lecture notes on quantum field theory - wuhan...
TRANSCRIPT
Draft for Internal Circulations:v1: Spring Semester, 2012, v2: Spring Semester, 2013v3: Spring Semester, 2014, v4: Spring Semester, 2015
Lecture Notes on Quantum Field Theory
Yong Zhang1
School of Physics and Technology, Wuhan University
(Spring 2015)
Abstract
These lectures notes are written for both third-year undergraduate studentsand first-year graduate students in the School of Physics and Technology,University Wuhan. They are mainly based on lecture notes of Sidney Colemanfrom Harvard and lecture notes of Michael Luke from Toronto and Peskin &Schroeder’s standard textbook, so I do not claim any originality. These notescertainly have all kinds of typos or errors, so they will be updated from timeto time. I do take the full responsibility for all kinds of typos or errors (besideserrors in English writing), and please let me know of them.
The third version of these notes are typeset by a team including:
Kun Zhang2, Graduate student (Id: 2013202020002),Participant in the Spring semester, 2012 and 2014
Yu-Jiang Bi3, Graduate student (Id: 2013202020004),Participant in the Spring semester, 2012 and 2014
The fourth version of these notes are typeset by
Kun Zhang, Graduate student (Id: 2013202020002),Participant in the Spring semester, 2012 and 2014-2015
1yong [email protected] [email protected]@hotmail.com
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Draft for Internal Circulations:v1: Spring Semester, 2012, v2: Spring Semester, 2013v3: Spring Semester, 2014, v4: Spring Semester, 2015
AcknowledgementsI thank all participants in class including advanced undergraduate students,first-year graduate students and French students for their patience and persis-tence and for their various enlightening questions. I especially thank studentswho are willing to devote their precious time to the typewriting of these lecturenotes in Latex.
Main References to Lecture Notes
* [Luke] Michael Luke (Toronto): online lecture notes on QFT (Fall, 2012);
the link to Luke’s homepage
* [Tong] David Tong (Cambridge): online lecture notes on QFT (October 2012);
the link to Tong’s homepage
* [Coleman] Sidney Coleman (Harvard): online lecture notes;
the link to Coleman’s lecture notes.
the link to Coleman’s teaching videos;
* [PS] Michael E. Peskin and Daniel V. Schroeder:
An Introduction to Quantum Field Theory, 1995 Westview Press;
* [MS] Franz Mandl and Graham Shaw:
Quantum Field Theory (Second Edition), 2010 John Wiley & Sons, Ltd.
* [Zhou] Bang-Rong Zhou (Chinese Academy of Sciences):
Quantum Field Theory (in Chinese), 2007 Higher Education Press;
Main References to Homeworks
* Luke’s Problem Sets #1-6 or Tong’s Problem Sets #1-4.
Research Projects
* See Yong Zhang’s English and Chinese homepages.
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To our parents and our teachers!
To be the best researcher is to be the best person first of all:
Respect and listen to our parents and our teachers always!
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Quantum Field Theory (I) focuses on Feynman diagrams and basic concepts.
The aim of this course is to study the simulation of both quantum field theoryand quantum gravity on a quantum computer.
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Contents
I Quantum Field Theory (I): Basic Concepts and Feynman Diagrams 10
1 Overview of Quantum Field Theory 11
1.1 Definition of quantum field theory . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Introduction to particle physics and Feynman diagrams . . . . . . . . . . . 12
1.2.1 Feynman diagrams in the pseudo-nucleon meson interaction . . . . . 12
1.2.2 Feynman diagrams in the Yukawa interaction . . . . . . . . . . . . . 17
1.2.3 Feynman diagrams in quantum electrodynamics . . . . . . . . . . . 19
1.3 The canonical quantization procedure . . . . . . . . . . . . . . . . . . . . . 21
1.4 Research projects in this course . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 From Classical Mechanics to Quantum Mechanics 23
2.1 Classical particle mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 Lagrangian formulation of classical particle mechanics . . . . . . . . 23
2.1.2 Hamiltonian formulation of classical particle mechanics . . . . . . . 24
2.1.3 Noether’s theorem: symmetries and conservation laws . . . . . . . . 24
2.1.4 Exactly solved problems . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.5 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.6 Scattering theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.7 Why talk about classical particle mechanics in detail . . . . . . . . . 28
2.2 Advanced quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.1 The canonical quantization procedure . . . . . . . . . . . . . . . . . 28
2.2.2 Fundamental principles of quantum mechanics . . . . . . . . . . . . 29
2.2.3 The Schroedinger, Heisenberg and Dirac picture . . . . . . . . . . . 30
2.2.4 Non-relativistic quantum many-body mechanics . . . . . . . . . . . . 32
2.2.5 Compatible observable . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.6 The Heisenberg uncertainty principle . . . . . . . . . . . . . . . . . . 33
2.2.7 Symmetries and conservation laws . . . . . . . . . . . . . . . . . . . 34
2.2.8 Exactly solved problems in quantum mechanics . . . . . . . . . . . . 34
2.2.9 Simple harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.10 Time-independent perturbation theory . . . . . . . . . . . . . . . . . 36
2.2.11 Time-dependent perturbation theory . . . . . . . . . . . . . . . . . . 36
2.2.12 Angular momentum and spin . . . . . . . . . . . . . . . . . . . . . . 38
2.2.13 Identical particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.14 Scattering theory in quantum mechanics . . . . . . . . . . . . . . . . 39
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3 Classical Field Theory 40
3.1 Special relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.1 Lorentz transformation . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.2 Four-vector calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.3 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.4 Mass-energy relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.5 Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.6 Classification of particles . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Classical field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.1 Concepts of fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.2 Electromagnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.3 Lagrangian, Hamiltonian and the action principle . . . . . . . . . . 47
3.2.4 A simple string theory . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2.5 Real scalar field theory . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.6 Complex scalar field theory . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.7 Multi-component scalar field theory . . . . . . . . . . . . . . . . . . 51
3.2.8 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2.9 General relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Symmetries and conservation laws 55
4.1 Symmetries play a crucial role in field theories . . . . . . . . . . . . . . . . 55
4.2 Noether’s theorem in field theory . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Space-time symmetries and conservation laws . . . . . . . . . . . . . . . . . 57
4.3.1 Space-time translation invariance and energy-momentum tensor . . . 57
4.3.2 Lorentz transformation invariance and angular-momentum tensor . . 59
4.4 Internal symmetries and conservation laws . . . . . . . . . . . . . . . . . . . 62
4.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.2 Noether’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.3 SO(2) invariant real scalar field theory . . . . . . . . . . . . . . . . . 63
4.4.4 U(1) invariant complex scalar field theory . . . . . . . . . . . . . . . 65
4.4.5 Non-Abelian internal symmetries . . . . . . . . . . . . . . . . . . . . 66
4.5 Discrete symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5.1 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5.2 Time reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5.3 Charge conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 Constructing Quantum Field Theory 69
5.1 Quantum mechanics and special relativity . . . . . . . . . . . . . . . . . . . 69
5.1.1 Natural units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1.2 Particle number unfixed at high energy (Special relativity) . . . . . 70
5.1.3 No position operator at short distance (Quantum mechanics) . . . . 71
5.1.4 Micro-Causality and algebra of local observables . . . . . . . . . . . 72
5.1.5 A naive relativistic single particle quantum mechanics . . . . . . . . 73
5.2 Comparisons of quantum mechanics with quantum field theory . . . . . . . 74
5.3 Definition of Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.4 Rotation invariant Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.4.1 Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.4.2 Occupation number representation (ONR) . . . . . . . . . . . . . . . 76
5.4.3 Hints from SHO (Simple Harmonic Oscillator) . . . . . . . . . . . . 77
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5.4.4 The operator formalism of Fock space in a cubic box . . . . . . . . . 78
5.4.5 Drop the box normalization and take the continuum limit . . . . . . 79
5.5 Lorentz invariant Fock spce . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.5.1 Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.5.2 Definition of Lorentz invariant normalized states . . . . . . . . . . . 80
5.5.3 Lorentz invariant normalized state . . . . . . . . . . . . . . . . . . . 80
5.5.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.6 Canonical quantization of classical field theory . . . . . . . . . . . . . . . . 82
5.6.1 Classical field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.6.2 Canonical quantization . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.6.3 Basic calculation on Hamiltonian . . . . . . . . . . . . . . . . . . . . 85
5.6.4 Vacuum energy and normal ordered product . . . . . . . . . . . . . 86
5.6.5 Micro-causality (Locality) . . . . . . . . . . . . . . . . . . . . . . . . 88
5.7 Remarks on canonical quantization . . . . . . . . . . . . . . . . . . . . . . . 90
5.8 Symmetries and conservation laws . . . . . . . . . . . . . . . . . . . . . . . 91
5.8.1 U(1) invariant quantum complex scalar field theory . . . . . . . . . 91
5.8.2 Discrete symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.9 Return to non-relativistic quantum mechanics . . . . . . . . . . . . . . . . . 93
6 Feynman Propagator, Wick’s Theorem and Dyson’s Formula 94
6.1 Retarded Green function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.1.1 Real scalar field theory with a classical source . . . . . . . . . . . . . 94
6.1.2 The retarded Green function . . . . . . . . . . . . . . . . . . . . . . 95
6.1.3 Analytic formulation of DR(x− y) . . . . . . . . . . . . . . . . . . . 96
6.1.4 Canonical quantization of a real scalar field theory with a classicalsource . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.2 Advanced Green function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.3 The Feynman Propagator DF (x− y) . . . . . . . . . . . . . . . . . . . . . . 99
6.3.1 Definition with contour integrals . . . . . . . . . . . . . . . . . . . . 100
6.3.2 Definition of Feynman propagator with iε→ 0+ prescription . . . . 101
6.3.3 Definition of Feynman propagator with time-ordered product . . . . 102
6.3.4 Definition of Feynman propagator with contractions . . . . . . . . . 103
6.4 Wick’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.4.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.4.2 Example for Wick’s theorem n = 3 . . . . . . . . . . . . . . . . . . . 105
6.5 Interaction Picture (Dirac Picture) . . . . . . . . . . . . . . . . . . . . . . . 106
6.5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.5.2 Dirac picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.6 Dyson’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.6.1 Unitary evolution operator . . . . . . . . . . . . . . . . . . . . . . . 108
6.6.2 Time dependent permutation theory . . . . . . . . . . . . . . . . . . 108
6.6.3 Dyson’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7 Scattering Matrix, Cross Section & Decay Width 112
7.1 Scattering matrix (operator) . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.1.1 Ideal model for scattering process . . . . . . . . . . . . . . . . . . . . 112
7.1.2 Scattering operator S . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.2 Calculation of two-“nucleon” scattering amplitude . . . . . . . . . . . . . . 113
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7.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.2.2 Two-“nucleon” scattering matrix . . . . . . . . . . . . . . . . . . . . 114
7.2.3 Computing methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.3 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.3.1 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.3.2 Correspondence between algebra and diagrams . . . . . . . . . . . . 115
7.3.3 Conventions of drawing external lines . . . . . . . . . . . . . . . . . 115
7.3.4 Conventions on diagrams vertices & internal lines . . . . . . . . . . . 117
7.3.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.4 Feynman rules in coordinate space-time (FRA) . . . . . . . . . . . . . . . . 118
7.5 Feynman rules in momentum space (FRB) . . . . . . . . . . . . . . . . . . 121
7.5.1 Calculation hint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.5.2 Feynman rules in momentum space . . . . . . . . . . . . . . . . . . . 122
7.6 Feynman rules C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.6.1 Calculation hint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.6.2 Feynman rules C (simplified FRB) . . . . . . . . . . . . . . . . . . . 122
7.7 Application of FRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.8 Remarks on Feynman propagator . . . . . . . . . . . . . . . . . . . . . . . . 126
7.9 Cross sections & decay widths – measurable quantities in high energy physics126
7.9.1 QFT in a box with volume V = L3 . . . . . . . . . . . . . . . . . . . 126
7.9.2 Calculation of differential probability . . . . . . . . . . . . . . . . . . 129
7.9.3 Cross section & decay width . . . . . . . . . . . . . . . . . . . . . . 129
7.9.4 Calculation of Cross-sections and Decay-widths . . . . . . . . . . . . 130
8 Dirac Fields 134
8.1 The Dirac equation and Dirac algebra . . . . . . . . . . . . . . . . . . . . . 134
8.1.1 The Dirac algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.1.2 Two widely used representations of the Dirac algebra . . . . . . . . 135
8.1.3 The Lagrangian formulation . . . . . . . . . . . . . . . . . . . . . . . 136
8.1.4 Plane wave solutions of Dirac equation in Dirac representation . . . 136
8.2 Dirac equation and Clifford algebra . . . . . . . . . . . . . . . . . . . . . . . 138
8.2.1 The Clifford algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.2.2 Dirac equation with γµ matrices . . . . . . . . . . . . . . . . . . . . 139
8.2.3 Plane wave solutions of Dirac equation with γµ matrices . . . . . . . 139
8.3 Lorentz transformation and parity of Dirac bispinor fields . . . . . . . . . . 140
8.3.1 Classification of quantities out of Dirac bispinor fields under Lorentztransformation and parity . . . . . . . . . . . . . . . . . . . . . . . . 140
8.3.2 Explicit formulation of D(Λ) . . . . . . . . . . . . . . . . . . . . . . 142
8.4 Canonical quantization of free Dirac field theory . . . . . . . . . . . . . . . 144
8.4.1 Canonical quantization . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.4.2 Fock space with the Fermi-Dirac statistics . . . . . . . . . . . . . . . 145
8.4.3 Symmetries and conservation laws: energy, momentum and charge . 146
8.5 Fermion propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
8.6 Feynman diagrams and Feynmans rules for fermion fields . . . . . . . . . . 149
8.6.1 FDs & FRs for vertices . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.6.2 Internal lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.6.3 External lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.6.4 Combinational factor (Symmetry factor) . . . . . . . . . . . . . . . . 151
8.7 Special Feynman rules for fermions lines . . . . . . . . . . . . . . . . . . . . 151
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8.7.1 Mapping from matrix to number . . . . . . . . . . . . . . . . . . . . 1518.7.2 Feynman rules for a single fermion line from initial state to the final
state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.7.3 Relative sign between FDs . . . . . . . . . . . . . . . . . . . . . . . . 1528.7.4 Minus sign from a loop of fermion lines . . . . . . . . . . . . . . . . 152
8.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1528.8.1 First order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1528.8.2 Second order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538.8.3 Fourth order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.9 Spin-sums and cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558.9.1 Spin-sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558.9.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
9 Quantum Electrodynamics 1589.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1589.2 Representative scattering processes in QED . . . . . . . . . . . . . . . . . . 1599.3 The calculation of the differential cross section of the Bhabha scattering in
second order of coupling constant . . . . . . . . . . . . . . . . . . . . . . . . 162
II Renormalization and Symmetries 166
10 Notes on BPHZ Renormalization 167
III Path Integrals and Non-Abelian Gauge Field Theories 218
IV The Standard Model and Particle Physics 219
V Integrable Field Theories and Conformal Field Theories 220
VI Quantum Field Theories in Condense Matter Physics 221
VII General Relativity, Cosmology and Quantum Gravity 222
VIII Supersymmetries, Superstring and Supergravity 223
IX Quantum Field Theories on Quantum Computer 224
X Quantum Gravity on Quantum Computer 225
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Part I
Quantum Field Theory (I): BasicConcepts and Feynman Diagrams
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Lecture 1 Overview of Quantum Field Theory
1.1 Definition of quantum field theory
Def 1: QFT=Special Relativity + Quantum Mechanics.
Note: QFT is a comprehensive subject for undergraduate student to revisit what theyhave learnt.
Preliminaries to QFT1PPq@@R
Quantum Mechanics
Special Relativity
Electrodynamics
Classical Mechanics
Special relativity
Gravity
PPq
1General relativity
Quantum mechanics
PPq
1Quantum gravity
Quantum field theory
Quantum gravity
PPq
1Superstring theory
Note: If quantum mechanics is changed, the entire modern physics has to be changed aswell.
Def 2: QFT=Quantization of Classical Field Theory.
E.g. 1: Relativistic quantization of Electrodynamics: Quantum Electrodynamics.
E.g. 2: Non-Relativistic quantization of Electromagnetic Field: Quantum Optics.
Def 3: Quantum many-particle system with unfixed particle-number.
Particle # fixed Particle # un-fixed
⇓ ⇓Non-relativistic QFT: Relativistic QFT:
Condensed matter physics High energy physics
E.g. 1: In high energy physics, the particles can be created or annihilated, so thatthe particle number is unfixed, which is described by relativistic QFT.
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E.g. 2: In condensed matter physics, particle number is a conserved quantity, whichis described by non-relativistic QFT.
Def 4: Quantum field theory is the present language in which the laws of nature arewritten.
Note 1: Particle/wave duality equals particle/field duality. Particles are describedby four dimensional vector (E, ~p), and waves are characterized by (ν, λ) or (ω, k).The particle wave duality allows the relations
E = hν = ~ω, |~p| = h
λ= ~k, (1.1.1)
where h = 2π~.
Note 2: In QFT, each type of fundamental particle is a derived object of quantizationof the associated field.
Photon ⇐⇒ EM field;
electron ⇐⇒ Dirac field;
God Particle(Higgs) ⇐⇒ Scalar field;
Quark ⇐⇒ Quark field.
Note 3: In QM, identical particles are indistinguishable, which is a kind of assump-tion. In QFT, particles of the same type are indistinguishable, because they areassociated with the same quantum field.
1.2 Introduction to particle physics and Feynman diagrams
Classification of particles.mass, charge, spin, parity, lifetime.particle and antiparticle.fermion and boson.lepton and hadrons.hadrons: meson and baryon.baryons: nucleon and hyperons.particle creation and annihilation: quantum field theories.interactions: gravity; electromagnetic; weak; strong.standard model: unification of electromagnetic; weak; strong.string theory: unification of gravity; electromagnetic; weak; strong.high energy experiments.cross section and lifetime.
1.2.1 Feynman diagrams in the pseudo-nucleon meson interaction
This course focuses on perturbative quantum field theory using Feynman diagrams. Thekey words of this course includes Feynman Diagrams, Feynman Rules and Feynman Inte-grals. To compute Feynman diagrams, one has to map a Feynman diagram to a Feynmanintegral with Feynman rules.
The “Nucleon” (pseudo-nucleon) meson interaction as a toy model to understand Yukawainteraction and QED.
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The model is characterized by the Lagrangian
L = L0 + Lint, (1.2.1)
where L0 is the free part included the free “nucleon” L(ψ) and “meson” L(φ),
L0(ψ) =1
2∂µψ
†∂µψ −m2ψ†ψ; (1.2.2)
L0(φ) =1
2∂µφ∂
µφ− µ2φ2. (1.2.3)
And the interaction part has the formalism Lint = −gψ†ψφ. The parameter g is namedas coupling constant. Note that we also have the Hamiltonian formalism H0 = L0 andHint = gψ†ψφ.
Table 1.1: “Nucleon”-meson interaction
“Nucleon” Anti-“Nucleon”Meson
Pseudo-nucleon Anti-pseudo-nucleon
“N” “N” φ
something something π0
spin-0 spin-0 spin-0
mass “m” mass “m” mass “µ”
charge +1 charge −1 charge 0
complex scalar fields ψ(x), ψ∗(x) real scalar φ∗(x) = φ(x)
E.g.1. Decay of a meson into “nucleon”-“anti-nucleon”.
φ(k) −→ “N”(p) + “N”(q). (1.2.4)
“N”
“N”p
q
k
φ
(1.2.5)
The decay amplitude is calculated as
S(φ→ “N” + “N”) = (2π)4δ4(k − p− q)× iA, (1.2.6)
where the delta function is required for energy-momentum conservation.
In the way of computing the amplitude iA, we can apply the time-dependent per-turbation theory with Dyson’s series to derive the Feynman diagrams and the as-sociated Feynman rules. The amplitude iA can be expanded as
iA = iA(1) + iA(2) + · · ·+ iA(n) + · · · , (1.2.7)
13
where the label n stands for the number of constant ~ and unusually set as ~ = 1.In the first order iA(1), it is related to the Feynman diagram (1.2.5). The Feynmanrules summarize as: the external lines represented for meson, “nucleon” and anti-“nucleon” contribute the factor 1. The interaction vertex gives rise to the factor−ig. Therefore, the first order amplitude is
iA(1) = 1× 1× 1× (−ig) = −ig. (1.2.8)
E.g.2. “Nucleon”-“nucleon” scattering.
“N”(p) + “N”(q) −→ “N”(p′) + “N”(q′). (1.2.9)
p1
p2
p′1
p′2
At the time t = −∞, two-incoming “nucleons” are freely moving. After scattering,t = +∞, two particles are freely moving again. The grey region represents for thescattering interaction. The scattering matrix S has the formalism
S(|p1, p2〉 → |p′1, p′2〉) = (2π)4δ(p′1 + p′2 − p1 − p2)iA. (1.2.10)
In interaction picture, the initial and final state take the form
|ψ(t = −∞)〉 = |p1, p2〉, |ψ(t = +∞)〉 = |p′1, p′2〉. (1.2.11)
Then applying the time-dependent perturbation to derive the Dyson’s series
UI =+∞∑
n=0
U (n)(tf , ti), (1.2.12)
the scattering operation (matrix) can be expressed as
S = UI(+∞,−∞) =
+∞∑
n=0
S(n) = 1 +
+∞∑
n=1
S(n), (1.2.13)
S(|p1, p2〉 → |p′1, p′2〉) = 〈p′1, p′2|p1, p2〉+
+∞∑
n=1
〈p′1, p′2|S(n)|p1, p2〉. (1.2.14)
The expansion of the amplitude can be calculated iA(0) = iA(1) = 0 and
(2π)4δ(p′1 + p′2 − p1 − p2)iA(2) = 〈p′1p′2|S(2)|p1p2〉. (1.2.15)
With Wick’s theorem, Feynman diagrams and Feynman rules can be determined.
14
q − q′
q′q
p p′
“N”
“N”
“N”
“N”
+ (p′ ↔ q′)iA(2) =
The internal line stands for the Feynman propagator for meson, and contributesthe factor i
(p1−p′1)2−µ2 .
iA(2) = 1× 1× (−ig)× i
(p1 − p′1)2 − µ2× (−ig)× 1× 1 + (p1 ↔ p′1)
= (−ig)2
(i
(p1 − p′1)2 − µ2+
i
(p1 − p′2)2 − µ2
). (1.2.16)
E.g.3. Anti-“nucleon”-anti-“nucleon” scattering.
“N”(p) + “N”(q) −→ “N”(p′) + “N”(q′). (1.2.17)
q − q′
q′q
p p′
“N”
“N”
“N”
“N”
+ (p′ ↔ q′)
E.g.4. “Nucleon”-anti-“nucleon” scattering.
“N”(p) + “N”(q) −→ “N”(p′) + “N”(q′). (1.2.18)
“N” ‘N”
“N” “N”
p− p′
p′p
q q′
+ p+ q
p′p
q′q
“N”“N”
“N”“N”
E.g.5. “Nucleon”-meson scattering.
“N”(p) + φ(q) −→ “N”(p) + φ(q′). (1.2.19)
p+ q
q′q
p′p
φφ
“N”“N”
+ (q ↔ q′)
15
E.g.6. Anti-“nucleon”-meson scattering.
“N”(p) + φ(q) −→ “N”(p) + φ(q′). (1.2.20)
p+ q
q′q
p′p
φφ
“N”“N”
+ (q ↔ q′)
E.g.7. Nucleon-anti-nucleon annihilation.
“N”(p) + “N”(q) −→ φ(p′) + φ(q′). (1.2.21)
p− p′
p′p
q q′
“N” φ
“N” φ
+ (p′ ↔ q′)
E.g.8. Meson-meson scattering.
φ(p) + φ(q) −→ φ(p′) + φ(q′). (1.2.22)
k + p
k − q
k k + p− p′
p′p
q q′
Remark: about two-particle scattering at order O(g2).
1). Virtual particle on internal line is meson.
“N” + “N” −→ “N” + “N”; (1.2.23)
“N” + “N” −→ “N” + “N”; (1.2.24)
“N” + “N” −→ “N” + “N”. (1.2.25)
2). Virtual particle on internal line is nucleon.
“N” + “N” −→ φ+ φ; (1.2.26)
“N” + φ −→ “N” + φ; (1.2.27)
“N” + φ −→ “N” + φ. (1.2.28)
16
1.2.2 Feynman diagrams in the Yukawa interaction
“Nucleon-meson” interaction as the Yukawa interaction is the standard model of particlephysics!
Table 1.2: The Yukawa interaction
“Nucleon” Anti-“Nucleon” Meson
“N” “N” φ
proton anti-proton π0-meson
spin- 12
spin- 12
spin-0
mass “m” mass “m” mass “µ” m
charge +1 charge −1 charge 0
Dirac field (spinor-field) ψ(x) real scalar φ∗(x) = φ(x)
E.g.1. The decay of a meson.
φ(p+ q) −→ N(p, r) +N(q, s). (1.2.29)
N
Np, r
q, s
p+ q
φ
p+ qφ particle with four momentum p+ q incoming.
p, routgoing nucleon with four momentum p and spin r, r = 1
2 , − 12 .
q, soutgoing anti-nucleon with four momentum q and spin s, s = 1
2 , − 12 .
The outline arrows represents for the momentum and the inline arrow charactersthe charge flow.
E.g.2. Nucleon-nucleon scattering.
N(p, r) +N(q, s) −→ N(p′, r′) +N(q′, s′). (1.2.30)
q − q′
q′, s′q, s
p, r p′, r′
N
N
N
N
+ (q′, s′ ↔ p′, r′)
17
E.g.3. Anti-nucleon-anti-nucleon scattering.
N(p, r) +N(q, s) −→ N(p′, r′) +N(q′, s′). (1.2.31)
q − q′
q′, s′q, s
p, r p′, r′
N
N
N
N
+ (q′, s′ ↔ p′, r′)
E.g.4. Nucleon-anti-nucleon scattering.
N(p, r) +N(q, s) −→ N(p′, r′) +N(q′, s′). (1.2.32)
p+ q
p′, r′p, r
q′, s′q, s
NN
NN
+(p, r ↔ q′, s′)
N N
N N
p− p′
p′, r′p, r
q, s q′, s′
E.g.5. Nucleon-meson scattering.
N(p, r) + φ(q) −→ N(p′, r′) + φ(q′). (1.2.33)
p+ q
q′q
p′, r′p, r
φφ
NN
+ (q ↔ q′)
E.g.6. Anti-nucleon-meson scattering.
N(p, r) + φ(q) −→ N(p′, r′) + φ(q′). (1.2.34)
p+ q
q′q
p′, r′p, r
φφ
NN
+ (q ↔ q′)
E.g.7. “Nucleon”-anti-“nucleon” annihilation.
N(p, r) +N(q, s) −→ φ(p′) + φ(q′). (1.2.35)
18
p− p′
p′p, r
q, s q′
N φ
N φ
+ (p′ ↔ q′)
E.g.8. Meson-meson scattering.
φ(p) + φ(q) −→ φ(p′) + φ(q′). (1.2.36)
k + p
k − q
k k + p− p′
p′p
q q′
1.2.3 Feynman diagrams in quantum electrodynamics
Table 1.3: Comparison between Yukawa interaction and QED
Matter Mediator (Interaction)
”Nucleon”-meson interaction ”Nucleon” (spinless, complex scalar) meson (spinless, real scalar)
Yukawa interaction Nucleon (spin- 12, spinor ) meson (spinless, real scalar)
QED electron (spin- 12, spinor) photon (spin-1, vector)
Example: typical Feynman diagrams in quantum electrodynamics.
a). The Moller scattering between two electrons.
e− + e− −→ e− + e−. (1.2.37)
p− p′
p′, s′p, s
q, r q′, r′
e−
e−
e−
e−
µ
ν− (p′, s′ ↔ q′, r′)
19
The on-line arrow represents an electron and the out-line for momentum labeled byp, q, p′ and q′, which have the form (Ep, ~p). Parameters r, s, r′ and s′ represents forthe spin degree of the electron. For example, r = 1
2 stands for spin up and r = −12 for
spin down. The wave line stands for a virtual photon in the scattering process. Suchthe diagram means that two-electron scattering is performed by an interchange of avirtual photon. Note that electron is a quanta of Dirac field and photon is a quantaof electrodynamics field.
Such two Feynman diagrams are indistinguishable in physics because two outgoingelectrons (p′, s′) and (q′, r′) are identical particles and we have the minus sign due toFermi-Dirac statistics.
Similarly, the scattering process between two positrons:
e+ + e+ −→ e+ + e+. (1.2.38)
p− p′
p′, s′p, s
q, r q′, r′
e+
e+
e+
e+
µ
ν− (p′, s′ ↔ q′, r′)
b). The Bhabha scattering between an electron and a positron.
e− + e+ −→ e− + e+. (1.2.39)
e− e−
e+ e+
p− p′
p′, s′p, s
q, r q′, r′
−(q, r ↔ p′, s′)
p+ q
p′, s′p, s
q′, r′q, r
e−e−
e+e+
c). The pair annihilation between an electron and a positron into two photons.
e− + e+ −→ 2γ. (1.2.40)
e−
e+
γ
γ
p− p′
p, s
q, r
p′, εν1
q′, εµ2
µ
ν
+ (p′, ε1 ↔ q′, ε2)
d). The Compton scattering between an electron and a photon.
e− + γ −→ e− + γ. (1.2.41)
20
p+ q
q′, εµ2q, εν1
p′, s′p, s
γγ
e−e−
ν µ + (q, ε1 ↔ q′, ε2)
e). Two-Photon scattering.
γ + γ −→ γ + γ. (1.2.42)
−→
1.3 The canonical quantization procedure
Canonical quantization is a procedure of deriving QM (quantum mechanics) (or QFT(quantum field theory)) from the Hamiltonian formulation of CPM (Classical particlemechanics)(or CFT (classical field theory)).
CPM CFT
QM QFT
H H
H H
−→ Continuum limit
Quantization
Note 1: In classical mechanics, we both have Lagrangian formalism L(q, q) and Hamil-tonian formalism H(q, p) to characterize the equation of motion.
Note 2: Non-relativistic quantum mechanics prefers the Hamiltonian formulation, forexample the Schrodinger equation is described by Hamiltonian.
Note 3: In relativistic QFT, the Hamiltonian formalism is associated with the canonicalquantization and the Lagrangian formalism is associated with the path integralformalism.
Note 4: The two formalisms are essentially equivalent, but they are used in differentcircumstances.
1.4 Research projects in this course
1. Conventional projects in quantum field theoriesTraditional research in (high energy physics) HEP:
21
• Particle physics: the study of phenomenology in experiments;
• Doing experiments (in CERN);
• QFT: theoretical study in high energy physics.
2. A millennium problem in quantum field theoryAn Open Problem in QFT :
One of Seven Millennium Problems Wikipedia : Prove that Yang-Mills theory (Non-Abelian Gauge Field Theory) actually exists and has a unique ground state.
Note 1: The Yang-Mills theory was proposed by C.N. Yang and Mills in 1954, whenYang was 32 years old.
Note 2: The Yang-Mills theory is Non-abelian gauge field theory, which is the theoret-ical formulation of the Standard Model.
3. New projects in quantum field theoryNowadays quantum mechanics has been updated with quantum computation and in-
formation.
Note 1: Quantum information and computation (QIC) represents a further develop-ment of quantum mechanics.
Note 2: QIC focuses on fundamental principles and logic of quantum mechanics.
Note 3: QIC is a new type of advanced quantum mechanics.
Note 4: QIC is the study of information processing tasks that can be accomplishedusing quantum mechanical systems (or using fundamentals of quantum mechan-ics).
Question: ?=Special Relativity+ Quantum Information and Computation.
22
Lecture 2 From Classical Mechanics to
Quantum Mechanics
2.1 Classical particle mechanics
2.1.1 Lagrangian formulation of classical particle mechanics
In a system of n-particles, the state is characterized by the generalized coordinates q1, q2, · · · , qnand their time derivations q1, q2, · · · , qn. The Lagrangian is defined as
L(q1 · · · qn, q1 · · · qn) = T − V. (2.1.1)
And the action is defined as
S =
∫ t2
t1
dtL. (2.1.2)
Note: L does not have explicit time dependence,
∂L
∂t= 0, (2.1.3)
which is associated with the conservation of energy.
Action principle which has another names including Hamilton’s principle, the principleof least action and variational principle: for arbitrary variation qa → qa(t) + δqa(t) withfixed boundaries δqa(t1) = δqa(t2) = 0, we have stationary action, namely δS = 0.
δS =
∫ t2
t1
dt
(∂L
∂qaδqa +
∂L
∂qaδqa
)(2.1.4)
with∂L
∂qaδqa =
d
dt
(∂L
∂qaδqa
)− d
dt
(∂L
∂qa
)δqa, (2.1.5)
we have
δS =
∫ t2
t1
dt
(∂L
∂qa− d
dt
(∂L
∂qa
))δqa +
∂L
∂qaδqa
∣∣∣∣t2
t1
. (2.1.6)
Due fixed boundary condition and action principle, we can derive
∂L
∂qa=
d
dt(∂L
∂qa), (2.1.7)
which is the equation of motion, also named as Euler-Lagrangian equation.
23
2.1.2 Hamiltonian formulation of classical particle mechanics
Canonical momentum:
pa =∂L
∂qa, a = 1, 2, · · · , n. (2.1.8)
Hamiltonian:
H(q1 · · · qn, p1 · · · pn) =∑
a
paqa − L(q1 · · · qn, q1 · · · qn). (2.1.9)
Note: Hamiltonian is not a function of qa, and it is a function of qa and pa.
Proof.
dH =∑
a
(padqa + qadpa −
∂L
∂qadqa −
∂L
∂qadqa
)
=∑
a
(qadpa −
∂L
∂qadqa
)+∑
a
(padqa −
∂L
∂qadqa
)
=∑
a
(qadpa −
∂L
∂qadqa
). (2.1.10)
Hamilton’s equations:
qa =∂H
∂pa, pa = −∂H
∂qa. (2.1.11)
Optional problem: Derive Hamilton’s equations.Optional problem: Derive Newtonian mechanics in both Lagrangian formulation and
Hamiltonian formulation.
2.1.3 Noether’s theorem: symmetries and conservation laws
We have two different variations:
• δqa: variation for deriving EoM;
• Dqa: variation for symmetries without specifying EoM.
Def 2.1.1. A symmetry is a transformation to keep physics (EoM) unchanged. A trans-formation Dqa is called symmetry iff
DL =dF
dt(2.1.12)
for some F (qa, qa, t) with arbitrary qa(t) which may not satisfy the EoM.
Remark:
DS =
∫ t2
t1
dtDL = F (t2)− F (t1), (2.1.13)
S′ = S +DS, (2.1.14)
δS′ = δS + δ(DS) = δS + δF (t2)− δF (t1), (2.1.15)
because the boundary terms at t1 and t2 are fixed, therefore the EoM is unchanged.
24
Thm 2.1.3.1 (Noether’s Theorem (Particle mechanics)). For every continuous symmetry,there is a conserved quantity.
Proof.
1). On the one hand, without EoM, we have DL = dF/dt for ∀qa, qa.
2). On the another hand, with EoM and canonical momentum,
DL =∑
a
(∂L
∂qaDqa +
∂L
∂qaDqa)
=∑
a
(paDqa + paDq
a)
=d
dt(∑
a
paDqa),
(2.1.16)
we can derive the conserved quantity denoted as Q
Q =∑
a
paDqa − F, (2.1.17)
d
dt(∑
a
paDqa − F ) =
dQ
dt= 0. (2.1.18)
E.g.1. Space translation invariance // momentum conservation.
L =1
2m1q
21 +
1
2m2q
22 − V (q1 − q2). (2.1.19)
The space translation with infinitesimal constant α is defined as
q1 → q1 + α, Dq1 = α;
q2 → q2 + α, Dq2 = α.(2.1.20)
And the Lagrangian is unchanged under the space translation, namely
L′ =1
2mq2
1 +1
2mq2
2 − V (q1 − q2) = L, (2.1.21)
thus
DL = 0⇒ F = 0. (2.1.22)
Therefore, the conserved quantity can be defined as
Q = p1Dq1 + p2Dq
2 = α(p1 + p2), (2.1.23)
d
dt(p1Dq1 + p2Dq2) =
dP
dt= 0, (2.1.24)
where P is the total momentum which is conserved, i.e., P = p1 + p2.
25
E.g.2. Time translation invariance// energy conservation.
The time translation with infinitesimal constant α is defined in the following way
t→ t+ α, (2.1.25)
qa(t)→ qa(t+ α), (2.1.26)
L(t)→ L(t+ α), (2.1.27)
Dqa = qa(t+ α)− qa = αdqa
dt+O(α2), (2.1.28)
thus
DL = L(t+ α)− L(t) = αdL
dt+O(α2)⇒ F = αL. (2.1.29)
Therefore, the conserved quantity Q is given by
Q = p1Dq1 + p2Dq2 − αL= α(p1q1 + p2q2 − L)
= αH,
(2.1.30)
dQ
dt= α
dH
dt= 0 =⇒ H = const, (2.1.31)
namely the Hamiltonian (energy) is conserved.
2.1.4 Exactly solved problems
4. The Kepler problem with the inverse-square law of force.
L =1
2m(r2 + r2θ2)− V (r), (2.1.32)
where V (r) is the potential energy denoted as
V (r) = −αr
(2.1.33)
and α is a constant.
4. The simple Harmonic oscillator.
L =1
2mq2 − 1
2kq2, (2.1.34)
from which the EoM can be derived as
mq + kq = 0 (2.1.35)
or
q + w2q = 0, (2.1.36)
where w =√
km .
26
2.1.5 Perturbation theory
In perturbation theory, the Hamiltonian can be written as
H(qa, pa, t) = H0(qa, pa, t) +H ′int(qa, pa, t), (2.1.37)
where H0 is the Hamiltonian that the EoM can be exactly solved and H ′int is the smallperturbation term. Therefore, the total Hamiltonian H can be solved in perturbationtheory.
2.1.6 Scattering theory
The procedures of describing scattering phenomena are the same whether themechanics is classical or quantum.
—Goldstein
The quantity jin, incident density (flux density), is defined as the incident particlenumber per unit time and per unit area, where the unit area is normal to the incidentdirection. For jout, it is the number of scattering particles per unit time. The total crosssection σ describes the scattering process in the way
jinσ = jout, (2.1.38)
and note that the total cross section has the dimension of area.
The differential cross section dσdΩ is defined as
jindσ
dΩ=djout
dΩ, (2.1.39)
hence djout
dΩ specifies the number of particles scattering into per solid angle where
dΩ = sin θdθdϕ = 2π sin θdθ. (2.1.40)
The angle θ is the degree between the incident particles and the scattered particles, namedas scattering angle. Usually in scattering process, the central force is symmetrical aroundthe incidental axis, therefore the angle ϕ in solid angle can be integrated out as 2π.
E.g. in classical particle mechanics, in the Rutherford scattering experiment, due to theconservation of particle number
djout = jin2πbdb = jin
(dσ
dΩ
)|2π sin θdθ|, (2.1.41)
where quantity b is defined as the perpendicular distance between the incidental particleand the center of force, also called as impact parameter. Absolute value is required forthe positivity of particle number.
In repulsive interaction:dσ
dΩ=
bdb
sin θdθ; (2.1.42)
In attractive interaction:dσ
dΩ= − bdb
sin θdθ. (2.1.43)
27
2.1.7 Why talk about classical particle mechanics in detail
Topics in the course of QFT:
1). Canonical quantization of QFT: Hamiltonian formulation.
2). Symmetries and conservation laws: Lagrangian formulation.
3). Perturbative QFT: Feynman diagrams.
4). Calculation of cross section in High energy scattering experiments.
2.2 Advanced quantum mechanics
Quantum mechanics is defined as non-relativistic quantum particle mechanics. Advancedquantum mechanics present contents between quantum mechanics for undergraduate stu-dents and quantum field theory for graduate students.
2.2.1 The canonical quantization procedure
Heisenberg’s quantum mechanics is derived from the canonical quantization procedure ofthe Hamiltonian formulation of classical particle mechanics.
The Possion brackets is defined as
f, g ≡∑
a
(∂f
∂pa
∂g
∂qa− ∂f
∂qa
∂g
∂pa
), (2.2.1)
with which the EoM can be rewritten as
qa(t) = H, qa =∂H
∂pa; (2.2.2)
pa(t) = H, pa = −∂H∂qa
. (2.2.3)
Note thatdA
dt=∂A
∂t+ H,A, (2.2.4)
where A is the observable other than H.
4. Canonical quantization procedure.
Step 1:
(qa(t), pa(t)) −→ (qa(t), pa(t)). (2.2.5)
Replace classical variables (qa(t), pa(t)) with operator-valued function of time sat-isfying the commutation relations
[qa(t), qb(t)] = 0 = [pa(t), pa(t)], (2.2.6)
[qa(t), pb(t)] = iδab = i~δab. (2.2.7)
28
Step 2:H(q, p) −→ H(q, p). (2.2.8)
Quantum Hamiltonian H has the same form of classical Hamiltonian H exceptthat it is a function of qa and pb.
Note: The canonical quantization suffers from the ordering ambiguity, for instancep2q 6= pqp 6= qp2, but they are the same in the classical physics limit, namely~→ 0, such as
〈p2q〉 = 〈pqp〉 = 〈p〉2〈q〉 = p2q. (2.2.9)
Step 3: Heisenberg’s equation of motion:
dqa(t)
dt= i[H, qa(t)] =
∂H
∂pa, (2.2.10)
dpa(t)
dt= i[H, pa(t)] = −∂H
∂qa. (2.2.11)
For the time dependent observable A(t):
dA(t)
dt=∂A
∂t+ i[H, A]. (2.2.12)
Step 4: Prove the Hamiltonian H is bounded from below, namely, that ground state existsand the Hilbert space can be constructed.
Note 1: Heisenberg’s matrix (operator) quantum mechanics was originally derived viacanonical quantization procedure.
Note 2: Canonical quantization procedure makes the relationship between CPM and QMclear.
2.2.2 Fundamental principles of quantum mechanics
4.
Static part*-HHj
A: Observable.
|ψ〉: State.
H : Hilbert space.
4.
Dynamic part*
HHj
Unitary evolution:
|ψ(t)〉 = U(t)|ψ(0)〉, i~ ∂∂t |ψ(t)〉 = H|ψ(t)〉.
Non-unitary evolution:
Quantum measurement, wave function collapse, |ψ〉 → |ψ′〉.
The standard quantum mechanics:
• System: Hilbert space H.
Note: A quantum binary digit is a two-dimensional Hilbert space.
29
• State: A closed system is described by a pure vector |ψ〉 ∈ H obeying the linearsuperposition principle, namely
a|ψ1〉+ b|ψ2〉 ∈ H, a, b ∈ C. (2.2.13)
Note: Quantum computer is powerful mainly due to the superposition principlewhich allows a kind of parallel computation on a single quantum computer.
• Observable O. Hermitian operator: O† = O.
• Unitary evolution of a state vector in a closed system:
|ψ(t)〉 = U(t)|ψ(0)〉, U †(t)U(t) = Id. (2.2.14)
E.g. A unitary evolution of a closed system is governed by the Schrodinger equation
i~∂
∂t|ψ(t)〉 = H|ψ(t)〉 (2.2.15)
with H† = H.
Note: A quantum gate is a unitary transformation acting on quantum binary digits.
• Non-unitary evolution of state vector, also named as quantum measurement, quan-tum jump, quantum transition, wave function collapse and information loss.
Note: Quantum measurement is not well defined in the viewpoint of an expert againstQM, but is the main computing resource in quantum information & computation.
• The composite system of A and B, i.e., HA and HB, is described by the tensorproduct HA ⊗HB, namely |ψ〉A ⊗ |ψ〉B ∈ HA ⊗HB.
Note: Quantum entanglement arises in the quantum composite system, which dis-tinguishes classical physics from quantum physics and it plays the crucial role inquantum information & computation.
2.2.3 The Schroedinger, Heisenberg and Dirac picture
In quantum mechanics, the probability or matrix element or probability amplitude isa realistic observed quantity, but quantum mechanics is directly described by the statevectors and observables. So that a picture defines a choice of state vectors and observablesto preserve the probability amplitude.
4.
QM: Shrodinger picture :XXz
|ψS(0)〉 → |ψS(t)〉.OS(t) = OS(0).
4.
QFT: Heisenberg picture :XXz
|ψH(0)〉 = |ψH(t)〉.OH(0)→ OH(t).
30
Transition amplitude is unchanged in any picture,
〈ψS(t)|OS |ψS(t)〉 = 〈ψH(t)|OH |ψH(t)〉. (2.2.16)
• The Schrodinger pictureOS(t) = OS(t0); (2.2.17)
|ψS(t)〉 = U(t, t0)|ψS(t0)〉. (2.2.18)
• The Heisenberg picture
|ψH(t)〉 = |ψH(t0)〉; (2.2.19)
OH(t) = U †(t, t0)OH(t0)U(t, t0). (2.2.20)
Table 2.1: The Shrodinger, Heisenberg and Dirac picture
Schrodinger Heisenberg Dirac (Interaction)
state vector
time-dependent
time-dependent time-independent H = H0 +H ′int
i~ ∂∂t|ψ(t)〉S = H|ψ(t)〉S |ψ(t)〉H = |ψ(0)〉H i~ ∂
∂t|ψ(t)〉I = HI|ψ(t)〉I
HI(t) = eiH0tH ′inte−iH0t
observabletime-independent time-dependent time-dependent
OS(t) = OS(0) i ddtOH = [OH, H] i d
dtOI = [OI, H0]
initial valueOS(0) = OH(0) = OI(0)
|ψS(0)〉S = |ψS(0)〉H = |ψS(0)〉I
probability amplitude S〈ψ(t)|OS|ψ(t)〉S = H〈ψ(t)|OH|ψ(t)〉H = I〈ψ(t)|OI|ψ(t)〉I
Note 1: Non-relativistic quantum mechanics prefers the Schrodinger picture.
Note 2: Heisenberg’s picture is good for relativistic QFT.
Note 3: Interaction picture is good for time-dependent perturbation theory. For example,perturbative QFT which is a collection of Feynman diagrams.
Note 4: In the case of time-independent Hamiltonian, we have the following equations indifferent pictures.
|ψ(t)〉S = e−iH(t−t0)|ψ(t0)〉S; (2.2.21)
OH(t) = eiHtOH(0)e−iHt; (2.2.22)
OI(t) = eiH0tOI(0)e−iH0t; (2.2.23)
|ψ(t)〉S = e−iH0t|ψ(t)〉I, (2.2.24)
where the free Hamiltonian H0 can be exactly solved.
Prove
id
dt|ψ(t)〉I = HI(t)|ψ(t)〉I. (2.2.25)
31
Proof.
id
dt|ψ(t)〉I = i
d
dt
(eiH0t|ψ(t)〉S
)
= −H0eiH0t|ψ(t)〉S + eiH0ti
d
dt|ψ(t)〉S
= −H0|ψ(t)〉I + eiH0t(H0 +H ′int)|ψ(t)〉S= −H0|ψ(t)〉I + eiH0tH0|ψ(t)〉S + eiH0tH ′int|ψ(t)〉S= eiH0tH ′inte
−iH0t|ψ(t)〉I= HI(t)|ψ(t)〉I. (2.2.26)
In the interaction picture, the equation of time evolution
id
dt|ψ(t)〉I = HI(t)|ψ(t)〉I (2.2.27)
can be solved in the Dyson’s series. Introduce the unitary operator UI(t, t0) satisfying
U †I UI = Id and UI(t0, t0) = 1. And the equation of time evolution can be rewritten as
i~d
dtUI(t, t0) = HI(t)UI(t, t0). (2.2.28)
Integrate out above equation, we can obtain
UI(t, t0) = UI(t0, t0) + (− i~
)
∫ t
t0
dt1HI(t1)UI(t1, t0), (2.2.29)
and then we can perform the iteration method to obtain the relation
UI(t, t0) = 1 ++∞∑
n=1
(− i~
)n∫ t
t0
dt1
∫ t1
t0
dt2 · · ·∫ tn−1
t0
dtnHI(t1)HI(t2) · · ·HI(tn). (2.2.30)
Note: In QFT, the Dyson’s series have another compact formulation which is used toderive Feynman diagrams.
2.2.4 Non-relativistic quantum many-body mechanics
An N -particle system with
• Particle number = N , same mass m;
• External potential: U(x);
• Inter-particle potential: V (~xi − ~xj), 1 < j < k < N .
The equation of motion has the form
i~∂
∂tψ(t, ~x1, · · · , ~xn) = Hψ(t, ~x1, · · · , ~xn), (2.2.31)
with the Hamiltonian H given by
H =N∑
j=1
(− ~2
2m∇2j + U(xj)) +
N∑
j=1
j−1∑
k=1
V (~xj − ~xk). (2.2.32)
32
Note 1: Particle number is fixed, namely N is a conserved quantity, [H,N ] = 0.
Note 2: Position operators X1, X2, · · · , Xn are well defined;
Note 3: Momentum operators P1, P2, · · · , Pn are well defined and other observables areindependent coordinates ~x1, ~x2, · · · , ~xn.
However, quantum field theory tells another story,
1. Particle number is not-fixed;
2. Position operators X1, X2, · · · , Xn do not exist;
3. Observable O(t, ~x) depends on space-time.
Table 2.2: Comparison between Quantum Mechanics and Quantum Field Theory
Quantum Mechanics Quantum Field Theory
Particle Number Fixed Un-fixed
time parameter label/parameter label/parameter
Position operator Xi well-defined No Xi because particles can be destroyed
Position parameter ~xi well-defined ~xi well-defined as labels or parameters
Momentum operator Pi well-defined Pi well-defined
Observable independent of coordinate ,O(t) dependent on (t, ~x), O(t, ~x)
Special reality No Yes
Lorentz transformation No Yes
ψ(t, ~x1, ~x2, · · · , ~xn) wave function (first quantization) quantum field operator (second quantization)
2.2.5 Compatible observable
Observables A and B are called compatible when
[A,B] = AB −BA = 0, (2.2.33)
and incompatible when [A,B] 6= 0.
Note: The concept of compatible observable is associated with micro-causality in QFT.
2.2.6 The Heisenberg uncertainty principle
The Heisenberg’s uncertainty relation
〈(∆A)2〉〈(∆B)2〉 ≥ 1
4|〈[A,B]〉|2 , (2.2.34)
where∆A = A− 〈A〉, (2.2.35)
〈(∆A)2〉 = 〈A2〉 − 〈A〉2. (2.2.36)
E.g. ∆x∆px ≥ ~4 . When ∆x → 0, ∆px → +∞, which implies px → +∞. Small distance
means high energy physics in which particles can be created and annihilated, so that theposition of a particle becomes as non-sense.
Note: the uncertainty relation ∆x∆px ≥ ~4 is meaningful in QM, but questionable in QFT.
33
2.2.7 Symmetries and conservation laws
Symmetry in quantum mechanics is a transformation D, iff [D,H] = 0. The conservedcharge is denoted as Q, which is a constant of the motion, namely dQ
dt = 0, then [Q,H] = 0.
E.g. D is a unitary transformation, and we can construct the transformation as
D = 1− i
~εQ+O(ε2). (2.2.37)
The unitary constraint of the transformation D requires the generator Q as a Hermitianoperator, namely Q† = Q.
Note: In the Schrodinger picture, the equation of time evolution has the form
i~∂
∂t|ψ(t)〉 = H|ψ(t)〉. (2.2.38)
Because the symmetry transformation D preserves the EoM, we have
i~∂
∂t(D|ψ(t)〉) = H(D|ψ(t)〉). (2.2.39)
If D−1 exists and D is irreverent of time, then
i~∂
∂t|ψ(t)〉 = D−1HD|ψ(t)〉 = H|ψ(t)〉, (2.2.40)
which implies [D,H] = 0.
Note: In QM, symmetries and conservation laws can be helpful in some sense, for example,the degeneracies of quantum states due to symmetry, but in modern QFT, symmetries andconservation laws play the essential (key) roles which guide us to specify the formulationof the action.
2.2.8 Exactly solved problems in quantum mechanics
Examples of exactly solved problems in quantum mechanics:
e.g.1: Free particles.
e.g.2: Many problems in one dimension.
e.g.3: Transmission-reflection problem.
e.g.4: Harmonic oscillator.
e.g.5: The central force problem.
e.g.6: The Hydrogen atom problem.
Note: In QFT, most problems can be solved in a perturbative approach using Feynmandiagram.
34
2.2.9 Simple harmonic oscillator
The Hamiltonian of the simple harmonic oscillator takes the form
H =~P 2
2m+
1
2mω2X2, ω 6= 0. (2.2.41)
where the position operator Xi and momentum operator Pj obey the commutative relation
[Xi, Pj ] = i~δij . (2.2.42)
Introduce the Canonical transformation:
p =P√mω
, q =√mωX, (2.2.43)
and rewrite the Hamiltonian asH =
ω
2(p2 + q2), (2.2.44)
where the lower indices of X,P, q, p are neglected for simplicity or one can understand theharmonic oscillator as a one-dimensional harmonic oscillator.
With the new defined raising and lowering operators denoted as
a =q + ip√
2, a† =
q − ip√2, (2.2.45)
with[a, a†] = 1, (2.2.46)
the Hamiltonian can be rewritten as
H = ω(a†a+1
2). (2.2.47)
The number operator is defined asN = a†a, (2.2.48)
thus
H = ω(N +1
2). (2.2.49)
Due to 〈ψ|a†a|ψ〉 > 0, define the ground state |0〉 as a|0〉 = 0, and the first excitedstate as |1〉 = a†|0〉.
H|0〉 = ω(N +1
2)|0〉 =
1
2ω|0〉 = E0|1〉, (2.2.50)
H|1〉 = ω(N +1
2)|1〉 =
3
2ω|1〉 = E1|1〉. (2.2.51)
The gap of eigenvalue isE1 − E0 = ω 6= 0. (2.2.52)
The eigenstate |n〉 is denoted as
|n〉 =1√n!
(a†)n|0〉 (2.2.53)
with the eigenvalue En = (n+ 12)~ω, namely H|n〉 = En|n〉.
Note: Free quantum field theory (QFT without interaction) can be regarded as a collectionof an infinite number of simple harmonic oscillator.
35
2.2.10 Time-independent perturbation theory
The full Hamiltonian can be expanded as two parts
H = H0 + λH ′int, (2.2.54)
where λ is a small real parameter in the range 0 < λ < 1 and H ′int is timeless. For the
free Hamiltonian H0, it has the eigenstate |n(0)〉 with the eigenvalue E(0)n , namely
H0|n(0)〉 = E(0)n |n(0)〉. (2.2.55)
And for the full Hamiltonian H, we define
H|n〉 = En|n〉. (2.2.56)
We can expand the eigenstate and eigenvalue of the full Hamiltonian in terms of the powerof parameter λ
|n〉 = |n(0)〉+ λ|n(1)〉+ λ2|n(2)〉+ · · · ; (2.2.57)
En = E(0)n + λ∆(1)
n + λ2∆(2)n + · · · . (2.2.58)
where the energy shift ∆n is defined as
∆n ≡ En − E(0)n . (2.2.59)
In the perturbative theory, the result formula of the eigenstate and eigenvalue of fullHamiltonian show as
|n〉 = |n(0)〉+ λ∑
k 6=n|k0〉〈k
0|H ′int|n(0)〉E
(0)n − E(0)
k
+ · · · ; (2.2.60)
En = E(0)n + λ〈n(0)|H ′int|n(0)〉+ λ2
∑
k 6=n
∣∣〈n(0)|H ′int|k(0)〉∣∣2
E(0)n − E(0)
k
+ · · · . (2.2.61)
Note: When E(0)n −E(0)
k is very small, the high energy physics arises naturally where QFThas to be considered.
2.2.11 Time-dependent perturbation theory
The full Hamiltonian can be split as
H = H0 +H ′int(t), (2.2.62)
where H ′int(t) is time-dependent.
In the interaction (Dirac) picture, the observable obeys the equation
dOI(t)
dt=
1
i~[OI , H0], (2.2.63)
and the state follows the Shrodinger-like equation
i~d
dt|ψI(t)〉 = H ′I(t)|ψI(t)〉 (2.2.64)
36
with H ′I(t) = eiH0tH ′inte−iH0t.
The state |ψI(t)〉 can be expanded in terms of the basis of free Hamiltonian, namely |n〉with H0|n〉 = E
(0)n |n〉,
|ψI(t)〉 =∑
n
Cn(t)|n〉, (2.2.65)
where Cn(t) = 〈n|ψI(t)〉.
Note:
|ψS(t)〉 = e−iH0t|ψI(t)〉 =∑
n
Cn(t)e−iE(0)
n t~ |n〉; (2.2.66)
i~d
dtCn(t) =
∑
m
〈n|H ′int|m〉eiωnmtCm(t) (2.2.67)
with
ωnm =E
(0)n − E(0)
m
~= −ωmn. (2.2.68)
Note: The equation of time evolution of state vector in interaction picture, i.e., i~ ddt |ψI(t)〉 =
H ′I(t)|ψI(t)〉, can be solved in Dyson’s series in time-dependent perturbative theory whichis used to derive Feynman diagrams in time-dependent perturbative QFT.
Denote the initial state |ψI(t0)〉 as |i〉 and introduce the time evolution operator UI(t, t0),
|ψI(t)〉 = UI(t, t0)|i〉 =∑
n
|n〉〈n|UI(t, t0)|i〉, (2.2.69)
where
UI(t, t0) = 1− i
~
∫ t
t0
HI(t1)dt1 + (− i~
)2
∫ t
t0
dt1
∫ t1
t0
dt2HI(t1)HI(t2) + · · · . (2.2.70)
Cn(t) as the matrix element of time evolution operator, namely Cn(t) = 〈n|UI(t, t0)|i〉,can take the perturbation expansion:
Cn(t) = C(0)n + C(1)
n + C(2)n + · · · , (2.2.71)
where
C(0)n = δni; (2.2.72)
C(1)n = − i
~
∫ t
t0
〈n|HI(t1)|i〉dt1
= − i~
∫ t
t0
eiωnit1〈n|H ′int|i〉dt1; (2.2.73)
C(2)n = · · · . (2.2.74)
The transition probability can be calculated as
P (|i〉 → |n〉) =∣∣∣C(1)
n + C(2)n + · · ·
∣∣∣2
(2.2.75)
for n 6= i.
37
2.2.12 Angular momentum and spin
The angular momentum operator denoted as ~J = (J1, J2, J3) satisfies the commutativerelation
[Ji, Jj ] = i~εijkJk, (2.2.76)
which is the Lie algebra of SU(2).
• The angular momentum composes the orbital angular momentum ~L = ~x × ~p andspin angular momentum ~S = ~
2~σ, namely ~J = ~L+ ~S.
• The spinor representation of SU(2) group defines the spin angular momentum inQM, such as |j,m〉 with m = −j,−j + 1, · · · , j.
* Spin-12 system, j = 1
2 (electron, position, proton, neutron): |12 , 12〉 stands for
spin-up state; |12 ,−12〉 stands for spin-down state.
* Spin-0 system: j = 0, |00〉.* Spin-1 system: j = 1, |1,−1〉, |1, 0〉, |1, 1〉.
Note: In QFT, the spin of a particle is defined as a spinor representation of the Lorentzgroup SO(1, 3) ∼= SU(2)⊗ SU(2) arising from the special relativity.
2.2.13 Identical particles
Principle: Identical particles can not be distinguished in QM.
Note: Such the principle can be explained in QFT.
• Bosons: The system of bosons are totally symmetrical under the exchange of anypair
Pij |N bosons〉 = +|N bosons〉, (2.2.77)
where Pij interchanges between arbitrary pair.
• Fermions: The system of fermions are totally anti-symmetrical under the exchangeof any pair
Pij |N fermions〉 = −|N fermions〉. (2.2.78)
E.g.1: two-boson system: |k′〉|k′〉, |k′′〉|k′′〉, 1√2(|k′〉|k′′〉+ |k′′〉|k′〉).
E.g.2: two-fermion system: 1√2(|k′〉|k′′〉 − |k′′〉|k′〉).
Spin-statistics theorem: half-integer spin particles are fermions and integer spin particlesare bosons.
Note 1: In QFT, the spin-statistics theorem can be verified.
Note 2: In canonical quantization, fermionic field theories are quantized with anti-commutatorA,B = AB + BA and bosonic field theories are quantized with commutator[A,B] = AB −BA.
38
2.2.14 Scattering theory in quantum mechanics
Consider the scattering problem in QM with central force interaction, with the incidentflux denoted as ~jin and outgoing flux denoted as ~jout, the differential cross section dσdefines as
|~jout| = dσ|~jin|. (2.2.79)
The scattering flux describes the scattered particle in per solid angle, namely |~jout| =|~jscatt|r2dΩ. Then the differential cross section takes the form
dσ
dΩ=|~jscatt|r2
|~jin|, (2.2.80)
and the total cross section can be integrated out σ =∫dσdΩdΩ.
After the scattering, the free particle takes the asymptotical wave function expressed as
eik·r → eik·r + f(θ)eikr
r, (2.2.81)
where the function f(θ) is named as scattering amplitude and it can be proved that thesquare of absolute value of f(θ) equals to the differential cross section, namely
dσ
dΩ= |f(θ)|2. (2.2.82)
Consider the symmetry in scattering.
• Two-spinless-boson scattering:
dσ
dΩ= |f(θ) + f(π − θ)|2. (2.2.83)
• Two-spinless-fermion scattering:
dσ
dΩ= |f(θ)− f(π − θ)|2. (2.2.84)
• Two-electron unpolarized beam scattering:
dσ
dΩ=
1
4|f(θ) + f(π − θ)|2 +
3
4|f(θ)− f(π − θ)|2. (2.2.85)
The factor 14 contributes from the spin-singlet state which is spin-antisymmetric,
therefore |f(θ) + f(π − θ)| is associated with a space -symmetric function. Andthe factor 3
4 contributes from spin-triplet states which are spin-symmetric, therefore|f(θ)− f(π − θ)| is associated with the space-anti-symmetric function.
39
Lecture 3 Classical Field Theory
About Lecture II:
• [Luke] P.P. 40-59;
• [Tong] P.P. 15-19;
• [Coleman] P.P. 40-69;
• [PS] P.P. 17-19;
• [Zhou] P.P. 25-33.
3.1 Special relativity
3.1.1 Lorentz transformation
Setup 1: Two inertial frames O and O′,
O : (ct, x, y, z)→ O′(ct′, x′, y′, z′). (3.1.1)
Setup 2: Lorentz transformation (LT):
ctxyz
Λµν−→
ct′
x′
y′
z′
, (3.1.2)
where µ and ν are index labels, µ, ν = 0, 1, 2, 3.
Assumptions 1: Light speed in vacuum is constant, namely c|o = c|o′ ;
Assumptions 2: Equation of motion is invariant under LT in two frames,
· · · = 0Λ−→ · · · = 0. (3.1.3)
Lorentz invariance:
(4s)2 = ct2 − x2 − y2 − z2 = ct′2 − x′2 − y′2 − z′2. (3.1.4)
Eg. 1: Lorentz boost along x-axis, β = vc , γ = 1√
1−β2, Lorentz boost along x-axis:
40
-
6
--
6
x
t
x′
t′
vx
Λµν =
γ −βγ 0 0−βγ γ 0 0
0 0 1 00 0 0 1
. (3.1.5)
Eg. 2: Lorentz rotation around z-axis,
Λµν =
1 0 0 00 cos θ − sin θ 00 sin θ cos θ 00 0 0 1
. (3.1.6)
Physics in special relativity:
Eg. 1: Time dilation;
Eg. 2: Length contraction;
Eg. 3: Causality = Locality = unchanged time-ordering.
3.1.2 Four-vector calculation
Three-vectors:~r = (x, y, z); ~r · ~r = x2 + y2 + z2; (3.1.7)
~p = (px, py, pz); ~p · ~p = p2x + p2
y + p2z; (3.1.8)
~∇ = (∂x, ∂y, ∂z); ~∇ · ~∇ = ∂2x + ∂2
y + ∂2z . (3.1.9)
Contra-variant four vectors:
• Space-time vector:
xµ = (t, ~r) = (x0, x1, x2, x3) = (t, x, y, z); (3.1.10)
• Energy-momentum vector:pµ = (E, ~pc); (3.1.11)
• Four-derivative:
∂µ ≡∂
∂xµ= (
∂
c∂t,∂
∂x,∂
∂y,∂
∂z). (3.1.12)
Covariant four vectors:
• Space-time vector:xµ = (t,−~r) = (t,−x,−y,−z); (3.1.13)
• Energy-momentum vector:pµ = (E,−~pc); (3.1.14)
41
• Four-derivative:
∂µ ≡ ∂
∂xµ= (
∂
c∂t,− ∂
∂x,− ∂
∂y,− ∂
∂z). (3.1.15)
xµ = ηµνxν = (t,−~r) = (t,−x,−y,−z)
= (x0, x1, x2, x3).(3.1.16)
Minkowski space-time metric denoted as ηµν represented for the flat space-time withoutgravity, expressed as
ηµν =
1 0 0 00 −1 0 00 0 −1 00 0 0 −1
, (3.1.17)
with its inverse ηνσ satisfying
ηµνηνσ = δ σ
µ (unit matrix). (3.1.18)
Contravariant vector and covariant vector is transformed via Minkowski metric:
xµ = ηµνxν , xµ = ηµνxν ; (3.1.19)
pµ = ηµνpν , pµ = ηµνpν ; (3.1.20)
∂µ = ηµν∂ν , ∂µ = ηµν∂ν . (3.1.21)
Inner product is invariant quantity under LT:
x2 = xµxµ = ηµνxµxν = ct2 − x2 − y2 − z2. (3.1.22)
p2 = pµpµ = p20 − ~p2c2 = E2 − ~p2c2 = m0c
4, (3.1.23)
where quantity m0 is denoted as mass at rest frame, therefore we have the mass-energyrelation
E2 = m20c
4 + ~p2c2, (3.1.24)
and the mass deficient violation relation
∆E = ∆mc2. (3.1.25)
3.1.3 Causality
Causality = Locality = unchanged time-ordering.
The space-time can be divided into three regions:
4s2 > 0, time-like region;4s2 = 0, light-cone;4s2 < 0, space-like region;
where
4s2 = c2(4t)2 −4x2 −4y2 −4z2. (3.1.26)
42
x
t 4s2 > 0
4s2 < 0
4s2 = 0
Thm 1: Time ordering is unchanged in time-like region;
Thm 2: Time ordering can be violated in space-like region.
Example: Mother was born at the event point O1 and the child is born at O2, where t2 > t1.In the space-like region, after the Lorentz boost, the time ordering can be changed, namelyt2 < t1, shown as
x
t
O1O2
t1: Mother born
t2: Child born
t2 > t1
x
t
O1
O2
t2 < t1
Λ
Note: The causality means that no instantaneous interaction in space-like region.
3.1.4 Mass-energy relation
Mass-energy relation:E = mc2 = γm0c
2. (3.1.27)
Optional problem: derive ∆E = ∆mc2.
Note 1: Energy and matter can be converted to each other.e.g. Atom bomb; Nuclear power station.
Note 2: At v ≈ c, particle number is not fixed.
e.g. 1.p (proton) + p −→ p+ p+ π0 (meson). (3.1.28)
e.g. 2.p+ p −→ p+ p+ p+ p (anti-proton). (3.1.29)
Fermilab, Stanford.
e.g. 3.p+ p −→ · · ·+ · · ·+ Higgs particle. (3.1.30)
LHC, CERN.
43
3.1.5 Lorentz group
Definition: Lorentz transformation matrix Λ = Λµν defined by x′µ = Λµνxν satisfyingx′µx′µ = xνxν , namely
ηµνΛµαΛνβxαxβ = ηαβx
αxβ, ηµνΛµαΛνβ = ηαβ. (3.1.31)
Note: In the infinitesimal transformation
Λνδ = δνδ + ωνδ + o(ω2), (3.1.32)
where ωνδ is small quantity. And we can check the following relations
(Λ−1)νδ = δνδ − ωνδ + o(ω2); (3.1.33)
ωµν = −ωνµ. (3.1.34)
Proof.
ηµν(δνδ + ωνδ)(δµρ + ωµρ) = ηρδ =⇒ ωµν = −ωνµ. (3.1.35)
E.g.1. Lorentz boost along x-direction with y− z fixed, in the infinitesimal formalism, wehave
ω01 = −ω10, ω01 = ω1
0. (3.1.36)
Λµν =
γ −βγ 0 0−βγ γ 0 0
0 0 1 00 0 0 1
vxc
small−−−−−→ δµν +
0 −vxc 0 0
−vxc 0 0 0
0 0 0 00 0 0 0
+ o((
vxc
)2).
(3.1.37)Note that the infinitesimal term in above Lorentz transformation stands for a sym-metric matrix.
E.g.2. Lorentz rotation around z-axis, in the infinitesimal formalism, we have
ωij = −ωji, ωij = −ωji (3.1.38)
with i, j = 1, 2, 3.
Λµν =
1 0 0 00 cos θ − sin θ 00 sin θ cos θ 00 0 0 1
θ small−−−−→ δµν +
0 0 0 00 0 −θ 00 θ 0 00 0 0 0
+o(θ2). (3.1.39)
Note that the infinitesimal term in above Lorentz transformation stands for ananti-symmetric matrix.
Thm 3.1.5.1. The set of all Lorentz transformations Λ = Λµν is a group called the Lorentzgroup denoted as SO(1, 3).
Proof. See group theory.
44
Note 1. If det(Λ) = 1, the Lorentz group is called proper Lorentz group denoted asSO+(1, 3). The determinant of the Lorentz transformation is important becauseof the Jacobian determinant which specifies the transformation of integral mea-sure under Lorentz transformation, namely dt′dx′dy′dz′ = det(Λ)dtdxdydz. Notethat dx4 is a Lorentz transformation invariant.
Note 2. If Λ00 ≥ 1 and det(Λ) = 1, the Lorentz group is called proper orthochronous
Lorentz group denoted as SO↑+(1, 3). Λ00 ≥ 1 means time ordering or time arrow
is unchanged under Lorentz transformation.
Note 3. Real physics prefers the proper and orthochronous Lorentz group which is calledthe Lorentz group in the course.
Note 4. SO(1, 3) ∼= SU(2)⊗SU(2). The angular momentum state |j,m〉 where j is integeror half-integer forms a representation of the SU(2) group, so the representationof the Lorentz group is denoted as |j1,m1〉 ⊗ |j2,m2〉 or |j1j2,m1m2〉.
3.1.6 Classification of particles
At high energy physics, particles including electrons, photons, etc, are classified as repre-sentations of the Lorenz group, labeled by (j1, j2), where j1, j2 are integers or half integersdenoting angular momentum.
Table 3.1: Classification of fundamental particles representation
Particles Fields Spin /Statistics LG representation
Messon or Higgs Scalar, φ(x) Spinless /Boson Invariant, (0, 0)
Photon Vector, Aµ(x) Spin 1 /Boson Covariant, (1, 0) or (0, 1)
Electron Spinor, ψ(x) Spin 12 /Fermion Spinor transformation, ( 1
2 , 0)⊕ (0, 12 )
3.2 Classical field theory
Before the special relativity, the concept of field is a kind of mathematical object, butafterwards, it becomes a real physical object.
3.2.1 Concepts of fields
Definition. 1. Field is a physical quantity depends on every space-time point. E.g. ~E(t, ~x) =~E~x(t), ~B(t, ~x) = ~B~x(t), where ~x is not dynamical variable thus becomes aslabel.
Definition. 2. A field is an object with infinity number of degrees of freedom, becauset ∈ R+, ~x ∈ R3.
Definition. 3. A field represents an interaction between particles, which can not propagateinstantaneously. And interaction between particles is mediated by fields orassociated quanta.
45
e−
e−
e−
e−γ
Photon: quanta of E.M. fields
Photone− e−
No instantaneous interaction
Two-electron interacts by the interchange of virtual photon.
Definition. 4. The dynamics of fields at space-time point completely depends on physicsaround the point, so mathematics of local fields prefer partial differentialequation, such as Maxwell equations.
Definition. 5. Particle-Wave duality. Particle-Field duality.
3.2.2 Electromagnetic fields
Note: Be careful of notations, because of SI unit, Gauss unit and natural unit. In naturalunit, we set c = ε0 = µ0 = 1.
Maxwell equations in vacuum:
∇ · ~E = 0;∂ ~E
∂t= ∇× ~B; (3.2.1)
∇ · ~B = 0;∂ ~B
∂t= −∇× ~E.
Let us prove electrodynamics is a Lorentz covariant theory. Namely, we show Maxwellequations have a Lorentz covariant form.
In Gauge potential, with ~A denoted the magnetic potential and ϕ denoted the electricpotential, the four-dimensional potential is defined as
Aµ(t, ~x) = (ϕ, ~A). (3.2.2)
with
~E = −∇ϕ− ∂ ~A
∂t; ~B = ∇× ~A. (3.2.3)
Define the field strength:Fµν = ∂µAν − ∂νAν . (3.2.4)
Note thatF 0i = −Ei; F ij = −εijkBk (3.2.5)
with i, j = 1, 2, 3. Therefore Fµν has the matrix expression
Fµν =
0 −Ex −Ey −EzEx 0 −Bz ByEy Bz 0 −BxEz −By Bx 0
. (3.2.6)
Fµν = ηµaηνbFab. (3.2.7)
46
• The Maxwell equations
∇ · ~B = 0,∂ ~B
∂t= −∇× ~E, (3.2.8)
are equivalent with the Bianchi identity
∂µFνλ + ∂νFλµ + ∂λFµν = 0. (3.2.9)
• The Maxwell equations
∇ · ~E = 0,∂ ~E
∂t= ∇× ~B, (3.2.10)
have other form shown as∂µF
µν = 0. (3.2.11)
3.2.3 Lagrangian, Hamiltonian and the action principle
• The field is denoted as φa(t, ~x), a = 1, 2, · · · , N .
• The state is described by φa(t, ~x) and ∂µφa(t, ~x).
• The Lagrangian density is the function of φa(t, ~x) and ∂µφa(t, ~x), namely L(φa, ∂µφa).
• The action is defined as S =∫d4xL(φa, ∂µφa).
Note: d4x, L and S are both Lorentz invariant scalars.
Under infinitesimal variation φa → φa + δφa with fixed boundary δφa(t1) = δφa(t2) = 0,the action principle states δS = 0, which is equivalent with the equation of motion.
0 = δS =
∫d4x(
∂L∂φa
δφa︸ ︷︷ ︸
I
+∂L
∂(∂µφa)δ(∂µφa)
︸ ︷︷ ︸II
). (3.2.12)
Because of
II =∂L
∂(∂µφa)∂µ(δφa)
=∂µ(∂L
∂(∂µφa)δφa)− ∂µ(
∂L∂(∂µφa)
)δφa,
(3.2.13)
so
0 = δS =
∫d4x
[(∂L∂φa− ∂µ(
∂L∂(∂µφa)
)
)δφa + ∂µ(
∂L∂(∂µφa)
δφa)
]. (3.2.14)
The 4-total derivative term∫d4x∂µ( ∂L
∂(∂µφa)δφa) means the surface term after integration
which is supposed to vanish under boundary condition. And because δφa is arbitraryvariation, so
∂L∂φa
= ∂µ(∂L
∂(∂µφa)), (3.2.15)
which is called equation of motion or Euler-Lagrangian equation.
Define the conjugate momentum as
Πa =∂L∂φa
, φa =∂
∂tφa(t, ~x). (3.2.16)
47
The Hamiltonian density has the formalism
H = Πaφa − L. (3.2.17)
And Hamiltonian has the expression
H =
∫d3xH. (3.2.18)
3.2.4 A simple string theory
Parameters L, ρ and T denote the string length, mass density and the string tension respec-tively. The boundary points are fixed, namely 0 ≤ x ≤ L. The transverse displacementquantity is denoted as φ(t, ~x). The corresponding Lagrangian density is
L =1
2ρφ2 − 1
2Tφ2
x, (3.2.19)
where φ = ∂∂tφ and φx = ∂xφ. The equation of motion can be calculated in the way
∂L∂φ
= 0,∂L∂φ
= ρφ,∂L∂φx
= −Tφx, (3.2.20)
ρφ− Tφxx = 0, φ− (T
ρ)φxx = 0. (3.2.21)
Define the canonical momentum
Π =∂L
∂φ= ρφ. (3.2.22)
The Hamiltonian density is
H = Πφ− L =1
2ρφ2 +
1
2Tφ2
x ≥ 0. (3.2.23)
Thm 3.2.4.1. Such the string φ(t, x) is equivalent to an infinite set of decoupled harmonicoscillators.
Proof. Assume
φ(t, x) =
√2
a
+∞∑
n=1
sinnπx
aqn(t). (3.2.24)
φ =
√2
a
+∞∑
n=1
sinnπx
aqn(t); (3.2.25)
φx =
√2
a
+∞∑
n=1
nπ
acos
nπx
aqn(t). (3.2.26)
(3.2.27)
So
L =
∫ a
0dxL =
+∞∑
n=1
(1
2ρq2n −
1
2T (nπ
a)2q2
n
). (3.2.28)
The equation of motion can be derived in the way
∂L
∂qn= ρqn,
∂L
∂qn= −T (
nπ
a)2qn, (3.2.29)
qn + w2nqn = 0 (3.2.30)
with wn =√
Tρnπa .
48
3.2.5 Real scalar field theory
Real scalar field (Klein-Gorden field) φ(t, ~x) is described by the Lagrangian density
L =1
2(∂µφ∂
µφ− µ2φ2)
=1
2(φ2 − (∇φ)2 − µ2φ2) (3.2.31)
with µ2 > 0. The corresponding canonical momentum is
Π =∂L∂φ
= φ, (3.2.32)
and the Hamiltonian density is
H =1
2(φ2 + (∇φ)2 + µ2φ2) ≥ 0. (3.2.33)
Equation of motion can be found in the way
∂L∂φ
= −µ2φ,∂L
∂(∂µφ)= ∂µφ, (3.2.34)
(+ µ2)φ = 0, (3.2.35)
which is called the Klein-Gordon equation associated with the real scalar field.
Plain-wave solution of Klein-Gordon equation has the formalism
φ(t, ~x) =
∫d4k
(2π)3/2δ(k2 − µ2)e−ik·xφ(k). (3.2.36)
The integration is in four dimension, namely d4k = dk0d3~k.
k2 − µ2 = k20 − w2
~k, w~k = ~k2 + µ2, (3.2.37)
where w~k is associated with the frequency. Substitute the solution into the differentialequation, we can find the term
(+ µ2)e−ik·x = (−k2 + µ2)e−ik·x, (3.2.38)
and with delta function, we have δ(k2 − µ2)(k2 − µ2) = 0. Note that
δ(k2 − µ2) =1
2w~kδ(k0 + w~k) +
1
2w~kδ(k0 − w~k), (3.2.39)
which implies the negative energy.
With interaction, the Lagrangian has the expression
L = L0 + Lint, (3.2.40)
where L = −V (φ). And the equation of motion becomes as
(+ µ2)φ = − ∂
∂φV (φ). (3.2.41)
E.g. V (φ) = λ4!φ
4 with λ as the coupling constant,
(+ µ2)φ = − λ3!φ3. (3.2.42)
Note: Up to now, no analytic solution to the equation of motion with interaction was beenfound.
49
3.2.6 Complex scalar field theory
• Two real scalar fields φ1 and φ2 are described by the Lagrangian
L = L0(φ1) + L0(φ2)− V (φ1, φ2), (3.2.43)
where the free part of the Lagrangian has the formalism
L0(φi) =1
2∂µφi∂
µφi −1
2m2φ2
i , (3.2.44)
And for instance the interaction part has the expression
V (φ1, φ2) = g(φ21 + φ2
2), (3.2.45)
where g is the coupling constant.
• Two-component real scalar field vector ~φ is defined as
~φ =
(φ1
φ2
), ~φT =
(φ1 φ2
). (3.2.46)
The Lagrangian can be rewritten as
L =1
2∂µ~φ
T∂µ~φ− 1
2m2~φT ~φ− V (~φT , ~φ). (3.2.47)
• Complex scalar fields are defined as
ψ(x) =1√2
(φ1 + iφ2), (3.2.48)
ψ†(x) =1√2
(φ1 − iφ2). (3.2.49)
which give the Lagrangian as
L = ∂µψ†∂µψ −m2ψ†ψ − V (ψ†, ψ). (3.2.50)
The equations of motion are expressed as
(+m2)ψ = −∂V∂ψ
; (3.2.51)
(+m2)ψ† = − ∂V∂ψ†
. (3.2.52)
Canonical momentums show as
Πψ =∂L∂ψ
= ψ†; Πψ† =∂L∂ψ†
= ψ, (3.2.53)
which give rise the Hamiltonian as
H = Πψψ + Πψ†ψ† − L (3.2.54)
= ψ†ψ +∇ψ†∇ψ +m2ψ†ψ + V (ψ†, ψ) ≥ 0. (3.2.55)
50
Note. “Nucleon”-meson theory, which is also named as the scalar Yukawa theory. Thelagrangian density shows as
L = Lφ + Lψ − gψ†ψφ. (3.2.56)
ψ: complex scalar field.
“nucleon”=pseudo-nucleon = spinless with charge.
Note. proton and anti-proton are nucleons with spin 1/2.
φ: real scalar fields.
meson = spinless without charge.
g: coupling constant. [ψ] = [φ] = [g] = 1 in mass dimension.
The equations of motion are
(+ µ2)φ = −gψ†ψ;
(+m2)ψ = −gψ†φ.(3.2.57)
3.2.7 Multi-component scalar field theory
• Define the N -component real scalar field vector as
~φ =
φ1
φ2...φN
; ~φT =
(φ1 φ2 · · · φN
). (3.2.58)
L =1
2
N∑
a=1
(∂µφa∂µφa − µ2φaφa)− V (
N∑
a=1
φ2a)
=1
2(∂µ~φ
T∂µ~φ−m2~φT ~φ)− V (~φT ~φ). (3.2.59)
• Define the N -component complex scalar field vector as
~ψ =
ψ1
ψ2...ψN
; ~ψ† =
(ψ†1 ψ†2 · · · ψ†N
). (3.2.60)
L =1
2
N∑
a=1
(∂µψ†a∂
µψa −m2ψ†aψa)− V (
N∑
a=1
ψ†aψa)
=1
2(∂µ ~ψ
†∂µ ~ψ −m2 ~ψ† ~ψ)− V (~ψ† ~ψ). (3.2.61)
51
3.2.8 Electrodynamics
With the four-dimension potential Aµ = (ϕ, ~A), the electromagnetic tensor Fµν is definedas Fµν = ∂µAν − ∂νAµ. Under the gauge transformation Aµ → Aµ + ∂µJ
µ, Fµν is gaugeinvariance. For redundant degrees of freedom of electromagnetic field, we have two kindsof gauge fixing shown as
Lorentz Gauge: ∂µAµ = 0; (3.2.62)
Coloumb Gauge: ∇ · ~A = 0. (3.2.63)
In vacuum, the electromagnetic field is governed by the Lagrangian
L = −1
4FµνF
µν
= −1
4(∂µAν − ∂νAµ)Fµν
= −1
2∂µAνF
µν . (3.2.64)
The equation of motion is found in the way
∂L∂(∂µAν)
= −Fµν , ∂L∂Aν
= 0, (3.2.65)
∂µFµν = 0. (3.2.66)
The time components of the canonical momenta are
Πµ =∂L
∂(∂0Aµ). (3.2.67)
Note that Π0 = 0 and Πi = ∂L∂(∂0Ai)
= Ei. The Hamiltonian density can be derived as
H = ΠµAµ − L =1
2( ~E2 + ~B2) +∇ϕ · ~E. (3.2.68)
Note:
L = −1
4FµνF
µν =1
2( ~E2 − ~B2); (3.2.69)
H =
∫d3xH =
1
2
∫d3x( ~E2 + ~B2) ≥ 0; (3.2.70)
~P =
∫d3x~E × ~B. (3.2.71)
With current, the Electric magnetic field is governed by the Lagrangian
L = −1
4FµνF
µν − JµAµ, (3.2.72)
where Jµ is four-dimension current Jµ = (ρ,~j). The equation of motion can be derived as
∂µFµν = Jν . (3.2.73)
52
The equation of motion can be reformulated as
∂µ(∂µAν − ∂νAµ) = Jν ; (3.2.74)
Aν − ∂ν(∂µAµ) = Jν . (3.2.75)
In Lorentz gauge, we have the equation
Aν = Jν , (3.2.76)
namely∂2
∂t2ϕ−∇2ϕ = ρ;
∂2
∂t2~A−∇2 ~A = ~j. (3.2.77)
Note that the equation is Lorentz covariant.
Note 1: The retarded solution of the equation of motion shows as
ϕ(t, ~r) =
∫ρ((t− v
c ), ~x′)
4π~rd3~x′. (3.2.78)
Note 2: To solve the equation of motion in electromagnetic field theory, the Green functionusually is applied. In quantum field theory, the Feynman propagator is also Greenfunction.
Note 3: The current conservation can be derived from the equation of motion, namely
∂µFµν = Jν =⇒ ∂µJ
µ = 0,∂ρ
∂t+∇ ·~j = 0 (3.2.79)
by the relation
∂ν∂µFµν = ∂νJ
ν = 0. (3.2.80)
And the conserved quantity Q can be defined as
Q ≡∫d3xρ (3.2.81)
due to dQdt = 0.
Note 4: Conservation law in field theory is described by the local current conservation.
Note 5: Photon is massless. If photon is massive, then it is described by the Proca La-grangian
L = −1
4FµνF
µν − JµAµ +µ2
2AµA
µ. (3.2.82)
3.2.9 General relativity
Physical quantities defined in general relativity:
• Metric field: gµν(x);
• Christoffel connection: Γijk(gµν(x));
• Curvature tensor: Rαµβγ(Γijk);
53
• Ricci tensor: Rµν = gαβRαµβν ;
• Scalar curvature: R = gµνRµν .
Action in vacuum takes the formula
S =
∫d4x√−gR/16πG (3.2.83)
with g = det(gµν) and G as the gravity constant. And the equation of motion can bederived
Rµν −1
2gµν = 0. (3.2.84)
Note 1: With matter, the equation of motion becomes as
Rµν −1
2gµν = 8πGTµν , (3.2.85)
where Tµν is the energy-momentum tensor.
Note 2: Quantum gravity is the theory of quantization of the general relativity, which isan open probelm in 21st century.
54
Lecture 4 Symmetries and conservation laws
About Lecture II:
• [Luke] P.P. 40-59;
• [Tong] P.P. 15-19;
• [Coleman] P.P. 40-69;
• [PS] P.P. 17-19;
• [Zhou] P.P. 25-33.
4.1 Symmetries play a crucial role in field theories
Lagrangian density in real scalar field with interaction takes the form
L =1
2∂µφ∂
µφ− 1
2µ2φ2 − λ
4!φ4, (4.1.1)
and the equation of motion is
(+ µ2)ϕ = − λ3!ϕ3. (4.1.2)
Up to now, no analytic solution has been found for this equation. Without solutions ofequation of motion, one can perform following ways to understand the physics
Way-out 1: Perturbative approach (Feynman diagram);
Way-out 2: Symmetries and conservation laws (without solving equation of motion);
Way-out 3: Numerical simulation (Lattice QCD);
Way-out 4: · · · · · · .
Note: Symmetries in quantum field theory are almost able to determine the formulationof the action, e.g., the Yang-Mills theory.
4.2 Noether’s theorem in field theory
We have two different variations:
• δqa, δφ(x): variation for deriving equation of motion;
• Dqa, Dφ(x): variation for symmetries without specifying equation of motion.
55
The fields are denoted as φa(t, ~x). The state is determined by φa(x) and first derivation∂µφa(x), namely by coordinate and momentum. The Lagrangian takes the form L =L(φa, ∂µφa) without explicit dependence on x. And the action can be obtained S =∫d4xL(φa, ∂µφa).
Def 4.2.1 (Symmetry). A transformation in field theory is called a symmetry iff
DL = ∂µFµ (4.2.1)
for some Fµ(φa, ∂µφa), which must be hold for arbitrary φa(x), but not necessarily satis-fying equation of motion.
Remark:
DS =
∫d4xDL =
∫d4x∂µF
µ
=
∫d3xF 0
∣∣∣∣t2
t1
+
∫dt
∮~F · d~S, (4.2.2)
S′ = S +DS (4.2.3)
where the time boundary is fixed and spatial boundary vanishes, thus two terms do nothave contribution in deriving equation of motion, namely δS′ = δS.
Thm 4.2.0.1 (Noether’s theorem). For every continuous symmetry, there is a conservedcurrent.
Proof.
1). From the definition of symmetry: DL = ∂µFµ.
2). From Hamilton’s principle:
DL =∑
a
(∂L∂φa
Dφa +∂L
∂(∂µφa)D(∂µφa))
=∑
a
(∂µ(∂L
∂(∂µφa))Dφa +
∂L∂(∂µφa)
∂µ(Dφa))
= ∂µ(∑
a
∂L∂(∂µφa)
Dφa),
(4.2.4)
therefore
=⇒ ∂µ(∑
a
∂L∂(∂µφa)
Dφa − Fµ) = 0. (4.2.5)
Define the conserved current as
Jµ =∑
a
∂L∂(∂µφa)
Dφa − Fµ, (4.2.6)
and it turns out
∂µJµ = 0. (4.2.7)
56
E.g. In Electromagnetic theory, the four-dimension current is defined as Jµ = (ρ,~j), whereρ is charge density and ~j is current. The current conservation ∂µJ
µ = 0 gives rise to therelation
∂ρ
∂t+∇ ·~j = 0. (4.2.8)
We can define the conserved quantity Q as
Q =
∫d3xρ (4.2.9)
due todQ
dt=
∫
Vd3x
∂ρ
∂t=
∫
Vd3x(−∇ ·~j) = −
∮
V
~j · d~S = 0. (4.2.10)
4.3 Space-time symmetries and conservation laws
4.3.1 Space-time translation invariance and energy-momentum tensor
Space-time translation takes the form
x→ x′ = x− ε, (4.3.1)
where ε is an infinitesimal constant. And the physical system described by the field isunchanged, namely
φa(x)→ φ′a(x′) = φa(x), (4.3.2)
L′(x′) = L(x), (4.3.3)
where x and x′ respectively denote the representations of a space-time point in two coor-dinate systems. The variations of Dφa and DL can be calculated as
Dφa(x) = φ′a(x)− φa(x)
= φa(x+ ε)− φa(x)
= εµ∂µφa(x) + o(ε2),
(4.3.4)
DL = L′(x)− L(x)
= L(x+ ε)− L(x)
= εµ∂µL+ o(ε2).
(4.3.5)
According to the definition 4.2.1 of symmetry, the function Fµ of space-time transforma-tion takes the form
Fµ = εµL. (4.3.6)
From Noether’s theorem 4.2.0.1, the conserved current can be defined
Jµ =∑
a
∂L∂(∂µφa)
Dφa − Fµ
=∑
a
∂L∂(∂µφa)
(εν∂νφa)− εµL
= εν
[∑
a
∂L∂(∂µφa)
∂νφa − ηµνL]
= εν(Tµ)ν ,
(4.3.7)
57
Tµν = (Tµ)ν , (4.3.8)
Tµν =∑
a
∂L∂(∂µφa)
∂νφa − ηµνL, (4.3.9)
where Tµν is named as energy momentum tensor. The conservation law expresses as
∂µJµ = 0 =⇒ ∂µT
µν = 0. (4.3.10)
Note: Tµν is crucial in general relativity.
Table 4.1: Energy momentum tensor Tµν
∂µ(Tµ)ν = 0 εν Charge density and current
ε = (1, 0, 0, 0) T 00: energy density(T 0)ν : charge density
Tµ0: energy term T 0j : momentum density
ε = (0, ~x) T i0: energy current(T i)ν : current
Tµj : momentum term T ij : momentum current
E.g. In scalar field theory
L =1
2∂µφ∂
µφ− 1
2µ2φ2, (4.3.11)
we have
T 00 = (T 0)0
=∑
a
∂L∂φa
φa − L
= H,
(4.3.12)
where H is energy density, and
H =
∫d3xT 00 =
∫d3xH, (4.3.13)
which implies the energy is conserved quantity.
T 0i = (T 0)i
=∑
a
∂L∂φa
∂iφa − 0
=∑
a
Πa∂iφa
= pi,
(4.3.14)
where pi is momentum density, and
P i =
∫d3xT 0i =
∫d3xpi, (4.3.15)
which implies the momentum is conserved quantity. Notice the conceptual differencebetween canonical momentum and conserved momentum.
58
4.3.2 Lorentz transformation invariance and angular-momentum tensor
Def 4.3.1. The constant Lorentz transformation is denoted as Λ = Λµν satisfying
ηρσ = ηµνΛµρΛνσ, (4.3.16)
where ηρσ is Minkowski flat space-time metric defined in (3.1.17).
Lemma 4.3.2.1. The infinitesimal Lorentz transformation takes the form
Λµν = ηµν + ωµν + o(ω2) (4.3.17)
with infinitesimal constant ωµν satisfying
ωµν = −ωνµ. (4.3.18)
Details and examples see previous subsection 3.1.5.
Note 1.
Λ−1µν = ηµν − ωµν + o(ω2). (4.3.19)
Note 2.
(Λx)µ = xµ + ωµνxν + o(ω2); (4.3.20)
(Λ−1x
)µ
= xµ − ωµνxν + o(ω2). (4.3.21)
Definition of the scalar field theory under the Lorentz transformation
•x→ x′ = Λx; (4.3.22)
•φ′(x′) = φ(x); (4.3.23)
L′(x′) = L(x). (4.3.24)
•φ′(x) = φ(Λ−1x); (4.3.25)
L′(x) = L(Λ−1x). (4.3.26)
Note: The scalar field φ(x), Lagrangian density L(φ, ∂µφ), integral measure d4x and nat-urally the action S =
∫d4xL are both scalars under Lorentz transformation.
Remark:
The same physics
Frame 1 Frame 2x x′
59
Apply the Noether’s theorem to derive the conserved current.
Dφ(x) = φ′(x)− φ(x)
= φ(Λ−1x)− φ(x)
= φ(xρ − ωρσxσ + o(ω2))− φ(x)
= −ωρσxσ∂ρφ(x)
= ωσρxσ∂ρφ(x). (4.3.27)
Note that ωρσ = −ωσρ.
DL = L′(x)− L(x)
= L(Λ−1x)− L(x)
= ωσρxσ∂ρL
= ∂ρ(ωσρxσL). (4.3.28)
From L = ∂µFµ, the quantity F can be found as
Fµ = ηµρωσρxσL, (4.3.29)
and the associated conserved current can be defined
Jµ =∑
a
∂L∂(∂µφa)
Dφa − Fµ
=∑
a
∂L∂(∂µφa)
ωσρxσ∂ρφa − ηµρωσρxσL
= ωσρ
∑
a
∂L∂(∂µφa)
xσ∂ρφa − ηµρxσL
= ωσρ
xσ
(∑
a
∂L∂(∂µφa)
∂ρφa − ηµρL)
= ωσρ xσTµρ . (4.3.30)
Note that xσTµρ is an anti-symmetric tensor, therefore we can symmetrize the term xσTµρ
to eliminate the redundant zero term.
xσTµρ =1
2(xσTµρ + xρTµσ)︸ ︷︷ ︸
symmetry
+1
2(xσTµρ − xρTµσ)︸ ︷︷ ︸
anti-symmetry
(4.3.31)
Thus the conserved current can be written as
Jµ =1
2ωσρ(x
σTµρ − xρTµσ). (4.3.32)
Define the angular-momentum tensor as
(Mµ)σρ = Mµσρ = xσTµρ − xρTµσ (4.3.33)
satisfying the conserved relation∂µM
µσρ = 0. (4.3.34)
Note 1.
60
µ: Current and charge indexσ, ρ: Lorentz transformation index
Mµσρ:σ = 0, ρ = i: Lorentz boost indexσ, ρ = i, j: Lorentz rotation index
E.g. when µ = 0, σ, ρ = i, j,
M0ij = xiT 0j − xjT 0i. (4.3.35)
Recall that T 0i is momentum density. Then M0ij can be viewed as angularmomentum density, namely
J ij =
∫d3xM0ij
=
∫d3x(xiT 0j − xjT 0i)
(4.3.36)
Note 2.Mµσρ = xσTµρ − xρTµσ (4.3.37)
which represents the orbital angular-momentum for spinless particle without spinangular-momentum.
E.g. 1: Angular-momentum for Lorentz boost σ = 0, ρ = i.
The current is denoted as Mµ0i, then the conserved charge can be calculated inthe way J0i =
∫d3xM00i.
J0i =
∫d3xM00i
=
∫d3x(x0T 0i − xiT 00)
= tP i −∫d3xxiT 00, (4.3.38)
where T 0i is the momentum density, i.e., P i =∫d3xT 0i, and T 00 is the mass-
energy density. The conserved relation ddtJ
0i = 0 gives rise to
d
dt(tP i −
∫d3xxiT 00) = 0, (4.3.39)
P i + tdP i
dt=
d
dt
∫d3xxiT 00, (4.3.40)
where the momentum is conserved quantity, i.e., dP i
dt = 0. Therefore
P i =d
dt
∫d3xxiT 00. (4.3.41)
E.g. T 00 = ρ, where ρ is the mass density, which gives rise to
P i = constant =d
dt
∫d3xρxi (4.3.42)
suggesting that the center of mass moves steadily.
61
E.g. 2: Point-particle in General Relativity.
Particle trajectory is denoted as
ξµ(τ) = (ξ(τ), ~ξ(τ)), (4.3.43)
where τ is proper time, and space-time point is denoted as xµ = (t, ~x).
Einstein equation gives rise toRµν ∝ Tµν (4.3.44)
where
Tµν(t, ~r) = m0
∫dτδ(4)(xµ − ξµ(τ))
dξµ(τ)
dτ· dξ
ν(τ)
dτ, (4.3.45)
T 0ν(t, ~r) = m0δ(3)(~r − ~ξ(t))dξ
ν
dτ, (4.3.46)
T 0i = P iδ(3)(~r − ~ξ(t)). (4.3.47)
Momentum:
Pµ =
∫d3xT 0µ(t, ~r), (4.3.48)
Angular-momentum:
J ij =
∫d3x(xiT 0j − xjT 0i)
= ξiP j − ξjP i.(4.3.49)
4.4 Internal symmetries and conservation laws
4.4.1 Definitions
∆. Active and passive transformation.
Field (system) Space-time (frame)
Active transformation changed fixed
Passive transformation fixed changed
∆. Classification of symmetries.
Space-time symmetry Internal symmetry
Continuous symmetrySpace-time transformation U(1) symmetry for
Lorentz transformation charge conservation
Discrete symmetryParity
Charge conjugationTime reversal
∆. Space-time symmetry and internal symmetry. .
∆. Comparison between space-time symmetry and internal symmetry.
62
Field Space-time
Space-time symmetry changed changed
Internal symmetry changed fixed
Space-time symmetry Internal symmetry
Space-time Changed Fixed
Field Changed Changed
Example: Space-time translation, U(1) symmetries,
symmetry Lorentz transformation U(1) ∼= SO(2)
Example: ∂µTµν = 0 (Electric)Charge
conservation law ∂µMµρσ = 0 ∂ρ
∂t +∇ ·~j = 0
4.4.2 Noether’s theorem
Note: Noether’s theorem only applies to continuous symmetry.
Noether’s theorem for internal and continuous symmetry.
x → x, (4.4.1)
φ(x) → φ′(x), (4.4.2)
L → L′, (4.4.3)
which gives rise toDφ(x) = φ′(x)− φ(x), (4.4.4)
DL = L′ − L = ∂µFµ = 0, (4.4.5)
so that Fµ is a constant and can be set as zero. So the conserved current is defined as
Jµ =∂L
∂(∂µφa)Dφa, (4.4.6)
Q =
∫d3xJ0 =
∫d3x (
∂L∂φa
Dφa) =
∫d3x (ΠaDφ
a). (4.4.7)
4.4.3 SO(2) invariant real scalar field theory
The Lagrangian takes the form
L = L0(φ1) + L0(φ2)− V (φ1, φ2), (4.4.8)
where φ1 and φ2 are real scalar fields with same m2. The free part of the Lagrangianshows as
L0(φi) =1
2∂µφi∂
µφi −1
2m2φ2
i (4.4.9)
and the interaction part has the formalism
V (φ1, φ2) = g(φ21 + φ2
2), (4.4.10)
where g is the coupling constant.
63
Thm 4.4.3.1. Lagrangian L (4.4.8) is SO(2) rotation invariance.
(φ′1φ′2
)= R
(φ1
φ2
)(4.4.11)
Proof. SO(2) stands for 2 dimensional special orthogonal group. R ∈ SO(2) has thematrix representation shown as
R =
(cosλ sinλ− sinλ cosλ
)(4.4.12)
where λ is constant and RTR = RRT = 112.
With the vector ~φ defined as
~φ =
(φ1
φ2
), (4.4.13)
the Lagrangian can be rewritten as
L =1
2∂µ~φ
T∂µ~φ− 1
2m2~φT ~φ− g~φT ~φ. (4.4.14)
Define the after rotation field as ~φ′ = R~φ and ~φ′T = ~φTRT , therefore
~φ′T ~φ′ = ~φTRTR~φ = ~φT ~φ; (4.4.15)
∂µ~φ′T∂µ~φ′ = ∂µ~φ
TRTR∂µ~φ = ∂µ~φT∂µ~φ. (4.4.16)
Thus
L(~φ′, ∂µ~φ′) = L(~φ, ∂µ~φ). (4.4.17)
Apply the Noether’s theorm.
With
DL = L′ − L = 0, (4.4.18)
from DL = ∂µFµ, we can find Fµ is constant. Rewrite the rotation matrix R into
infinitesimal formalism
R =
(cosλ sinλ− sinλ cosλ
)
= 112 +
(0 λ−λ 0
)+ o(λ2). (4.4.19)
Therefore
φ′1 = φ1 + λφ2 + o(λ2); (4.4.20)
φ′2 = φ2 − λφ1 + o(λ2), (4.4.21)
which give rise to
Dφ1 = φ′1 − φ1 = λφ2; (4.4.22)
Dφ2 = φ′2 − φ2 = −λφ1. (4.4.23)
64
The conserved current can be calculated as
Jµ =∂L
∂(∂µφ1)Dφ1 +
∂L∂(∂µφ2)
Dφ2
= λ(∂µφ1φ2 − ∂µφ2φ1) (4.4.24)
with the associated conserved charge θ
θ =
∫d3J0 =
∫d3(φ1φ2 − φ1φ2). (4.4.25)
In classical theory, we do not know the exact meaning of the conserved charge θ.
4.4.4 U(1) invariant complex scalar field theory
Introduce the new complex field ψ(x)
ψ(x) =1√2
(φ1 + iφ2); (4.4.26)
ψ†(x) =1√2
(φ1 − iφ2), (4.4.27)
which give the Lagrangian (4.4.8) as
L0 = ∂µψ†∂µψ −m2ψ†ψ − V (ψ†, ψ). (4.4.28)
with V (ψ†ψ) = gψ†ψ. The SO(2) rotation of the field doublet (φ1, φ2) becomes the U(1)transformation of the complex field ψ(x), namely
ψ′(x) =1√2
(φ′1 + iφ′2)
=1√2
(cosλφ1 + sinλφ2 + i(− sinλφ1 + cosλφ2))
=1√2
(e−iλφ1 + ie−iλφ2)
= e−iλψ(x); (4.4.29)
ψ′†(x) =1√2
(φ′1 − iφ′2)
=1√2
(cosλφ1 + sinλφ2 − i(− sinλφ1 + cosλφ2))
=1√2
(eiλφ1 − ieiλφ2)
= eiλψ†(x). (4.4.30)
(4.4.31)
U(1) invariance of Lagrangian L with eiλ ∈ U(1), i.e.,
ψ′†(x)ψ′(x) = ψ†(x)eiλe−iλψ(x) = ψ†(x)ψ(x). (4.4.32)
The conserved current can be calculated as
Jµ =∂L
∂(∂µψ)Dψ +
∂L∂(∂µψ†)
Dψ†
= iλ((∂µψ)ψ† − (∂µψ†)ψ) (4.4.33)
with the charge densityJ0 = i(ψψ† − ψ†ψ). (4.4.34)
65
4.4.5 Non-Abelian internal symmetries
The standard model of particle physics includes non-Abelian gauge field theories.
Note:
Lie group Product Example
Abelian group R1R2 = R2R1 SO(2) ∼= U(1)
Non-Abelian group R1R2 6= R1R2 SO(N) with N ≥3, U(N) with N ≥ 2
E.g. 1. SO(N) with N ≥ 3 invariant quantum field theory.
~φ =
φ1
φ2...φN
, ~φT =
(φ1, φ2, · · · φN
). (4.4.35)
L =1
2∂µ~φ
T∂µ~φ− 1
2m2~φT ~φ− V (~φT , ~φ). (4.4.36)
The after rotation field is denoted as ~φ′ = R~φ and ~φ′T = ~φTRT , where R ∈ SO(N)with RTR = RRT = 11. Note that R1R2 6= R2R1 in general.
The Lagrangian L (4.4.36) is invariant under transformation R due to
~φ′T ~φ′ = ~φTRTR~φ = ~φT ~φ; (4.4.37)
∂µ~φ′T∂µ~φ′ = ∂µ~φ
TRTR∂µ~φ = ∂µ~φT∂µ~φ. (4.4.38)
In the case N = 3, apply the Noether’s theorem, we can find the conserved currentsdenoted as
Jµ12 = (∂µφ1)φ2 − (∂µφ2)φ1; (4.4.39)
Jµ13 = (∂µφ1)φ3 − (∂µφ3)φ1; (4.4.40)
Jµ23 = (∂µφ2)φ3 − (∂µφ3)φ2. (4.4.41)
E.g. 2. U(N) with N ≥ 2 invariant quantum field theory.
~ψ =
ψ1
ψ2...ψN
, ~ψ† =
(ψ†1, ψ†2, · · · ψ†N
). (4.4.42)
L =1
2∂µ ~ψ
†∂µ ~ψ − 1
2m2 ~ψ† ~ψ − V (~ψ†, ~ψ). (4.4.43)
The after rotation field is denoted as ~ψ′ = U ~ψ and ~ψ′† = ~ψ†U †, where U ∈ U(N)with U †U = UU † = 11. Note that U1U2 6= U2U1 in general.
The Lagrangian L (4.4.43) is invariant under transformation U due to
~ψ′† ~ψ′ = ~ψ†U †U ~ψ = ~ψ† ~ψ; (4.4.44)
∂µ ~ψ′†∂µ ~ψ′ = ∂µ ~ψ
†U †U∂µ ~ψ = ∂µ ~ψ†∂µ ~ψ. (4.4.45)
66
4.5 Discrete symmetries
Noether’s theorem is only defined for continuous symmetries. For discrete symmetry, noconservation law can be defined. However, discrete symmetries play important roles inquantum field theories.
4.5.1 Parity
A parity transformation (space reflection or mirror reflection) is defined as
t→ t, ~x→ −~x. (4.5.1)
In four dimensional vector, it can be characterized as xµ → xµ.
The scalar field is classified by
φ(t, ~x)P−→ φ′(t, ~x) = ±φ(t,−~x), (4.5.2)
where “ + ” sign represents for scalar and “− ” represents for pseudo-scalar.
E.g. 1. For N -component real scalar field, see previous subsection 3.2.7, it transformedunder the parity operation in the way
~φ(t, ~x)P−→ ~φ′ = R~φ(t,−~x), (4.5.3)
where R satisfies some constraints, for example RTR = RRT = 11.
E.g. 2. In the case of electromagnetic field Aµ(x) in subsection 3.2.2, the parity transfor-mation works as
Aµ(t, ~x)P−→ A′µ(t, ~x) = Aµ(t,−~x). (4.5.4)
4.5.2 Time reversal
The time reversal transformation is defined as
t→ −t, ~x→ ~x. (4.5.5)
In four dimensional vector, we have xµ → −xµ.
E.g. 1. Real scalar field theory:
φ(t, ~x)T−→ φ′(t, ~x) = φ(−t, ~x). (4.5.6)
Pseudo scalar field theory:
φ(t, ~x)T−→ φ′(t, ~x) = −φ(−t, ~x). (4.5.7)
E.g. 2. Electromagnetic field:
Aµ(t, ~x)T−→ A′µ(t, ~x) = Aµ(−t, ~x). (4.5.8)
Note: Parity and time reversal transformations are Lorentz transformation with
det(P ) = det(T ) = −1, (4.5.9)
namely P, T do not belong to the SO↑+(1, 3) group instead of P, T ∈ SO(1, 3), details referto Lorentz group in subsection 3.1.5.
67
4.5.3 Charge conjugation
Charge conjugation is a special operation defined in quantum field theory, and arises froma transformation between particles and anti-particles.
• For real scalar field φ(x), charge conjugation acts as
φ(x)C−→ φ′(x) = φ(x), (4.5.10)
which implies the anti-particle of meson is itself, namely π0 C−→ π0.
• For the real complex field ψ(x), charge conjugation gives rise to
ψ(x)C−→ ψ′(x) = ψ†(x); (4.5.11)
ψ†(x)C−→ ψ′†(x) = ψ(x). (4.5.12)
Note 1. Local quantum field theory (describing standard model) is invariant under thejoint action of C, P and T , which is called the CPT theorem.
Note 2. Under charge conjugation, the electromagnetic field (describing the photon) istransformed as
Aµ(x)C−→ A′µ(x) = −Aµ(x) (4.5.13)
with the “−” sign due to the CPT theorem.
68
Lecture 5 Constructing Quantum Field Theory
About Lecture III:
• [Luke] P.P. 20-39;
• [Tong] P.P. 7-37;
• [Coleman] P.P. 17-40;
• [PS] P.P. 15-29;
• [Zhou] P.P. 50-70.
5.1 Quantum mechanics and special relativity
5.1.1 Natural units
• SI (International system of units).
Primary units:
Meter(m) → LengthSecond(s)→ TimeKilogram(kg)→ Mess
E = ML2
T 2 = [Joule];
1eV = 1.602× 10−19J ;
1GeV = 1.602× 10−10J = 109eV ;
V = LT , c = 2.998× 108m · s−1;
~ = h2π = 1.054× 10−34J · s = 6.577× 10−25GeV · s.
Fine structure constant: α = 14πε0
e2
~c = 1137.04 .
• Natural units (NU)
Set the basic constants as ~ = 1, c = 1, which gives rise to
~ = 1↔ [time] =1
GeV∼ 6.577× 10−25sec;
~c = 1↔ [length] =1
GeV∼ 1.973× 10−16m = 197fm.
(5.1.1)
We can choose the dimension of energy or mass as the basic dimension, [E] = GeV =[mass], to perform the dimension analysis, and the dimension of quantity X is defined as
[X] =[massd
]= [md] = [Ed] = d. (5.1.2)
69
E.g.[c] = [~] = [k] = 0, [e] = 0; (5.1.3)
[time] = [length] =
[1
GeV
]= −1. (5.1.4)
Optional Problem: What’s the Plank mass and its unit?
5.1.2 Particle number unfixed at high energy (Special relativity)
In special relativity, see previous subsection 3.1, we have mass-energy relation
E = mc2 = γm0c2. (5.1.5)
Optional problem: derive ∆E = ∆mc2.
Note 1. The mass-energy relation (5.1.5) tells energy and matter can be converted to eachother such as atom bomb and nuclear power station.
Note 2. At v ≈ c, particle number is not fixed.
E.g. 1.p (proton) + p −→ p+ p+ π0 (meson). (5.1.6)
E.g. 2.p+ p −→ p+ p+ p+ p (anti-proton). (5.1.7)
Fermilab, Stanford.
E.g. 3.p+ p −→ · · ·+ · · ·+ Higgs particle. (5.1.8)
LHC, CERN.
Optional problem: Calculate the threshold energy EL of the incident particle at which allparticles after collision are at rest in the CoM (center-of-mass frame).
E.g. 1.
(EL, ~pLc)Proton Target proton
(mp, 0)Before collision
at the lab frame
p
p
π0
After collision
at the CoM frame
EL =(mπ0 + 2mp)
2c4 − 2m2pc
4
2mpc2
=(0.134 + 0.938× 2)2 − 2× 0.9382
2× 0.938
= 1.22GeV,
(5.1.9)
70
EL = γmpc2 =
1√1− (vL/c)2
mpc2, (5.1.10)
vLc
=
√1− (
mpc2
EL)2 ≈ 0.639. (5.1.11)
E.g. 2.
p+ p −→ p+ p+ p+ p, (5.1.12)
(EL +mpc2)− (~pLc+ 0)2 = (4mpc
2)2 − 0, (5.1.13)
EL = 7mpc2, (5.1.14)
vLc
=
√1− (
1
7)2 ≈ 0.989. (5.1.15)
E.g. 3.
EL ≈ 7TeV ≈ 7000mpc2, (5.1.16)
vLc
=
√1− (
1
7000)2 ≈ 0.999 · · · 9. (5.1.17)
5.1.3 No position operator at short distance (Quantum mechanics)
A particle of mass m can not be localized in a region smaller than its Compton wavelengthλc, which is defined as
λc =~mc
=~cmc2
. (5.1.18)
For example, we have a particle in a container with the reflection wall
L λc
The Heisenberg uncertainty relation 4x · 4p ≥ ~2 gives rise to
4x ∼ L, 4p ∼ ~2L. (5.1.19)
On the other hand, the mass-energy relation E =√m2c4 + p2c2 suggests
4E ≈ 4p · c =~c2L. (5.1.20)
When
4E ≥ 2mc2 = mc2 +mc2, (5.1.21)
namely
L ≤ ~4mc
=λc4, (5.1.22)
particle-antiparticle pair can be created.
71
L λc/4
E.g. Trap an electron in a container, the typical length scale L is the Bohr radii,
L =~2
mc2≈ 0.529× 10−8cm. (5.1.23)
The Compton wavelength of electron can be calculated as
λe =~cmec2
=197fm
0.511≈ 4× 10−11cm. (5.1.24)
Therefore we can see L λe, so it is safe enough to apply non-relativistic quantummechanics in condensed matter physics.
5.1.4 Micro-Causality and algebra of local observables
Two observables OA(t1, ~x1) and OB(t2, ~x2) should be commutative at space-like separatedregion, namely
[OA(t1, ~x1), OB(t2, ~x2)] = 0, (x1 − x2)2 < 0, (5.1.25)
because special relativity requires that particles in space-like regions can not have in-stantaneous interference with each other. In other words, interaction between particles ispropagated via fields.
e− e−
Interaction in space-like region via fields
Note: A commutative commutator between two operators OA and OB means that theyhave common eigenstates, so quantum measurements by OA and OB are not affected witheach other, i.e.,
|ψ〉 |ψ0〉 |ψ0〉.OA OB- -
For example, Alice and Bob are space-like separated, (xA−xB)2 < 0 with the observables
OA = σAZ , OB = σBX , (5.1.26)
which are the Pauli matrices, so OA and OB are space-time independent operators andthey have global action in the entire space-time. Experiments between Alice and Bob canbe perform in the following steps.
Step 1. Alice has the spin-up eigenstate | ↑〉 satisfying σAZ | ↑〉 = | ↑〉.
72
Step 2. Bob performs the measurement σBX which gives rise to
| ↑〉 −→ 1√2
(| ↑〉 ± | ↓〉). (5.1.27)
Step 3. Alice performs the measurement and then immediately realizes what Bob has doneon her spin due to
[σAZ , σBX ] 6= 0, (5.1.28)
which violates causality.
5.1.5 A naive relativistic single particle quantum mechanics
Inside the light-cone (time-like region) is the physical region and particles can not traveloutside the light-cone. In non-relativistic quantum mechanics, however, particles cantravel faster than the speed of light c and can be found outside the light-cone.
E.g. The spinless, free moving particle is described by the Hamiltonian
H =~P 2
2m, (5.1.29)
where the position operator and momentum operator satisfy the commutative relation
[Xi, Pj ] = i~δij , i, j = 1, 2, 3. (5.1.30)
The equation of motion follows the Schrodinger equation expressed as
i~∂
∂t|ψ(t)〉 = H|ψ(t)〉. (5.1.31)
Recall that we have
1 =
∫d3~k|~k〉〈~k|, 〈~k|~x〉 =
1
(2π)3/2e−i
~k·~x. (5.1.32)
Suppose at t = 0, the particle at ~x = 0 with |ψ(0)〉 = |~x = 0〉. After time t, the stateevolves as |ψ(t)〉, and we can compute the probability amplitude of finding this particleat ~x,
〈~x|ψ(t)〉 = 〈~x|e−iHt|ψ(0)〉
=
∫d3~k〈~x|e−iHt|~k〉〈~k|~x = 0〉
=
∫d3~k
(2π)3e−i
~k2/2mtei~k·~x
= (m
2πit)3/2eim~x
2/2t,
(5.1.33)
so the probability amplitude from point (t = 0, ~x = 0) to arbitrary place (t, ~x) like theregion outside the light-cone is non-zero, which violates causality.
A Naive single particle relativistic quantum mechanics can be constructed as
H =
√~P 2 +m2, H|~k〉 =
√~k2 +m2|~k〉. (5.1.34)
73
The probability amplitude can be calculated as
〈~x|ψ(t)〉 =
∫d3~k
(2π)3e−i√~k2+m2tei
~k·~x (5.1.35)
= ie−mr
2π2r
∫ ∞
mzdze−(z−m)r sinh
√z2 −m2t. (5.1.36)
Optional problem: Derive relation (5.1.36) from (5.1.35).
Note 1. Outside the light-cone, r = |~x| > t, we can find
〈~x|ψ(t)〉 6= 0, (5.1.37)
which still violates causality.
Note 2. As r 1/m, e−mr/r → 0, the probability amplitude becomes as
〈~x|ψ(t)〉 = 0, (5.1.38)
therefore at the length scale greater than the Compton wavelength, the probabilityamplitude outside the light cone is almost zero, due to no relativistic correctionsat that scale.
5.2 Comparisons of quantum mechanics with quantum fieldtheory
Quantum Mechanics Quantum Field TheoryParticle Number Fixed Un-fixedPosition operator Defined Not defined, labels/Parameters
Momentum operator Defined DefinedObservable Hermitian, space independent Hermitian, space dependentCausality No Yes
Time Label/Parameter Label/Parameter
5.3 Definition of Fock space
Definition: Fock space is the Hilbert space for any number of particles defined by
F =
+∞∑
n=0
⊕H(n) = H(0) ⊕H(1) ⊕ · · · (5.3.1)
with H(n) denotes an n-particle Hilbert space.
N ote 1: Fock space is the state space of QFT.
N ote 2: Refer to “Second Quantization” in Advanced Quantum Mechanics.
N ote 3:
74
Construction of QFT
Direct construction(Second quantization)
Canonical quantization
of classical filed theory
Rotation invariantFock space
Lorentz invariantFock space
5.4 Rotation invariant Fock space
5.4.1 Fock space
We consider free moving spinless particles, e.g. meson and Higgs, which are scalars underLorentz transformations and obey the Boson statistics. Observables are the Hamiltonian
H and the momentum ~P .The H(0) denotes an no-particle space, called the vacuum. The vacuum state, |vac〉,
instead of |0〉 in Luke’s notes, satisfies the normalization condition
〈vac|vac〉 = 1, (5.4.1)
with no observed energy and no observed momentum
H|vac〉 = 0, ~P |vac〉 = 0. (5.4.2)
The H(1) denotes a one-particle space. State: |~k〉,
〈~k|~k′〉 = δ(3)(~k − ~k′), (5.4.3)
and Observable: H, ~P~P |~k〉 = ~k|~k〉, (5.4.4)
H|~k〉 = ω~k|~k〉, ω~k =
√~k2 +m2. (5.4.5)
Note: Spatial rotation
~kR−−→ ~kR = R~k, R ∈ SO(3), (5.4.6)
with RTR = 113×3, and spatial rotation invariant one-particle state
|~k〉 O(R)−−−−→ |~kR〉 = O(R)|~k〉 = |R~k〉, (5.4.7)
with O†(R)O(R) = O(R)O†(R) = 11.Rotation invariant integral measure,
∫d3~k =
∫d3~kR, (5.4.8)
and rotation invariant scalar product,
〈~kR|~k′R〉 = 〈~k|~k′〉 = δ3(~k − ~k′), (5.4.9)
75
and rotation invariant completeness relation,∫d3~k|~k〉〈~k| = 1 =
∫d3~kR|~kR〉〈~kR|. (5.4.10)
H(2): two-particle space. The state |~k1,~k2〉 of two identical or indistinguishable parti-cles satisfying the Boson statistics
|~k1,~k2〉 = |~k2,~k1〉. (5.4.11)
with observables H and ~P ,
~P |~k1,~k2〉 = (~k1 + ~k2)|~k1,~k2〉, (5.4.12)
H|~k1,~k2〉 = (ω~k1+ ω~k2
)|~k1,~k2〉. (5.4.13)
The completeness relation in Fock space:
11 = |vac〉〈vac|+∞∑
n=1
1
n!
∫d3~k1d
3~k2 · · · d3~kn|~k1~k2 · · ·~kn〉〈~k1
~k2 · · ·~kn|. (5.4.14)
Note: An n-particle wave-function is complicated
|ψn(t)〉 =
∫d3~k1d
3~k2 · · · d3~kn |~k1~k2 · · ·~kn〉〈~k1
~k2 · · ·~kn|ψn(t)〉, (5.4.15)
due to 3n integral variables.
5.4.2 Occupation number representation (ONR)
Step 1: Confine the system in a periodic cubical box with side L.
L
L
L
System
Step 2: Due to translation invariance, allowed values of ~k are discrete with the form,
~k = (2π
Lnx,
2π
Lny,
2π
Lnz), nx, ny, nz ∈ Z. (5.4.16)
2π/L
2π/L
kx
ky
O
76
Step 3: n(~k) denotes the particle number with momentum ~k,
|n(·)〉 = |n(~k1), n(~k2), · · · 〉. (5.4.17)
Step 4: N(~k) denotes the occupation number operator with momentum ~k satisfying
N(~k1)|n(·)〉 = n(~k1)|n(·)〉,N(~k2)|n(·)〉 = n(~k2)|n(·)〉.
(5.4.18)
Step 5: Hamiltonian and momentum opertors,
H =∑
~k
ω~kN(~k), ~P =∑
~k
~kN(~k). (5.4.19)
Note: |n(·)〉 is still in the state formalism.
5.4.3 Hints from SHO (Simple Harmonic Oscillator)
Statement:
Occupation number representation of Fock space
Representation of infinite system of independent Harmonic oscillators
one to one correspondence
Theorem: A SHO has a unique ground state |0〉 and there is a gap between |0〉 and itsfirst excited state.
Proof.
H =~P 2
2m+
1
2mω2X2, ω 6= 0. (5.4.20)
[Xi, Pj ] = i~δij . (5.4.21)
Canonical transformation:
p =P√mω
, q =√mωX, (5.4.22)
=⇒ H =ω
2(p2 + q2) (5.4.23)
where the lower indices of X,P, q, p are neglected for simplicity or one can understand ourharmonic oscillator as a one-dimensional harmonic oscillator.
Define raising and lowering operators
a =q + ip√
2, a† =
q − ip√2, (5.4.24)
q =1√2
(a+ a†), (5.4.25)
where
[a, a†] = 1, H = ω(a†a+1
2), (5.4.26)
77
N = a†a, H = ω(N +1
2). (5.4.27)
Due to 〈ψ|a†a|ψ〉 > 0, define
a|0〉 = 0, |0〉 : ground state, (5.4.28)
|1〉 = a†|0〉, |1〉 : first excited state, (5.4.29)
H|1〉 = ω(N +1
2)|1〉 =
3
2ω|1〉 = E1|1〉, (5.4.30)
H|0〉 = ω(N +1
2)|0〉 =
1
2ω|0〉 = E0|1〉, (5.4.31)
The gap:E1 − E0 = ω 6= 0, (5.4.32)
|n〉 =1√n!
(a†)n|0〉. (5.4.33)
Note: In SHO, “N” denotes the excited energy level; in ONR, “N” denotes the occu-pation number of particles.
5.4.4 The operator formalism of Fock space in a cubic box
Define creation and annihilation operators: a†~k, a~k, satisfying
[a~k, a†~k′
] = δ~k,~k′ , (5.4.34)
[a~k, a~k′ ] = [a†~k, a†~k′
] = 0. (5.4.35)
Vacuum state,a~k|vac〉 = 0, for ∀~k. (5.4.36)
One-particle state,|~k〉 = a†~k|vac〉. (5.4.37)
Two-particle state,|~k1,~k2〉 = a†~k1
a†~k2|vac〉. (5.4.38)
Hamiltonian,
H =∑
~k
ω~ka†~ka~k. (5.4.39)
Momentum,~P =
∑
~k
~ka†~ka~k. (5.4.40)
Particle number,
N =∑
~k
a†~ka~k. (5.4.41)
Define N(~k) = a†~ka~k, then
N =∑
~k
N(~k), (5.4.42)
78
H =∑
~k
ω~kN(~k), (5.4.43)
~P =∑
~k
~kN(~k). (5.4.44)
Check:
〈~k|~k′〉 = 〈vac|a~ka†~k′|vac〉 = 〈vac|[a~k, a
†~k′
]|vac〉
= 〈vac|vac〉δ~k,~k′ = δ(3)~k,~k′
.(5.4.45)
H|~k〉 = Ha†~k|vac〉 = [H, a†~k]|vac〉
= [∑
~k′
ω~k′a†~k′a~k′ , a
†~k]|vac〉
= ω~ka†~k|vac〉 = ω~k|~k〉.
(5.4.46)
5.4.5 Drop the box normalization and take the continuum limit
L −→∞
L
L
[a~k, a†~k′
] = δ(3)(~k − ~k′), (5.4.47)
〈~k|~k′〉 = δ(3)(~k − ~k′), (5.4.48)
H =
∫d3~k ω~k a
†~ka~k, (5.4.49)
P =
∫d3~k ~k a†~ka~k. (5.4.50)
Note: Any observables in free QFT can be written in terms of creation and annihilationoperators.
Note: a~k, a†~k
are operator-valued distributions, i.e. only integrations with a~k, a†~k
aremeaningful in mathematics.
Note: Why a†~k and a~k, because they represent particle creation and annihilation inexperiments.
Note: Quantum free field theory is a collection of independent harmonic oscillators.
QFT Harmonic oscillators
a~k annihilation operator lowering operator
a†~k creation operator rasing operator
a†~ka~k particle number excitation energy level
H0 = 12ω vacuum energy zero-point energy
Lowest energy state |vac〉 ground state |0〉
79
5.5 Lorentz invariant Fock spce
5.5.1 Lorentz group
A Lorentz transformation is generated by both Lorentz boosts and rotations. The Lorentzgroup is a set of all Lorentz transformations. The proper Lorentz group is the Lorentzgroup with det(Λ) = 1; the proper orthochronous Lorentz group is the Lorentz group withdet(Λ) = 1 and Λ0
0 > 0. Usually, the proper orthochronous Lorentz group is called theLorentz group for simplicity.
5.5.2 Definition of Lorentz invariant normalized states
The Lorentz transformation Λ
kΛ−−→ kΛ = Λk, (5.5.1)
with
k′µ = Λµνkν , k = (k0, k1, k2, k3), (5.5.2)
and the corresponding unitary transformation U(Λ)
|k〉 U(Λ)−−−−→ |k′〉 = U(Λ)|k〉 = |kΛ〉, (5.5.3)
with
U †U = UU † = 11, (5.5.4)
Define the Lorentz invariant one-particle state
|k〉 ≡ (2π)3/2√
2ω~k|~k〉, (5.5.5)
where ω~k =√~k2 +m2. The factor
√2ω~k is the compensation factor from Lorentz boost
and (2π)3/2 is from the convention of the Feynman rules for the Feynman diagrams.
The normalization condition,
〈k|k′〉 = (2π)32ω~kδ3(~k − ~k′), (5.5.6)
and the completeness relation
∫d3~k|~k〉〈~k| = 1 =
∫d3~k
(2π)3
1
2ω~k|k〉〈k|. (5.5.7)
5.5.3 Lorentz invariant normalized state
Theorem: |k〉 = (2π)3/2√
2ω~k|~k〉 is Lorentz invariant.
Proof.
Step 1: Verify∫d3~k2ω~k
is a Lorentz transformation invariant integral measure.
The four dimensional integral measure d4k = dk0dk1dk2dk3 is Lorentz invariant dueto
d4kΛ = det(Λ)d4k = d4k (5.5.8)
where det(Λ) is the Jacobian determinant.
80
The k2 = m2 is called on-shell mass condition
δ(k2 −m2) = δ(k20 − ω2
~k)
=1
2ω~k(δ(k0 + ω~k) + δ(k0 − ω~k)).
(5.5.9)
which is Lorentz invariant.The step function,
θ(k0) = 1, for k0 > 0. (5.5.10)
is Lorentz invariant.
~k
k0
δ(k2 −m2)θ(k0)
So we have ∫d4kδ(k2 −m2)θ(k0) =
∫d3~k
2ω~k
∣∣∣∣∣k0=ω~k
, (5.5.11)
which is LT invariant.Step 2: The completeness relation is LT invariant.
U(Λ)1U †(Λ) = U(Λ)
∫d3~k|~k〉〈~k|U †(Λ) = 1, (5.5.12)
|k〉 = (2π)3/2√
2ω~k|~k〉, (5.5.13)
=⇒ U(Λ)
∫d3~k
2ω~k2ω~k|~k〉〈~k|U
†(Λ)
=
∫d3~kΛ
2ω~kΛ
U(Λ)2ω~k|~k〉〈~k|U†(Λ)
=1
(2π)3
∫d3~kΛ
2ω~kΛ
|kΛ〉〈kΛ|
=1
(2π)3
∫d3~k
2ω~k|k〉〈k|
= 1.
(5.5.14)
Step 3: Check the normalization condition
〈k|k′〉 = (2π)32ω~kδ3(~k − ~k′), (5.5.15)
1
(2π)3
∫d3~k
2ω~k︸ ︷︷ ︸1©
〈k|k′〉︸ ︷︷ ︸3©
= 1︸︷︷︸2©, (5.5.16)
where 1© and 2© are LT invariant, so 3© is LT invariant, 1©+ 2©→ 3©.
81
5.5.4 Notation
Rotation invariant state,
|~k〉 = a†~k|vac〉, (5.5.17)
[a~k, a†~k′
] = δ3(~k − ~k′), (5.5.18)
a~k = a(~k), a†~k = a†(~k). (5.5.19)
Lorentz invariant state,
|k〉 = (2π)3/2√
2ω~k|~k〉
= (2π)3/2√
2ω~ka†~k|vac〉
= β†k|vac〉,
(5.5.20)
where
βk = (2π)3/2√
2ω~ka~k, (5.5.21)
[βk, β†k′ ] = (2π)32ω~kδ
3(~k − ~k′). (5.5.22)
Introduce
α(~k) =a(~k)
(2π)3/2√
2ω~k, (5.5.23)
[α~k, α†~k′
] =1
(2π)32ω~kδ3(~k − ~k′). (5.5.24)
5.6 Canonical quantization of classical field theory
Canonical quantization is a procedure of deriving QM (quantum mechanics) (or QFT(quantum field theory)) from the Hamiltonian formulation of CPM (Classical particlemechanics)(or CFT (classical field theory)).
Diagrammatic definition:
CPM CFT
QM QFT
H H
H H
−→ Continuum limit
Quantization
82
5.6.1 Classical field theory
What is a field? Field is a quantity defined on every space-time point.
Lagrangian density: L(φa, ∂µφa), φa = φa(t, ~x), a = 1, 2, · · · , N .
Lagrangian:
L(t) =
∫d3xL(φa, ∂µφ
a). (5.6.1)
Action:
S =
∫ t2
t1
dtL(t) =
∫d4xL(φa, ∂µφ
a). (5.6.2)
Note: L and S are Lorentz invariant scalar under proper & orthochronous Lorentztransformation.
Action principle:
Arbitrary variation: φa → φa + δφa.
Fixed boundary: δφa(t1) = δφa(t2) = 0, φa(t,±∞) = 0.
Stationary action: δS = 0.
Equations of motion or Euler-Lagrangian equations.
∂L∂φa
= ∂µ(∂L
∂(∂µφa)) (5.6.3)
Conjugate momentum:
Πa =∂L∂φa
, φa =∂
∂tφa(t, ~x). (5.6.4)
Hamiltonian density:
H = Πaφa − L. (5.6.5)
Hamiltonian:
H =
∫d3xH. (5.6.6)
Eg. Real scalar field theory in 4 dimensional space-time.
L =1
2(∂µφ∂
µφ−m2φ2)
=1
2(φ2 − (∇φ)2 −m2φ2),
(5.6.7)
where m2 > 0.
Π =∂L∂φ
= φ, (5.6.8)
H =1
2(φ2 + (∇φ)2 +m2φ2) > 0. (5.6.9)
Equation of motion,∂L∂φ
= −m2φ,∂L
∂(∂µφ)= ∂µφ, (5.6.10)
(+m2)φ = 0, (5.6.11)
which is called the Klein-Gordon equation associated with the Klein-Gordon field.
83
5.6.2 Canonical quantization
Step 1: φa(t, ~x),Πb(t, ~y)→ φa, Πb, satisfying the equal time commutation relations,
[φa(t, ~x), φb(t, ~y)] = 0 = [Πa(t, ~x), Πb(t, ~y)], (5.6.12)
[φa(t, ~x), Πb(t, ~y)] = iδabδ(3)(~x− ~y). (5.6.13)
Step 2: Hamiltonian operator:
H =
∫d3xH(φa, Πa). (5.6.14)
Step 3: EoM in Heisenberg picture:
∂φa∂t
= i[H, φa],
∂Πa
∂t= i[H, Πa].
(5.6.15)
Eg. Real scalar field operators, φ(t, ~x) and Π(t, ~x) =˙φ(t, ~x),
H =1
2
∫d3x
[Π2(t, ~x) + (∇φ(t, ~x))2 +m2φ2(t, ~x)
]. (5.6.16)
EoM:
˙φ(x) = Π(x),
˙Π(x) = ∇2φ−m2φ.
(5.6.17)
Optional homework : derive 5.6.17.
¨φ = ∇2φ−m2φ =⇒ (+m2)φ(t, ~x) = 0. (5.6.18)
Plane-wave solution of the K-G equation:
φ(t, ~x) =
∫d3~k
[α(~k)e−ik·x + α†(~k)eik·x
], (5.6.19)
where φ = φ†, k · x = ω~k − ~k · ~x.
Theorem:
[φ(t, ~x), Π(t, ~y)] = iδ(3)(~x− ~y)
⇐⇒[α(~k), α†(~k′)] =1
(2π)32ω~kδ(3)(~k − ~k′). (5.6.20)
Optional homework : prove 5.6.20. See Luke’s notes at P.P. 33.
α(~k) =1
2
∫d3x
(2π)3
[φ(~x) +
i
ω~kΠ(~x)
]e−i
~k·~x, (5.6.21)
a(~k) = (2π)3/2√
2ω~kα(~k), (5.6.22)
[a(~k), a†(~k′)] = δ(3)(~k − ~k′). (5.6.23)
84
• α(~k): Fourier expansion of φ(t, ~x);
• β(k): Lorentz invariant state, i.e. |k〉 = β†(k)|vac〉;
• a(~k): Creation or annihilation operator.
Note: Why does relativistic quantum field theory prefer the Heisenberg picture?(1). The particle creation or annihilation can be easily described in the operator for-
malism. (2). The equations of motion or the Heisenberg equations are Lorentz covariant,for examples, the Klein-Gordon equation, the Maxwell equations and the Dirac equa-tion. (3). The classical-quantum correspondence is obvious. The expectation value of theHeisenberg equations at ~→ 0 is the Hamilton’s equations.
5.6.3 Basic calculation on Hamiltonian
Theorem:
H =1
2
∫d3x(φ2 + (∇φ)2 +m2φ2)
⇐⇒H =1
2
∫d3kω~k(a(~k)a†(~k) + a†(~k)a(~k)).
(5.6.24)
Optional homework : prove 5.6.24.
φ(t, ~x) =
∫d3k
(2π)3/2
1√2ω~k
(a(~k)e−ik·x + a†(~k)eik·x). (5.6.25)
Π(t, ~x) = φ(t, ~x) =
∫d3k
(2π)3/2(−i√ω~k2
)(a(~k)e−ik·x − a†(~k)eik·x). (5.6.26)
H = I+II:
I =1
2
∫d3xΠ2 =
1
2
∫d3xφ2(t, ~x)
=1
2
∫d3x
∫d3k
(2π)3/2
∫d3k′
(2π)3/2(−i√ω~k2
)(−i√ω~k′
2)
×(a(~k)e−ik·x − a†(~k)eik·x
)(a(~k′)e−ik
′·x − a†(~k′)eik′·x).
(5.6.27)
I = 1©+ 2©+ 3©+ 4©:
1© =1
2
∫d3x
(2π)3
∫d3k
∫d3k′(−i)2
√ω~kω~k′
2a(~k)a(~k′)e−ik·x−ik
′·x. (5.6.28)
Because ofe−ik·x−ik
′·x = e−i(ω~k+ω~k′ )tei(~k+~k′)·~x, (5.6.29)
∫d3x
(2π)3e−i(
~k+~k′)·~x = δ(3)(~k + ~k′), (5.6.30)
∫d3xδ(3)(x) = 1, (5.6.31)
so
1© =1
2
∫d3k
∫d3k′δ(3)(~k + ~k′)(−i)2
√ω~kω~k′
2a(~k)a(~k′)e−i(ω~k+ω~k′ )t
=1
2
∫d3k − 1
2ω~ka(~k)a(−~k)e−2iω~kt.
(5.6.32)
85
2© =1
2
∫d3x
(2π)3
∫d3k
∫d3k′(−i)2
√ω~kω~k′
2a(~k)(−a†(~k′))e−ik·x+ik′·x
=1
2
∫d3k
1
2ω~ka(~k)a†(~k).
(5.6.33)
3© =1
2
∫d3x
(2π)3
∫d3k
∫d3k′(−i)2
√ω~kω~k′
2(−a†(~k))a(~k′)eik·x−ik
′·x
=1
2
∫d3k
1
2ω~ka
†(~k)a(~k).
(5.6.34)
4© =1
2
∫d3x
(2π)3
∫d3k
∫d3k′(−i)2
√ω~kω~k′
2(−a†(~k))(−a†(~k′))eik·x+ik′·x
=1
2
∫d3k − 1
2ω~ka
†(~k)a†(−~k)e2iω~kt.
(5.6.35)
I = 1©+ 2©+ 3©+ 4©
=1
2
∫d3k
1
2ω~k(−ω2
~ka(~k)a(−~k)e−2iω~kt − ω2
~ka†(~k)a†(−~k)e2iω~kt
+ ω2~ka(~k)a†(~k) + ω2
~ka†(~k)a(~k)).
(5.6.36)
II =1
2
∫d3x(∇φ∇φ+m2φ2)
=1
2
∫d3k
1
2ω~k(~k2 +m2)(a(~k)a†(~k) + a†(~k)a(~k)
+ a(~k)a(−~k)e−2iω~kt + a†(~k)a†(−~k)e2iω~kt).
(5.6.37)
H =I + II
=1
2
∫d3k
1
2ω~k(a(~k)a†(~k)(ω2
~k+ ~k2 +m2) + a†(~k)a(~k)
× (ω2~k
+ ~k2 +m2) + a(~k)a(−~k)e−2iω~kt(−ω2~k
+ ~k2 +m2)
+ a†(~k)a†(−~k)e2iω~kt(−ω2~k
+ ~k2 +m2)
=1
2
∫d3k ωk (a(~k)a†(~k) + a†(~k)a(~k)),
(5.6.38)
where[a(~k), a†(~k′)] = δ(3)(~k − ~k′). (5.6.39)
5.6.4 Vacuum energy and normal ordered product
H =1
2
∫d3kω~k(2a
†(~k)a(~k) + δ(3)(0))
=
∫d3kω~ka
†(~k)a(~k) +1
2
∫d3kω~kδ
(3)(0).
(5.6.40)
86
Evac ≡〈vac|H|vac〉
=1
2
∫d3kω~kδ
(3)(0).(5.6.41)
δ(3)(~k) =
∫d3x
(2π)3ei~k·~x, (5.6.42)
δ(3)(0) =
∫d3x
(2π)3=
V
(2π)3. (5.6.43)
Evac = (1
2
∫d3k
√~k2 +m2)× V
(2π)3. (5.6.44)
Vacuum energy density:
ρvac =EvacV
=1
2
∫d3k
(2π)3
√~k2 +m2. (5.6.45)
• V → +∞, Evac → +∞, infra-red divergence.
• V fixed, ρvac → +∞, ultra-violet divergence.
Remark 1: Vacuum energy is absolute energy. An observer is only concerned aboutenergy difference.
Remark 2: Without gravity, the vacuum energy Evac can not be detected.
Google/Wiki
a. Casimir force. The vacuum can be disturbed.
b. Cosmological constant problem. The observed vacuum energy os theUniverse is almost zero, so there is a contradiction between theoretical cal-culation and experiment:
+∞ = ρtherovac ≫ ρexpervac ≈ 0. (5.6.46)
Definition: Normal-ordered product,
: a†(~k)a(~k) := : a(~k)a†(~k) :
=a†(~k)a(~k).(5.6.47)
1. Inside normal ordering,: · · · · · · :, (5.6.48)
operators are commutative.
2. In the result, creation operators are always on LHS, annihilation operators are onRHS.
: H :=1
2
∫d3kω~k : a†~ka~k + a~ka
†~k
:
=
∫d3kω~ka
†~ka~k.
(5.6.49)
Evac = 〈vac| : H : |vac〉 = 0. (5.6.50)
87
5.6.5 Micro-causality (Locality)
Theorem: in free scalar field theory, the equal time commutator [φ(t, ~x), φ(t, ~y)] = 0 bythe canonical quantization ensures the micro-causality,
[φ(x), φ(y)] = 0, for (x− y)2 < 0. (5.6.51)
Lemma a: φ(t = 0, ~x)|vac〉 creates a particle at position ~x.
Proof.
φ(t, ~x) =
∫d3k
(2π)32ω~k(β(k)e−ik·x + β†(k)eik·x), (5.6.52)
where|k〉 = β†(k)|vac〉, (5.6.53)
〈p|k〉 = (2π)32ω~kδ(3)(~p− ~k). (5.6.54)
φ(0, ~x)|vac〉 =
∫d3k
(2π)32ω~ke−ik·x|k〉. (5.6.55)
〈p|φ(0, ~x)|vac〉 =
∫d3k
(2π)32ω~ke−i
~k·~x〈p|k〉
=e−i~p·~x.
(5.6.56)
In non-relativistic QM,X|~x〉 = ~x|~x〉, (5.6.57)
P |~p〉 = ~p|~p〉, (5.6.58)
〈~p|~x〉 =1
(2π)3/2e−i~p·~x. (5.6.59)
Hence φ(t = 0, ~x)|vac〉 can be viewed as a particle at position ~x.
Lemma b: D(x− y) = 〈vac|φ(x)φ(y)|vac〉 denotes the amplitude of creating a particleat y and destroying it at x, namely the particle propagating from y to x.
D(x− y) =
y
x
Proof.φ(t, ~x) = φ(+)(t, ~x) + φ(−)(t, ~x), (5.6.60)
with
φ(+)(t, ~x) =
∫d3k
(2π)3/2√
2ω~ka(~k)e−ik·x, (5.6.61)
φ(−)(t, ~x) =
∫d3k
(2π)3/2√
2ω~ka†(~k)eik·x, (5.6.62)
where φ(+)(t, ~x) with positive energy and φ(−)(t, ~x) with negative energy.
88
D(x− y) =〈vac|φ(+)(x)φ(−)(y)|vac〉=〈vac|[φ(+)(x), φ(−)(y)]|vac〉
=
∫d3kd3k′
(2π)3√
2ω~k2ω~k′[a(~k), a†(~k′)]e−ik·x+ik′·y
=
∫d3k
(2π)3√
2ω~ke−ik·(x−y)
6=0.
(5.6.63)
Note: In space-like region, (x− y)2 < 0, D(x− y) 6= 0 violates causality.
Lemma c: D(x− y) is a Lorentz invariant scalar.
Proof.
D(x− y) =
∫d3k
(2π)32ω~ke−ik·(x−y), (5.6.64)
D(Λ(x− y)) = D(x− y). (5.6.65)
Lemma d: D(x− y)|tx=ty= D(y − x)|tx=ty
.
Proof.
(D(x− y)−D(y − x))tx=ty
=
∫d3k
(2π)32ω~k(ei
~k·(~x−~y) − e−i~k·(~x−~y))
=0
(5.6.66)
Lemma e: for space-like separated events, (x − y)2 < 0, there always exists Λ givingrise to tx = ty.
Proof.
t
y x
tx > ty
t
y x
tx = ty
Λ
89
Lemma f: (D(x− y)−D(y − x))(x−y)2<0 = 0.
Proof.
(D(x− y)−D(y − x))(x−y)2<0
Λ−−→(D(Λ(x− y))−D(Λ(y − x)))tx=ty = 0.(5.6.67)
Proof. Proof of theorem5.6.5.
[φ(x), φ(y)] =[φ+(x), φ−(y)] + [φ−(x), φ+(y)]
=D(x− y)−D(y − x).(5.6.68)
[φ(x), φ(y)](x−y)2<0 = 0. (5.6.69)
N ote 1: Micro-causality is violated in the sense of D(x − y) 6= 0, but it is obeyed for[φ(x), φ(y)](x−y)2<0 = 0, i.e. in the sense of quantum measurement.
N ote 2:
D(x− y) =
y
x
Particle
(eg. Electron)
D(y − x) =
x
y
Anti-particle
(eg. Positron)
Micro-causality predicts the existence of anti-particle for every particle.
5.7 Remarks on canonical quantization
Advantages
i. Particle interpretation. (Hamiltonian has a discrete spectrum of particle.)
ii. Particle creation & annihilation.
iii. Classical-quantum correspondence.
Hamilton’s equations⇐⇒ Heisenberg equations
Disadvantages
i. No explicit Lorentz-invariance in Hamiltonian formulation, e.g. equal-time com-mutator depending on time.
ii. Infinity large vacuum energy.
Comparison of canonical quantization with path integral quantization
90
Canonical Path
Hamiltonian formalism Lagrangian formalismFormalism
Operator-valued function Ordinary-valued function
Particle interpretation Explicit Not explicit
Classical-quantum Hamilton’s equationscorrespondence ⇐⇒Heisenberg equations
Action/Action principle
Lorentz covariant Not explicit Explicit
Vacuum energy +∞ +∞
5.8 Symmetries and conservation laws
5.8.1 U(1) invariant quantum complex scalar field theory
SO(2) invariant quantum real scalar field theoryConsider a free field theory without the interacting part V (φ1, φ2) and apply the canon-
ical quantization procedure to obtain
φ1 =
∫d3~k1
(2π)3/2√
2ω~k1
[a~k1
e−ik1·x + a†~k1eik1·x
]; (5.8.1)
φ2 =
∫d3~k2
(2π)3/2√
2ω~k2
[a~k2
e−ik2·x + a†~k2eik2·x
], (5.8.2)
where [a~ki , a
†~ki
]= 1, i = 1, 2. (5.8.3)
The vacuum is defined bya~ki |vac〉 = 0, (5.8.4)
and one-particle states are defined by
|~k, 1〉 = a†~k1|vac〉, (5.8.5)
|~k, 2〉 = a†~k2|vac〉. (5.8.6)
The conserved charge Q has the form
Q = i
∫d3~k(a†~k1
a~k2− a†~k2
a~k1) (5.8.7)
which has no explicit physical interpretation. With new creation and annihilation opera-tors
b~k =a~k1
+ ia~k2√2
, b†~k =a†~k1− ia†~k2√
2; (5.8.8)
c~k =a~k1− ia~k2√
2, c†~k =
a†~k1+ ia†~k2√
2, (5.8.9)
the conserved charge Q has a new form
Q =
∫d3~k(b†~kb~k − c
†~kc~k)
= Nb −Nc
= (+1)Nb + (−1)Nc.
(5.8.10)
91
We have b-type of particles and c-type of particles:
b~k|vac〉 = c~k|vac〉 = 0, (5.8.11)
|~k, b〉 = b†~k|vac〉, |~k, c〉 = c†~k|vac〉 (5.8.12)
and a b-type particle has charge +1, a c-type particle has charge −1:[Q, b†~k
]= (+1)b†~k,
[Q, c†~k
]= (−1)c†~k. (5.8.13)
Q|~k, b〉 = +1|~k, b〉, Q|~k, c〉 = −1|~k, c〉. (5.8.14)
Introduce new fields which are complex scalar fields,
ψ(x) =1√2
(φ1 + iφ2), (5.8.15)
ψ†(x) =1√2
(φ1 − iφ2). (5.8.16)
which give the free part of the Lagrangian
L0 = ∂µψ†∂µψ −m2ψ†ψ. (5.8.17)
The SO(2) rotation of the field doublet (φ1, φ2) becomes the U(1) transformation ofthe field ψ(x),
ψ′(x) =1√2
(φ′1 + iφ′2)
=1√2
(cosλφ1 + sinλφ2 + i(− sinλφ1 + cosλφ2))
= e−iλψ(x);
and (5.8.18)
ψ†(x)′ = eiλψ†(x). (5.8.19)
U(1) invariance of L0 with eiλ ∈ U(1):
ψ′†(x)ψ′(x) = ψ†(x)eiλe−iλψ(x) = ψ†(x)ψ(x). (5.8.20)
Canonical quantization of the above complex scalar field theory,
ψ(x) =
∫d3~k
(2π)3/2√
2ω~k(b~ke
−ik·x + c†~keik·x), (5.8.21)
ψ†(x) =
∫d3~k
(2π)3/2√
2ω~k(b†~ke
ik·x + c~ke−ik·x). (5.8.22)
which gives the current
Jµ =∂L
∂(∂µψ)Dψ +
∂L∂(∂µψ†)
Dψ†
= iλ((∂µψ)ψ† − (∂µψ†)ψ),
(5.8.23)
with the charge densityJ0 = i(ψψ† − ψ†ψ), (5.8.24)
giving the charge,
Q =
∫d3xJ0 =
∫d3~k(b†~kb~k − c
†~kc~k). (5.8.25)
92
5.8.2 Discrete symmetries
5.9 Return to non-relativistic quantum mechanics
93
Lecture 6 Feynman Propagator, Wick’s
Theorem and Dyson’s Formula
About Lecture IV:
• [Luke] P.P. 65-80;
• [Tong] P.P. 38-41 & 47-48;
• [Coleman] P.P. 70-82;
• [PS] P.P. 29-33 & 77-90;
• [Zhou] P.P. 70-72 & 127-146.
6.1 Retarded Green function
6.1.1 Real scalar field theory with a classical source
Def. 1. Free theory: Lagrangian with quadratic terms of fields.
Eg.
L0 =1
2∂µφ∂
µφ− 1
2m2φ2. (6.1.1)
Def. 2. Interaction theory: Lagrangian with non-quadratic terms of fields.
Eg.
L = L0 − Lint, Lint =λ
4!φ4, (6.1.2)
where λ is coupling constant.
Model:L = L0 + ρ(x)φ(x), (6.1.3)
where ρ(x) is a real number function which can be turned on and off.
ρ(x)t>tf = ρ(x)t<ti = 0. (6.1.4)
ti tf
ρ(x)t<ti = 0 ρ(x)t>tf = 0
t
ρ(x)
94
Note: ρ(x) is not a field operator, just an ordinary function.
Πφ =∂L∂φ
= φ, (6.1.5)
H = Πφφ− L =1
2φ2 +
1
2(∇φ)2 +
1
2m2φ2 − ρ(x)φ(x). (6.1.6)
EoM:∂L∂φ
= −m2φ+ ρ(x), (6.1.7)
∂µ(L
∂(∂µφ)) = φ, (6.1.8)
(+m2)φ = ρ(x). (6.1.9)
To understand the meaning of the source term ρ(x), see: in Electrodynamics
∇2ϕ = −4π
ερ (6.1.10)
where ϕ is electric potential determined by charge-density distribution ρ.
6.1.2 The retarded Green function
EoM:(+m2)φ = ρ(x). (6.1.11)
Boundary condition:φ(x)|t=−∞ = φ0(x), (6.1.12)
where φ0(x) is free field since t→ −∞, there is no interaction.Solution:
φ(x) = φ0(x) + i
∫d4yDR(x− y)ρ(y), (6.1.13)
where DR(x− y) is the retarded Green function satisfying
(+m2)DR(x− y) = −iδ(4)(x− y) (6.1.14)
with DR(x− y) = 0, x0 < y0.Note: The term “Retarded” means that the future is determined by the past, see:
x
t
I
III
IIIII: forward light-cone, time-like region
II: outside light-cone, space-like region
III: backward light-cone, time-like region
in which the origin denotes y and the forward light-cone denotes x, i.e,
DR(x− y)
= 0 Domain II, III.6= 0 Domain I, x0 > y0.
(6.1.15)
95
6.1.3 Analytic formulation of DR(x− y)
Definition.
DR(x− y) = θ(x0 − y0)[φ(x), φ(y)]
= θ(x0 − y0)〈vac|[φ(x), φ(y)]|vac〉x0>y0
===== D(x− y)−D(y − x)
x0>y0
=====
∫d3~k
(2π)3
(e−ik·(x−y)
2w~k− eik·(x−y)
2w~k
)(6.1.16)
where
D(x− y) =
∫d3~k
(2π)3
e−ik·(x−y)
2w~k. (6.1.17)
Thm.
DR(x− y) = θ(x0 − y0)
∫
CR
d4k
(2π)4
i
k2 −m2e−ik·(x−y)
≡ θ(x0 − y0)
∫d3~k
(2π)3
[∮
CR
dk0
2π
ie−ik0(x0−y0)
(k0 − w~k)(k0 + w~k)
]ei~k·(~x−~y)
(6.1.18)
in which the contour CR is given by
Re(k0)
Im(k0)
−w~k w~k
CR
Proof. Since(+m2)DR(x− y) = −iδ(4)(x− y), (6.1.19)
we have
DR(x− y) =
∫d4k
(2π)4
i
k2 −m2eik·(x−y). (6.1.20)
But the above integral is not well defined due to two poles,
1
k2 −m2=
1
k20 − w2
~k
=1
(k0 − w~k)(k0 + w~k). (6.1.21)
besides the retarded condition
DR(x− y) = 0, x0 < y0 (6.1.22)
has to be satisfied. In the following, we construct two contour integrals and apply theresidue theorem.
At x0 < y0, we choose the contour CR′ on the upper half plane:
96
Re(k0)
Im(k0)
−w~k w~k
C′R
No contribution
=⇒Re(k0)
Im(k0)
−w~k w~k
to compute
IA =
∮
C′R
dk0
2π
i
(k0 − w~k)(k0 + w~k)e−ik
0(x0−y0). (6.1.23)
because
e−i(iImk0)(x0−y0) Imk0→+∞−−−−−−−−→ eImk0(x0−y0) → 0. (6.1.24)
Since no pole terms in C ′R, with the residue theorem, we have
IA =
∮
C′R
dk0
2π
i
(k0 − w~k)(k0 + w~k)e−ik
0(x0−y0) = 0 (x0 < y0), (6.1.25)
=⇒ DR(x− y) = 0. (6.1.26)
At x0 > y0, we choose the contour CR on the lower half plane:
Re(k0)
Im(k0)
−w~k w~k
CR
Integral vanishing on boundary
=⇒Re(k0)
Im(k0)
−w~k w~k
to compute
IR =
∮
CR
dk0
2π
i
(k0 − w~k)(k0 + w~k)e−ik
0(x0−y0), (6.1.27)
because
e−i(iImk0)(x0−y0) Imk0→−∞−−−−−−−−→ eImk0(x0−y0) → 0. (6.1.28)
Apply the residue theorem to compute the contour integral IR to obtain
IR =i
2π(−2πi)
e−ik0(x0−y0)
2w~k
∣∣∣∣∣k0=w~k
+i
2π(−2πi)
eik0(x0−y0)
−2w~k
∣∣∣∣∣k0=−w~k
=e−ik
0(x0−y0)
2w~k
∣∣∣∣∣k0=w~k
− eik0(x0−y0)
2w~k
∣∣∣∣∣k0=−w~k
,
(6.1.29)
97
so that
DR(x− y)x0>y0
=====
∫d3~k
(2π)3
(e−iw~k(x0−y0)
2w~k− eiw~k(x0−y0)
2w~k
)ei~k·(~x−~y)
x0>y0
=====
∫d3~k
(2π)3
(e−ik·(x−y)
2w~k− eik·(x−y)
2w~k
)
= θ(x0 − y0)[φ(x), φ(y)]
= θ(x0 − y0)〈vac|[φ(x), φ(y)]|vac〉.
(6.1.30)
6.1.4 Canonical quantization of a real scalar field theory with a classicalsource
With the analytical form of the retarded Green function, we have
φ(x) = φ0(x) + i
∫d4y
∫d3~k
(2π)3
1
2w~kθ(x0 − y0)
(e−ik·(x−y) − eik·(x−y)
)ρ(y). (6.1.31)
As x0 → +∞, we have ρ(x0) = 0, θ(x0 − y0) = 1 so that
φ(x)x0→+∞
======= φ0(x) + i
∫d4y
∫d3~k
(2π)32w~k
(e−ik·(x−y) − eik·(x−y)
)ρ(y). (6.1.32)
Define
ρ(k) =
∫d4yeik·yρ(y), ρ(−k) = ρ∗(k), (6.1.33)
where k2 = m2 and ρ(k) is a known and fixed function. We have
φ(x)x0→+∞
=======φ0(x) + i
∫d3~k
(2π)32w~k
(e−ik·(x−y)ρ(k)− eik·(x−y)ρ(−k)
)
x0→+∞=======
∫d3~k
(2π)3/2√
2w~k
(b~ke−ik·x + b†~ke
ik·x) (6.1.34)
where
b~k = a~k +i
(2π)3/2√
2w~kρ(k), (6.1.35)
satisfying[b~k, b
†~k′
] = δ(3)(~k − ~k′). (6.1.36)
As x0 → +∞, the Hamiltonian has the form
: H :=
∫d3~x : H :=
∫d3~kw~kb
†~kb~k, (6.1.37)
and the vacuum energy can be calculated,
Evac = 〈vac| : H : |vac〉
=
∫w~k
d3~k
(2π)32w~k|ρ(k)|2
≡∫w~kdN(~k) =⇒ N ≡
∫dN(~k),
(6.1.38)
where dN(~k): the expectation value of the number of particles with momentum ~k and N :the expectation value of the total number of particles.
98
6.2 Advanced Green function
It is defined in the way
(x +m2)DA(x0 − y0) = −iδ4(x− y),DA(x− y) = 0, x0 > y0 (6.2.1)
in which the past is determined by the future, see the diagrammatic interpretation,
x
t
I
III
IIIII: forward light-cone: time-like region
II: outside light-cone: space-like region
III: backward light-cone: time-like region
where the origin denotes y and the backward-light cone denotes x,
DA(x− y)
= 0 Domain I, II.6= 0 Domain III, x0 < y0.
(6.2.2)
The analytical formulation of the advanced Green function DA(x− y) has the form
DA(x− y) = θ(y0 − x0)[φ(x), φ(y)], (6.2.3)
which can be defined with the contour integral
DA =
∫
CA
d4k
(2π)4
i
k2 −m2e−ik·(x−y), (6.2.4)
with the contour CA is given by
Re(k0)
Im(k0)
−w~k w~k
CA
6.3 The Feynman Propagator DF (x− y)
We use the notation, DF (x−y) to denote the Feynman propagator. It is a Green function
(x +m2)DF (x− y) = −iδ(4)(x− y). (6.3.1)
which has the analytical form given by
DF (x− y) = θ(x0 − y0)D(x− y) + θ(y0 − x0)D(y − x). (6.3.2)
99
so that the Feynman propagator is neither retarded nor advanced Green function and itis basic ingredient in Feynman diagrams for Feynman integrals.
Remark: Various approaches to defining the Feynman propagator.
1 Time-ordering with the step function;
2 Contour integrals;
3 iε→ 0+;
4 Time-ordered product;
5 Wick’s theorem.
6.3.1 Definition with contour integrals
Thm.
DF (x− y) =
∫
C1,C2
d4k
(2π)4
i
k2 −m2e−ik·(x−y) (6.3.3)
where
x0 > y0, clockwise contour C1 on the lower half-plane.x0 < y0, anti-clockwise contour C2 on the upper half-plane.
Proof.In the condition x0 > y0, we perform the contour integral with the clockwise contour C1,
Re(k0)
Im(k0)
−w~kw~k
C1
=⇒Re(k0)
Im(k0)
w~k
because
Im(k0)→ −∞ : e−i(iIm(k0))(x0−y0) = eIm(k0)(x0−y0) → 0. (6.3.4)
With the residue theorem,
∮
C1
dk0
2π
ie−ik0(x0−y0)
(k0 + ω~k)(k0 − ω~k)
=(−2πi)
2ω~k
i
2πe−iω~k(x0−y0) =
e−iω~k(x0−y0)
2ω~k, (6.3.5)
where the pole is k0 = w~k so
DF (x− y)x0>y0
=====
∫d3~k
(2π)3
e−ik·(x−y)
2ω~k= θ(x0 − y0)D(x− y). (6.3.6)
In the condition y0 > x0, we perform the anti-clockwise C2 contour integral,
100
Re(k0)
Im(k0)
−w~kw~k
C2
=⇒Re(k0)
Im(k0)
−w~k
because
Im(k0)→ +∞ : e−i(iIm(k0))(x0−y0) = eIm(k0)(x0−y0) → 0. (6.3.7)
We apply the residue theorem to the above contour integral to obtain
∮
C2
dk0
2π
ie−ik0(x0−y0)
(k0 + ω~k)(k0 − ω~k)
=(2πi)
−2ω~k
i
2πeiω~k(x0−y0) =
eiω~k(x0−y0)
2ω~k, (6.3.8)
with the pole at k0 = −w~k, so
DFy0>x0
=====
∫d3~k
(2π)3
eik·(x−y)
2ω~k= θ(y0 − x0)D(y − x). (6.3.9)
6.3.2 Definition of Feynman propagator with iε→ 0+ prescription
Thm.
DF (x− y) =
∫d4k
(2π)4
i
k2 −m2 + iεe−ik·(x−y), (6.3.10)
where ε→ 0+.
Proof. Poles distribution. Because ε is infinitesimal quantity, we have
[k0 − (ω~k − iε)][k0 + (ω~k − iε)]
=(k0)2 − (ω~k − iε)2
=(k0)2 − ω2~k
+ iε,
(6.3.11)
1
k2 −m2 + iε=
1
k0 − (ω~k − iε)· 1
k0 + (ω~k − iε). (6.3.12)
Re(k0)
Im(k0)
−w~k + iε
w~k − iε
101
6.3.3 Definition of Feynman propagator with time-ordered product
Thm.DF (x− y) = 〈vac|Tφ(x)φ(y)|vac〉 (6.3.13)
where “T” stands for “time-ordering”.
Tφ(x)φ(y) =
φ(x)φ(y) x0 > y0
φ(y)φ(x) x0 < y0 (6.3.14)
Remark: Earliest time on the rightmost;Next to earliest on the next to rightmost;Latest time on the leftmost;Next to latest on the next to leftmost.
Tφ(x1)φ(x2) · · ·φ(xn) =
t1 > t2 > · · · > tn φ(x1)φ(x2) · · ·φ(xn)t2 > t1 > · · · > tn φ(x2)φ(x1) · · ·φ(xn)
......
tn > tn−1 > · · · > t1 φ(xn)φ(xn−1) · · ·φ(x1)
(6.3.15)
Proof.
DF (x− y) =θ(x0 − y0)〈vac|φ(x)φ(y)|vac〉+ θ(y0 − x0)〈vac|φ(y)φ(x)|vac〉=θ(x0 − y0)D(x− y) + θ(y0 − x0)D(y − x).
(6.3.16)
Note 1: φ(x)φ(y) are scalar fields, i.e. spinless, and they obey Bosonic statistics, i.e.inside time ordering, we have
Tφ(x)φ(y) = Tφ(y)φ(x). (6.3.17)
Note 2: x0 = y0, ~x 6= ~y is space-like region, we define
θ(x0 − y0) + θ(y0 − x0) = 1. (6.3.18)
DF (x− y)|tx=ty= 〈vac|φ(x)φ(y)|vac〉. (6.3.19)
Note 3: x0 = y0, ~x = ~y, Feynman propagator is ill-defined.
DF (x− y) = DF (0) = 〈vac|φ(x)φ(y)|vac〉
=
∫d3~k
(2π)3
1
2w~k→ +∞.
(6.3.20)
Note 4:
x0 > y0:
y
x
Amplitude of propagation
of particle from y to x.
x0 < y0:
x
y
Amplitude of propagation
of particle from x to y.
Note 5: DF (x− y) is defined for free theories, not for interaction theories.
102
6.3.4 Definition of Feynman propagator with contractions
Thm.
DF (x− y) = φ(x)φ(y)
= Tφ(x)φ(y)− : φ(x)φ(y) :(6.3.21)
which is an example of Wick’s theorem.
〈vac|DF (x− y)|vac〉 = 〈vac|Tφ(x)φ(y)|vac〉 − 〈vac| : φ(x)φ(y) : |vac〉, (6.3.22)
=⇒ DF (x− y) = 〈vac|Tφ(x)φ(y)|vac〉. (6.3.23)
Proof.
φ(x) = φ+(x) + φ−(x) (6.3.24)
where φ+(x) is positive frequency part associated with annihilation operator and φ+(x)is negative frequency part associated with creation operator.
Normal ordering:
: φ(x)φ(y) := : (φ+(x) + φ−(x))(φ+(y) + φ−(y)) :
=φ+(x)φ+(y) + φ−(x)φ−(y) + φ−(x)φ+(y) + φ−(y)φ+(x).(6.3.25)
Time ordering:
• x0 > y0
T [φ(x)φ(y)] =φ(x)φ(y)
=(φ+(x) + φ−(x))(φ+(y) + φ−(y))
=φ+(x)φ+(y) + φ−(x)φ−(y) + φ−(x)φ+(y) + φ+(x)φ−(y).
(6.3.26)
So we have
T [φ(x)φ(y)]− : φ(x)φ(y) :x0>y0
===== φ+(x)φ−(y)− φ−(y)φ+(x)
= [φ+(x), φ−(y)]
= 〈vac|[φ+(x), φ−(y)]|vac〉= 〈vac|φ(x)φ(y)|vac〉= θ(x0 − y0)D(x− y).
(6.3.27)
• x0 < y0
T [φ(x)φ(y)]− : φ(x)φ(y) := θ(y0 − x0)D(y − x). (6.3.28)
From above all, we have the conclusions
T [φ(x)φ(y)] =: φ(x)φ(y) : +DF (x− y); (6.3.29)
〈vac|T [φ(x)φ(y)]|vac〉 = DF (x− y). (6.3.30)
103
6.4 Wick’s theorem
6.4.1 Theorem
Time-ordered product T [φ(x1)φ(x2) · · ·φ(xn)] can be expressed as a summation of prod-ucts of Feynman propagator (or contractions) and normal-ordered products.
For simplicity, change the notation
φi = φ(xi). (6.4.1)
T (φ1φ2 · · ·φn) = : φ1φ2 · · ·φn : no-contraction
+φ1φ2 : φ3 · · ·φn :+C2
n terms with one contraction
only one contraction
+φ1φ2 φ3φ4 : φ5 · · ·φn :+1
2C2nC
2n−2 terms with two contractions
two contractions
+ · · ·+all possible terms with [n/2]− 1 contractions
+φ1φ2 φ3φ4 · · ·φn−1φn+ · · ·
n is even with n/2 contractions
+φ1φ2 φ3φ4 · · ·φn−2φn−1 φn+ · · ·
n is odd with (n− 1)/2 contractions
(6.4.2)
Proof. Induction proof.
1. n = 1, trivial, Tφ(x) =: φ(x) :.
2. n = 2, Feynman propagator.
T (φ1φ2) =: φ1φ2 : +φ1φ2 . (6.4.3)
3. Assume Wick’s theorem is OK for n− 1, then denote
T (φ1φ2 · · ·φn−1) = w(φ1φ2 · · ·φn−1). (6.4.4)
Suppose tn = min(t1, t2, · · · , tn).
T (φ1φ2 · · ·φn) =w(φ1φ2 · · ·φn−1)φn
=T (φ1φ2 · · ·φn−1)(φ+n + φ−n )
=T (φ1φ2 · · ·φn−1)φ+n + φ−n T (φ1φ2 · · ·φn−1)
+ [T (φ1φ2 · · ·φn−1), φ−n ]
(6.4.5)
where the term T (φ1φ2 · · ·φn−1)φ+n + φ−n T (φ1φ2 · · ·φn−1) are all possible contractions
without φn and the term [T (φ1φ2 · · ·φn−1), φ−n ] are all possible contractions with φn.
104
6.4.2 Example for Wick’s theorem n = 3
Supposet3 = min(t1, t2, t3), (6.4.6)
so we have
T (φ1φ2φ3) = T (φ1φ2)φ3
= T (φ1φ2)φ+3 + φ−3 T (φ1φ2)︸ ︷︷ ︸
I
+ [T (φ1φ2), φ−3 ]︸ ︷︷ ︸II
. (6.4.7)
I =T (φ1φ2)φ+3 + φ−3 T (φ1φ2)
= : φ1φ2 : φ+3 + φ1φ2 φ
+3 + φ−3 : φ1φ2 : +φ1φ2 φ
−3
=φ1φ2 φ3+ : φ1φ2φ+3 : + : φ−3 φ1φ2 :
=φ1φ2 φ3+ : φ1φ2φ3 :,
(6.4.8)
II =[Tφ1φ2, φ−3 ]
=[: φ1φ2 :, φ−3 ]
=[φ+1 φ
+2 + φ−1 φ
−2 + φ−1 φ
+2 + φ−2 φ
+1 , φ
−3 ]
=[φ+1 φ
+2 , φ
−3 ] + [φ−1 φ
+2 , φ
−3 ] + [φ−2 φ
+1 , φ
−3 ]
=[φ+1 , φ
−3 ]φ+
2 + [φ+2 , φ
−3 ]φ+
1 + [φ+2 , φ
−3 ]φ−1 + [φ+
1 , φ−3 ]φ−2
=φ1φ3 φ2 + φ2φ3 φ1,
(6.4.9)
From above all, we verify the theorem.
e.g.1.
T (φ1φ2φ3) =: φ1φ2φ3 : +φ1φ2φ3 +φ1φ2φ3 +φ1φ2φ3 . (6.4.10)
e.g.2.
T (φ1φ2φ3φ4) =: φ1φ2φ3φ4 : +φ1φ2 : φ3φ4 : + : φ1φ2 : φ3φ4
+ φ1φ4 : φ2φ3 : + : φ1φ4 : φ2φ3 +φ1φ3 : φ2φ4 : + : φ1φ3 : φ2φ4
+ φ1φ2 φ3φ4 +φ1φ3 φ2φ4 +φ1φ4 φ2φ3 .(6.4.11)
one-contraction: C24 = 6.
two-contractions: 12C
24C
22 = 3.
Remarks:
1).〈vac|T (φ1φ2φ3)|vac〉 = 0. (6.4.12)
〈vac|T (φ1φ2φ3φ4)|vac〉 =
1 2
3 4
1 2
3 4
1 2
3 4
+ +
(6.4.13)
105
φiφj =i j (6.4.14)
2).
: φ1φ2φ3φ4 : =: φ1φ2 : φ3φ4 (6.4.15)
where : φ1φ2 : is an operator, and φ3φ4 is a number.
3). Lemma. [: φ1φ2 · · ·φn :, φ
(−)n+1
]=:[φ1φ2 · · ·φn, φ(−)
n+1
]: . (6.4.16)
4). For different fields,
[φ(x1), ψ(x2)] = 0, (6.4.17)
φ(x1)ψ(x2) = 0. (6.4.18)
5). Normal ordering & linear combination.
: aa† + a†a := 2a†a, (6.4.19)
=⇒: aa† − a†a =: aa† − aa† := 0, (6.4.20)
: Number := 0. (6.4.21)
6.5 Interaction Picture (Dirac Picture)
6.5.1 Motivation
Interaction Field Theory.
H = H0 +Hint (6.5.1)
where H is total Hamiltonian, H0 is free particle’s Hamiltonian, and Hint is interactionpart.
Fact 1. EoM of interacting fields can not be exactly solvable in most cases.
Fact 2. EoM of free fields can be exactly solved and creation and annihilation operatorscan be well-defined.
Way out:
Choose a suitable framework in which field operators have evolution of free operators.
106
6.5.2 Dirac picture
A picture defines a choice of operators and state vectors to preserve matrix elements orprobability amplitude.
e.g. Shrodinger picture.
OS(t) = OS(0),
i ddt |ψ(t)〉S = H|ψ(t)〉S .(6.5.2)
where H is total Hamiltonian.
Interaction picture, OI(t), |ψ(t)〉I .
Constraint 1.
S〈ψ(t)|OS |ψ(t)〉S =I 〈ψ(t)|OI(t)|ψ(t)〉I . (6.5.3)
Constraint 2.
OS(0) = OI(0), (6.5.4)
|ψ(0)〉S = |ψ(0)〉I . (6.5.5)
Constraint 3.
OI(t) = eiH0tOSe−iH0t, (6.5.6)
|ψ(t)〉I = eiH0t|ψ(t)〉S , (6.5.7)
=⇒ id
dtOI(t) = [OI(t), H0] . (6.5.8)
OI(t) is determined by free Hamiltonian.
Thm.
id
dt|ψ(t)〉I = HI(t)|ψ(t)〉I (6.5.9)
with HI(t) = eiH0tHinte−iH0t.
Proof.
id
dt|ψ(t)〉I = i
d
dt(eiH0t|ψ(t)〉S)
= −H0eiH0t|ψ(t)〉S + eiH0t(i
d
dt|ψ(t)〉S)
= −H0eiH0t|ψ(t)〉S + eiH0t(H0 +Hint)|ψ(t)〉S
= −H0eiH0t|ψ(t)〉S + eiH0t(H0 +Hint)e
−iH0t|ψ(t)〉I= eiH0tHinte
−iH0t|ψ(t)〉I= HI(t)|ψ(t)〉I .
(6.5.10)
Remarks:
Hint = 0, (6.5.11)
=⇒ |ψ(t)〉I = |ψ(t)〉H . (6.5.12)
107
6.6 Dyson’s formula
Problem to solve
id
dt|ψ(t)〉I = HI |ψ(t)〉. (6.6.1)
6.6.1 Unitary evolution operator|ψ(t)〉I = UI(t, t0)|ψ(t0)〉I ,|ψ(t0)〉I = |ψ(t0)〉S . (6.6.2)
Change the problem into
i ddtUI(t, t0) = HI(t)UI(t, t0),UI(t0, t0) = 1.
(6.6.3)
Note: scattering operator (S-matrix).
S = U(+∞,−∞). (6.6.4)
6.6.2 Time dependent permutation theory
Recursive solution
UI(t, t0) = 1− i∫ t
t0
dt1HI(t1)UI(t1, t0). (6.6.5)
First order approximation
U(1)I (t, t0) = 1− i
∫ t
t0
dt1HI(t1), (6.6.6)
U(0)I (t1, t0) = 1. (6.6.7)
Second order approximation
U(2)I = 1− i
∫ t
t0
dt1HI(t1)U(1)I (t1, t0)
= 1 + (−i)∫ t
t0
dt1HI(t1) + (−i)2
∫ t
t0
dt1
∫ t1
t0
dt2HI(t1)HI(t2).
(6.6.8)
N -order approximation
U(n)I (t, t0) = 1 +
N∑
n=1
(−i)n∫ t
t0
dt1
∫ t1
t0
dt2 · · ·∫ tN−1
t0
dtnHI(t1)HI(t2) · · ·HI(tN ). (6.6.9)
t0 tn tn−1 tt1t2· · · (6.6.10)
N → +∞, tn → t0, we have U(tN , t0) = 1.When N → +∞, we have exact solution.
UI(Tf , Ti) = 1 +
+∞∑
n=1
(−i)n∫ Tf
Ti
dt1
∫ t1
Ti
dt2 · · ·∫ tN−1
Ti
dtnHI(t1)HI(t2) · · ·HI(tN ). (6.6.11)
108
6.6.3 Dyson’s formula
Thm.
UI(Tf , Ti) = T exp(−i∫ Tf
Ti
dtHI(t))
= 1 ++∞∑
n=1
(−i)nn!
∫ Tf
Ti
dt1
∫ Tf
Ti
dt2 · · ·∫ Tf
Ti
dtnT (HI(t1)HI(t2) · · ·HI(tN )),
(6.6.12)
where “T” denotes time ordering.
Remark:
1). Inside time ordering, operators are commutative.
T (HI(t1)HI(t2)) = T (HI(t2)HI(t1)). (6.6.13)
2). T (HI(t1)HI(t2)) is a symmetric function under t1 ↔ t2.
T (HI(t1)HI(t2) · · ·HI(tn)) is a symmetric function under permutation (t1, t2, · · · , tn),#SN = N !.
Proof. order by order proof.
1). n = 0,
U(0)I (Tf , Ti) = 1. (6.6.14)
2). n = 1,
U(1)I (Tf , Ti) = −i
∫ Tf
Ti
dt1HI(t1). (6.6.15)
No time ordering.
3). n = 2,
U(2)I (Tf , Ti) = (−i)2
∫ Tf
Ti
dt1
∫ t1
Ti
dt2HI(t1)HI(t2)
t1>t2===== (−i)2
∫ Tf
Ti
dt1
∫ t1
Ti
dt2T (HI(t1)HI(t2))
= (−i)2 1
2!
∫ Tf
Ti
dt1
∫ t1
Ti
dt2
︸ ︷︷ ︸I
(2)1
+
∫ Tf
Ti
dt2
∫ t2
Ti
dt1
︸ ︷︷ ︸I
(2)2
T (HI(t1)HI(t2)).
(6.6.16)
where 12! is normalization factor.
Integration domain I(2)1 and I
(2)2 .
109
t1
t2
Ti Tf
Ti
Tf
I(1)2
t1
t2
Ti Tf
Ti
Tf
I(2)2
Integration domain I(2) = I(2)1
⋃I
(2)2 .
t1
t2
Ti Tf
Ti
Tf
I2
U(2)I (Tf , Ti) =
(−i)2
2!
∫ Tf
Ti
dt1
∫ Tf
Ti
dt2T (HI(t1)HI(t2)). (6.6.17)
4). N -th order,
U(n)I (Tf , Ti) = (−i)N
∫ Tf
Ti
dt1
∫ t1
Ti
dt2 · · ·∫ tN−1
Ti
dtnHI(t1)HI(t2) · · ·HI(tn)
t1>t2>···>tN========== (−i)N∫ Tf
Ti
dt1
∫ t1
Ti
dt2 · · ·∫ tn−1
Ti
dtN
︸ ︷︷ ︸I
(N)1
T (HI(t1)HI(t2) · · ·HI(tn))
=(−i)4
N !
∫ ∫· · ·∫
︸ ︷︷ ︸I
(N)1
+
∫ ∫· · ·∫
︸ ︷︷ ︸I
(N)2
+ · · ·+∫ ∫
· · ·∫
︸ ︷︷ ︸I
(N)N !
× T (HI(t1)HI(t2) · · ·HI(tn))
=(−i)4
N !
∫ Tf
Ti
dt1
∫ Tf
Ti
dt2 · · ·∫ Tf
Ti
dtN
︸ ︷︷ ︸I(N)
T (HI(t1)HI(t2) · · ·HI(tn)).
(6.6.18)
110
6.6.4 Examples
Ex.1. When Hamiltonian at different time is commutative,
HI(t1)HI(t2) = HI(t2)HI(t1). (6.6.19)
Time ordering becomes unnecessary.
UI(Tf , Ti) = exp
[−i∫ Tf
Ti
dt′HI(t′)]. (6.6.20)
Note: micro-causality.[HI(t, ~x), HI(t, ~y)] = 0, (6.6.21)
where HI(t, ~x) =∫d3~xH(t, ~x) and HI(t, ~y) =
∫d3~yH(t, ~y). And it suggests that it
in space-like region.
Ex.2. When HI(t) is independent of time, i.e. HI(t) = HI , we have
UI(Tf , Ti) = exp(−i(Tf − Ti)HI). (6.6.22)
111
Lecture 7 Scattering Matrix, Cross Section &
Decay Width
About Lecture V:
• [Luke] P.P. 65-105;
• [Tong] P.P. 48-75;
• [Coleman] P.P. 70-139;
• [PS] P.P. 70-115;
• [Zhou] P.P. 127-170.
7.1 Scattering matrix (operator)
7.1.1 Ideal model for scattering process
H = H0 + f(t)Hint
where f(t) is a ”turning on and off” function, vanishing in the far past and far future.
4 4T
00
t
f(t)
T → +∞, 4→ +∞, 4T→ 0.
Note 1. Before and after scattering, only eigenstates of H0 are allowed.
Note 2. There is one-to-one correspondence between asymptotic eigenstates of H andeigenstates of H0.
f(−∞) = 0t = −∞ Black box
f(t) = 1 f(+∞) = 0t = +∞
S-matrix
Scatteringarea
T
|i〉
...
|f〉
112
7.1.2 Scattering operator S
H = H0 +Hint, (7.1.1)
id
dt|ψ(t)〉I = HI(t)|ψ(t)〉I . (7.1.2)
In the far past (without subscription I): |i〉 = |ψ(−∞)〉 is eigenstate of H0.In the far future:
|ψ(+∞) = U(+∞,−∞)|ψ(−∞)〉= S|ψ(−∞) = S|i〉, (7.1.3)
where S = U(+∞,−∞) is scattering operator matrix (S-matrix).The probability of finding |f〉, eigenstate of H0, in t = +∞,
〈f |ψ(−∞)〉 = 〈f |S|i〉 = Sfi (7.1.4)
where Sfi is S-matrix element.
7.2 Calculation of two-“nucleon” scattering amplitude
7.2.1 Model
“Nucleon”-meson theory, which is also named as Scalar Yukawa theory.The lagrangian density
L = Lφ + Lψ − gψ†ψφ. (7.2.1)
ψ: complex scalar field.
“nucleon”=pseudo-nucleon = spinless with charge.
Note. proton and anti-proton are nucleons with spin 1/2.
φ: real scalar fields.meson = spinless without charge.
g: coupling constant. [ψ] = [φ] = [g] = 1 in mass dimension.
EoM. + µ2φ = −gψ†ψ+m2ψ = −gψ†φ
(7.2.2)
In Dirac’s Picture.
ψ(x) =
∫d3~k
(2π)32ω~k(βke
−ik·x + γ†keik·x). (7.2.3)
βk = (2π)3/2√
2ωkb~k,
γ~k = (2π)3/2√
2ω~kc~k.(7.2.4)
φ(x) =
∫d3~k
(2π)3/2√
2ω~k(a~ke
−ik·x + a†~keik·x). (7.2.5)
113
7.2.2 Two-“nucleon” scattering matrix
N(P1) +N(P2) −→ N(P ′1) +N(P ′2). (7.2.6)
p1
p2
p′1
p′2
t
pi: momentum assignment.
Arrow on line: charge denoting nucleon.
Time arrow: LHS −→ RHS.
Incoming two-particle states.
|i〉 = |p1, p2〉 = β†(p1)β†(p2)|vac〉. (7.2.7)
Outgoing two-particle states.
|f〉 = |p′1, p′2〉 = β†(p′1)β†(p′2)|vac〉. (7.2.8)
Note: Assume 〈f |i〉 = 0.
Scattering operator:
S = T
[exp(−i
∫ +∞
−∞d4xgψ†(x)ψ(x)φ(x))
]. (7.2.9)
Calculation: 〈f |S|i〉.Up to order of g.
g(0) = 1, 〈f |S(0)|i〉.g(1) = g, 〈f |S(1)|i〉.g(i) = gi, 〈f |S(i)|i〉.
7.2.3 Computing methods
1. Application of Wick’s theorem (rigorous & complicate).
2. Feynma diagrams & Feynman rules (intuitive & guesswork).
3. 1 & 2.
114
7.3 Feynman diagrams
7.3.1 Main theorem
The algebraic terms in 〈f |S−1|i〉 are one to one correspondence with Feynman diagrams,namely
〈f |S − 1|i〉 =(2π)4δ(4)(PI − PF )iA=Sum of all connected, amputated Feynman diagrams
with PI incoming and PF outgoing
(7.3.1)
wherePI =
∑
i
pi, PF =∑
f
pf . (7.3.2)
Remark 1. Connected diagrams have no disjoint sub-diagrams.
Remark 2. External lines are amputated, because of no contribution.
=
7.3.2 Correspondence between algebra and diagrams
Feynman diagrams 〈f |S − 1|i〉External lines |i〉, |f〉
Vertices Hint
Internal lines Contractions (Feynman propagator)
Remark.
Feynmandiagrams
⇐⇒FRA
FRB
FRC
⇐⇒Feynmanintegrals
elementS-matrix
Mapping rules
FRA. Feynman rules in coordinate space-time.
FRB. FR in momentum space.
FRC . Simplified FR in moment space.
7.3.3 Conventions of drawing external lines
1.
• LHS for initial state.
• RHS for final state.
115
• Middle part for interaction.
• Time from left to right.
p1
p2
p′1
p′2
t+∞−∞
|f〉|i〉
2. Solid lines for “nucleons”.
ψ or ψ†
Dotted lines for “meson”.
φ
3. One line for one-particle.
4. Each line with directed momentum.
p1 p′1
k
5. Arrows on the solid lines for charge of “nucleon”. No arrows on dotted lines for nocharge of meson.
Arrow Initial Final(charge) (incoming) (outgoing)
Q = −1, anti-nucleon Q = +1, nucleonOutgoing arrow
N N
Q = +1, nucleon Q = −1, anti-nucleonIncoming arrow
N N
No arrow Q = 0, meson Q=0, meson
incoming
NQ = 1
outcoming
NQ = −1
outcoming
N Q = 1
incoming
N Q = −1
N +N −→ N +NQ+Q = 0 = Q+Q
116
N
N
φ
φ −→ N +N
0 = Q+Q
p1
N
p2
N
p′1
N
p′2
N
N +N −→ N +NQ+Q = −2 = Q+Q
p1
φ
p2
φ
p′1
φ
p′2
φ
φ+ φ −→ φ+ φ
0 + 0 = 0 + 0
Rough rules
Same Opposite
Charge arrow +1 −1
Momentum N N
7.3.4 Conventions on diagrams vertices & internal lines
ψ
ψ†
φ
xVertex: −ig
∫d4xψ†(x)ψ(x)φ(x)
Internal lines (contraction)
x1 x2: Dψ
F (x1 − x2) = ψ(x1)ψ†(x2) .(7.3.3)
x1 x2: Dφ
F (x1 − x2) = φ(x1)φ(x2) .(7.3.4)
The solid lines do not specify online arrow, which depends on the external lines.
7.3.5 Example
N(p1) +N(p2) −→ N(p′1) +N(p′2). (7.3.5)
〈f |S(0)|i〉 = 〈f |1|i〉 = 0. (7.3.6)
117
p′1p1
p2 p′2
〈f |S(1)|i〉 =No enough internal lines toform a connected diagram
=
= 0
(7.3.7)
p′1p1
p2 p′2
x1
x2
N
N
N
N
〈f |S(2)|i〉 = + (p′1 ←→ p′2)
(7.3.8)
7.4 Feynman rules in coordinate space-time (FRA)
1. Draw all possible connected diagrams.
2. For each external line
xp
incoming−→ ψ(x)|p〉 ≡ 〈vac|ψ(x)|p〉 = e−ip·x
(7.4.1)
xp′
outgoing−→ 〈p′|ψ(x) ≡ 〈p′|ψ(x)|vac〉 = eip
′·x(7.4.2)
3. For internal line
x
ψ†
ψ
−→ (−ig)∫d4x
(7.4.3)
4. For internal line
x1 x2−→ φ(x1)φ(x2)
(7.4.4)
118
x1 x2−→ ψ(x1)ψ(x2)
(7.4.5)
5. Divided by symmetric factor1
S=
1
n!×m (7.4.6)
where 1/n! is from Taylor expansion of S matrix, and m is the number of same Feynmandiagrams which is due to permutation of vertexes or Bose statistics.
Example. N +N −→ N +N .
x1
x2
p′1p1
p2 p′2
〈f |S(2)|i〉 = 1S [ +(p′1 ↔ p′2)]
(7.4.7)
〈f |S(2)|i〉 =(−ig)
∫d4x1(−ig)
∫d4x2e
−ip1·x1e−ip2·x2eip′1·x1eip
′2·x2 φ(x1)φ(x2)
+ (p′1 ↔ p′2)
=(−ig)2
∫d4x1d
4x2
(e−i(p1−p′1)·x1e−i(p2−p′2)·x2 + (p′1 ↔ p′2)
)φ(x1)φ(x2) .
(7.4.8)
Note: 1S = 1
2! × 2!.
Algebraic proof of this example.
Proof.
|i〉 = |p1, p2〉 = (2π)3√
2w~p1
√2w~p2
b†(~p1)b†(~p2)|vac〉; (7.4.9)
|f〉 = |p′1, p′2〉 = (2π)3√
2w~p′1
√2w~p′2b
†(~p′1)b†(~p′2)|vac〉. (7.4.10)
S = T exp
[−i∫ +∞
−∞d4xgψ†(x)ψ(x)φ(x)
]. (7.4.11)
Compute 〈f |S|i〉.
a. Zero-th order of g.
〈f |S(0)|i〉 = 〈f |i〉 = 0 (7.4.12)
which is from experiment setup.
119
b. First order of g.
S(1) = −ig∫d4xψ†(x)ψ(x)φ(x). (7.4.13)
〈f |S(1)|i〉 = −ig∫d4x〈f |ψ†(x)ψ(x)φ(x)|i〉 = 0. (7.4.14)
Analysis|i〉 = β†β†|vac〉; 〈f | = 〈vac|ββ. (7.4.15)
Thus we need ββ to cancel β†β† and β†β† to cancel ββ. β†β†ββ are needed.
ψ ∼ β + γ†; ψ† ∼ β† + γ. (7.4.16)
ψ†ψ ∼ β†β + γβ + β†γ† + γγ†. (7.4.17)
〈f |S(1)|i〉 ∼ 〈f |β†β|i〉 = 0. (7.4.18)
c. Second order of g.
〈f |S(2)|i〉 =(−ig)2
2!
∫d4x1d
4x2〈f |T (ψ†(x1)ψ(x1)φ(x1)ψ†(x2)ψ(x2)φ(x2))|i〉
=(−ig)2
2!
∫d4x1d
4x2〈f | : ψ†(x1)ψ(x1)ψ†(x2)ψ(x2) : |i〉φ(x1)φ(x2) .
(7.4.19)
Trick 1. Apply the Wick’s theorem.
Trick 2. Insert |vac〉〈vac| to replace normal ordering.
〈f |S(2)|i〉 =(−ig)2
2!
∫d4x1d
4x2〈f |ψ†(x1)ψ(x1)|vac〉〈vac|ψ†(x2)ψ(x2)|i〉
× φ(x1)φ(x2) .
(7.4.20)
Trick 3. Contraction between fields and states.
〈vac|ψ(x1)ψ(x2)|i〉=〈vac|ψ(x1)ψ(x2)|p1, p2〉=〈vac|ψ(x1)|p1〉〈vac|ψ(x2)|p2〉+ 〈vac|ψ(x1)|p2〉〈vac|ψ(x2)|p1〉
≡ψ(x1)|p1〉ψ(x2)|p2〉+(p1 ↔ p2).
(7.4.21)
e.g.
ψ(x1)|p1〉 =
∫d3~k
(2π)3
e−ik·x
2w~k〈vac|β(k)β†(p1)|vac〉. (7.4.22)
〈vac|ψ(x1)ψ(x2)|i〉 = e−ip1·x1−ip2·x2 + (p1 ↔ p2). (7.4.23)
〈f |ψ†(x1)ψ†(x2)|vac〉 =〈p′1|ψ†(x1)|vac〉〈p′2|ψ†(x2)|vac〉+ (p′1 ↔ p′2)
=eip′1·x1+ip′2·x2 + (p′1 ↔ p′2).
(7.4.24)
120
Trick 4. Symmetry under interchange x1 ↔ x2.
For the interchange x1 ↔ x2, we have
d4x1d4x2 = d4x2d
4x1; (7.4.25)
φ(x1)φ(x2) = φ(x2)φ(x1) . (7.4.26)
Thus
〈f |S(2)|i〉 =(−ig)2
2!
∫d4x1d
4x2 φ(x1)φ(x2)
× (e−ip1·x1−ip2·x2 + (x1 ↔ x2))(eip′1·x1+ip′2·x2 + (p′1 ↔ p′2))
=(−ig)2
∫d4x1d
4x2 φ(x1)φ(x2)
× e−ip1·x1−ip2·x2(eip′1·x1+ip′2·x2 + (p′1 ↔ p′2))
(7.4.27)
7.5 Feynman rules in momentum space (FRB)
7.5.1 Calculation hint
φ(x2)φ(x1) =
∫d4k
(2π)4
i
k2 −m2 + iεeik·(x1−x2). (7.5.1)
〈f |S(2)|i〉 =(−ig)2
∫d4x1d
4x2
∫d4k
(2π)4
i
k2 −m2 + iεeik·(x1−x2)
× e−ip1·x1−ip2·x2(eip′1·x1+ip′2·x2 + (p′1 ↔ p′2))
=(−ig)2
∫d4x1d
4x2
∫d4k
(2π)4
i
k2 −m2 + iε
× (ei(p′1+k−p1)·x1ei(p
′2−k−p2)·x2 + (p′1 ↔ p′2))
=(−ig)2
∫d4x1d
4x2
∫d4k
(2π)4
i
k2 −m2 + iε
× ((2π)4δ(4)(p′1 + k − p1)(2π)4δ(4)(p′2 − p2 − k) + (p′1 ↔ p′2)).
(7.5.2)
k
p′1p1
p2 p′2
〈f |S(2)|i〉 = +(p′1 ↔ p′2)
(7.5.3)
121
7.5.2 Feynman rules in momentum space
1. Draw all possible connected diagrams.
2. For each external line, assign the factor 1.
Note: External lines have no contribution, so they are amputated.
3. For each vertex, assign the factor (−ig)(2π)4δ(4)(p′1 + k − p1).
k
p′1p1
−→ (−ig)(2π)4δ(4)(p′1 + k − p1)
(7.5.4)
4. For internal line
k−→
∫d4k
(2π)4i
k2−m2+iε (7.5.5)
5. Symmetry factor.
7.6 Feynman rules C
7.6.1 Calculation hint
Integrate over internal momentum k.
〈f |S(2)|i〉 =(2π)4δ(4)(p1 + p2 − p′1 − p′2)(−ig)2
(i
(p1 − p′1)2 − µ2+
i
(p2 − p′2)2 − µ2
)
+ (p′1 ↔ p′2).
(7.6.1)
p1 − p′1
p′1p1
p2 p′2
〈f |S(2)|i〉 = +(p′1 ↔ p′2)
(7.6.2)
7.6.2 Feynman rules C (simplified FRB)
41 Draw all possible connected Feynman diagrams with momentum assign.
41′ Impose energy momentum at each vertex.
41′′ Write down the factor for the total energy-momentum conservation.
(2π)4δ(4)(PF − PI)
42 For each external line, assign “1”.
122
43 For each vertex, assign “−ig”.
44 For momentum determined internal line
p1 − p′1
−→ i(p1−p′1)2−µ2
(7.6.3)
44′ For momentum in-determined internal line
k−→
∫d4k
(2π)4i
k2−µ2+iε (7.6.4)
45 Symmetry factor.
7.7 Application of FRC
e.g.1. Decay of a meson into “nucleon”-“anti-nucleon”.
φ −→ N +N. (7.7.1)
To compute 〈NN |S|φ〉.
N
Np1
p2
p
−→ −igS(1):
(7.7.2)
In center of mass frame, ~p = 0 and ~p1 + ~p2 = 0.
〈NN |S(1)|φ〉 = (2π)4δ(4)(p1 + p2 − p)(−ig). (7.7.3)
Remark 1: Feynman diagrams.
1. Diagrammatic representation tool of S-matrix element.
2. (Feynman) Diagrammatic movement of particles in space time. (physics)
Remark 2: Feynman rules.
1. FRA: in coordinate space-time.
2. FRB: integrate out vertex.
3. FRC : integrate out internal momentum.
123
e.g.2.
φ(p1) + φ(p2) −→ φ(p′1) + φ(p′2). (7.7.4)
To compute 〈φφ|S|φφ〉, at least fourth order is non-zero.
〈φφ|S(4)|φφ〉 6= 0. (7.7.5)
k + p1
k − p2
k k + p1 − p′1
p′1p1
p2 p′2 (7.7.6)
iA(4) =(−ig)4
∫d4k
(2π)4
i
(k2 −m2 + iε)
i
((k2 + p1)2 −m2 + iε)
× i
((k + p1 − p′1)2 −m2 + iε)
i
((k − p2)2 −m2 + iε)
(7.7.7)
Remark:1
S=
1
n!×m. (7.7.8)
In this example, n = 4 and m = 4!, which is due to the permutations of four vertexesx1, x2, x3 and x4.
x2x1
x3 x4
e.g.3.
N(p1) +N(p2) −→ N(p′1) +N(p′2). (7.7.9)
In second order O(g2).
N N
N N
p1 − p′1
p′1p1
p2 p′2
〈f |S(2)|i〉 = + p1 + p2
p′1p1
p′2p2
NN
NN
(7.7.10)
124
iA(2) = (−ig)2
[i
(p1 − p′1)2 − µ2+
i
(p1 + p2)2 − µ2
]. (7.7.11)
The symmetry factor S is
S =1
2!× 2! = 1. (7.7.12)
e.g.4.N(p1) +N(p2) −→ φ(p′1) + φ(p′2). (7.7.13)
p1 − p′1
p′1p1
p2 p′2
N φ
N φ
〈f |S(2)|i〉 = + (p′1 ↔ p′2)
(7.7.14)
iA(2) = (−ig)2
[i
(p1 − p′1)2 −m2+
i
(p1 − p′2)2 −m2
]. (7.7.15)
e.g.5.N(p1) + φ(p2) −→ N(p′1) + φ(p′2). (7.7.16)
N φ
φ N
p1 − p′2
p′2p1
p2 p′1
〈f |S(2)|i〉 = + p1 + p2
p′2p1
p′1p2
φN
Nφ
(7.7.17)
iA(2) = (−ig)2
[i
(p1 − p′2)2 −m2+
i
(p1 + p2)2 −m2
]. (7.7.18)
Note: no iε term in the result, because we do not have integrals.
Remark: about two-particle scattering at order O(g2).
1). Virtual particle on internal line is meson.
N +N −→ N +N ; (7.7.19)
N +N −→ N +N ; (7.7.20)
N +N −→ N +N. (7.7.21)
2). Virtual particle on internal line is nucleon.
N +N −→ φ+ φ; (7.7.22)
N + φ −→ N + φ; (7.7.23)
N + φ −→ N + φ. (7.7.24)
125
Note: the charge conservation due to U(1) invariance favours the above process, butforbids
N +N −→ φ+ φ. (7.7.25)
7.8 Remarks on Feynman propagator
1). Internal lines k2 6= m2, due to −∞ < kµ < +∞.
The Feynman propagator represents the propagation of a virtual particle with k2 6= m2.
DF (x− y) =
y
x
2).
DF (x− y) is invariant under
Lorentz transformation DF (Λ(x− y));space-time translation DF (x+ a− y − a);space-time reflection DF (y − x).
Hence Feynman integrals also have such symmetries.
Note: Hamiltonian formulation of QFT is not explicitly Lorentz invariant, but Feyn-man rules with Feynman diagrams define an explicitly Lorentz invariant perturbativeQFT.
7.9 Cross sections & decay widths – measurable quantitiesin high energy physics
Question: Afi ∝ δ(4)(PF − PI).
• Scattering amplitude ∝ δ(4)(PF − PI) =⇒∫δ = 1;
• Scattering probability ∝ (δ(4)(PF − PI))2 =⇒∫δ2 =∞.
Answer 1. Take the box normalization.
Answer 2. Take the wave-packet.
7.9.1 QFT in a box with volume V = L3
L
L
L
126
Periodic boundary condition:
p1 =2π
Ln1, p2 =
2π
Ln2, p3 =
2π
Ln3. (7.9.1)
Lemma 1. ∑
~p
=V
(2π)3
∫d3~p. (7.9.2)
Proof.
4p14p24p3 =(2π)3
L34n14n24n3, (7.9.3)
=⇒∫dp1dp2dp3 =
(2π)3
L3
∑
~p
. (7.9.4)
Lemma 2.
limV→+∞
V
(2π)3δ~pi,~pj = δ(3)(~pi − ~pj), (7.9.5)
where δ~pi,~pj is Kronecker delta function
δ~pi,~pj =
1, ~pi = ~pj0, ~pi 6= ~pj
. (7.9.6)
Proof. ∑
~pj
δ~pi,~pjf(~pj) = f(~pi), (7.9.7)
∫d3~pδ(3)(~pi − ~pj)f(~pj) = f(~pi) =
(2π)3
V
∑
~pj
δ(3)(~pi − ~pj)f(~pj), (7.9.8)
=⇒ δ~pi,~pj =(2π)3
Vδ(3)(~pi − ~pj). (7.9.9)
Lemma 3.
limV,T→+∞
[δ
(4)V T (p− p′)
]2=
V T
(2π)4δ(4)(p− p′), (7.9.10)
where V is space box, and T is time box.
L
L
L T2
−T2
V : T :
space box time box
127
Proof.V
(2π)3δ~p,~p′ = δ(3)(~p− ~p′). (7.9.11)
Square itV 2
(2π)6δ~p,~p′ =
[δ(3)(~p− ~p′)
]2, (7.9.12)
due toδ~p,~p′δ~p,~p′ = δ~p,~p′ . (7.9.13)
Replace 7.9.5 in the former formula,
V
(2π)3δ(3)(~p− ~p′) =
[δ(3)(~p− ~p′)
]2. (7.9.14)
SimilarlyT
(2π)δ(p0 − p′0) =
[δ(p0 − p′0)
]2. (7.9.15)
=⇒ V T
(2π)4δ(4)(p− p′) =
[δ(4)(p− p′)
]2. (7.9.16)
Box-normalization [a~pi , a
†~pj
]= δ~pi,~pj , (7.9.17)
[a~pi , a~pj
]= 0 =
[a†~pi , a
†~pj
]. (7.9.18)
Fock space|~p〉 = a†~p|vac〉, (7.9.19)
|~p, ~q〉 = a†~pa†~q|vac〉. (7.9.20)
Field operators
φ(t, ~x) =∑
~p
1√2V p0
(a~pe−ip·x + a†~pe
ip·x). (7.9.21)
Note:
φ(t, ~x)|vac〉 =∑
~p
1√2V p0
eip·x|~p〉. (7.9.22)
Remark 1.[φ(x), φ(y)]
V→+∞======= D(x− y)−D(y − x). (7.9.23)
Remark 2.
a(~p) =
√V
(2π)3/2a~p. (7.9.24)
[a(~p), a†(~q)
]= δ(3)(~p− ~q). (7.9.25)
Feynman rules in a box.|~p〉 = a†~p|vac〉. (7.9.26)
φ(t, ~x) =∑
~p
1√2V p0
(a~pe−ip·x + a†~pe
ip·x). (7.9.27)
External lines add a factor 1√2V p0
and other lines (vertexes) are unchanged.
128
7.9.2 Calculation of differential probability
Reformulation of S-matrix in a box.
〈f |S − 1|i〉 = iAV Tfi (2π)4δ(4)V T (PF − PI)
∏
f
1√2wfV
∏
i
1√2wiV
. (7.9.28)
|〈f |S − 1|i〉|2 = |AV Tfi |2(2π)8|δ(4)V T (PF − PI)|2
∏
f
1
2wfV
∏
i
1
2wiV. (7.9.29)
Summation over momentum of outgoing particle.
Lemma.
S ×∑
~pf
= S × V
(2π)3
∫d3~pf . (7.9.30)
S is symmetry factor due to statistics.
S =1∏imi!
(7.9.31)
if there are mi identical particles in the final state.
Define the new symbol wV T ,
wV T = |AV Tfi |2(2π)8|δ(4)V T (PF − PI)|2
∏
i
1
2wiV
∏
f
d3~pf(2π)32wf
× S. (7.9.32)
whered3~pf
(2π)32wfis Lorentz measure.
With Lemma 3,
wV T = |AV Tfi |2(2π)4δ(4)(PF − PI)× V T ×∏
i
1
2wiV
∏
f
d3~pf(2π)32wf
× S
= |AV Tfi |2 × V T ×∏
i
1
2wiV×D
(7.9.33)
where
D = (2π)4δ(4)(PF − PI)∏
f
d3~pf(2π)32wf
× S. (7.9.34)
7.9.3 Cross section & decay width
Differential cross section
dσ ≡ differential probability
unit time× 1
unit flux
=wV TT× 1
F.
(7.9.35)
1). Only with one incident particle, i.e. particle decay.
129
~v
...
F1 =|~v|V. (7.9.36)
2). With two incident particles.
v1
v2
...
F2 =|~v1 − ~v2|
V. (7.9.37)
e.g. 1. Differential decay width of a rest particle, E = M . Change the notation,dσ → dΓ.
dΓ =wV TT
=1
2EV|Afi|2 × V ×D
=1
2M|Afi|2 ×D.
(7.9.38)
e.g. 2.1 + 2 −→ 1′ + 2′ + · · ·+ n′. (7.9.39)
dσ =wV TT× 1
F2
=|Afi|24E1E2
× 1
|~v1 − ~v2|×D.
(7.9.40)
7.9.4 Calculation of Cross-sections and Decay-widths
For two-particle final states in CoM
1 + 2 + · · · → 1′ + 2′. (7.9.41)
PI = p1 + p2 + · · · = PF = p′1 + p′2. (7.9.42)
In CoM~p′1 + ~p′2 = 0 =⇒ |~p′1| = |~p′2|. (7.9.43)
ET = E1 + E2 + · · · = E′1 + E′2 (7.9.44)
and ET is a fixed number.
130
Then:δ(4)(p′1 + p′2 − PI) = δ(3)(~p′1 + ~p′2)δ(E′1 + E′2 − ET ). (7.9.45)
Lemma:
D(1 + 2 + · · · → 1′ + 2′) =1
S
|~p′1|016π2
1
ETdΩ′1. (7.9.46)
where |~p′1|0 is the solution of ET = E′1 + E′2 and dΩ′1 is the solid angle, i.e. dΩ′1 ≡d cos θ′1dϕ
′1.
Proof.
D =1
S
∫d3~p′1
(2π)32E′1
∫d3~p′2
(2π)32E′2(2π)4δ(4)(p′1 + p′2 − PI)
=1
S
∫d3~p′1
(2π)22E′12E′2δ(E′1 + E′2 − ET )
=1
S
∫ |~p′1|2d|~p′1|dΩ′1(2π)22E′1E
′2
δ(E′1 + E′2 − ET ).
(7.9.47)
Since
δ(E′1 + E′2 − ET ) = δ(√|~p′1|2 +m2 +
√|~p′2|2 +m2 − ET )
= δ(f(|~p′1|))
=1
f ′(|~p′1|)δ(|~p′1|0 − |~p′1|),
(7.9.48)
f ′(|~p′1|) =∂(E′1 + E′2 − ET )
∂(|~p′1|)=
∂E′1∂(|~p′1|)
+∂E′2∂(|~p′2|)
, (7.9.49)
E′21 = |~p′1|2 +m2, E
′22 = |~p′2|2 +m2, (7.9.50)
f ′(|~p′1|) =|~p′1|E′1
+|~p′2|E′2
= |~p′1|(1
E′1+
1
E′2) =|~p′1|ETE′1E
′2
. (7.9.51)
Then
D =1
S
∫ |~p′1|2d|~p′1|dΩ′14π24E′1E
′2
× E′1E′2
|~p′1|ETδ(|~p′1| − |~p′1|0) =
1
S× 1
16π2
|~p′1|0ET
dΩ′1. (7.9.52)
e.g.1 Decay width φ→ N + N .
N
Np′1
p′2
p
φ
A(1)fi = −ig. (7.9.53)
In CoM, φ particle is at rest, i.e. ~p = 0 = ~p′1 + ~p′2.
131
(µ, 0) = (√|~p′1|2 +m2, ~p′1) + (
√|~p′2|2 +m2, ~p′2), (7.9.54)
ET = µ = 2√|~p′1|2 +m2, (7.9.55)
|~p′1|0 =
õ2
4−m2. (7.9.56)
Γ(1)(φ→ N +N)|CoM =1
2µ
∫|A(1)
fi |2D
=1
2µ
∫| − ig|2 × |~p′1|0
16π2ETdΩ′1
=g2
32π2µ
õ2
4−m2
∫dΩ′1
=g2
8πµ2
õ2
4−m2.
(7.9.57)
Where S = 1, because N , N are not identical particles.
Note: Particle lifetime is τ = 1T ∝ 1
g2 , where g is coupling constant. g is weaker, τis longer.
e.g.2 Cross-section for 1 + 2→ 1′ + 2′ in CoM.
(dσ)CoM =1
4E1E2|Afi|2
1
|~v1 − ~v2|× 1
16π2× |~p
′1|0dΩ′1ET
× 1
S, (7.9.58)
(dσ
dΩ′1)CoM =
|Afi|264π2
|~p′1|0E1E2|~v1 − ~v2|ET
× 1
S, (7.9.59)
(dσ
d cos θ′1)CoM =
|Afi|232π
|~p′1|0E1E2|~v1 − ~v2|ET
× 1
S. (7.9.60)
~v1 ≡~p1
γm=~p1
E1, ~v2 ≡
~p2
γm= − ~p1
E1, (7.9.61)
|~v1 − ~v2| = |~p1|0(1
E1+
1
E2) = |~p1|0
ETE1E2
, (7.9.62)
where E1 + E2 = ET = E′1 + E′1, and |~p1|0 is the solution of E1 + E2 − ET = 0.
Then we have
1
|~v1 − ~v2|ETE1E2=
1
ETE1E2× 1∣∣∣ ~p1
E1+ ~p2
E2
∣∣∣
=1
ETE1E2× 1
|~p1|0E1+E2E1E2
=1
E2T |~p1|0
,
(7.9.63)
(dσ
dΩ′1
)
CoM
=|Afi|64π2
× 1
E2T
× |~p′1|0|~p1|0
× 1
S. (7.9.64)
132
In the special caseN +N −→ N +N, (7.9.65)
we have m1 = m2 = m′1 = m′2, and |~p1|0 = |~p′1|0,
=⇒(dσ
dΩ′1
)
CoM
=|Afi|64π2
× 1
E2T
× 1
S. (7.9.66)
1S = 1
2! for NN → NN , 1S = 1
1 for NN → NN .
133
Lecture 8 Dirac Fields
About Lecture VI:
• [Luke] P.P. 125-167;
• [Tong] P.P. 81-123;
• [Coleman] P.P. 221-337;
• [PS] P.P. 35-76 & 115-123;
• [Zhou] P.P. 34-45 & 81-104.
8.1 The Dirac equation and Dirac algebra
Dirac fields describe dynamics of massive spin-1/2 particles like electrons, protons, and soon. In non-relativistic quantum mechanics, Pauli realized that an electron is described bya two-component column vector
ψ =
(∗∗
)(8.1.1)
with Pauli matrices 112 and σx, σy, σz. In relativistic quantum field theory, Dirac realizedthat we had to introduce a 4-component column vector, called the Dirac field,
ψ =
∗∗∗∗
(8.1.2)
to describe the motion of electron with Dirac matrices β, ~α = (α1, α2, α3) which are 4× 4matrices.
8.1.1 The Dirac algebra
The principles of writing the Dirac equation give rise to the Dirac algebra generated byDirac matrices β, ~α = (α1, α2, α3).
1). The Lorentz invariance of the Dirac equation
(i∂
∂t+ i~α · ~∇− βm)ψ(t, ~x) = 0, (8.1.3)
implies that ~α, β are constant matrices and that both the time derivative and spatialderivatives are first-order derivatives.
134
2). Hamiltonian is a Hermitian operator. With the Dirac equation
i∂
∂tψ(t, ~x) = (−i~α · ~∇+ βm)ψ(t, ~x), (8.1.4)
and the canonical quantization in quantum mechanics
i∂
∂t↔ H, −i~∇ ↔ ~p, (8.1.5)
the Hamiltonian has the form
H = −i~α · ~∇+ βm = ~α · ~p+ βm. (8.1.6)
That Hamiltonian is a Hermitian operator suggests that
H† = H ⇐⇒ ~α† = ~α, β† = β. (8.1.7)
3). The mass-energy relation E2 = m2 + ~p2 leads to the following calculation,
i∂
∂tψ = Hψ = Eψ = (~α · ~p+ βm)ψ ⇒ Eψ = (~α · ~p+ βm)ψ (8.1.8)
(~α · ~p+ βm)(~α · ~p+ βm)ψ = E2ψ, (8.1.9)
αipiαjpj + αipiβm+ βmαjpj +m2β2 = E2, (8.1.10)
αiαj + αjαi2
pipj +mpi(αiβ + βαi) +m2β2 = E2 = δijpipj +m2. (8.1.11)
which derives the algebraic relations of the Dirac algebra,
αiαj + αjαi = 2δij ;αiβ + βαi = 0;β2 = 112.
(8.1.12)
Note: α2i = 11, β2 = 11 and
tr(αi) = tr(β2αi) = tr(βαiβ) = −tr(αi) = 0 =⇒ tr(αi) = tr(β) = 0, (8.1.13)
so αi and β are Hermitian traceless square identity matrices.
8.1.2 Two widely used representations of the Dirac algebra
1). The Weyl representation.
~αW =
(~σ 00 −~σ
), βW =
(0 112
112 0
). (8.1.14)
Note: The Weyl representation presents a suitable description on physics of Neutrino.
2). The Dirac representation (Standard representation).
~αD =
(0 ~σ~σ 0
), βD =
(112 00 −112
). (8.1.15)
135
With the unitary matrix,
H =1√2
(112 112
112 −112
), (8.1.16)
these two representations are related to each other via the similarity transformation
~αD = H~αWH†, βD = HβWH
†. (8.1.17)
Thm. The Dirac algebra generated by ~αi, β is invariant under any unitary transfor-mation U ,
~α′ = U~αU †, β′ = UβU † (8.1.18)
satisfying the same algebraic constraints defining the Dirac algebra.
8.1.3 The Lagrangian formulation
The Lagrangian density is
L = ψ†(i∂
∂t+ i~α · ~∇− βm)ψ (8.1.19)
and its component formalism is
L = ψ†a(i∂
∂tδab + i(~α)ab~∇− βabm)ψb (8.1.20)
where ψ†a is the component of a 1× 4 matrix, ψa is the component of a 4× 1 matrix and(~α)ab, βab is in 4× 4 matrix, and ψ is called bispinor.
EoM is derived from the Euler-Lagrangian equation
∂L∂ψ†a
= 0, (8.1.21)
which gives rise to
=⇒ (i∂
∂t+ i~α · ~∇− βm)ψ = 0. (8.1.22)
8.1.4 Plane wave solutions of Dirac equation in Dirac representation
The Dirac equation has the form in the momentum representation
i∂
∂tψ = (~αD · ~p+ βDm)ψ, (8.1.23)
with p0 =√|~p|2 +m2. Substitute the positive-frequency solution
ψ(+)(x) = u~pe−ip·x, (8.1.24)
into the Dirac equation, we have
i∂
∂tψ(+)(x) = p0ψ(+)(x) = (~αD · ~p+mβD)ψ(+)(x), (8.1.25)
which is(p0 − ~αD · ~p)u~p = mβDu~p. (8.1.26)
136
Substitute the negative-frequency solution
ψ(−)(x) = v~peip·x, (8.1.27)
into the Dirac equation, we have
i∂
∂tψ(−)(x) = −p0ψ(−)(x) = (~αD · ~p+mβD)ψ(−)(x), (8.1.28)
which is(p0 + ~αD · ~p)v~p = −mβDv~p. (8.1.29)
In the rest frame, i.e. ~p = 0, p0 = m,
p0u~p = mβDu~p;p0v~p = −mβDv~p.
=⇒βDu~p = u~p;βDv~p = −v~p,
(8.1.30)
which have the following linearly independent solutions spanning the four dimensionalvector space,
u(1)~p=0 =
√2m
1000
, u
(2)~p=0 =
√2m
0100
, (8.1.31)
v(1)~p=0 =
√2m
0010
, v
(2)~p=0 =
√2m
0001
. (8.1.32)
Note:√
2m is the normalization factor which is very meaningful in deriving Feynmanrules of Dirac fields or in calculating the cross section.
In the frame of px = py = 0, pz 6= 0, E = p0,
(E − (αD)3pz)u~p = mβDu~p;(E + (αD)3pz)v~p = −mβDv~p,
(8.1.33)
which have the following linearly independent solutions spanning the four dimensionalvector space,
u(1)pz =
√E +m
0√E −m
0
, u(2)
pz =
0√E +m
0
−√E −m
, (8.1.34)
v(1)pz =
√E −m
0√E +m
0
, v(2)
pz =
0
−√E −m0√
E +m
. (8.1.35)
Note: pz = 0, E = m, the above solutions return to the solutions in the rest frame.Thm.
u(r)†~p=0βDu
(s)~p=0 = u
(r)†~p βDu
(s)~p = 2mδrs; (8.1.36)
v(r)†~p=0βDv
(s)~p=0 = v
(r)†~p βDv
(s)~p = −2mδrs, (8.1.37)
which are Lorentz invariant scalars.
137
8.2 Dirac equation and Clifford algebra
The Clifford algebra is isomorphism to the Dirac algebra, and with the Clifford algebra,the Dirac equation has an explicit Lorentz covariant formalism which is widely used inrelativistic quantum field theory.
8.2.1 The Clifford algebra
Def. Define γµ matrix as
γ0 = β, γi = βαi, i = 1, 2, 3, (8.2.1)
satisfying
γµ, γν = 2gµν = 2
1−1
−1−1
, (8.2.2)
γµ = gµνγν , (8.2.3)
and the algebra generated by the γµ matrices is called the Clifford algebra.
The γµ matrices have the following interesting properties:
1).
(γ0)† = γ0, (γi)† = −γi. (8.2.4)
2).
(γ0)2 = 114, (γi)2 = −114. (8.2.5)
3).
(γµ)† = gµνγν = γµ. (8.2.6)
4). The Dirac adjoint of the γµ matrices are defined as
γµ ≡ γ0(γµ)†γ0, γµ = γµ. (8.2.7)
5). The rank-2 anti-symmetric tensor, σµν = i2 [γµ, γν ], have
σ00 = 0, σ0i = iαi, σij = −iαiαj . (8.2.8)
6). The fifth Gamma matrix γ5 given by γ5 = 14!εµναβγ
µγνγαγβ with εµναβ is thetotally anti-symmetric tensor, has
γ5 = γ5, (γ5)2 = 114, γ5 = γ†5 = −γ5. (8.2.9)
Note 1: γ5 = iγ0γ1γ2γ3 = −iα1α2α3.
Note 2: The γ5 has the respective forms in the Dirac representation and the Weylrepresentation,
(γ5)D =
(0 112
112 0
), (γ5)W =
(112 00 −112
). (8.2.10)
138
8.2.2 Dirac equation with γµ matrices
In the first place, we reformulate the Lagrangian with γµ matrices,
L = ψ†(i∂
∂t+ i~α · ~∇− βm)ψ = (ψ†γ0)γ0(i
∂
∂t+ i~α · ~∇− βm)ψ. (8.2.11)
Secondly, with the Dirac adjoint ψ = ψ†γ0, we have
L = ψγ0i∂
∂t+ ψi~γ · ~∇−mψψ. (8.2.12)
Thirdly, with
γµ = (γ0, γi), ∂µ = (∂
∂t,∂
∂xi), (8.2.13)
we haveL = ψ(iγµ∂µ −m)ψ = ψa((iγ
µ)ab∂µ −mδab)ψb. (8.2.14)
The Euler-Lagrangian equation, EoM, is the Dirac equation
∂L∂ψ
= 0 =⇒ (iγµ∂µ −m)ψ = 0. (8.2.15)
Notation: With the Dirac slash a/ ≡ γµaµ, b/ ≡ γµbµ satisfying a/b/+b/a/ = 2a ·b, we define∂/ ≡ γµ∂µ, to reformulate the Dirac equation as
(i∂/−m)ψ = 0 (8.2.16)
which is obviously covariant under Lorentz transformations.
8.2.3 Plane wave solutions of Dirac equation with γµ matrices
Substitute the positive-frequency and negative-frequency solutionsψ(+)(x) = u
(r)~p e−ip·x;
ψ(−)(x) = v(r)~p eip·x
r = 1, 2 (8.2.17)
into the Dirac equation, we have
(p/−m)u(r)~p = 0;
u(r)~p (p/−m) = 0.
(p/+m)v
(r)~p = 0;
v(r)~p (p/+m) = 0.
(8.2.18)
with p/ = γµpµ and u(r)~p = u
(r)†~p γ0.
The set of u(r)~p , r = 1, 2 and v
(s)~p , s = 1, 2 forms a kind of the orthonormal basis in the
four-dimensional Hilbert space:
Thm. 1. The completeness relation. ∑2
r=1 u(r)~p u
(r)~p = p/+m;
∑2s=1 v
(s)~p v
(s)~p = p/−m.
(8.2.19)
Thm. 2. The orthonormality condition.u
(r)~p u
(s)~p = 2mδrs = −v(r)
~p v(s)~p ;
u(r)~p v
(s)~p = 0.
(8.2.20)
139
8.3 Lorentz transformation and parity of Dirac bispinor field-s
A Lorentz transformation Λµν given by
Λ : xµ → x′µ = Λµνxν (8.3.1)
satisfies x2 = x′2 or ΛT ηΛ = η with η denoting the Minkowski metric.The Dirac bispinor field ψ(x) transforms under the Lorentz transformation
ψ(x)Λ−−→ ψ′(x) = D(Λ)ψ(Λ−1x), (8.3.2)
the associated component formalism,
ψ′a(x) = D(Λ)abψ(λ−1x)b. (8.3.3)
where the D(Λ) is called the bispinor representation of Lorentz group.The Hermitian conjugation of the Dirac bispinor field ψ(x) tranforms
ψ†(x)Λ−−→ ψ†(Λ−1x)D†(Λ); (8.3.4)
and its Dirac adjoint
ψ(x)Λ−−→ ψ†(Λ−1x)D†(Λ)γ0 = ψ
†(Λ−1x)D(Λ); (8.3.5)
with D(Λ) = γ0D†(Λ)γ0.The Parity means
P : (t, ~x)→ x′µ = (t,−~x) = Pxµ, (8.3.6)
which gives rise to
(∂
∂t,∂
∂x) −→ (
∂
∂t,− ∂
∂x). (8.3.7)
The Dirac bispinor field ψ(x), its Hermitian conjugation, and its Dirac adjoint trans-form respectively under parity
ψ(x)P−−→ γ0ψ(Px); (8.3.8)
andψ†(x)
P−−→ ψ†(Px)γ0; (8.3.9)
andψ(x)
P−−→ ψ(Px)γ0. (8.3.10)
Note that Gamma matrices γµ are unchanged under Lorentz transformation and parity.
8.3.1 Classification of quantities out of Dirac bispinor fields under Lorentztransformation and parity
1). Scalar field quantity is invariant under Lorentz transformation (LT) and parity.
Example: ψ(x)ψ(x) is a scalar field quantity.
About Lorentz transformation of ψ(x)ψ(x):
ψ(x)ψ(x)Λ−−→ ψ
′(x)ψ′(x) =ψ(Λ−1x)D(Λ)D(Λ)ψ(Λ−1x)
=ψ(Λ−1x)ψ(Λ−1x)(8.3.11)
140
if and only ifD(Λ)D(Λ) = 114. (8.3.12)
About parity of ψ(x)ψ(x):
ψ(x)ψ(x)P−−→ ψ
′(x)ψ′(x) =ψ(Px)γ0γ0ψ(Px)
=ψ(Px)ψ(Px)(8.3.13)
which does not impose any constraints on the Dirac bispinor representation D(Λ).
2). Vector field quantity is covariant under LT and invariant under parity.
Note: parity of vector V µ:
(V 0, ~V )P−−→ (V 0,−~V ), (8.3.14)
V µ P−−→ VµP−−→ V µ. (8.3.15)
Example: ψ(x)γµψ(x) is a vector field quantity:
ψ(x)γµψ(x)Λ−−→ψ′(x)γµψ′(x) = ψ(Λ−1x)D(Λ)γµD(Λ)ψ(Λ−1x)
=Λµνψ(Λ−1x)γνψ(Λ−1x),(8.3.16)
which gives rise to the constraint equation on the Dirac bispinor representation D(Λ):
D(Λ)γµD(Λ) = Λµνγν . (8.3.17)
About parity of ψ(x)γµψ(x),
ψ(x)γµψ(x)P−−→ψ′(x)γµψ′(x) = ψ(Px)γ0γµγ0ψ(Px)
=ψ(Px)γµψ(Px).(8.3.18)
3). Tensor field quantity.
When the constraint equations 8.3.12 and 8.3.17 are satisfied, we have
ψ(x)γµγνψ(x)Λ−−→ ΛµαΛνβψ(Λ−1x)γαγβψ(Λ−1x). (8.3.19)
Example for antisymmetric tensor field quantity:
Tµν = ψ(x)σµνψ(x), σµν =i
2[γµ, γν ]. (8.3.20)
4). Pseudo-scalar quantity is invariant under LT and “−” under parity.
Example: ψ(x)γ5ψ(x) is a pseudo-scalar field quantity.
ψ(x)γ5ψ(x)Λ−−→ψ(Λ−1x)D(Λ)γ5D(Λ)ψ(Λ−1x)
=ψ(Λ−1x)γ5ψ(Λ−1x),(8.3.21)
which imposes the constraint equation on the Dirac bispinor representation D(Λ):
D(Λ)γ5D(Λ) = γ5 (8.3.22)
which can be derived from (8.3.17) due to det(Λ) = 1.
About parity of ψ(x)γ5ψ(x), we have
ψ(x)γ5ψ(x)P−−→ψ(Px)γ0γ5γ0ψ(Px)
=− ψ(Px)γ5ψ(Px).(8.3.23)
141
5). Pseudo-vector, called axial vector, is covariant under LT and “−” under parity.
Note: parity of axial vector:
AµP−−→ −Aµ. (8.3.24)
Example ψ(x)γµγ5ψ(x) is a pseudo-vector field quantity.
ψ(x)γµγ5ψ(x)Λ−−→ Λµνψ(Λ−1x)γνγ5ψ(Λ−1x), (8.3.25)
when the constraint conditions (8.3.12) and (8.3.17) are satisfied.
About parity of ψ(x)γ5ψ(x), we have
ψ(x)γµγ5ψ(x)P−−→ψ(Px)γ0γµγ5γ0ψ(Px)
=− ψ(Px)γµγ5ψ(Px).
(8.3.26)
In summary, we derive two constraint equations: (8.3.12) and (8.3.17) for the Diracbispinor representation:
D(Λ)D(Λ) = 114; (8.3.27)
D(Λ)γµD(Λ) = Λµνγν ; (8.3.28)
which are to be solved in the following subsection.
8.3.2 Explicit formulation of D(Λ)
The Lorentz transformation Λ satisfying
ΛT ηΛ = η, (8.3.29)
has a solution give by
Λ = exp(− i2ωρσJ
ρσ), (8.3.30)
with ωρσ = −ωσρ, where the orbital angular-momentum tensor Jρσ has the form
(Jρσ)αβ = i(δραδσβ − δρβδσα) (8.3.31)
satisfying the algebra generated by the generators of the Lorentz group.
The Dirac spinor representation D(Λ) defined as
ψa(x)D(Λ)−−−→ ψ′a(x) = D(Λ)abψb(Λ
−1x). (8.3.32)
has the form
D(Λ) = exp(− i2ωρσS
ρσ), (8.3.33)
where the spin angular-momentum tensor Sρσ is given by
Sρσ =i
4[γρ, γσ] =
1
2σρσ (8.3.34)
with
σ0i = iαi, σij = −iαiαj (8.3.35)
142
in the Dirac algebra, which satisfy the algebra generated by the generators of the Lorentzgroup. As an exercise, we prove the lemma to be used soon.
Lemma 1.
[γµ, Sρσ] = [Jρσ]µνγν . (8.3.36)
Also, we have another lemma:
Lemma 2.For the Lorentz boost: D†(Λ) = D(Λ);For the Lorentz rotation: D†(Λ) = D−1(Λ).
Proof. To prove Lemma 2, we perform the following calculation step by step.
Step 1. In the Weyl representation, the spin angular-momentum tensor Sρσ has theform
S0i =i
2
(σi 00 −σi
), (8.3.37)
Sij = εijkLk, Lk =
1
2
(σk 00 σk
). (8.3.38)
where σi is the Pauli matrix.
Step 2. Consider the Lorentz boost in the i-th direction, and denote
~φ = (ω0i) = (φ, 0, 0) (8.3.39)
where φ is a real number, so the Dirac spinor representation D(Λ) has the form
D(Λ) = exp(−iφiS0i) = exp(1
2~φ · ~α)
=
(exp
~φ·~σ2 0
0 exp −~φ·~σ2
).
(8.3.40)
Step 3. Consider the Lorentz rotation around the k-th direction, namely, take
ωij = εijkθk, (8.3.41)
with the antisymmetric rank-three tensor εijk and real numbers θk, so the Dirac spinorrepresentation D(Λ) has the form
D(Λ) = exp(− i2ωijεijkLk) = exp(−i~θ · ~L)
=
(exp −i
~θ·~σ2 o
0 exp −i~θ·~σ2
).
(8.3.42)
Note: D(Λ) is a 4× 4 matrix with two 2× 2 block matrices.
Now, we prove that the Dirac spinor representation (8.3.33) satisfies the two constraintequations: (8.3.12) and (8.3.17), derived in the last subsection.
Thm. 1. D(Λ) = D−1(Λ).
143
Proof.D(Λ) = γ0D†(Λ)γ0. (8.3.43)
For the Lorentz boost,
D(Λ) = γ0D†(Λ)γ0 = γ0D(Λ)γ0
= γ0 exp(1
2~φ · ~α)γ0 = exp(−1
2~φ · ~α)
= D−1(Λ)
(8.3.44)
where the anticommutator γ0, αi = 0 has been used since γ0 = β.For the Lorentz rotation,
D(Λ) = γ0D†(Λ)γ0 = γ0D−1(Λ)γ0
= γ0 exp(i~θ · ~L)γ0 = exp(i~θ · ~L)
= D−1(Λ),
(8.3.45)
where the commutator [αiαj , γ0] = 0 has been exploited.
Thm. 2. D(Λ)γµD(Λ) = Λµνγν .
Proof. In first order of ωρσ, we have
(1 +i
2ωρσS
ρσ)γµ(1− i
2ωρσS
ρσ) = (1− i
2ωρσJ
ρσ)µνγν (8.3.46)
which gives rise to the Lemma 1. In higher order expansion of ωρσ, we do similar calcula-tion.
8.4 Canonical quantization of free Dirac field theory
Spin-statistics theorem (Fierz, 1939; Pauli, 1940; Schwinger 1950):Particles with integral spin (spin 0, spin 1, · · · ) are quantized as Bosons with commu-
tators; particles with half integral spin (spin 1/2, spin 3/2, · · · ) are quantized as fermionswith anti-commutators.
The Dirac field theory describes the dynamics of spin-1/2 fermions, so the associatedcanonical quantization makes use of anti-commutators.
8.4.1 Canonical quantization
The free Dirac field ψ(t, ~x), with four component-fields ψa(t, ~x), a = 1, 2, 3, 4, has theLagrangian
L = ψ(iγµ∂µ −m)ψ
= iψ†aψa + iψaγiab∂iψb −mψaψa,
(8.4.1)
so that the conjugate momentum of the component-field ψa(t, ~x) is given by
Πψa ≡∂L∂ψa
= iψ†a. (8.4.2)
In accordance with the Spin-statistics theorem, we have the equal-time anti-commutatorsgiven by
144
(I). The equal-time anti-commutators of Dirac component-fields.
ψa(t, ~x), ψ†b(t, ~y) = δabδ(3)(~x− ~y), (8.4.3)
ψa(t, ~x), ψb(t, ~y) = 0 = ψ†a(t, ~x), ψ†b(t, ~y). (8.4.4)
(II). The equal-time anti-commutators of creation and annihilation operators.
The plane-wave solution of the free Dirac equation has the form
ψ(t, ~x) =2∑
r=1
∫d3~p
(2π)3/2
1√2w~p
(b(r)~p u
(r)~p e−ip·x + c
(r)†~p v
(r)~p eip·x), (8.4.5)
with its Dirac conjugation
ψ(t, ~x) =2∑
r=1
∫d3~p
(2π)3/2
1√2w~p
(b(r)†~p u
(r)~p eip·x + c
(r)~p v
(r)~p e−ip·x), (8.4.6)
where b(r)~p , c
(r)~p are annihilation operators and b
(r)†~p , c
(r)†~p v
(r)~p are creating operators,
r = 1 for the spin-up polarization and r = 2 for the spin-down polarization.
The equal-time anti-commutators of creation and annihilation operators have theform
b(r)~p , b(s)†~p′ = δrsδ
(3)(~p− ~p′) = c(r)~p , c
(s)†~p′ , (8.4.7)
b(r)~p , b(s)~p′ = 0 = b(r)†~p , b
(s)†~p′ , (8.4.8)
c(r)~p , c
(s)~p′ = 0 = c(r)†
~p , c(s)†~p′ . (8.4.9)
Theorem. Two types of anti-commutative relations (I) and (II) are equivalent.
8.4.2 Fock space with the Fermi-Dirac statistics
Let us construct the Fock space of fermions defined as particles obeying the Fermi-Diracstatistics.
1). Vacuum state |vac〉 (no-particle state) defined as
b(r)~p |vac〉 = 0 = c
(s)~p |vac〉, (8.4.10)
where the b(r)~p operator is about the annihilation of particle (such as electron) with spin
r, and the c(s)~p operator is about the annihilation of anti-particle (such as positron)
with spin s, and the label r = s = 1 represents spin up and the label r = s = 2represents spin down.
2). One-particle state. Rotation invariant one-particle state defined as
|~p, r〉 = b(r)†~p |vac〉, (8.4.11)
and Lorentz invariant one-particle state defined as
|p, r〉 = (2π)3/2√
2w~p|~p, r〉. (8.4.12)
The normalization condition of one-particle state can be calculated
〈~p′, s|~p, r〉 = 〈vac|b(s)~p′ , b(r)†~p |vac〉 = δrsδ
(3)(~p− ~p′). (8.4.13)
Similarly, one-antiparticle state can be defined too.
145
3). Multi-particle state. For example, a two-particle state defined as
|~p1, r1; ~p2, r2〉 = b(r1)†~p1
b(r2)†~p2|vac〉
= −b(r2)†~p2
b(r1)†~p1|vac〉
= −|~p2, r2; ~p1, r1〉(8.4.14)
which gives rise to |~p, r; ~p, r〉 = 0, interpreted as The Pauli exclusion principle: Twoidentical fermions are forbidden in the same quantum state.
8.4.3 Symmetries and conservation laws: energy, momentum and charge
Energy, momentum and charge in the free Dirac theory are physical quantities becausethey are conserved quantities associated with symmetries.
1). Energy-momentum tensor Tµν due to the invariance of the action under the space-time translation transformation
δψ(x) = aµ∂µψ(x) (8.4.15)
has the form
Tµν ≡ ∂L∂(∂µψ)
∂νψ = ψiγµ∂νψ. (8.4.16)
As the upper index µ = 0, T 0ν is the charge density of the form
T 0ν = iψ†∂νψ (8.4.17)
in which the case of ν = 0 denotes the energy density (the Hamiltonian density)
T 00 = H = iψ†ψ; (8.4.18)
and the other case of ν = i denotes the momentum density
T 0i = −iψ†∂iψ. (8.4.19)
2). Total energy and total momentum.
The energy density T 00 is the Hamiltonian density,
H = Πψψ − L= iψ†ψ − ψ(iγµ∂µ −m)ψ
= iψ†ψ,
(8.4.20)
and the total energy is the total Hamiltonian
H =
∫d3~xH =
∫d3~x iψ†ψ
=2∑
r=1
∫d3~p E~p(b
(r)†~p b
(r)~p − c
(r)~p c
(r)†~p )
=2∑
r=1
∫d3~p E~p(b
(r)†~p b
(r)~p + c
(r)†~p c
(r)~p ) + 2
∫d3~p E~pδ
(3)(0),
(8.4.21)
146
where the infinity can be removed by the normal ordering defined as
: H :=
2∑
r=1
∫d3~p E~p(b
(r)†~p b
(r)~p + c
(r)†~p c
(r)~p ). (8.4.22)
Note: Inside the normal ordering of fermion operators (also in the time ordering offermion operators), fermion operators are anti-commutative of the form
: c(r)~p c
(r)†
~p := − : c(r)†
~p c(r)~p := −c(r)†
~p c(r)~p . (8.4.23)
The total momentum is calculated as follows with the help of the normal ordering offermion operators,
: P i : =
∫~x : T 0i :=
∫d3~x : ψ†(−i ∂
∂xiψ) :
=2∑
r=1
∫d3~p pi(b
(r)†~p b
(r)~p + c
(r)†~p c
(r)~p ).
(8.4.24)
3). Charge due to the U(1) invariance. The U(1) transformation given by
δψ(x) = −iαψ(x); (8.4.25)
δψ†(x) = iαψ(x) (8.4.26)
gives rise to the conserved current
jµ ≡ ∂L∂(∂µψ)
δψ = ψγµψ, (8.4.27)
with the charge density from the µ = 0 case
j0 = ψγ0ψ = ψ†ψ, (8.4.28)
and the total charge
Q =
∫d3~x ψ†ψ
=2∑
r=1
∫d3~p (b
(r)†~p b
(r)~p − c
(r)†~p c
(r)~p ).
(8.4.29)
A Dirac particle has +1 charge due to the commutative relation
[Q, b(r)†~p ] = (+1)b
(r)†~p , (8.4.30)
and a Dirac anti-particle has −1 charge due to the commutative relation
[Q, c(r)†~p ] = (−1)c
(r)†~p , (8.4.31)
both of which can be used to test the charge number of a given multi-particle state inthe Fock space.
147
8.5 Fermion propagator
Inside time-ordering and normal ordering, exchange of two fermion operators introducesa minus sign.
e.g. 1.T (ψ1ψ2) = −ψ2ψ1; (8.5.1)
: ψ1ψ2 := − : ψ2ψ1 : . (8.5.2)
e.g. 2.
T (ψ1ψ2ψ3) =: ψ1ψ2ψ3 : +ψ1ψ2 ψ3 + ψ1 ψ2ψ3−ψ1ψ3 ψ2. (8.5.3)
e.g. 3.
: ψ1ψ2ψ3 := − : ψ1ψ3 ψ2 := −ψ2 ψ1ψ3 = −ψ1ψ3 ψ2. (8.5.4)
e.g. 4.
: ψ1ψ2ψ3ψ4 :=: ψ1ψ4 ψ2ψ3 : (−1)2. (8.5.5)
Sαβ ≡ ψα(x)ψβ(y) = Tψα(x)ψβ(y)− : ψα(x)ψβ(y) :, (8.5.6)
Sαβ = 〈vac|Tψα(x)ψβ(y)|vac〉. (8.5.7)
When x0 > y0,
ψα(x)ψβ(y) = 〈vac|ψα(x)ψβ(y)|vac〉= 〈vac|ψ(+)
α (x)ψ(−)β (y)|vac〉
=
∫d3~pd3~p′
(2π)3
1√2E~p
√2E~p′
2∑
r,s=1
u(r)~p,αu
(s)~p′,βe
−ip·x+ip′·y〈vac|b(r)~p b(s)†~p′ |vac〉
=
∫d3~p
(2π)32E~p
2∑
r=1
u(r)~p,αu
(r)~p,βe
−ip·(x−y)
=
∫d3~p
(2π)32E~p(p/+m)αβe
−ip·(x−y)
= (iγµ∂µ +m)αβ
∫d3~p
(2π)32E~pe−ip·(x−y)
= (iγµ∂µ +m)αβθ(x0 − y0)〈vac|φ(x)φ(y)|vac〉,
(8.5.8)
where the Dirac field has been connected with real scalar field theory, and we have
ψα(x)ψβ(y) = (iγµ∂µ +m)αβ φ(x)φ(y)
= (iγµ∂µ +m)αβ
∫d4p
(2π)4
i
p2 −m2 + iεe−ip·(x−y)
=
∫d4p
(2π)4
i(p/+m)αβp2 −m2 + iε
e−ip·(x−y).
(8.5.9)
Note:p2 = p/p/, p2 −m2 = (p/+m)(p/−m). (8.5.10)
148
ψ(x)ψ(y) =
∫d4p
(2π)4
i
p/−m+ iεe−ip·(x−y), (8.5.11)
ψ(x)ψ(y) = 0 = ψ(x)ψ(y), (8.5.12)
ψ(x)φ(y) = 0, (8.5.13)
φ(x)φ(y) =
∫d4p
(2π)4
ie−ip·(x−y)
p2 −m2 + iε. (8.5.14)
8.6 Feynman diagrams and Feynmans rules for fermion field-s
Nucleon-meson theory:
The Lagrangian is
L = ψ(iγµ∂µ −m)ψ +1
2(∂µφ)2 − 1
2µ2φ2 − gψΓψφ, (8.6.1)
where Γab = δab is called Yukawa interacton, Γab = (iγ5)ab is called Pseudo Yukawainteraction.
The interaction Hamiltonian is
Hint = gψaΓabψbφ (8.6.2)
where g is coupling constant and φ(x) is pseudo scalar field satisfied
φ(Px) = −φ(x). (8.6.3)
Dyson’s formula:
S = U(+∞,−∞) = T exp(−i∫d4xHint)
= 1 ++∞∑
n=1
(−i)nn!
∫d4x1d
4x2 · · · d4xnT (Hint(x1)Hint(x2) · · ·H(xn)).
(8.6.4)
Feynman diagrams*-HHj
Vertex: interaction;
Internal line: Feynman propagator;
External line: incoming or outgoing particles.
Feynman rules*-HHj
FRA: in coordinate space time;
FRB: integrate out vertices;
FRC: integrate out possible internal momentum.
149
8.6.1 FDs & FRs for vertices
Hint = igψΓψφ. (8.6.5)
−igΓ −igΓ
The arrow on the line denotes the charge flow.In FRC, assign the factor −igΓ to each vertex.
8.6.2 Internal lines
DF = ip2−µ2+iε
p
DF = i(p/+m)p2−m2+iε
p
DF = i(−p/+m)p2−m2+iε
p
8.6.3 External lines
In FRC, assign factor “1” to incoming or outgoing meson fields.Incoming fermions line:
~p,s
xFRA : u
(s)~p e−ip·x;
FRB&FRC : u(s)~p . (4× 1)
ψ(x)|p, s〉 ≡ 〈vac|ψ(s)(x)|p, s〉
= 〈vac|∫
d3~p′
(2π)3
1√2ω~p′
2∑
r=1
b(r)~p′ u
(r)~p′ e−ip′·x(2π)3
√2ω~pb
(s)†~p |vac〉
=
∫d3~p′
(2π)3
1√2ω~p′
u(r)~p′ e−ip′·x(2π)3
√2ω~pδrsδ
(3)(~p− ~p′)
= u(s)~p e−ip·x.
(8.6.6)
Incoming anti-fermion line:
~k,s
x FRC : v(s)~k. (1× 4)
Outgoing fermion line:
~p,s
x FRC : u(s)~p . (1× 4)
150
Outgoing anti-fermion line:
~k,s
x FRC : v(s)~k. (4× 1)
8.6.4 Combinational factor (Symmetry factor)
1
S=
1
n!× n! = 1 (8.6.7)
where n! is permutations of vertices, and 1n! comes from Dyson’s formula. Once 1
S = 1,positions of vertices in FDs are fixed.
8.7 Special Feynman rules for fermions lines
8.7.1 Mapping from matrix to number
Scattering amplitude is a number.
Feynman propagator SF 4× 4Vertex −igΓab 4× 4
u&v 4× 1External line
u&v 1× 4
(1× 4)(4× 4)(4× 4) · · · (4× 4)(4× 1)
row matrix u&v iΓ SF column matrix u&v
8.7.2 Feynman rules for a single fermion line from initial state to the finalstate
~p,r
~p− ~q1~p− ~q1 − ~q2 ~p′,s
~q1
~q2 ~q3
u(s)~p′ (−igΓ)SF (~p− ~q1 − ~q2)(−igΓ)SF (~p− ~q1)(−igΓ)u
(r)~p
1× 4 4× 4 4× 4 4× 4 4× 4 4× 4 4× 1
Remark 1: The arrow on the fermion line must flow consistently through the diagram,due to charge conservation.
Remark 2: The order of matrix multiplication is given by starting at the end of anarrow and working back to the start.
151
8.7.3 Relative sign between FDs
e.g. 1.
1 3
42
1 3
42
+(1↔ 3)
FDs with same fermion lines but two boson lines exchanged, then they differ by a“+” sign due to Bose statistics.
e.g. 2.
1
2
3
4
−(3↔ 4)
4
3
1
2
FDs width two fermion lines exchanged, then they differ by a relative sign “−”.
8.7.4 Minus sign from a loop of fermion lines
~p
~q − ~p
φ φ−igΓ−igΓ
~q
: ψ(x1)(−igΓ)ψ(x1)φ(x1)ψ(x2)(−igΓ)ψ(x2)φ(x2) :
=− (−igΓ)ψ(x1)ψ(x2)(−igΓ)ψ(x2)ψ(x1) : φ(x1)φ(x2) :
=− (−igΓ)SF (x1 − x2)(−igΓ)SF (x2 − x1) : φ(x1)φ(x2) :
=Tr((−igΓ)SF (x1 − x2)(−igΓ)SF (x2 − x1)) : φ(x1)φ(x2) : .
(8.7.1)
Remark: Loop of fermion line is the trace with an additional“−” sign
− Tr((−igΓ)SF (~p)(−igΓ)SF (~q − ~p)). (8.7.2)
8.8 Examples
8.8.1 First order
φ −→ N +N. (8.8.1)
152
N v(s)−~p 4× 1
N u(r)~p 1× 4
~p, r
−~p, s
φ
iA(1)fi = u
(r)~p (−igΓ)v
(s)−~p. (8.8.2)
8.8.2 Second order
1).
N + φ −→ N + φ. (8.8.3)
~q′~q
~p′, r′~p, r
φφ
NN u(r′)~p′u
(r)~p
+ (q ↔ q′)
iA(2)fi = u
(r′)~p′ (−igΓ)(SF (~p+ ~q) + SF (~p− ~q′))(−igΓ)u
(r)~p . (8.8.4)
2).
N + φ −→ N + φ. (8.8.5)
~q′~q
~p′, r′~p, r
φφ
NN v(r′)~p′v
(r)~p
+ (q ↔ q′))−(
iA(2)fi = −
v
(r)~p (−igΓ)(SF (~p+ ~q) + SF (~p− ~q′))(−igΓ)v
(r′)~p′
. (8.8.6)
There is a minus overall sign, due to the exchange of fermion operators.
Remark: overall sign has no contribution to cross section.
: ψ(x1)Γψ(x1)φ(x1)ψ(x2)Γψ(x2)φ(x2) : |N(p, r)〉
=(−1)3 : Γψ(x1)ψ(x2) Γψ(x2)φ(x1)φ(x2) : ψ(x1)|N(p, r)〉(8.8.7)
3).
N +N −→ N +N. (8.8.8)
153
~q − ~q′
~q′, s′~q, s
~p, r ~p′, r′
N
N
N
N
− (~q′, s′ ↔ ~p′, r′))−(
iA(2)fi =− (−ig)2
×
(u(s′)~q′ Γu
(s)~q )(v
(r)~p Γv
(r′)~p′ )DF (~q − ~q′)− (u
(r′)~p′ Γu
(s)~q )(v
(r)~p Γv
(s′)~q′ )DF (~q − ~p′)
.
(8.8.9)
: ψ(x1)Γψ(x1)φ(x1)ψ(x2)Γψ(x2)φ(x2) : |N(p1, r1),N(p2, r2)〉
=− : ψ(x1)Γφ(x1)ψ(x2)Γφ(x2) : ψ(x1)ψ(x2)|N(p1, r1),N(p2, r2)〉 .(8.8.10)
4).N +N −→ N +N. (8.8.11)
~p+ ~q
~p′, r′~p, r
~q′, s′~q, s
NN
NN
−(~p, r ↔ ~q′, s′)
N N
N N
~p− ~p′
~p′, r′~p, r
~q, s ~q′, s′
iA(2)fi =(−ig)2
×
(v(s)~q Γu
(r)~p )(u
(r′)~p′ Γv
(s′)~q′ )DF (~p+ ~q)− (v
(s)~q Γv
(s′)~q′ )(u
(r)~p Γu
(r′)~p′ )DF (~p− ~p′)
.
(8.8.12)
8.8.3 Fourth order
φ+ φ −→ φ+ φ. (8.8.13)
~k + ~p
~k − ~q
~k ~k + ~p− ~p′
~p′~p
~q ~q′
φφ
φφ
154
iA(4)fi = −(−ig)4
∫d4k
(2π)4Tr(SF (~k)ΓSF (~k + ~p)ΓSF (~k − ~q)ΓSF (~k + ~p− ~p′)Γ). (8.8.14)
8.9 Spin-sums and cross section
1 + 2 −→ 1′ + 2′. (8.9.1)
In CoM frame
~p
~q = −~p
~p′
~q′
θ
(dσ
dΩ′1
)
CoM
=1
64π2(E1 + E2)2× |~p
′||~p| |Afi|
2 (8.9.2)
where
dΩ′1 = d cos θdϕ. (8.9.3)
8.9.1 Spin-sums
It is not easy to measure the spin of particles in high energy experiments.
• Unpolarized incoming fermion particles and averaging over all incoming polariza-tions.
• All polarized outgoing particles sum over all final polarizations.
(dσ
dΩ′1
)
CoM
=1
64π2(E1 + E2)2× |~p
′||~p|
1
2
2∑
r,s=1
|Afi|2 . (8.9.4)
Lemma.
X =1
2
2∑
r,s=1
∣∣us(~p′)Gur(~p)∣∣2 , (8.9.5)
X =1
2Tr(p/′ +m)G(p/+m)G
), (8.9.6)
where G is product of gamma matrices and G = γ0G†γ0.
Proof.
X =1
2
2∑
r,s=1
(us(~p
′)Gur(~p)) (us(~p
′)Gur(~p))†, (8.9.7)
155
(us(~p
′)Gur(~p))†
=(u†s(~p
′)γ0Gur(~p))†
= u†r(~p)G†(γ0)†us(~p′)
= ur(~p)Gus(~p′)
(8.9.8)
with G = γ0G†γ0 and (γ0)† = γ0.
X =1
2
2∑
r,s=1
us(~p′)Gur(~p)ur(~p)Gus(~p′), (8.9.9)
X =1
2
2∑
r,s=1
usα(~p′)Gαβurβ(~p)urγ(~p)Gγδu
sδ(~p′)
=1
2
2∑
r,s=1
(usδ(~p
′)usα(~p′))Gαβ
(urβ(~p)urγ(~p)
)Gγδ.
(8.9.10)
Noted that r, s are spin index, and α, β, γ, δ are matrix component.
Because of completeness relation
2∑
s=1
usδ(~p′)usα(~p′) = (p/′ +m)δα, (8.9.11)
2∑
r=1
urβ(~p)urγ(~p) = (p/+m)βγ , (8.9.12)
thus we have
X =1
2(p/′ +m)δαGαβ(p/+m)βγGγδ
=1
2Tr(
(p/′ +m)G(p/+m)G).
(8.9.13)
Corollary.
1
2
2∑
r,s=1
∣∣vs(~p′)Gur(~p)∣∣2 =
1
2Tr(p/′ −m)G(p/+m)G
); (8.9.14)
1
2
2∑
r,s=1
∣∣us(~p′)Gvr(~p)∣∣2 =
1
2Tr(p/′ +m)G(p/−m)G
); (8.9.15)
1
2
2∑
r,s=1
∣∣vs(~p′)Gvr(~p)∣∣2 =
1
2Tr(p/′ −m)G(p/−m)G
). (8.9.16)
156
8.9.2 Examples
See Luke’s notes P.P. 166-167.
iA(2)fi = ig2u
(s)~p′ γ5
[(p/+ q/+m)
(p+ q)2 −m2 + iε+
(p/− q/′ +m)
(p− q′)2 −m2 + iε
]γ5u
(r)~p
= ig2u(s)~p′ γ
µqu(r)~p F (p, p′, q)
(8.9.17)
where
F (p, p′, q) =1
2p · q + µ2 + iε+
1
2p′ · q − µ2 + iε. (8.9.18)
1
2
2∑
r,s=1
∣∣∣iA(2)fi
∣∣∣2
=1
2g4F (p, p′, q)2qµqνTr
((p/′ +m)γµ(p/+m)γν
)
= 2g4F (p, p′, q)2[2(p′ · q)(p · q)− p · p′µ2 +m2µ2
].
(8.9.19)
157
Lecture 9 Quantum Electrodynamics
About Lecture VII:
• [Luke] P.P. 168-193;
• [Tong] P.P. 124-140;
• [Coleman] empty;
• [PS] P.P. 123-174;
• [Zhou] P.P. 141-181.
• [MS] P.P. 99-160.
This semester, we have spent much time in both technique details and solutions tohomeworks, so that we only have a little time in Quantum Electrodynamics (QED), whichis the key topic of this course. Fortunately, we have prepared everything for perturbativeQED in sense of Feynman diagrams. Recall that we have discussed the pseudo-Yukawatheory (describing the interaction between pseudo-nucleons (complex scalar particles) andreal scalar particles) and the Yukawa theory (describing the interaction between nucleons(spin-1/2 particles) and real scalar particles), both of which present a good guidance forperturbative QED with the interaction between electrons (spin-1/2 particles) and photons(real vector particles).
9.1 Model
The Lagrangian of Quantum Electrodynamics (QED) is
L = −1
4FµνF
µν + ψ(iγµ∂µ −m)ψ + LI , (9.1.1)
in which the interaction Lagrangian density is LI = −eψγµψAµ, and the vector field Aµrepresents electromagnetic fields and the bispinor field ψ represent the Dirac field. Notethat the photon as quanta of electromagnetic fields Aµ has the following properties,
1). scalar-like; massless;
2). with vector index; polarization up or down.
The interaction Hamiltonian density is HI = eψγµψAµ with coupling constant e, andthe scattering matrix via the Dyson’s formula is
S(+∞,−∞) = T exp(−i∫HId4x). (9.1.2)
Feynman rules for Feynman diagrams including vector fields in QED:
158
1). Interaction vertex. Assign the factor −ieγµ at each vertex.
µ
−ieγµ
ψ
ψ
Aµ
2). Internal line. Feynman propagator of vector fields.
Aµ(x)Aν(y) = 〈vac|TAµ(x)Aν(y)|vac〉. (9.1.3)
k
νµ−igµνk2+iε
3). External lines.
Incoming photon with the polarization vector ε(r)µ (k).
k, µ
Aµ|k, r〉 = ε(r)µ (k)e−ik·x . (9.1.4)
Outgoing photon with the polarization vector ε(r)∗ν (k).
k, ν
〈k, r|Aν(x) = ε(r)∗ν (k)eik·x. (9.1.5)
The polarization sum is the completeness relation of the polarization vectors,
2∑
r=1
ε(r)µ ε(r)∗
ν = −gµν (9.1.6)
where r is the polarization parameter and µ, ν are the space-time indices.
9.2 Representative scattering processes in QED
The following Feynman diagrams are new things from the viewpoint of QED, but areindeed the closest relatives of our old friends which we have already studied in detail inboth the pseudo-Yukawa theory and the Yukawa theory.
159
a). The Moller scattering between electrons.
3. e− + e− −→ e− + e−. (9.2.1)
2. N +N −→ N +N. (9.2.2)
1. “N” + “N” −→ “N” + “N”. (9.2.3)
p− p′
p′, s′p, s
q, r q′, r′
e−
e−
e−
e−
µ
ν− (p′, s′ ↔ q′, r′)
iA(2)fi = (−ie)2
(u
(s′)~p′ γ
µu(s)~p
) −igµν(p− p′)2
(u
(r′)~q′ γ
νu(r)~q
)− (−ie)2(r′ ↔ s′, p′ ↔ q′)
= (−ie)2(−i)
(u
(s′)~p′ γ
µu(s)~p
)(u
(r′)~q′ γµu
(r)~q
)
(p− p′)2− (r′ ↔ s′, p′ ↔ q′)
.
(9.2.4)
Similarly, we can compute the scattering process between positrons:
3. e+ + e+ −→ e+ + e+. (9.2.5)
2. N +N −→ N +N. (9.2.6)
1. “N” + “N” −→ “N” + “N”. (9.2.7)
b). The Bhabha scattering between an electron and a positron.
3. e− + e+ −→ e− + e+. (9.2.8)
2. N +N −→ N +N. (9.2.9)
1. “N” + “N” −→ “N” + “N”. (9.2.10)
e− e−
e+ e+
p− p′
p′, s′p, s
q, r q′, r′
−(q, r ↔ p′, s′)
p+ q
p′, s′p, s
q′, r′q, r
e−e−
e+e+
iA(2)fi = −(−ie)2(−i)
(u
(s′)~p′ γ
µu(s)~p
)(v
(r)~q γµv
(r′)~q′
)
(p− p′)2−
(v
(r)~q γµu
(s)~p
)(u
(s′)~p′ γµv
(r′)~q′
)
(p+ q)2
(9.2.11)
160
where the minus sign in front of (−ie)2 denotes the overall minus sign.
Note that the following notes are about the lecture given on 6/17, 2014.
c). The pair annihilation between an electron and a positron into two photons.
3. e− + e+ −→ 2γ. (9.2.12)
2. N +N −→ 2φ. (9.2.13)
1. “N” + “N” −→ 2φ. (9.2.14)
e−
e+
γ
γ
p− p′
p, s
q, r
p′, εν1
q′, εµ2
µ
ν
+ (p′, ε1 ↔ q′, ε2)
iA(2) = (−i)(−ie)2v(r)~q
[γµ(p/− p/′ +m)γν
(p− p′)2 −m2+γν(p/− q/′ +m)γµ
(p− q′)2 −m2
]u
(s)~p εµ∗2 εν∗1 . (9.2.15)
d). The Compton scattering between an electron and a photon.
3. e− + γ −→ e− + γ. (9.2.16)
2. N + φ −→ N + φ. (9.2.17)
1. “N” + φ −→ “N” + φ. (9.2.18)
(9.2.19)
p+ q
q′, εµ2q, εν1
p′, s′p, s
γγ
e−e−
ν µ + (q, ε1 ↔ q′, ε2)
iA(2) = −(−ie)2(−i)u(s′)~p′
[γµ(p/+ q/+m)γν
(p+ q)2 −m2+γν(p/− q/′ +m)γµ
(p− q′)2 −m2
]u
(s)~p εµ∗2 εν1 (9.2.20)
where the minus sign in front of (−ie)2 denotes the overall minus sign.
e). Two-Photon scattering.
2. γ + γ −→ γ + γ. (9.2.21)
1. φ+ φ −→ φ+ φ. (9.2.22)
−→
161
iA(4) ∝∫
d4k
(2π)4
1
k4−→ +∞ (9.2.23)
in which the integral becomes the infinity, so only regularized and renormalized Feynmanintegrals make sense in quantum field theory.
9.3 The calculation of the differential cross section of theBhabha scattering in second order of coupling constant
The Bhabha scattering has the form of reaction,
e+(~p1, r1) + e−(~p2, r2) −→ e+(~p′1, s1) + e−(~p′2, s2), (9.3.1)
which gives rise to two Feynman diagrams in second order of coupling constant
e− e−
e+ e+
p1 − p′1
p′2, s2p2, r2
p1, r1 p′1, s1
− p1 + p2
p′2, s2p2, r2
p′1, s1p1, r1
e−e−
e+e+
The associated Feynman integral has the form
iA(2) = iAa + iAb. (9.3.2)
with
Aa = −ie2[us2(~p′2)γαur2(~p2)
] 1
(p1 − p′1)2
[vr1(~p1)γαvs1(~p′1)
]; (9.3.3)
Ab = ie2[us2(~p′2)γαvs1(~p′1)
] 1
(p1 + p2)2[vr1(~p1)γαur2(~p2)] . (9.3.4)
In the center-of-mass frame, the kinematics of the two-particle collision given by
p1 = (E, ~p) p2 = (E,−~p)
p′1 = (E, ~p′)
p′2 = (E,−~p′)
θ e−e+
e+
e−
p1 · p′1 = E2 − |~p|∣∣~p′∣∣ cos θ; (9.3.5)
p1 · p′2 = E2 + |~p|∣∣~p′∣∣ cos θ; (9.3.6)
p1 · p2 = E2 + |~p|2 ; (9.3.7)
p′1 · p′2 = E2 +∣∣~p′∣∣2 , (9.3.8)
andET = E1 + E2 = 2E, E2
T = 4E2 = (p1 + p2)2, (9.3.9)
162
and∣∣~p′∣∣ = |~p| ⇐⇒ me+ = me− ,
p1 · p′1 = E2 − ~p · ~p′ = E2 − |~p|2 cos θ,
(p1 − p′1)2 = −(~p− ~p′)2 = −4 |~p| sin2 θ
2, (9.3.10)
the differential cross section of the Bhabha scattering with polarized particles in secondorder of coupling constant has the form
(dσ
dΩ′1
)
CoM
=1
64π2E2T
|Afi|2|~p′||~p|
=1
256E2π2|Afi|2 .
(9.3.11)
In high energy particle physics, we usually do not measure the spin polarizations ofincoming or outgoing particles, so we perform the spin average on incoming particles andthe spin sum on outgoing particles, namely,
1
2
∑
r1
1
2
∑
r2
∑
s1
∑
s2
=1
4
∑
r1,r2,s1,s2
(9.3.12)
which gives rise to the following the scattering amplitude
∣∣∣iA(2)fi
∣∣∣2
=1
4
∑
r1r2s1s2
|iAa + iAb|2 = (Xa +Xb +Xab +X∗ab), (9.3.13)
with
Xa =1
4
∑
r1,r2,s1,s2
|Aa|2 ; (9.3.14)
Xb =1
4
∑
r1,r2,s1,s2
|Ab|2 ; (9.3.15)
Xab =1
4
∑
r1,r2,s1,s2
AaA∗b ; (9.3.16)
X∗ab =1
4
∑
r1,r2,s1,s2
A∗aAb. (9.3.17)
For example, let us calculate Xb in detail:
Ab = ie2[us2(~p′2)γαvs1(~p′1)
] 1
(p1 + p2)2[vr1(~p1)γαur2(~p2)] . (9.3.18)
and its complex conjugation
A∗b = A†b = −ie2[vs1(~p′1)γβus2(~p′2)
] 1
(p1 + p2)2
[ur2(~p2)γβvr1(~p1)
], (9.3.19)
withγβ ≡ γ0γ†βγ
0 = γβ, γβ ≡ γ0(γβ)†γ0 = γβ. (9.3.20)
has the form
A∗b = −ie2[vs1(~p′1)γβus2(~p′2)
] 1
(p1 + p2)2
[ur2(~p2)γβvr1(~p1)
](9.3.21)
163
so that the Xb has the form as the product of traces of gamma matrices
Xb =1
4
e4
(p1 + p2)4AαβB
αβ, (9.3.22)
where the spin sum is performed to derive the following trace formulaes:
Aαβ =∑
s1,s2
(us2(~p′2)γαvs1(~p′1)
) (vs1(~p′1)γβus2(~p′2)
)
= Tr((p/′2 +m)γα(p/′1 −m)γβ
),
(9.3.23)
andBαβ = Tr
((p/1 −m)γα(p/2 +m)γβ
). (9.3.24)
With the trace properties of gamma matrices, the Aαβ is expressed as
Aαβ = Tr((p/′2γα +mγα)(p/′1γβ −mγβ)
)
= Tr(p/′2γαp/
′1γβ −mp/′2γαγβ +mγαp/
′1γβ −m2γαγβ
)
= Tr(p/′2γαp/
′1γβ)−m2Tr(γαγβ)
= pµ2′pν1′(2gµα2gνβ − 2gµν2gαβ + 2gµβ2gαν)− 4m2gαβ
= 4(p′1αp
′2β + p′2αp
′1β − (m2 + p′1 · p′2)gαβ
).
(9.3.25)
and similarly, the Bαβ has the form
Bαβ = 4(pα1 p
β2 + pα2 p
β1 − (m2 + p1 · p2)gαβ
). (9.3.26)
With the kinematics of the two-particle scattering in the center-of-mass frame, the Xb
is calculated as
Xb =e4
4(p1 + p2)4× 16×
[(p1 · p′1)(p2 · p′2) + (p′1 · p2)(p1 · p′2)
− (m2 + p1 · p2)p′1 · p′2 + (p1 · p′2)(p′1 · p2) + (p2 · p′2)(p1 · p′1)
− (m2 + p1 · p2)p′2 · p′1 − p1 · p2(m2 + p′1 · p′2)− (m2 + p′1 · p′2)p1 · p2
+4(m2 + p′1 · p′2)(p1 · p2 +m2)],
(9.3.27)
and it can simplified as
Xb =e4
4(p1 + p2)4× 32×
[(p1 · p′1)(p2 · p′2)
+ (p1 · p′2)(p2 · p′1) + (p′1 · p′2)m2 +m2p1 · p2 + 2m4].
(9.3.28)
which has a further simplified form
Xb =e4
2E4
((E2 − |~p|
∣∣~p′∣∣ cos θ)2 + (E2 + |~p|
∣∣~p′∣∣ cos θ)2
+ m2(E2 +∣∣~p′∣∣2) +m2(E2 + |~p|2) + 2m4
).
(9.3.29)
In the approximation E me, E ≈ |~p′| = |~p|, the Xb can be eventually simplified as
Xb =e4
E4
(E4 + |~p|2
∣∣~p′∣∣2 cos2 θ +O(m2E2)
)
=e4
(1 + cos2 θ +O(
m2
E2)
) (9.3.30)
164
which gives rise to the part of the differential cross section,
(dσ
dΩ′1
)
Xb
∝ 1
256π2E2e4
(1 + cos2 θ +O(
m2
E2)
)
=1
16E2α2
(1 + cos2 θ +O(
m2
E2)
),
(9.3.31)
with the fine structure constant α = e2/4π or α = 1/137.Similarly, we can calculate Xa,
Xa =1
4
e4
(p1 − p′1)4
∑
r1,r2,s1,s2
A′αβB′αβ
=1
4
e4
(p1 − p′1)4× 128E4 ×
(1 + cos4 θ
2+O(
m2
E2)
)
=1
4
e4
16E4 sin4 θ2
× 128E4 ×(
1 + cos4 θ
2+O(
m2
E2)
)
=2e4
sin4 θ2
(1 + cos4 θ
2+O(
m2
E2)
).
(9.3.32)
and the associated part of the differential cross section has the form
(dσ
dΩ′1
)
Xa
∝ 1
8E2α2 1 + cos4 θ
2
sin4 θ2
. (9.3.33)
Furthermore, we can calculate Xab
Xab =−2e4
sin2 θ2
(cos4 θ
2+O(
m2
E2)
), (9.3.34)
which happens to be a real number so Xab = X∗ab in sense of the approximation of O(m2
E2 ).As a conclusion, the differential cross section of the Bhabha scattering in second order
of coupling constant has the form
(dσ
dΩ′1
)=
α2
8E2
(1 + cos4 θ
2
sin4 θ2
+1 + cos2 θ
2− 2 cos4 θ
2
sin2 θ2
+O(m2
E2)
)(9.3.35)
which has been verified in experiments, see the reference book [MS].
165
Part II
Renormalization and Symmetries
166
Lecture 10 Notes on BPHZ Renormalization
A .pdf file is attached for the time being. Later on, it will be replaced by a .tex file.
167
Draft for Internal Circulations:v1: August 19, 2003, Universitat Leipzig, Germany.
v2: September 4, 2004, Institute of Theoretical Physics, Chinese Academy of Sciences.
v3: Almost unchanged from v2, April 16, 2015.
Notes on BPHZ Renormalization1 2
Yong Zhang3
School of Physics and Technology, Wuhan University
(April 2015)
Abstract
This article gives an introduction to the BPHZ renormalization in Zimmerman-n’s sense. The historical development of the BPHZ renormalization is firstlyreviewed, and then as an example, a massive scalar model defined by the BPHZrenormalization is given. Afterwards, the Zimmermann‘s forest formula, thequantum action principle, the Zimmermann identities, the Callan–Symanzikequation, the algebraic renormalization procedure, charge constructions vi-a the LSZ reduction procedure, and the Connes–Kreimer Hopf algebra arepresented.
1A part of my thesis for a PhD. degree (2003) in Universitat Leipzig, Germany.2These notes are uploaded online for who are interested in theoretical research of quantum field theory3yong [email protected].
1
Contents
1 Notations 3
2 Introduction 62.1 The history of the BPHZ renormalization . . . . . . . . . . . . . . . . . . . 72.2 The massive ϕ4 model via the BPHZ renormalization . . . . . . . . . . . . . 8
3 The Zimmermann’s forest formula 113.1 A simple introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 A general calculation procedure . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Examples in two-loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 The quantum action principle 194.1 The equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 The renormalized version in higher orders . . . . . . . . . . . . . . . . . . . 21
5 The Callan–Symanzik equation 245.1 The Zimmermann identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Deriving the Callan–Symanzik equation . . . . . . . . . . . . . . . . . . . . 285.3 The different representations of the Callan–Symanzik equation . . . . . . . 31
6 The algebraic renormalization procedure 336.1 The construction of the Slavnov–Taylor identity . . . . . . . . . . . . . . . . 346.2 The consistency condition to all orders . . . . . . . . . . . . . . . . . . . . . 36
7 Current constructions and charge constructions 377.1 The conformal algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.2 Current constructions and charge constructions . . . . . . . . . . . . . . . 38
8 The Connes–Kreimer Hopf algebra 418.1 The axioms of the Hopf algebra . . . . . . . . . . . . . . . . . . . . . . . . . 418.2 An example: the Connes–Kreimer Hopf algebra . . . . . . . . . . . . . . . . 43
9 Acknowledgements 47
2
1 Notations
Following general conventions, we would like to choose the natural units: ~ = 1, c = 1.But in the algebraic renormalization procedure, it is both reasonable and prudent to keep~ in all formulas. On the one hand, in quantum field theories with spontaneously brokensymmetries such as the standard model, an expansion in loop numbers is a better choicethan in coupling constants. On the other hand, the quantum action principle representedby the 1PI Green functions is a suitable formulation when it is applied to the problemof “renormalization and symmetries”. Furthermore, the loop number in 1PI diagrams isright the power counting of ~, as suggests that keeping ~ in formulas makes much clear theinduction proof based on the loop expansions. Moreover, in the sense of the dimensionalanalysis, keeping ~ is a good way of checking calculation if right or not.
The metric ηµν in four dimensional flat space-time is chosen as
(ηµν) = diag(1, −1,−1, −1). (1)
The Feynman propagator in the article is given by
∆F (x1 − x2) = i~∫
d4k
(2π)4e−i k(x1−x2) Dk, (2)
where ϵ is a positive parameter and the symbol Dk is denoted by
Dk =1
k2 − m2 + iϵ. (3)
• Z [J ], ZC [J ], and Γ[ϕ]
The generating functional of the general Green functions, Z [J ], is expanded as thepower series of a given external source J(x),
Z [J ] =
∞∑
n=0
1
n!
(i
~
)n ∫d4x 1
∫d4x 2 · · ·
∫d4x n
J(x1)J(x2) · · · J(xn)⟨0|Tϕ(x1)ϕ(x2) · · · ϕ(xn)|0⟩, (4)
where a n-point general Green function is denoted by
Gn(x1, x2 · · · xn) = ⟨0|Tϕ(x1)ϕ(x2) · · ·ϕ(xn)|0⟩. (5)
The generating functional of the connected Green functions, ZC [J ], is defined by
ZC [J ] := −i~ lnZ [J ], (6)
which can also be expanded as
ZC [J ] =
∞∑
n=1
1
n!
(i
~
)n−1 ∫d4x 1
∫d4x 2 · · ·
∫d4x n
J(x1)J(x2) · · · J(xn)⟨0|Tϕ(x1)ϕ(x2) · · · ϕ(xn)|0⟩c, (7)
where a n-point connected Green function is denoted by
Gn(x1, x2 · · ·xn)c = ⟨0|Tϕ(x1)ϕ(x2) · · ·ϕ(xn)|0⟩c. (8)
3
The generating functional of the 1PI(one particle irreducible) Green functions, Γ[ϕ],also called the vertex functional, is defined by the Legendre transformation,
Γ[ϕ] := ZC [J ] −∫
d4xJ(x) ϕ(x), (9)
which defines ϕ(x) and J(x) by
ϕ(x) =δZC [J ]
δJ(x), J(x) = − δΓ
δϕ(x). (10)
The generating functional of the 1PI(one particle irreducible) Green functions, Γ[ϕ],can also be expanded as a power series of the classical field ϕ(x) by
Γ[ϕ] =
∞∑
n=2
1
n!
∫d4x 1
∫d4x 2 · · ·
∫d4x n
ϕ(x1)ϕ(x2) · · · ϕ(xn)⟨0|Tϕ(x1)ϕ(x2) · · · ϕ(xn)|0⟩1PI . (11)
where a n-point 1PI Green function is denoted by
Γn(x1, x2 · · ·xn) = ⟨0|Tϕ(x1)ϕ(x2) · · · ϕ(xn)|0⟩1PI . (12)
• Z [J, ρ], ZC [J, ρ], and Γ[ϕ, ρ]
The generating functional of the general Green functions Z [J, ρ] with insertions ofcomposite operators QA(x) is defined by
Z [J, ρ] := ⟨0|T expi
~
∫d4xJ(x)ϕ(x) + ρA(x)QA(x)|0⟩ (13)
where the symbol A denotes quantum numbers or space-time indices.The generating functional of the connected Green functions, ZC [J, ρ], with insertions
of composite operators QA(x), is defined by
ZC [J, ρ] := −i~ lnZ [J, ρ]. (14)
The generating functional of the 1PI(one particle irreducible) Green functions, Γ[ϕ, ρ]with insertions of composite operators QA(x), is defined by the transformation,
Γ[ϕ, ρ] := ZC [J, ρ] −∫
d4xJ(x) ϕ(x), (15)
which also defines ϕ(x) and J(x) by
ϕ(x) =δZC [J, ρ]
δJ(x), J(x) = −δΓ[ϕ, ρ]
δϕ(x). (16)
• Insertions of composite operators QA(x)
An insertions of composite operators QA(x) into the general Green function are definedby
QA1(y1) · QA2(y2) · · ·QAm(ym) · Gn(x1, x2, · · · xn)
:= ⟨0|T QA1(y1)QA2(y2) · · · QAm(ym)ϕ(x1)ϕ(x2) · · ·ϕ(xn)|0⟩
=(−i~)n+mδn+m
δρA1(y1)δρA2(y2) · · · δρAm(ym)δJ(x1)δJ(x2) · · · δJ(xn)Z [J, ρ]|J=ρ=0. (17)
4
An insertion of composite operators QA(x) into the connected Green function aredefined by
QA1(y1) · QA2(y2) · · · QAm(ym) · Gn(x1, x2, · · ·xn)c
:= ⟨0|T QA1(y1)QA2(y2) · · · QAm(ym)ϕ(x1)ϕ(x2) · · · ϕ(xn)|0⟩c
=(−i~)n+m−1δn+m
δρA1(y1)δρA2(y2) · · · δρAm(ym)δJ(x1)δJ(x2) · · · δJ(xn)ZC [J, ρ]|J=ρ=0. (18)
An insertions of composite operators QA(x) into the 1PI Green function are definedby
QA1(y1) · QA2(y2) · · ·QAm(ym) · Γn(x1, x2, · · · xn)
:= ⟨0|T QA1(y1)QA2(y2) · · · QAm(ym)ϕ(x1)ϕ(x2) · · ·ϕ(xn)|0⟩1PI
=(−i~)m−1δn+m
δρA1(y1)δρA2(y2) · · · δρAm(ym)δϕ(x1)δϕ(x2) · · · ϕ(xn)Γ[ϕ, ρ]|J=ρ=0. (19)
In the simplest case, a insertion of one composite operator, QA(x) ·Γ can be expandedas
QA(x) · Γ = QA(x) + O(~ QA(x)), (20)
where the second term denotes non-local contributions from loop Feynman diagrams.Relating insertions of quantum composite operators to variations of external fields is avery pragmatic approach, which brings much convenience to deal with quantum correctionsto classical composite operators and hence makes possible the algebraic renormalizationprinciple: For example, it is a necessary way to construct the linearized Slavnov–Tayloridentity operator in a non-abelian gauge field theory.
• Quantum action principle
With insertions of composite operators into the 1PI Green function, the quantumaction principle can be represented by
δΓ[ϕ, ρ]
δρA(x)= ∆A(x) · Γ[ϕ, ρ],
δΓ[ϕ, ρ]
δϕa(x)= ∆a(x) · Γ[ϕ, ρ],
∂Γ[ϕ, ρ]
∂λ= ∆λ · Γ[ϕ, ρ],
ϕb(x)δΓ[ϕ, ρ]
δϕa(x)= ∆a
b (x) · Γ[ϕ, ρ],
δΓ[ϕ, ρ]
δρA(x)
δΓ[ϕ, ρ]
δϕa(x)= ∆a
A(x) · Γ[ϕ, ρ], (21)
where the symbol ϕa denotes the field component.In the BPHZ renormalization, a quantum composite operator represented by a corre-
sponding normal product, the quantum action principle is denoted by
δΓ[ϕ, ρ]
δρA(x)= [
δΓeff
δρA(x)] · Γ[ϕ, ρ],
5
δΓ[ϕ, ρ]
δϕa(x)= [
δΓren
δϕa(x)] · Γ[ϕ, ρ],
∂Γ[ϕ, ρ]
∂λ= [
∂Γren
∂λ] · Γ[ϕ, ρ],
ϕb(x)δΓ[ϕ, ρ]
δϕa(x)= [ϕb(x)
δΓren
δϕa(x)] · Γ[ϕ, ρ]. (22)
• Γ, Γeff , Γren and Γint
In our case, the four symbols Γ, Γeff , Γren and Γint are often used. The Γ denotesthe generating functional of vertex functions; the Γeff denotes the effective action, whichis needed to discuss the structure of vacuum in models with spontaneously broken sym-metries; the Γeff can be obtained by calculating Γ because they are different faces of thesame thing. The Γren denotes the local part of the Γeff(or Γ) is obtained by consideringcontributions only from tree Feynman diagrams, and is applied to generate insertions ofnormal products in the quantum action principle in the BPHZ renormalization. The Γint
denotes a non-free part of the Γren, namely interaction terms in the classical action Γcl
plus all counter terms, and is used to generate all possible vertices in the Gell-Mann– Lowformulae.
However, in [38], [45] and [46], for a perturbative quantum field theory defined by thealgebraic renormalization procedure, the symbols Γ, Γeff , and Γint are always used: theΓeff plays the same role as the Γren in our case and it is always finite in the BPHZ renor-malization. Moreover, for a perturbative quantum field theory defined by the dimensionalregularization and the dimensional renormalization in [32], the symbols Γ, Γeff , Γren andΓint are used and they can be introduced similar to those in our case. But the Γren is notfinite since it includes counter terms with poles, and is also not a local part of Γ(Γeff) sincethe latter one is always finite.
Notes
The above notations have been almost presented in [45] and [46]. But there are somedifferences. In [45], the Euclidean space-time is taken from the begin to the end, and in[46], the Planck constant ~ is not shown explicitly in all formulas. In addition, here, it isspecially emphasized that insertions of quantum composite operators can be treated byintroducing external fields.
2 Introduction
The BPHZ renormalization denotes a procedure developed by Bogoliubov, Parasiuk, Heppand Zimmermann in a series papers, see [1], [2], [3], [4] and [5], which equips a perturbativequantum field theory with exact rules to compute the renormalized Green functions andderive relations among them. It can be divided into the two parts: BPH renormalizationand Zimmermann’s recipe. Although the BPH renormalization is by itself a rigorousand complete renormalization scheme, in this article the BPHZ renormalization will beintroduced based on the Zimmermann’s papers.
In the following, a historical development of the BPHZ renormalization will be firstlyreviewed in a simple way, and then naturally induces an outline of the article. Secondly,as an example, a massive ϕ4 model will be defined by applying the BPHZ renormalization.
6
2.1 The history of the BPHZ renormalization
In the BPHZ renormalization, the Zimmermann’s forest formulae in [3], the momentumsubtraction scheme, and the normal product algorithm in [4] and [5], are the most fun-damental elements, from which the Zimmermann’s identities in [5], the Callan–Symanzikequation in [6] and the quantum action principle in [7] and [8], can be derived.
Before Zimmermann’s original work on the forest formulae, at least the three thingshad been well prepared: in the practical calculation it was realized that the momentumsubtraction procedure makes the divergent Feynman integral convergent, see [9], [10] and[1]; the power counting theorem had been proved in the Euclidean space-time, see [11],in order to obtain the asymptotic behaviour of the Feynman integral in the high energy-momenta domain; the R operation completely solved the nonlocal problem induced byoverlapping divergences and could be used to define the finite S-matrix, see [1] and [2].Therefore, first of all, the proof for the power counting theorem would be simplified in [12]and had to be generalized to be also right in the Minkowski’s space-time in [13]. Second,a solution of the R operation without a recursive recipe was found out and had to beproved to make the divergent Feynman integral convergent by applying the momentumsubtraction procedure to the unrenormalized Feynman integrand, see [3]. Third, the laterprogress showed that this type of solution together with other regularization schemeswas also successful, for example see the case of the dimensional regularization and thedimensional renormalization in [14] and [15]. Hence in a common sense they are calledZimmermann’s forest formulae.
The BPHZ renormalization is different from others such as the dimensional renor-malization mainly because of the momentum subtraction scheme. It is a normal Taylorexpansion up to the order specified by the subtraction degree. On the one hand, therefore,no any regularization schemes are involved, and then only finite coefficients appear in therenormalized action Γren. That is to say that only physical parameters will be treated andthe bare ones are not needed. Naturally, those physical ones are fixed by the renormaliza-tion conditions. On the other hand, furthermore, it is controlled by the subtraction degreebeing a positive number not less than the superficial divergence degree. This suggests thatthe oversubtraction problem can be well treated. This results in the Zimmermann’s iden-tities, see [4], the best formulae relating the case of the subtraction and that of the oversubtraction.
The normal product algorithm is combining the momentum subtraction scheme withthe forest formulae to define the insertion of the renormalized composite operators into therenormalized Green function in a rigorous way from the mathematical point, see [4] and[5]. With its help, the equations of motion could be expressed in a renormalized version,the Zimmermann’s identities was derived, and the operator product expansions could beproved, see [16]. It was improved in [17] and [18] and used to derive the Callan–Symanzikequation in [6]. Furthermore, in [6], [17] and [18] it was realized that derivations of pa-rameters or linear transformations of fields on the vertex functional Γ are correspondingto local insertions of normal products into Γ, which had been concluded in the quantumaction principle represented by several differential equations. Moreover, the Ward identi-ties for linear transformations, such as the U(1) gauge transformation in QED, could beconstructed to all orders of the Planck constant ~ in the BPHZ renormalization, see [19],[20] and [21], as shed a light on the subject of “renormalization and symmetries”.
The quantum action principle for non-linear transformations was proved in [7, 8], whichsuggested that symmetries represented by non-linear transformations could be well treatedin the BPHZ renormalization. For example, in the non-abelian gauge field theory with
7
the spontaneously broken symmetries, the Slavnov–Taylor identity can be constructedto all orders of ~, see [22] and [23]: the point is applying the quantum action principlerepresented by the normal product algorithm to change the problem of “renormalizationand symmetries” into the algebraic one of how to solve the cohomology of local inte-grals with constraints from the power counting theorem, discrete symmetries and so on.Later progress shows that the quantum action principle could be also derived in otherrenormalization schemes, see [14], [24], [25], [26]. Hence the approach of constructing therenormalized Slavnov–Taylor identity or the renormalized Ward identity could be arguedto be independent of renormalization procedures and regularization schemes, and thencalled in a common sense by the algebraic renormalization procedure. In addition, via theLSZ reduction procedure in [27], charges responsible for the BRST transformations wasconstructed and the unitarity of the S-matrix could be proved in [28] and [29].
The BPHZ renormalization invented in [5] was mainly devised for massive models andbased on the momenta subtraction at the zero-momenta subtraction point. Its generalizedversion in [30] was also suitable for massless models, called by the BPHZL renormaliza-tion. It had been also applied in supersymmetrical field theories, in [31]. Afterwards,it didn’t make a big progress for a long time. There are at least two types of reasons.The algebraic renormalization procedure is more interesting since physical results have tobe independent of choices of renormalization schemes. Furthermore, for the high energyphenomenology physics, the momenta subtraction scheme can’t compete with the dimen-sional regularization in [32]. However, until now, the BPHZ renormalization is still thebest approach of showing the quantum action principle by the normal product algorithm.Recently, moreover, Connes and Kreimer discovered the Hopf algebra behind the Zimmer-mann’s forest formulae, see [33] and [34], which means that a lot of things in the BPH orBPHZ renormalization need to be revived.
The outline of the article is given in the following. Firstly, as an example, the massiveϕ4 model will be defined by the BPHZ renormalization, Secondly, the forest formulaewill be explained by examples in two-loop. Thirdly, the quantum action principle will beintroduced and its proof in [7] will be sketched. Then the Zimmermann’s identities willbe introduced by examples and used to derive the Callan–Symanzik equation. Fourthly,the algebraic renormalization procedure will be introduced in a common way, currentconstructions and charge constructions via the LSZ reduction will be given in detail.Lastly, the Connes–Kreimer Hopf algebra will be reviewed in the Zimmermann’s sense.
2.2 The massive ϕ4 model via the BPHZ renormalization
As an example, the well-defined massive ϕ4 model in the BPHZ renormalization is intro-duced. Namely, the renormalized action, the renormalization conditions, the Zimmerman-n’s forest formula, the renormalized Green function, the renormalized Green function withthe insertions of the normal products, the quantum action principle and the Zimmermannidentities are presented.
The renormalized action Γren can be regarded as the sum of the free part Γ0 and theinteraction part Γint,
Γren = Γ0 + Γint = −z ∆1 − a∆2 − ρ ∆4. (23)
In the tree approximation, the coefficients z, a, ρ are specified by
z(0) = 1, a(0) = m2, ρ(0) = λ, (24)
8
where the upper indices denote the power counting of ~ in this section. The normalproducts ∆1, ∆2, ∆4 are given by
∆1 =
[∫d4x 1
2ϕ(x)2ϕ(x)
]
4
, (25)
∆2 =
[∫d4x 1
2ϕ2(x)
]
4
, (26)
∆4 =
[∫d4x
1
4!ϕ4(x)
]
4
. (27)
The free part Γ0 is given by −∆1 − m2∆2 determining the propagator. The renormalizedLagrangian density is chosen to be
Lren = −12 z ϕ2ϕ − 1
2 aϕ2 − 1
4!ρϕ4, (28)
but it can be changed by adding total derivatives. On the other hand, the renormalizedaction Γren is also the sum of the classical action Γcl and all the possible local countertermsΓcounter, namely,
Γren = Γcl + Γcounter = −∫
(12 ϕ(2 + m2)ϕ +
1
4!λϕ4) + O(~). (29)
In higher orders, the coefficients z, a, ρ are decided by the renormalization conditions
Γ2(p,−p) |p2=m2 = 0, (30)
∂p2Γ2(p, −p) |p2=µ2 = 1, (31)
Γ4(p1, p2, p3, p4) |Q = −λ, (32)
where m is the physical mass, µ is the normalization mass denoting the renormalizationscale, λ is the physical coupling constant and Q is the set given by
Q = pi | p2i = µ2, (pi + pj)
2 =4
3µ2; i = j; i, j = 1, 2, 3, 4. (33)
Γ2(p, −p) and Γ4(p1, p2, p3, p4) are the two-point 1PI Green function and the four-point1PI Green function respectively. Here the rule of the Fourier transformation of an ordinaryfunction F , such as a Green function or an 1PI Green function, between the momentumspace and the coordinate space has been chosen as
F (p1, p2, · · · , pn) = (2 π)4 δ4
(n∑
i=1
pi
)F (p1, p2, · · · , pn)
=
∫d4x1 d4x2 · · · d4xn ei
∑nj=1 pj ·xj F (x1, x2, · · · , xn). (34)
The Zimmermann’s forest formula was proposed in [3]. It denotes a procedure forobtaining a renormalized Feynman integrand RΓ(p, k) from a unrenormalized Feynmanintegrand IΓ(p, k), namely
RΓ(p, k) =∑
U∈FSΓ
∏
γ∈U
(−tδγSγ) IΓ(p, k), (35)
9
where U is a forest, F is a set of all possible renormalization forests, SΓ or Sγ are sub-stitution operators, tδγ is a Taylor subtraction operator cut off by a subtraction degreeδγ , the argument p denotes a set of external momenta and the argument k denotes a setof independent internal momenta, and in the product
∏γ∈U (−tδγSγ) an element on the
right hand side is either disjoint with or a subset of one on the left hand side.A renormalized Green function in the BPHZ renormalization was defined in [4] and
[5]. A unrenormalized Green function is given in the Gell-Mann– Low formulation, andits renormalized version is directly defined as its finite part in the BPHZ renormalization,namely,
⟨Tϕ(x1)ϕ(x2) · · · ϕ(xn)⟩:= BPHZ finite part of
⟨Tϕ0(x1)ϕ0(x2) · · · ϕ0(xn)ei Γ0
int⟩/⟨Tei Γ0
int⟩. (36)
A renormalized Green function with insertions of normal products is given by
∏
i
Nδi[Qi(yi)] · Gn(x1, x2, · · · , xn)
:= ⟨T∏
i
Nδi[Qi(yi)] ϕ(x1)ϕ(x2) · · · ϕ(xn)⟩
= BPHZ finite part of
⟨T∏
i
Nδi
[Q0
i (yi)]ϕ0(x1)ϕ0(x2) · · · ϕ0(xn)e
i Γ0
int⟩/⟨Tei Γ0
int⟩, (37)
where a normal product Nδi
[Q0
i (yi)]denotes vertices in Feynman graphs to be treated with
a subtraction degree δi. The symbols with the upper index 0 are defined in free quantumfield theory: the Gell-Mann– Low formula is calculated by an approach of diagram-by-diagram, and each diagram is corresponding to a unrenormalized Feynman integral byordinary Feynman rules. For convenience, the subtraction degree of a normal product willnot be explicitly shown in some cases.
The quantum action principle was derived in the BPHZ renormalization, see [6], [7] and[8]. It means that variations of parameters or fields of a Green function can be representedby appropriate local insertions of normal products. Its differential formalism is given inthe following way,
∂AΓ = [∂AΓren]4 · Γ, A = m,µ, λ, (38)
ϕ(x)δΓ
δϕ(x)=
[ϕ(x)
δΓren
δϕ(x)
]
4
· Γ, (39)
where the lower indices of normal products are the subtraction degrees used in the BPHZrenormalization.
The Zimmermann’s identities were proved in [4] and [5]. They relate the subtractionand the over-subtraction in the BPHZ renormalization. They are given by
Nδ [Q] · Γ =∑
i
uiNχ [Qi] · Γ, (40)
where the sum is over all possible normal products of the over-subtraction degree χ withthe same quantum numbers, and δ is the subtraction degree, with χ>δ. The coefficientsui are determined by normalization conditions on insertions of composite operators.
The involved notations in the article are given in the appendix.
10
Notes
The BPH renormalization has been reviewed in [35]. The BPHZ renormalization wasreviewed in [16], [36] and [37], and was recently introduced in [38].
Besides the BPH renormalization and the BPHZ renormalization, there are also oth-er renormalization procedures: the Epstein- Glaser renormalization in [39]; the analyticrenormalization in [24]; the dimensional regularization and the dimensional renormaliza-tion in [32], reviewed in [40]; the differential renormalization in [41]; and the renormal-ization group differential equation approach in [42]. They are all invented for defining aperturbative quantum field theory, but the first four schemes are based on the expansionof diagram-by-diagram and the last one is on the expansion of scale-by-scale.
3 The Zimmermann’s forest formula
The Zimmermann’s forest formula plays the key role in the BPHZ renormalization. In thissection, it will be firstly introduced by an example and then some necessary remarks aregiven. Furthermore, a general procedure applying it to calculate a renormalized Feynmanintegral is presented, as will be explained in details by examples in two-loop. In addition,only connected graphs are considered.
3.1 A simple introduction
The Zimmermann’s forest formula can be observed in a heuristic way, because it has tosolve problems in non-overlapping divergent cases and is a solution of the R-operation inthe BPHZ renormalization. Firstly, a given divergent Feynman graph without overlappingdivergences can be made convergent by a natural strategy in [9], namely by making its alldivergent subgraphs convergent and then making itself finite. For example, a divergentFeynman graph γ, in the massive ϕ4 model in four-dimension, will be treated and it hastwo disjoint divergent subgraphs γ1 and γ2, see Figure 1.
Figure 1: A graph γ and its two disjoint divergent subgraphs γ1 and γ2
Its unrenormalized Feynman integral is given by∫∫
d4k1d4k2
(2π)8Dp−k1Dk1Dp−k2Dk2 , (41)
and the corresponding convergent Feynman integral is obtained by∫∫
d4k1d4k2
(2π)8(1 − t
dγp )
×[(1 − tdγ1pγ1 )Dpγ1−k1Dk1 ]pγ1=p[(1 − t
dγ2pγ2 )Dpγ2−k2Dk2 ]pγ2=p, (42)
11
where the symbols dγ , dγ1 and dγ2 respectively denote the superficial divergent degrees ofthe corresponding graphs γ, γ1 and γ2, and the action of the Taylor subtraction operator
tdγp acting on an arbitrary rational function f(p) is given by,
tdγp f(p) = f(0) + pµ∂pµf(0) + · · · +
pdγ
(dγ)!∂dγ f(0), (43)
similarly for tdγ1pγ1 and t
dγ2pγ2 . Secondly, expanding the above convergent integral (42) partly,
the result is∫∫
d4k1d4k2
(2π)8[1 + (−t
dγ1pγ1 ) + (−t
dγ2pγ2 ) + (−t
dγ1pγ1 )(−t
dγ2pγ2 )]
×Dpγ1−k1Dk1Dpγ2−k2Dk2 |pγ1=pγ2=p
+(−tdγp ) [1 + (−t
dγ1pγ1 ) + (−t
dγ2pγ2 ) + (−t
dγ1pγ1 )(−t
dγ2pγ2 )]
×Dpγ1−k1Dk1Dpγ2−k2Dk2 |pγ1=pγ2=p, (44)
which suggests a recursive procedure in the BPH renormalization, namely, the R-operation:obtaining a divergent graph without subdivergences by subtracting overall divergences ofits all divergent subgraphs; obtaining an overall divergences of a graph by firstly makingit free from subdivergences and then taking a remaining divergent part; and obtaininga finite graph by subtracting its overall divergence, see [10] and [1]. Thirdly, expandingthe above convergent Feynman integral (44) completely, the corresponding renormalizedFeynman integral is obtained by
1 + (−tdγp )Sγ + Sγ(−t
dγ1pγ1 ) + Sγ(−t
dγ2pγ2 )
+(−tdγp )Sγ(−t
dγ1pγ1 ) + (−t
dγp )Sγ(−t
dγ2pγ2 ) + Sγ(−t
dγ1pγ1 )(−t
dγ2pγ2 )
+(−tdγp )Sγ(−t
dγ1pγ1 )(−t
dγ2pγ2 ) Dpγ1−k1Dk1Dpγ2−k2Dk2 , (45)
where the substitution operator Sγ is defined by
Sγ : pγ1 −→ pγ1(p), pγ2 −→ pγ2(p), (46)
namely,Sγ f(pγ1) = f(p), Sγ f(pγ2) = f(p). (47)
Introducing a forest set as
∅, γ, γ1, γ2, γ1, γ, γ2, γ, γ1, γ2, γ1, γ2, γ, (48)
the Zimmermann’s forest formula comes out and its general formalism is given by
RΓ(p, k) =∑
U∈FSΓ
∏
γ∈U
(−tδγSγ) IΓ(p, k), (49)
which was firstly presented in [3]. It denotes a procedure of obtaining a renormalizedFeynman integrand RΓ(p, k) from a unrenormalized Feynman integrand IΓ(p, k).
To apply the Zimmermann’s forest formula in the calculation, several things have tobe made clear. First, in the BPHZ renormalization, a graph is defined by a set of itsall lines, and so its subgraph is obtained by taking one subset. However, in the BPHrenormalization in [1] and the Epstein- Glaser renormalization in [39], a graph is definedby a set of its all vertices. Naturally for the same graph, the number of subgraphs in the
12
latter case is less than in the BPHZ renormalization. Second, a subtraction degree of acomposite operator in a graph is different from that in its subgraph. In a 1PI subgraph,some of all lines connecting a composite operator in a graph become external and are notinvolved in a renormalization procedure, see [4] and [7]. Third, a substitution operator hasto be chosen to obey the admissible substitution rules in the next section, which is requiredby the proof applying both the forest concept and the momenta subtraction procedure,see [3].
3.2 A general calculation procedure
In order to calculate a renormalized Feynman integral corresponding to a given Feynmandiagram in the BPHZ renormalization, the following steps have to be carried out.
• Step I. Specify a formula calculating subtraction degrees;
• Step II. Find out all possible forests;
• Step III. Calculate admissible momenta assignments;
• Step VI. Specify substitution operators satisfying the admissible rule;
• Step V. Apply the Zimmermann’s forest formula to obtain a renormalized Feynmanintegrand;
• Step VI. Calculate a renormalized Feynman integral.
For the massive ϕ4 model in the BPHZ renormalization, a subtraction degree of aconnected Feynman graph γ with N external legs, is given by
δγ = 4 − N +∑
i
(δi − 4), (50)
where the symbol δi denotes a subtraction degree of a normal product at the i-th insertionpoint as a vertex. In the case without insertions of normal products, it is simplified as thesuperficial divergence degree dγ , namely,
δγ = dγ = 4 − N. (51)
Furthermore, when subtraction degrees of normal products are their mass dimensions, thesubtraction degree δγ is also the same as the superficial divergence degree dγ . Moreover,a divergent graph means its non-negative subtraction degree.
A forest denoted by U is a set in which each element is one divergent 1PI subgraphand every two elements are not overlapping. This shows that two 1PI graphs µ and τ inthe forest U are either disjointed or nested, namely,
µ ∩ τ = ∅, or µ ⊂ τ, or µ ⊃ γ. (52)
Two special forests of the divergent graph γ are the empty set ∅ and the graph itself,γ. The set of all possible forests denoted by F is involved in the Zimmermann’s forestformula. Obtaining F is a big task: first, all possible divergent 1PI subgraphs have tobe found out; second, all possible non-overlapping combinations among them have to beworked out. For example, in cases of two-loop, the involved forest sets have been obtained,see the following subsection.
13
In the following, an algorithm of calculating admissible momenta assignments of alldivergent graphs in a forest is given. Firstly, notations will be specified. Secondly, how toobtain admissible momenta assignments is explained. Thirdly, the admissible substitutionrule is defined, and then an algorithm calculating substitution operators is presented.
Once an oriented Feynmangraph γ is given, its admissible momenta assignment isalso fixed. It has Vγ vertexes and V e
γ external vertexes connecting external lines. It hasmγ independent oriented loops, denoted respectively by C1, C2, · · · Cmγ . The symbol pγ
a
denotes the external momentum flow at the external vertex Va, which is a variable ofa Taylor subtraction operator. The symbol lγabσ denotes the σ-th oriented line from thevertex Va to the vertex Vb. It also represents the momentum flow in the line and can bedivided into a sum of the external line momentum pγ
abσ and the no-source momentum kγabσ,
lγabσ = pγabσ + kγ
abσ. (53)
In the oriented graph γ, the symbol lγabσ takes −lγbaσ. The “no-source” means that anyvertex is not the source of this type of momenta, namely
∑
b,σ
kabσ = 0, (54)
for a fixed vertex point a. Furthermore, every internal line lγabσ is assigned the resistancevariable rabσ, taking values of zero or one. In addition, in the massive ϕ4 model, themaximum of the symbol σ is three since vacuum graphs are not involved.
Determining the admissible momenta assignment is to calculate pγabσ and kγ
abσ. Theexternal line momenta flow pγ
abσ are obtained by solving the following equations,
pγ
a =∑
b,σ pγabσ, a = 1, 2, · · · , V e
γ − 1,
0 =∑
lγabσ∈Cirabσ pγ
abσ, i = 1, 2, · · ·mγ , rabσ = 0 or 1.. (55)
in which it is best to set rabσ zero on the grounds that in a loop the resistance of at leastone line is 1, and once rabσ fixed its value holds for all graphs including the line labσ, sothat it doesn’t need the upper index γ. The no-source momenta flow kγ
abσ are chosen in adifferent way. Each loop has its own momentum flow, namely,
kγ1 , kγ
2 , · · · , kγmγ
, (56)
which are integral variables in the measure of a Feynman integral. The kγabσ can be
constructed by
kγabσ =
∑
lγabσ∈Ci
kγi −
∑
lγbaσ∈Cj
kγj . (57)
Similarly, an admissible momenta assignment of a subgraph τ is calculated by the sameprocedure, that is to say that the formulas (55), (56) and (57) are available except thatthe upper index γ is replaced with τ .
For a divergent graph γ and its divergent subgraph τ , the substitution operator Sγ isdefined by
Sγ : pτabσ −→ pγ
abσ, kτabσ −→ kγ
abσ, τ ⊂ γ. (58)
The admissible substitution rule in the Zimmermann’s recipe requires that the substitutionoperator has to satisfy
kτabσ = kτ
abσ(kγ), τ ⊂ γ, (59)
14
which means that a momenta assignment of a graph can’t be chosen in an arbitrary way.Based on the fact,
lγabσ = lτabσ, (60)
the external momenta flow pτa is a function of variables pγ , kγ , namely,
pτa =
∑
b,σ
lτabσ =∑
labσ∈τ
lγabσ
= pγa −
∑
labσ∈γ−τ
lγabσ = pτa(p
γ , kγ), (61)
where the vertex a is fixed and the γ − τ denotes a difference of two sets. Hence theadmissible substitution operator Sγ can be constructed in the following way,
Sγ :
pτ
abσ = pτabσ(pτ ) = pτ
abσ(pγ , kγ)kτ
abσ = lγabσ − pτabσ = kτ
abσ(kγ). (62)
Afterwards, a renormalized Feynman integral can be computed from a renormalizedFeynman integrand, see the following subsection.
3.3 Examples in two-loop
To explain the procedure of applying the Zimmermann’s forest formula to obtain a renor-malized Feynman integral proposed in the above subsection, seven examples will be givenas follows. They treat 1PI self-energy two-loop Feynman graphs with an insertion of anormal product or without in the massive ϕ4 model in the BPHZ renormalization. Thenormal product is either 1
2 [ϕ2]2 or 12 [ϕ2]4, and the latter one is with an over-subtraction
degree.As the first example, the sunset diagram γ is a best choice because it is a typical
two-loop Feynman diagram with overlapping divergences. It consists of three internallines, l121, l122, l123, connecting two external vertexes V1, V2, and has three 1PI divergentsubgraphs: γ1, γ2, γ3, see Figure 2.
l121
l123
l122 lγ1
122
lγ1
123
lγ2
121
lγ2
123
lγ3
121
lγ3
122
Figure 2: A sunset diagram γ and its divergent subgraphs γ1, γ2, γ3
The forest set F is found out by trying all possible non-overlapping combinations offour divergent 1PI graphs γ, γ1, γ2, γ3, and the result is obtained by
∅, γ1, γ2, γ3, γ, γ1, γ, γ2, γ, γ3, γ. (63)
The resistance of each line is chosen as
r121 = 0, r122 = 1, r123 = 1. (64)
15
The external momentum flow at the vertex V1 is noted by the symbol p. The admissiblemomenta assignments of the graph γ and its subgraphs γ1, γ2, γ3 are obtained by solvingthe corresponding equations (55) and using the construction (57), and the results are givenas follows,
l121 = p + k1 + k2
l122 = −k2
l123 = −k1
;
lγ1122 = pγ1
122 + 12(k1 − k2)
lγ1123 = pγ1
123 − 12(k1 − k2)
;
lγ2121 = pγ2
121 + k1
lγ2123 = −k1
;
lγ3121 = pγ3
121 + k2
lγ3122 = −k2
, (65)
which have considered the following substitution operator Sγ given by
Sγ : kγ1122 = 1
2(k1 − k2), kγ2121 = k1, kγ3
121 = k2, (66)
showing that the given momenta assignments are indeed admissible.The unrenormalized Feynman integrands in the forest formula are given by
Iγ = Dp+k1+k2Dk1Dk2 ,
Iγ1Iγ/γ1= Dp+k1+k2Dl
γ1122
Dlγ1123
,
Iγ2Iγ/γ2= D−k2Dp
γ2122+k1
D−k1 ,
Iγ3Iγ/γ3= D−k1Dp
γ3121+k2
Dk1 . (67)
The renormalized Feynman integrand is obtained by,
Rγ = (1 − t2p) Sγ(Iγ +
3∑
i=1
(−t0pγi )IγiIγ/γi)
= (1 − t2p)Dp+k1+k2 [Dk2Dk1 − D212 (k1−k2)
], (68)
where the Taylor subtraction operators t0pγi and t2p are defined in (43). The renormalizedFeynman integral is calculated by
1
6(~λ)2
∫∫d4k1d
4k2
(2π)8Rγ
=1
6(~λ)2
∫∫d4k1d
4k2
(2π)8
(Dp+k3 − Dk3 − 12 pµ pν∂µ∂νDk3)Dk1Dk3−k1 , (69)
where in the second line the integral variable k3 replaces k1 + k2 and some calculation hasbeen already carried out.
The second example is the sunset diagram γ with an insertion of the normal product12 N [ϕ2]2, in order to show that a divergent Feynman diagram with insertions are treatedthe same as one without insertions. It is specified by the four internal lines, l131, l321,l122, l123, which connects three internal vertex points, V1, V2, V3, and the vertex V3 is theinsertion point, see Figure 3. The subtraction degrees of the graph γ and its divergent1PI subgraph γ1 are calculated by applying (50),
δγ = dγ = 0, δγ1 = dγ1 = 0. (70)
16
l131
l123
l122
l321
lγ1
122
lγ1
123
Figure 3: A sunset graph with an insertion 12 N [ϕ2]2 and its divergent subgraph γ1
Then the forest set F is given as follows,
∅, γ1, γ, γ1, γ. (71)
The external momentum flows at the vertex V1 and the vertex V3 are respectivelydenoted by the symbols p and q. The resistance of each line is assumed to be
r131 = r321 = 0, r122 = r123 = 1. (72)
Then the admissible momenta assignment is obtained by applying the given algorithm andis given by
l131 = p + k1 + k2
l321 = p + q + k1 + k2
l122 = −k2, l123 = −k1;
lγ1122 = pγ1
122 + 12(k1 − k2)
lγ1123 = pγ1
123 − 12(k1 − k2),
(73)
which determines the renormalized Feynman integral given by
1
6(i~λ)2
∫∫d4k1d
4k3
(2π)8(Dp+k3Dp+q+k3 − D2
k3)Dk1Dk3−k1
− 1
6(i~λ)2
∫∫d4k1d
4k2
(2π)8(Dp+k1+k2Dp+q+k1+k2 − D2
k1+k2)D2
12 (k1−k2)
(74)
Furthermore, replacing the local insertion 12 [ϕ2]2 with the local integral insertion
∫12 [ϕ2]2,
the corresponding renormalized Feynman integral will be the first term in the above ex-pression, the variable q being zero.
In the third example, the sunset graph with an insertion of the normal product 12 [ϕ2]4.
Its subtraction degree is the same as in the first example, and hence they have the sameforest set. Its admissible momenta assignment is the same as in the second example. Therenormalized Feynman integral is given by
1
6(i~λ)2
∫∫d4k1d
4k3
(2π)8(1 − t2p,q)Dp+k3Dp+q+k3Dk1Dk3−k1 . (75)
In the fourth example, the scoop diagram γ will be treated. It has three internal lines,connecting one internal vertex and one external vertex. It has two bubbles, and the one isconstructed by an internal line connecting an internal vertex to itself, called by a tadpole.It has two 1PI subgraphs γ1, γ2. See Figure 4. The subtraction degrees of the graph γand its subgraphs γ1, γ2 are given by
dγ = 2, dγ1 = 0, dγ2 = 2. (76)
17
The forest set F is the same as in (48). The external momenta flow at the external vertexof γ is zero due to the conservation of the energy-momentum, and the same holds forγ1 and γ2. Then the admissible momenta assignment can be obtained by only takingloop momenta k1, k2, without considering external momenta. The renormalized Feynmanintegral is obtained by applying the forest formula,
1
4(~λ)2
∫∫d4k1d
4k2
(2π)8(1 − t2p)Sγ(1 − t0pγ1 )(1 − t2pγ2 )D
2k1
Dk2 = 0, (77)
which suggests that in the BPHZ renormalization procedure all Feynman diagrams withtadpoles do not make contributions. The reason is that the momenta subtraction schemehires external momenta which are not needed in tadpoles. This is consistent with resultsfrom an ordinary normal ordering recipe in quantum field theory.
γ
γ1γ2
Figure 4: A scoop diagram γ and its two divergent subgraphs γ1, γ2
The fifth example will be chosen as the scoop diagram with the insertion of a normalproduct, either 1
2 [ϕ2]2 or 12 [ϕ2]4, and the insertion point is not standing in a tadpole. Hence
the final result is still zero. See Figure 5.
γ1 γ2
Figure 5: A scoop diagram with an insertion and its one subgraph being a tadpole
The sixth example is the scoop diagram with the insertion of the normal product12 [ϕ2]2 but without a tadpole. The graph γ is constructed by four internal lines l121, l122,l231, l232, connecting two external vertexes V1, V3 and one internal vertex V2. It has two1PI subgraphs γ1 and γ2. See Figure 6. The subtraction degrees of the graph γ and itssubgraph γ1, γ2 are given by
dγ = dγ1 = dγ2 = 0. (78)
The forest set F is the same as in the fourth example. The external momentum flow atthe vertex V1 is denoted by the symbol p. The resistance of each line is chosen as
r121 = r231 = 0, r122 = r232 = 1. (79)
18
l121
l122
l231
l232
γ
lγ1
121
lγ1
122
γ1
lγ2
232
lγ2
231
γ2
Figure 6: A scoop diagram with an insertion and no subgraphs being tadpoles
The admissible momenta assignment containing the substitution rules are given by
l121 = p + k1 = lγ1
121
l122 = −k1 = lγ1121
;
l231 = p + k2 = lγ2
231
l232 = −k2 = lγ2232
, (80)
and the renormalized Feynman integral is calculated by
1
4(i~λ)2
∫d4k1d
4k2
(2π)8(1 − t0p)Sγ
×(1 − t0pγ1 )(1 − t0pγ2 )Dpγ1+k1D−k1Dpγ2+k2D−k2
=1
4(i~λ)2
∫∫d4k1d
4k2
(2π)8
(Dp+k1D−k1 − Dk1D−k1)(Dp+k2D−k2 − Dk2D−k2). (81)
In the end, all the above examples in the subsection are taken from [44].
Notes
The Zimmermann’s forest formula had been reviewed in [16], [36], [37] and [38]. Hereexplaining how to apply it in the calculation is a point.
4 The quantum action principle
The quantum action principle is always represented by differential equations in whichvariations of parameters or fields on the Green function are related to local insertions ofnormal products. In the BPHZ renormalization, the first type of differential equationsare given by
∂AΓ = [∂AΓren]4 · Γ, A = m, µ, λ, (82)
while the second type of differential equations are
δϕδΓ
δϕ= [δϕ
δΓren
δϕ]4 · Γ, (83)
where δϕ describes all general transformations. First, δϕ being constants, such as space-time translations, (83) denotes the equation of motion in quantum field theory. Second,δϕ being linear functionals of ϕ, such as the conformal transformations, (83) denotes linear
19
transformations of the Green function. Third, δϕ being non-linear functionals of ϕ, such asthe BRST transformation or supersymmetric transformations in the Wess–Zumino gauge,(83) denotes non-linear transformations of the Green function.
In the first subsection, the one differential equation representing the quantum actionprinciple will be related to the equation of motion in quantum field theory. In the secondsubsection, the strategy of proving the quantum action principle will be presented anddelicate points will be explained by simple examples.
4.1 The equation of motion
As an example, the differential equation will be treated in detail,
δΓ
δϕ(x)= [
δΓren
δϕ(x)] · Γ, (84)
which relates the variation of a classical field ϕ(x) on the vertex functional Γ to a localinsertion into the vertex functional Γ.
It can be firstly checked in the classical approximation. The vertex functional Γ is alsothe effective action and its classical approximation is the classical action Γcl,
Γ = Γcl + O(~), (85)
and thus its left hand side is given by
δΓ
δϕ(x)=
δΓcl
δϕ(x)+ O(~). (86)
Its right hand side is given by
[δΓren
δϕ(x)] · Γ =
δΓren
δϕ(x)+ O(~), (87)
due to the fact that in the classical approximation the local contribution of the insertionis the composite operator itself in the classical approximation.
With the generating functional Z[J ] of the general Green functions, (84) is representedby
[δΓren
δϕ(x)] · Z[J ] + J(x)Z[J ] = 0. (88)
which has a unrenormalized version by
[δΓcl
δϕ(x)] · Z + J(x)Z[J ] = 0. (89)
The above equation can be derived by applying the action principle to the path integralrepresentation of the generating functional Z[J ]
Z[J ] =
∫D ϕ exp
i
~Γcl[ϕ, J ], (90)
in which the symbol Γcl[ϕ, J ] is given by
Γcl[ϕ, J ] = Γcl[ϕ] +
∫d4x J(x)ϕ(x). (91)
20
The generating functional Z[J ] being invariant by infinitesimally shifting the integralvariable ϕ(x),
ϕ′ = ϕ + δϕ, Dϕ′ = Dϕ, Z ′[J ] = Z[J ]. (92)
it induces the equation of motion in a quantum sense,
δΓcl[ϕ, J ]
δϕ(x)|ϕ=−i~ δ
δJ(x)Z[J ] = 0, (93)
which becomes (89) when the insertion is given by
δΓcl[ϕ]
δϕ(x)|ϕ=−i~ δ
δJ(x)Z[J ] =
δΓcl[ϕ]
δϕ(x)· Z[J ]. (94)
With the exponential factor Γcl replaced with Γren, (93) is a renormalized version of theequation of motion.
Furthermore, in the canonical quantization, the Dyson- Schwinger equation determinesdynamics. Its popular unrenormalized version is given by
(2x + m2)ϕcl(x) = J(x) − ~2 λ
3!ϕ3
cl +λ
3!
δ2ϕcl(x)
δ2J(x)+ i ~
λ
2ϕcl(x)
δϕcl(x)
J(x), (95)
in which the classical field ϕcl(x) is defined by
ϕcl(x) =δZC [J ]
δJ(x). (96)
With the general Green functions instead of the connected Green functions, the Dyson-Schwinger equation has a simpler form,
(2x + m2)(−i~δ
δJ(x))Z[J ] = J(x)Z[J ] − λ
3!(−i~)3
δ3
δ3J(x)Z[J ], (97)
which is the same as (93). Therefore, the quantum action principle indeed takes root inthe action principle.
4.2 The renormalized version in higher orders
In this subsection, the proof of the quantum action principle will be sketched.The first type of differential equations (82) can be represented by the Green functions,
−i~ ∂AGn = [∂AΓren]4 · Gn, A = m,µ, λ. (98)
When A is parameters µ and λ, the above equations can be proved in the following way,
−i~ ∂AGn = −i~ ∂ABPHZ finite part of
⟨Tϕ0(x1)ϕ0(x2) · · · ϕ0(xn)ei Γ0
int⟩/⟨Tei Γ0
int⟩= BPHZ finite part of
⟨T∂AΓ0intϕ0(x1)ϕ0(x2) · · ·ϕ0(xn)e
i Γ0
int⟩/⟨Tei Γ0
int⟩= BPHZ finite part of
⟨T∂AΓ0renϕ0(x1)ϕ0(x2) · · · ϕ0(xn)e
i Γ0
int⟩/⟨Tei Γ0
int⟩= [∂AΓren]4 · Gn, (99)
21
in which the point is that the involved differential operations commute with the momentasubtraction procedure in the BPHZ renormalization. When A is the physical mass, inaddition, the differentiation operation acting on propagators have to be involved.
For the second type of differential equations sedieq, the proof in [7, 8] will be sketched.The given differential equation can be represented by the general Green functions,
−⟨0|T [δϕ(x)(2x + m2)ϕ(x)]ϕ(x1) · · ·ϕ(xn)|0⟩
−i~n∑
i=1
δ4(x − xi)⟨0|T [δϕ(x)]ϕ(x1)ϕ(x2) · · · ϕ(xi) · · ·ϕ(xn)|0⟩
= (a − m2)⟨0|T [ϕ(x)δϕ(x)]ϕ(x1) · · · ϕ(xn)|0⟩+ (z − 1)⟨0|T [δϕ(x)2xϕ(x)]ϕ(x1) · · · ϕ(xn)|0⟩+
ρ
3!⟨0|T [δϕ(x)ϕ3(x)]ϕ(x1) · · ·ϕ(xn)|0⟩. (100)
To prove the above the equality in the BPHZ renormalization, the strategy of perturbativeexpansion on diagram-by-diagram is used. The general Feynman diagram is a researchobject. Here, On its both sides, two general Feynman diagrams are considered and haveto be proved to have the same unrenormalized Feynman integrals, the same forest setsand the same external momenta assignments.
When δϕ being constant, the corresponding differential equation in the general Greenfunctions is given by
−⟨0|T [(2x + m2)ϕ(x)]ϕ(x1) · · · ϕ(xn)|0⟩
−i~n∑
i=1
δ4(x − xi)⟨0|T ϕ(x1)ϕ(x2) · · · ϕ(xi) · · ·ϕ(xn)|0⟩
= (a − m2)⟨0|T ϕ(x)ϕ(x1) · · · ϕ(xn)|0⟩+ (z − 1)⟨0|T [2xϕ(x)]ϕ(x1) · · · ϕ(xn)|0⟩+
ρ
3!⟨0|T [ϕ3(x)]ϕ(x1) · · ·ϕ(xn)|0⟩ (101)
where the symbol ϕ means that ϕ(xi) is not included. On its left hand side, one connectedFeynman graph is picked up and it includes an external vertex point x. When an externalline connecting the point x and the other external point xi is amputated, the factorδ4(x − xi) comes from the calculation,
−(2x + m2)∆F (x − xi)G(xi) = i~ δ(x − xi)G(xi) (102)
When an external line connecting the point x and the interaction vertex 14!ϕ
4(z) in fourequivalent ways is amputated, see Figure 7, a new interaction vertex 1
3!ϕ3(z) is created by
−(2x + m2)(− i
~)
∫d4z
3!∆F (x − z)G(z) =
1
3!G(x), (103)
where G is a Feynman integral excluding the propagator from x to z.Considering renormalized interaction vertexes for counter terms, the differential equa-
tion (101) can be proved in the level of the unrenormalized Feynman integral. Furthermore,applying the forest formulae, it is also proved because a line connecting the point x andthe other is always external which does not affect the power counting of divergence degree.
When δϕ being linear transformations of ϕ, the corresponding differential equation
J(x)δZ[J ]
δJ(x)= [ϕ(x)
δΓren
δϕ(x)]4 · Z[J ], (104)
22
Figure 7: A new vertex 13!ϕ
3(z) generated by amputating one external line
is represented in the general Green function,
−⟨0|T [ϕ(x)(2x + m2)ϕ(x)]4ϕ(x1) · · ·ϕ(xn)|0⟩
−i~n∑
i=1
δ4(x − xi)⟨0|T ϕ(x)ϕ(x1)ϕ(x2) · · · ϕ(xi) · · ·ϕ(xn)|0⟩
= (a − m2)⟨0|T [ϕ2(x)]4ϕ(x1) · · · ϕ(xn)|0⟩+(z − 1)⟨0|T [ϕ(x)2xϕ(x)]4ϕ(x1) · · · ϕ(xn)|0⟩+
ρ
3!⟨0|T [ϕ4(x)]4ϕ(x1) · · · ϕ(xn)|0⟩. (105)
Similar to what was done in the above example, both sides of (105) have the same un-renormalized Feynman integral. Involving the forest formula, several points have to makeclear. First, the oversubtraction is necessary for defining the amputation. The Feynmangraph with the insertion [ϕ2(x)]4 instead of [ϕ2(x)]2 has the same forest set as the onewith the insertion [ϕ(x)2xϕ(x)]4. Second, two types of the Feynman graphs on the lefthand side must make no contribution because no corresponding parts on the right handside. The first case is that two or more than two lines connect the vertex x and the otherone. Its counter part contains the tadpole and vanishes in the BPHZ renormalization.The second case is that there is a renormalized 1PI subgraph including the vertex pointx and the other one but excluding the line between them. This renormalized subgraphplus that line is still a renormalized one so both will be mapped to the same part afterthe amputation, which makes ambiguous. Third, choosing the resistance of an amputatedline zero, the external momentum at the vertex point x will only be in the amputated line,then both sides have the equivalent admissible momenta assignments.
In the following, two examples are given to declare (105). The first example is usedto emphasize the importance of the oversubtraction. It is on the four-point graph in one-loop with one insertion. Obviously, without the oversubtraction, the amputation can’t becarried out in the renormalized level. See Figure 8, the corresponding Feynman integralis given by
L.H.S = (− i
~λ)2(−i~)
4∏
i=1
∆F (xi) exp
4∑
i=1
ipix
(i~)2∫
d4k
(2π)4(Dk Dp1+p2−k − Dk Dk). (106)
The second example treats the case of two lines connecting the vertex point x and theother one. It is natural to obtain a tadpole. See Figure 9, its unrenormalized Feynman
23
x
+x
+x
+x
=x
Figure 8: An example to emphasize the importance of the oversubtraction
integral is given by
1
SL(− i
~λ)
∫d4z ∆F (x1 − z)∆F (x2 − z)∆F (x − z)(2x + m2)∆F (x − z)
=1
SL(−λ)∆F (x1 − x)∆F (x2 − x)∆F (x − x), (107)
where the symmetry factors SL and SR can be calculated by
1
SL= 1,
1
SR= 2. (108)
x
+x
+x
=xFigure 9: A tadpole obtained by amputating one external line
Notes
The quantum action principle had been proved in [7, 8] in the BPHZ renormalization; in[14], [43] in the dimensional regularization and the dimensional renormalization scheme;in [24] in the analytical renormalization; in [25] in the BPH renormalization; in [26] inthe Wilson-Polchinski renormalization.
5 The Callan–Symanzik equation
The Callan–Symanzik equation was originally obtained in [53, 54]. It denotes broken scaleinvariance, the generator of scale transformation being represented by m∂m+µ∂µ. Then itcan be also derived by applying both the quantum action principle and the Zimmermannidentities, see [17]. In the first subsection, the Zimmermann identities will be introducedby examples. In the second subsection, the Lownstein’s recipe of deriving the Callan–Symanzik equation in the BPHZ renormalization will be sketched. In the third subsection,the Callan–Symanzik equations in different formalisms will be shown up.
24
5.1 The Zimmermann identities
In the BPHZ renormalization, the same composite operator can be assigned differentsubtraction degrees. Applying the forest formulae, the normal subtraction and the over-subtraction can be related in the Zimmermann identity, expanding one insertion with thelower subtraction into linear combination of all admissible insertions with the same highersubtraction degree. Here many examples are given in detail to show the Zimmermannidentities.
One Zimmermann identity used in the article is given by
∆d · Γ = ∆2 · Γ + u∆1 · Γ + t ∆4 · Γ, (109)
where ∆d denotes the same composite operator as ∆2 but has the subtraction degree 2,
∆2 =
∫[12ϕ2]2, ∆d =
∫[12ϕ2]4. (110)
Its coefficients u, t are decided either by using the normalization conditions of insertionsof composite operators or by testing special Feynman diagrams. Here, the latter methodwill be applied to determine coefficients up to orders of ~.
• Determining u(0), t(0).
First, the zero order values of the coefficients, u(0), t(0), comes out by testing tree Feyn-man diagrams or convergent ones. In order to determine u(0), the following Zimmermannidentity can be chosen,
(∆d · Γ)(0)2 = (∆2 · Γ)
(0)2 + u(0) (∆1 · Γ)
(0)2 + t(0) (∆4 · Γ)
(0)2 , (111)
where the upper indices in the brackets denote the power countings of ~. The simpleanalysis of the relevant tree Feynman diagrams results in
(∆d · Γ)(0)2 = (∆2 · Γ)
(0)2 = 1, (∆1 · Γ)
(0)2 = −p2, (∆4 · Γ)
(0)2 = 0, (112)
which suggests that u(0) is zero. In order to determine t(0), the following Zimmermannidentity can be chosen,
(∆d · Γ)(0)4 = (∆2 · Γ)
(2)4 + u(0) (∆1 · Γ)
(0)4 + t(0) (∆4 · Γ)
(0)4 , (113)
which implies that t(0) is zero by the results from tree diagram calculation,
(∆d · Γ)(0)4 = (∆2 · Γ)
(2)4 = (∆1 · Γ)
(0)4 = 0, (∆4 · Γ)
(0)4 = 1. (114)
In order to determine u(0), t(0) at the same time, the following Zimmermann identityup to ~ can be considered,
(∆d · Γ)(1)6 = (∆2 · Γ)
(1)6 + u(0) (∆1 · Γ)
(1)6 + u(1) (∆1 · Γ)
(0)6
+ t(0) (∆4 · Γ)(1)6 + t(1) (∆4 · Γ)
(0)6 . (115)
The relevant Feynman diagram given in Figure 10, it shows that the subtraction degrees
of (∆d · Γ)(1)6 and (∆d · Γ)
(1)6 are respectively given by −2 and −4, then the above identity
can be simplified as
(∆d · Γ)(1)6 = (∆2 · Γ)
(1)6 . (116)
25
Figure 10:
Diagram 10.By simple calculation, the values of the involved insertions are given by
(∆4 · Γ)(0)6 = (∆1 · Γ)
(0)6 = 0, (∆4 · Γ)
(1)6 = 0, (∆1 · Γ)
(1)6 = 0, (117)
which means that u(0) and t(0) have to be zero.
• Determining u(1), t(1).
In order to determine u(1), the following Zimmermann identity up to one-loop can bechosen,
(∆d · Γ)(1)2R = (∆2 · Γ)
(1)2R + u(1) (∆1 · Γ)
(0)2 + t(1) (∆4 · Γ)
(0)2 , (118)
where the symbol R denotes the renormalized contribution. The involved typical Feynmandiagram is shown in Figure 11.
Figure 11:
Since it is similar to the tadpole, it vanishes under the BPHZ renormalization,
(∆d · Γ)(1)2R = (∆2 · Γ)
(1)2R = 0, (119)
which suggestsu(1) = 0. (120)
In order to determine t(1), the following Zimmermann identity up to one-loop can bechosen as
(∆d · Γ)(1)4 = (∆2 · Γ)
(1)4R + t(1) (∆4 · Γ)
(0)4 , (121)
in which (∆d · Γ)(1)4 is given by
(∆d · Γ)(1)4 = i~
λ2
2
3∑
i=1
∫d4k
(2π)4(D2
k DPi−k + Dk D2Pi−k), (122)
26
where P1, P2, P3 are specified by
P1 = p1 + p2, P2 = p1 − p3, P3 = p1 − p4. (123)
Its diagrammatic representation is shown in Figure 12.
∆d
=∆2
+ t(1)
Figure 12:
Therefore, t(1) is obtained by
t(1) = 3 i~λ2
∫d4k
(2π)4D3
k = ~3λ2
2× 1
16m2 π2. (124)
• Determining u(2).
In order to determine u(2), the following Zimmernmann identity up two orders can bechosen,
(∆d · Γ)(2)2R = (∆2 · Γ)
(2)2R + u(2) (∆1 · Γ)
(0)2 + t(1) (∆4 · Γ)
(1)2 , (125)
which is simplified as
(∆d · Γ)(2)2R = (∆2 · Γ)
(2)2R + u(2) (−p2), (126)
since (∆4 · Γ)(1)2 only has the tadpole contribution. The typical Feynman diagram is shown
in Figure 13.
Figure 13:
The corresponding Feynman integral for (∆d · Γ)(2)2R given by
(∆d · Γ)(2)2R = 1
2(i~λ)2∫
d4k1
(2π)4
∫d4k3
(2π)4(D2
p+k3− D2
k3)Dk3−k1Dk1 , (127)
the coefficient u(2) can be calculated by
u(2) =1
16(~λ)2
∫d4k1
(2π)4
∫d4k3
(2π)4(2k3 D2
k3)Dk3−k1Dk1 . (128)
27
In addition, the local Zimmermann identity is most used to derive the conformal trans-formations of the S-matrix in [44]. Here, two relevant examples are calculated. The firstone is given by
([12ϕ2(x)]2 · Γ)(1)4R = ([12ϕ2(x)]4 · Γ)
(1)4R + t(1)([
1
4!ϕ4(x)]4 · Γ)
(0)4 , (129)
whose integral version has been shown in (121). The second one is given by
([12ϕ2(x)]2 · G)(1)2R = ([12ϕ2(x)]4 · G)
(1)2R + v
(1)k ([122xϕ2(x)]4 · G)
(0)2 , (130)
where v(1)k is obtained by simple calculation,
v(1)k = i~
λ
2(a − 4 b) = ~
λ
6× 1
32 m2 π2
a =
∫d4k
(2π)41
(k2 − m2 + iϵ)3; bηµν =
∫d4k
(2π)4kµkν
(k2 − m2 + iϵ)4. (131)
Its diagrammatic representation is shown in Figure 14.
[12φ2]2
=[12φ2]4
+ v(1)k
[122φ2]4
Figure 14:
The Zimmermann identity is a special property owned by the BPHZ renormalizationprocedure. This means that in other regularization or renormalization schems, the similarthing is impossible or difficult to obtain. It solves the problem of how to relate the normalsubtraction and the oversubtraction. The main reason is that in the momenta subtrac-tion procedure the subtraction is exactly controlled by the subtraction degree. It can beproved by applying the forest formulae to insertions of involved composite operators inthe momentum space and carefully treating additional forests due to the oversubtraction.
Notes
The Zimmermann identity can be used to prove the OPE(operator product expansion)in a perturbative but rigorous sense, see [5]. It is also applied to study anomalies orbreakings in abelian gauge theories, see in [19, 20, 21], but in complicated field theoriessuch as non-abelian gauge field theories, the algebraic renormalization procedure is moresuitable to apply, see [22, 23].
5.2 Deriving the Callan–Symanzik equation
In the BPHZ renormalization, the Callan–Symanzik equation can be derived in a rigoroussense. Combing the quantum action principle and the Zimmermann identity, the following
28
five equations comes out,
m∂mΓ = [−m∂mz ∆1 − m∂ma∆2 − m∂mρ∆4] · Γ;µ∂µΓ = [−µ∂µz ∆1 − µ∂µa ∆2 − µ∂µρ ∆4] · Γ;∂λΓ = [−∂λz ∆1 − ∂λa∆2 − ∂λρ ∆4] · Γ;12NΓ = [−z ∆1 − a∆2 − 2ρ ∆4] · Γ;∆d · Γ = [∆2 + u ∆1 + t ∆4] · Γ;
(132)
where the parameters u, t are decided by normalization conditions on the insertions of therelevant composite operators. Since there only have three independent insertions ∆1, ∆2,∆4, one constraint equation, the Callan–Symanzik equation, is obtained by
(m∂m + βλ∂λ − 1
2γN)Γ = αm ∆d · Γ, (133)
where the symbol “m∂m” denotes m∂m+µ∂µ, the classical approximation of the parameter
αm is given by α(0)m =−2m2 and N , ∆d are respectively denoted by
N =
∫d4x ϕ(x)
δ
δϕ(x), ∆d =
[∫d4x 1
2ϕ2(x)
]
2
. (134)
Furthermore, the following three constraint conditions have been imposed,
αm = −2 a − βλ ∂λa + γ a,αm u = −βλ ∂λz + γ z,αm t = −βλ ∂λρ + 2γ ρ.
(135)
There are two ways to determine βλ, γ and αm, either solving the constraint equationsorder by order or test special Feynman diagrams. Both will be used in order to give theCallan–Symanzik equation a solid grounding.
To calculate β(0)λ , γ(0) and α
(0)m , the following information is needed,
a(0) = m2, u(0) = t(0) = 0, z(0) = 1, ρ(0) = λ, (136)
which are applied to the constraint equations to obtain
α(0)m = −2 a(0) + γ(0) a(0),
0 = γ(0)
0 = −β(0)λ + 2γ(0) λ
(137)
Solving the above equations, the results are given by
α(0)m = −2 a(0) = −2 m2, β
(0)λ = γ(0) = 0. (138)
On the other way, they are also decided by choosing the suitable 1PI Green functions. For
example, applying the Callan–Symanzik equation to Γ(0)2 given by
Γ(0)2 = p2 − m2, (139)
the values of α(0)m , γ(0) are obtained. Testing with Γ
(0)4 given by
Γ(0)4 = −λ, (140)
the value of β(0)λ will be known.
29
To calculate β(1)λ , γ(1) and α
(1)m , the input is given by
a(1) = z(1) = u(1) = 0, ρ(1) = 0, t(1) = 0. (141)
It induces the constraint equations,
α(1)m = γ(1) a(0),
0 = γ(1),
αm(0) t(1) = −β
(1)λ + 2γ(1) λ.
(142)
which are solved to give
β(1)λ = 2 m2 t(1), γ(1) = α(1)
m = 0. (143)
Similarly, testing with Γ(1)2 which is zero, α
(1)m has to vanish. Testing with Γ
(1)4R given by
Γ(1)4R = −i~
λ2
2
3∑
i=1
∫d4k
(2π)4(Dk DPi−k − D2
k), (144)
the corresponding Callan–Symanzik equation up one-loop is
m∂mΓ(1)4R + β
(1)λ (∂λ Γ)
(0)4 − 2γ(1) Γ
(0)4 = α(0)
m (∆d · Γ)(1)4R. (145)
Due to the equation
m∂mΓ(1)4R = α(0)
m (∆2 · Γ)(1)4R, (146)
the known information in the above subsection about the Zimmermann identity can be
applied to get the values of β(1)λ and γ(1).
In order to get β(2)λ , γ(2) and α
(2)m , the constraint equations up to two-loop have to be
used,
α(2)m = −2 a(2) + γ(2) a(0),
αm(0) u(2) = γ(2),
αm(0) t(2) = −β
(1)λ ∂λρ(1) + 2γ(2) λ.
(147)
Testing Γ(2)2R, the Callan–Symanzik equation up two-loop comes out,
m∂mΓ(2)2R − γ(2) Γ
(0)2 = α(0)
m (∆d · Γ)(2)2R + α(2)
m . (148)
Applying again the relevant Zimmermann identity and the equation
m∂mΓ(2)2R = α(0)
m (∆2 · Γ)(2)2R, (149)
the values of γ(2) and α(2)m are given by
γ(2) = −2m2 u(2), α(2)m = −2m4 u(2), (150)
which induce one resulta(2) = 0. (151)
About the values of β(2)λ and t(2), Γ
(2)4R has to be tested.
In the quantum field theory, there are two types of differential equations: the one isthe renormalization group equation and the other is the Callan–Symanzik equation. The
30
first one is a homogeneous equation, and the latter one is inhomogeneous. They bothtreat scaling transformations, but the first one is related to the renormalization scale andthe second is with the physical mass. They both use the beta function and the gammafunction but give them different meanings in massive models.
The Callan–Symanzik equation was derived in several ways. In [53] it was realizedby applying the skeleton expansions of the Green functions to study broken scale invari-ance. In [54] it was obtained by adding one mass counter term with a finite coefficientin the action equivalent to changing the renormalization scale. The additional mass termrelates two renormalized Green functions by the wavefunction renormalization, but oneis independent of its coefficient and the other one is not. Taking the differentiation onthis additional parameter then setting zero, the original Callan–Symanzik equation comesout after dividing one normalization factor. Both are very intuitive and indeed based onprevious research on the renormalization group differential equation. In models definedby regularization schemes, see [40], physical parameters and bare parameters are needed,and they are related by the wavefunction renormalization. The theory represented by thebare parameters is independent of the renormalization scale µ. Taking the differentiationof µ, the renormalization group differential equation is obtained.
In [6], the Callan–Symanzik equation can be derived in some simpler models definedby the BPHZ renormalization. But in models such as non-abelian gauge field theories, thetypical algebraic renormalization strategy is suitable: expanding the insertion given by thedifferential operator m∂m into equivalent basis which are chosen by constraint conditionsfrom the Slavnov–Taylor identity, the gauge fixing condition and the ghost equation, see[45].
Notes
The Callan–Symanzik equation is helpful to understand the quantum field theory. It wasused to study scaling breakings in [53], and the topic of the small distance behaviour,OPE, the asymptotic expansion in [54]. It was also used to simplify the renormalizabilityproof by combining it with the Weinberg power-counting theorem in [47]. Furthermore, itwas applied to prove the non-renormalization theorem in [48].
5.3 The different representations of the Callan–Symanzik equation
The Callan–Symanzik equation can be shown up in different ways. It can be represented bythe general Green function or the connected Green function or the 1PI Green function. Itis necessary to derive the transformation laws among the different formalisms. In addition,the skeleton expansion of the general Green function is used in the proofs.
• Among Γ, ZC and Z
The Callan–Symanzik equation is denoted by the generating functional of 1PI Greenfunctions,
m∂mΓ + βλ∂λΓ − 12γNΓ = αm∆d · Γ. (152)
By using the Legendre transformation (9) and definitions of insertions (18), (19), theCallan–Symanzik equation denoted by the generating functional of the connected Greenfunctions, is obtained by
m∂mZC + βλ∂λZC + 12γ
∫d4xJ(x)
δ
δJ(x)ZC = αm∆d · ZC . (153)
31
By using the definition (14) and definitions of insertions (17), (18), the Callan–Symanzik equation denoted by the generating functional of the general Green functions,is given by
m∂mZ + βλ∂λZ + 12γ
∫d4xJ(x)
δ
δJ(x)Z =
i
~αm∆d · Z . (154)
• Between G2 and Γ2
Assuming the Callan–Symanzik on the two-point general Green function G2 given by
m∂mG2(x, y) + βλ∂λG2(x, y) + γG2(x, y) =i
~αm∆d · G2(x, y), (155)
we have to prove that the Callan–Symanzik on the two-point general Green function Γ2
is bym∂mΓ2(x, y) + βλ∂λΓ2(x, y) − γΓ2(x, y) = αm∆d · Γ2(x, y). (156)
Proofs are as follows. From (10) we derive∫
d4z G2(x, z)Γ2(z, y) = i~δ4(x − y). Usingthe differential operator m∂m + βλ∂λ acting on both sides of the above equation, we get
∫d4z G2(x, z) (m∂mΓ2(x, z) + βλ∂λΓ2(x, z) − γΓ2(z, y))
= − i
~
∫d4z ∆d · G2(x, z) Γ2(z, y). (157)
Then applying the relation between ∆d · G2 and ∆d · Γ2,
∆d · G2(x, z) =
∫d4z1
∫d4z2 G2(x, z1)G2(z, z2)∆d · Γ2(z1, z2), (158)
we get the result same as (156).
• Between Gn and Γn
Similar to what we did, we have to prove that the Callan–Symanzik on the n-point1PI Green function Γn is given by
m∂mΓn + βλ∂λΓn − nγΓn = αm∆d · Γn, (159)
Assuming that the Callan–Symanzik on the n-point general Green function Gn is givenby
m∂mGn + βλ∂λGn + nγGn =i
~αm∆d · Gn (160)
Proofs are sketched as follows. With our notations, we expand Gn and ∆d ·Gn respec-tively as
Gn =i
~
∫G2 G2 · · · G2︸ ︷︷ ︸
n
Γn + · · · , (161)
∆d · Gn =
∫G2 G2 · · · G2︸ ︷︷ ︸
n
∆d · Γn
+ni
~
∫(∆d · G2) G2 · · · G2︸ ︷︷ ︸
n−1
Γn + · · · (162)
32
where the symbol∫
denotes the multi-variables integration and the symbol · · · denotes theother unwritten terms which do not affect our proof. Applying the differential operatorm∂m + βλ∂λ acting on both sides of the equation (161), using the Callan–Symanzikequation (160) and the equation (162), then comparing the terms with the same order ofi~ , we get the result like (159).
In the above, the transformation rules are obtained by applying the Legendre trans-formation to the generating functionals Γ, ZC [J ] and Z[J ]. The two things are used. Theinsertion is defined by introducing the external field; the skeleton expansion is needed. Infact, changing rules are carried out on the variation of the external field, instead of theinsertion itself.
The different formalism of the Callan–Symanzik equation are all useful when the alge-braic renormalization procedure is applied. First, the standard model requires spontaneoussymmetry breakings, which prefers loop expansions instead of perturbative expansions ofcoupling constants. In the loop expansion of a 1PI Green function, the power counting ofthe Planck constant is the same as the counting of the loop number, as can simplify thealgebraic renormalization procedure which constructs the Slavnov–Taylor to all orders ina perturbative sense. Second, the Feynman rules are invented particular for the generalGreen functions. In calculation, the formalisms given by the 1PI Green functions haveto be transformed to those by the general Green functions. Third, the quantum actionprinciple represented by the 1PI Green functions is clearer, but is proved in the generalGreen functions.
Notes
For example, the different formalisms of the Callan–Symanzik equation are used often in[44].
6 The algebraic renormalization procedure
In this section, the algebraic renormalization procedure will be introduced based on [49].It has several things different from other reviews on the same subject, for example [45, 46].First, it aims at presenting a general review independent of specific models. Second, ittries to introduce the linearized Slavnov–Taylor operator SΓ in a natural way from thebeginning, which is argued from the non-linear property of the Slavnov–Taylor identity,and plays the key role in the construction. Third, it gives a complete procedure of therecursive construction, which consists of checking in first order, assumption in n-th orderand construction in n + 1-th order.
The algebraic renormalization procedure was invented to treat the problem related torenormalization and symmetry in the perturbative quantum field theory. A given regu-larization procedure or subtraction scheme ensures one theory finite but may break itssymmetries. However, in the physical sense, some symmetries such as gauge ones rep-resented by the Slavnov–Taylor identities, must survive under renormalization. Further-more, in some models such as chiral gauge field theories or supersymmetry gauge fieldtheories, no known approaches preserve both renormalizability and symmetries at thesame time. Therefore, one type of recipe is adding non-invariant counter terms to cancelbreaking terms induced by the renormalization procedure, in other words, the algebraicrenormalization procedure is the program of how to constructing suitable non-invariantterms. It makes a judgement whether the anomaly exists, and obtains its name by chang-
33
ing the problem to the algebraic one, namely, solving the cohomology of a given nilpotentoperator.
The first subsection will make a sketch of how to construct of the Slavnov–Tayloridentity to all orders of ~. The second subsection will give an example of how to applythe algebraic constraint.
6.1 The construction of the Slavnov–Taylor identity
Assuming the Slavnov–Taylor identity is in the classical level, one question is proposed as
Can the Slavnov–Taylor identity survive the quantization?
Before construction, assume that the answer has been reached, namely,
S(Γ) = 0. (163)
Involving additional quantum correction, the new vertex functional Γε expands in the form
Γε = Γ + ε ∆, (164)
where ϵ is an infinitesimal constant. Replacing Γε with Γ in the Slavnov–Taylor identity,it results in
S(Γε) = S(Γ) + ε SΓ ∆ + O(ε2), (165)
where the linearized Slavnov–Taylor operator SΓ is introduced. The following constraintcondition by
SΓ∆ = 0. (166)
means the Slavnov–Taylor identity invariant under the new quantum correction at leastin first order of ϵ
Furthermore, the linearized Slavnov–Taylor operator SΓ satisfies two equations givenby
SΓ S(Γ) = 0, ∀ Γ, (167)
SΓ SΓ = 0, ∀ Γ, if S(Γ) = 0. (168)
The first identity is proved by using the statistics conventions in non-abelian gauge fieldtheories. The second one denotes the nilpotency of the functional operator SΓ, and it isproved by both applying the spin-statistics theorem and the Slavnov–Taylor identity. Forexample, in the classical case, there are
b S(Γcl) = 0, b2 = 0, S(Γcl) = 0, (169)
where the symbol b is defined by
b := SΓcl. (170)
In the following, the construction starts. As a first step, the Slavnov–Taylor identityhas to be realized in one-loop order. Applying the quantum action principle, the possiblebreaking of the Slavnov–Taylor identity is
S(Γ) = ∆ · Γ, (171)
34
which is independent of choices of regularization schemes and renormalization proceduresin a nonperturbative sense. The insertion ∆ · Γ is at least in one-loop order since theclassical Slavnov–Taylor identity is given. The linearized Slavnov–Taylor operator beinga local functional, a integral of x-dependent terms, the insertion ∆ · Γ can be representedby
∆ · Γ =
∫d4 x∆I(x) · Γ, (172)
where ∆I(x) is a local polynomial. In a perturbative quantum field theory, the insertion∆ · Γ has an expansion in ~,
∆ · Γ = ~ ∆class + O(~2), (173)
where the ∆class is an integral of x-dependent local field polynomial and the second termdenotes non-local contributions at least including one loop. Hence, the possible breakingof the Slavnov–Taylor identity is
S(Γ) = ~ ∆class + O(~2). (174)
Applying SΓ on both sides of the above equation, the constraint equation or the con-sistency condition on the ∆class is
b ∆class = 0, (175)
where the approximation is used
SΓ = b + O(~). (176)
Due to the nilpotency of the operator b, one type of solutions of the consistency condition(175) have the form
∆class = b ∆′class = b
∫d4x∆′
I(x), (177)
where ∆′I(x) is the local field polynomial with the dimension 4, the positive parity and the
zero ghost number in the ordinary gauge field theory. Shifting the given vertex functionalΓ like the following,
Γ −→ Γ − ~ ∆′class + O(~2) (178)
by adding ~ ∆′class into the renormalized action Γren, the Slavnov–Taylor identity is con-
structed up to first order of ~, namely,
S(Γ − ~ ∆′class) = ~ ∆class − ~ b ∆′
class + O(~2) = O(~2), (179)
where the equation (165) has been used. However, it is possible that solutions like (177)are not general, that is to say that there may exist one solution which is not the action ofthe operator b on a local polynomial. In the latter case, the construction fails.
In the second step of the construction, the recursive procedure is used. Assuming theproblem has been solved to order of ~n−1,
S(Γ) = O(~n), (180)
the construction of the Slavnov–Taylor identity up to order of ~n is equivalent to lookingfor one suitable local counter terms ∆ satisfying,
S(Γ − ~n ∆) = O(~n+1). (181)
Hence, a general procedure is summarized in the following.
35
1. With the quantum action principle, the possible breaking of the Slavnov–Tayloridentity is given by the insertion of the local functional,
S(Γ) = ~n ∆ · Γ = ~n ∆ + O(~n+1). (182)
2. The consistency condition is obtained by
b ∆ = 0, (183)
by applying SΓ on both sides of (182) and taking the approximation formula (176).
3. Solve the consistency condition (183) to get the general solution like
∆ = b ∆ + rA, (184)
where the coefficient r is a number and A is not the solution of the b-variation type.
4. In the case that either no solutions are like the A term or symmetries force r tovanish, the vertex functional Γ is shifted to Γ − ~n ∆, which satisfies
S(Γ − ~n ∆) = S(Γ) − ~n ∆ + O(~n+1) = O(~n+1). (185)
Therefore, the Slavnov–Taylor identity is constructed up to order of ~n. Otherwise,the Slavnov–Taylor identity is broken by anomaly.
In the algebraic renormalization procedure, an insertion has to be an integral of alocal field polynomial, and has suitable quantum numbers required by symmetries orphysical ansatz and has to satisfy the dimensional constraint due to the power-countingrenormalizability. They are enough to solve the consistency condition. Without knowingthe insertion exactly, therefore, the Slavnov–Taylor identity can be constructed to allorders. In addition, in simpler models such as abelian gauge theories, an insertion can becalculated by using the renormalized action Γren.
6.2 The consistency condition to all orders
In simpler models defined by the BPHZ renormalization, an insertion can be exactlycalculated. To realize symmetries or consistency conditions in this case, the procedure israther different from the construction of the Slavnov–Taylor identity to all orders. Oneexample is given in the following.
In the massive scalar field theory treated in [49], the Callan–Symanzik operator C andthe improved Ward identity operator WK , are commutative in the classical approximation,
C(0) = m∂m, (WK)(0) = WK , [m∂m, WK ] = 0. (186)
One problem is how to keep the commutativity to all orders of ~, namely, under quantumcorrections there is the commutator by
[C, WK ]Γ = 0. (187)
Assuming it has been realized in order of ~(n), one constraint equation on coefficientsin the renormalized action Γren has been known. It collects all relevant coefficients up to
36
order of ~n. Considering the constraint equation in order of ~(n+1) and expanding it inorders of ~, the following equation is obtained by
n+1∑
k=0
(WK)(n+1−k)(CΓ)(k) =
n+1∑
k=0
C(n+1−k)(WKΓ)(k). (188)
Applying the quantum action principle in the BPHZ renormalization,
(CΓ)(n+1) = (CΓren)(n+1) +
n∑
k=0
([CΓren](k) · Γ)(n+1−k)
(WΓ)(n+1) = (WΓren)(n+1) +
n∑
k=0
([WΓren](k) · Γ)(n+1−k),
(189)
the above equation (188) is collecting all the coefficients up to order of ~(n+1). So thereare two equations: one is known; the other needs verified. The strategy is applying theknown one to the other to get the one new constraint equation on the coefficients in orderof ~(n+1). In the case that there are some parameters to be decided, at least one parametercan be fixed to all orders by using the new constraint one.
Notes
In [22, 23], the algebraic renormalization procedure was used to construct non-abeliangauge field theories with spontaneous symmetry breakings. In [50], it was used to constructsupersymmetrical field theories defined in the superspace and in [55, 56] to constructsupersymmetrical field theories in the Wess–Zumino gauge. In [51], it was applied toconstruct the standard model to all orders of ~.
7 Current constructions and charge constructions
In the above several sections, the basic ingredients of the BPHZ renormalization procedurehave been introduced. Besides them, the LSZ reduction procedure and the algebraicrenormalization procedure are also used to treat the conformal transformations of theS-matrix, see [44]. In this section, the charge constructions via the LSZ reduction aretreated in detail. The first subsection presents the conformal algebra in the classical case.The second subsection shows how to construct the currents responsible for the conformaltransformations and the charges for the Poincare transformations.
7.1 The conformal algebra
The generators of the conformal algebra in the four dimension space-time are denotedby Pµ for the translation transformation, Lµν for the Lorentz transformation, D for thedilatation transformation and Kµ for the special conformal transformation, which satisfythe following Lie algebra,
[Pµ, Pν ] = 0, (190)
[Pµ, Lρσ] = ηµρ Pσ − ηµσ Pρ, (191)
[Pµ, D] = Pµ, (192)
37
[Pµ,Kλ] = 2 ηµλ D + 2 Lλµ, (193)
[Lµν , Lρσ] = ηνρ Lµσ − ηµσ Lρν + ηµρ Lνσ − ηνσ Lµρ, (194)
[D, Lµν ] = 0, (195)
[Lµν ,Kλ] = ηνλ Kµ − ηµλ Kν , (196)
[D, D] = 0, (197)
[D, Kλ] = Kλ, (198)
[Kσ,Kλ] = 0. (199)
The infinitesimal transformations generated by the above algebra are denoted by
δϕαa(x) = Σα
β(x, ∂x)ϕβa(x), (200)
which are shown in detail respectively by
δT ϕαa(x) = aµ∂µϕα
a(x); (201)
δLϕαa(x) = ωµν(xµ∂ν − xν∂µ)ϕα
a(x) − i ωµν [Σµν ]αβϕβ
a(x); (202)
δDϕαa(x) = ϵ(d + xµ∂µ)ϕα
a(x); (203)
δKϕαa(x) = αµ
((2xµxν − ηµνx
2)∂ν + 2 d xµ
)ϕα
a(x)
−2i αµxν [Σµν ]αβϕβ
a(x), (204)
where [Σµν ]αβ is the spin matrix; d is the dimension of the field ϕα
a(x); and aµ, ωµν , ϵ,αµ are the infinitesimal constant parameters.
7.2 Current constructions and charge constructions
the S-matrix can be defined in the LSZ reduction procedure. With this approach, theinformation on the operator formalism can be recovered, such as current constructions,charge constructions and quantum transformations of quantum fields.
For simplification, the moment constructions of the local Ward identity operators canbe chosen as follows,
wTµ (x) = ∂µϕ(x)
δ
δϕ(x)− 1
4∂µ
(ϕ(x)
δ
δϕ(x)
), (205)
wLµν(x) = xµw
Tν (x) − xνw
Tµ (x), (206)
wD(x) = xµwTµ (x), (207)
wKν (x) = (2xµxν − ηµ
ν x2)wTµ (x). (208)
Applying the quantum action principle in the BPHZ renormalization procedure, theenergy-momentum tensor Tµν can be calculated from
wTµ (x) · Γ = −[∂ν Tµν(x)] · Γ. (209)
Then the breaking of the conformal invariance are controlled by the insertion of the traceof the energy-momentum tensor, namely,
wDΓ = −∂ν [Dν ] · Γ + [T νν ] · Γ, (210)
wKν Γ = −∂µ [Kµν ] · Γ + 2xν
[Tµ
µ
]· Γ, (211)
38
where the current Dν is xµTµν for the dilatation transformation and the current Kµν is
(2xνxζ − ηζ
νx2)Tµζ for the special conformal transformation.Define the local Ward identity for the space-time translation in the generating func-
tional Z[J ] by
wTµ (x)Z[J ] :=
(J(x)∂µ
δ
δJ(x)− 1
4∂µ(J(x)
δ
δJ(x))
)Z[J ]
=i
~∂ν [Tµν ] · Z[J ]. (212)
It can be realized in the Green function via
wTµ (x)Gn =
n∑
l=1
δ(x − xl)∂xµGn(x, x1, · · · , xl, · · · , xn)
−1
4
n∑
l=1
∂xµ (δ(x − xl)Gn(x, x1, · · · , xl, · · · , xn))
=i
~∂ν [Tµν ] · Gn(x1, · · · , xn), (213)
where xl indicates that xl is missing in the string of variables. Furthermore, the localWard identity for space-time translations in the momentum space is given by
wTµ (p)Gn = −i
n∑
l=1
((pµ + pl,µ) − 1
4pµ
)Gn(p + pl, p1, · · · , pl, · · · , pn)
=i
~(−ipν)[Tµν(p)] · Gn(p1, · · · , · · · , pn), (214)
which can be transfered into the form
i
~(−ipν)[Tµν(p)] · Sn(p1, · · · , · · · , pn)
= −i
n∑
l=1
((pµ + pl,µ) − 1
4pµ
)(−ir−1
2 )nn∏
j=1
(p2j − m2)
×Gn(p + pl, p1, · · · , pl, · · · , pn)|P . (215)
In the on-shell limit and in case of p being nonzero and the right hand side being zero,the conservation of the energy-momentum tensor is obtained by
pν Tµν(p) = 0, (216)
which is given in coordinate space by
∂ν Tµν = 0. (217)
Hence the four-momentum charge Pµ is defined as
Pµ :=
∫d3x Tµ0. (218)
Here all involved operators are defined in the asymptotic Hilbert space H satisfying
H = Hin = Hout. (219)
39
Similarly, for the other conformal transformations, the corresponding expression canbe also set up
∂aMµνa = 0,
∂µDµ = T νν ,
∂µKµν = 2xν Tµµ , (220)
where Mµνa, Dµ, and Kνµ are respectively given by
Mµνa = xµTνa − xν Tµa, Dµ = xν Tµν , Kνµ = (2xνxζ − ηζ
νx2)Tµζ . (221)
Then the charge Mµν for the Lorentz rotations is denoted by∫
d3xMµν0 . But the chargesfor both the dilatation transformation and for the special conformal transformation cannotbe easily found out.
To derive the quantum transformations of the quantum field operator Φ, the LSZreduction procedure can be applied on the both sides of the local Ward identity (214),namely,
i
~(−ir−1
2 )nn∏
j=2
(p2j − m2)(−ipν)[Tµν(p)] · Gn(p1, · · · , · · · , pn)|P
= −i
n∑
l=1
((pµ + pl,µ) − 1
4pµ
)(−ir−1
2 )nn∏
j=2
(p2j − m2)
× Gn(p + pl, p1, · · · , pl, · · · , pn)|P . (222)
In the on-shell limit, the above formalism is related to
−i
((pµ + pl,µ) − 1
4pµ
)Φ(p + p1) =
i
~(−ipν)T
(Tµν(p)Φ(p1)
), (223)
where the symbol T denotes the time-ordering defined in the coordinate space. Then theaction of the local Ward identity operator for space-time translations on the quantum fieldoperator Φ(x) in coordinate space is given by
wTµ (x)Φ(x1) := ∂x
µΦ(x)δ4(x − x1) − 1
4∂x
µ(Φ(x)δ4(x − x1))
=i
~∂νT
(Tµν(x)Φ(x1)
). (224)
The similar equations for the other conformal transformations can be also obtained by
(xµwTν (x) − xνw
Tµ (x))Φ(x1) =
i
~∂aT
(Mµνa(x)Φ(x1)
), (225)
xµwTµ (x)Φ(x1) =
i
~∂µT
(Dµ(x)Φ(x1)
)
− i
~T(T ν
ν (x)Φ(x1))
, (226)
(2xνxµ − ηµ
ν x2)wTµ (x)Φ(x1) =
i
~∂µT
(Kµν(x)Φ(x1)
)
− i
~2xνT
(Tµ
µ (x)Φ(x1))
. (227)
40
The quantum transformations of the quantum field Φ for space-time translations andLorentz rotations are obtained by integrating
∫ x0+ε
x0−εdx0
1
∫d3x1 (228)
on both sides of the above equations (224) and (225), namely,
δTµ Φ := ∂µΦ =
i
~[Pµ, Φ],
δLµν Φ := (xµ∂ν − xν∂µ)Φ =
i
~[Mµν , Φ]. (229)
In the cases of the dilatation transformation and the special conformal transformation,the quantum transformations in the free(or asymptotic free) field theory are constructedby
δDϕin(x) := (1 + x∂x)ϕin(x) =i
~[D, ϕin(x)],
δKν ϕin(x) :=
((2xνx
µ − ηµν x2)∂µ + 2xν
)ϕin(x) =
i
~[Kν , ϕin(x)], (230)
where D is the charge for the dilatation transformation and Kν is the charge for the specialconformal transformation.
The charge constructions are important in the perturbative quantum field theory de-fined by the algebraic renormalization procedure. There are two types of reasons. The oneis fundamental. As in quantum mechanics there are both the operator formulation andthe wavefunction one, the quantum field theory also has its representation of operators inthe canonical quantization procedure and the formulation of ordinary functionals in thepath integral theory. Operator is convenient to present the interpretation of the quantumphysics, and functional is suitable to practical calculation. The algebraic renormalizationprocedure only deals with Green functions, so no operators are involved. The other oneis pragmatic. Since the algebraic renormalization procedure aims at keeping symmetriesunder renormalization, the charges responsible for the symmetries can be constructed inprinciple.
Notes
In [28, 29], charges responsible for BRST transformations were constructed and used toprove the unitarity of the S-matrix. In [52], with the above method, supersymmetrytransformation of quantum fields are derived.
8 The Connes–Kreimer Hopf algebra
This section is taken from [57]. The axioms of the Hopf algebra will be firstly introduced,and then the Connes–Kreimer will be sketched in Zimmermann’s sense.
8.1 The axioms of the Hopf algebra
The Hopf algebra is a bialgebra with a linear antipode map. The bialgebra is both analgebra and a coalgebra in a compatible way. The coalgebra is a vector space with a
41
coassociative linear coproduct map. The algebra is a vector space with an associativelinear product map and a linear unit map. The vector space is a set over a field togetherwith a linear addition map.
We denote the set by H, the field by F , the addition by +, the product by m, theunit map by η, the coproduct by ∆, the counit by ε, the antipode by S, the identity mapby Id, and the tensor product by ⊗. Then the vector space is given in terms of the triple(H, +;F) where the field F with unit 1.
Let (H, +,m, η, ∆, ε, S; F) be a Hopf algebra over F . The Hopf algebra has to satisfythe following seven axioms,
(1) m(m ⊗ Id) = m(Id ⊗ m),(2) m(Id ⊗ η) = Id = m(η ⊗ Id),
m : H ⊗ H → H,m(a ⊗ b) = ab, a, b ∈ H,η : F → H,
(231)
which denote the associative product m and the linear unit map η in the algebra (H,+,m, η; F)over F respectively;
(3) (∆ ⊗ Id)∆ = (Id ⊗ ∆)∆,(4) (Id ⊗ ϵ)∆ = Id = (ϵ ⊗ Id)∆,
∆ : H → H ⊗ H,ε : H → F ,
(232)
which denote the coassociative coproduct ∆ and the linear counit map ε in the coalgebra(H, +, ∆, ε; F) over F and where we used F ⊗ H=H or H ⊗ F=H;
(5) ∆(ab) = ∆(a)∆(b),(6) ε(ab) = ε(a)ε(b),
∆(e) = e ⊗ e, ε(e) = 1, a, b, e ∈ H,(233)
which are the compatibility conditions between the algebra and the coalgebra in the bial-gebra (H, +, m, η, ∆, ε; F) over F , claiming that the coproduct ∆ and the counit ϵ arehomomorphisms of the algebra (H, +,m, η; F) over F with the unit e;
(7) m(S ⊗ Id)∆ = η ϵ = m(Id ⊗ S)∆, (234)
which is the antipode axiom that can be used to define the antipode.With the algebra (H, +, m, η; F) and the coalgebra (H, +, ∆, ε; F), a convolution ⋆
can be defined in the vector space End(H,H), consisting of all linear maps from H to H,namely,
f ⋆ g := m(f ⊗ g)∆, f, g ∈ End(H,H). (235)
(End(H,H), +, ⋆, η ε; F) is an algebra over F . The convolution can also be defined bythe character ϕ of algebra. The character ϕ is a nonzero linear functional over the algebraand is a homomorphism of the algebra satisfying ϕ(ab)=ϕ(a)ϕ(b). The correspondingconvolution between two characters ϕ and φ, is defined by
ϕ ⋆ φ := m(ϕ ⊗ φ)∆, ϕ, φ are characters. (236)
The Hopf algebra is said to be commutative if we have the equality m = m P ,where the flip operator P is defined by P (a ⊗ b)=b ⊗ a. Similarly, the Hopf algebra(H,m, ∆, η, ϵ, S) is not cocommutative if P ∆ =∆.
An algebra is graded if it can be written as the vector sum of subsets: H = H0 ⊕H1 ⊕H2 ⊕ · · ·. If h ∈ Hm, we denote the grading of h by grad(h)=m. The grading must becompatible with the product: if h1 ∈ Hm and h2 ∈ Hn, then (h1 · h2) ∈ Hm+n.
42
8.2 An example: the Connes–Kreimer Hopf algebra
As an example, we sketch the Connes–Kreimer Hopf algebra in Zimmermann’s sense givenin [3], in order to show the general procedure of realizing the Hopf algebra in perturbativequantum field theory.
First of all, some notations have to be declared.
The connected Feynman graph is constructed from vertices, external lines and internallines between the two vertices. Denote the set of all the lines of the Feynman graph Γ byL(Γ), the set of all external lines of Γ by Le(Γ), and the set of all internal lines of Γ byLi(Γ), then
L(Γ) = Le(Γ) ∪ Li(Γ). (237)
Denote the set of all vertices of Γ by V(Γ), where only the vertices related to at least twolines are included. In the case of BPHZ, the Feynman graph Γ is completely fixed by L(Γ)and V(Γ) is the set of all vertices attaching to the lines in L(Γ).
Subgraph. The subgraph γ is determined by L(γ), where L(γ)⊆L(Γ). The connectedsubgraph γ is expanded by L(γ) and V(γ), where V(γ)⊆V(Γ).
Disjoint subgraphs. The graphs γ and γ′ are disjoint if Li(γ) ∩ Li(γ′)=∅. Denotethe disjoint subgraphs γ1, γ2 · · · γc by
s = (γ1, γ2 · · · γc), (238)
where every two graphs are disjoint. The set of all possible s in Γ is denoted by S,
S = s | Li(γj) ∩ Li(γk) = ∅, L(γj) ⊆ L(Γ), j = k; j, k = 1, 2 · · · c, (239)
where S=∅ is allowed. The symbol∑
s∈ S denotes the summation over all possible dis-joint connected subgraphs and the symbol
∑′′s∈ S denotes the same except for s=(Γ) and
s=∅. The symbol∑′
s∈ S denotes the same except for s=(Γ). Under renormalization, onlydivergent graphs and divergent 1PI subgraphs are involved.
Reduced subgraph. Denote the disjoint union among connected subgraphs by
δ = γ1γ2 · · · γc. (240)
The reduced subgraph Γ/δ is obtained by contracting each connected parts of δ to a point,namely,
L(Γ/δ) = L(Γ) \ Li(δ), (241)
V(Γ/δ) = V(Γ) \ V(δ) ∪ V 1, V 2 · · · V c, (242)
where V i, i=1, 2 · · · c, are the new vertices from the contraction. Here A \ B denotes thedifference between two sets A and B. And Γ/δ denotes the reduced diagram Γ/γ1γ2 · · · γc.
Feynman rules. The Feynman rules are applied to get unrenormalized Feynmanintegrand, or unrenormalized Feynman integral from the Feynman graph. Construct theFeynman rules as a map, from the set of Feynman graphs to the set of the unrenormalizedFeynman integrand(integral),
ϕ : Γ −→ ϕ(Γ), (243)
where the external lines do not contribute to the map. For the disjoint union γ1γ2, wehave a homomorphism,
ϕ(γ1γ2) = ϕ(γ1)ϕ(γ2), (244)
43
thus the Feynman rules can be regarded as a character of the algebra of Feynman graphs.Feynman rules are unchanged for the subgraph, but for the reduced graph, the factor of1 has to be arranged to the new vertices from the contraction.
In order to see the translation between the BPHZ and the Connes–Kreimer Hopfalgebra, we sketch the recursive formalism of BPHZ.
For the unrenormalized Feynman integrand(integral) IΓ(KΓ, qΓ), where KΓ denoteinternal momenta and qΓ denote external momenta in the Feynman graph Γ. The globaldivergence OΓ is defined by
OΓ = −R(RΓ), (245)
where R is the renormalization map to extract the relevant divergent part from the RΓ,given by
RΓ = IΓ(KΓ, qΓ) +
∑
s∈ S
′′IΓ/γ1γ2···γc
(K, q)
c∏
τ=1
Oγτ (Kγτ , qγτ ), (246)
in which RΓ does not contain any subdivergences and it is defined by the recursiveprocedure. The renormalized Feynman integrand(integral) related to IΓ is obtained by
RΓ = RΓ + OΓ = RΓ − R(RΓ), (247)
namely,
RΓ = IΓ(KΓ, qΓ) +
∑
s∈ S
′IΓ/γ1γ2···γc
(K, q)
c∏
τ=1
Oγτ (Kγτ , qγτ )
=∑
s∈ SIΓ/γ1γ2···γc
(K, q)
c∏
τ=1
Oγτ (Kγτ , qγτ ). (248)
Now let us come to the Connes–Kreimer Hopf algebra denoted by
(H,+,m, η, ∆, ε, S; F). (249)
Its all elements are given by
H : the set generated by 1PI Feynman diagrams;+ : the linear combination;F : the complex number C with the unit 1;m : the disjoint union, m(Γ1 ⊗ Γ2) = Γ1 ∪ Γ2;e : the empty set: ∅ and η(1) = e;∆ : ∆(Γ) = Γ ⊗ e + e ⊗ Γ+∑′′
s∈ S γ1γ2 · · · γc ⊗ Γ/γ1γ2 · · · γc;ε : ε(e) = e; ε(Γ) = 0, if Γ = e;S : S(Γ) = −Γ −∑′′
s∈ S S(γ1γ2 · · · γc)Γ/γ1γ2 · · · γc.
44
The following axioms need to be checked,
(1) m(m ⊗ Id) = m(Id ⊗ m),(2) m(Id ⊗ η) = Id = m(η ⊗ Id),(3) (∆ ⊗ Id)∆ = (Id ⊗ ∆)∆,(4) (Id ⊗ ϵ)∆ = Id = (ϵ ⊗ Id)∆,(5) ∆(Γ1Γ2) = ∆(Γ1)∆(Γ2),
∆(e) = e ⊗ e, Γ1, Γ2 ∈ H,(6) ε(Γ1Γ2) = ε(Γ1)ε(Γ2), ε(e) = 1, Γ1, Γ2 ∈ H,(7) m(S ⊗ Id)∆ = η ϵ = m(Id ⊗ S)∆.
(250)
Axioms (1) and (2) are satisfied by the choice of the disjoint union as the product, forit is commutative. Axiom (7) is automatically satisfied, since it was used to define theantipode. The proof for the coassociativity of the coproduct is sketched as follows.
Proof. The coassociativity means that we have to prove
(∆ ⊗ Id)∆(Γ) − ∆(Γ) ⊗ e =
(Id ⊗ ∆)∆(Γ) − ∆(Γ) ⊗ e, (251)
namely we have to prove
∑
γ⊂Γ
∑
γ′1⊆γ
γ′1 ⊗ γ/γ′
1 ⊗ Γ/γ =
∑
γ′2⊂Γ
∑
γ′′⊆Γ/γ′2
γ′2 ⊗ γ′′ ⊗ (Γ/γ′
2)/γ′′. (252)
Then we come to the proof:1. The divergent subgraph γ′ of the divergent subgraph γ of Feynman graph Γ is still
the divergent subgraph in Γ, γ ⊂ Γ. Then we can choose γ′1=γ′
2. Denote γ′1, γ′
2 by γ′.2. The reduced graph γ/γ′ obtained by contracting γ′ to a point in the divergent
subgraph γ of the Feynman graph Γ is still the divergent subgraph of Γ/γ′. Then we canchoose γ′′=γ′ to obtain Γ/γ=(Γ/γ′
2)/γ′′.From the above axioms, we can get
S(γ1γ2) = S(γ2)S(γ1), (253)
then the antipode can be written as
S(Γ) = −Γ −∑
s∈ S
′′S(γ1)S(γ2) · · · S(γc)Γ/γ1γ2 · · · γc. (254)
Combining the Feynman rules, the character ϕ of the Hopf algebra and the antipode,we get a new character ϕ S, namely,
ϕ S = −ϕ(Γ) −∑
s∈ S
′ϕ S(γ1γ2 · · · γc)ϕ(Γ/γ1γ2 · · · γc). (255)
Define the convolution between two characters ϕ and ϕ S by
(ϕ S ⋆ ϕ)(Γ) := m(ϕ ⊗ ϕ)(S ⊗ Id)∆(Γ), (256)
which satisfies (ϕ S ⋆ ϕ)(Γ)=0 provided that Γ=e.
45
Introduce the renormalization map R to deform the character ϕ S, we get,
SR(Γ) = −R[ϕ(Γ)] −
R[∑
s∈ S
′′SR(γ1γ2 · · · γc)ϕ(Γ/γ1γ2 · · · γc)
]. (257)
SR is also a character,SR(γ1γ2) = SR(γ1)SR(γ2), (258)
if the renormalization map R satisfies the requirements
R(γ1γ2) + R(γ1)R(γ2) = R(γ1R(γ2)) + R(R(γ1)γ2), (259)
where R(γ)=Rϕ(γ). The corresponding proof can be realized by an iterative procedure.It is remarkable that
SR(Γ) = R(ϕ S)(Γ), (260)
which implies that SR is not a trivial character and we can not use the Hopf algebradirectly in problems involving renormalization.
Now we are able to construct the translation between the BPHZ and the Connes–Kreimer Hopf algebra with the help of
OΓ ⇐⇒ SR(γ), (261)
RΓ ⇐⇒ SR ⋆ ϕ(Γ). (262)
In order to relate the Connes–Kreimer Hopf algebra to BPHZ in Zimmermann’s sense,we choose the renormalization map R(γ) as Rγ(γ) and specify it as
Rγ = (−tγ)Sγ , (263)
where tγ is the Taylor subtraction operator cut off by the divergence degree of the Feynmangraph γ, and Sγ is the substitution operator defined by
Sµ : Kγ −→ Kγ(Kµ),
qγ −→ qγ(Kµ, qµ),
for γ ⊂ µ, (264)
SΓ : Kγ −→ Kγ(KΓ),
qγ −→ qγ(KΓ, qΓ),
for γ ⊂ Γ. (265)
Since the case Rγ1γ2(γ1γ2)=Rγ1(γ1)Rγ2(γ2) is explicit, we can also define the deformedcharacter SR. But we still have SR(Γ) = R(ϕ S)(Γ) due to the fact that
RΓ(Rγ(γ)IΓ/γ
)= Rγ(γ) RΓ/γ(Γ/γ). (266)
Notes
Recent development on the subject of the Connes–Kreimer Hopf algebra can be found in[58], [59] and [60].
46
9 Acknowledgements
I thank Dirk Kreimer, Manfred Salmhofer, Klaus Sibold and Raimer Wulkenhaar forhelpful discussions on renormalization theories.
The DFG is acknowledged for the financial support. The author was supported byGraduiertenkolleg “Quantenfeldtheorie: Mathematische Struktur und physikalische An-wendungen”, University Leipzig, for three years.
References
[1] N. Bogoliubov, O. Parasiuk, “On the Multiplication of the Causal Function in Quan-tum Theory of Fields”, Acta Math. 97 (1957) 227.
[2] K. Hepp, “Proof of the Bogoliubov-Parasiuk Theorem on Renormalization”, Com-mun. Math.Phys. 2 (1966) 301.
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Part III
Path Integrals and Non-AbelianGauge Field Theories
218
Part IV
The Standard Model and ParticlePhysics
219
Part V
Integrable Field Theories andConformal Field Theories
220
Part VI
Quantum Field Theories inCondense Matter Physics
221
Part VII
General Relativity, Cosmology andQuantum Gravity
222
Part VIII
Supersymmetries, Superstring andSupergravity
223
Part IX
Quantum Field Theories onQuantum Computer
224
Part X
Quantum Gravity on QuantumComputer
225