example the energymomentum can be expressed as an integral of the local energy momentum tensor.

P µ = (P 0,P ) =

∫d3xθµ0(x, t) (18.1)

where θµν(x), the energy momentum tensor, is a local operator:

[θµν(x), θρσ(y)] = 0 (x− y)2 > 0 . (18.2)

The treatment of scattering in quantum field theory which I describe in this chapter fol-lows the method of Francis Low, developed in the late 1950’s. His key innovation was therecognition that wave packets must smear in time as well as space.

Single Particle States |p, a〉, with 〈p, a|p ′, b〉 = δabδ3(p − p ′) are characterized by pos-

sessing a unique energy for each momentum:

P µ|p, a〉 =(√

m2a + p 2,p

)|p, a〉. (18.3)

The space-time picture of scattering processes requires the use of wave packet single particlestates:

|f, a〉 ≡∫d3pf(p )|p, a〉. (18.4)

Their physical properties follow by considering for any local operator Ω(x),

〈f, a|Ω(x)|f, a〉 =

∫d3p′d3pf ∗(p ′)f(p )ei(p−p

′)·x〈p′, a|Ω(0)|p, a〉, (18.5)

169

where we used translation invariance in space and time Ω(x) = e−iP ·xΩ(0)eiP ·x. This meanswe are restricting consideration to vanishing external fields. Assuming f is peaked at p0,over an interval small compared to important variations of 〈p ′, a|Ω(0)|p, a〉, we have

〈f, a|Ω(x)|f, a〉 ≈ 〈p0, a|Ω(0)|p0, a〉[∫

d3p′f ∗(p′)e−i(p′·x−ωa(p′)t

)][∫

d3pf(p)ei(p·x−ωa(p)t

)]≈ |f(x, t)|2〈p0, a|Ω(0)|p0, a〉. (18.6)

We can select f(p) so that at some particular time, t,

f(x, t) =

∫d3pf(p)ei

(p·x−ωa(p)t

)(18.7)

is confined to some spatial volume V (t). As time evolves this volume will move and spread.The center of the packet will follow a straight line trajectory at the group velocity

vg =dωa(p)

dp

∣∣∣∣p=p0

=p0

ωa(p0)= v0 (18.8)

and spreading will be negligible over a time interval ωa(p0)/(∆p)2 < ∆x/∆v where ∆pis the width of the peak in fa at p0.Creation Operators for Single Particle States

We assume that for each single particle state, a, there is a local operator Ωa(x) such that

〈0|Ωa(0)|p, a〉 6= 0 (18.9)

Ωa must carry the quantum numbers of a, but is otherwise unspecified. A given Ω maycouple to several particles. Introduce wave packets fa(p ) peaked about some momentumwith a narrow width ∆p assumed much smaller than any variation in matrix elements orωa(p ). These packets are selected so that

fa(x, t) =

∫d3p

(2π)3fa(p )ei(p·x−ωa(p )t) (18.10)

is confined to a volume Va(t) = 0 (1/∆p3) with negligible spreading in the interval −T/2 <t < +T/2. Introduce also a switching function F (t) with shape

T /20-T /2 t

F(t)

0


and normalized by

∞∫−∞

dtF (t) = F (0) = 1 (18.11)

We stipulate that 1/T0 is much smaller than any important momentum variation in matrixelements or the function ωa(p ). In particular, 1/T0 is much smaller than any mass differences.This means that F (ω) the Fourier transform of F can be chosen sharply peaked in ω aboutω = 0. The width of this peak is only limited by ∆ω > 1/T0. Then, we define (for−T/2 < t0 < T/2)

A†a(t0) ≡∫d 4xfa(x, t)F (t− t0)Ω†a(x) (18.12)

The essential point here is that∫d4x forces A†a(t0) to create energy-momentum in a narrow

range about(p0, ωa(p0)

). It then follows:

(1) A†a(t0) creates a particle, a, in a wave packet fa at t0.

(2) Aa(t0) destroys a particle, a, in a wave packet fa at t0.

Proof:

〈0|Afa(t0)|p, b〉 =

∫d 4xf ∗a (x, t)F (t− t0)〈0|Ωa(0)|p, b〉eipb·x (18.13)

=

∞∫−∞

dtf ∗a (p )ei(ωa(p )−ωb(p )

)tF (t− t0)〈0|Ωa(0)|p, b〉 (18.14)

= f ∗a (p )〈0|Ωa(0)|p, b〉ei(ωa(p )−ωb(p )

)t0 F(ωa(p )− ωb(p )

)︸︷︷︸6= 0 only when ma = mb

≈ δabf∗a (p )〈0|Ωa(0)|p0, b〉 (18.15)

The last approximate equality assumes each mass value is non degenerate, and is based onthe narrow peaking of F . The equality is exact as long as the support of F lies entirelywithin the gap between ωa and any other energy available to the system. The fundamentalassumption is that each single particle for which such a creation operator can be defined hasa mass separated by a finite gap from any other mass. If the state b is a multiparticle state,its energy is not discrete but must in any case be greater than the sum of the masses of theparticles it describes. This minimum multiparticle energy is also assumed to be separated bya gap from ma. One can handle degeneracies in mass by carefully choosing the operators Ωa

so that they decouple from the single particle states degenerate with particle a. Clearly theconstruction does not completely succeed if there are massless particles in the theory becausethen a multiparticle state containing particle a and several low energy massless particles can


have an energy arbitrarily close to ma. In fact, in this case it is experimentally impossibleto give a definite meaning to “single particle state” so the limitations of the theory areappropriate.Normalization:

〈0|Aa(t0)A†a(t0)|0〉 ≈∫d3p〈0|Aa(t0)|p, a〉〈p, a|A†a(t0)|0〉

=

∫d3p∣∣fa(p )

∣∣2∣∣〈0|Ωa(0)|p, a〉∣∣2 (18.16)

≈∣∣〈0|Ωa(0)|p0, b〉

∣∣2 since fa is sharply peaked.

The first approximate equality follows because after insertion of a complete set of statesbetween A and A†, the smearing functions in the definition of the latter focus the energyand momentum of the intermediate states to be (within the allowed windows ∼ (1/T0,∆p))those of the single particle state a. Thus only those give a significant contribution. The lastapproximate equality is based on the narrow peaking of the wave packet fa.

18.1 Multiparticle States

We can use our creation operators to construct multiparticle states. For example

A†a(t0)A†b(t0)|0〉 (18.17)

creates a two particle state. Of course, there may be interactions between the two particles,so this interpretation is only meaningful when packet, a, is spacelike separated from packet ,b. In this case, we have[

A†a(t0), A†b(t0)]

= 0 (Since [Ωa(x),Ωb(y)] = 0 for (x− y)2 > 0) (18.18)

from which it follows that the norm of the two particle state factorizes into the product ofsingle particle norms:

〈0|Aa(t0)Ab(t0)A†a(t0)A†b(t0)|)〉 = 〈0|Aa(t0)A†a(t0)Ab(t0)A†b(t0)|0〉≈ 〈0|Aa(t0)A†a(t0)|0〉〈0|Ab(t0)A†b(t0)|0〉

Furthermore, for any operator of the form Ω =∫d3xf(x)Ω(x, t0), we have

〈0|Aa(t0)Ab(t0)ΩA†a(t0)A†b(t0)|0〉 ≈ 〈0|Aa(t0)ΩA†a(t0)|0〉〈0|Ab(t0)A†b(t0)|0〉+〈0|Aa(t0)A†a(t0)|0〉〈0|Ab(t0)ΩA†b(t0)|0〉

This follows since we can write effectively

Ω ∼∫Va(t0)

d3xf(x)Ω(x, t0) +

∫Vb(t0)

d3xf(x)Ω(x, t0) (18.19)

and the two pieces act independently on particles a and b. Since all the observables of aquantum field theory are local fields such as Ω(x), this justifies the multiparticle interpreta-tion. It should be clear that one can extend this construction to states with any number ofspatially separated particles.


18.2 Reduction Formulae

Assume −T2< t1, t2, . . . , tn <

T2

TAa

(T

2

)Ω1(x1) . . .Ωn(xn)

− T

Ω1(x1) . . .Ωn(xn)Aa

(−T

2

)=

T2∫

−T2

dt0d

dt0TAa(t0)Ω1(x1) . . .Ωn(xn) (18.20)

=

T2∫

−T2

dt0

∫d4xF (t− t0)

∂

∂tf ∗a (x, t)TΩa(x)Ω1(x1) . . .Ωn(xn

=

T2∫

−T2

dt0

∫d4xF (t− t0)f ∗a (x, t)

(iωa(−i∇) +

∂

∂t

)TΩa(x)Ω1(x1) . . .Ωn(xn)

TA†a

(T

2

)Ω1(x1) . . .Ωn(xn)

− T

Ω1(x1) . . .Ωn(xn)A†a

(−T

2

)=

T2∫

−T2

dt0

∫d4xF (t− t0)

∂

∂tfa(x, t)TΩ†a(x)Ω1(x1 . . .Ωn(xn) (18.21)

=

T2∫

−T2

dt0

∫d4xF (t− t0)fa(x, t)

(−iωa(i∇) +

∂

∂t

)TΩ†a(x)Ω1(x1) . . .Ωn(xn) .

In these reduction formulae we have used the fact that by construction fa satisfies a Schrodingerequation with hamiltonian ωa(−i∇) =

√m2a −∇2. Spatial integration by parts then allows

the spatial derivatives to be transferred to Ω(x).

18.3 Single Particle States are Handled Consistently

A single particle state prepared at early times should remain a single particle state for alltimes. This follows from the reduction formulae:

〈0|Aa(T

2

)A†b

(−T

2

)|0〉 = 〈0|Aa

(−T

2

)A†b

(−T

2

)|0〉 (18.22)

+

∫ T2

−T2

dt0

∫d 4xF (t− t0)f ∗a (x, t)

(∂

∂t+ iωa(−i∇)

)〈0|Ωa(x)A†b

(−T

2

)|0〉


But 〈0|Ωa(x)A†b (−T/2) |0〉 ≈∫d3p〈0|Ωa(0)|p, b〉〈p, b|A†b

(−T

2

)|0〉e+i

(p·x−ωb(p )t

)So the second term becomes∫ T

2

−T2

dt0

∫ ∞−∞

dt

∫d3pf ∗a (p )F (t− t0)i (ωa(p)− ωb(p)) ei

(ωa(p )−ωb(p )

)t

〈0|Ωa(0)|p, b〉〈p, b|A†b(−T

2

)|0〉

=

∫d3p

∫ T2

−T2

dt0ei(ωa(p)−ωb(p)

)t0f ∗a (p )F

(ωa(p )− ωb(p )

)i(ωa(p)− ωb(p)

)〈0|Ωa|p, b〉〈p, b|A†b|0〉 ≈ 0

Since F ≈ 0 if b 6= a and the r.h.s. is identically zero if a = b because of the factorωa(p)− ωb(p). The approximate equalities become exact in the limit

T, T0 →∞ with1

δm T0 T T 2

0m (18.23)

(The latter inequality insures negligible wave packet spread.)We therefore obtain the required result:

〈0|Aa(T

2

)A†b

(−T

2

)|0〉 ≈ lim〈0|Aa

(−T

2

)A†b

(−T

2

)|0〉

≈∣∣∣〈0|Ωa(0)|p0, a〉

∣∣∣2δab (18.24)

where the approximate equality becomes exact as T0, T →∞.

18.4 Two Particle Scattering Amplitudes

The scattering process will consist of preparation at a very early time −T ′/2 of two separatedsingle particle wave packets with well defined momentum, aimed to collide at roughly timet ≈ 0, and the subsequent observation at a much later time +T

2of two well separated wave

packets.Let fa, fb describe the incoming packets with momentum pa and pb respectively; fc, fd

the outgoing packets with momentum pc,pd. The information we want is contained in thematrix element

〈0|Ac(T

2

)Ad

(T

2

)A†a

(−T

′

2

)A†b

(−T

′

2

)|0〉

≈ 〈0|Ωc|pc〉〈0|Ωd(0)|pd〉〈pa|Ω†a(0)|0〉〈pb|Ω†b(0)|0〉 (18.25)∫

d3p′∫d3q′

∫d3p

∫d3qf ∗c (p′)f ∗d (q′)fa(p )fb(q )〈p′, c; q′, d|S|p, a; q, b〉,


which can be taken as the definition of the S-matrix. Now, we apply the Reduction Proce-dure.

〈0|Ac(T

2

)Ad

(T

2

)A†a

(−T

′

2

)A†b

(−T

′

2

)|0〉

= 〈0|Ac(T

2

)Ad

(−T

2

)A†a

(−T

′

2

)A†b

(−T

′

2

)|0〉 (18.26)

+

∫ T2

−T2

dt10

∫d 4x1F (t1 − t10)f ∗d (x1)

(∂

∂t1+ iωd(i∇1)

)(18.27)

〈0|Ac(T

2

)Ωd(x1)A†a

(−T

′

2

)A†b

(−T

′

2

)|0〉 (18.28)

The first term on the RHS contains the amplitude that nothing happens. The final packetd has been extrapolated back as a free particle to time −T/2. If no scattering has occurredthis extrapolated packet can overlap with initial packet a or packet b but not both since aand b are spatially separated. Thus Ad(−T/2) will commute with Aa(−T ′/2) or Ab(−T ′/2)or both. Both possibilities are included by writing

〈0|Ac(T

2

)Ad

(−T

2

)A†a

(−T

′

2

)A†b

(−T

′

2

)|0〉

≈ δma,md〈0|Ac(T

2

)A†b

(−T

′

2

)|0〉〈0|Ωd(0)|pd〉〈pa|Ω†a(0)|0〉

∫d3pf ∗d (p )fa(p )

+δmb,md〈0|Ac(T

2

)A†a

(−T

′

2

)|0〉〈0|Ωd(0)|pd〉〈pb|Ω

†b(0)|0〉

∫d3pf ∗d (p )fb(p )

≈[δma,mdδmb,mc

∫d3pf ∗d (p )fa(p )

∫d3pf ∗c (p )fb(p ) (18.29)

+δma,mcδmb,md

∫d3pf ∗d (p )fb(p )

∫d3pf ∗c (p )fa(p )

]×〈0|Ωd(0)|pd〉〈0|Ωc(0)|pc〉〈pa|Ω†a(0)|0〉〈pb|Ω

†b(0)|0〉

where δmm′ = 0 if m 6= m′, 1 if m = m′. Of course if the extrapolated packet overlaps withneither a nor b this term contributes nothing, as the above formula states because then fdwill be orthogonal to both fa and fb.

In the 2nd term we now reduce particle c:

〈0|Ac(T

2

)Ωd(x1)A†a

(−T

′

2

)A†b

(−T

′

2

)|0〉

≈ 〈0|Ωd(x1)Ac

(−T

2

)A†a

(−T

′

2

)A†b

(−T

′

2

)|0〉

+

∫ T2

−T2

dt(2)0

∫d 4x2F (t2 − t20)f ∗c (x2)

(∂

∂t2+ iωc(i∇2)

)(18.30)

×〈0|T [Ωc(x2)Ωd(x1)]A†a

(−T

′

2

)A†b

(−T

′

2

)|0〉


The first term will give negligible contribution because the factor(∂∂t

+ iωd(−i∇1))

will yield

a factor i(ωd +ωc−ωa−ωb

)which will vanish within the support of the factor F (ωd +ωc−

ωa − ωb). (A nonvanishing contribution would require either c = a or c = b, because Ac isa destruction operator and then the support of F requires d = b or respectively d = a.) Inthis second reduction we have tacitly written Ac

(T2

)Ωd(x1) = T

[Ac(T2

)Ωd(x1)

]which is

not strictly true since Ac(T2

)can involve Ωc(y) at some times earlier than t1. However the

integrands include wave packets which are spacelike separated for times near T2. Since Ωc

and Ωd commute at spacelike separations no mistake is made by this procedure. Similarlythe replacement

T

[Ωd(x)Ac

(−T

2

)]= Ωd(x)Ac

(−T

2

)is validated because the wave packets extrapolated back to times near −T

2will again be

space-like separated. (We assume packets are aimed to overlap at times near 0.)To continue reducing, we need to replace

T [Ωc(x2)Ωd(x1)]A†a(−T ′/2)

byT [Ωc(x2)Ωd(x1)A†a(−T ′/2)]

. The above argument could be applied for nonforward but not quite for forward scatteringbecause then one of the final packets extrapolated to time −T/2 could well overlap one ofthe initial packets. However, by choosing T ′/2 > T/2+T0, we force −T ′/2 to be earlier thaneither t1 or t2 and likewise T ′/2 to be later than t1 or t2. Then the required replacementsare valid. As long as −T/2 is well before and T/2 well after the collision, such a choice iscompletely satisfactory.

With this in mind we can reduce particles a and b. We drop immediately terms whereA† stands next to 〈0|, because they could only produce a negative energy state which weassume does not exist. The final result is:

〈0|Ωc(0)|pc〉〈0|Ωd(0)|pd〉〈pa|Ω†a(0)|0〉〈pb|Ω†b(0)|0〉

∫d3p′ d3q′

∫d3p d3q

×f ∗c (p ′ )f ∗d (q ′ )fa(p )fb(q )〈p ′, c; q ′, d|(S − I)|p, a; q, b〉

≈∫ T/2

−T/2dt

(1)0 dt

(2)0

∫ T ′/2

−T ′/2dt

(3)0 dt

(4)0

∫d4x1d

4x2d4x3d

4x4F (t1 − t10)F (t2 − t20)F (t3 − t30)F (t4 − t40)

f ∗d (x1)f ∗c (x2)fa(x3)fb(x4)

(∂

∂t1+ iωd

)(∂

∂t2+ iωc

)(∂

∂t3− iωa

)(∂

∂t4− iωb

)(18.31)

〈0|T [Ωc(x2)Ωd(x1)Ω†a(x3)Ω†b(x4)]|0〉 (18.32)

Now define

T [q2q1; q3q4] ≡∫d4x1d

4x2d4x3d

4x4e−i(q2·x2+q1·x1−q3·x3−q4·x4)

〈0|T [Ωc(x2)Ωd(x1)Ω†a(x3)Ω†b(x4)]|0〉 (18.33)


and the r.h.s. becomes∫ T/2

−T/2dt10dt

20

∫ T ′/2

−T ′/2dt30dt

40

∫ ∏m

d4qm(2π)4

F (ωd − q10)F (ωc − q2

0)F (q30 − ωa)F (q4

0 − ωb)

f ∗d (q1)f ∗c (q2)fa(q3)fb(q4)i4(ωd − q01)(ωc − q0

2)(ωa − q03)(ωb − q0

4)

ei(ωd−q01)t10ei(ωc−q

02)t20e−i(ωa−q

03)t30e−i(ωb−q

04)t40T [q2, q1; q3, q4]

∼∫

d3q1d3q2d

3q3d3q4

(2π)3(2π)3(2π)3(2π)3f ∗d (q1)f ∗c (q2)fa(q3)fb(q4), T, T ′ →∞

lim (ωd − q01)(ωc − q0

2)(ωa − q03)(ωb − q0

4)T [q2, q1; q3, q4], (18.34)

where the limit taken is q01 → ωd, q

02 → ωc, q

03 → ωa, q

04 → ωb, and is forced by delta functions

arising from the large T, T ′ limits of the integrals over the tk0. We see that there is nonzeroscattering only if T has poles in all of the q0

i at the energies of the respective incoming andoutgoing particles.

Taking the limit of infinitely narrow packets we may express our result as follows

T [p′, q′; p, q]−→p0′→ωcq0′→ωdp0→ωaq0→ωb

[(2π)3]4(2ωa)2ωb)(2ωc)(2ωd)〈0|Ωc(0)|p ′〉〈0|Ωd(0)|q ′〉

×〈p|Ω†a(0)|0〉〈q|Ω†b(0)|0〉 × −ip′2 +m2

c

−iq′2 +m2

d

−ip2 +m2

a

−iq2 +m2

b

×[〈p ′, c; q ′, d|S|p, a; q, b〉 − δacδbdδ3(p ′ − p )δ3(q ′ − q )

−δadδbcδ3(p− q ′)δ3(p ′ − q )]

(18.35)

Note that translation invariance implies that T has an overall factor of (2π)4δ4(p′+q′−p−q)expressing energy momentum conservation. It should now be clear how the generalization toan arbitrary number of incoming and outgoing particles should be expressed. The F.T. of theT.O.P. of interpolating fields will have a pole factor −i/(q2

a+m2a) associated with each exter-

nal line. The coefficient of all these pole factors is proportional to the desired scattering am-plitude. Furthermore, the factors of proportionality are clear: a factor 2ωa(2π)3〈pa|Ωa(0)|0〉for each incoming line and a factor 2ωa(2π)3〈0|Ω†a(0)|pa〉 for each outgoing line.

We are left with the task of computing the matrix elements of the interpolating fieldsbetween the vacuum and one particle states. This information is contained in the F.T. ofthe two point functions∫

d4xe−ix·q〈0|T [Ωa(x)Ω†b(0)]|0〉

=∑λ,p

〈0|Ωa(0)|λ, p〉〈λ, p|Ω†b(0)]|0〉(2π)3δ(p− q)−i

p0 − q0 − iε(18.36)

±∑λ,p

〈0|Ω†b(0)|λ, p〉〈λ, p|Ωa(0)]|0〉(2π)3δ(p+ q)−i

p0 + q0 − iε


The sums over states include all states with any number of particles. But the single particlestates are singled out by their unique values of energy for fixed three momentum. Thusonly these states will produce poles in the variable q0. (Multiparticle states produce cuts).Single particle contributions in the first term produce poles at positive values q0 = +ωc(q)whereas in the second term they produce poles at negative values q0 = −ωc(q). The statescontributing to the second term are the charge conjugates (antiparticles) of those contributingto the second term. Focusing on the positive energy poles, we see that the single particle cgives a pole with structure

−iq2 +m2

c − iε(2π)32ω(q)〈0|Ωa(0)|c, q〉〈c, q|Ω†b(0)]|0〉. (18.37)

Coosing a = b = c this reads, in the case that Ωc is a scalar operator:

−iq2 +m2

c − iε(2π)32ω(q)〈0|Ωc(0)|c, q〉〈c, q|Ω†c(0)]|0〉 ≡ −iZc

q2 +m2c − iε

, (18.38)

from which we learn that

|〈0|Ωc(0)|c, q〉| =

√Zc

(2π)3/2√

2ωc(18.39)

If ωc is a fermion field, Lorentz invariance dictates that the matrix element has the struc-ture 〈0|Ωc|q〉 ∝ u(q), so the corresponding Zc is defined by setting the left side equal to−iZc/(mc + γ · q). Correspondingly,

〈0|Ωc|q〉 =

√Zc

(2π)3/2√

2ωcu(q).

In either case, the treatment of external lines in perturbation theory of the scattering ampli-tude, is to drop all external line corrections, replace the external line propagator with

√Z

times the appropriate wave function (1 for scalars; u, u, v, or v for fermions; ε or ε∗ forphotons).


Chapter 19

One Loop Corrections in QED

19.1 Photon self energy

Recall that iin our discussion of vacuum polarization we calculated the matrix element〈0|Tjµ(x)jν(0)|0〉 which also gives the one loop correction to the photon propagator. Weneed its Fourier transform

iΠµν = i2∫d4xe−iq·x 〈0|Tjµ(x)jν(0)|0〉 = (qµqν − q2ηµν)iΠ(q2) (19.1)

Π(q2) =Q2

0

2π2

∫ 1

0

dxx(1− x) lnΛ2

m2 + x(1− x)q2 − iε(19.2)

Feeding this into the calculation of the photon propagator

DF =−iqµqν

q4+−i(ηµν − qµqν/q2)

q2

[1− Π(q2) + · · ·

](19.3)

→ −iqµqν

q4+−i(ηµν − qµqν/q2)

q2(1 + Π(q2))(19.4)

If Π is taken as the sum of all 1 particle irreducible two photon diagrams, this expressionis exact! This is because in terms of this Π the rest of the two photon propagator is ageometric sum. We shall see that the absence of corrections to the longitudinal part of thephoton propagator is a consequence of gauge invariance.

19.2 Proper Vertex Function

Give the photon a small mass λ as an infrared cutoff. Then in Feynman gauge the propervertex is:

Γµ(p′, p)− γµ = −iQ20

∫d4k

(2π)4

γρ(m− γ · (p′ − k))γµ(m− γ · (p− k))γρ

(k2 − λ2 − iε)((p′ − k)2 +m2 − iε)((p− k)2 +m2 − iε)(19.5)

179

The first step is to combine denominators using

1

D1D2 · · ·Dn

=

∫ ∞0

dT1 · · · dTne−∑k TkDk

=

∫T n−1dT

∫ 1

0

dx1 · · · dxnδ(1−∑k

xk)e−T

∑k xkDk

= Γ(n)

∫ 1

0

dx1 · · · dxnδ(1−∑k

xk)1

[∑

k xkDk]n(19.6)

The denominators of the Λ integrand are combined with x2 = x, x3 = y and x1 = 1− x− y:

1

D1D2D3

= 2

∫x+y≤1

dxdy1

[k2 − iε+ λ2(1− x− y)− 2k · (xp′ + yp)]3

= 2

∫x+y≤1

dxdy ∗ 1

[(k − xp′ − yp)2 − iε+ λ2(1− x− y)− (xp′ + yp)2]3(19.7)

where in the last line we completed the square. We can simplify

(xp′ + yp)2 = −m2(x2 + y2) + 2xyp′ · p = −m2(x+ y)2 − xyq2 (19.8)

where q = p′−p is the momentum transfer. Define a new integration variable k = k−xp′−ypin terms of which the numerator of the integrand becomes

γρ(m− γ · ((1− x)p′ − yp− k))γµ(m− γ · ((1− y)p− xp′ − k))γρ

= γρ(m− γ · ((1− x)p′ − yp))γµ(m− γ · ((1− y)p− xp′))γρ + γρk · γγµk · γγρ

+ Linear terms in k (19.9)

Inside the k integral the terms linear in k give zero and kµkν can be replaced by ηµν k2/4.

Hence the numerator can be replaced with

γρ(m− γ · ((1− x)p′ − yp))γµ(m− γ · ((1− y)p− xp′))γρ + k2γµ (19.10)

Only the last term of this expression contributes to the ultraviolet divergence of Γµ.

19.2.1 UV divergences

Inserting the last term of the numerator into the integral for Γµ − γµ, we find

Γµuv − γµ = −2iQ20

∫dxdy

∫d4k

(2π)4

k2γµ

[k2 − iε+ λ2(1− x− y) +m2(x+ y)2 + xyq2]3(19.11)

Now use ∫d4k

(2π)4

k2

[k2 + A− iε]3=

i

16π2

[ln

Λ2

A− 3

2

](19.12)

to arrive at

Γµuv − γµ = γµQ2

0

8π2

∫dxdy

(ln

Λ2

λ2(1− x− y) +m2(x+ y)2 + xyq2− 3

2

)(19.13)


19.2.2 UV finite remainder

Use ∫d4k

(2π)4

1

[k2 + A− iε]3=

i

32π2A(19.14)

to evaluate the k integral for the remainder of the vertex part. For this we rearrange

γρ(m− γ · ((1− x)p′ − yp))γµ(m− γ · ((1− y)p− xp′))γρ (19.15)

= 2m2γµ − 4m[(1− 2x)p′ + (1− 2y)p]µ + 2γ · [(1− y)p− xp′]γµ[(1− x)p′ − yp)γ

If we sandwich Γµ between on shell spinors u′Γµu satisfying u′(m+γ ·p′) = 0 and (m+γ ·p)u =0, use p = p′−q or p′ = p+q as appropriate, and use the symmetry of the rest of the integrandunder x↔ y, we can replace the numerator with

2m2γµ − 4m(1− x− y)[p′ + p]µ

+2[−(1− x− y)m− (1− y)q · γ]γµ[−(1− x− y)m+ (1− x)q · γ]

= 2m2(1 + (1− x− y)2)γµ − 4m(1− x− y)[p′ + p]µ − 2(1− x)(1− y)q2γµ

−2m(1− x− y)[(1− x)γµq · γ − (1− y)q · γγµ] (19.16)

under these same circumstances we can eliminate the p+ p′ term using

[q · γ, γµ] → (−m− p · γ)γµ − γµ(m+ p′ · γ) = −4mγµ + 2(p+ p′)µ (19.17)

with which the numerator can be replaced with

2m2((x+ y)2 − 2(1− x− y)− q2(1− x)(1− y))γµ +m(1− x− y)(x+ y)[γµ, q · γ]

Putting every thing together we get

u′Γµ(p′, p)u ≡ u′[γµF1(q2) +

[γµ, q · γ]

4mF2(q2)

]u (19.18)

F1(q2) = 1 +Q2

0

8π2

∫dxdy

(ln

Λ2

λ2(1− x− y) +m2(x+ y)2 + xyq2− 3

2

+m2((x+ y)2 − 2(1− x− y)− q2(1− x)(1− y))

λ2(1− x− y) +m2(x+ y)2 + xyq2

)(19.19)

F2(q2) =Q2

0

4π2

∫dxdy

m2(x+ y)(1− x− y)

λ2(1− x− y) +m2(x+ y)2 + xyq2(19.20)

The UV divergences is entirely in F1(q2). We define f1(0) ≡ Z−11 :

1

Z1

= 1 +Q2

0

8π2

∫dxdy

(ln

Λ2

λ2(1− x− y) +m2(x+ y)2− 3

2

+m2((x+ y)2 − 2(1− x− y))

λ2(1− x− y) +m2(x+ y)2

)= 1 +

Q20

8π2

∫ 1

9

udu

(ln

Λ2

λ2(1− u) +m2u2− 3

2+m2(u2 − 2(1− u))

λ2(1− u) +m2u2

)(19.21)


then we can write

F1(q2) =1

Z1

+ f1(q2)− f1(0) (19.22)

with the UV divergence residing in the first term. In contrast F2(q2) is both UV and IRfinite. In particular

F2(0) =Q2

0

4π2

∫ 1

0

dum2u2(1− u)

λ2(1− u) +m2u2(19.23)

→ Q20

4π2

∫ 1

0

du(1− u) =Q2

0

8π2≈ α

2π, λ = 0 (19.24)

19.2.3 Anomalous magnetic moment of the electron

We already know that before radiative corrections the g factor for the electron is 2. To seewhat radiative corrections do, we use (19.17) to write

u′γµu = u′[

(p+ p′)µ

2m+

1

4m[γµ, q · γ]

]u (19.25)

The second term is reponsible for the spin magnetic moment which we know correpondsto g = 2. By comparing coefficients it follows that radiative corrections give g − 2 =2F2(0) ≈ α/π a result first obtained by Schwinger. Experiment gives an extremely precisemeasurement for g − 2 shich agrees with QED calculated to order αn.

19.3 Self energy of the electron

The fermion self-energy part Σ is defined as i times the sum of all one particle irreducibletwo point function. If it were known exactly, the exact fermion propagator would be

S ′F =−i

m+ γ · p

[1− Σ

1

m+ γ · p+ · · ·

]=

−im+ γ · p+ Σ(p)

(19.26)

and hence the physical mass of the fermion is determined by finding p such that m+γ ·p+Σ(p)has a zero eigenvalue. To 1 loop order we have

−iΣ = Q20

∫d4k

(2π)4

γρ(m− γ · (p− k))γρ

(k2 + λ2 − iε)(m2 + (p− k)2 − iε)+O(Q4

0) (19.27)

= −iQ20

∫ 1

0

dx

∫d4kE(2π)4

4m+ 2γ · ((1− x)p− k)

[k2 + λ2(1− x) + xm2 + x(1− x)p2]2

Σ =Q2

0

8π2

∫ 1

0

dx(2m+ (1− x)γ · p) lnΛ2e−1

λ2(1− x) + xm2 + x(1− x)p2(19.28)


To interpret this result, first imagine that we have Σ to all orders. Although Σ is a matrix,by Lorentz covariance it muat be a function of the single matrix γ · p. Moreover, sincep2 = −(γ · p)2, all the p dependence is through γ · p. If M is the physical mass, we mayexpand m+ γ · p+ Σ in a power series in M + γ · p, starting at linear order:

m+ γ · p+ Σ = (M + γ · p)(1 + Σ′) +O((M + γ · p)2) (19.29)

where Σ′ is the first derivative of Σ regarded as a function of γ · p evaluated at γ · p = −M .With the one loop approximation to Σ, we have

m−M +Q2

0

8π2

∫ 1

0

dx(2m−M(1− x)) lnΛ2e−1

λ2(1− x) + xm2 − x(1− x)M2= 0 (19.30)

which determines the mass shift

M −m =Q2

0

8π2

∫ 1

0

dxM(1 + x)) lnΛ2e−1

λ2(1− x) + x2M2+O(Q4

0) (19.31)

where we used M −m = O(Q20) to set m = M on the right side.

Σ′1loop =Q2

0

8π2

∫ 1

0

dx

[(1− x) ln

Λ2e−1

λ2(1− x) + x2M2− 2

x(1− x)(1 + x)M2

λ2(1− x) + x2M2

](19.32)

This result is intimately related to F1(0) = 1/Z1. To see this write

(1− x) = x+ 1− 2x = x+d

dx[x(1− x)] (19.33)

and integrate the second term by parts:

Σ′1loop =Q2

0

8π2

∫ 1

0

dx

[x ln

Λ2e−1

λ2(1− x) + x2M2+ x(1− x)

−λ2 − 2M2

λ2(1− x) + x2M2

]=

Q20

8π2

∫ 1

0

dxx

[ln

Λ2

λ2(1− x) + x2M2− 2 +

M2x2 − 2M2(1− x)

λ2(1− x) + x2M2

](19.34)

=1

Z1

− 1− Q20

32π2(19.35)

This is not an fluke. The identity

∂

∂pµ1

m+ γ · p= − 1

m+ γ · pγµ

1

m+ γ · p(19.36)

allowed Ward to prove that

∂Σ

∂pµ= Γµ(p, p)− γµ, Ward Identity (19.37)

To one loop order this result is almost obvious, provided the loop momentum is assignedsuch that the external momentum p is routed to always follow the electron line. Our 1 loopresult disagrees with the identity only by an additive constant, which can be attributed tothe linear divergence of the self-energy loop integral. We shall see that the Ward identitycan be derived from gauge invariance or current conservation. Recall that the photon selfenergy calculation also showed a discrepancy with gauge invariance.


19.4 Ward Identities in QED

19.4.1 Coordinate space derivation

The Ward identities are generally a consequence of gauge invariance. In QED gauge in-variance implies em current conservation ∂µj

µ = 0 as an operator statement in Heisenbergpicture. To obtain the implication of this for amplitudes, we insert the current into the timeordered product of a string of charged fields and take its divergence outside the time orderingsymbol, e.g.:

∂µ⟨0|T [jµ(x)ψ(y)ψ(z)]|0

⟩=

⟨0|T [∂µj

µ(x)ψ(y)ψ(z)]|0⟩

(19.38)

+δ(x0 − y0)⟨0|T [[j0(x), ψ(y)]ψ(z)]|0

⟩+ δ(x0 − z0)

⟨0|T [ψ(y)[j0(x), ψ(z)]]|0

⟩where the terms on the second line come from the time derivatives acting on the time orderingsymbol. Those terms involve equal time commutators of the charge density with local fields:

δ(x0 − y0)[j0(x), ψ(y)] = δ4(x− y)Qψψ(x) (19.39)

For the case of the Dirac electron, Qψ = e and Qψ = −e. So the identity becomes

∂µ⟨0|T [jµ(x)ψ(y)ψ(z)]|0

⟩= eδ(x− y)

⟨0|T [ψ(x)ψ(z)]|0

⟩− eδ(x− z)

⟨0|T [ψ(y)ψ(z)]|0

⟩= eδ(x− y)S ′F (x− z)− eδ(x− z)S ′F (y − x) (19.40)

where S ′F is the exact (all orders) propagator for the dirac field. Fourier transforming bothsides w.r.t. x, y, z gives

iqµ∫d4xeix·q−iy·p

′+iz·p ⟨0|T [jµ(x)ψ(y)ψ(z)]|0⟩

= e[S ′F (p)− S ′F (p′)](2π)4δ(q − p′ + p)

We can recognize the left side as S ′F (p′)eΓν(p′, p)S ′F (p)D′F (p′ − p)ρνΠµρ so we can write the

Ward identity as

iqµΓµ(p′, p) = i(m+ γ · p′ + Σ(p′))− i(m+ γ · p+ Σ(p)) (19.41)

where we used qµΠνµ = 0.

19.4.2 Ward Identities from Diagrams in momentum space

An external photon in a diagram is always attached to a charged line. Replacing the polar-izatioin vector with kµ, the photon momentum should give zero for any scattering amplitude.The attachment vertex γµ is sandwiched between two porpagators so we observe

−im+ γ · (p+ k)

k · γ −im+ γ · p

=−i

m+ γ · (p+ k)(m+ (p+ k) · γ − (m+ γ · k)

−im+ γ · p

= −i( −im+ γ · p

− −im+ γ · (p+ k)

SF (p+ k)k · γSF (p) = −i(SF (p)− SF (p+ k)) (19.42)


The external photon line in a diagram contributing to the proper vertex function attachesto a charged line which could start and end on the external fermion lines or could closeon itself as a fermion loop. In the first case the factors associated with the charged line inkµΓµ(p+ k, p) take the form

SF (p+ k)γµnSF (p+ k +n−1∑j=1

kj)γµn−1 · · · γµlSF (p+ k +

l−1∑j=1

kj)k · γ

SF (p+n−1∑j=1

kj)γµl−1SF (p+

l−2∑j=1

kj) · · · γµ1SF (p) =

−iSF (p+ k)γµnSF (p+ k +n−1∑j=1

kj)γµn−1 · · · γµl [SF (p+

l−1∑j=1

kj)

−SF (p+ k +n−1∑j=1

kj)]γµl−1SF (p+

l−2∑j=1

kj) · · · γµ1SF (p) (19.43)

where the kj are the momenta of various internal photon lines that attach to the chargedline, satisfying

∑nj=1 kj = 0. In this example the external vertex is between l − 1 and l but

one must of course sum over all attachment locations. The right side of the above equation isthe difference of two expressions which could be a charged line contributing to a propagatorexcept that in the first expression the momentum k is added to the momenta of all thepropagators to the left of γµl and the second expression has k added to the momenta of allthe propagators to the left of γµl−1 . When this difference is summed over all attachmentlocations (i.e. the choice of l) these terms will cancel in pairs, leaving uncancelled theexpressions

−i[SF (p)γµnSF (p+n−1∑j=1

kj)γµn−1 · · · γµ2SF (p+ k1)γµ1SF (p)− (19.44)

SF (p+ k)γµnSF (p+ k +n−1∑j=1

kj)γµn−1 · · · γµ2SF (p+ k + k1)γµ1SF (p+ k)]

The first expression in square brackets can be identified as the corresponding charged linein a diagram contributing to the full propagator S ′F (p) whereas the second expression wouldcontribute similarly to S ′(p+ k). If the vertex attaches to a closed fermion loop the expres-sions would be traced and the momentum p woud be integrated. If the change of variablesp→ p+ k is valid the two terms on the right side would cancel, leaving no contribution andthe Ward identity would be established. Anomalies to the Ward identity emerge if that pintegal is linearly divergent or worse.


19.5 Charge and mass renormalization to 1 Loop

We have seen that the physical mass of a charged fermion is in general different from theLagrangian mass m, called the bare mass. In the following we call the physical mass m andthe bare mass m0. A good systematic way to take this into account is to write m0 = m− δmand break the mass term in the Lagrangian into two pieces:

Lm = −mψψ + δmψψ (19.45)

and include the second term in the interaction, which adds a two point vertex to the Feynmanrules. Then as one executes perturbation theory one chooses δm so that to each order thephysical mass stays equal to m.

Suppose we have managed to calculate Σ to all orders. Then the exact propagator canbe written

S ′F =−i

m0 + γ · p+ Σ(p)=

−i(m+ γ · p)(1 + Σ′) +O((m+ γ · p)2)

≡ −iZ2

m+ γ · p+O(1) (19.46)

Z2 is called the electron wave function renormalization, just as we have called Z3 = 1/(1 +Π(0)) the photon wave function renormalization. If we normalize each of the external prop-agators entering the vertex to unity, this means we absorb the Z factors in the definition ofthe coupling constant:

Γµ(p′, p) ≡ Z2

√Z3Γµ(p′, p) (19.47)

→ Z2

√Z3

Z1

γµ, p′ → p (19.48)

Thus the physical charge, as measured, for example, in scattering processes is given by

e = e0Z2

√Z3

Z1

= e0

√Z3 (19.49)

because the Ward identity implies Z1 = Z2. Separately, Z1 and Z2 depend on the nature ofthe charged particle. The importance of the Ward identity is that the charge renormalizationis universal: every charged particle no matter what its properties is renormalized by the samefactor

√Z3.

After renormalization the physical effect of the corrections resides entirely in the newmomentum dependence on q2. At low energy and momenta we can expand in powers of q2.


For instance

F1(q2)− F1(0)

≈ Q20

8π2q2

∫dxdy

[− 2xy + 1− x− yλ2(1− x− y) +m2(x+ y)2

− m2xy((x+ y)2 − 2(1− x− y))

(λ2(1− x− y) +m2(x+ y)2)2

]≈ Q2

0

8π2q2

∫ 1

0

du

∫ u

0

dy

[−2(u− y)y + 1− uλ2(1− u) +m2u2

− m2(u− y)y(u2 − 2(1− u))

(λ2(1− u) +m2u2)2

]≈ Q2

0

8π2q2

∫ 1

0

du

[− u3/3 + u(1− u)

λ2(1− u) +m2u2− m2(u3/6(u2 − 2(1− u))

(λ2(1− u) +m2u2)2

]≈ Q2

0

8π2q2

∫ 1

0

du

[− 1

m2

[u

2− 2

3

]− u

λ2(1− u) +m2u2+

m2u3/3

[λ2(1− u) +m2u2]2

](19.50)

where we have set λ = 0 in the terms with no IR divergence. The remaining λ dependencemay be evaluated as follows:∫ 1

0

duu

λ2(1− u) +m2u2∼

∫ 1

ε

du1

m2u+

∫ ε

0

duu

λ2 +m2u2

∼ 1

m2ln

1

ε+

1

2m2lnm2ε2

λ2= ln

m

λ(19.51)∫ 1

0

dum2u3/3

(λ2(1− u) +m2u2)2∼

∫ 1

ε

du1

3m2u+

∫ ε

0

dum2u3/3

(λ2 +m2u2)2

∼ 1

3m2ln

1

ε+

1

6m2lnε2

λ2−∫ ε2

0

duλ2/6

(λ2 +m2u)2

∼ 1

3m2lnm

λ− λ2

6m2

(1

(λ2)− 1

λ2 +m2ε2

)∼ 1

3m2lnm

λ− 1

6m2(19.52)

Then

F1(q2)− F1(0) ≈ Q20

8π2

q2

m2

[−2

3lnm

λ+

1

4

](19.53)

This infrared divergence is occurring in the correction to elastic electron scattering. It isnoteworthy that the sign is negative. We shall find that if an unobserved low energy photonis also emitted, that inelastic event also has an infrared divergence, necessarily positive,and indeed exactly cancels this contribution to the elastic amplitude! The vertex correctionmodifies the Coulomb potential which binds the hydrogen atom together: it adds a V0δ(r)much as the Vacuum polarization does, Here V0 = O(α2/m2) ln(m/λ), But the relevantinfrared cutoff λ should be of order the typical electron momentum in the atom mα ratherthan the photon mass. This reasoning leads to an order of magnitude estimate shift ofthe s levels +mα5 ln(1/α) ( larger and in the opposite direction from the effect of vacuumpolarization.


19.6 Atomic bound states

QED includes much more than the interactions of electrons and photons. Electrons can bindto nuclei to form atoms which can in turn bind into molecules. In NR quantum mechaicswe use the Schrodinger equation to describe this physics. So we must see how this works inthe context of QED. We already know that the Dirac equation in the presence of a Coulombpotential gives a very good description of hydrogen. In this calculation we treat the Diracwave function as a (first quantized) Schrodinger wave function. In QED we have instead theDirac field operator ψ. Using Feynman diagrams we can study the interaction of an electronwith a very heavy positively charged nucleus. If we imagine the limit of infinite mass, it seemsintuitive that the effect of the nucleus could be replaced by the static Coulomb potentialcentered at the nucleus location. One can confirm this explicitly by studying Feynmandiagrams with one electron line one heavy nucleus line and the excchange of any number ofphotons. Here one can in first instance neglect vacuum po;arization and radiative corrections.But those can be put back in perturbatively and a consistent description is attained.

In this context it is very convenient to set up a new interation picture (“bound stateinteraction picture”) in which H0 is the Hamiltonian of free photons and the Dirac fieldequation including the Coulomb potential of the heavy nucleus as an external field. Inthis case the creation operators create an electron (or positron) in a Coulomb eigenstate,either a discrete bound state or an unbound state in the continuum. Then the spectrum ofH0 includes both bound and unbound electrons as well as free photons. Time dependentperturbation theory then puts back in radiative effects, order by order, including the physicsof the Lamb shift.

Unlike the NR Schrodinger equation the Dirac equation doesn’t have a useful multi-body generalization. This makes an elegant discussion of positronium. charmonium, etca challenge, because both particles have the same mass–far from one being very heavycompared to the other. One approach is to identify the bound states as poles in a partialsummation of Feynman diagrams–the Bethe-Salpeter equation.


Chapter 20

Soft Bremsstrahlung and Infrareddivergences

20.1 One Soft Photon

Let us consider the emission of an extra photon in the scattering of an electron in an externalpotential. To regulate infrared divergences we temporarily introduce a small photon massλ. To lowest order in the external potential the diagrams are:

p’+k

p

p’ k p’

p-k

p

k

Writing the Feynman amplitude for this process M = Q0MνAν(p′ + k − p), we have

Mν = iQ0u′[ε∗ · γm− γ · (p

′ + k)

m2 + (p′ + k)2γν + γν

m− γ · (p− k)

m2 + (p− k)2ε∗ · γ

]u (20.1)

= iQ0u′[

2ε∗ · p′ − ε∗ · γγ · k2p′ · k − λ2

γν + γν2ε∗ · p+ γ · kε∗ · γ)

−2p · k − λ2

]u. (20.2)

In this chapter we are mainly interested in the case of soft photon emission, since thatis intimately involved with the cancellation of infrared divergences in physical processes. Inthe limit of very small k, λ, the terms involving three gamma matrices are negligible andMν becomes a numerical momentum dependent factor

Q0

[2ε∗ · p′

2p′ · k − λ2− 2ε∗ · p

2p · k + λ2

]≡ Q0ε

∗ · J (20.3)

times the lowest order elastic scattering amplitude. If the soft photon is unobserved andits energy is smaller than the energy resolution ∆, one must include the contribution of

189

the emission when calculating the cross section for electron scattering. In this case wemust obviously sum over polarization and integrate over photon angles. The soft photoncontribution to the differential cross section is then approximately given by

dσ

dΩ1 soft photon≈ dσ

dΩElasticQ2

0

∫|k|≤∆

d3k

(2π)32ω

(J · J +

(kµ · J)2

λ2

), (20.4)

where we used the identity ∑Pol

εµε∗ν = ηµν +

kµkνλ2

(20.5)

to evaluate the sum over polarizations.Clearly J · J = O(1/k2) for small k. On the other hand, k · J = O(λ2/k), so the second

term in the integrand is a factor of λ2 smaller than the first, and can be dropped. Moreover,the smallest in magnitude k · p or k · p′ can be is mλ so the λ2 in the denominators of theexpression for J can be dropped, leading to

J · J ≈ − 2p′ · pp′ · kp · k

− m2

(p′ · k)2− m2

(p · k)2. (20.6)

Since the last two terms are special cases of the first, we only need to evaluate one integral:

K ≡∫|k|≤∆

d3k

(2π)32ω

1

p′ · kp · k. (20.7)

To extract the divergent contribution of this integral, first scale |k| → λ|q|. In the newvariables |q| ≤ ∆/λ. The divergence as λ → 0 then comes from the integration regionq0 ≤ |q| ≤ ∆/λ, where q0 >> 1 is fixed. For this region the q dependence is simply 1/q3 sothe q integral just gives a factor ln(∆/q0λ). Thus we have

K = Finite + ln(∆/λ)1

16π3

∫dΩk

1

(p′ · k− E ′)(p · k− E). (20.8)

Define

I(p′, p) ≡∫dΩk

1

(p′ · k− E ′)(p · k− E). (20.9)

Then

dσ

dΩ1 soft photon≈ Finite

+ ln(∆/λ)dσ

dΩElastic

Q20

16π3[−2p′ · pI(p′, p)−m2I(p′, p′)−m2I(p, p)]. (20.10)


The integral defining I can be simplified by combining the denominators with the Feynmantrick, and then choosing the z-axis for the polar angles to be parallel to the vector xp′ +(1− x)p. Then I reduces to

I(p′, p) = 4π

∫ 1

0

dx1

(xE ′ + (1− x)E)2 − (xp′ + (1− x)p)2. (20.11)

The denominator is the Lorentz scalar −(xp′ + (1 − x)p)2 which evaluates to (x2 + (1 −x)2)m2 − 2x(1− x)p′ · p. Using q2 ≡ (p′ − p)2 = −2m2 − 2p′ · p, we then obtain

I(p′, p) = 4π

∫ 1

0

dx1

m2 + x(1− x)q2, (20.12)

from which we see that I depends only on q2. Taking account of this information, we canwrite

dσ

dΩ 1 softphoton

≈ Finite + ln(∆/λ)dσ

dΩElastic

α

4π2[q2I(q2) + 2m2(I(q2)− I(0))]. (20.13)

What we have computed is the contribution to the scattering cross section for electronscattering with the emission of an unobservable soft photon. Adding this to the contributionwith no photon emission gives to this order the total electron scattering cross section

dσ

dΩ≈ α× Finite +

dσ

dΩElastic

×(

1 + ln(∆/λ)α

4π2[q2I(q2) + 2m2(I(q2)− I(0))]

). (20.14)

The unobserved photon emission contribution seems to make this cross section divergefor λ → 0. However, in this limit the elastic cross section is not directly measurable, sincesoft photon emission can not even in principle be experimentally vetoed. It is only a physicalquantity (in principle) for a non zero photon mass. We shall find that the one loop radiativecorrection to the elastic amplitude has a divergence as λ → 0 which precisely cancels theone in the above formula.

To see this cancellation, let’s examine the one loop vertex function in the presence of afinite photon mass.

Γµ(p′, p)− γµ =

−iQ20

∫d4k

(2π)4

γρ(m− (p′ − k) · γ)γµ(m− (p− k) · γ)γρ

(k2 + λ2 − iε)(k2 − 2k · p− iε)(k2 − 2k · p′ − iε). (20.15)

We are interested in the infrared divergence in this expression when λ→ 0. Since we want topresent the result in a form suitable for comparison to the photon emission amplitude, it isappropriate to first evaluate the k0 integration by contours. Examination of the denominatorsreveals six simple poles at the following values of k0:

±(√

k2 + λ2 − iε) (20.16)

E ±√E2 + k2 − 2k · p− iε E ′ ±

√E ′2 + k2 − 2k · p′ − iε.


Three poles are in the upper half plane and three in the lower half plane. We can close thecontour up or down, but the best choice is down in the lower half plane. The reason is thatonly one of the poles in the lower plane is near the origin for small k, the region responsiblefor the infrared divergence. The other two poles are far from the origin at small k and itis simple to see that their residues do not contribute to the divergence. Thus closing in thelower half plane we only need to consider the residue of the pole at k0 = +

√k2 + λ2 − iε.

Γµ(p′, p)− γµ ≈ Q20

∫d3k

(2π)32ω

γρ(m− p′ · γ)γµ(m− p · γ)γρ

(−λ2 − 2k · p)(−λ2 − 2k · p′), (20.17)

where we have dropped terms linear in k in the numerator, since they won’t contribute to theinfrared divergence. When the vertex is sandwiched between on-shell spinors, the numeratorsimplifies to 4p′ · p by moving γ · p′ to the left and γ · p to the right. To extract the divergentpart as λ → 0 we compute the integral over the region |k| ≤ k0 for some fixed k0. Thisintegral is identical to the one encountered in the soft photon calculation, so by comparison,we find

Γµ(p′, p) ≈ γµ(

1 +Q2

0

2(2π)3p · p′ ln k0

λI(q2)

). (20.18)

Finally, we have to separate from the vertex correction the part that should be absorbedinto 1/Z1. This is just the value of the correction at p′ = p, so we write (Recall 2p · p′ =−q2 − 2m2.)

Γµ(p′, p) =1

Z1

γµ(

1− α

8π2lnk0

λ[q2I(q2) + 2m2(I(q2)− I(0))]

)+α× Finite +O(α2). (20.19)

Of course, defining Z1 this way makes it depend on the infrared cutoff. In any regulationscheme respecting the Ward Identity, it will turn out that Z1 = Z2 so that charge renormal-ization will not be infrared sensitive, even though Z1, Z2 separately are. The elastic electroncross section involves the square of Γ, and since the infrared divergence occurs at order α, itis correct to this order to extract the divergent λ dependence of the elastic differential crosssection as an overall factor:

dσ

dΩElastic=

dσ

dΩ IR Finite

((1− α

8π2lnk0

λ[q2I(q2) + 2m2(I(q2)− I(0))]

)2

+O(α2)

=dσ

dΩ IR Finite

((1− α

4π2lnk0

λ[q2I(q2) + 2m2(I(q2)− I(0))]

)+O(α2). (20.20)

Inserting this information into the expression for the electron scattering cross section, we seethat the sensitivity to the small photon mass disappears:

dσ

dΩ≈ dσ

dΩ IR Finite

(1 + ln(∆/k0)

α

4π2[q2I(q2) + 2m2(I(q2)− I(0))]

)+α× Finite. (20.21)


Notice, however that the physical effect of soft photons remains, signaled by the unavoid-able sensitivity of the electron scattering cross section to the experimental energy resolution∆. There is no “infrared catastrophe,” i.e. the theory gives perfectly finite predictions forall physical measurements.


Chapter 21

Nonabelian Gauge Theory

In the mid 1950’s Yang and Mills invented an analogue of QED which replaced QED’s localphase symmetry, ψ(x) → eiQΛ(x)ψ(x), with a local SU(2) symmetry. Later it was realizedthat SU(2) could be any continuous group. To motivate the construction, first recall howthe local phase symmetry works. Terms in L with no derivatives, like the mass term ψψ areobviously invariance under an x dependent phase change. However terms with derivativesare not invariant:

ψγ · ∂ψ → ψγ · (∂ + iQ∂µΛ(x))ψ (21.1)

This motivates the introduction of a vector potential Aµ(x) via ∂µ → (∂µ − iQAµ) whichundergoes the gauge transformation Aµ → Aµ + ∂µΛ so that

ψγ · (∂ − iQA)ψ → ψγ · (∂ − iQA)ψ (21.2)

is invariant. To make the new gauge field dynamical we need gauge invariant terms con-taining derivatives of A. A simple derivative ∂µAν → ∂µAν + ∂µ∂νΛ is not invariant butthe antisymmetrized derivative Fµν = ∂µAν − ∂νAµ is invariant. Thus the QED Lagrangedensity

L = −1

4FµνF

µν − ψ (m− iγ · (∂ − iQA))ψ (21.3)

is properly gauge invariant. Notice that the commutator of the covariant derivatives Dµ =∂µ − iQAµ is proportional to Fµν : [Dµ, Dν ] = −iQFµν .

To achieve the Yang-Mills generalization, we consider an internal multiplet of fields putinto a column vector transforming under some unitary matrix representation of a continuoussymmetry group ψ → Ω(x)ψ. Ω(x) ∈ G. Continuous groups are mostly determined byinfinitesimal transformations I − iθaTa where Ta satisfy a Lie algebra

[Ta, Tb] = ifabcTc, fabc = −fbac, (21.4)

where fabc are the (real) structure constants of the group. For our discussion the Taare taken to be Hermitian matrices representing the Lie algebra, and the Lie “product”

195

[A,B] = AB−BA is just the commutator. The Hermiticity condition guarantees that finitegroup transformations e−iTaθ

aare unitary. The inner product of two elements of the Lie

algebra 〈A|B〉 = Tr(AB) is invariant under A,B → Ω†(A,B)Ω and it is convenient to takethe basis of generators orthonormal under this metric TrTaTb = δab/2. For such a basisifabc = 2Tr[Ta, Tb]Tc which is antisymmetric under the interchange of any pair of a, b, c. (Ina nonorthonormal basis the antisymmetry is only under a↔ b.)

Returning to the local gauge transformation of ψ, its derivative has the transform

∂µψ → Ω(x)(∂µ + Ω−1∂µΩ)ψ (21.5)

Now Ω−1∂µΩ is in the Lie algebra of G, generated by Ta, a = 1, ..., n. Next we introduce agauge field Aµ(x) =

∑a TaA

aµ(x) in the Lie algebra. Then the transform

Aµ → Ω(x)AµΩ−1(x) +1

ig(∂µΩ)Ω−1 (21.6)

makes ∆µψ ≡ (∂µ − igAµ)ψ transform like ψ:

Dµψ → Ω(x)Dµψ (21.7)

so that, for example, ψDµγµψ is invariant.

To construct gauge covariant derivatives of Aµ itself, we examine the commutator

[Dµ, Dν ] = −ig∂µAν + ig∂νAµ + (−ig)2[Aµ, Aν ] ≡ −igFµν (21.8)

noting that it produces a field strength Fµν which transforms as

Fµν → Ω(x)FµνΩ−1 (21.9)

so that the trace of any matrix product of the components Fµν is gauge invariant. In termsof the gauge field components Aaµ we have

Fµν =∑a

Ta(∂µAaν − ∂νAaµ)− ig

∑b,c

[TbTc]AbµA

cν

=∑a

Ta(∂µAaν − ∂νAaµ)− ig

∑b,c

ifabcTaAbµA

cν (21.10)

F aµν = ∂µA

aν − ∂νAaµ + gfabcA

bµA

cν (21.11)

The simplest gauge invariant Lagrangian only has the Aaµ as dynamical variables and isconventionally written, in analogy with QED, as

L = −1

4

∑a

F aµνF

aµν = −1

2TrFµνF

µν (21.12)

It is interesting that this simplest Lagrangian necessarily includes cubic and quartic termsin the vector potentials: the gauge particles interact with each other, not like photons!


To this simplest system one can add additional “matter” fields transforming under variousrepresentations R of the gauge group. For them the covariant derivative is DR = ∂ − igAR,where ARµ ≡

∑aA

aµT

Ra . Possible gauge invariant terms include

ψψ, ψiDRµ γ

µψ, φ†φ, (DRµ φ)†DRµφ, (φ†φ)2 (21.13)

and so on. Here TRa are the generators of the Lie algebra in the chosen representation.Note that once we normalize the defining generators Ta according to TrTaTb = δab/2, Thenormalization for other representations is determined TrTRa T

Rb ≡ T (R)δab. For the defining

representation T (R) = 1/2.

21.1 Some gauge groups and their representations

21.1.1 The Adjoint Representation

For any Lie group one can form a matrix representation from the structure constants. Thisfollows from the Jacobi identity for commutators:

0 = [Ta, [Tb, Tc]] + [Tc, [Ta, Tb]] + [Tb, [Tc, Ta]]

= −fbcdfadeTe − fabdfcdeTe − fcadfbdeT e

0 = fbcdfdea + fabdfdec + fcadfdeb (21.14)

Now consider the matrices M bkl = ifkbl and write out their commutators

[Ma,M b]kl = MaknM

bnl −M b

knManl

= −(fkanfnbl − fkbnfnal) = flknfnba = ifabnMnkl (21.15)

showing that Ta = Ma represents the Lie algebra of the given group. This universal rep-resentation is called the adjoint representation. The covariant derivative in the adjointrepresentation is

DµVa = ∂µV

a − igM bacA

bµV

c = ∂µVa + gfabcA

bµV

c (21.16)

For instance the covariant derivative on the field strength tensor is DµFaρσ = ∂µF

aρσ +

gfabcAbµF

cρσ. If we make a matrix V ≡

∑a V

a, then

DµV = ∂µV + g∑a

T afabcAbµV

c = ∂µV + g∑a

(−i[T b, T c]AbµV c

= ∂µV − ig[Aµ, V ] (21.17)

21.1.2 Examples

A gauge field theory can be constructed based on any continuous group. The original Yang-Mills work chose SU(2) which every physicist knows well, because it gives the action of the


rotation group on spin 1/2 particles. When SU(2) is regarded as an internal symmetry groupunrelated to rotations we call the Pauli matrix generators τ/2 rather than σ/2:

τ1 =

(0 11 0

), τ2 =

(0 −ii 0

), τ3 =

(1 00 −1

)[τa

2,τb2

]= iεabc

τ c

2(21.18)

so the structure constants of SU(2) are just εabc.

In general the group SU(N) is defined as the set of unitary N × N matrices with unitdeterminant. The corresponding Lie algebra is the set of N×N hermitian traceless matrices.The complete classification of continuous Lie groups includes the infinite sequences SU(N),SO(2N), SO(2N+1), and Sp(2N), together with five exceptional groups, G2, F4, E6, E7, E8.The dimension of a group is the number of infinitesimal generators. The rank of a group isthe maximal subset of mutually commuting generators. The subscripts on the exceptionalgroup names is the rank of that group. So SU(N) has dimension N2 − 1 and rank N − 1.SO(N) has dimension N(N − 1)/2, but SO(2N) and SO(2N + 1) both have rank N .

The standard model gauge group is SU(3)×SU(2)×U(1) Here SU(3), the gauge groupof the strong interactions, is the only one that may be new to you. The three factor groups,which are all simple groups, act independently and each have an independent coupling. Thewhole gauge group is not simple. Larger (simple;e) groups play a role in unification ideas,but will not figure in the physics of the standard model.

The defining generators of SU(3) can be taken to be Ta = λa/2, where λa are the Gell-Mann matrices

λk =

(τa 00 0

), a− 1, 2, 3, λ4 + iλ5 =

0 0 20 0 00 0 0

λ6 + iλ7 =

0 0 00 0 20 0 0

, λ8 =1√3

1 0 00 1 00 0 −2

, (21.19)

In addition to gauge fields a gauge theory can possess several fields transforming in variousrepresentations of the gauge group. For example the defining representation of SU(N), alsocalled the fundamental representation, is N dimensional, requiring N field components.Starting with any representation of a group Ta(R) satisfying [Ta(R), Tb(R)] = ifabxTc(R),−T ∗a (R) is also a representation, not necessarily a new one. For example if Ta(R)∗ = −Ta(R)or if U †Ta(R)U satisfies this condition the representation is called real. It is called pseudoreal if −T ∗a (R) = V †Ta(R)V . otherwise it is called complex. For SU(2) the fundamentalrepresentation is pseudoreal, but the fundamental representation of SU(N) is complex forN > 2. We call the fundamental irrep of SU(N) N and the complex conjugate representationN .


21.2 Path integrals for gauge theories and gauge fixing

Recall the path integral formula for time ordered products:

〈Φ〉 =

∫DAaµD(Fields) Φei

∫d4x(−FaµνFaµν/4+Lfields∫

DAaµD(Fields)ei∫d4x(−FµνFµν/4+Lfields

.

Here Φ just represents some string of gauge invariant operators. As we have already discussedfor QED the gauge invariance of the functional integrand means it is independent of one ofthe gauge field components, so both the numerator and denominator contain an infinite factorof the gauge group volume for each space time point. The factor formally cancels betweennumerator and denominator, but to make the cancellation more rigorous, it is best to fix thegauge by imposing a constraint on the gauge fields. One has a lot of flexibility in choosingthis constraint, but in our work we will use a generalized Lorentz gauge ∂ ·Aa(x)−fa(x) = 0.

Fadeev and Popov introduce such a constraint through the following device. Call thegauge transformed gauge field AΩ and define ∆FP [A] to satisfy

∆FP (A)

∫dΩ(x)δ(∂ · AaΩ(x)− fa(x)) = 1 (21.20)

The integration over the gauge group is of course a functional integral, not an ordinary one,and it is assumed to be invariant under the change of variables Ω(x) → U(x)Ω(x). Thisimplies that ∆FP is gauge invariant. Next insert the right side of this expression in the pathintegrals in both the numerator and denominator. Change variables in the gauge functionalintegral A→ AΩ−1 and use the gauge invariance of measure and observables to get

〈Φ〉 =∫dΩ(x)

∫DAaµD(Fields) Φ∆FP [A]δ(∂ · Aa(x)− fa(x))ei

∫d4x(−FaµνFaµν/4+Lfields∫

dΩ(x)∫DAaµD(Fields)∆FP [A]δ(∂ · Aa(x)− fa(x))ei

∫d4x(−FµνFµν/4+Lfields

.

The volume of the gauge group can now be cancelled, leaving a formally convergent integralover the gauge fields. The presence of the gauge fixing delta function is awkward in practicalcalculations. It can be finessed by noting that both numerator and denominator by gaugeinvariance should be independent of fa(x). Therefore we are free to multiply numeratorand denominator by e−i

∫d4xfa2/(2ξ) and integrate both over fa. When this is done the gauge

fixing delta function sets fa = ∂ · Aa inside the functional integral, leading to

〈Φ〉 =

∫DAaµD(Fields) Φ∆FP [A]ei

∫d4x(−FaµνFaµν/4−(∂·Aa)2/(2ξ)+Lfields∫

DAaµD(Fields)∆FP [A]ei∫d4x(−FµνFµν/4−(∂·Aa)2/(2α)+Lfields

.

21.2.1 The Fadeev-Popov Determinant

WE would like to put ∆FP [A] in a form suitable for practical calculations. We first notethat since it multiplies the gauge fixing delta function we only need it for A satisfying the


gauge condition. For A satisfying the gauge condition the integral over Ω(x) need onlyinclude infinitesimal gauge transformations Ω = I − iG AG = A − i[G,A] − g−1∂G orAa → Aa + fabcG

bAc − g−1∂Ga∫dΩδ(∂ · Aa − fa) =

∫dGδ(∂ · (−g−1(δab∂ − gfabcAc)Gb)

= det−1[−∂ · (δab∂ − gfabcAc) (21.21)

Then we infer

∆FP [A] = det[−∂ · (δab∂ − gfabcAc) (21.22)

Now we can represent the determinant as a Grassmann functional integral over two differentGrassmann variables;

det[−∂ · (δab∂ − gfabcAc) =

∫dBadCae−i

∫d4x∂Ba(x)·(∂Ca+gfabcA

bCc (21.23)

We note that DµCa = ∂µC

a + gfabcAbµC

c is just the covariant derivative of a gauge field inthe adjoint representation.

21.3 Feynman Rules for a Nonabelian Gauge Theory

As always the propagators are determined by the quadratic terms in the Lagrangian:

L0 = −1

2∂µA

aν(∂µA

aν − ∂νAaµ)− 1

2ξ(∂ · Aa)2 − ∂Ba(x) · ∂Ca

= −1

2∂µA

aν(∂µA

aν − ∂νAaµ)− 1− ξ

2ξ(∂ · Aa)2 − ∂Ba(x) · ∂Ca (21.24)

The gauge field propagator is δabDµν in momentum space where

(p2ηµν +1− ξξ

pµpν)Dνλ = −iηλµ. (21.25)

To solve put Dνλ = A(p2)ηνλ +B(p2)pνpλ and plug in

Aηµλp2 +Bp2pµpλ +

1− ξξ

pµ(A+Bp2)pλ = −iηµλ

A =−i

p2 − iε, (A+Bp2)(1− ξ) + ξBp2 = 0

⟨AaµA

bν

⟩ξ

= −iδabηµν + (ξ − 1)pµpν/p

2

p2 − iε(21.26)

The ghost propagator is easily read off:⟨CaBb

⟩=−iδabp2 − iε

(21.27)


The Feynman gauge is ξ = 1 and the Lorentz or Landau gauge is ξ = 0.The vertices are obtained from the cubic and quartic terms in L

L3A = −gfabc(∂µAaν − ∂νAaµ)AbµAcν (21.28)

From which we infer the cubic gauge vertex⟨Adκ(k1)Aeρ(k2)Afσ(k3)

⟩= −gfdef (k1ρηκσ − k1σηκρ + k2σηρκ − k2κηρσ

+k3κηρσ − k3ρηκσ)

= −gfdef (k1 − k3)ρηκσ + (k2 − k1)σηρκ

+(k3 − k2)κηρσ) (21.29)

The cubic ghost term is

L3G = −gfabc∂µBAbµCc (21.30)

which implies the vertex ⟨Cd(k1)Aeρ(k2)Bf (k3)

⟩= −gfdefk1ρ (21.31)

We must remember to treat the ghosts as fermions, meaning a factor of −1 for each ghostloop. Finally the quartic gauge term is

L4 = −g2

4fabcfadeA

bµA

cνA

dµAeν (21.32)

and the corresponding vertex is⟨AdκA

eλA

fρA

gσ

⟩= −ig2[fadefafg(ηκρηλσ − ηκσηλρ)

+fadffaeg(ηκληρσ − ηκσηλρ)+fadgfaef (ηκληρσ − ηκρηλσ)] (21.33)

Additional Dirac fermions in some representation of the group would add the vertex igTRaγµ.

21.4 Gauge Invariant Regulation Procedures

For practical calculations in quantum gauge field theories, it is sufficient to establish a sys-tematic procedure for regulating divergences compatible with gauge invariance. Such a pro-cedure will automatically supply the polynomial modifications needed for gauge invariance.We mention briefly the more popular regulators.Pauli-Villars Method. This method is particularly suited to Abelian gauge theories sinceit relies on the fact that the violations in gauge invariance are independent of the mass of thecharged fields. Thus if we introduce extra charged fields of large mass Mi which contribute todivergent diagrams with negative signs we can adjust the coefficients of their contributions


to render the loop integrals finite. Then the gauge violating pieces of each contribution willcancel, and the regulated calculation will be gauge invariant. One then lets Mi →∞ at theend of the calculation. This may still leave uv divergences, but only those compatible withgauge invariance as we found in the vacuum polarization calculation.Dimensional Regularization. The idea here is that the severity of uv divergences dependson the space-time dimension. In particular, the nature of uv gauge invariance violations isdifferent in each dimension. Thus if we can carry out the calculation in a way that appliesto general dimension, the violations of gauge invariance must disappear. This method willobviously not work in theories that can be defined only in particular dimensions. For examplethe alternating symbol εµ1···µD has a different tensor structure in each dimension. Thisloophole allows gauge invariance anomalies to creep in. A theory in which it appears in afundamental way can not be formulated in a general dimension.

To illustrate how dimensional regularization is used, we reconsider the vacuum polar-ization calculation. Let us first make some general remarks. In D dimensions the chargehas units [mass]2−D/2. Thus it is convenient to introduce a mass parameter µ to define adimensionless coupling qI in general D via

q2 ≡ q2Iµ

4−D. (21.34)

The identities for calculating traces of products of gamma matrices carry over to D dimen-sions except that the constant 4 = TrI is replaced by 2D/2. The Feynman trick for handlingdenominators of propagators is unchanged in general dimension, and the shift of integrationvariables to make the denominators depend only on the squares of the loop momenta is stillapplicable. But then averaging over directions of the loop momenta gives a D dependentfactor: For example 〈pµpν〉angles = ηµνp2/D. After taking all this into account, the vacuumpolarization calculation for general D, becomes

T µνD (k) = −2D/2q2Iµ

4−D

(2π)D

∫ 1

0

dx

∫dDp (21.35)

ηµν [p2((2/D)− 1)−m2 + x(1− x)k2]− 2x(1− x)kµkν

[m2 + p2 + x(1− x)k2]2(21.36)

Clearly we need to be able to do the integral∫dDppm

[p2 + A2]2= AD+m−4 2πD/2

Γ(D/2)

∫ ∞0

pD+m−1dp

[p2 + 1]2(21.37)

= πD/2AD+m−4 Γ((D +m)/2)Γ(2− (D +m)/2)

Γ(D/2)(21.38)

where we used the identity∫ ∞0

pm+D−1dp

[p2 + 1]2=

1

2

∫ 1

0

dxx1−(D+m)/2(1− x)((D+m)/2)−1 (21.39)

=Γ(2− (D +m)/2)Γ((D +m)/2)

2Γ(2). (21.40)


Note that the quantity ΩD ≡ 2πD/2/Γ(D/2) is just the value of the integral over all anglesin D dimensions. The following table lists ΩD for the 2 ≤ D ≤ 8.

Angular Integral for Various DimensionsD 2 3 4 5 6 7 8ΩD 2π 4π 2π2 8π2/3 π3 16π3/15 π4/3

Using these results to do the integrals, we obtain

T µνD (k) =q2I

2π2Γ(2−D/2)

∫ 1

0

dxx(1− x)(kµkν − k2ηµν)

[m2 + x(1− x)k2

2πµ2

](D−4)/2

.

As advertised the result is gauge invariant. The answer is finite as long as D < 4, but we seethat the gamma function has a pole as D → 4. This is how divergences appear in dimensionalregularization. To regain the result for 4 dimensions we have to write A(D−4)/2 ≈ 1+ D−4

2lnA

and the second term must be retained since Γ(2−D/2)(D − 4)/2→ −1 as D → 4:

T µνD (k) → q2I

2π2(kµkν − k2ηµν)

∫ 1

0

dxx(1− x)

[Γ(2−D/2)− ln

m2 + x(1− x)k2

2πµ2

].

The pole at D = 4 represents an infinity which has after renormalization the same fate asthe cutoff dependence did in our earlier calculation, namely it disappears after expressingmeasurable results in terms of measured parameters.

Since the pole at D = 4 corresponds to the logarithmic divergence of a direct cutoffprocedure, it is useful to establish the relation between the residue of the pole and thecoefficient of ln(Λ2). This follows from the simple integral∫ Λ

µ

dppD−5 =ΛD−4 − µD−4

D − 4(21.41)

∼

− 1D−4− lnµ Λ→∞, D < 4

ln Λµ

D → 4,Λ fixed,(21.42)

from which we see that the coefficient of ln(Λ2) is −(residue of pole)/2. This is of course inagreement with our two calculations of vacuum polarization.

21.4.1 Minimal subtraction

For practical calculations, dimensional regularization provides a systematic way to specifythe separation of the divergent parts which are absorbed into renormalization of parametersand the finite parts which characterize the physics.

The divergences in the integral of any Feynman diagram take the form of multi orderpoles at D = 4. Generally they can take the form of a sum

∞∑n=1

An(gI)

(4−D)n+ Finite (21.43)


where we can assume that the coefficients An(gI) are independent of D. Then the separationis unique. (Any coefficient An that depends on D can be Taylor expanded about D = 4modifying the coefficients for smaller n and the finite part, dropping the terms of order(D − 4)n+1.) The minimal subtraction prescription (MS) is to choose the countertermsto precisely cancel the multi-order poles and nothing more. In our example of vacuumpolarization the pole at D = 4 resides in

Γ(2−D/2) =2

4−DΓ(3−D/2)

∼ 2

4−D(Γ(1) + (2−D/2)Γ′(1) +O((D − 4)2)

)∼ 2

4−D− γ +O((D − 4)) (21.44)

Here γ ≡ −Γ′(1) ≈ 0.577 is the Euler-Mascheroni constant. For example the vacuumpolarization result is then

T µνD (k) (21.45)

→ q2I


∫ 1

0

dxx(1− x)

[2

4−D− γ − ln

m2 + x(1− x)k2

2πµ2

].

→ q2I


[1

6

2

4−D−∫ 1

0

dxx(1− x) lnm2 + x(1− x)k2

2πµ2e−γ

].

A modified version of MS called MS defines the finite part by scaling µ→ µeγ/(4π) so that

T µνD (k) (21.46)

→ q2I


[1

6

2

4−D−∫ 1

0

dxx(1− x) lnm2 + x(1− x)k2

µ2/2

].

In other words the relation between g and gI is

g2 = g2I

[µ2eγ

4π

]2−D/2

, MS (21.47)

Minimal subtraction in dimensional regularization is especially advantageous in theories withno mass scale. For example, the vacuum polarization with zero electron mass becomes

T µνD (k) → q2I

2π2(kµkν − k2ηµν)[

1

6

2

4−D−∫ 1

0

dxx(1− x) lnx(1− x)k2

µ2/2

]. (21.48)

Here we see a a characteristic feature of apparently scale invariant field theories:the scaleinvariance is broken by the UV divergences, which allows a nontrivial dependence on the


momenta of the process. The argument of the logarithm is made dimensionless with theunphysical scale µ, which cannot enter any physical quantity, The way this works is that therenormalized coupling gI is regarded as dependent on µ in just such a way that the physicalquantity is independent of µ. What this means is that before renormalization the theory hadonly one free parameter, the bare coupling. After renormalization the coupling is not reallyfree, A scale is determined by say the momentum at which the coupling is unity. That scalecancels out of all mass ratios, so in effect the theory has no free parameters.

21.5 UV divergences in nonabelian gauge theory

In this section we examine the structure of UV divergences in nonabelian gauge theories. Wefirst remark that the superficial degree of divergence of a given diagram is essentially thatof scalar electrodynamics: the cubic gauge particle vertex as well as the cubic ghost vertexhas one derivative and the quartic vertex has none. Therefore

D = D − D − 2

2EB +

D − 4

2(2V0 + V1)

→ 4− EB, D → 4 (21.49)

The most striking difference between nonabelian gauge theory and QED is that the cou-pling renormalization has the opposite sign to vacuum polarization in QED. The latter signcorresponds to a screening of the bare charge: the charge grows weaker at longer distances. Ifthe effect has the opposite sign it corresponds to antiscreening: the coupling grows larger atlonger distances. Conversely, in QED the charge gets larger at shorter distances whereas innonabelian gauge theory it weakens at shorter distances. These opposite effects are evidentin one loop corrections. However the calculations which reveal it are quite lengthy.

It is somewhat simpler to renormalize coupling of the gauge field to a Dirac fermion inthe fundamental representation of the gauge group than that of the self interaction of thegauge field. The relevant diagrams are shown at the end of this section.

The diagrams in the first row involve the same loop integrals as the corresponding ones inQED. To adapt them to nonabelian theory one replaces the QED charge with the generatorsTa of the gauge group. For example the fermion self energy bubble replaces e2 in QED byg2TaδabTb = g2

∑a T

2a = g2CF I the Casimir operator for the fundamental representation.

Remembering our normalization TrTaTb = δab/2 for the fundamental of SU(N) shows thatCF = (N2 − 1)/(2N). For the QED-like vertex diagrams the gauge group factors are∑

a

TaTbTa = CFTb + ifabcTcTa = CFTb +1

2ifabcifcadTd =

(CF −

CA2

)Tb. (21.50)

For SU(N) one can regard the adjoint plus a singlet as the tensor product of an N and anN . Then

(TNa + T Na )2 = 2CF + 2∑a

TNa TNa = CAPA + CSPS = CAPA (21.51)


Taking the trace of both sides leads to

2N2CF = CA(N2 − 1), CA = N. (21.52)

Apart from these color factors the loop integrals are the same as QED. If we calculate inLandau gauge ξ = 0 these two QED like diagrams do not contribute tho the UV renor-malization of the coupling. (Exercise). The fermion contribution to the gauge particle selfenergy is obtained from the QED calculation by replacing e2 with g2TrTaTb = δab/2.

In Landau gauge the coupling renormalization is determined by the three diagrams onthe second line. For now we focus on isolating the coefficient of (4 −D)−1 or alternativelythe coefficient of ln(Λ/µ) in these diagrams. We do them in reverse order.

21.5.1 Fermion Gluon Vertex

The value of the last diagram is, with all momenta outgoing and in Landau gauge,

(−i)3(ig)2gfabcTaTc

∫d4p

(2π)4

γρ(−(p+ k) · γγσ

k2(q + k)2(p+ k)2

(ηρρ′ −

(k + q)ρ(k + q)ρ′

(k + q)2

)(ησσ′ −

kσkσ′

k2

)(ησ

′

µ (k − q)ρ′ + ηρ′

µ (2q + k)σ′ − ηρ′σ′(q + 2k)µ) (21.53)

To capture the log divergence, we only need the leading term as k →∞:

(−i)3(ig)2gfabcTbTc

∫d4k

(2π)4

γρ(−k · γγσ

k6

(ηρρ′ −

kρkρ′

k2

)(ησσ′ −

kσkσ′

k2

)(ησ

′

µ kρ′ + ηρ

′

µ kσ′ − ηρ′σ′2kµ)

= (−i)3(ig)2gfabcTbTc

∫d4k

(2π)4

2kµγρk · γγσ

k6

(ηρσ −

kρkσ

k2

)= −ig3fabcTbTc

∫d4k

(2π)4

6kµk · γk6

=3g3

16π2fabcTbTcγµ ln

Λ

µ(21.54)

Now note

fabcTbTc =1

2fabc[Tb, Tc] = −i1

2fabcfcbdT

d =1

2i(TAb T

Ab )cdTd =

iCA2Ta (21.55)

so the last diagram on the second line has a log divergence

3g2CA32π2

igTaγµ lnΛ

µ(21.56)


21.5.2 FP ghost loop

Next we consider the second diagram on the first line, the contribution of the FP ghosts:

−(−i)2g2febdfdae

∫d4k

(2π)4

∫ 1

0

dxkµkν − x(1− x)qµqν[k2 + x(1− x)q2]2

∼ g2fbedfdea

∫d4k

(2π)4

∫ 1

0

dxk2ηµν/4− x(1− x)qµqν

k4

[1− 2x(1− x)

q2

k2

]∼ g2CA

∫d4k

(2π)4

[−ηµν

4k2+

1

12

(ηµνq2 + 2qµqν)

k4

](21.57)

The log divergence is in the second term in square brackets:

iCA12

g2

8π2ln

Λ

µ(q2ηµν + 2qµqν), FP ghosts (21.58)

21.5.3 Gluon loop

By far the most tedious diagram to evaluate is the first one on the second line, the gluonpropagator correction with gluons circulating in the loop. The upshot is that it adds to thefactor q2ηµν + 2qµqν in the ghost contribution the quantity

24(q2ηµν − qµqν) + q2ηµν − 4qµqν (21.59)

giving the net result

i26CA

12

g2

8π2ln

Λ

µ(q2ηµν − qµqν), gluon propagator (21.60)

When this corrected propagator is attached to the tree vertex one includes a gluon propa-gator and a factor of 1/2 (corresponding to

√Z3), giving the net result, remembering that

qµuγµTau = 0

igγµTa13CA

12

g2

8π2ln

Λ

µ(21.61)

Adding the result for the vertex correction, adds 3/4 to 13/12:

igγµTa22CA

12

g2

8π2ln

Λ

µ(21.62)

or written in terms of the renormalized coupling

gR = g

(1 +

11CA12

g2

4π2ln

Λ

µ

)(21.63)


Increasing the renormalization scale µ decreases the renormalized coupling. This is what ismeant by asymptotic freedom. Another way to write this is in terms of α = g2/(4π):

αR = α

(1 +

11CA6

α

πln

Λ

µ

)∼ 1

α−1 − (11CA/(6π)) ln(Λ/µ)(21.64)

Details of one loop gluon propagator. In Landau gauge we need a generalizationof the Feynman trick

1

A2B2= 2

∫ 1

0

dxx(1− x)

[Ax+B(1− x)]4(21.65)

Then the numerator of the loop integrand is

Nµν = [ηµρ(2q + k)σ − ηρσ(q + 2k)µ + ηµσ(k − q)ρ][ηνρ′(2q + k)σ′ − ηρ′σ′(q + 2k)ν + ηνσ′(k − q)ρ′ ][ηρρ

′(k + q)2 − (k + q)ρ(k + q)ρ

′][ησσ

′k2 − kσkσ′ ]

= [ηµρ2qσ − ηρσ(q + 2k)µ − ηµσ2qρ][ηνρ′2qσ′ − ηρ′σ′(q + 2k)ν − ηνσ′2qρ′ ][ηρρ

′(k + q)2 − (k + q)ρ(k + q)ρ

′][ησσ

′k2 − kσkσ′ ] (21.66)

In the second form we have used the transverse projectors to set kσ = kσ′ = 0 and kρ =kρ′ = −q. To simplify writing in what follows it is convenient to define

Rµρσ ≡ ηµρ2qσ − ηρσqµ − ηµσ2qρ (21.67)

so the numerator can be written

Nµν = [Rµρσ − 2ηρσkµ][Rνρ′σ′ − 2ηρ′σ′kν ]

[ηρρ′(k + q)2 − (k + q)ρ(k + q)ρ

′][ησσ

′k2 − kσkσ′ ] (21.68)

Meanwhile after combining denominators the denominator is

[(k + xq)2 + x(1− x)q2]4 (21.69)

Then after the change of variables k → k − xq, the terms odd in k can be dropped andaveraging over directions of k allows the replacements

〈kµkν〉 = k2ηµνD, 〈kµkνkρkσ〉 = k4ηµνηρσ + ηµρηνσ + ηµσηνρ

D(D + 2)(21.70)

The term in the numerator with 6 factors of k only has two free indices so we don’t need theaverage of six ks. That term reduces to

4kµkνk2(k2ηρρ − kρkρ) = 12k4kµkν → 12k6ηµν

D(21.71)


The full numerator will include terms of order k6, k4q2, k2q4.q6 but the log divergence iscontained only in the k4q2 term in the numerator and in the k6 term multiplied by −4x(1−x)q2/k2 coming from expanding the denominator in powers of q2/k2. The zeroth orderterm of this expansion is a quadratic divergence which dimensional regularization sets tozero. After keeping only the first two terms in this expansion the x integral is just that ofa polynomial which can be immediately done. After completing all these steps the resultquoted above is found.

Putting in the factors of (−i)2 from the two propagators and g2fbcdfdca = −CAg2δab fromthe two vertices, and the symmetry factor 1/2, the gluon propagator correction is so far

g2CA2

∫dDk

(2π)D

∫ 1

0

dxx(1− x)

[k2 + x(1− x)q2]4

[Rµρσ − 2ηρσ(k − xq)µ][Rνρ′σ′ − 2ηρ′σ′(k − xq)ν ][ηρρ

′(k + q(1− x))2 − (k + q(1− x))ρ(k + q(1− x))ρ

′]

[ησσ′(k − xq)2 − (k − xq)σ(k − xq)σ′ ] (21.72)

To capture the log divergence we simply collect terms quadratic in q. For instance

(Rµρσ + 2xηρσqµ)(Rνρ′σ′ + 2xηρ′σ′qν)(ηρρ′k2 − kρkρ′)(ησσ′k2 − kσkσ′)

→ k4

((ηρρ′ησσ′

(1− 2

D

)+

1

D(D + 2)(ηρσηρ′σ′ + ηρρ′ησσ′ + ηρσ′ησρ′)

)(Rµρσ + 2xηρσqµ)(Rνρ′σ′ + 2xηρ′σ′qν) (21.73)

After isolating the quadratic term in the numerator, one easily obtain the log divergence bysetting D = 4 and q2 = 0 in the denominator.


21.6 BRST Invariance

As we know from QED the gauge invariance implies that correlation functions satisfy theWard identities. They can be used to prove gauge independence as well as the decouplingof negative norm particle states present in covariant gauges. The corresponding identitiesknown as Ward-Takahashi identities in nonabelian gauge theories are considerably moreinvolved because unlike the photon, the gauge particles themselves are adjoints in the gaugegroup and there for interact with each other. Fortunately Becchi, Rouet, and Stora andindependently Tyutin discovered a remarkable symmetry (BRST) of the gauge fixed actioninvolving the FP ghost fields, which captures in an elegant way the important decouplingsand gauge independence required for consistency on any gauge theory.

Recall that an infinitesimal gauge transformation is given by

δAaµ = ∂µθa + gfabcA

bµθ

c (21.74)

which of course is not a symmetry of the gauge fixed action. The BRST discovery was thatif the gauge parameter is is replaced by the ghost field Ca, then one can find transformationsof the other fields so that the gauge-fixed action is invariant!

Denote the BRST transform of a field Φ by sΦ Then we have sAaµ = ∂µCa+gfabcA

bµC

c =Dabµ C

b. The gauge invariant part of L is clearly invariant. The gauge fixing term is notinvariant:

−s 1

2ξ(∂ · Aa)2 = −1

ξ∂ · Aa∂ ·DabCb → 1

ξ∂µ(∂ · Aa)DabµCb (21.75)

after an integration by parts. But this violation can be compensated by defining sBa =(∂ · Aa)/ξ. in the ghost Lagrangian. Finally the transform of Ca can be fixed by requiring

0 = s(Dabµ C

b) = Dabµ sC

b + gfacb(DcdCd)Cb (21.76)

The antisymmetry of f allows the replacement

(DcdCd)Cb → 1

2

((DcdCd)Cb − (DbdCd)Cc

)→ 1

2

((DcdCd)Cb + Cc(DbdCd)

)(21.77)

where the second replacement uses the Grassmann nature of C. Now we need the productlaw for covariant derivatives

Daeµ (febcC

bCc) = fabc(∂µCb)Cc + fabcC

b∂µCc) + gfahefebcA

hCbCc

= fabc(Dbdµ C

d)Cc + fabcCbDcd

µ Cd (21.78)

where the last step uses the Jacobi identity. Now we can say that with sCe = −febcCbCc/2The gauge fixed Lagrangian is BRST invariant. In summary, the BRST transform is

sAaµ = ∂µCa + gfabcA

bµC

c = Dabµ C

b (21.79)

sBa =1

ξ(∂ · Aa) (21.80)

sCa = −1

2fabcC

bCc (21.81)


A remarkable property of the BRST transformation is that it is nilpotent: applied twice itgives zero:

s2Aaµ = Dabµ sC + gfabc(sA

b)Cc = 0 (21.82)

by the same calculation we did in showing the invariance of the FP ghost Lagrangian.

s2Ba =1

ξ∂µDab

µ Cb = 0 (21.83)

by the field equation for C. And

s2Ca = −1

2fabc(sC

b)Cc +1

2fabcC

bsCc

=1

4fabcfbdeC

dCeCc − 1

4fabcC

bfcdeCdCe

=1

2fabcfbdeC

dCeCc =1

6(fabcfbde + fabefbcd + fabdfbec)C

dCeCc = 0

by the Jacobi identity.The BRST transformation of the various fields in the system will be generated by an

operator Q in the operator formalism. This operator should commute with the Hamiltonian[Q,H] = 0 and all physical observables, and it should also be nilpotent Q2 = 0. BRSTinvariance implies the Ward-Takahashi identities. It is the Hilbert space realization of theredundancy of gauge fields that are pure gauge.

The state space includes the action of all four components of the gauge field Aaµ theghosts Ca and antighosts B2 on the vacuum. The physical states are required to satisfyQ|Phys〉 = 0. But any vector in state space of the form Q|Λ〉 is also annihilated by Q. Butdue to Q2 = 0 such states will be null, whereas a true physical state is required to havefinite norm. Thus any pair of states differing by Q|Λ〉 will be identified. States annihilatedby Q are said to be in the kernel of Q. So a physical state is in the kernel of Q mod Q|λ〉.It is appropriate to use the language of differential geometry: Q playing a role analogous tothe exterior derivative d. both of which are nilpotent. Then we can say that the space ofphysical states is the cohomology of Q, consisting of the kernel of Q mod Q|X〉.

21.7 Gauge theory of the Standard Model

Having learned how to describe a general gauge theory, we now turn to its major applicationin this course: the standard model of elementary particle physics. In this model the gaugegroup is the non simple group SU(3)× SU(2)× U(1). There is an independent gauge fieldfor each factor group as follows:

SU(3) : Aµ =∑a

λa

2Aaµ, F a

µν = ∂µAaν − ∂νAaµ + g3fabcA

bµA

cν

SU(2) : Wµ =∑a

τa2W aµ , W a

µν = ∂µWaν − ∂νW a

µ + g2εabcWbµW

cν

U(1) : Bµ, Bµν = ∂µBν − ∂νBµ (21.84)


Each gauge field has its own coupling constant g3, g2, g1 respectively. Notice that g2,3 arephysically meaningful in the pure gauge theory because SU(2, 3) are nonabelian. The cou-pling g1 of the abelian gauge group U(1) only becomes meaningful with the addition offurther charged fields.

The SU(3) gauge field mediates the strong interactions and couples only to quarks anditself. It is referred to as the color group. The electroweak gauge group, mediating theelectromagnetic and weak interactions, is SU(2)× U(1). The SU(2) factor is referred to asweak isospin and the U(1) factor as weak hypercharge. This terminology is borrowed fromthe familiar isospin and hypercharge of strong interactions.

The electroweak part of the standard model gauge group was strongly motivated by thephenomenology of the electromagnetic and weak interactions of the elementary particles.In the 1950’s and 60’s studies of nuclear beta decay, e.g. n → p + e + ν and muon decayµ → e + νe + νµ established that the weak interactions had a current-current structurestrongly suggesting that they were mediated by a charged heavy vector boson, called W±

µ .This together with the long understood mediation of electromagnetic interactions by thephoton Aµ suggested a gauge group with at least a three dimensional adjoint representation.For a time in the early 1960’s physicists attempted to do with just SU(2). But Salam andWeinberg soon (1967) built a successful model based on SU(2) × U(1) which turned outto be the right answer. Their model had an additional neutral vector boson, which wouldmediate neutral current weak interactions. The experimental discovery of neutral currentinteractions (several years after the theory was proposed) clinched the Weinberg-Salam modelas the gauge theory of electroweak interactions. As we shall see the photon field is not simplythe Bµ field but is a linear combination of Bµ with W 3

µ .

The electroweak fields couple to quarks and leptons as well as to each other. The gaugepart of the Lagrangian is

Lgauge = −1

4BµνB

µν − 1

4F aµνF

aµν − 1

4W aµνW

aµν (21.85)

The fermionic “matter fields” are organized in three families, each consisting of a left handedSU(2) doublet of quarks, a left handed SU(2) doublet of leptons, with the correspondingright-handed fermions in SU(2) singlets. The quarks are fundamental triplets under SU(3)whereas all of the leptons are singlets under SU(3). The pattern of irreducible representationassignments for each family are tabulated below.


Lepton irrepsParticle SU(3) SU(2) U(1)(νee

)L

1 2 −1

eR 1 1 −2νeR 1 1 0

Quark irrepsParticle SU(3) SU(2) U(1)(ud

)L

3 2 1/3

uR 3 1 4/3dR 3 1 −2/3

where the subscript L on a fermion field ψ means (1 − γ5)ψ/2 and the subscript R means(1 + γ5)ψ/2. The assignment of different gauge group representations to the left and rightcomponents of a Fermi field characterizes the standard model as chiral. Dirac mass termsare therefore forbidden in the standard model. A Majorana mass term is allowed for aright handed neutrino νR, should it exist, because that field is a gauge singlet under thewhole gauge group. Indeed if that Majorana mass is sufficiently large, it would “explain”why such a particle has not been seen. The doublet left handed pattern of couplings wasstrongly motivated by the beta decay processes already mentioned. For the baryons, thedoublet structure was initially taken to be (p, n) but once the quark model was establishedthe composite nucleons (p, n) were replaced by the more “fundamental” quarks (u, d).

The assignments of U(1) charge, called weak hypercharge, shown in the table is dictatedby the electric charge each fermion should carry. It is normalized according to the relationQ = (τ3 +Y )/2, which will later be seen to be a consequence of the Higgs mechanism. So theleft handed neutrino (T3 = +1/2) and its lepton partner (T3 = −1/2, which have 0 and −1units of charge, would have Y = −1, whereas the right handed electron would have Y = −2.The righthanded neutrino, should it exist would have Y = 0. For the quarks, the up typequark has electric charge 2/3 corresponding to Y = 1/3 for the left handed one and Y = 4/3for the right handed one. The right handed down quark has Y = −2/3. There are two morelepton/quark families (νµ, µ), (c, s) and (ντ , τ), (t, b) with identical pattern of electroweakirreps.

As presented so far this model seems far from reasonable: it has 12 massless gaugeparticles, and a number of necessarily massless chiral fermions. The only massless gaugeboson in nature is the photon. The only near massless fermions are the neutrinos, but thepresence of neutrino oscillations provides evidence that even the neutrinos have a (tiny)nonzero mass. So we need another ingredient to give masses to all but one particle in thestandard model. This missing ingredient is the Higgs mechanism, which uses additionalscalar fields with nonzero constant values in the vacuum. If they are judiciously coupledto the fields we have encountered so far, the mass parameters will be proportional to theseconstant values.

To motivate these couplings consider a potential electron mass term eLeR + eReL. Since


the left handed electron is a member of an SU(2) doublet and the righthanded one anSU(2) singlet, this is not gauge invariant because it has weak isospin 1/2 and nonzero weakhypercharge. If we introduce in the model an SU(2) doublet scalar field φ = (φ1, φ2), wecan form an SU(2) singlet

λφ†eR

(νeLeL

)= λφ∗1eRνeL + λφ∗2eReL (21.86)

so an effective mass λ 〈φ∗2〉 for the electron would arise if 〈φ∗2〉 6= 0 and 〈φ∗1〉 = 0. So thissimple way of giving mass to fermions requires an SU(2) doublet scalar field.

Now examine the scalar kinetic term of such an SU(2) doublet, assumed to be a singletunder SU(3), −(Dµφ)†Dµφ, where

Dµφ = (∂µ − ig2Wµ − ig1(Yh/2)Bµ)φ (21.87)

Since this is the first appearance of g1 we may define Yh = −1 as a convention. Then if〈φ〉 ≡ v is a constant spinor, the shift φ = v + φ leads to

Dµφ = (∂µ − ig2Wµ + i(g1/2)Bµ)φ+ (−ig2Wµ + ig1Bµ/2)v (21.88)

Then

(Dµφ)†Dµφ = ∂φ†∂φ+ ∂φ† · (−ig2W + ig1B/2)v + v†(ig2W − ig1B/2) · ∂φ

+v†(g2

2

4W a2 +

g21

4B2 − g1g2

2taW a ·B)

)v + cubic terms (21.89)

and the last term is a quadratic form in the gauge fields:

v†(g2

2

4W a2 +

g21

4B2 − g1g2

2taW a ·B)

)v (21.90)

It is a basic property of the doublet representation of SU(2) that any spinor is the eigenvectorof n · τ with eigenvalue +1 for some unit vector n. We can then fix our 3-axis in the ndirection. Then the above quadratic form becomes v†v times

g22

4(W 2

1 +W 22 ) +

g22

4W 2

3 +g2

1

4B2 − g1g2

2B ·W3

=g2

2

4(W 2

1 +W 22 ) +

1

4(g2W3 − g1B)2 (21.91)

With our conventions the component of the scalar doublet we have taken to be a nonzeroconstant is the upper one with t3 = +1/2. Three of the gauge bosons gain a mass: W 1,W 2

or W± = (W 1± iW 2)/√

2 gain a mass squared of M2W = v†vg2

2/2 and the linear combination

g2W3µ − g1Bµ ≡ g2(W 3

µ −Bµ tan θW ) =g2

cos θW(W 3

µ cos θW −Bµ sin θW )

≡ g2

cos θWZµ (21.92)


gains a mass squared M2Z = M2

W/ cos2 θW . The orthogonal linear combination

Aµ = Bµ cos θW +W 3µ sin θW , tan θW =

g1

g2

(21.93)

remains massless and can be identified with the photon. (The orthogonality is dictated bythe desire to keep the quadratic derivative terms of the the gauge fields diagonal.) We cannow confirm the formula for the charge by eliminating W3 and B in favor of A, Z in thegauge couplings:

g2W3T 3 + g1B

Y

2

= g2T3(Z cos θW + A sin θW ) + g1

Y

2(−Z sin θW + A cos θW )

= A

(g1Y

2cos θW + g2T3 sin θW

)+ Z

(g2T

3 cos θW − g1Y

2sin θW

)= eA

(T 3 +

Y

2

)+ Zg2 cos θW

(T3 −

Y

2tan2 θW

)(21.94)

where we have identified the electric charge e = g2 sin θW .After the Higgs mechanism we have a massless photon and 8 massless gluons. The reason

we don’t see massless gluons has to do with the confinement phenomenon, not the Higgsmechanism. According to the confinement hypothesis only color singlet bound states havefinite energy. Quarks and gluons carrying nonsinglet color have infinitely long flux tubesattached which makes them infinitely massive.

21.7.1 Gauge-fixing with Higgs mechanism

Fixing the ξ gauge for the SU(3) gauge fields is done as usual. But the electroweak quadraticterms after the shift include mixing between the scalar and gauge fields Let’s collect theundesirable quadratic terms in the Lagrangian after the shift φ→ v + φ.

1

2

[(∂ ·B)2 + (∂ ·W a)2

]+ φ†Tav(−ig2∂ ·W a) + (φ†vig1∂ ·B/2)

+ig2∂ ·W av†φ− ig1∂ ·B/2)v†φ (21.95)

’t Hooft had the clever idea to choose gauge fixing terms

− 1

2ξ[∂ ·B + iξg1(φ†v − v†φ)/2]2

− 1

2ξ[∂ ·W a − ig2ξ(φ

†Tav − v†Taφ]2 (21.96)

The added terms are engineered so that the cross terms cancel the scalar-gauge mixingterms! For calculations ξ = 1 has obvious advantages because then the gauge propagators


are simply −iηµν/(k2 +M2). In that case what remains of the gauge fixing term gives mass

to φ2 equal to the W boson mass, and gives mass to the imaginary part of φ1 equal to theZ boson mass. These masses squared are proportional to ξ when ξ 6= 1.

Finally, to construct the FP ghost terms we calculate the change of the gauge fixingcondition under an infinitesimal gauge transformation:

∆Bµ = ∂µη. ∆W aµ = Dab

µ θb

∆φ = −ig2Ta(v + φ)θa + ig1

2(v + φ)η (21.97)

The terms involving v in the FP determinant, expressed in terms of FP ghost fields, lead tomasses for the FP ghost fields which match the ones for the scalar fields discussed above.In particular when ξ = 1 there is a scalar and FP ghost field for W 1,W 2, Z with matchingmasses. There is only a massless FP ghost field but no scalar for the em field A. For eachof the gauge fields the FP ghosts cancel two polarizations, leaving 3 for each massive gaugefield and only two for the photon field, as required by Lorentz invariance. The scalar fieldserves as the 0 helicity polarization needed for a massive vector particle.

21.7.2 Massive vector boson couplings

We have determined the couplings of the neutral Z boson in the process of determining thephoton couplings. Since charge assignments are easier to remember than hypercharge, it isuseful to give it in two forms:

g2 cos θW

(T3 −

Y

2tan2 θW

)= g2 cos θW

(T3(1 + tan2 θW −Q tan2 θW

)=

g2

cos θW

(T3 −Q sin2 θW

)(21.98)

It is noteworthy that the relative couplings depend on θW , unlike the electric charge.The massive W couplings are g2(T1W

1 + T2W2), but it is useful to express the W field

in terms of charge eigenstates W = (W 1 + iW 2)/√

2 or W 1 = (W + W †)/√

2 and W 2 =(W −W †)/(i

√2). Then

g2(T1W1 + T2W

2) =g2√

2(WT− +W †T+) (21.99)

where T± = T1± iT2 are the standard raising and lowering operators for SU(2). For instanceon the electron doublet

T+

(νee

)L

=1− γ5

2

(e0

), T−

(νee

)L

=1− γ5

2

(0νe

)(21.100)

so we can write out the W couplings to a lepton doublet

g2

2√

2

[νlW

† · γ(1− γ5)l + lW · γ(1− γ5)νl]

(21.101)


In the early days (1930’s - 1970’s) of interpreting the weak interactions, the energies were solow e.g. tenths of a GeV, that the momentum dependence of the vector boson propagatorwas negligible: it could be replaced by −iηµν/M2

W . The physics could be completely under-stood by ignoring the W boson and replacing the W coupling with an effective four fermioninteraction

GF√2

∑l

(νlγµ(1− γ5)l)∑l′

(lγµ(1− γ5)νl) =GF√

2JµJ

†µ, GF =g2

2

8M2W

This precise form was determined through painstaking study of muon and nuclear beta decay.In the standard model the coupling to quarks was analogous:

g2

2√

2

[ulW

† · γ(1− γ5)d+ dW · γ(1− γ5)u]

(21.102)

which is valid for massless quarks. As we shall see once fermions are given mass there willbe a flavor mixing matrix.

21.7.3 Fermion masses

As we have seen, fermion (Dirac) mass terms are forbidden in the standard model because ofthe chiral coupling patterns. But with the Higgs scalar having a vacuum expectation value,one can form gauge invariant Yukawa coupling terms which act like mass terms for constantVEV’s. To do this for all the necessary mass terms we shall use not only φ = (φ0, φ−), whichhas Y = −1, but also its charge conjugate φc = −iσ2φ

∗ = (−φ∗−, φ∗0), which has Y = +1.(To see that φc transforms the same way under SU(2) as φ, note that (e−iσ·ξ)∗ = σ2e

−iσ·ξσ2)Let the doublets of the standard model be labeled Ei for the lepton doublets and Qi for

the quark doublets. Then the most general Yukawa interaction terms can be written

GeijE

ieRjφc +GνijE

iνRjφ+GdijQ

idRjφc +GuijQ

iuRjφ+ h.c. (21.103)

It can be shown (Exercise) that any complex matrix can be brought to diagonal form withnonnegative eigenvalues by a pair of unitary matrices: Ga = U †aG

DIAGa Va. Now suppose

〈φ〉 = (v, 0) so 〈φc〉 = (0, v) with v real. Then define φ = 〈φ〉+ φ. Then the terms involvingv become

vGeij eLieRj + vGν

ij νLiνRj + vGdij dLidRj + vGu

ijuLiuRjφ+ h.c. (21.104)

Now diagonalize each Ga = U †aDaV and absorb U and V into a redefinition of the corre-sponding fermion field faR → V a†faR f

aL → Ua†faL. This redefinition of fermion fields leaves the

kinetic part of the Lagangian invariant. However the interaction terms are not all invariantunder this field redefinition:

1) The SU(3) gauge couplings are family blind.∑

i qL,Ri γ · AqL,Ri is invariant.

2) The photon couplings are invariant because each U, V associated with each charge.


3) The part of the Z couplings proportional to charge Q is invariant for the same reasonthe photon couplings are. The T3 couplings are invariant because the L fermions areall in doublets. Before charm was discovered, there was a doublet partner for only oneof the charge −1/3 quarks.

4) The charged W couples fermions of different charge, for which the mass diagonalizationmatrix is different. Thus

uLiγ ·WdLi → uLi(UuUd†)ijγ ·WdLj

νLiγ ·WeLi → νLi(UνU e†)ijγ · edLj (21.105)

[If the neutrinos were massless one could choose Uν to cancel U e†, with no flavormixing.] The matrix UCKM = Uu

LUd†L , for Cabibbo-Kobayashi-Maskawa, allows the

charged currents to change quark flavor. The corresponding matrix in the leptonsector requires massive neutrinos, for which there is now indirect evidence.

The possibility of giving a Majorana mass to νiR has been mentioned. If this is combined withthe Dirac mass term coming from the Yukawa interactions one has the following neutrinomass terms

GνijE

iνRjφ+1

2νiTR Mij(iγ2β)νjR + h.c. (21.106)

If all eigenvalues of M are large we can ignore the kinetic terms of νR amd integrate out theνR fields,

νjTR = −Ekφiβγ2(GνM−1)kj (21.107)

generating the terms

−1

2ETkL φT †[Gν∗M∗−1Gν∗T ]kjφ

†β (21.108)

Putting φ = (v, 0) and real leads to the neutrino mass matrix v2Gν∗M∗−1Gν∗T .


Chapter 22

Systematics of Renormalization

22.1 Renormalized perturbation theory

In this chapter we try to explain the general procedures to follow in establishing the successof the renormalization program. Rather than attempting a completely general approachwe will restrict attention to the important example of QED. We have in fact carried outrenormalization through one loop explicitly so now we want to sketch how the programworks at higher orders.

Recall that in QED the superficially divergent 1PIR graphs are the photon self energy(vacuum polarization) Π the electron self energy Σ and the electron photon vertex. Let’sfirst introduce the notion of a skeleton graph for a process that is not one of these, in whichcorrections to vertices and propagators are shrunk to a point. Then one gets the completegraph to a given order by inserting the complete propagators and vertices (to the desiredorder) rather than the lowest order ones.

Suppose we know:

a) D′ = Z3(uv Finite) + gauge terms

b) S ′ = Z2(uv finite)

c) Γ = Z−11 (uv finite)

d) a), b), c) have no worse large momentum behavior than their zeroth order versions,modulo logarithms. (Weinberg’s Theorem).

Now each vertex in the skeleton has a factor of e0, so the Z factors can be absorbed byrenormalizing the charge as eR = e0Z2

√Z3Z

−11 . If the original graph is primitively convergent

then this procedure renders all skeleton subgraphs convergent and then d) allows us toconclude that the overall graph is convergent.

But what about a), b), and c)? Define a 2 particle irreducible e+e− kernel. Then Γsatisfies an in homogeneous integral equation with K as the kernel of the structure

Γµ = γµ +

∫ΓµS ′S ′K =

1

Z1

γµ + ∆

∫ΓµS ′S ′K (22.1)

221

Now the ingredients of the second term are all of lower order so with the high momentumassumption a single subtraction makes the last integral convergent times Z−1

1 to lower order.This establishes c).

For QED one can define ∂Σ/∂pµ = Γ by the Ward identity: S ′, D′,Γ ok to order αn−1

implies Γ ok to order αn which by WI implies S ′ ok to order αn. The WI allows us to bypassthe problem of overlapping divergences that we must face in attacking Π.

For the vacuum polarization, gauge invariance reduces the superficial degree of divergencefrom D = 2 to D = 0:

D′ = −i[ξqµqν

q4+ Z3

ηµν − qµqν/q2

q1(1 + e20Z3(Π(q2)− Π(µ2))

](22.2)

So must prove [Π(q2)−Π(µ2)] is finite. The problem is overlapping divergences. For examplethe two loop diagrams correcting lowest order vacuum polarization.

22.1.1 Two loop example

∫d4pd4k

1

k2Trγµ

1

m+ γ · pγλ

1

m+ γ · (p− k)γν

1

m+ γ · (p− k − q)γλ

1

m+ γ · (p− q)

The graph displays only one possible vertex correction. But there are actually two disjointvertex divergences: one with k → ∞ with p fixed and a second with k → ∞ with p − kfixed. Since they are disjoint regions of integration they both contribute and at this order areadditive. Thus even though the graph displays only one vertex correction, their are actuallydivergences available for both! As one goes to higher order it gets more and more challengingto disentangle the overlapping divergences, but it turns out to work in the end.

Another approach is to fashion an analogue of the Ward identity which handles theproblem for the electron self energy. One doesn’t have a Ward identity but one can constructa new object (

q · ∂∂q− 2

)Πµν(q) (22.3)

which has the character of a forward vertex function like Γ(p, p). Then overlapping diver-gences are absent. One constructs a proof that it is rendered finite after renormalization andthen recovers Πµν by integration.

22.2 Renormalization group of Gell-Mann and Low

Once one accepts the facts of renormalization it turns out that one can get a surprisingamount of information just by consistency of the scheme. It is simplest to apply this notionin a renormalizable theory with no mass scale, e.g. QED with me = 0. But it could equallybe QCD with all quark masses set to zero, or indeed pure nonabelian gauge theory. Also the


neglect of masses is only a technical simplification, which can be removed with a slightly morecomplicated procedure: Equivalently one is studying any renormalizable theory at momentamuch larger than all masses.

Focussing first on QED (QCD will be the subject of the next chapter), consider thevacuum polarization function Π(q2). Since it is dimensionless the naive expectation is thatit must be independent of q, the only dimensionful scale available. But UV divergencesinvalidate this conclusion. Since me = 0 it makes no sense to renormalize at q = 0. Instead,we pick an arbitrary mass µ and make the necessary subtraction at q2 = µ2. We have

D′Fµν =−iηµν

q2(1 + e20Π(q2))

(22.4)

and define the renormalized charge as e2(µ) = e20/(1 + e2

0Π(µ2)). Then we shall examine themomentum dependent quantity

d(q2, e2(µ), µ2) =e2(µ)

1 + e2(µ)(Π(q2)− Π(µ2))=

e20

1 + e20Π(q2)

(22.5)

The far right side makes no reference to the scale µ Thus d is independent of µ which issimply a reflections of the success of renormalization: changing µ is just a change in thevalue of q at which e is defined. Therefore e must depend on µ according to[

µ∂

∂µ+ µ

de2

dµ

∂

∂e2

]d(q2, e2(µ), µ) = 0 (22.6)

Now define the Gell-Mann-Low function

ψ(e2, µ) ≡ µde2

dµ= ψ(e2) (22.7)

by dimensional analysis1. But d is itself dimensionless so it must be a function only of q/µand e. So we write d(q2/µ2, e2) and note by definition that d(1, e2) ≡ e2. The Gell-Mann-Lowrenormalization group equation can now be written(

−q ∂∂q

+ ψ(e2)∂

∂e2

)d(q, e2) = 0. (22.8)

or, defining t by dt = de2/ψ(e2) with e2(0) = e2, the equation reads2(∂

∂t− ∂

∂ ln q

)d = 0 (22.9)

1Standard dimensional analysis is allowed now because the renormalization process has removed UVdivergences

2Note that ψ(e2) = 0 identically, would imply that d is independent of q, the consequence of naive scaleinvariance.


whose general solution is d = f(t + ln(q/µ)). f can be determined by setting q = µ whered(µ2, e2(t)) = e2(t), so d(q2/µ2, e2(t)) = e2(t+ ln(q/µ)). We can now set t = 0 to get

d(q2/µ2, e2) = e2(ln q/µ), lnq

µ=

∫ e2(ln q/µ)

e2

dx

ψ(x)(22.10)

We notice that we can calculate ψ(e2) by calculating d(q2) in perturbation theory, takingits derivative with respect to ln q and setting q = µ. Recalling our lowest order result forvacuum polarization

e2(Π(q2)− Π(µ2)) =e2

2π2

∫ 1

0

dxx(1− x) lnm2 + x(1− x)µ2

m2 + x(1− x)q2→ e2

12π2lnµ2

q2

d(q2, e2) = e2

(1− e2

12π2lnµ2

q2

)+O(e6) (22.11)

from which

ψ(e2) =e4

6π2+O(e6) (22.12)

It is important to appreciate that solving the RG equations with ψ(e2) truncated at somefinite order gives some (but of course not all) information about higher order contributions tothe quantity calculated. This is evident from our QED example. Look at the exact solutionfor e2(ln q/µ) with ψ(x) = bx2:

b lnq

µ=

∫ e2(ln q/µ)

e2

dx

x2=

1

e2− 1

e2(ln q/µ)(22.13)

e2(ln q/µ) =e2

1− be2 ln q/µ(22.14)

The right side has contributions from all orders in perturbation theory, the expansion param-eter being e2 ln q/µ. We sometimes call this a leading log approximation. it can be trustedif 1 ln q/µ . 1/e2., because it neglects terms of order (e2)n(ln q/µ)n−1.

Another way to see this is to examine how the equation is solved in perturbation theory.We try a double expansion in α = e2/(4π) and t ≡ ln q/µ:

d =∑m,n

cmnαntm (22.15)

dd

dt=

∑m,n

mcmnαntm−1 = (b1α

2 + b2α3 + · · · )

∑m,n

ncmnαn−1tm (22.16)

Coming back to the exact equation, we ask what kind of behavior can we expect as ln q/µ→∞. This requires that the integral

∫dx/ψ(x) diverge. There are several possibilities:

a) If e2 →∞, i.e. the effective charge blows up at short distances, then ψ(x) ≤ x as x→∞.


b) Another way is for e2 to approach a zero of ψ as ln q/µ→∞. Suppose there is a linearzero: ψ(x) ∼ (x− x0)ψ′(x0 Then

ln q/µ ≈∫ x0−ε

e2

dx

ψ(x)+

1

ψ′(x0)lnx0 − e2(q)

ε(22.17)

e2(q) ≈ x0 − Aeψ′(x0) ln q/µ (22.18)

The approach to x0 as q → ∞ requires that ψ′(x0) < 0. In that case we say that x0

is an UV stable fixed point. A theory with this property would “predict” the value ofthe coupling. If ψ′(x0) > 0 it would be called an IR stable fixed point.

c) What about e2 → 0? The integral certainly diverges there. Suppose ψ(x)→ bx2. Then

lnq

µ∼ 1

be2− 1

be2(q)

e2(q) ∼ 1

1/e2 − b ln q/µ(22.19)

For the approach to take place in the UV, must have b < 0! this is the famous caseof asymptotic freedom, a feature of nonabelian gauge theory! We say that QED has aUV unstable fixed point at the origin. Massless QED has an IR stable fixed point atthe origin,


Chapter 23

Renormalization and Short DistanceProperties of QCD

In this chapter we focus on QCD neglecting quark masses, so the classical theory is scaleinvariant. As we have seen this scale invariance is broken by UV divergences and dimensionalanalysis has to be supplemented by the renormalization group.

23.1 Scaling properties of Green functions in massless

gauge theory.

In our discussion of QED we have applied RG ideas to an actual physical quantity: e20 times

the photon propagator. The presence of e20 made the quantity the actual measured charge

at a given value of q. Had we considered the propagator itself, renormalization would meanthat D′/Z3 is what must be finite.

In nonabelian gauge theories it is not so easy to find a gauge invariant physical quan-tity. The simplest might be correlation functions of singlet operators like

∑a F

a2. Insteadwe consider a general (non gauge invariant) correlation function. Then the statement ofrenormalization is

Gbare(q1, . . . , qn; p1, . . . , pm; g0, ξ) = Zn/2G Z

m/2Q Gr(q, p; gµ, ξµ) (23.1)

Then the Callan-Symanzik RG equation is[µ∂

∂µ+ µ

dgµdµ

∂

∂g+ µ

dξµdµ

∂

∂ξ+n

2µd lnZGdµ

+m

2µd lnZqdµ

]Gr = 0. (23.2)

Renormalization has removed all divergences, so including the scale µ we may now applyordinary dimensional analysis to define:

µdgµdµ

= β(gµ, ξµ), µdξµdµ

= δ(gµ, ξµ) (23.3)

1

2µd lnZGdµ

= −γG(gµ, ξµ),1

2µd lnZqdµ

= −γq(gµ, ξµ) (23.4)

227

Now we can regard the parameters µ, γ, ξ in Gr as independent provided Gr is subject tothe constraint [

µ∂

∂µ+ β

∂

∂g+ δ

∂

∂ξ− nγG −mγq

]Gn,mr = 0. (23.5)

One of the Ward identities says that the longitudinal component of the gluon propagator isnot corrected by interactions:

qµD′µν = qµDbareµν =

−iq2ξqν . (23.6)

Inserting this in the CS equation for G2,0r yields the relation

δ(g, ξ)−iqνq2

= 2γG(g, ξ)ξ−iqνq2

(23.7)

giving δ = 2ξγG. This is a situation where Landau gauge (ξ = 0) simplifies life since thenδ = 0 and the ξ term is absent from the CS equation.

23.2 Calculation of β, γg, γq

The coefficient functions can be extracted from the RG equation by expanding, sayG2,0r , G0,2

r , G3,0r

in perturbation theory. In doing so keep in mind that they will differ from one calculationscheme to another depending on what one takes as a measure of g. Notice the following:

1) Coupling renormalization occurs only at order g3 and higher, because any bare vertex isindependent of µ.

2) Thus δ, γG, γQ,which start at order g2 can be obtained in lowest order from G2,0r and G0,2

r .From calculations in ξ gauge we obtain for SU(N)

γG =g2

16π2

[(13

6− ξ

2

)Nc −

4Nf

6

](23.8)

γq = −ξ Ng2

16π2(23.9)

(The δ∂/∂ξ term is of order g4

To compute β we examine G1,2 the quark gluon vertex, obtaining

β = −Ng3

16π2

(11

3− 2Nf

3N

)+O(g5) (23.10)

[The earlier calculation was in Landau gauge (ξ = 0).] The γ’s depended on ξ but thatdependence cancels in the determination of β


23.2.1 Non universality of β, α

Suppose we consider an alternate renormalization scheme. We assume that the two prescrip-tions agree at lowest order. Then

g′(g) = g + a1g3 + a2g

5 + · · · (23.11)

β(g) = −bg − cg5 + dg7 + · · · (23.12)

Then we calculates

β′(g′) ≡ µdg′

dµ= β(g)

[1 + 3a1g

2 + 5a2g4 + · · ·

]= −bg3 − cg5 − 3a1bg

5 +O(g7) = −b(g3 + 3a1g5)− cg′5 +O(g′7)

= −b(g + a1g3)3 − cg′5 +O(g′7) = −bg′3 − cg′5 +O(g′7) (23.13)

So the first two terms of β are the same in any scheme. As an exercise you can show thatthe terms beyond the first two are different.

23.2.2 The Nature of the Callan-Symanzik Equation

Let us examine the CS equation for an observable G(p, g, µ) with γ = 0 that depends ononly one momentum. Its perturbation expansion would have the form

G =∞∑n=0

g2n

n∑m=0

anm

(lnp

µ

)m(23.14)

Plugging this expansion in the CS equation and equating to zero the net coefficient of eachdistinct power of g and ln(p/µ) relates all of the anm with m 6= 0 to an0 . For example supposeβ(g) = −bg3 exactly. Then one finds

anm = (−2b)m(n− 1

m

)an−m0

G =∞∑n=0

g2n

n−1∑m=0

an−m0

(n− 1

m

)(−2b ln

p

µ

)mG =

∞∑m=0

∞∑n=m+1

g2nan−m0

(n− 1

m

)(−2b ln

p

µ

)mG =

∞∑m=0

∞∑k=1

g2(k+m)an0

(k +m− 1

m

)(−2b ln

p

µ

)mG =

∞∑k=1

ak0

∞∑m=0

g2(k+m)(−)m(−km

)(−2b ln

p

µ

)mG =

∞∑k=1

ak0

(g2

1 + 2bg2 ln(p/µ)

)k≡

∞∑k=1

ak0(g2(p2))k (23.15)


where we have defined the running coupling g2(p2) = g2/[1+2bg2 ln(p/µ)]. So in this simpli-fied example the renormalization group sums the powers of ln’s by replacing the expansionparameter g2 with the running coupling g2(p). Here b > 0 corresponds the asymptoticfreedom.

23.3 High momentum behavior

Now let us return to the RG equation for a general (renormalized) QCD Green’s function(assuming masses are negligible) Gn,m(qi, pi, g, ξ, µ) with n gluon legs and m quark legs:(

µ∂

∂µ+ β

∂

∂g+ 2ξγG

∂


)Gn,m = 0 (23.16)

For dimensional reasons β and γ can only depend on g and ξ. Gn,m is of course the F.T. ofa correlation function of the gluon and quark fields. Its physical mass dimension is (in Dspacetime dimensions)

dG = nD − 2

2+m

D − 1

2−D(n+m) +D (23.17)

because the gluon field A has dimension (D−2)/2, the quark field has dimension (D−1)/2,The F.T. of each field adds a dDx and we remove the momentum conserving delta functionfrom the definition of G. Since we have removed all infinities by renormalization, at theexpense of introducing the fake parameter µ we can apply standard dimensional analysis toscale out the overall µ dependence. First of all introduce an overall momentum scale Q bywriting qi = Qqi, pi = Qpi. Then we can write

Gn,m = µdGGn,m(qi, pi; g, ξ,Q/µ). (23.18)

So one can replace derivatives w.r.t. µ by derivatives w.r.t. Q:(−Q ∂

∂Q+ dG + β

∂

∂g+ 2ξγG

∂


)Gn,m = 0 (23.19)

Next define a new variable t by the differential equations;

dg

dt= β(g(t), ξ(t)), g(0) = g

dξ

dt= 2ξ(t)γG(g(t), ξ(t)), ξ(0) = ξ. (23.20)

Then the RG equation becomes(−Q ∂

∂Q+∂

∂t+ dG − nγG −mγq

)Gn,m = 0 (23.21)


which has the general solution

Gn,m = F (t+ ln(Q/µ)) exp

∫ t

0

dt′[−dG + nγG +mγq]

(23.22)

The function F is of a single variable and can be evaluated by setting t = 0: F (ln(Q/µ) =Gn,m(Q/µ, g, ξ), so

Gn,m(Q/µ, g(t), ξ(t)) = Gn,m(Qet/µ, g, ξ) exp

∫ t

0

dt′[−dG + nγG +mγq]

Finally set µ = Q to obtain

Gn,m(et, g, ξ) = Gn,m(1, g(t), ξ(t)) exp

∫ t

0

dt′[dG − nγG −mγq]

(23.23)

from which we get

Gn,m(Q/µ, g, ξ) = µdGGn,m(1, g(lnQ/µ), ξ(lnQ/µ))

× exp

∫ lnQ/µ

0

dt′[dG − nγG(t′)−mγq(t′)]

= QdGGn,m(1, g(lnQ/µ), ξ(lnQ/µ))

× exp

−∫ lnQ/µ

0

dt′[nγG(t′) +mγq(t′)]

(23.24)

This formula shows how to calculate the large Q behavior of a Green’s function with knowl-edge of the large t behaviors of g(t), ξ(t), knowledge of G(1, g, ξ) and γG(g, ξ), γq(g, ξ). Theonly approximations made in this formula are the neglect of quark masses. With asymptoticfreedom (b > 0) we can use perturbation theory to analyze the large Q behavior of Green’sfunctions, because then g(lnQ)→ 0 as Q→∞.

To see how this works in a simplified context, suppose that g = g(0) 1. Then g(t)for t > 0 is even smaller. If perturbation theory is valid at t = 0 it is even more valid ast increases. In this situation we should be able to just keep the first few terms in the RGfunctions. Let’s truncate at two terms for β:

β(g) = −bg3 − cg5 +O(g7) (23.25)

Since the first two terms are universal we expect b, c to be independent of ξ. Then we canattempt to solve the equation for g with this truncated β:

dt = − dg

bg3 + cg5= −dg

g3

1

b+ cg2

=1

2d

1

g2

1/g2

b/g2 + c=

1

2bd

1

g2− c

2b2d ln(b/g2 + c)

t =1

2b

[1

g2(t)− 1

g2(0)

]− c

2b2lnb/g2(t) + c

b/g2(0) + c

1

g2(t)= 2bt+

1

g2(0)+c

blng2(0)

g2(t)+c

blnb+ cg2(t)

b+ cg2(0)(23.26)


To leading order we have

g2(ln(Q/µ)) ∼ 1

2b ln(Q/µ) + 1/g2≡ 1

2b ln(Q/Λ)(23.27)

where Λ = µ2e−1/(2bg2) is an RG invariant characteristic scale of QCD. That scale replacesthe coupling g as a parameter. This is sometimes known as dimensional transmutation.Putting in the QCD values for b and c we find

Ng2

4π2→ 6

(11− 2nf/N)

1

lnQ/ΛQCD

+O

(ln lnQ

(lnQ)2

), Q→∞ (23.28)

To complete the evaluation of Gn,m at large Q we can use the lowest order results for theanomalous dimensions.

γG ≈ dg2, γq ≈ fξg2 (23.29)

to evaluate ∫ lnQ/µ

0

dt′g2(t′) =

∫dg′

g′2

β(g′)≈ −1

b

∫dg′

g′≈ − 1

2blng2(ln(Q/µ))

g2

23.4 Composite Operators

The physics of the color SU(3) part of the standard model, the strong interactions is generallynot tractable analytically, because our tools are limited to perturbation theory. We havelearned how to calculate renormalized correlation functions of quark and gluon fields, andto use the renormalization group plus asymptotic freedom to learn something about theirhigh momentum scaling behavior. We can not directly translate these results to experimentsbecause we can’t form beams of quarks or beams of gluons. We can only form beams ofprotons, electrons, and indirectly beams of unstable particles like the muon or neutron.

To describe hadron scattering, using the reduction formalism, instead of quark and gluonoperators, we must use composite operators that connect the one hadron state to the vacuum.For example the proton is thought to be a composite containing two up quarks and one downquark. This suggests that the operator εαβγu

α(x)uβ(x)uγ(x) would be appropriate for thereduction formalism. We would then assume that⟨

0|εαβγuα(0)uβ(0)uγ(0)|proton⟩6= 0 (23.30)

and use the reduction formalism to define the proton proton S-matrix. Notice that we takeall fields to be at the same space-time point. This is not only for simplicity, but also tomaintain gauge invariance. The requirement that the three quark field connects the protonstate to the vacuum does not in any way imply that the proton has only three quarks: itsstate vector could also contain quark anti quark pairs and gluons. One could use a 6 quark


operator to define scattering of deuterons, say. By using different flavor combinations onecould describe other baryons, e.g. the Ω− would couple to εαβγs

αsβsγ.To include mesons in the scattering process, we could use operators like qf1(x)Γqf2(x).

For instance ⟨0|d(0)γgu(0)|π+

⟩6= 0 (23.31)

would enable the inclusion of the π+ meson in the scattering process. In order to include highspin mesons in the scattering we could employ derivatives of operators such as qDµ1 · · ·Dµnqin the reduction formalism. In this choice we use covariant derivatives so that the operatorretains its gauge invariance. Each operator we use requires its own wave function renormal-ization factor ZΩ which, in the renormalization group equation means its own anomalousdimension γΩ(g). These factors for actual hadronic on shell states would require findingsay the proton pole in the F.T. of

⟨0|TΩ(x)Ω†(0)|0

⟩, a task beyond our tools in pertur-

bation theory. But as far as handling the UV divergences is concerned one can choose torenormalization at any scale µ.

Among all of the operators we might choose to describe mesons, conserved currents arespecial. One advantage is that the scale of the operator is fixed by the conserved charge ofthe state examined.

It is particularly fruitful to study hadronic physics through their interaction with leptons.Consider for instance scattering of electrons by protons, which involves hadronic matrixelements of the electromagnetic current operator. Suppose, for instance we are interested inthe electromagnetic properties of the proton. So we are interested in the matrix elementsof the quark parts of the current operator jµ = e0

∑f Qf qfγ

µqf . The quark model suggeststhat the proton is a bound system of two u quarks and one d quark. But we can’t calculatethe wave functions since the quarks interact strongly. But we can nonetheless use symmetryprinciples to narrow our ignorance. Lorentz invariance and current conservation tell us that

〈p′|jµ(0)|p〉 = uλ′

[γµF1(q2) +

[γµ, q · γ]

mp

F2(q2)

]uλ (23.32)

Here q is the momentum transfer q = p′−p. So the two “form factors” F1, F2 summarize ourignorance. We can’t compute these form factors, but we can measure them in electron andphoton scattering, because the current matrix element is just the proton vertex for (virtual)photon emission. Such measurements show that the form factor falls exponentially with qfor a while, but at very high q becomes power behaved. We can interpret this matrix elementas the F.T. of the M.E. of the current operator:∫

d4xeix·q 〈p′|jµ(x)|p〉 = (2π)4δ(p′ − p− q) 〈p′|jµ(0)|p〉 (23.33)

In the limit q → 0 the left side for µ = 0 is proportional to the matrix element of thetotal charge which is just 1 in units of e. From which we learn that F1(0) = 1. Similarconsideration relate F2(0) to the magnetic moment of the proton. For many important EW


processes q is so small that it is accurate to simply replace the form factors by their valuesat q = 0.

One might hope to use perturbation theory at very high q since the QCD coupling is weakthen. However, according to the RG, strict perturbation theory is only valid if all momentaare large and space-like, which is not true of the proton momenta which are on the protonmass shell. If we think of the proton as 3 quarks in a bag (e. g. MIT bag model) a highmomentum virtual photon will strike one of the quarks hard giving it large momentum. Theonly way the proton can recoil is for this momentum to be shared by all three quarks whichrequires at least two gluon exchanges. In this way one can make plausible assertions of thepower behavior of the form factors at high q, but one gets no information of the coefficientof that behavior.

More hopeful is the process called deep inelastic electroproduction, in which a high energyelectron is scattered with large angle off a proton, and one doesn’t observe the final hadrons:in this sense it is a total cross section related by the optical theorem to the imaginary partof the F.T. of the two current matrix element: 〈p|Tjµ(x)jν(0)|p〉.

An even simpler example is the process e+e− → hadrons. We can relate the total crosssection of this process via the optical theorem to the F.T. of 〈0|TjµEM(x)jνEM(0)|0〉While wecan’t calculate this VEV in perturbation theory at all q, we can describe its large q behaviorin perturbation theory.

In this relatively simple case let’s consider how this works in practice. To be clear wewill work only to lowest order in QED perturbation theory, but (in principle) to allorders in the strong interactions. Before looking at the asymptotics at large q, let’srecall that gauge invariance tells us that

i

∫d4x 〈0|TjµEM(x)jνEM(0)|0〉 = (qµqν − q2ηµν)e2

0Π(q2) +O(e40) (23.34)

We factor out the e20 from Π and consider Π to zeroth order in e0. It therefore starts with

the quark loop with no gluon exchange, and in QCD perturbation theory one adds diagramswith more gluons and quark loops.

Π(q2) requires a single subtraction to make it finite. Its normalization is fixed by thecharge of the quarks (relative to the electron charge). For that reason there is no multiplica-tive renormalization, i.e. the anomalous dimension is zero. Thus we can write

Π(q2) = Π(q2) + Π(µ2), Π(µ2) = 0 (23.35)

where Π is finite when expressed in terms of g(µ). Since Π is independent of µ we have theRG equation

µ∂Π

∂µ= 0 = µ

∂Π

∂µ+ β(g2)

∂Π

∂g2+ µ

∂Π

∂µ(23.36)


By dimensional analysis we can write

µ∂Π

∂µ= ε(g) (23.37)

µ∂Π

∂µ+ β(g2)

∂Π

∂g2= −ε(g) (23.38)

To compute ε we use the fact that Π(µ2) = 0, so that ε = −µ∂Π(q2)/∂µ evaluated at q2 = µ2.Write q = Qq where q2 = 1. Then by dimensional analysis we can replace ∂/∂µ by

−∂/∂Q. Introducing t by β(g(t)) = dg(t)/dt as before, we have(∂

∂t− ∂

∂ lnQ

)(Π(Q, g) +

∫ t

0

dt′ε(g(t′))

)= 0

Π(Q, g) +

∫ t

0

dt′ε(g(t′)) = F (t+ lnQ) (23.39)

Setting t = 0, giving F (lnQ) = Π(Q/µ, g) and then restoring it gives F (t + lnQ) =Π(Qet/µ, g). Then settingQ = µ and remembering Π(µ, g(t)) = 0 gives Π(et, g) =

∫ t0dt′ε(g(t′))

or

Π(Q/µ, g) =

∫ lnQ/µ

0

dt′ε(g(t′)). (23.40)

Asymptotic freedom then allows us to use perturbation theory to calculate Q→∞.To get the first two terms in ε we need to calculate Π to two loops. At one loop we simply

multiply ordinary vacuum polarization by N∑

iQ2i to get

Π1 = − N

12π2

∑i

Q2i ln

q2

µ2(23.41)

The two loop diagrams are just the corresponding QED versions (which we can read off fromJost and Luttinger) times N2−1

2

∑iQ

2i , which is N times the sum of squared charges times

the Casimir operator in the fundamental representation:

Π2 = −N2 − 1

2

g2

64π4

∑i

Q2i ln

q2

µ2(23.42)

Setting N = 3 for QCD this gives

Π =1

4π2

∑i

Q2i

(1 +

g2

4π2

)lnµ2

q2+O(g4) (23.43)

ε = −µ ∂

∂µΠ = − 1

2π2

∑i

Q2i

(1 +

g2

4π2

)+O(g4) (23.44)


Therefore

Π

(Q

µ, g

)= − 1

2π2

∑i

Q2i

∫ lnQ/µ

0

dt′[1 +

g2(t′)

4π2+O(g4)

](23.45)

Assuming µ,Q are large enough so that g(t′) is small over the whole integration range wecan use the one loop running coupling

g2(t′)

4π2=

2

(11− 2Nf/3)t′ + 8π2/g2(23.46)∫ t

0

dt′g2(t′)

4π2=

2

11− 2Nf/3ln

(11− 2Nf/3)t+ 8π2/g2

8π2/g2

=2

11− 2Nf/3ln

g2

g2(t)(23.47)

So finally

Π

(Q

µ, g

)≈ − 1

2π2

∑i

Q2i

[1

2lnQ2

µ2+

2

11− 2Nf/3ln

g2

g2(Q)

](23.48)

valid for q2 →∞. This conclusion is solid for q large and spacelike. To apply calculations likethis to an actual experiment, we must assume that the above formula also gives the correctasymptotic behavior for large negative Q2. The total cross section for e+e− → hadronsis proportional to the Imaginary part of Π which is non zero with the above approximateformula for negative (timelike) Q2. Putting Q2 → |Q2|eiπ gives

Im Π =1

2π2

∑i

Q2i

[π

2+

2

11− 2Nf/3Im ln

g2

g2(Q)

](23.49)

To evaluate the imaginary part of the second term we note that ln(A+ iB) = ln√A2 +B2 +

i arctanB/A, so its imaginary part is arctan(B/A).

g2

g2(q)=

(11− 2Nf/3)(ln |Q2|/µ2 + iπ) + 16π2/g2)

16π2/g2(23.50)

for which

B

A=

π(11− 2Nf/3)

(11− 2Nf/3) ln |Q2|/µ2 + 16π2/g2

Im Π ≈ 1

4π

∑i

Q2i

[1 +

4

(11− 2Nf/3) ln |Q2|/µ2 + 16π2/g2

](23.51)

where we used B A to replace arctan(B/A) with B/A. For a quick normalization werecognize that the corresponding quantity for e+e− → µ+µ− is 1/12π, so the ratio

R =σhadrons

σµ+µ−≈ 3

∑i

Q2i

[1 +

4

(11− 2Nf/3) ln |Q2|/Λ2QCD

](23.52)

≈ 3∑i

Q2i

[1 +

g2(|Q2|)4π2

](23.53)


where ΛQCD = µ2e−16π2/g2/(11−2Nf/3) is a fundamental scale of QCD. It is an RG invariant!The first term in brackets is the parton model result and the second RG improved QCDcorrections.

If we look back and keep the N dependence, we find

R = N∑i

Q2i

[1 +

3

8

(1− 1

N2

)Ng2(|Q2|)

4π2

](23.54)

A couple of points to note. The rough agreement of R with experiment requires N = 3 andso represents a measurement of the number of colors. A useful analysis tool of QCD is tostudy it in the large N limit as suggested by ‘t Hooft: N →∞ with Ng2 fixed. For N = 3,R/N differs from the N →∞ limit by about 10%.

Our calculation of the prediction for R breaks into two steps. (1) Use asymptotic freedomto calculate Π at large positive Q2 (so q is spacelike). (2) Continue the asymptotic result toQ2 < 0 and calculate the imaginary part to get R. In retrospect it is clear that if one simplycalculates the total cross section for e+e− → qq + gluons with 0 or 1 additional gluon in thefinal state in perturbation theory, and then substitutes the running coupling g2(Q) for g2,one would obtain the same answer.

23.4.1 Interpreting naive perturbation theory in QCD

On the one hand we have a more or less solid way to connect the physics of asymptoticfreedom with observed processes: first get asymptotics at large spacelike momenta and thencontinue the result to time like momenta which is where the physics happens. On the otherhand this procedure produces the result more physically in terms of the production of quarksand gluons, in spite of the fact that it is hadrons not quarks and gluons that are actuallyproduced. By focussing on inclusive processes (e.g. total cross sections) we avoid confrontingquarks and gluons. This is the rule: don’t look to closely at the final state.

But there must be some truth to the quark and gluon picture of the final state. This leadsto the idea that hadrons will be produced in “jets” that reflect their origin as a producedpair of quarks. Thus we say that a pair of high energy quarks are really produced but overtime the quarks find antiquarks (or vice versa) and fragment into a shower of hadrons thatthat remembers the parent quark’s momentum. If no gluons are produced with the quarksthe hadrons would distribute into two back to back jets. If a hard gluon is produced alongwith the quarks, the final state would mimic three jets. Each produced quark or gluon withsufficient energy will fragment into a jet of hadrons. We summarize a typical sequence ofevents:

1. qq produced at a point with large momenta, so αs(Q) 1

2. As qq separate αs increases.

3. At R & 1/ΛQCD perturbation ceases validity.

4. Each quark (or gluon) → a jet of hadrons


5. Jet momentum remembers momentum of parent quark or gluon.

Just as in QED, there are IR divergences in the emission of soft or collinear gluons individualcontributions, but they cancel, leaving O(αs) corrections. Non collinear high momentumgluons get interpreted as gluon jets. The dictionary is that jet scattering cross sections onaverage are the same as the cross sections of the parent quark or gluon.

23.5 Operator product expansion (OPE)

Correlators of fundamental operators are made finite by wave function renormalization:Γbare = Zn/2ΓRen. But a more complicated operator may introduce new infinities thatrequire additional renormalization. For example inserting an operator φ2 is equivalent to thezero separation limit of two elementary φ’s.

Our discussion of e+e− → hadrons is an example of a more general technique, the operatorproduct expansion (OPE). We related the desired cross section to the imaginary part of a twocurrent correlation function, and then by analyzing Q→∞ with use of the renormalizationgroup obtained a prediction for the cross section at high energy. In coordinate space Q→∞corresponds to the separation of the current coordinates going to zero. K. Wilson proposedthat in a general context one could make a short distance expansion of the product of twolocal operators of the form

Ω1(x)Ω2(0) ∼∑n

Cn(x)Ωn(0), x→ 0 (23.55)

Here one imagines that n labels all the possible local operators. We expect the limit x→ 0to be divergent, and the proposal asserts that all divergences are captured by the c-numbercoefficients Cn(x). Operators of higher dimension will have less singular coefficients. In oursimple example we only used the VEV of two currents, rather than the actual operators, andsince the expectation was in the vacuum, theQ→∞ limit only involved the identity operatorin the OPE. Wilson’s proposal is much more powerful because the same coefficient functionswill appear no matter what matrix element is evaluated. Although physically plausible, theoperator product expansion in this most powerful form is not completely proven.

Although we will not go into details, the OPE is used to analyze deep inelastic scatteringby taking the OPE of two currents (very schematically)∫

d4xeiq·x〈p|ψ(x)γµψ(x)ψ(0)γνψ(0)|p〉

∼∑〈p|ψ(0)γµ1D

µ2 · · ·Dµsψ|p〉Cs(q2)qµ1 · · · qµs (23.56)

∼∑(

q · pq2

)sAs(q

2/µ2, g2) (23.57)

The (engineering) dimension of an operator on the right side is 3 + s− 1 = 2 + s, and it willalso have an anomalous dimension γs(g

2). The coefficient function must scale as (qµ/q2)s


and will inherit the anomalous dimension −γs(g2) because the operators on the left have noanomalous dimension. then the terms are all multiplied by (q·p/q2)s which is held fixed in thedeep inelastic scattering experiment. It is fashionable to define the twist of an operator as itsdimension minus its spin, so all of the operators we listed have twist 2. In the deep inelasticscaling limit they all contribute, but higher twist operators get suppressed as q2 → ∞.There are then an infinite number of operators that will contribute. For kinematic reasons−2q ·p/q2 > 1. To see this write 2q ·p = (q+p)2−q2 +m2

p so −2q ·p/q2 = 1− (q+p)2/q2 ≥ 1because (q + p)2 < 0 because q + p must be time-like. A few terms will dominate only inthe unphysical domain −2p · q q2, which can be reached by analytic continuation viadispersion relations. We will look at these issues more closely next semester.


Chapter 24

Spontaneous Global SymmetryBreaking: Chiral Dynamics

24.1 Effective Action and Potential

The concept of effective action is a very useful tool to discuss symmetry breaking in quantumfield theory. We start by constructing it in scalar quantum field theory, but it will becomeclear that its scope is much more general than that. So let’s imagine a Lagangian for anynumber of scalar fields of the form

L = − !

2

∑a

(∂φa)2 − V (φ) (24.1)

where we put no limitations on V (φ) for the moment. We first define a generating functionfor correlators of these fields:

Z(J) =

∫Dφei

∫d4x(L+Ja(x)φa(x) ≡ eiW (J). (24.2)

Correlators of any number of φ’s are obtained by taking that number of functional derivativesof Z w.r.t. the J ’s. These correlators include disconnected diagrams as well as connectedones. Functional derivatives of W (J) give connected correlators.

Next we define an effective field ϕ by the formula

ϕa(x) ≡ 1

iZ

δZ

δJa(x)=

δW

δJa(x)(24.3)

. By this definition it is clear that ϕ has the interpretation of the expectation of the scalarfield φa in the presence of the source J . In the limit J → 0 this is simply the VEV of φa. Thefinal step in the construction of the effective action is to perform a Legendre transformation,defining the effective action

Γ(ϕ) ≡ W (J(ϕ))−∫d4x

∑a

ϕaJa (24.4)

241

The point of the Legendre transform is to remove the implicit dependence of Γ on ϕ throughJ so that

δΓ

δϕa(x)= −Ja +

δJbδϕa

[δW

δJb− ϕb

]= −Ja(x) (24.5)

We call Γ(ϕ) the effective quantum action, because when the source Ja = 0, Γ is stationaryunder variations of the effective field. Solutions of this action principle are possible resultsfor the vacuum expectation value of the quantum field ϕ = 〈0|φ|0〉

Although we have gone through this construction using the scalar field theory as a crutch,it should be clear that every step of the construction goes through if the source is coupledto a composite local field. for example the qi(I ± γ5)qj in QCD. Then the interpretation ofφ±ij would be 〈0|qi(I ± γ5)qj|0〉.

For the case of scalar field theory, the effective action Γ(ϕa is the generating functionalfor the 1PIR Feynman diagrams. To see this we first note the pair of relations

i 〈φa(x)φb(y)〉c =δ2W

δJa(x)δJb(y)=δϕb(y)

δJa(x)(24.6)

δ2Γ

δϕa(x)δϕb(y)= − δJb(y)

δϕa(x)(24.7)

which together show that∫d4z∑c

δ2Γ

δϕa(x)δϕc(z)i 〈φc(z)φb(y)〉c = −δabδ(x− y). (24.8)

This means that −iδ2Γ/δϕa(x)δϕb(y) is the inverse connected two point function, which iswhat we mean by the 1PIR two point function. Calculating higher functional derivativesof Γ, one finds that they are 1PIR. Carrying on this analysis to higher order shows thatthe connected diagrams are interpreted as trees, with the 1PIR diagrams as vertices and〈φa(x)φb(y)〉 = −iδ2W/δJa(x)δJb(y) as the propagator.

In this regard it is helpful to regard perturbation theory as a semiclassical approximation.To see this, restore the ~ dependence in the path integral, by multiplying the action by1/~, so the limit ~ → 0 is a stationary phase approximation. On the other hand onecan then rescale the fields by φ →

√~φ, to make the quadratic derivative terms in the

Lagrangian, which determine the perturbation propagator, independent of ~. Alter thisrescaling, V (φ)→ V (

√~φ), so an expansion in powers of ~ is also an expansion in the cubic

and higher terms in the Lagrangian, a term φn → ~(n−2)/2. Linear terms in φ are removedby the stationary phase condition applied at leading order. This point of view shows thatthe zeroth order approximation to the effective action is just the classical action:

δ

δJa(x)

(∫d4xL(φ) +

∫d4xJa(x)φa(x))

)= φa(x) (24.9)

so in the classical limit ϕa = φa so that

Γ = W − ϕaJa =

∫d4xL (24.10)


precisely the classical action!

24.1.1 Effective potential

For the purposes of studying the vacuum, which is assumed to be Poincare invariant, weshould find that 〈φa〉 = ϕa = constant. In that case the effective action is proportional tothe space time volume, and we define the effective potential V(ϕ) by Γ = −(V T )V(ϕ). Thevalue the VEV actually assumes must be a stationary point of V

dVdϕ

= 0 (24.11)

for ϕa = va implies that 〈φa〉 = va.The effective potential is the first term in and expansion of the effective action about

derivatives of ϕ:

Γ(ϕ) =

∫d4x

[−V(ϕ)− Z(ϕ)(∂ϕ)2 + · · ·

](24.12)

More generally, if Ja(x) is time independent the path integral at large times projectsonto the ground state of the source dependent Hamiltonian

HJ = H0 −∫d3xJa(x)φa(x) (24.13)

and W (J) → −2TEG(J). If there are more than 1 stationary points of V , the theory willchoose the one that is the absolute minimum. In the classical limit this simply says that weshould do perturbation theory with φ = v+ φ with v the minimum of the classical potential.

A final general point. By construction the effective action and effective potential willpossess the same symmetries under transformations of ϕa as the quantum theory has underthe corresponding transformations of φa. Since φa does not have to be one of the fundamentalfields of the theory, identifying these symmetries is not completely straightforward. Becauseof this The effective potential is the perfect language for the discussion of spontaneoussymmetry breaking. This is especially so in quantum field theories whose UV divergencescan spoil symmetries.

It is important to appreciate that the dependence of Γ on ϕ is obtained dynamically bythe introduction of the source. One is not always guaranteed the existence of such a sourcefor all values of ϕ. This is seen by inspecting the construction in the classical approximationfor the quartic potential

V (φ) =λ

4!(φ2 − a2)2 (24.14)

Then J = dV/dφ = (λ/6)φ(φ2 − a2). Inverting this equation to get φ(J) is ambiguousbecause it is multi-valued. The dynamics from the source is given by W (J) which can bereconstructed from V :

−W (J) = V (φ(J))− φ(J)V ′(φ(J)) (24.15)


When J varies from −∞→ 0, φ(J) varies from −∞τ0−a, and when J varies from 0→ +∞,φ varies from a → +∞. These two curves span the whole range of J following the trueminimum of −W (J). One can of course continue the relation φ until the point where v′′ = 0which follows a curve above the ground state curve (this is analogous to supercooling orsuperheating in thermodynamics). There is a third branch connecting the inflection pointson which the unstable φ = 0 point lies.

24.1.2 Feynman diagrams for the effective potential

When the order parameter is the expectation of a fundamental scalar field, s systematicprocedure to calculate the effective potential in perturbation theory is the expansion innumber of loops. To set this up we first notice that

eiΓ[ϕ] =

∫Dφei

∫d4x(L[φ]+J(φ−ϕ) =

∫Dφei

∫d4x(L[ϕ+φ]+Jφ (24.16)

Then we can expand L[ϕ + φ] in a power series in φ with ϕ dependent coefficients anddevelop the logarithm of the path integral as a sum of connected 1PIR vacuum Feynmandiagrams. The cancellation of the 1 particle reducible diagrams determines the choice ofsource J . But in practice one simply drops the source term together with all one particlereducible diagrams. For the effective potential, one choose ϕ =constant:

L[ϕ+ φ] + φJ = −V (ϕ)− 1

2(∂φ)2 − 1

2V ′′(ϕ)φ2 + · · · (24.17)

the linear term in φ is dropped because it would contribute a reducible diagram (or would becancelled by choice of J), If we neglect the cubic and higher powers of φ, The path integralis a Gaussian and leads to

−iV TV(ϕ) = −iV TV (ϕ)− 1

2Tr ln

(−∂2 − V ′′(ϕ)

)+ · · · (24.18)

To calculate the Tr ln we can go to momentum space with Wick rotation:

Tr ln(−∂2 − V ′′(ϕ)

)= iV T

∫d4p

(2π)4ln(p2 +m2(ϕ)) (24.19)

where we defined m2(ϕ) ≡ V ′′(ϕ).


24.1.3 Symmetries of the effective action

Suppose the action and path integral measure are invariant under φa → Mabφb. We claimthat this will imply that the effective quantum action is symmetric under ϕa → Mabϕb.Carrying out such a transformation in the path integral with a source Ja shows that

eiW (J) =

∫eiS+iJ ·φ =

∫eiS+iJ ·Mφ = eiW (JM) (24.20)

so W [J ] = W [JM ]. Then

Γ[ϕ] = W [J ]− Jaϕa = W [JM ]− (JM)a(M−1ϕ)a = Γ[M−1ϕ] (24.21)

where we used the fact that M−1ϕ is the effective field for W [MJ ]. To see this differentiatethe relation W [J ] = W [JM ]:

ϕa ≡δW [J ]

δJa= Mabϕ

Mb , ϕMb = (M−1)baϕa. (24.22)

In other words, at least for homogeneous linear symmetries of the fundamental action, theeffective action has the same symmetries. In the case of constant ϕ, this same symmetryextends to the effective potential.

24.1.4 Spontaneous Symmetry breaking

Symmetry plays a central role in our understanding of particle physics. In many casesthe symmetry is only approximate because breaking terms in the Lagrangian happen tohave small coefficients. But there are symmetries that we take to be exact. these includespacetime symmetries such as Poincare invariance. But the discrete spacetime symmetrieslike parity, time reversal and charge conjugation are broken in Nature. In gauge theories,gauge invariance is essential for self-consistency.

The apparent breaking of symmetry can be real, due to terms in the Quantum effectiveaction that don’t respect it, or spontaneous, meaning that the quantum action is invariantbut the vacuum state is not. Technically this means that the minimum of the quantumeffective potential ϕa 6= 0. We have already encountered spontaneous symmetry breakingin the Higgs sector of the standard model, without fully recognizing it. When we assumedthat the scalar field had a nonzero vacuum expectation value, we were tacitly assumingthat spontaneous breaking was taking place. In this case the relevant symmetry was theSU(2)× U(1) gauge symmetry. So let’s begin our study with that example.

The standard model Higgs field was taken to transform as a doublet under the SU(2)factor and to have weak hypercharge Y = −1. Therefore the components of the doubletare complex. If the vacuum were invariant under the symmetry, we would have 〈0|φ|0〉 =⟨0|U †φU |0

⟩which would imply that 〈0|φ|0〉 = 0 since U †φU has different components than

φ. Let us consider the renormalizable Lagrangian for the scalar field

L = −∂φ† · ∂φ− λ

4(φ†φ− v2)2 (24.23)


The zeroth order effective potential is

V(ϕ) =λ

4(ϕ†ϕ− v2)2 (24.24)

which clearly is minimized when ϕ†ϕ = v2. For definiteness consider the solution

ϕ =

(v0

)(24.25)

with v real. We then proceed by shifting φ1 = v + φ1 and φ2 = φ2, so that

φ†φ− v2 = v(φ1 + φ∗1) + φ†1φ1 + φ†2φ2 (24.26)

It follows that

V (φ) =λ

4

[v(φ1 + φ∗1) + φ†1φ1 + φ†2φ2

]2

(24.27)

= v2λ(Re φ1)2 + cubic and quartic (24.28)

This shows that the real part of φ1 gains a mass, while the imaginary part of φ1 and bothreal and imaginary parts of φ2 remain massless. To identify the mass of Re φ1, we note thatthe derivative terms involving φ can be written

−∂φ† · ∂φ = −(∂Re φ1)2 − (∂Re φ2)2 − (∂Im φ1)2 − (∂Im φ2)2 (24.29)

From which it follows that the mass squared of Re φ1 is m2h = v2λ. This is the Higgs particle,

whose measured mass of mh ≈ 125GeV gives information about the quartic coupling λ. Thethree massless scalars play the role of the zero helicity states of the massive vector bosonsof the standard model.

The existence of massless scalars whenever a continuous symmetry is spontaneously bro-ken is the content of the Goldstone theorem and they are called Goldstone bosons, or moreproperly Nambu-Goldstone bosons. The reasoning behind the theorem is simple using theconcept of the effective action and potential. In the scalar field example we just discussed,there are four real scalar fields φa, which we can assemble into a 4 component vector andthe potential is a function only of its squared length,

∑4a=1 φ

2a. If it has a minimum with

φ0 6= 0, then any O(4) rotation of φ0 is also a minimum. An infinitesimal rotation of φ0

defines directions perpendicular to φ0 along which the potential doesn’t change. Scalar fieldsin those directions will not have a mass term.

The Goldstone theorem requires a broken continuous symmetry. For example a single realscalar field could have a symmetry under φ→ −φ for instance the potential (λ/4!)(φ2 − a2)has two minima at φ = ±a. Since those minima are discrete there is no implication ofmassless particles. In terms of the fluctuation field φ = φ− v the potential has a mass term,a cubic term, and a quartic term. the content of the discrete symmetry is a specific relationbetween the three coefficients of these terms.


24.2 Goldstone Bosons and Soft Pion Theorems

24.2.1 Goldstone’s Theorem

Suppose there is a vacuum state with 〈0|φa|0〉 = va 6= 0. Then δΓ/δϕa = 0 for φa = va =constant. Suppose there is a continuous symmetry of Γ under ϕ → ϕ − iεGϕ. Then,restricting to constant fields it follows that

−iεGabϕb∂V∂ϕa

= 0 (24.30)

for all ϕ. Differentiating w.r.t. ϕ gives

Gac∂V∂ϕa

+Gabϕb∂2V

∂ϕc∂ϕa= 0 (24.31)

again for all ϕ. Setting ϕ = v makes the first term zero, so we conclude that

Gabvb ∂2V∂ϕc∂ϕa

∣∣∣∣ϕ=v

= 0 (24.32)

Thus Gabvb is a zero eigenvector of the second derivative matrix of the effective potential,

which is intuitively the mass term. More precisely, in terms of the effective action

∂2V∂ϕc∂ϕa

= limq→0

∫d4xeiq·x

δ2Γ

δϕc(x)δϕa(0)(24.33)

which is the q → 0 limit of the F.T. of the inverse propagator. The existence of a zeroeigenvector of the latter implies that

∫d4xeiq·x 〈φa(x)φb(0)〉 has a pole at q2 = 0. That

is, there are massless particles (Nambu-Goldstone bosons) coupling to the field φa. Thisconclusion depends on Gabv

b 6= 0, i.e. that va is not invariant under the symmetry.The number of NGB’s depends on the symmetry and the nature of the symmetry break-

ing. For example if the symmetry is O(N), with φa transforming as a vector, There will beN − 1 NGB’s. We can count these as follows: there are N(N − 1)/2 generators of O(N).Of these the vector v is invariant under (N − 1)(N − 2)/2 generators. The difference N − 1is the number of generators such that Gv 6= 0. More generally we can say that if G is thesymmetry group and H is the subgroup of G that is unbroken, then the NGB’s are in 1-1correspondence with the generators in G/H1.

24.2.2 SSB and matrix elements of operators.

By definition the order parameter ϕa changes under the symmetry, so the correspondingoperator satisfies

[Ga, φb] = (T a)bcφc (24.34)

1The meaning of the coset G/H is that two elements g1, g2 of G are identified if g1 = g2h with h ∈ H.


But in field theory symmetry generators are integrals of time components of conserved cur-rents, so we expect the local version

[j0a(x, 0), φb(0)] = (T a)bcφc(0)δ(x). (24.35)

And current conservation leads to

qµ

∫d4xe−iq·x 〈0|Tjµa (x)φb(0)|0〉 = −i

∫d4xe−iq·xδ(t) 〈0|[jµa (x), φb(0)]|0〉

= −i(T a)bc 〈0|φc(0)|0〉 (24.36)

If there were no massless particles the left side would go to 0 as q → 0, implying that〈0|φc(0)|0〉 = 0. Conversely if 〈0|φc(0)|0〉 6= 0, the left side must have singularities as q → 0,and hence massless particles (GB’s).

Recall our discussion of the reduction formalism where we considered single particlecontributions to two point functions:∫

d4xe−iq·x 〈0|Tjµa (x)φb(0)|0〉 =∑λ

〈0|jµa (0)|λ〉〈λ|φb(0)|0〉 −i(2π)32q0

q2 +m2λ

(24.37)

On general grounds we can write

〈0|jµa (0)|λ〉 =iqµFaλ

(2π)3/2√

2q0, 〈λ|φb(0)|0〉 =

√Zbλ

(2π)3/2√

2q0(24.38)

and the left side of the conservation identity reads

∑λ

q2Faλ√Zbλ

q2 +m2λ

→∑mλ=0

Faλ√Zbλ = −i(T a)bcvc (24.39)

as q → 0. We see that the low energy couplings of the massless NGB’s are linked to theorder parameter. Note that consistency demands that the NGB’s couple both to the orderparameter and to the current for the symmetry. So one could choose either field to use inthe reduction formalism to derive scattering amplitudes for the NGB’s.

Matrix elements of the current 〈β|jµa (0)|α〉, where |β〉, |α〉 are general outgoing or incom-ing multiparticle states, have massless poles in q2 = (pβ − pα)2 of the form

qµ

q2FaB 〈β,B|α〉 (24.40)

where B is the NGB. Here −i 〈β,B|α〉 ≡ MBβα has the interpretation as the S-matrix for

NGB emission during the core process α→ β. We can write the current matrix element

〈β|jµa (0)|α〉 =iFaBq

µ

q2MB

βα +Naµβα (24.41)


where N has no pion pole. Then the conservation of the current implies

iFaBMBβα + qµN

aµβα = 0. (24.42)

The generic consequence of this equation is that NGB emission vanishes at low energy q → 0.However there are exceptions to this statement: when the current attaches to an externalline in a scattering amplitude the propagator between the attachment point and the rest ofthe diagram has a singularity as q → 0.

1

(q + p)2 +m2=

1

q2 + 2q · p∼ 1

2p · q. (24.43)

24.2.3 Approximate symmetries

We shall be considering approximate symmetries in QCD. So we need to develop a treatmentof spontaneous breaking of an approximate symmetry. The upshot will be that the NG bosonsgain a small mass, and are then called pseudo NGB’s. We use the conceptual framework ofthe effective potential. Write the latter as

V(ϕ) = V0(ϕ) + V1(ϕ) (24.44)

where V0 is symmetric and V1 breaks the symmetry, and we assume breaking is small. Weassume V ′0 = 0 for some nonzero ϕ0. Then we search for a minimum of V at ϕ = ϕ0 + ϕ1,with ϕ1 ϕ0. Working to first order in the symmetry breaking we get the condition

∂2V0

∂ϕa∂ϕb

∣∣∣∣ϕ0

ϕ1b = −∂V1

∂ϕa

∣∣∣∣ϕ0

(24.45)

This is an inhomogeneous linear equation for ϕ1. But the matrix on the left has zeroeigenvalues by the Goldstone theorem. The zero eigenvectors are just Tabϕ

b0. Taking the

inner product of these eigenvectors with the left and right side of the equation leads to therequirement that

∂V1

∂ϕa

∣∣∣∣ϕ0

Tabϕb0 = 0. (24.46)

This is the vacuum alignment condition: of the many choices one can make for ϕ0, one mustpick the one that aligns with the symmetry breaking term in this sense. The manifold ofpossible vacua is given by the symmetry transformations applied to any particular vacuumUϕ0 The alignment condition can be interpreted as finding the U which minimizes V1(Uϕ0).

24.2.4 (Approximate) Symmetries of QCD

As the main application of these ideas we finally turn to QCD. Starting with the standardmodel with Higgs mechanism implemented, we can diagonalize the fermion mass terms,


generating the CKM flavor mixing matrix. We then isolate QCD by taking MW ,Mz →∞.This just means we only consider QCD processes with energy and momenta much smallerthan the vector boson masses. In that limit the interactions between leptons and quarks ispurely electromagnetic, which we also neglect in first instance. That leaves us with “pure”QCD, interacting quarks and gluons, with dynamics governed by the Lagangian

LQCD = −1

4F aµνF

aµν +∑f

Qf (−mf + iγ ·D)Qf (24.47)

There is strong evidence that each quark has a different nonzero mass parameter whichlimits the symmetries to U(1)6 generated by the number operators of each kind of quark.But the masses of the quarks range widely from the up and down quarks of mass a fewMeV to the top quark with mass of 175 GeV! From asymptotic freedom we can define thebasic hadronic scale as ΛQCD which from high momentum experiments is estimated to beseveral hundred MeV (depending on the renormalization scheme chosen). It is certainly agood approximation to set mu = md = 0, and perhaps not so bad an approximation to alsoset ms = 0. So we will study QCD with nf − 2 or nf = 3 massless quarks. We will alsoassume P,C, T invariance. We have all together Nf = 6 flavors of quark, nf of which aretreated initially as massless.

24.2.5 Hypothesis of chiral SSB

With nf massless quarks one can identify a matrix scalar order parameter σkl =⟨0|LkRl|0

⟩with L and R transforming under independent nf × nf unitary matrices σkl → (U †LσUR)kl.Thus QCD has an apparent U(nf )× U(nf ) symmetry in this approximation. We can treateach U(nf ) symmetry as U(1)×SU(nf ). We shall find that one combination of the U(1)’s isanomalous, meaning that quantization destroys that symmetry, and the other combination,namely [U(1)× U(1)]diagonal is not spontaneously broken. So we will focus on the SU(nf )×SU(nf ) chiral symmetry. The fundamental hypothesis is that, with nf massless quarks inQCD, SU(nf ) × SU(nf ) is spontaneously broken to [SU(nf ) × SU(nf )]diagonal = SU(nf ).This means that we assume that the effective potential in this approximation

V0(σ) = V0(U †LσUR) (24.48)

has a minimum at

σkl = Sδkl, S = S∗ 6= 0 (24.49)

where the Kronecker delta ensures that U †σU = σ and the hypothesis that σ is real isnecessary if parity is not also spontaneously broken. Since the nf quarks actually have smallmasses, we should take them into account. The mass terms in the Lagangian of QCD are

−∑f

mf qfqf = −∑f

mf (LfRf + RfLf ) (24.50)


so we make the hypothesis that the symmetry breaking term in the Effective potential is

V1 = Tr[Mσ + σ†M†] (24.51)

By a chiral transformation we can bring an arbitrary complex mass matrix M to diagonalform with positive entries times a common phase factor. So wolog we can take Mkl =eiαmkδKl, which we shall do. For parity and charge conjugation invariance, α = 0, π, i.e. theoverall phase is just a sign.

Next we address the alignment problem. Before symmetry breaking the vacuum orderparameter could be SU †LUR ≡ SU , and we should choose U to minimize

V1 = STr[MU + U †M] = Seiγ∑k

mk(U + U †)kk (24.52)

Where U ∈ SU(nf ). If we assume P and C invariance. Seiγ is real but can be either positiveor negative. If it is negative the minimum is evidently realized with U = I. If it is positiveand nf is even, the minimum is realized with U = −I because then −I ∈ SU(nf ). Howeveris nf is odd, −I is not in SU(nf ) and the alignment solution is more complicated! We shallassume that Seiγ < 0 for QCD with 3 light quarks.

24.2.6 Masses of Nambu-Goldstone bosons in QCD.

From our general discussion the NGB’s of chiral SSB are determined by the symmetrytransformations that change 〈σkl〉:

〈σkl〉 → σ(U †LUR)kl (24.53)

Writing U †LUR = U → I − iεaGa, the Ga are the generators of SU(nf ) which are n2f − 1 in

number, so there are that many NGB’s. For nf = 2 (only u, d massless these are just thethree pions π0, π±. For nf = 3 (s is also massless) add the K±, K0, K0, η to the three pions.

In building the low energy effective action, we can use an SU(nf ) matrix U = e−iπaλa/F as

the NGB effective field, transforming under SU(nf )×SU(nf ) by U → U †LUUR. To constructan invariant Γ(U) need at least 2 derivatives, which is uniquely.

Γ0(U) = −F2

4Tr∂µU

†∂µU = = −F2

4Trλaλb

∂µπa∂µπbF 2

+ · · ·

= −1

2

∑a

(∂πa)2 + · · · (24.54)

where the dots stand for terms cubic and higher in the NGB fields, πa. Similarly the sym-metry breaking mass term in the effective potential is

V1 = −|S|∑k

mk(U + U †)kk = −2|S|∑k

mk +|S|F 2

∑k

mk(λaλb)kkπaπb + · · · (24.55)


In the case of nf = 2 the λ’s are just the Pauli matrices τa and because πaπb is symmetricwe can replace

τaτb →1

2τa, τb = δab (24.56)

and the mass term becomes

−2|S|∑k

mk +|S|(mu +md)

F 2π2a (24.57)

so all three pions get the same mass m2π = 2|S|(mu +md)/F

2.The case nf = 3 gets more complicated because the Gell-Mann matrices satisfy

λa, λb =4

3δab + 2dabcλc (24.58)

The only λ’s with diagonal entries are λ3, λ8 and we need∑k

mkλ3kk = mu −md,∑k

mkλ8kk =1√3

(mu +md − 2ms) (24.59)

Putting these into the mass term gives

V1 = −2|S|∑k

mk (24.60)

+|S|F 2

[2

3δab∑k

mk + dab3(mu −md) + dab8mu +md − 2m8√

3

]πaπb

Looking up the d coefficients we find

dab3πaπb =2√3π3π8 +

1

2(π2

4 + π25 − π2

6 − π27) (24.61)

dab8πaπb =1√3

(π21 + π2

2 + π23 − π2

8)− 1

2√

3(π2

4 + π25 + π2

6 + π27) (24.62)

Collecting all the quadratic terms in the effective potential gives

|S|F 2

[2

3

∑a

π2a

∑k

mk + (mu −md)

(2√3π3π8 +

1

2(π2

4 + π25 − π2

6 − π27)

)+mu +md − 2ms

3

((π2

1 + π22 + π2

3 − π28)− 1

2(π2

4 + π25 + π2

6 + π27)

)]=|S|F 2

[(mu +md)(π

21 + π2

2 + π23) +

mu +md + 4ms

3π2

8 + (mu −md)2√3π3π8

+(mu +ms)(π24 + π2

5) + (md +ms)(π26 + π2

7)

](24.63)


We identify (π1± iπ2)/√

2 with π±, (π4± iπ5)/√

2 with K±, and (π6± iπ7)/√

2 with K0, K0.π3 is π0 and π8 is η. Because of the mixing term π3π8 these are not strictly mass eigenstates,But the effect of mixing is exceedingly small, of order (mu − md)

2.ms ∼ m4π/m

2K , so we

neglect it in the following,Neglecting this mixing effect we can simply read off the predictions for the meson masses:

m2π =

2|S|F 2

(mu +md), m2K± =

2|S|F 2

(mu +ms)

m2K0 =

2|S|F 2

(md +ms), m2η =

2|S|3F 2

(mu +md + 4ms) (24.64)

These results predict one relation between the meson squared masses, namely

2m2K0 + 2m2

K± − 3m2η −m2

π = 0 (24.65)

In addition the calculations give estimates for the ratio of quark masses. For this purpose wetake seriously the K+/K0 mass difference of a few MeV which is comparable to the π+/π0

mass difference, which the present physics does not allow for. It is plausible that thesedifferences are due to EM corrections. If we add EM couplings to the QCD Lagrangian, in thelimit that u, d, s are massless, we notice that because the d and s quarks have the same charge,which is half the u quark charge, there remains a chiral (SU(2) × U(1)) × (SU(2) × U(1))symmetry, which ,means that 4 NGB’s remain, namely π0, η,K0, K0 will stay massless evenafter EM corrections. So only π±, K± can get a mass. But these mesons are related by theresidual SU(2) so if no mass terms are included they would get a common mass. We don’tcompute it but add it as a parameter ∆ = m2

π+ −m2π0 . Then we add ∆ to the right side of

the two formulas for m2K± and m2

π± . There is still only one relation because we are fittingone more mass with one new parameter:

2m2K0 + 2m2

K± − 3m2η − 2m2

π± +m2π0 = 0 (24.66)

Then we get, for example,

mu

ms

=2m2

π0 −m2π± +m2

K0 −m2K±

m2K0 +m2

K± −m2π±

≈ 0.027 (24.67)

md

ms

=m2π± +m2

K0 −m2K±

m2K0 +m2

K± −m2π±≈ 0.050 (24.68)

The existence of the relation, which is not experimentally perfect, means that equally validexpressions obtained by using it can give different estimates for these quark mass ratios.

24.2.7 Low energy pion scattering

We can use the effective action to calculate the low energy limit of pion pion scattering.Here we use the spontaneous breaking of chiral symmetry for the case nf − 2. We need to


expand the effective action to order π4a.

U = I − i 1

Fπaτa −

1

2F 2π2a +

iπaτa6F 3

π2a +

(π2a)

2

24F 4

U + U † = 2I − 1

F 2π2a +

(π2a)

2

12F 4

∂µU = −i 1

F∂µπaτa −

1

F 2πa∂µπa +

i∂µπaτa6F 3

π2a +

iπaτa3F 3

πa∂µπa

Tr∂µU†∂µU =

2

F 2(∂πa)

2 +2

F 4(πa∂µπa)

2 − 2

3F 4(∂µπa)

2π2a −

4

3F 4(πa∂µπa)

2

=2

F 2(∂πa)

2 +2

3F 4(πa∂µπa)

2 − 2

3F 4(∂µπa)

2π2a (24.69)

Putting everything together gives the effective action:

Γ(πa) = −1

2(∂πa)

2 − m2π

2π2a −

1

6F 2(πa∂µπa)

2 +1

6F 2(∂µπa)

2π2a

+m2π

24F 2(π2

a)2 +O(π6) (24.70)

The pion scattering amplitude predicted by this action is simply the sum of quartic vertices.Let the pion momenta and isospin labels be pi, ai, i = 1, · · · , 4. Then the last quartic termproduces

im2π

3F 2(δa1a2δa3a4 + δa1a3δa2a4 + δa1a4δa2a3) (24.71)

The first quartic term produces

−i 1

3F 2[δa1a2δa3a4i(p1 + p2) · i(p3 + p4) + δa1a3δa2a4i(p1 + p3) · i(p2 + p4)

+δa1a4δa3a2i(p1 + p4) · i(p3 + p2)]

= −i 1

3F 2[δa1a2δa3a4(p1 + p2)2 + δa1a3δa2a4(p1 + p3)2 + δa1a4δa3a2(p1 + p4)2]

= i1

3F 2[δa1a2δa3a4s+ δa1a3δa2a4u+ δa1a4δa3a2t] (24.72)

Finally the second quartic vertex produces

i1

3F 2[δa1a2δa3a4i

2(2p1 · p2 + 2p3 · p4) + δa1a3δa2a4i2(2p1 · p3)

+2(p2 · p4) + δa1a4δa3a2i2(2p1 · p4 + p2 · p3]

= 2i1

3F 2(−)[δa1a2δa3a42p1 · p2 + δa1a3δa2a42p1 · p3 + δa1a4δa3a22p1 · p4]

= 2i1

3F 2[δa1a2δa3a4(s− 2m2

π) + δa1a3δa2a4(u− 2m2π) + δa1a4δa3a2(t− 2m2

π)]


The sum of these three vertices gives the pion-pion scattering amplitude:

M = i1

F 2

×[δa1a2δa3a4(s−m2π) + δa1a3δa2a4(u−m2

π) + δa1a4δa3a2(t−m2π)] (24.73)

Let’s pause to review the validity of the conclusion that this formula gives he exact lowenergy pion scattering amplitude in pure QCD. Recall that Γ is the quantum effective actionwhose tree approximation gives the exact quantum action. We did not calculate Γ directly,but made the hypothesis that chiral SU(2) × SU(2) was spontaneously broken to diagonalSU(2). Based on that hypothesis we then wrote the most general effective action involvingthe NGB’s of the broken symmetry (pions) with the minimum number (2) of derivatives.The last condition means its validity is low energies. Finally we introduced a mass termwhich broke the chiral symmetry, assuming it was linear in quark masses. These are theconditions for the validity of the prediction.

So far the parameter F seems to be an undetermined free parameter which could only becalculated if we could treat QCD nonperturbatively. While this is true, we can measure itsvalue in a process independent of pion scattering, namely the weak decay π → µ + νµ. Tounderstand this point recall the way theNGB effective field transforms under SU(2)×SU(2):

U(x) = e−iτaπa/F → U †LU(x)UR. (24.74)

Specifically consider the transformation generated byQa5 ≡ Qa

R−QaL =

∫d3x

∑f qfγ

0γ5(τa/2)qf .

These transformations satisfy UL = U−1R = U †R. Parameterizing UR = e−ξaτa/2, the transfor-

mation of U(x) is

e−iτaπa/F → e−ξaτa/2e−iτ

aπa/F e−ξaτa/2 = e−iτaπa/F−iξaτa+O(ξπ) (24.75)

by Baker-Hausdorff. This means that, among other effects, the chiral transformation shiftsπa → πa +Fξa. But this shift has to be produced by −i[πa, ξbQb

5], which implies that A05 has

a term linear in πz, namely

A0a = Fπa +O(π2

a), Aµ = −F∂µπa +O(πaa2) (24.76)

This term produces a non zero matrix element

〈0|Aµa |πb,p〉 = −iFpµ δab

(2π)3/2√

2p0(24.77)

The axial current Aµa is a component of the current that couples to the electroweak bosons,and hence can be measured in electroweak processes. In particular the process π− → µ−+ νµis controlled by F , which experiment then fixes to be around 93 MeV. With this value lowenergy pion scattering is completely fixed!


24.3 Adler-Weisberger and Other Sum Rules

24.3.1 Soft pion scattering off nucleons

Here we attempt to introduce baryons (3 quark hadrons) into the effective action formalismwith the goal of deriving low energy amplitudes for NGB-baryon scattering. Spin 1/2 baryonsfall into octets under SU(3)×SU(3) (p, n,Σ±,0,Λ,Ξ0.−), which is not the fundamental irrepof SU(3). In contrast, the proton and neutron are doublets under SU(2) × SU(2). SinceSU(2) doublets are the fundamental irrep of SU(2), the construction of the effective actionis simpler in that case, so we discuss it first.

We introduce an effective nucleon field as as an SU(2) doublet N = (p, n). Then thechiral group sends NL → ULNL and NR → URNR, so that NLU(π)NR and NRU

†(π)NL arecandidate chiral invariants to include in the effective action, where U(π) − e−iτaπa . We canwrite a parity invariant combination using γ5:

NLU(π)NR + NRU†(π)NL = Neiγ5τaπaN (24.78)

Additional chiral invariants include Niγ · ∂N , which we include with the canonical nor-malization and Nγµ(e−iγ5τaπa∂µe

iγ5τaπa)N .This exhausts the possibilities bilinear in N andcontaining at most one derivitive. Introduce the notation U5 = eiγ5τaπa/F so the completelow energy effective action is

Γ(U,N) = −F2

4Tr∂µU

†∂µU + |S|TrM(U + U †) (24.79)

+Niγ · ∂N −mNNeiγ5τaπaN − iCNγµ(e−iγ5τaπa∂µe

iγ5τaπa)N

We shall analyze pion-nucleon scattering which involves up to powers of πa multiplying thenucleon bilinear. So we expand

U5 = I +i

Fγ5τaπa −

1

2F 2π2a +O(π3

a) (24.80)

U †5∂µU5 =

(I − i

Fγ5τaπa

)(i

Fγ5τa∂πa −

1

2F 2∂π2

a

)=

i

Fγ5τa∂µπa −

1

2F 2∂π2

a +1

F 2τaπaτb∂µπb

=i

Fγ5τa∂µπa +

1

F 2iεabcτcπa∂µπb (24.81)

To this order the nucleon bilinear in the effective action is

N

[iγ · ∂ −mN −

imN

Fγ5τaπa +

C

Fγµγ5τa∂µπa +

mN

2F 2π2a +

C

F 2εabcτcπaγ

µ∂µπb

]N

Let us first discuss the pion-nucleon vertex implied by this action. It is given by

u(p′)

(mN

Fγ5τa −

C

Fγµγ5τai(p

′ − p)µ)u(p) = u(p′)

(mN(1 + 2C)

Fγ5

)τa (24.82)


from which we infer the pion-nucleon coupling constant to be

GπN =mN(1 + 2C)

F(24.83)

To relate C to a measurable property of the nucleon, consider the nucleon matrix elementof the axial current Aµa

∑f qfγ

µγ5taqf ,

〈N ′|Aµa |N〉 = u′ta[−γµγ5gA(q2) + qµγ5g2(q2) + iqν [γµ, γν ]γ5g3(q2)]u (24.84)

The hypothesis of SSB for chiral symmetry says that in the limit mu = md = 0 this currentis conserved, so in this limit we should have

u′τa[−q · γγ5gA(q2) + q2γ5g2(q2)]u = 0 (24.85)

Using the Dirac equation this requires −2mNgA(q2) + q2g2(q2) = 0. Now gA is measured innuclear beta decay (for which q ≈ 0) to be roughly 1.2, far from zero. So conservation requiresthe presence of the pion as a NGB, which contributes to the g2 term as g2(q2) = 2GπNFπ/q

2.So we arrive at the famous Goldberger-Treiman relation

GπN =mNgAF

, Goldberger− Treiman (24.86)

Comparing to our single soft pion relation we see that C = (gA − 1)/2.We next turn to two soft pions so we can derive a prediction for low energy pion nucleon

scattering. We just have to use our effective Lagrangian to calculate the tree diagrams forthis process. First we look at the diagrams that don’t involve the C vertices:

u′−im2

N

F 2

[γ5τb

mN − γ · (p+ q)

−m2π + 2p · q

γ5τa + γ5τamN − γ · (p− q′)−m2

π − 2p · q′)γ5τb − δab

1

mN

]u

= u′−im2

N

F 2

[τb

γ · q−m2

π + 2p · qτa + τa

−γ · q′

−m2π − 2p · q′

τb − δab1

mN

]u (24.87)

The low energy limit is the limit of threshold, where all spatial components of momenta arezero and the time components are the masses:

u′[τb

−γ0mπ

−m2π − 2mπmN

τa + τaγ0mπ

−m2π + 2mπmN

τb − δab1

mN

]u

= u′−imN

F 2

[τbτa

1

2 +mπ/mN

+ τaτb1

2−mπ/mN

− δab]u

= u′−imN

F 2

[δab

m2π/m

2N

4−m2π/m

2N

+ iεbacτc−2mπ/mN

4−m2π/m

2N

]u

=mπ

2F 2εabcu

′τcu (1 +O(mπ/mN)) (24.88)


We next show that the total contribution of the vertices involving C is smaller than theseexpression by a factor of mπ/mN . To simplify the argument we evaluate the amplitudes atthreshold. We first look at the diagrams of order C2:

u′−iC2

F 2(q′ · γ)γ5τb

mN − γ · (p+ q)

−m2π + 2p · q

(−q · γγ5τau

→ u′iC2

F 2(mπγ

0)γ5τbmN + γ0(mN +mπ)

−m2π − 2mNmπ

(mπγ0γ5τau

= u′−iC2

F 2mπτb

mN − γ0(mN +mπ)

−m2π − 2mNmπ

mπτau

= u′−iC2

F 2τb

m2π

mπ + 2mN

τau (24.89)

where we used the Dirac equation at threshold, namely (γ0 − 1)u = 0. So this diagram andthe one with meson legs crossed are down by a factor mπ/mN ≈ 1/7 as claimed. To completethe argument we examine the diagrams linear in C.

The four diagrams involving cubics are similar

u′−iCmN

F 2(q′ · γ)γ5τb

mN − γ · (p+ q)

−m2π + 2p · q

(γ5τa)u →imπC

2F 2u′τbτau

u′−iCmN

F 2γ5τb

mN − γ · (p+ q)

−m2π + 2p · q

(−q · γγ5τa)u →imπC

2F 2u′τbτau

u′−iCmN

F 2γ5τb

mN − γ · (p− q′)−m2

π − 2p · q′(q′ · γγ5τa)u → − imπC

2F 2u′τaτbu

u′−iCmN

F 2(−q · γ)γ5τb

mN − γ · (p− q′)−m2

π + 2p · q(γ5τa)u → − imπC

2F 2u′τaτbu

These four diagrams together give at threshold

imπC

F 2u′[τb, τa]u(1 +O(

mπ

mN

)) = 2mπC

F 2εabcu

′τcu(1 +O(mπ

mN

)) (24.90)

This is still linear in mπ, but we have one more diagram linear in C, the four particle vertex

iC

F 2εabcu

′(−iq′ − iq) · γτcu→ −2mπC

F 2εabcu

′τcu (24.91)

which cancels the terms linear in mπ from the other four diagram.she upshot is that C =(gA− 1)/2 only enters low energy pion nucleon scattering at order m2

π. So the C = 0 resultsare the leading contribution. To recapitulate, we have obtained

MπN =mπ

F 2εabcu

′tcu (1 +O(mπ/mN)) (24.92)

This result was first obtained by Steven Weinberg. To interpret the isospin dependence, werecall that the nucleon is an iso doublet I = 1/2 and the pion is an isovector. The pion-nucleon system can then have I = 3/2, 1/2. We can then interpret εacb = i(Tc)ab, with T the


isospin generator for I = 1. Then we have

(Tc + tc)2 = 2 +

3

4+ 2Tctc =

15/4 I = 3/2

3/4 I = 1/2

2Tctc =

1 I = 3/2

−2 I = 1/2(24.93)

so we can also cast the results as

M3/2 = −imπmN

F 2, M3/2 = 2i

mπmN

F 2(24.94)

The conventional πN scattering amplitude f is related to M by f = −iM/(8π(mπ +mN)) which at 0 energy is known as the scattering length. With this prediction a3/2 =−mπ/(8πF

2) ≈ −0.075/mπ and a1/2 = mπ/(4πF2) ≈ +0.15/mπ compared to −0.101± .004

and 0.173± .003.

24.3.2 Dispersion relation for πN scattering

Scattering amplitudes seem to be analytic functions of the energy. This is demonstrably truefor non relativistic potential scattering and is also true in QFT order by order in perturbationtheory. What is less rigorously known is the location of all possible singularities and the highenergy behavior of scattering amplitudes. Experience with perturbation theory associatesat least some singularities with concrete physical facts. the existence of stable particles isresponsible for simple poies in the amplitudes. Also thresholds for the production of particlesare responsible for branch points. A sound working hypothesis is “maximal” analyticity,namely, that the only singularities are attributed to such physical effects. Here we use thishypothesis and Cauchy’s theorem to derive a formula for pion-nucleon scattering in theforward direction in terms of its assumed singularities.

We first specify the kinematics of the process. Let pµ, qµ be the initial momenta of thenucleon and pion respectively. In the forward direction these are also the final momenta.The Mandelstam invariants are s = −(p + a)2, t = 0, and u = −(p − q)2. s and u can beexpressed in terms of the invariant

s = m2π +m2

N + 2mNν, u = m2π +m2

N − 2mN (24.95)

The energy variable ν is the energy of the pion in the nucleon rest frame. For simplicity weshall assume isospin symmetry is exact mu = md. We use the Cartesian basis for the pionisospin πa, a = 1, 2, 3. Then s-u crossing symmetry says that Mba(ν) =Mab(−ν).

The threshold for πN scattering in the s-channel is at s = (mπ +mN)2 which translatesto ν = mπ. Because of crossing symmetry this pole also appears in the u-channel u =(mπ + mN)2 or ν = −mπ. Thus we cut the ν plane from ν = mπ to ν = ∞ and fromν = −∞ to ν = −mπ. The poles corresponding to the stable nucleon are at s = m2

N or atν = −m2

π/(2mN) and at u = m2N or ν = m2

π/(2mN).


To parameterize the pole singularities we write down the Feynman amplitude for the treeprocess, with the physical values for the pion-nucleon coupling and nucleon and pion masses.The s-channel pole is

(iGπN)2uiγ5τb−i(mN − γ · (p+ q))

m2N + (p+ q)2

iγ5τau

= G2πN u−i(mN + γ · (p+ q))

−m2π − 2mNν

τbτau = uiG2

πNγ · qm2π + 2mNν

τbτau

=−iG2

πN2mNν

m2π + 2mNν

τbτa →iG2

πNm2π

m2π + 2mNν

φ†τbτaφ (24.96)

where in the last step we replaced 2mNν in the numerator by its value −m2π at the pole.

In the previous step we used uγµu = 2pµ for the forward spinor matrix element. In thefinal expression φ is the isospin spinor for the initial nucleon. The u-channel nucleon pole isobtained by the substitution ν, a, b→ −ν, b, a. The sum of the two pole terms is

iG2πNm

2π

m2π + 2mNν

φ†τbτaφ+iG2

πNm2π

m2π − 2mNν

φ†τaτbφ

=iG2

πNm2π

m4π − 4m2

Nν2φ†[τbτa(m

2π − 2mNν) + τaτb(m

2π + 2mNν)

]φ

=2iG2

πNm2π

m4π − 4m2

Nν2φ†[m2πδab −mNν[τb, τa]

]φ (24.97)

We next use Cauchy’s theorem to derive a dispersion relation. Pick a point ν in the complexν plane and write

Mba(ν) =1

2πi

∮C

dν ′

ν ′ − νMba(ν

′) (24.98)

Here C is a closed contour encircling ν in a counter-clockwise direction and entirely withinthe domain of analyticity of M. Now deform C until it hugs all the singularities outsideof C. Assuming maximal analyticity, the final contour will consist of large semicircles atinfinity, clockwise circles about the isolated poles, A contour just below the positive real axisfrom +∞ to mπ encircling mπ and returning to = ∞ just above the real axis. And finallya similar contour running from −∞ to −mπ just above the real axis and back to −∞ justbelow the real axis. Assuming the large semicircles give nothing we arrive at the formula

Mba(ν) =2iG2

πNm2π

m4π − 4m2


]φ

+1

2πi

∫ ∞mπ

dν ′

ν ′ − ν[Mba(ν

′ + iε)−Mba(ν′ − iε)]

+1

2πi

∫ −mπ−∞

dν ′

ν ′ − ν[Mba(ν

′ + iε)−Mba(ν′ − iε)] (24.99)


We can change variables in the integral of the last line ν ′ → −ν ′ so that both integrals goover the same region, and also use crossing symmetry Mba(−ν) =Mab(ν):

Mba(ν) =2iG2

πNm2π

m4π − 4m2


]φ

+1

2πi

∫ ∞mπ

dν ′(ν ′ + ν)

ν ′2 − ν2[Mba(ν

′ + iε)−Mba(ν′ − iε)]

+1

2πi

∫ ∞mπ

dν ′(ν ′ − ν)

ν ′2 − ν2[−Mab(ν

′ − iε) +Mab(ν′ + iε)] (24.100)

In obtaining this result we made an assumption that the semi-circular contours at infinitycontributed nothing: we assumed Mba → 0 as ν → ∞. What if this doesn’t hold? Thenwe can write a dispersion relation for M/νn instead, where n is large enough to drop thesemicircular contours. The price paid for this extra convergence is a less powerful dispersionrelation, because the new pole at ν = 0 introduces terms involving M and up to n − 1derivatives of M. Experimental indications are that n = 1 should suffice. The resultingdispersion relation is referred to as a subtracted dispersion relation, because it amounts tostarting with the unsubtracted dispersion relation and subtracting from both sides the firstfew terms of its Taylor series about ν = 0.

Lets write the dispersion relation with 1 subtraction (n = 1):

Mba(ν)−Mba(0) =2iG2

πNm2π

m4π − 4m2


]φ− 2iG2

πNφ†δabφ

+1

2πi

∫ ∞mπ

dν ′ν(ν ′ + ν)

ν ′(ν ′2 − ν2)[Mba(ν

′ + iε)−Mba(ν′ − iε)]

+1

2πi

∫ ∞mπ

dν ′ν(ν − ν ′)ν ′(ν ′2 − ν2)

[Mab(ν′ + iε)−Mab(ν

′ − iε)] (24.101)

It is noteworthy that one can write separate dispersion formulas for the parts of M evenand odd under ν → −ν.

M+ba(ν) =Mba(0) +

2iG2πNm

2π

m4π − 4m2

Nν2φ†[m2πδab

]φ− 2iG2

πNφ†δabφ

+1

2πi

∫ ∞mπ

dν ′ν2

ν ′(ν ′2 − ν2)[Mba(ν

′ + iε)−Mba(ν′ − iε)]

+1

2πi

∫ ∞mπ

dν ′ν2

ν ′(ν ′2 − ν2)[Mab(ν

′ + iε)−Mab(ν′ − iε)] (24.102)

M−ba(ν) =

2iG2πNm

2π

m4π − 4m2

Nν2φ† [−mNν[τb, τa]]φ

+1

2πi

∫ ∞mπ

dν ′ν

ν ′2 − ν2[Mba(ν

′ + iε)−Mba(ν′ − iε)]

− 1

2πi

∫ ∞mπ

dν ′ν

ν ′2 − ν2[Mab(ν

′ + iε)−Mab(ν′ − iε)] (24.103)


The final step is to use the optical theorem to relate the discontinuities Mba(ν′ + iε) −

Mba(ν′ − iε) to pion nucleon total cross sections. The optical theorem is a consequence of

unitarity of the S-matrix. We write S = I +M and impose S†S = I:⟨f |M †|i

⟩+ 〈f |M |i〉 = −

⟨f |M †M |i

⟩(24.104)

The relation of M to the Feynman amplitude by

〈f |M |i〉 = (2π)4δ(Pf − Pi)Mfi

∏k∈i.f

√1

(2π)32Ek(24.105)

Put this into the unitarity equation:

M∗ii +Mii = −

∑f

∏k∈.f

1

(2π)32Ek(2π)4δ(Pf − Pi)|Mfi|2 (24.106)

= −σtotali 4E1E2|v1 − v2| = −4|E2p1 − E1p2|σtotal

i

= −4|E2p1 − E1p2|σtotali → −4m2|p1||σtotal

i (24.107)

where the last line is in the Lab frame. M has the property of being imaginary on the realaxis between the branch points. This property extends to the condition M(ν)∗ = −M(ν∗)off the real axis, so one can write the discontinuity as

Mii(ν′ + iε)−Mii(ν

′ − iε) = Mii(ν′ + iε) +Mii(ν

′ + iε)∗

= −4m2|p1|σtotali (24.108)

by the optical theorem. It is important for the optical theorem that the left side involvesforward amplitudes. In the Cartesian basis that would mean a = b. But the charged pionstate is |π±〉 = (|π1〉 ± i|π2〉)/

√2 in the initial state, but 〈π±| = (|π1〉 ∓ i|π2〉)/

√2 in the

final state. So the desired forward amplitude is actuallyM1∓i2,q∓i2/2. The upper sign is theamplitude for π+N scattering and the lower sign for π−N scattering.

So we can now give the dispersion relation for π+N scattering by choosing b = (1−i2)/√

2and a = (1 + i2)/

√2 for which δab = 1 and [τb, τa] = i([τ1, τ2]− [τ2, τ1])/2 = −2τ3.

M+π+(ν) =Mπ+(0) +

2iG2πNm

2π

m4π − 4m2

Nν2φ†[m2π

]φ− 2iG2

πNφ†φ

+−4mN

2πi

∫ ∞mπ

dν ′ν2√ν ′2 −m2

π

ν ′(ν ′2 − ν2)[σπ+ + σπ− ] (24.109)

M−π+(ν) =

2iG2πNm

2π

m4π − 4m2

Nν2φ† [2mNντ3]φ

+−4mN

2πi

∫ ∞mπ

dν ′ν√ν ′2 −m2

π

ν ′2 − ν2[σπ+ − σπ− ] (24.110)


24.3.3 Adler-Weisberger sum rule

Although the original derivation of the AW sum rule was somewhat different, it is illumi-nating, as stressed by S. Weinberg, to obtain it by combining the soft pion results with thedispersion relation for pion nucleon scattering. The former gave the threshold limit

Mba =mπ

2F 2εabcu

′τcu =mπmN

F 2εabcφ

†τcφ =imπmN

2F 2φ†[τb, τa]φ (24.111)

which gives the low energy limit of M−ba.

On the other hand the threshold limit of the dispersion relation is

M+ba(mπ) =Mba(0) +

2iG2πN

m2π − 4m2

N

φ†[m2πδab

]φ− 2iG2

πNφ†δabφ

+1

2πi

∫ ∞mπ

dν ′m2π

ν ′(ν ′2 −m2π)

[Mba(ν′ + iε)−Mba(ν

′ − iε)]

+1

2πi

∫ ∞mπ

dν ′m2π

ν ′(ν ′2 −m2π)


′ − iε)] (24.112)

M−ba(mπ) =

2iG2πN

m2π − 4m2

N

φ† [−mNmπ[τb, τa]]φ

+1

2πi

∫ ∞mπ

dν ′mπ

ν ′2 −m2π


′ − iε)]

− 1

2πi

∫ ∞mπ

dν ′mπ

ν ′2 −m2π


′ − iε)] (24.113)

the celebrated Adler-Weisberger sum rule follows by setting the left side of the second sumrule to its low energy prediction:

imπmN

2F 2φ†[τb, τa]φ =

ig2AmπmN

2F 2φ†[τb, τa]φ

+1

2πi

∫ ∞mπ

dν ′mπ

ν ′2 −m2π


′ − iε)]

− 1

2πi

∫ ∞mπ

dν ′mπ

ν ′2 −m2π


′ − iε)] (24.114)

where we neglected mπ2 compared to m2N and used the Goldberger-Treiman relation GπN =

gAmN/F . Now apply the optical theorem to the case for π+N scattering leads to

imπmN

2F 2φ†(−2τ3)φ =

ig2AmπmN

2F 2φ†(−2τ3)φ

+−4mNmπ

2πi

∫ ∞mπ

dν ′√ν ′2 −m2

π

[σπ+ − σπ− ] (24.115)

Or, rearranging,

(g2A − 1)φ†τ3φ =

2F 2

π

∫ ∞mπ

dν ′√ν ′2 −m2

π

(σπ+ − σπ−) (24.116)


If the nucleon is a proton φ†τ3φ = +1 so in that case we have

g2A = 1 +

2F 2

π

∫ ∞mπ

dν ′√ν ′2 −m2

π

(σπ+p − σπ−p) , A−W (24.117)

Scattering on a neutron in isolation would be nearly impossible, but could be disentangledfrom neutron rich nuclei. In any case isospin invariance predicts that the replacement σπ+p−σπ−p → σπ−n − σπ+n on the right side would be equally valid.

24.4 Effective Lagrangians

24.4.1 Loop effects from chiral Lagrangian

By construction the tree diagrams of the exact effective action include all quantum effectsdue to loops. However in our treatment of spontaneous symmetry breakdown, we made useof a low energy approximation by expanding the effective action in powers of derivatives.However, this expansion would be technically invalid if the Green functions at low energiescontains IR logarithms. In the presence of SSB of chiral symmetry these are called chirallogs. One can either abandon the derivative espansion idea or reinterpret it.

The idea of the reinterprtation is that such IR logs can be taken into account by expandingthe Lagrangian in powers of derivatives, but treat the massless fields e.g. NGB’s as fullquantum fields, including multiloop diagrams, which will supply the IR logs. That is wewrite

eiΓ(ϕ) =

∫Dϕei

∫d4xLeff (ϕ) (24.118)

where Leff is the most general power series in derivatives of ϕ consistent with the sym-metries. Such a definition produces a nonrenormalizable QFT, with an infinite number ofcounterterms. But the IR logs are controlled by low energy couplings of massless particlesand their leading behavior is not subject to the infinite ambiguity.

24.4.2 Baryons and SU(3)× SU(3).

We have devoted considerable time to the case of nf = 2 massless quarks for which thelightest baryons (3 quark hadrons) are the proton and neutron with spin 1/2 which belongto a doublet under SU(2), as do the u, d quarks. This coincidence leads to a relativelysimple effective Lagrangian for NGB’s interacting with baryons. In the case of nf = 3massless quarks, where the chiral symmetry is SU(3)× SU(3) the lightest spin 1/2 baryonsfall into an octet of SU(3):

JP = (1/2)+ : p, n,Λ0,Σ±,0,Ξ−,0 (24.119)

The next heavier baryons have spin 3/2 and fall into a decuplet:

JP = (3/2)+ : ∆++,+,0,−,Σ±,0∗,Ξ0,−,Ω− (24.120)


In the quark model the proton is uud the neutron is udd, the Λ0 is uds the Σ’s areuus, uds, dds, the Ξ’s are uss, dss, the ∆’s are uuu, uud, udd, ddd, the Σ∗’s are uus, uds, dds,the Ξ∗’s are uss, dss and the Ω− is sss. The masses of the baryons grow with the numberof strange quarks due to the small ratio mu,d/ms 1.

As we have seen in the meson sector the lightest odd parity spin zero mesons, which serveas the NGB’s, fall into an octet. In the quark model there are actually nine qq mesons, theninth being an SU(3) singlet. We have for spin 0 mesons

JP = 0− : K+,0, π±,0, η0, K−,0 (24.121)

with the singlet denoted η′0. Its mass is significantly heavier than the octet masses, makingit a poor NGB. We shall see that the UA(1) “symmetry” is actually broken by anomalies(quantum effects. For the vector mesons

JP = 1− : K+,0∗, ρ±,0, ω0, K−,0∗ (24.122)

The ninth vector meson φ0 has properties consistent with an ss bound state, with the ω0

with properties as if made up only of up and down quarks.In terms of quarks, the K,K∗ are su, sd, the K, K∗ are us, ds, the π, ρ are ud, uu− dd, du.

However the η, η′ system is very different from the ω, φ system: the η′ is nearly an SU(3)singlet (uu+ dd+ ss)), with the η nearly a member of an SU(3) octet (uu+ dd− 2ss)). Incontrast the ω behaves like uu+ dd and the φ like ss.


In the case nf = 2 we introduced an isospin 1/2 nucleon doublet field N into the effectivechiral lagrangian as in (24.79). For nf = 3 we can represent the baryon octet fermion fieldsby traceless 3× 3 matrices, BL, BR, with L,R chiral transformations

BL → U †LBLUL, BR → U †RBRUR (24.123)

Then of course TrBLγ ·DBL + TrBRγ ·DBR is invariant, but TrBLBR is not. But rrecallingthe transformation properties of our NGB effective field U = e−iλaπa/F → U †LUUR, we findthat TrU †BLUBR is invariant. This last term gives an effective mass to the baryon even ifthe 3 quarks are exactly massless.

In considering the quark model of hadrons, we should keep in mind that if we consider thequarks as actual particles rather than as permanently confined constituents, the phenomenonof spontaneous breaking of chiral symmetry could give effective quark fields a mass, just likeit gives the baryons mass. We could call this the constituent mass to distinguish it fromthe bare mass which is very small. If this constituent mass is of order MN/3, these effectivequarks could be treated nonrelativistically, and the nonrelativistic quark model could achievesome respectability.

24.5 Review of Standard Model Lagrangian

In this model the gauge group is the non simple group SU(3)× SU(2)× U(1). There is anindependent gauge field for each factor group as follows:

SU(3) : Aµ =∑a

λa

2Aaµ, F a

µν = ∂µAaν − ∂νAaµ + g3fabcA

bµA

cν

SU(2) : Wµ =∑a

τa2W aµ , W a

µν = ∂µWaν − ∂νW a

µ + g2εabcWbµW

cν

U(1) : Bµ, Bµν = ∂µBν − ∂νBµ (24.124)

The coupling g1 of the abelian gauge group U(1) only becomes meaningful with the additionof further charged fields.

The SU(3) gauge field mediates the strong interactions and couples only to quarks anditself. It is referred to as the color group. The electroweak gauge group, mediating theelectromagnetic and weak interactions, is SU(2)× U(1). The SU(2) factor is referred to asweak isospin and the U(1) factor as weak hypercharge. This terminology is borrowed fromthe familiar isospin and hypercharge of strong interactions. Last semester we spent a gooddeal of time studying QCD with the electro weak interactions turned off. This semester wewill concentrate mostly on the electroweak aspects of hadrons, quarks, and leptons.

A salient feature of the electro weak theory is that fermions of opposite chirality areassigned different electroweak quantum numbers summarized by the table


Lepton irrepsParticle SU(3) SU(2) U(1)(νee

)L

1 2 −1

eR 1 1 −2νeR 1 1 0

Quark irrepsParticle SU(3) SU(2) U(1)(ud

)L

3 2 1/3

uR 3 1 4/3dR 3 1 −2/3

where the subscript L on a fermion field ψ means (1 − γ5)ψ/2 and the subscript R means(1 + γ5)ψ/2. This means that Dirac mass terms in the Lagrangian are not gauge invariantand hence are forbidden. Moreover, This feature raises the specter of anomalies of gaugeinvariance, ruining the consistency of the theory beyond tree diagrams.

For this reason we begin this semester with a close study of gauge invariance anomalies,and their relevance to the consistency of the standard model.


Chapter 25

Ward Identities, UltravioletDivergences and Gauge InvarianceAnomalies

25.1 Ward Identities

One might think that gauge-fixing would exhaust the implications of gauge invariance oncorrelation functions. This is not really true because physics must not depend on whichgauge is chosen, and this gauge independence implies many relationships among differentcorrelation functions known as Ward identities.

Recalling the gauge fixing procedure, it is clear that a necessary condition for gaugeindependence is that the part of the path integral describing the coupling of charged fieldsto the gauge fields, 〈out|in〉A must be invariant under a gauge change of A:

Dµδ

δAµ(z)〈out|in〉A = 0. (25.1)

Here we have written the condition for a general gauge theory; for QED D = ∂. If thisis inserted for the 〈out|in〉A factor in the path integral for a correlation function of gaugeinvariant operators, it becomes the statement that the correlator of the divergence of thegauge current with any set of gauge invariant operators is conserved (abelian case) or relatedto another correlator (nonabelian case).

If gauge noninvariant operators also appear in the correlator, more contributions appear.As an important example of this, take the case of additional charged fields in QED. Thenthe gauge transform induces a change in the phase of each charged field ψ → eiQ0Λψ. Forexample, if only ψ, ψ appear we find

∂µ⟨jµ(z)ψ(x)ψ(y)

⟩= −Q0[δ(x− z)− δ(y − z)]

⟨ψ(x)ψ(y)

⟩. (25.2)

The VEV (vacuum expectation) on the left hand side of this Ward identity is simply relatedto the vertex to which a photon propagator couples. In fact, all diagrams in which the current

273

couples to a closed charged field line contribute nothing to the l.h.s.1 so we can apply theidentity to the one photon line irreducible vertex. To get an identity for the completely oneparticle irreducible vertex Γ, we write

i⟨jµ(z)ψ(x)ψ(y)

⟩1γIR

=

∫d4x′d4y′

⟨ψ(x)ψ(x′)

⟩i⟨jµ(z)ψ(x′)ψ(y′)

⟩1PIR

⟨ψ(y′)ψ(y)

⟩(25.3)

≡∫d4x′d4y′

⟨ψ(x)ψ(x′)

⟩iQ0Γ(z, x′, y′)

⟨ψ(y′)ψ(y)

⟩. (25.4)

We state the relevant identity in momentum space, for which we define⟨ψ(x)ψ(y)

⟩=

∫d4p

(2π)4eip·(x−y) −i

m0 + γ · p+ Σ(p), (25.5)

where Σ is the electron’s proper self energy part. In momentum space, the Ward identitythen reads:

(p′ − p)µΓµ(p′, p) = γ · (p′ − p) + Σ(p′)− Σ(p). (25.6)

This is actually a generalized form of the original Ward Identity which follows from thegeneralized one plus an assumption on the absence of certain infrared singularities.

To get it, take the derivative of the above equation with respect to p′, and set p′ = p.Then, assuming that (p′ − p)µ∂/∂p′νΓµ → 0 in this limit, we obtain

Γµ(p, p) = γµ +∂Σ(p)

∂pµ. (25.7)

An example of the sort of singularity that would invalidate this derivation would be a con-tribution to Γµ of the form

(p′ − p)µγ · (p′ − p)(p′ − p)2

. (25.8)

Such a singularity would imply a zero mass bound state coupled to the current, and can beshown to be absent in QED to any finite order in perturbation theory.

As a simple application of the Ward identity, we use it to show that the only contributionto charge renormalization comes from corrections to the photon propagator. We define chargeby the photon coupling at zero photon momentum. Define Z1 by

Γµ(p, p) =1

Z1

γµ +O(m+ γ · p). (25.9)

By the definition of Z2,

m0 + γ · p+ Σ(p) =1

Z2

(m+ γ · p) +O(m+ γ · p)2. (25.10)

1This follows simply from the gauge invariance of 〈out|in〉A.


Then the Ward identity implies Z1 = Z2. Clearly the renormalized (physical) charge isrelated to the bare one by

Q =Z1

Z2

√Z3Q0 =

√Z3Q0 (25.11)

by the Ward identity.Recall also that the Ward identity for diagrams with only photon external lines required

that the vacuum polarization tensor was transverse:

Πµν(q) = −(q2ηµν − qµqν)Π(q2). (25.12)

Π is defined so that the sum of all 1PIR 2 photon diagrams is given by iQ20Πµν(q). Thus the

photon two point function is given by

〈Aµ(x)Aν(0)〉 =

∫d4q

(2π)4eiq·x

[−iαqµqν

q4+−iq2

(ηµν −

qµqνq2

)1

1 +Q20Π(q2)

].

We note two consequences. (1) The longitudinal part of the two point function is notcorrected. (2) The photon mass remains zero unless Π(q2) has a pole at q2 = 0. We can readoff the value of Z3 = 1/(1 +Q2

0Π(0)).

25.2 Ultraviolet Divergences and Gauge Invariance

In our calculation of vacuum polarization, we found that ultraviolet divergences spoiled thegauge invariance of the calculation. In this chapter we look into this question more closely.Our discussion generalizes that of K. Johnson, Brandeis Lectures, 1964.

The focus will be on the vacuum persistence amplitude of a charged field in the presenceof a fixed external gauge field. This quantity is a basic ingredient of the path historyquantization of gauge fields in interaction with charged fields, and as such must be gaugeinvariant for consistency. As long as the gauge field path integral is amenable to dimensionalregularization, difficulties with gauge invariance must, in fact reside in the charged fieldsector which is the focus of this chapter.

Let us first sketch the important consequences of gauge invariance. Of course, gaugeinvariance demands that, in addition to the persistence amplitude itself, the outin matrixelement of every gauge invariant observable should likewise be invariant under gauge trans-formations. For QED (the abelian case), the current operator jµ(x) is one such observable.In the nonabelian case the current operator jµa (x) is covariant rather than invariant undergauge transformations since gauge transformations act on the index a.

If we return to the variational equation for 〈out|in〉

δ 〈out|in〉 = i

∫d4xδAµ(x)〈out|jµ(x)|in〉 (25.13)

we see that gauge invariance of 〈out|in〉 under infinitesimal gauge transformations that vanishsufficiently rapidly at infinity is equivalent to current conservation ∂µj

µ = 0 in the abelian


case. (In the nonabelian case the requirement is Dµjµ = 0.) More is in fact true: since

the conservation laws hold for arbitrary A, all terms with the same power of A must cancelamong themselves.

We first try to understand how ultraviolet divergences can spoil current conservation.Consider the representation of the current matrix element in terms of the Green function

〈out|jµ(x)|in〉〈out|in〉

= −qTr[γµSF (x, x;A)]. (25.14)

The two arguments of the Green function are coincident, but it is apparent from the freeDirac propagator that SF (x− y) ∼ 1/(x− y)3 at short distances. The fact that A 6= 0 is notgoing to alter this fact. To exert a bit more caution, we should consider the Green functionSF (x, y;A) at slightly separated points and study what happens as y → x. The first thingto notice is that SF is not gauge invariant because the Dirac field ψ → eiqΛ(x)ψ under gaugetransformations2, so

SF (x, y;A)→ eiq[Λ(x)−Λ(y)]SF (x, y;A). (25.16)

Superficially, one would think that the above expression for the current matrix element wouldbe gauge invariant because the prefactor formally goes to 1 as y → x. But the approachto unity is only linear so the fact that SF has a cubic short distance singularity means thatgauge noninvariance can occur with quadratic or lower singularities. Thus point splittingalone will not yield a gauge invariant definition of the current matrix element let alone thevacuum persistence amplitude.

To obtain a gauge invariant definition of the current, we first notice that∫ y

x,C

dξµ∂µΛ(ξ) = Λ(y)− Λ(x). (25.17)

Thus

SF (x, y;A,C) ≡ eiq∫ yx,C dξ

µAµ(ξ)SF (x, y;A) (25.18)

is gauge invariant, albeit path dependent3. We can attempt to define the current matrixelement as the coincident point limit of SF averaged in some suitable way over C. The result

2In the nonabelian case the Dirac field transforms as ψ → Ω(x)ψ, where Ω(x) is an element of thenonabelian gauge group. Then the following equation is replaced by

SF (x, y;A)→ Ω(x)SF (x, y;A)Ω†(y). (25.15)

As y → x this approaches a similarity transformation, but only linearly, so divergences spoil this property.3In the nonabelian case, we can construct the matrix P expig

∫ yx,C

dξµAµ(ξ), where the P denotes pathordering: matrices associated with “later” points on the path C always stand to the left of “earlier” ones.This matrix transforms under gauge transformations as

P expig∫ y

x,C

dξµAµ(ξ) → Ω(y)P expig∫ y

x,C

dξµAµ(ξ)Ω†(x). (25.19)


will certainly be gauge invariant, and with a bit of luck might also be conserved. If so wecan then define 〈out|in〉 from this construction through the variational equation.

If we consider the weak field expansion for SF (x, y;A), the divergence gets one degree lesssevere with each extra factor of A. Thus the field independent piece has a cubic divergence,the linear term a quadratic divergence, the quadratic term a linear divergence and thecubic term only a logarithmic divergence. Thereafter all the terms are finite. Since themodification to S in S vanishes linearly, one only needs to keep terms up to order A3 in themodification factor. Actually, charge conjugation invariance makes alternate terms vanish.The electromagnetic current 〈out|jµ|in〉 should have only odd powers of A, whereas the axialcurrent jµ5 ≡ ψγ5γ

µψ should have only even powers of A in its outin matrix element4. Thuswe tentatively define

Jµ(x) ≡ 〈out|jµ(x)|in〉〈out|in〉

(25.23)

= −q limε→0

1

2TrγµS(x− ε/2, x+ ε/2;A)− γµS(x− ε/2, x+ ε/2;−A)

Jµ5 (x) ≡ 〈out|jµ5 (x)|in〉〈out|in〉

(25.24)

= − limε→0

1

2Trγ5γ

µS(x− ε/2, x+ ε/2;A) + γ5γµS(x− ε/2, x+ ε/2;−A)

To check current conservation we need the Green function equations

(1

iγ · ∂x +m− qγ · A(x))SF (x, y;A) =

SF (x, y;A)(−1

iγ ·←−∂ y +m− qγ · A(y))− iδ(x− y), (25.25)

Then

SF (x, y;A,C) ≡ Peig∫ yx,C

dξµAµ(ξ)SF (x, y;A) (25.20)

transforms by the similarity transformation SF (x, y;A,C)→ Ω(y)SF (x, y;A,C)Ω†(y) under gauge transfor-mations

4The currents in a nonabelian gauge theory carry a gauge symmetry index, jµa = ψλaγµψ, and jµ5a =

ψλaγ5γµψ, where λa is the generator of the gauge group in the representation carried by ψ. Their outin

matrix elements can be defined by

Jµa (x) = − limε→0 TrλaγµS(x− ε/2, x+ ε/2;A) (25.21)

Jµ5a(x) = − limε→0 Trλaγ5γµS(x− ε/2, x+ ε/2;A) (25.22)


from which we get5

∂µTr[γµS(x− ε/2, x+ ε/2;A)] = (25.28)

iqTr[(Aµ(x− ε/2)− Aµ(x+ ε/2))γµS(x− ε/2, x+ ε/2;A)]. (25.29)

(25.30)

To calculate the gradient of the modification factor, we must first specify what happens tothe integration contour under differentiation. Under the change x → x + δx, we globallytranslate C parallel to itself by the same amount. We use Stokes theorem in the form∮

dξµAµ(ξ) =

∫dσµνFµν , (25.31)

where the integral on the r.h.s. is over a surface spanning the closed curve implicit in theline integral on the l.h.s6. To be completely explicit about the conventions in this identityparameterize the surface by ξµ(σ, τ) with σ and τ each ranging from 0 to 1, with σ labelingthe abscissa and τ the ordinate. Then

dσµν =1

2dσdτ

(∂ξµ

∂σ

∂ξν

∂τ− ∂ξν

∂σ

∂ξµ

∂τ

), (25.34)

and with these parameters, the line integral runs around the boundary of the unit square ofparameter space in a counter-clockwise direction. The factor in parentheses in (25.34 is ofcourse a total divergence:

2

(∂ξµ

∂σ

∂ξν

∂τ− ∂ξν

∂σ

∂ξµ

∂τ

)= ∂

∂σ

(ξµ ∂ξ

ν

∂τ− ξν ∂ξµ

∂τ

)(25.35)

+ ∂∂τ

(ξν ∂ξ

µ

∂σ− ξµ ∂ξν

∂σ

), (25.36)

5In the nonabelian case the gauge fields are of course matrices and the appropriate equation is

∂µTr[λaγµS(x− ε/2, x+ ε/2;A)] = (25.26)

iqTr[λa(Aµ(x− ε/2)−Aµ(x+ ε/2)λa)γµS(x− ε/2, x+ ε/2;A)]. (25.27)

Then the coincident point limit gives formal covariant conservation rather than ordinary conservation.6A gauge covariant form of Stokes theorem can be given in the nonabelian case for an infinitesimal closed

loop

Peig∮dξµAµ(ξ) ≈ I + ig

∫dσµνFµν , infinitesimal loop, (25.32)

where Fµν = ∂µAν − ∂νAµ − ig[Aµ, Aν ] is the nonabelian field strength. (The quadratic term in F comesfrom (ig)2

∮dξµ

∮dξ′νP [Aµ(ξ)Aν(ξ′)] which for an infinitesimal loop at z over which A is constant is just

(ig)2Aµ(z)Aν(z)∮dξµξν = −(ig)2[Aµ, Aν ]δσµν .) Another version of this is a statement of how the path

ordered phase changes under an infinitesimal deformation of the curve C spanning an infinitesimal surfaceelement δσµν at the point z:

δzPeig

∫dξµAµ(ξ) = igP [δσµνFµν(z)eig

∫dξµAµ(ξ)] (25.33)

where it is understood that F (z) is included in the path ordering.


which leads to the useful identity

∫dσµν =

1

2

∮ξµdξν = −1

2

∮ξνdξµ, (25.37)

with the contour for the r.h.s. enclosing the surface on the l.h.s. in a counterclockwise sense.

δ

1

1

C C + x

To define the derivative of the modification factor, take the left vertical boundary ofparameter space to map onto the curve C, the right vertical boundary to map onto C + δx,the curve rigidly translated by the amount δx. The top and bottom boundaries then givethe displacements of the end points of the curve x + ε/2 and x − ε/2 respectively. UsingStokes theorem on this closed contour then gives7

∂µ

∫Cx

dξ · A = Aµ(x+ ε/2)− Aµ(x− ε/2) +

∫C

dξνFµν(ξ). (25.39)

7For the nonabelian case we get

∂µPeig

∫dξµAµ = ig(Aµ(x+ ε/2)Peig

∫dξµAµ − Peig

∫dξµAµAµ(x− ε/2) + P [

∫dξνFµν(ξ)eig

∫dξµAµ ]).(25.38)


Combining these results then gives the conservation laws8

∂µJµ(x) = −iq2 lim

ε→0

1

2

∫C

dξνFµν(ξ)Trγµ(S(x− ε/2, x+ ε/2;A)

+S(x− ε/2, x+ ε/2;−A)) (25.44)

∂µJµ5 (x) = im lim

ε→0Trγ5(S(x− ε/2, x+ ε/2;A) + S(x− ε/2, x+ ε/2;−A))

−iq limε→0

1

2

∫C

dξνFµν(ξ)Trγ5γµ(S(x− ε/2, x+ ε/2;A)

−S(x− ε/2, x+ ε/2;−A)) (25.45)

Now we consider whether we can choose the contour to give conserved currents. Weneed to consider the behavior of the factors multiplying

∫F . Since the latter is of order ε

only the singular parts of these factors need be retained. First consider TrγµSF . Since itis gauge invariant it must be a vector formed from εµ, Fµν and derivatives of F . The fieldindependent term must be of the form εµf(ε2) where f behaves as 1/ε4. Because of theantisymmetry of F , it will drop out if we specify the integration contour to be a straight lineconnecting the two endpoints (εµενFµν = 0). The term linear in A was a priori quadraticallydivergent, but in S this is reduced by one power of ε because a factor of momentum mustbe provided to form F out of A. Technically the quadratic and linear divergences in theunadjusted vacuum polarization T µν(k) turn out to be proportional to (1− iε · k) εµεν/ε4 inpoint splitting regularization and therefore give zero when contracted with Fµνε

ν . Thus thelinear term in the “vector” current is at worst logarithmically divergent.

The quadratic term in A is a priori linearly divergent but gauge invariance in the abeliancase requires two factors of momentum to be used in forming F 2 so it and all higher powersmust be finite. In the nonabelian case a quadratic term with no derivatives could conceivablybe needed to complete the nonabelian field strength that might appear in the linear termin A. But as we note below, the divergence in the “vector” linear term is only logarithmicand so does not contribute. The divergence in the “axial” linear term is linear and then thequadratic term provides the rest of the nonabelian field strength.

In summary, provided we take a straight line contour, the only term of relevance in theabelian case is the linear one. It doesn’t contribute in the divergence of the electromagnetic

8We quote here the nonabelian results:

∂µJµa (x) = −ig lim

ε→0Tr[λa, Aµ(x+ ε/2)]γµS(x− ε/2, x+ ε/2;A)

−ig limε→0

Trλaγµ

∫C

dξνP [Fµν(ξ)eig∫dξµAµ ]S(x− ε/2, x+ ε/2;A) (25.40)

∂µJµ5a(x) = 2im lim

ε→0Trλaγ5S(x− ε/2, x+ ε/2;A) (25.41)

−ig limε→0

Tr[λa, Aµ(x+ ε/2)]γ5γµS(x− ε/2, x+ ε/2;A) (25.42)

−ig limε→0

Trλaγ5γµ

∫C

dξνP [Fµν(ξ)eig∫dξµAµ ]S(x− ε/2, x+ ε/2;A) (25.43)


current because of odd charge conjugation. Even if we don’t explicitly enforce charge conju-gation by making J manifestly odd in A, we know from our vacuum polarization calculationthat the linear term is only logarithmically divergent since the adjustments for gauge invari-ance are included when we use S. With point splitting regularization the linear divergencecan be present, but its coefficient vanishes for “vector” currents. This is important in thenonabelian case because then the charge conjugation properties are no longer so simple. Itis sufficient to know that the term is only log divergent to prove that it won’t contribute tothe “vector” conservation law. Thus the e.m. current and, more generally, the “vector” non-abelian currents, defined through S with a straightline contour, are conserved after ε → 0,and S therefore determines a gauge invariant persistence amplitude.

The same cannot be said about the axial current which is even under charge conjugation.In that case the linear term of Trγ5γ

µS does contribute a linearly divergent factor to cancelthe factor of ε implicit in

∫F , leaving a finite anomalous contribution. This is the celebrated

axial anomaly to which we shall return below.

We have shown that choice of a straight line contour suffices to make SF yield a conservede.m. current. However a direction εµ is singled out breaking Lorentz invariance. Thus,it is convenient to average over all directions. This averaging procedure preserves gaugeinvariance and current conservation, since those features hold for each fixed direction. Sinceall terms of order A4 and higher are independent of ε as ε → 0 we only need to apply thisaveraging procedure for the linear and cubic terms. ( The constant and quadratic termsvanish by charge conjugation (Furry’s theorem).) The linear term is the one relevant tovacuum polarization, so let’s look at that one in detail.

The linear terms that come from the modification factor

iq

∫ x+ε/2

x−ε/2dξµAµ(ξ) ≈ iqε · A+

iq

24(ε · ∂)2ε · A (25.46)

are multiplied by

Tr[γµSF (ε)] = −4i

∫d4p

(2π)4e−ip·ε

pµ

p2 +m2 − iε(25.47)

= 4∂

∂ε

∫d4p

(2π)4

e−ip·ε

p2 +m2 − iε≡ εµ

ε4f(m2ε2), (25.48)

(25.49)

where f is regular and nonvanishing at zero argument. In the average over directions of εthe following replacements take place

εµεν → 1

4ηµνε2 (25.50)

εµενερεσ → ε4

24(ηµνηρσ + ηµρηνσ + ηνρηµσ) (25.51)


Thus ⟨iq

∫ x+ε/2

x−ε/2dξµAµ(ξ)TrγµSF

⟩≈

iqf(m2ε2)

ε4

(ε2

4Aµ +

ε4

242(∂2Aµ + 2∂µ∂ · A)

). (25.52)

These terms represent the adjustment that should be made to the linear term in A (whosecoefficient is related to the vacuum polarization T µν(k)). We see that they have the quali-tative appearance of the adjustment we actually had to make in our original calculation: aquadratically divergent constant times ηµν and a finite second order polynomial in k. Thedetailed coefficients cannot be compared because the cutoff procedure was different in thatcalculation.

The term of order A2 is absent in Jµ because of charge conjugation (Furry’s theorem).Finally there will be a finite adjustment to the term of order A3 that arises from the orderk term in the modification factor times the order 3 − k term in SF (A) for k = 1, 2, 3. Thismodification enters the fourth order term in 〈out|in〉. The unmodified value of this termturns out to be uv finite but not gauge invariant. In summary, we have seen that a carefulgauge invariant definition of the e.m. current has led to modifications in the calculation ofonly the first few terms in the weak field perturbation series for 〈out|in〉, specifically thequadratic and order four terms. All higher terms are gauge invariant without modification.

25.2.1 Gauge Invariant Regulation Procedures

For practical calculations in quantum gauge field theories, it is unnecessary to carry outthis detailed procedure. It is sufficient to establish a systematic procedure for regulatingdivergences compatible with gauge invariance. Such a procedure will automatically supplythe polynomial modifications needed for gauge invariance. We mention briefly the morepopular procedures.

Pauli-Villars Method. This method is particularly suited to Abelian Gauge theories sinceit relies on the fact that the violations in gauge invariance are independent of the mass of thecharged fields. Thus if we introduce extra charged fields of large mass Mi which contribute todivergent diagrams with negative signs we can adjust the coefficients of their contributionsto render the loop integrals finite. Then the gauge violating pieces of each contribution willcancel, and the regulated calculation will be gauge invariant. One then lets Mi →∞ at theend of the calculation. This may still leave uv divergences, but only those compatible withgauge invariance as we found in the vacuum polarization calculation.

Dimensional Regularization. The idea here is that the severity of uv divergences dependson the space-time dimension. In particular, the nature of uv gauge invariance violations isdifferent in each dimension. Thus if we can carry out the calculation in a way that appliesto general dimension, the violations of gauge invariance must disappear. This method willobviously not work in theories that can be defined only in particular dimensions. For example


the alternating symbol εµ1···µD has a different tensor structure in each dimension. A theoryin which it appears in a fundamental way can not be formulated in a general dimension.

To illustrate how dimensional regularization is used, we reconsider the vacuum polar-ization calculation. Let us first make some general remarks. In D dimensions the chargehas units [mass]2−D/2. Thus it is convenient to introduce a mass parameter µ to define adimensionless coupling qI in general D via

q2 ≡ q2Iµ

4−D. (25.53)

The identities for calculating traces of products of gamma matrices carry over to D dimen-sions except that the constant 4 = TrI is replaced by 2D/2. The Feynman trick for handlingdenominators of propagators is unchanged in general dimension, and the shift of integrationvariables to make the denominators depend only on the squares of the loop momenta is stillapplicable. But then averaging over directions of the loop momenta gives a D dependentfactor: For example 〈pµpν〉angles = ηµνp2/D. After taking all this into account, the vacuumpolarization calculation for general D, becomes

T µνD (k) = −2D/2q2Iµ

4−D

(2π)D

∫ 1

0

dx

∫dDp (25.54)

ηµν [p2((2/D)− 1)−m2 + x(1− x)k2]− 2x(1− x)kµkν

[m2 + p2 + x(1− x)k2]2(25.55)

Clearly we need to be able to do the integral∫dDppm

[p2 + A2]2= AD+m−4 2πD/2

Γ(D/2)

∫ ∞0

pD+m−1dp

[p2 + 1]2(25.56)

= πD/2AD+m−4 Γ((D +m)/2)Γ(2− (D +m)/2)

Γ(D/2)(25.57)

where we used the identity∫ ∞0

pm+D−1dp

[p2 + 1]2=

1

2

∫ 1

0

dxx1−(D+m)/2(1− x)((D+m)/2)−1 (25.58)

=Γ(2− (D +m)/2)Γ((D +m)/2)

2Γ(2). (25.59)

Note that the quantity ΩD ≡ 2πD/2/Γ(D/2) is just the value of the integral over all anglesin D dimensions. The following table lists ΩD for the 2 ≤ D ≤ 8.

Angular Integral for Various DimensionsD 2 3 4 5 6 7 8ΩD 2π 4π 2π2 8π2/3 π3 16π3/15 π4/3


Using these results to do the integrals, we obtain

T µνD (k) =q2I

2π2Γ(2−D/2)

∫ 1

0

dxx(1− x)(kµkν − k2ηµν)

[m2 + x(1− x)k2

2πµ2

](D−4)/2

.

As advertised the result is gauge invariant. The answer is finite as long as D < 4, but we seethat the gamma function has a pole as D → 4. This is how divergences appear in dimensionalregularization. To regain the result for 4 dimensions we have to write A(D−4)/2 ≈ 1+ D−4

2lnA

and the second term must be retained since Γ(2−D/2)(D − 4)/2→ −1 as D → 4:

T µνD (k)→ q2I


∫ 1

0

dxx(1− x)

[Γ(2−D/2)− ln

m2 + x(1− x)k2

2πµ2

].

The pole at D = 4 represents an infinity which has after renormalization the same fate asthe cutoff dependence did in our earlier calculation, namely it disappears after expressingmeasurable results in terms of measured parameters.

Since the pole at D = 4 corresponds to the logarithmic divergence of a direct cutoffprocedure, it is useful to establish the relation between the residue of the pole and thecoefficient of ln(Λ2). This follows from the simple integral∫ Λ

µ

dppD−5 =ΛD−4 − µD−4

D − 4(25.60)

∼

− 1D−4

Λ→∞, D < 4

ln Λµ

D → 4,Λ fixed,(25.61)

from which we see that the coefficient of ln(Λ2) is −(residue of pole)/2. This is of course inagreement with our two calculations of vacuum polarization.

25.3 Lattice Gauge Theory

In this chapter we have taken for granted that UV divergences do not ruin the path integralfor the gauge fields themselves: we have concentrated on gauge anomalies found in the matter(non gauge field) sector of the theory. As far as anyone can tell this presumption is justifiedin perturbation theory, because of the existence of gauge invariant regulation procedures,especially dimensional regularization.

But at a deeper level, the path integral for the continuum theory must be regarded asa formal object. Replacing spacetime with a finite four dimensional lattice gives a concretedefinition of the path integral as well as a systematic cutoff of the UV and IR divergences.The problem, solved by Ken Wilson, is to formulate a lattice path integral that is exactlygauge invariant. This involves a new description of the gauge degrees of freedom.

In our discussion of UV divergences and gauge invariance we made use of the concept ofthe path dependent group element

Φ(x, y;C) = P exp

i

∫ y

x,C

dξµAµ

(25.62)


which is gauge covariant Φ→ U †(y)ΦU(x) under the gauge transformation U(x). It followsthat TrΦ(x, x;C) is gauge invariant. Wilson’s idea was to choose lattice variables for thegauge field to be just such variables.

To adapt this idea to a D-dimensional rectangular lattice of points (called sites), imagineeach nearest neighbor pair of sites connected by a straight line called a link. Then to eachlink L of the lattice, we assign a group element UL. The gauge transformations are thengroup elements VS assigned to each site, If L is the link from site S to site,S ′, then the gaugetransformation is UL → V †S′ULVS. If L is the reversed link from S ′ to S, we define UL′ ≡ U †L,which is seen to be consistent.

The various matter fields in the theory will be assigned to the sites of the lattice: ψS.Their gauge transformations are simply ψS → URψS, where US(R) is the gauge transforma-tion at site S in the representation R carried by ψ..

To define a lattice version of Ψ pick two sites on the lattice, and an ordered sequence oflinks that make up a path C from S to S ′. Then

Ψ →∏L∈C

UL (25.63)

where the UL are ordered from the one connected to S to the one connected to S ′. If S = S ′

(the path C is closed), then TrΨ is gauge invariant.The integration variables of the gauge path integral are chosen to be the UL. A gauge

group invatiant measure is the the Haar group inariant measure. Since we are mostly in-terested in compact gauge groups, the domain of integration for each UL is compact. If thelattice is also finite, no gauge fixing is necessary, since the integral over the gauge groupisitself finite. The gauge group factor therefore rigorously cancels with no delicacy betweenthe numerator and denominator of a correlation function of observables.

The guideline for choosing a lattice action is that it be a gauge invariant function of thelink variables UL, that it be real, and that it yuelds that standard lagrangian −F 2/4 in theformal continuum limit. With a rectangular lattice, the simplest closed path is a squareenclosed by four links. Such a square is called a plaquette. A gauge invariant variable forplaquette P is then defimed as

TrUL1UL2U†l′1U †(L′2) ≡ TrUP (25.64)

where L′1 is translated one lattice step in the L2 direction from L1, and L′2 is the linktranslated one lattice step in the −L1 direction from L2. Then a simple proposal for thelattice gauge action is

S =∑P

Tr(UP + U †P ) (25.65)

where the second term in the summand can be regarded as associated with the plaquette Pobtained from P by reversing the direction of all the links in P . If we understand that thesum over P includes P as well as P , the second term would be automatically included in


the first.It is convenient to keep both terms explicitly in discussion of the formal continuumlimit.

To study the continuum limit write each link variable UL − eiaA)L where a is the latticespacing. Then

UP = eiaA1eiaA2e−iaA′1e−iaA

′2

= eiaA1+iaA2−a2[A1,A2]/2+O(a3)e−iaA′1−iaA′2−a2[A′1,A

′2]/2+O(a3)

= eiaA1+iaA2−a2[A1,A2]/2−iaA′1−iaA′2−a2[A′1,A′2]/2+a2[A1+A2,A′1+A′2]/2+O(a3)

= eia2(∂1A2−∂2A1+i[A1,A2])+O(a3) (25.66)

To obtain the last line we used A′1 − A1 = a∂2A1 + O(a2), and A′1 − A1 = a∂2A1 + O(a2).Notice that the quantity in parentheses in the exponent is precisely the nonabelian fieldstrength! Since UP is a unitary matrix, the entire exponent is antihermitian including allorders in powers of a. Thus upon expanding UP + U †P in powers of the exponent, all oddpowers cancel between the two terms as the even terms add. Thus

Tr(UP + U †P ) = 2TrI − a4Tr (∂1A2 − ∂2A1 + i[A1, A2])2 +O(a5) (25.67)

summing over all plaquettes then produces the formal continuum limit∫d4xTrFµνF

µν+constant.

25.4 Chiral Anomalies

The existence of a gauge invariant regularization scheme such as dimensional regularizationassures the absence of anomalies (violations) of gauge invariance. Our careful analysis of thee.m. current showed that it is indeed possible to define 〈out|in〉 in a gauge invariant way forthe case of the electromagnetic field. But our parallel discussion of the axial current jµ5 =[ψ, γ5γ

µψ] showed that it is not conserved for m = 0, contrary to what the c-number Diracequation would lead us to believe. The significance of this is that whereas we can consistentlycouple the quantum Dirac field to electromagnetism through jµAµ, the gauge coupling jµ5Aµviolates gauge invariance, and would lead to inconsistencies. Our point splitting definitionof the currents shows the presence of the anomaly and allows us to compute it. One can alsounderstand why the popular regularization schemes I have described fail to forbid anomalies.The Pauli-Villars scheme requires the addition of massive fermions which explicitly violatechiral invariance, and dimensional regularization gives no method for defining εµνρσ or γ5.

The possibility of anomalies in axial gauge couplings puts constraints on viable theoriesof the weak interactions which violate parity conservation through just such couplings. Theway parity violation enters the standard electroweak theory is by assigning left and righthanded fermions to different representations of the electro-weak gauge group SU(2)×U(1).This is of course possible only if explicit mass terms are not included in the Hamiltonian.Thus the I ± γ5 projections of the Dirac field for each fermion couple in different ways tothe gauge fields. What our discussion shows is that such a scheme would be inconsistent fora single fermion. The way the electroweak theory escapes this difficulty is by a cancellationof the anomalies between the contributions of different fermions.


Let us return to the axial current conservation law (25.45.

∂µJµ5 (x) = −2mJ5 − iq lim

ε→0

1

2

∫C

dξνFµν(ξ)

Trγ5γµ(S(x− ε/2, x+ ε/2;A)− S(x− ε/2, x+ ε/2;−A)), (25.68)

where we have defined J5 as the suitably regularized version of

〈out|[ψ, iγ5ψ]|in〉/ 〈out|in〉 (25.69)

given by the first term on the r.h.s. of (25.45). We would like to extract the explicitcontribution of the anomaly which arises from the linearly divergent term in Trγ5γ

µS(x −ε/2, x + ε/2;A), which resides (in the abelian case) in the term with only one power of A.When we expand S(x− ε/2, x+ ε/2;A) in A, the order zero term vanishes because one can’tform an axial vector from the only available four vector εµ. The linear term in A wouldappear to be quadratically divergent, but a momentum factor must be used along with εµ

to form a pseudo two index tensor, so the divergence is only linear. The quadratic term inA is also linearly divergent, but in the case of an abelian gauge field Bose symmetry killsthis leading linear divergence. (We have already noted that since the linear divergence inthis term would not involve derivatives of A it would be inconsistent with abelian gaugeinvariance. It does give a contribution to the anomaly in the nonabelian case, where itis needed to complete the nonabelian field strength whose derivative terms come from thelinear term. Since we only need keep the linearly divergent term, the modification factor canbe dropped. Thus, for the abelian case we only need to extract the linearly divergent pieceof

Trγ5γµS(x− ε/2, x+ ε/2;A) (25.70)

≈ iq

∫d4yTrγ5γ

µSF (x− y − ε/2)γ · A(y)SF (y − x− ε/2) (25.71)

≈ −iq∫

d4k

(2π)4A(k)νe

i(x+ε/2)·k∫d4p

(2π)4e−iε·p

Trγ5γµ(m− γ · p)γν(m− γ · (p− k))

(m2 + p2 − iε)(m2 + (p− k)2 − iε). (25.72)

The trace of γ5 times fewer than 4 gamma matrices vanishes and

Trγ5γµγργνγσ = −4iεµρνσ. (25.73)

Thus the trace in the numerator gives simply +4iεµρνσpρkσ. Furthermore, the leading diver-gence as ε→ 0 coming from the integral over p is independent of k and m so the latter can


be set to zero, and we only need to evaluate∫d4p

(2π)4

pρ(p2 − iε)2

e−iε·p =−iερ

2

∫d4p

(2π)4

1

p2 − iε(25.74)

=ερ2

∫d4pE(2π)4

1

p2(25.75)

=ερ2

∫d4pE(2π)4

∫ ∞0

dTe−Tp2−iε·p (25.76)

=ερ

32π4

∫ ∞0

dTπ2

T 2e−ε

2/4T (25.77)

=ερ

8π2ε2(25.78)

where we did a Wick rotation in the second line and used a simple representation for 1/p2

in the third line.Collecting these results we obtain

Trγ5γµS(x− ε/2, x+ ε/2;A) (25.79)

∼ 4q

∫d4k

(2π)4A(k)νe

ix·kεµρνσkσερ

8π2ε2(25.80)

= −iqεµρνσ∂σAν(x)ερ

2π2ε2(25.81)

= iqεµρνσFνσ(x)ερ

4π2ε2(25.82)

This can now be substituted into our expression for ∂µJµ(x), which gives after averaging

over directions of ε

∂µJµ5 (x) = −2mJ5 +

q2

16π2εµρνσFνσ(x)Fµρ(x) (25.83)

= −2mJ5 +α0

4πεµρνσFµρ(x)Fνσ(x) (25.84)

More generally it is clear from our general remarks that these results generalize in thenonabelian case to

DµJµ5a(x) = −2mJ5a +

g2

16π2εµρνσTr[λaFµρ(x)Fνσ(x)]. (25.85)

25.4.1 Ambiguities

Having obtained the anomalous divergence law for a gauge invariant current, we now examineother definitions of the current useful in certain contexts. First consider the expansion of anunmodified current in powers of the gauge potential described by the series of diagrams.

〈out|jµ|in〉〈out|in〉

= (25.86)


which have degrees of divergence 3, 2, 1, 0,−1 respectively.

Note that the terms that are involved in the anomaly are parity violating containing anodd number of γ5’s and start at 3rd order. Considering the l.h.s. of the divergence laworder by order in A, we see that it is a sum of the divergence of the nth order term and,in the nonabelian case, terms with A multiplying the n − 1th order term. Each of thesediagrams naturally has a cyclic symmetry in the labels a, µ, x of the n vertices. Our gaugeinvariant construction of the current evidently does not reflect this symmetry in the termswith an odd number of γ5’s, because the anomaly is present in only one of the vertices. Thisdifference reflects a fundamental ambiguity in the definition of divergent diagrams, whichin four dimensions includes only those of order ≤ 4. In momentum space, a diagram withdegree of divergence D is ambiguous up to the addition of a polynomial in the externalmomenta of order D. Accordingly the parity violating ambiguities are in the triangle andsquare diagrams and are εµνρσ(αp1 + βp2)ρ and αεµνρσ respectively. Our construction, withthe anomalies absent from all but one vertex, is related to the cyclic symmetric definition bythe addition of such polynomials. A potentially confusing point is that the pentagon diagramis finite and unambiguous but the r.h.s. of the divergence law contains terms quartic in A.This is explained by the fact that the l.h.s. contains a contribution from the square diagram.Changing the square diagram by a term proportional to αεµνρσ adds a term of exactly thestructure of the quartic term on the r.h.s. of the divergence law. Thus depending on theresolution of the ambiguities in the triangle and square, there may or may not be a quarticterm on the r.h.s. Similarly the cubic term on the r.h.s. is influenced by the ambiguityresolutions in both the square and triangle diagram.

25.4.2 Physical Consequences of the Chiral Anomaly

The existence of the chiral anomaly has two sorts of ramifications. The more fundamentalis the constraints it puts on the sorts of gauge fields that can be consistently coupled tofermions. But even if the chiral current is not coupled to a gauge field, it is still an observableof the theory, which would be a conserved current for massless fermions in the absence of theanomaly. The anomaly breaks this conservation law in a way that becomes experimentallysignificant for very light fermions. The classic example is the decay π0 → γ+ γ which wouldbe oversuppressed by the small up and down quark masses were it not for the anomaly.

We first consider the limitations imposed on gauge couplings. The gauge fields mediating


the weak interactions must couple differently to left and right handed fermions

ψR ≡I + γ5

2ψ ψL ≡

I − γ5

2ψ. (25.87)

This is an experimental necessity. The standard electroweak theory, based on the nonabeliangauge group SU(2) × U(1), achieves this requirement by assigning left handed fermions todoublets under SU(2) while the right-handed fermions are singlets under SU(2). The twotypes of fermion have different nonzero weak hypercharges y under the U(1). The ordinaryelectric charge is related to y and the weak isospin I3, one of the three generators of theSU(2), according to the formula

Q = I3 +y

2. (25.88)

For example, the first generation of fermions consists of the electron e, the electron neutrino νethe up quark u and the down quark d. The neutrino and left handed electron form a doublet

l1L =

(νeeL

). Since the neutrino is neutral with I3 = 1/2 and the electron has charge −1 with

I3 = −1/2, we have yl = −1. In the standard model there is no right-handed neutrino andthe right handed electron, being an SU(2) singlet has I3 = 0 and hence yeR = −2. The up anddown quarks have charge +2/3 and −1/3 respectively. Their lefthanded components are anSU(2) doublet with I3 = +1/2,−1/2 respectively, and accordingly carry weak hyperchargeyqL = +1/3. Their right handed components are singlets and hence have yuR = +4/3 andydR = −2/3. There are at least two more generations which seem to repeat the pattern ofthe first only differing in masses, which of course cannot arise from explicit mass terms whichwould violate gauge invariance.

Now we consider the limitations imposed by the anomaly. Since the couplings to left andright handed fermions are different the currents that must be conserved for gauge invarianceare ψλa(I ± γ5)γµψ, separately for λa = I. When λa = τa ∈ SU(2), only the left handedcurrent couples to the gauge fields, and that’s the one that must be conserved. Whendefined in terms of the Green’s functions for the Dirac equation, these are just Jµa ± Jµ5a.Since the gauge couplings conserve handedness for massless fermions, it is not necessary toinclude the (I ± γ5)/2 in the coupling of each gauge field. The anomaly is thus proportionalto Tr[λaε

µρνσFµρ(x)Fνσ(x)]. The field strengths can be expanded in terms of the matricesF =

∑a Faλa, so the vanishing of the anomaly requires9∑

Trλaλb, λc = 0 (25.93)

9The cancellation condition is certainly enough for the vanishing of the anomaly in the conservation ofthe currents constructed to be gauge covariant as we have done. This construction is not quite suited forthe derivation of 〈out|in〉 when anomalies are present. To see this note that in the expansion of J in powersof the gauge field, the coefficient Γn of n− 1 powers of A is an n current amplitude in which the vertex of Jhas been singled out. Because the anomaly is present only in this vertex the Γn is evidently not symmetricin the n currents as the coefficient of n powers of A in the expansion of ln 〈out|in〉 must be. Therefore toconstruct 〈out|in〉 from its variational equation, one must first symmetrize each Γn in the labels of the ncurrents. Since different powers of A are symmetrized differently, this process destroys the gauge covarianceof the current. The term in the anomaly quadratic in A retains its structure but is now distributed equally


where the sum is over the contribution of all fermions coupling to the gauge field underexamination.

We first notice that when a, b, c all refer to the SU(2) matrices the contribution van-ishes: Trτaτb, τc = 2δbcTrτa = 0. When two refer to SU(2) and one to U(1) there is apotential anomaly proportional to

∑L yLTrτaτb = 2δab

∑L yL where the sum is over all the

hypercharges of the left handed doublets. The contribution where only one index refers toSU(2) is clearly zero and we are left with the case where all indices refer to U(1). Thenboth left and right handed fermions contribute, but with opposite signs, so this contributionis proportional to ∑

R

y3R −

∑L

y3L = 0. (25.94)

It is fortunate that the fermion content of the standard model required by experimentsatisfies the constraints on hypercharges we have just obtained. If we substitute the relationbetween electric charge and the weak hypercharges into the constraints, the first just requiresthat the charges of all the components of the lefthanded doublets sum to zero. Thus for thefirst generation this is realized because there are three “colors” for each quark: −1+3(2/3−1/3) = 0. The second generation consisting of the muon, muon neutrino, charmed quarkand strange quark, has gauge couplings identical to the first and so the contribution tothe anomaly from them also cancels. The third generation, follows the same pattern. thelast member, the charged 2/3 top quark, has recently (Spring 1995) been discovered at theTeVatron at Fermilab. There has long been evidence for the τ lepton, its neutrino, and thebottom quark (with charge −1/3). We can regard the required cancellation of anomalies asa prediction of the existence of the top quark, which has now been confirmed. The recentTevatron experiments measure its mass to be 176±13 GeV, almost two orders of magnitudelarger than all other quarks and leptons.

We have yet to consider the “cubic” constraint from anomaly cancellation,∑

R y3R −∑

L y3L = 0. It is helpful to express this constraint also in terms of the ordinary electric

among the 3 vertices, so is multiplied by a factor of 1/3. The symmetrization process modifies the structureof the higher terms: in particular the quartic term disappears and the anomaly then reads

DµJµL,Ra(x) = ± g2

48π2εµρνσTr[λa(2∂µAρ(x)∂νAσ(x)− ig∂µ(Aρ(x)Aν(x)Aσ(x)))]. (25.89)

The cancellation condition would seem to only insure the vanishing of the first term. The second term wouldseem to require the additional condition

Trλaλ[bλcλd] = 0, (25.90)

where the indices enclosed in square brackets are completely antisymmetrized. However this condition isautomatically satisfied if the first is:

λ[bλcλd] = λb[λc, λd] + λd[λb, λc] + λc[λd, λb] (25.91)

=1

2λb, [λc, λd]+

1

2λd[λb, λc]+

1

2λc[λd, λb], (25.92)

by virtue of the Jacobi identity.


charge. ∑R

(2QR)3 −∑L

(2QL − 2I3)3 = 0 (25.95)

8(∑R

Q3R −

∑L

Q3L) + 8

∑L

(3Q2LI3L − 3QLI

23L + I3

3L) = 0. (25.96)

The last term in the third sum vanishes within each doublet and the second term is propor-tional to

∑LQL because (I3)2 = 1/4, a constant for all terms. The contribution to the first

term in the third sum from each doublet is (Q↑2 − (Q↑ − 1)2)/2 which is just equal to thesum of the two charges of the doublet. Thus the whole third sum is proportional to

∑LQL

which vanishes by the first “linear” constraint:∑R

y3R −

∑L

y3L = 8(

∑R

Q3R −

∑L

Q3L) + 6

∑L

QL. (25.97)

Thus the new information in the “cubic” constraint reduces to∑R

(QR)3 =∑L

(QL)3. (25.98)

In the standard model each charged particle state has both a left and right handed component(this means it is possible for all charged particles to gain a (Dirac) mass), and this constraintis automatically satisfied. This left-right symmetry of nonzero charge assignments is anexample of a vector-like Q. To define this “vector-like” property, first enumerate all of thefields according to their L components. Thus we think of the right-handed fields as thecharge conjugates of left handed fields: R = L′c, so that QL′ = −QR. In this new labelingthe anomaly cancellation conditions read∑

L∈Doublet

QL = 0 (25.99)∑Q3L = 0 (25.100)

Then the charge operator Q is vector-like if for each non-vanishing charge Q > 0 there are anequal number of left-handed fields with charge Q and −Q. Thus the cubic equation above isautomatically satisfied if Q is vector-like. In the new notation the most general mass termis of the form ∑

k,n

mk,nLTk iγ

2γ0Ln + h.c. (25.101)

If Q is vector-like it is therefore possible to have a charge conserving mass for every fieldof non-zero charge. If Q is not vector-like, there must remain at least one massless chargedfield.

But notice that the anomaly constraint could also have been satisfied in other moreintricate ways, which would necessarily mean that Q is not vector-like, entailing the pre-diction of massless charged fermions, which experiment strongly contradicts. The fact that


the standard model fails to rule out such a possibility a priori, has led many theorists tothe idea of grand unification: that the gauge group is a simple or semi-simple one (no U(1)factors) and SU(2) × U(1) is just the remnant “low” energy group. One consequence ofthis hypothesis is the quantization of charge. In particular the hypercharge assignments willbe determined by the representations to which each fermion is assigned. These will varywith the unification group and representation choice. Since the electric charge is propor-tional to a generator of a semi-simple group, its trace must be zero which means in ourcontext that

∑LQL = 0, a constraint satisfied in all unification schemes and satisfied by

the fermions of the standard model (again because Q is vector-like). It does not rule outmassless charged particles completely, but it does put an additional restriction on how theycan arise. But as pointed out by Alvarez-Gaume and Witten, this constraint also followsfrom the requirement that the fermions can consistently couple to (classical) gravity. Wewon’t show this, but there is a chiral anomaly for gravitons completely analogous to thatfor gauge particles. We don’t need its explicit form since the graviton couples “univer-sally”. Thus this anomaly is just proportional to Trλa and vanishes for the SU(2) currentand for the U(1) current gives

∑L yL −

∑R yR = 2(

∑LQL −

∑RQR) =

∑L′ QL′ = 0,

where the last sum is over all left-handed fields in the new labeling scheme. Thus thisargument for grand unification is weakened. Similarly, the consistent coupling of the elec-troweak U(1) current to QCD requires an anomaly cancellation which holds if and only if∑

L∈quarksQL −∑

L∈quarksQR =∑

quarksQL′ = 0. These three constraints, viz. the cubicelectroweak, gravitational, and QCD anomaly cancellation go some distance to forcing thevector-like character of Q.

In fact, if they are applied within a single “generation” ((ν, e)L, (u, d)L, eR, uR, dR), thevector-like character of Q follows from the single further assumption that precisely one ofthe leptons is neutral. If the neutrino’s charge is fixed to be zero, the charge assignmentto every other member of the generation is uniquely fixed by anomaly cancellation (Gengand Marshak, Minahan, Ramond and Warner) to be the standard one. More generally, withQν 6= 0 anomaly cancellation implies the following assignments:

Anomaly Free Charge Assignments: Qν 6= 1/2ν eL eR uL dL uR dRQν Qν − 1 2Qν − 1 1

2+ 1−2Qν

2Nc−1

2+ 1−2Qν

2Nc±1−2Qν

2+ 1−2Qν

2Nc∓1−2Qν

2+ 1−2Qν

2Nc

Anomaly Free Charge Assignments: Qν = 1/2ν eL eR uL dL uR dR+1/2 −1/2 0 1/2 −1/2 QuR — −QuR


In the case Qν 6= 1/2, 0, 1 none of the charge assignments allows a mass for even a singlemember of the generation: the entire generation must be massless! Qν = 0 gives the standardassignments and Qν = 1 the charge conjugate standard assignments, both cases allowing a∆IW = 1/2 mass for e, u, and d. The case Qν = 1/2 forbids a ∆IW = 1/2 mass for theelectron, but the up and down quarks can have a ∆IW = 1/2 mass if QuR = 1/2. In thiscase a ∆IW = 1 mass for the (ν, e) doublet is possible. All of these possible mass termsviolate the gauge symmetry and are forbidden unless the gauge symmetry is spontaneouslybroken (Higgs mechanism). Mass terms in the quark sector must be ∆IW = 1/2 (to be colorsinglets) and can arise if an IW = 1/2 Higgs scalar develops a vacuum expectation valueand has a Yukawa coupling to the quarks. In the cases Qν = 0, 1, a mass for the electroncan arise from the same mechanism. However in the case Qν = 1/2, one would also need anIW = 1 Higgs scalar to give a mass to a lepton.

25.4.3 Consequences of Anomalies that Don’t Cancel

We have discussed above the restrictions arising when anomalies must cancel. Now weturn to their consequence when they are allowed. This is when the axial currents underexamination are not coupled to gauge fields. For example the quark part of the axial currentis not coupled to a gauge field. (Parts of it contribute to the electroweak currents but thosealso include the leptons and their anomaly is cancelled between quarks and leptons.) Thusthe neutral component

jµ35 = qτ3

2γ5γ

µq =1

2(uγ5γ

µu− dγ5γµd) (25.102)

of the axial isospin current for up and down quarks has an axial anomaly, in the approxi-mation of massless quarks,

∂µjµ35 = Nc(

4

9− 1

9)α0

8πεµνρσFµνFρσ =

Ncα0

24πεµνρσFµνFρσ, (25.103)

where F is the e.m. field strength. The numerical factors are explained as follows: Theup and down quarks couple with opposite signs to j35 and the contribution of each to theanomaly is the square of the charge. The overall factor of Nc = 3 is for the number ofcolors of each quark. This equation implies a nonvanishing matrix element of jµ35 betweenthe vacuum and a two photon state. Now neglect the weak interactions, keeping stronginteractions to all orders. Since the strong and electromagnetic interactions conserve parity,this matrix element, 〈0|jµ35|γγ〉, must be a pseudo three index tensor Xµρσ where ρ and σare the Lorentz indices describing the polarization of the photons, which carry momentak1, k2 respectively. EM gauge invariance then requires k1ρX

µρσ = k2σXµρσ = 0, and Bose

statistics for photons requires symmetry under ρ, k1 ↔ σ, k2. There is essentially only onesuch pseudotensor (up to terms proportional to kρ1 or to kσ2 which decouple from physical


photons) that can be formed from the epsilon symbol and k1, k2 since10:

εµρστ (k1 − k2)τ +2(k1 + k2)ρk1τk2λε

µσλτ

(k1 + k2)2− 2(k1 + k2)σk1τk2λε

µρλτ

(k1 + k2)2= (25.106)

+2(k1 + k2)µk1τk2λε

ρσλτ

(k1 + k2)2. (25.107)

Note that any contribution of O(k) at small k is of necessity nonanalytic at zero k. Theanomaly equation fixes uniquely the coefficient of this nonanalytic term11. Thus we can write

〈k1, λ1; k2, λ2|jµ35|0〉 = iNcα0

24πεκνρσ〈k1, λ1; k2, λ2|FκνFρσ|0〉

(k1 + k2)µ

(k1 + k2)2, (25.108)

where we stress we have approximated the quark masses as zero. The presence of thesingularity at (k1 + k2)2 = 0 is a striking consequence of the anomaly. The source of thesingularity can be traced to the masslessness of the quarks. But if quarks are confinedthey can’t be responsible for the singularity in the exact amplitude. There are therefore twopossibilities (‘t Hooft): either some of the physical baryons are massless or there is a masslessscalar (to be identified with a Goldstone boson) coupling to the axial isospin current. Thelatter possibility seems to be the one realized in Nature, with the pion playing the role ofthe Goldstone boson:

〈q, π0|jµ35|0〉 =iqµfπ

(2π)3/2√

2ω. (25.109)

(The π0 is related by strong isospin to the π±. In the limit of exact isospin (mu = md andEM turned off) fπ0 = fπ± , and the latter can be independently measured in the weak decayprocess π− → µ− + νµ.) In that case the residue of the pole is (2π)3/2

√2ωfπq

µ times thetransition amplitude for the π0 to decay into two photons. In perturbation theory, this lattertransition amplitude is (−i) times the matrix element of the perturbation in the Hamiltoniandensity describing the electromagnetic interactions of hadrons, from which we conclude:

〈k1, λ1; k2, λ2|He.m.I (0)|q, π0〉 ≈ −

1

(2π)3/2√

2ω

Ncα0

24πfπεµνρσ〈k1, λ1; k2, λ2|FµνFρσ|0〉,(25.110)

10The following equation is a special case for k21 = k22 of the identity

(k1 + k2)µk1τk2λερσλτ = εµρστ (k1 − k2)τ

(k1 + k2)2

2(25.104)

+εµρστ (k1 + k2)τ(k22 − k21)

2+ (k1 + k2)ρk1τk2λε

µσλτ − (k1 + k2)σk1τk2λεµρλτ . (25.105)

This identity can be proved by checking it in the center of mass frame of the two photons k1 = −k2.11This is an essential aspect of the anomaly, reflecting the fact that it really can’t be removed by polynomial

adjustments to the definition of the current. If an analytic piece of the axial current could produce theanomaly, one could make an analytic adjustment to the definition of the current to remove it.


where the approximation is due only to the fact that the quarks and the pion are not exactlymassless. This corresponds to a term in the effective Lagrangian:

Ncα0

24πfππ0ε

µνρσFµνFρσ (25.111)

To the extent that this is a good approximation, we see that the anomaly controls thedecay π0 → 2γ. In fact, this approximation gives a good account of the experimental rateto within 20%. This success may be regarded as evidence for the three colors of quarks.Incidentally, the anomaly breaks the apparent chiral U(1) invariance remaining after elec-tromagnetic interactions have broken SU(2) × SU(2) by virtue of unequal up and downcharges. In particular the π0 will have a small squared mass of order α2 even if all quarkmasses are zero. This shift however is very small compared to the order α shift given to theπ+ and does not disturb the derivation of the Gell-Mann Okubo relation in the SU(3)×SU(3)case.

25.4.4 Mathematical Consequences of the Anomaly: Index Theo-rems

The chiral anomaly puts constraints on the eigenvalues of the Euclidean space Dirac operator(1/i)γ · D in the presence of sufficiently “nice” gauge fields. This differential operator isantihermitian, provided it acts on functions for which one can integrate by parts withoutkeeping surface terms. We assume that A is such that the Dirac operator has a complete setof eigenfunctions with this property. Then since it is antihermitian, the eigenvalues will bepurely imaginary. For each nonzero eigenvalue iλr there is another eigenvalue −iλr, becauseγ5 anticommutes with γµ: if ψr is the eigenvector for iλr then γ5ψr is the eigenvector for−iλr.

Now introduce the Euclidean Green function for m+ 1iγ ·D:

(m+ (1/i)γ ·D)SE(x, y;A) = δ(x− y). (25.112)

We can construct the gauge invariant SE in Euclidean space exactly as in Minkowski space.Then the quantity

Jµ5E = −Trγ5γµ limy→x

SE(x, y;A) (25.113)

has the anomalous conservation law

∂µJµ5E = 2imTrγ5 lim S − i g2

16π2εµνρσTrFµνFρσ. (25.114)

The extra factor −i in the second term on the r.h.s. appears because we have continuedthe Minkowski result to Euclidean space: x0 = −ix4, and every contravariant time indexgets this same −i. We could have used a “Euclidean” εE defined as the continuation ofthe Minkowski one so that ε1234

E ≡ iε1230 = −i, and with such a definition no −i would


appear. But the epsilon symbol is conventionally always understood to be real, and with theconvention ε1234 = +1 the −i must be explicitly included as written. Because m 6= 0 thereare no infrared singularities so if we integrate both sides over x, the l.h.s. will vanish and weget the identity

2m

∫d4xTrγ5 lim S − g2

16π2

∫d4xεµνρσTrFµνFρσ = 0. (25.115)

Now the quantity Trγ5S potentially has u.v. divergences, which would make the limitdelicate. However the γ5 requires multiplication by at least 4 gamma matrices to give anonvanishing trace. To get this many one has to go to the order A2 term in the weak fieldexpansion which is a priori linearly divergent. But this is the term with 5 gamma matricesand that trace vanishes: the term with 4 gamma matrices is only log divergent. Finally tosaturate the epsilon tensor one needs at least two vectors: one could be ε but the other mustbe an external momentum, which gives one further power of convergence, enough to makeit finite. Similarly, all higher terms are convergent. Thus S can be replaced by S, and thelimit y → x safely taken.

If the eigenfunctions of (1/i)γ ·D are complete, we can represent the Green function as

SE(x, y;A) =∑r

ψr(x)ψ†r(y)

m+ iλr(25.116)

and thus ∫d4xTrγ5S =

∑r

∫d4xψ†r(x)γ5ψr(x)

m+ iλr(25.117)

But all terms for which λr 6= 0 vanish because ψr and γ5ψr then have different eigenvaluesand so are orthogonal. Thus the sum is just over the values of r for which λr = 0. Wecan organize the zero eigenfunctions according to the eigenvalues of γ5 which are +1 and−1 Let n± be the number of zero eigenvalues with ±1 eigenvalue of γ5. Then we have∫d4xTrγ5S = n+−n−

mand finally the Atiyah-Singer index theorem

g2

32π2

∫d4xεµνρσTrFµνFρσ = (n+ − n−). (25.118)

We have not been precise about the conditions on A except to say that the Dirac operatormust possess a complete set of eigenfunctions. It is not hard to show that εF 2 is a totalderivative of a gauge noninvariant function. Thus if the A falls off at infinity sufficientlyrapidly we would expect the l.h.s. to vanish, so n+ = n−, with no conclusion about whetherthere are any zero eigenvalues of the Dirac operator. The proper condition is not thatA vanish at infinity, but rather that it approach a pure gauge there. Then the l.h.s. justmeasures the number of times this gauge function “winds around” the three-sphere at infinity.The manifold of SU(2) is the three sphere, so nontrivial windings are possible in that case,


but not in the U(1) case, when the manifold is a circle. (You can’t lasso a sphere.) Forsuch topologically nontrivial gauge fields the index theorem implies at least one vanishingeigenvalue for the Dirac operator. This has the nontrivial physical implication that thevacuum persistence amplitude, det γ · D, in the presence of such a field vanishes when themass of the field vanishes.


Chapter 26

Higgs Mechanism and CustodialSymmetry

26.1 New Look at the Higgs Sector

In the standard model the Higgs field is a complex gauge SU(2) doublet and SU(3) gaugesinglet,

φ =

(φ0

φ−

), Yφ = −1 (26.1)

φc = −iσ2φ∗ =

(−φ−∗φ0∗

), Yφc = +1 (26.2)

and the Higgs mechanism is in effect when 〈φ0〉 = v ≡ v/√

2 6= 01, and we may assume v isreal. The simplest scalar potential which dictates this nonzero VEV is

V (φ) =λ

4(φ†φ− v2)2. (26.3)

The symmetry of this potential is much bigger than the SU(2) × U(1) gauge group of theelectroweak theory. Indeed

φ†φ = |φ−|2 + |φ0|2 = φ−2r + φ−2

i + φ02r + φ02

i , (26.4)

where the subscripts r, i denote real and imaginary parts, is invariant under O(4) rotations!We are familiar with the isomorphism between O(4) and SU(2)×SU(2), the latter of whichis more convenient to employ in coupling the gauge fields of the standard model.

The SU(2)×SU(2) description of O(4) is obtained by defining a 2×2 matrix scalar field

√2Φ ≡ Iφ4 + iσ · φ =

(φ4 + iφ3 φ2 + iφ1

−φ2 + iφ1 φ4 − iφ3

)(26.5)

1In earlier chapters we defined 〈φ〉 = v instead of the more standard 〈φ〉 = v/√

2. In the following weshall use v to denote the standard choice. We will continue to use both v and v ≡ v

√2 in our narrative

299

where (φ4,φ) are the components of 4-vector. Then the O(4) transformation is implementedas

Φ → ULΦU †R (26.6)

where UL,R are independent SU(2) matrices. We can identify UL transformations with thegauge transformations under the SU(2) part of the electroweak gauge group if we identifythe electroweak Higgs doublets as

φ =

(φ0

φ−

)≡ 1√

2

(φ4 + iφ3

iφ1 − φ2

)(26.7)

φc = −iσ2φ∗ =

1√2

(iφ1 + φ2

φ4 − iφ3

). (26.8)

Then we can think of Φ as a row vector with column vector entries

Φ =(φ φc

)(26.9)

so that the transformation Φ→ U(x)Φ is just the electroweak SU(2) gauge transformation.Next we consider the action of the electroweak U(1) gauge transformations. These mul-

tiply φ and φc by opposite phases. In the matrix language this can be done by a specialSU(2)R transformation:

Φ → Φ

(e−iα(x) 0

0 eiα(x)

)=(e−iαφ eiαφc

)(26.10)

The diagonal matrix on the right is obviously unitary and has unit determinant so it is anelement of SU(2)R. If α(x) is the electroweak U(1) gauge transformation, it follows that theweak hypercharge is

Y =

(−1 00 1

)(26.11)

acting from the right.The construction of the covariant derivative of Φ is now evident:

DµΦ = ∂µΦ− ig2WµΦ− ig1BµΦY

2, (26.12)

so we can proceed to the standard model Lagrangian. To express the Higgs potential interms of Φ we note that

Φ†Φ =

(φ†φ φ†φcφ†cφ φ†cφc

)= Iφ†φ (26.13)

where we used φ†φc = 0 and φ†cφc = φ†φ. Taking the trace of both sides gives TrΦ†Φ = 2φ†φ.Putting everything together, we have

Lhiggs = −1

2Tr(DµΦ)†DµΦ− λ

4

(1

2TrΦ†Φ− v2

)2

(26.14)


Now the symmetry of the Higgs potential under SU(2)L × SU(2)R is manifest. If g1 = 0,the SU(2)R symmetry holds as a global symmetry of the derivative terms, because thenΦ → ΦU is not obstructed by the presence of the Y matrix. If g1 6= 0, the presence of Yin the derivative term spoils this global symmetry. This global symmetry has been nameda custodial symmetry, since by imposing it on generalizations of the standard model, someexperimentally well-established relationships of the standard model will be maintained inthe generalization.

26.2 Higgs Mechanism Revisited

Now let us see how the Higgs mechanism works in this new matrix language. Recall that inour original treatment, the VEV of the Higgs doublet was (φ0, φ−) = (v, 0), and we were freeto choose v to be real. In terms of the matrix Φ, the VEV would be Φ0 = 〈Φ〉 = vI. ThisVEV is obviously invariant under the diagonal SU(2) where UL = UR. There are of courseNGB’s described by the effective field ULΦ0UR = vULU

†R ≡ vU . Notice that this description

is parallel to our effective Lagrangian treatment of pions as NGB’s for spontaneously brokenchiral symmetry.

We gain some insight into the workings of the Higgs mechanism by writing the effectiveLagrangian for the NGB field:

DµU = v

[∂µU − ig2WµU − ig1BµU

Y

2

]= −ivU

[iU †∂µU + g2U

†WµU + g1BµY

2

](26.15)

But

W µU ≡ U †W µU +

i

g2

U †∂µU (26.16)

is just an SU(2) gauge transformation on W µ. This means that if we define a new fieldW ′ ≡ WU the matter and gauge part of the standard model Lagrangian will simply haveW → W ′ and the NGB term reduces to

−v2

2Tr(DµU)†DµU = −g

22v

2/2

2

[W ′2

1 +W ′22 +

(W ′3 cos θW −B sin θW )2

cos2 θW

]which provides the mass terms for W ′ and Z = W ′

3 cos θW − B sin θW with tan θW = g1/g2.Then MW = g2v/2 and, calling Z = W3 cos θW − B sin θW , MZ = MW/ cos θW . This is thesame mass term obtained by evaluating the Higgs part of the Lagrangian at Φ0.

Lhiggs(Φ0) = −v2

2Tr

(g2W

† + g1BY

2

)·(g2W + g1B

Y

2

)= −v

2

4

[g2

2(W 21 +W 2

2 ) + (g2W3 − g1B)2]

= −g22v

2/2

2

[W 2

1 +W 22 +

(W3 cos θW −B sin θW )2

cos2 θW

](26.17)


The fact that MZ = MW in the limit θW → 0 can be understood as a consequence of custodialSU(2)R, because for g1 = 0 SU(2)R is an exact global symmetry of the Gauge-Higgs system

To understand this statement we recall that W ′ is a combination of the original gauge fieldand the NGB field U . The construction of W ′ arranges the gauge transformation of W tocancel that of U , so that W ′ is “gauge invariant”. Although W was unaffected by the SU(2)Rcustodial symmetry, W ′ transforms under it as U †RW

′UR because U → UUR. Similarly, theprocess of converting to W ′ involves the matter fields being redefined as F ′ = U †F so theyalso transform under SU(2)R: F ′ → U †RF

′. Expressing the Lagrangian in terms of theseprimed fields, which can be thought of as choosing a gauge in which the NGB field U = I,one ends up with a gauge fixed (and therefore non gauge invariant) Lagrangian which in thelimit g1 = 0 and in the absence of the Yukawa couplings, enjoys a global SU(2)R symmetrythat implies important consequences.

We have seen that one consequence is MW = MZ , Another important one is the equalityof couplings to the W and Z gauge bosons. Recall that the gauge couplings to fermions inthe standard model Lagrangian are

g2fγ · (W1t1 +W2t2)I − γ5

2f + f

g2Z · γcos θW

(t3I − γ5

2−Q sin2 θW

)f (26.18)

The fact that the coefficient of the second term is 1 is a consequence of the custodial symmetrywhen θW = 0. When studying the experimental consequences of the standard model, it isuseful to introduce a parameter called ρ which is a measure of the strength of neutralcurrent couplings to charged current couplings. In the effective low energy Lagrangian, theinteractions are parameterized by

L′eff =GF√

2

[∑f1,f2

f1t+γ(1− γ5)f1 · f2t−γ(I − γ5)f2

+ρ

(∑f

f t3γ(I − γ5)f − Jem sin2 θW

)2 ](26.19)

In writing this expression, we remember that since MZ = MW/ cos θW , ρ = 1 at tree level.However if electroweak radiative corrections are incorporated in a ρeff the latter will onlybe 1 in the limit that the Yukawa couplings are zero and θW = 0. Any physics beyond thestandard model could also cause changes in ρ. Experiments show that ∆ρ = ρ − 1 ≈ 0.01to about 10%. This is consistent with the standard model violations of ρ = 1, the largestof which is due to the huge top quark mass, which contributes ∆ρ = 3GFm

2t/(8π

2√

2).This puts strong constraints on such new physics. Maintaining the custodial symmetry ingeneralizations helps satisfy these constraints.


26.3 Gauge-fixing the Standard Model v2

We now review the process of gauge-fixing using the matrix representation of the Higgs field.We focus on the electroweak part of the Lagrangian

Lstd = LGlue+matter −1

2TrF 2

W −1

4F 2B −

1

2Tr(DΦ)† ·DΦ− V(Φ) (26.20)

DµΦ = ∂µΦ− ig2WµΦ + ig1BµΦt3, V(Φ) =λ

4Tr

(1

2TrΦ†Φ− v2

)2

where we used Y/2 = −t3. Under an infinitesimal electroweak gauge transformation

∆Bµ = −∂µε, ∆Wµ = −DµG

∆Φ = −ig2GΦ + ig1εΦt3 (26.21)

The Higgs potential is minimized for Φ = vI so we change variables to Φ = Φ − vI, afterwhich

V(Φ) =λv2

16[Tr(Φ + Φ†)]2 +

λv

8TrΦ†ΦTr(Φ + Φ†) +

λ

16(TrΦ†Φ)2

−1

2Tr(DµΦ)†DµΦ = −1

2Tr(DµΦ)†DµΦ− v2

2Tr(g2W − g1Bt3)2

−v2

Tr(DµΦ)†(−ig2Wµ + ig1B

µt3)

−v2

Tr(ig2Wµ − ig1t3B

µ)DµΦ (26.22)

In the presence of SSB, ’t Hooft invented a gauge that simplifies the quadratic terms in theLagrangian. With strategic integration by parts the quadratic terms of the Higgs-gauge partof the Lagrangian are

−Tr∂µWν∂µW ν − 1

2∂µBν∂

µBν − 1

2Tr(∂µΦ)†∂µΦ− λv2

16Tr(Φ + Φ†)2

+Tr(∂ ·W )2 +1

2(∂ ·B)2 − v2

2Tr(g2W − g1Bt3)2

+iv

2Tr(Φ− Φ†)(g2∂ ·W − g1∂ ·Bt3) (26.23)

The awkward terms which make the kinetic terms non diagonal are

+Tr(∂ ·W )2 +1

2(∂ ·B)2 +

iv

2Tr(Φ− Φ†)(g2∂ ·W − g1∂ ·Bt3)

= Tr

(∂ ·W +

ig2v

4(Φ− Φ†

)2

+1

2

(∂ ·B − ig1v

2Trt3(Φ− Φ†)

)2

+g2

2v2

16Tr(Φ− Φ†)2 +

g21v

2

8

(Trt3(Φ− Φ†)

)2(26.24)


The terms on the last line give masses to the NGB’s. ’t Hooft’s idea was to choose gaugefixing terms to cancel the terms on the middle line. Define gauge-fixing functions

FB = ∂ ·B − ig1v

2Trt3(Φ− Φ†) (26.25)

FW = ∂ ·W +ig2v

4(Φ− Φ†) (26.26)

by adding the terms −TrF2W − (1/2)F2

B to the Lagrangian. This arranges propagators withmomentum independent numerators as in the Feynman gauge. A more general ξ gauge isobtained by using the gauge-fixing functions

F ξB = ∂ ·B − iξ g1v

2Trt3(Φ− Φ†) (26.27)

F ξW = ∂ ·W + iξg2v

4(Φ− Φ†) (26.28)

and adding −(1/ξ)TrF ξ2W − (1/(2ξ))F ξ2B to the Lagrangian. Then ξ = 1 reduces to the’t Hooft-Feynman gauge and ξ = 0 to the ’t Hooft-Landau gauge, with transverse gaugepropagators.

To obtain the ghost part of the Lagrangian we first subject the gauge-fixing functions toa gauge transformation:

∆FB = −∂2ε− ig1v

2Trt3(−ig2GΦ− ig2Φ†G+ ig1Φt3ε+ ig1t3Φ†) (26.29)

∆FW = −∂ ·DG+ig2v

4(−ig2GΦ− ig2Φ†G+ ig1εΦt3 + ig1εt3Φ†) (26.30)

Then the ghost Lagrangian is obtained by replacing G, ε by CW , CB respectively in The ∆F ’sand multiplying on left by BW , BB:

Lgh = ∂µBB∂µCB + 2∂µBWDµCW

−g1v

2BBTrt3

[(g2(CWΦ + Φ†CW )− g1(Φt3 + t3Φ†)CB

]+g2v

2TrBW

[g2(CWΦ + Φ†CW )− g1(Φt3 + t3Φ†)CB

](26.31)

We have already discussed the tree level mass terms for the gauge bosons, but we also haveto consider the NGB’s in the Higgs sector, the FP ghosts and of course the Higgs particle.The mass terms for the NGB’s are

g22v

2

16Tr(Φ− Φ†)2 +

g21v

2

8(Tr(Φ− Φ†))2 (26.32)

Recall Φ√

2 = Iφ4 + 2it · φ, which makes the derivative terms canonical

−1

2Tr∂Φ† · ∂Φ =

1

2(∂φ4)2 + (∂φ)2). (26.33)


Now plug Φ− Φ† = 4it · φ into the NGB mass terms to get

−g22v

2

4φ2 − g2

1v2

4φ2

3 = −g22v

2

4(φ2

1 + φ22)− (g2

1 + g22)v2

4φ2

3 (26.34)

We see that the charged NGB’s φ1 ± iφ2 have the same mass as the W gauge bosons, andthe neutral NGB φ3 has the same mass as the Z gauge boson.

The masses assigned to the FP ghosts are obtained by setting W = B = 0 and Φ = vIin the ghost Lagrangian:

∂µBB∂µCB + 2Tr∂µBW∂µCW + v2Tr(g2BW − g1t3BB)(g2CW − g1t3CB)

= ∂µBZ∂µCZ + ∂µBA∂µCA + 2Tr∂µBW1,2∂µCW1,2

+v2TrBW1,2CW1.2 +g2

2

2 cos2 θWBZCZ (26.35)

where BZ = BW3 cos θW −BB sin θW and CZ = CW3 cos θW −CB sin θW . So again the chargedghosts have the mass of the W and the neutral one has the mass of the Z. The ghosts forthe QED field, BA, CA, are not only massless bur free (non-interacting. This means that incalculating correlations of the interacting fields, they contribute nothing and can be ignored.

In summary the gauge/higgs sector of the standard model includes gauge bosons, NGB’s,and FP ghosts. In the ’t Hooft-Feynman gauge (ξ = 1), the numerator of the propagatorsfor these various particles is constant, and there are three tree level masses:

M2W =

g22v

2

2=g2

2v2

4: W µ

1,2, φ1,2, BW1,2 , CW1,2 (26.36)

M2Z =

M2W

cos2 θW: Zµ, φZ , BZ , CZ (26.37)

M2γ = 0; Aµ, BA, CA (26.38)

Each propagator is −1/(p2 + M2i ) for the spin 0 particles and this times ηµν for the vector

particles.The mass eigen fields are

W µ1,2, Zµ = W3 cos θW −Bµ sin θW , Aµ = Bµ cos θW +W µ

3 sin θW (26.39)

for the gauge bosons. For the higgs fields the mass eigenfields are linked to our matrix fieldΦ via Φ = (φ4 + 2it ·φ)/

√2. The NGB’s are φ and masses come entirely from the covariant

derivative terms of the Lagrangian. The higgs scalar is h = φ4 − v√

2 and its mass is givenby the quadratic term in the potential:

V(h,φ) =λv2

2h2 +

λv

2√

2h(h2 + φ2) +

λ

16(h2 + φ2)2 (26.40)

Thus the higgs mass is

m2h = λv2 =

λv2

2=

2λ

g22

M2W (26.41)

Since mh ≈ 125GeV and MW ≈ 80GeV this leads to am estimate of λ ≈ 1.2g22.


Chapter 27

Electroweak Interactions of Leptonsand Quarks

27.1 Vector Boson decay to Leptons and Quarks

309

27.2 Lepton or Quark Beta Decay


27.3 Scattering Processes


Chapter 28

Electroweak Interactions of Hadrons

28.1 Hadron Spectroscopy

321

28.2 Vector and Axial Symmetries and Electroweak

Processes


28.3 Semi-Leptonic Hadronic Decay Processes


28.4 Electroweak Processes involving NGB’s

We would like to extend our effective action description of NGB’s to include their electroweakdecays.


28.5 θ vacua and the Strong CP problem


Chapter 29

Quark Confinement

29.1 String Model of Hadrons

29.2 Lattice Gauge Theory

369

Chapter 30

Physics at High energy and LowMomentum transfer

371

Chapter 31

Beyond the Standard Model

373

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