proceedings of the seminar on modern quantum field theory - … · 2017. 1. 16. · the seminar was...

Proceedings of the Seminar onModern Quantum Field Theory

Spring Semester 2010, Tsinghua University

Edited by

Zhong-Zhi Xianyu

Center for High Energy Physics, Tsinghua University

Preface

This volume is a collection of lecture notes used in the seminar on modern quantum field the-ory during spring semester 2010 at Tsinghua University (THU). This journal club was originatedby my friend Long Zhang (nicknamed Big Tiger∼) and me, and participated by some undergrad-uate physics students from BNU, PKU and THU. The seminar was held weekly and consists of12 informal lectures given by our participants.

Since most of us had taken a course on quantum field theory before, we were aiming atachieving a better understanding of the theory as well as acquainting ourselves with some of itsapplications to various directions of modern physics. As a result, the topics vary from condensedmatter physics to high energy physics, as can be seen from the contents.

The volume contains 9 notes of the talks, provided kindly by the speakers. I choose to leaveeverything in its original form without any correction (except for adding a new reference to thenote of my own). One year has passed since then, and many things in this volume may look “toosimple” and “sometimes naıve” for us. However, at least in my opinion, this is probably the bestway to record the wonderful time we have shared together, which I will always treasure.

I would like to acknowledge all speakers who have made essential contributions to this vol-ume with their elaborate work for not only preparing the talks but also writing their lecture notesinto LATEXfiles. I would also like to thank all the participants, without whom the seminar wouldnever be successful. At the same time I have to apologize, as the editor, for the terrible decay ofthe appearance of this volume.

Zhong-Zhi Xianyu

August, 2011

iii

iv PREFACE

Contents

Preface iii

1 An Introduction to Chiral Anomaly by Zhong-Zhi Xianyu 11.1 Symmetries: Classical vs. Quantum . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 The Classical Case: Nother’s Theorem . . . . . . . . . . . . . . . . . . 11.1.2 The Quantum Case: Ward Identities . . . . . . . . . . . . . . . . . . . . 3

1.2 Anomalous Integral Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Chiral SU(N) theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Transformation of the Integral Measure . . . . . . . . . . . . . . . . . . 61.2.3 *Regularization Independence . . . . . . . . . . . . . . . . . . . . . . . 101.2.4 Fujikawa’s Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Some Related Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.1 Anomalous Ward Identities . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.2 Triangle Diagrams (Outline) . . . . . . . . . . . . . . . . . . . . . . . . 121.3.3 A Phenomenological Application . . . . . . . . . . . . . . . . . . . . . 12

2 A Brief Introduction to Loop Quantum Gravity and Its Applications by Biao Huang 152.1 Canonical Formulation of Classical General Relativity . . . . . . . . . . . . . . 15

2.1.1 Traditional Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.2 Ashtekar Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 LQG Kinematical Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.1 Cylindrical Functions and The Inner Product . . . . . . . . . . . . . . . 192.2.2 SU(2) Gauge Invariance: Spin Networks . . . . . . . . . . . . . . . . . 212.2.3 Diffeomorphism Invariance: s-Knot States . . . . . . . . . . . . . . . . . 22

2.3 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.1 Area and Volume Operators On The Kinematic Space . . . . . . . . . . 242.3.2 Dynamics: The Hamiltonian Constraint and Spinfoam Models . . . . . . 27

2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

v

vi CONTENTS

2.4.1 Black Hole Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 302.4.2 Loop Quantum Cosmology (LQC) . . . . . . . . . . . . . . . . . . . . . 32

3 Anomaly, Topology and Renormalization Group by Zhong-Zhi Xianyu 373.1 A Brief Review of Chiral Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Anomaly and the Index Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 Conformal Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.2 Local Structure of Conformal Group . . . . . . . . . . . . . . . . . . . . 433.3.3 Conformal Invariance in Classical Field Theory . . . . . . . . . . . . . . 44

3.4 Scale Anomaly and Renormalization Group . . . . . . . . . . . . . . . . . . . . 453.4.1 Scaling the Phi-Four Model . . . . . . . . . . . . . . . . . . . . . . . . 453.4.2 Ward Identities in Momentum Space . . . . . . . . . . . . . . . . . . . . 46

3.5 QED beta function from scale anomaly . . . . . . . . . . . . . . . . . . . . . . . 48

4 Peierls Transition in 1-Dimensional Lattice: Three Approaches by Long Zhang 514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Effective Lagrangian and Feynman Rules . . . . . . . . . . . . . . . . . . . . . 534.3 Renormalization Group Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 544.4 Effective Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Spin Connection and Local Lorentz Transformation by Xiao Xiao 595.1 Why spin connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3 Spinor field in curved spacetime and the local transformation property of spin

connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.4 Covariant exterior derivative, torsion and curvature . . . . . . . . . . . . . . . . 645.5 Calculating the curvature tensor: . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6 An Introduction to Dark Matter by Lan-Chun Lu 696.1 Cosmology evidence for the existence of DM . . . . . . . . . . . . . . . . . . . 69

6.1.1 Rotation curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.1.2 X Ray observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.1.3 Gravitational lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.1.4 Anisotropy of Cosmology Microwave Background . . . . . . . . . . . . 71

6.2 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.3 Models for Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.3.1 SUSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

CONTENTS vii

6.3.2 mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.4 Detection of DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.4.1 Direct detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.4.2 Indirect detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.4.3 Accelerator detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7 Quantum Geometric Tensor... by Ran Cheng 797.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.2 Formalism - A Physicist’s way . . . . . . . . . . . . . . . . . . . . . . . . . . . 817.3 Case One – Adiabatic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827.4 Case Two – non-Adiabatic System . . . . . . . . . . . . . . . . . . . . . . . . . 857.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

8 An Introduction to Linearized Gravity by Zhong-Zhi Xianyu 898.1 Relativity at Linear Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8.1.1 Expanding the Hilbert-Einstein Action around Flat Background . . . . . 898.1.2 Free Spin-2 Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908.1.3 Recovery of Some Classical Results . . . . . . . . . . . . . . . . . . . . 938.1.4 Pauli-Fierz Mass Term . . . . . . . . . . . . . . . . . . . . . . . . . . . 948.1.5 DVZ Discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948.1.6 Vainshtein Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

8.2 Geometric Effective Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948.2.1 Traditional Effective Potential Method . . . . . . . . . . . . . . . . . . . 948.2.2 Gauge-Dependence Problem of Coleman-Weinberg Model . . . . . . . . 968.2.3 Geometric Effective Potential . . . . . . . . . . . . . . . . . . . . . . . 978.2.4 Applications in Gravitational Theories . . . . . . . . . . . . . . . . . . . 102

9 On Chern-Simons Gauge Theory by Xiao Xiao 1039.1 Chiral Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039.2 Topologically Massive Theory And Higgs Mechanism . . . . . . . . . . . . . . 1049.3 Non-Abelian Chern-Simons Theory . . . . . . . . . . . . . . . . . . . . . . . . 1069.4 Induced Chern-Simons Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079.5 Gravity With Topological Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 1109.6 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

9.6.1 DERIVATION OF (10) . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

viii CONTENTS

CHAPTER 1An Introduction to Chiral Anomaly

Zhong-Zhi Xianyu

Department of Physics, Tsinghua University, Beijing 100084

(April 4, 2010)

1.1 Symmetries: Classical vs. Quantum

1.1.1 The Classical Case: Nother’s Theorem

A classical field theory is built up with the action principle. Once the action is given, ev-erything of the theory is determined. Here we assume that the action, as a functional of variouskinds of field ϕi, can be written as an integral of the Lagrangian over the space-time:

S[ϕ] =

∫d4xL[ϕ(x)]. (1.1)

A symmetry of the theory can be thought as a transformation on fields ϕi, that leave the actionS[ϕ] invariant. That is,

ϕ→ ϕ′ = ϕ+ δϕ; while δS = 0. (1.2)

We all know that δS = 0 holds for any small field transformations when the fields satisfy theEuler-Lagrangian equation. But in the case of a symmetry transformation, we demand that δS =

0 holds for all the field configurations, not only when the Euler-Lagrange equation is satisfied.

At present, we are interested in the continuous symmetry. In this case, the symmetry trans-formation can be parameterized by a set of continuous variables ϵa. A continuous symmetry isusually described by a Lie group

G = g ; g = exp(iϵaT a).

1

2 Chapter 1. An Introduction to Chiral Anomaly by Zhong-Zhi Xianyu

In which T a are generators of the group G. They form a basis of the corresponding Lie algebrag. We assume the group G are connected without loss of generality, since we are only concernedwith the continuous symmetry.

In particular, if under the symmetry transformation, the field ϕi(x) falls into some repre-sentation Ri of the group G, then the symmetry transformation on the field ϕi can be writtenas

gϕig−1 = Ri(g)ϕi. (1.3)

When the transformation parameters ϵa are small, we have

Ri(g) = Ri(eiϵaTa

) = 1 + iϵaRi(Ta) +O(ϵ2). (1.4)

Compared with (1.2), the symmetry transformation on field ϕi can be written as

δϕ = iϵaRi(Ta)ϕi ≡ iϵaδaϕi. (1.5)

In the following we will simply write T a instead of Ri(T a).There is a fundamental theorem on symmetries of an action, namely the Nother theorem. It

claims that, for each global continuous symmetry of the action, there exists a conserved currentjµ. Let us quickly review the proof of the theorem to make this statement clear.

We assume that, under a transformation parameterized by ϵa, the action transforms as

δS =

∫d4x∆a(x)ϵa. (1.6)

The transformation becomes a symmetry when ∆a(x) = 0. The purpose of introducing ∆a willbe clear in the future.

Now, if the parameter ϵa depend on the space-time coordinates, then the change of the actionwill be generally not in the form of (1.6), but has an addtional term. This term should vanishwhen ϵa do not vary with space-time variables, thus it must be proportional to ∂µϵa(x):

δS =

∫d4x∆a(x)ϵa(x) +

∫d4x jaµ(x)∂µϵ

a(x). (1.7)

Well, if we demand that the fields satisfy the Euler-Lagrangian equation, then δS = 0 will betrue for any variations δϕ on field configurations. In this case,

δS =

∫d4x[∆a(x)ϵa(x) + jaµ(x)∂µϵ

a(x)]= 0.

Perform integration by parts, we get

δS = −∫

d4x[∂µj

aµ −∆a(x)]ϵa(x) = 0.

1.1. Symmetries: Classical vs. Quantum 3

Don’t worry about the surface term. It can be made vanish by letting ϵa(x) vanish at space-timeinfinity. Now, since ϵa(x) are arbitrary, we conclude that

∂µjaµ(x) = ∆a(x). (1.8)

We see that ∆a(x) acts as a source of the current jaµ. Once it vanishes, or equivalently, once thetransformation becomes a symmetry, the current will be conserved:

∂µjaµ = 0. (1.9)

The current jaµ looks like a vector field. But this is not necessarily the case. Because we have saidnothing about the Latin indices. Just think about the energy-momentum tensor as the conservedcurrent associated with the space-time translations. So don’t judge a book by its cover.

Now we have finished the discussion about Nother’s theorem. We emphasize again, that infield configuration space, a symmetry can exist anywhere, but the corresponding Nother currentis only conserved on-shell. Thus we see that Nother theorem is purely a classical result.

1.1.2 The Quantum Case: Ward Identities

We said that Nother theorem is purely classical, because its statement only holds on-shell.When quantum fluctuations come in, we have to take these fluctuations into account, by perform-ing path integral over all physical field configurations, besides the on-shell configuration. Thuswe need to develop a quantum counterpart of Nother’s theorem. The result is a series of Wardidentities.

In a classical theory, we talk about the general coordinates, configuration space, and theaction, while in a quantum theory, we talk about the states, the Hilbert space, the operators andthe path integral. Thus, we’d better begin with a path integral, rather than an action:

Z[J ] =

∫Dϕ eiS[ϕ]+i

∫d4x Jiϕi . (1.10)

This path integral is a functional of various external source Ji, and is called the generating func-tional, in the sense that its nth functional derivative with Ji will gives out n-point Green func-tions:

⟨Tϕi1(x1) · · ·ϕin(xn)⟩ ≡1

Z[0]

∫Dϕ eiS[ϕ]ϕi1(x1) · · ·ϕin(xn)

=(−i)nδn

δJi1(x1) · · · δJin(xn)logZ[J ]

∣∣∣J=0

(1.11)

In which T represents the time ordered products. For instance,

Tϕ1(x)ϕ2(y) = ϕ1(x)ϕ2(y)θ(x0 − y0) + ϕ2(y)ϕ1(x)θ(y

0 − x0).


The ward identities are relations among various kinds of Green functions. To derive them,we begin with the following path integral:∫

DϕO(y1) · · ·O(yn)eiS[ϕ], (1.12)

where Oi(yi) (i = 1, · · · , n) are local operators, which are functionals of the field operators ϕj .We will then assume the transformation of the fields (1.5) be a symmetry of the quantum

theory. Equivalently, we will demand the integral measureDϕ to be invariant under the symmetrytransformation. In this case, we have:

0 =

∫Dϕ δ

[O1(y1) · · ·On(yn)eiS[ϕ]

]=

∫Dϕ[δO1(y1)O2(y2) · · ·On(yn) + · · ·

+O1(y1) · · ·On−1(yn−1)δOn(yn)]eiS[ϕ]

+ i

∫DϕO1(y1) · · ·On(yn)eiS[ϕ]δS[ϕ]. (1.13)

Note that

δOi(yi) = ϵa(yi)δaOi(yi) = δaOi(yi)

∫ddx ϵa(x)δ(x− yi), (1.14)

and

δS = −∫

ddx[∂µj

aµ(x)−∆a(x)]ϵa(x), (1.15)

we get

0 =

∫Dϕ∫

ddx ϵa(x)[δaO1(y1)O2(y2) · · ·On(yn) + · · ·

+O1(y1) · · ·On−1(yn−1)δaOn(yn)

]eiS[ϕ]

− i∫Dϕ∫

ddxO1(y1) · · ·On(yn)[∂µj

aµ(x)−∆a(x)]eiS[ϕ]. (1.16)

Now, since ϵa(x) is arbitrary, we conclude that

∂µ⟨jaµ(x)O1(y1) · · ·On(yn)

⟩=⟨∆a(x)O1 · · ·On

⟩− i⟨δaO1(y1)O2 · · ·On

⟩δ(x− y1)

− · · · − i⟨O1 · · ·On−1δ

aOn(yn)⟩δ(x− yn). (1.17)

This is the so called Ward identities.

1.2 Anomalous Integral Measure

1.2. Anomalous Integral Measure 5

After a rather long discussion on symmetries, let us begin the investigation on anomalies.At the very beginning, we may ask, what is an anomaly, and why there exist anomalies. In thissection, we hope to give a formal answer to these questions.

Loosely speaking, the term “anomaly” refers to the break down of a symmetry in a classicaltheory when quantum fluctuations enter. In light of discussions in the last section, we know thata classical theory respect a symmetry if the action functional keeps invariant under the symmetrytransformation, while a quantum theory respect a symmetry if the generating functional Z keepsinvariant under the transformation. Now, staring at the expression

Z =

∫Dϕ eiS[ϕ],

you may quickly realize that, if the action S[ϕ] respects a symmetry while the integral measureDϕ does not, the phenomenon of anomaly occurs. In fact, This intuitive observation can be madedefinite by an explicit calculation. This is due to Fujikawa’s elegant work. In the following, let’sreview Fujikawa’s argument in detail.

1.2.1 Chiral SU(N) theory

Fujikawa’s calculation was performed in the chiral SU(N) theory, which is given by theLagrangian

L = ψ(i/∂ −m)ψ, (1.18)

where ψ = (ψ1 · · ·ψN )T is column of N Dirac spinors.It’s not difficult to see that the Lagrangian respects the following global SU(N)V ×SU(N)A

symmetry, if m = 0: ψ → exp

[iαaT aV + iβaT aAγ5

]ψ;

ψ → ψ exp[− iαaT aV + iβaT aAγ5

].

(1.19)

Where TV and TA are two copies of matrices for generators of SU(N) group in their fundamentalrepresentation, indices a, b = 1, · · ·N2 − 1 count the generators. αa and βa are two sets ofarbitrary parameter used to parameterize the group elements. γ5 = iγ0γ1γ2γ3. Note that wedefine all T aV to commute with all T aA.

The Nother currents associated with this global symmetry arejaµV = ψγµT aV ψ;

jaµA = ψγµγ5TaAψ.

(1.20)

Till now we are talking about free spinors. However, it is often the case that spinors will becoupled to some gauge fields. For instance, electrons and quarks interact with photons, quarks in-teract with gluons, etc. It is well known that, in this case, we should replace the partial derivative


on spinors by covariant derivatives, as

∂µ → ∂µ − igAaµτa = ∂µ − iAµ, (1.21)

where Aaµ is gauge field. We have also introduced the compact notation:

Aµ ≡ gAaµτa. (1.22)

For convenience, we will treat them as external fields at the moment. It is very important to notethat, the gauge symmetry associated with Aaµ is NOT the global symmetry we have considered.They are different things. We have clarified this by introducing different notations for generatorsof these two different groups. For global symmetry we use T aV and T aA and for gauge symmetrywe use τa.

Of course the covariant derivative introduced above is only one way to couple the spinors togauge fields. There also exists another way to acquire such a coupling. That is:

∂µ → ∂µ − iAµγ5.

This coupling also respects all continuous symmetries, but it violates the discrete parity symme-try. Therefor it is very useful in standard model. Since the violation of parity symmetry in weakinteractions is a well known experimental fact. But we won’t consider this term further in thefollowing, only because the anomalies generated by such a coupling is complicated to study inmathematics.

Now the modified Lagrangian

L = ψ[i/∂ + /A−m

]ψ (1.23)

is invariant under the following local transformation, when m = 0:ψ → UV (x)UA(x)ψ, ψ → ψUA(x)U

†V (x);

Vµ → UV VµU−1V − i

g (∂µUV )U−1V ;

Aµ → UAAµU−1A − i

g (∂µUA)U−1A .

(1.24)

1.2.2 Transformation of the Integral Measure

Our task is quite clear now. Since we have guessed that the anomalies come from the non-trivial behavior of the path integral measure under the symmetry transformation, thus we’d bettercalculate the transformations of the integral measure explicitly.

Now let’s enter the details. In our case, the path integral can be written as

Z[I, I] =

∫DψDψ eiS[ψ,ψ]+i

∫d4x (ψI+Iψ), (1.25)


where S[ψ, ψ] is the action given by the integration of the Lagrangian (1.23), I and I are externalsources.

We will see in the following that it is convenient to Wick-rotate the theory into a Euclideanspace. In this case, we have:

t→ −ix4, γ0 → −iγ4. (1.26)

We still distinguish the upper index and the lower index, since the Euclidean metric in our con-vention is (−,−,−,−). Thus all quantities will change sign once one of their indices is raisedor lowered.

Let’s focus on the integral measure DψDψ now. From (1.24), Under the transformationSU(N)V × SU(N)A, we can find,

Dψ → Det (eiαT1+iβγ5T2)−1Dψ,

Dψ → Det (e−iαT1+iβγ5T2)−1Dψ.(1.27)

Where T1 and T2 are arbitrary generators of SU(N). The integral measure transforms with aninverse determinant because we are dealing with Grassmann numbers.

Then we conclude that

DψDψ → Det e−i2βγ5T2DψDψ. (1.28)

But we can not be satisfied unless the anomalous determinant det e−i2βγ5T2 is explicitly evalu-ated. Now let us calculate it. The calculation can be made more transparent if we expand thespinors ψ as

ψ(x) =∑n

anφn(x),

ψ(x) =∑m

φ†m(x)bm.

(1.29)

where an and bm are Grassmann coefficients, and φn(x) are eigenfunctions of the Dirac operator/D ≡ γµ(∂µ − igVµ),

/Dφn(x) = λnφn(x), (1.30)

with eigenvalues λn. Now the Wick rotation works. Since in our conventions, all γµ are anti-Hermitian, thus the Dirac operator /D is Hermitian. Then we can properly choose the eigenfunc-tions φn(x) to make them form a orthonormal basis for the space of functions,∫

d4xφ†m(x)φn(x) = δnm. (1.31)

It should be noted that the spectrum of /D is not needed to be discrete, as our notations here, whichis chosen purely for convenience.

Then, the integral measure can be properly defined as

DψDψ =∏m

bm∏n

an. (1.32)


Repeat the steps leading to (1.28), we get a better expression:∏m

bm∏n

an →[Det

∫d4xφ†

meiβγ5Tφn

]−2∏m

bm∏n

an (1.33)

Here and follows, we omit the subscript “2” of the generator T , which appears in (1.28), since itis of no use any more.

Then, for infinitesimal β, the determinant in (1.28) can be written as[Det

∫d4xφ†

meiβγ5Tφn

]−2

≃[Det

∫d4xφ†

m(1 + iβγ5T )φn

]−2

≃[Det exp

∫d4x iφ†

mβγ5Tφn

]−2

=exp[− i2

∫d4xβ Tr (φ†

mγ5Tφn)], (1.34)

where we have used log detA = tr logA, and log(1 + x) ≃ x for small x.It remains to evaluate the trace in the exponent:

Tr (φ†mγ5Tφn) = Tr

∑n

φ†nγ5Tφn. (1.35)

Since we know tr γ5 = 0, and ∑n

φ†n(x)φn(x) = δ(4)(0),

thus the trace (1.35) becomes an undefined quantity like 0 · ∞. I wish that you do not feelconfused by the equality like tr (γ5T ) = tr γ5 trT , since the matrix γ5 and the matrix T belongto different spaces.

To carry on the calculating furthermore, we need a proper regularization scheme. Here it is:

Tr (φ†mγ5Tφn) = lim

M→∞Tr∑n

φ†nγ5Te

−(λn/M)2φn

= limM→∞

Tr∑n

φ†nγ5Te

−(/D/M)2φn (1.36)

We have introduce a Gaussian cut-off as the regulator. Now we go into the momentum basis tocarry on the calculation. That is, we use plane waves instead of φn to expand quantities.


M→∞Tr

∫d4k

(2π)4e−ik·xγ5Te−(/D/M)2eik·x. (1.37)

Note that [Dµ,Dν ] = Fµν , where

Fµν = ∂µVν − ∂νVµ + [Vµ, Vν ],


we get/D2=(

12 γ

µ, γν+ 12 [γ

µ, γν ])DµDν

=DµDµ + 1

2 γµγνFµν .

Also using the follow relation that can be directly checked,

e−ik·xD2µeik·x = (ikµ +Dµ)

2,

we find (1.37) becomes

Tr (φ†mγ5Tφn)

= limM→∞

Tr

∫d4k

(2π)4γ5T exp

[− 1

M2

((ikµ +Dµ)

2 + 12 γ

µγνFµν)].

Rescale the momentum as kµ → kµ/M , then

Tr (φ†mγ5Tφn)

= limM→∞

M4 Tr

∫d4k

(2π)4γ5T exp

[− (ikµ +Dµ/M)2 − 1

2 γµγνFµν/M

2].

Now it’s clear that we should expand the exponent, up to O(M4). We also note that, in theseexpanded terms, only those who contribute four gamma matrices do not vanish when we taketrace of them together with γ5. Thus the non-vanishing terms are

Tr (φ†mγ5Tφn) =

12!

122 tr

(γ5Tγ

µγνγργσFµνFρσ) ∫ d4k

(2π)4e−k

µkµ . (1.38)

Noticing thattr(γ5Tγ

µγνγργσ)= 4iϵµνρσ,

where ϵµνρσ are total-antisymmetric with its indices, and ϵ1234 = 1, we get


1

32π2ϵµνρσ tr (FµνFρσT ). (1.39)

This is our final result. We see that under an infinitesimal SU(N)A transformation, the pathintegral measure transforms as

DψDψ → exp[− i

16π2

∫d4xβ(x)ϵµνρσ tr (FµνFρσT )

]DψDψ. (1.40)

Up to now, we have evaluated the infinitesimal SU(N)A transformation of the integral mea-sure explicitly, shown in (1.39). Now let us briefly summarize how this result is reached.

Our calculations begin with the transformation property of the integral measure (1.28). Whilewe are trying to make this expression more explicit, we find that it is in fact an indefinite one like0 · ∞. Thus, to evaluate this factor properly, we use a regulator e−(/D/M)2 , and express the trace


in the momentum basis, as showed in (1.37). The trace over plane wave modes are taken beforereaching the limit M →∞. The rest of the calculations are trivial then.

It seems that, the basis of eigenfunctions φn of /D is of no use, since all the calculations areperformed in the plane wave basis. In other word, we can begin directly from (1.37), and omitall the steps between (1.28) and (1.37). In fact, though φn seem to be useless now, they areimportant in following discussions.

1.2.3 *Regularization Independence

At this stage, you may wonder, whether our result (1.39) is unique. Since our start point issomething like 0 ·∞, then, to what extent is our result independent of the regularization process?

This problem can be answered in different ways. Now we will give a weak answer, and leavea more general discussion to following sections.

Our weak answer is that, the result (1.39) is independent of the choice of regulator, as afunction of /D/M . More accurately, the calculations above can be performed by using a generalfunction f(/D/M) instead of the Gaussian regulator. The only properties we need f(x) to fulfillis that it should be a smooth function, with

f(0) = 1, f(∞) = 0, and xf ′(x) = 0 for x = 0,∞.

That is, f should decrease rapidly enough as x→∞.With this assumptions on f(x), we can repeat the calculations in the last subsection by re-

placing e−(/D/M)2 with f(/D/M). Then (1.37) becomes


M→∞Tr

∫d4k

(2π)4e−ik·xγ5Tf(/D/M)eik·x, (1.41)

and (1.38) becomes


12!

122 tr

(γ5γ

µγνγργσTFµνFρσ) ∫ d4k

(2π)4f ′′(kµkµ). (1.42)

Where ∫d4kf ′′(k2) = 2π2

∫dk2f ′′(k2)k2 = 2π2

∫dtf ′′(t)t.

Integrating by parts, ∫d4kf ′′(k2) = −2π2

∫ ∞

0

f ′(t) = 2π2.

Thus we have ∫d4k

(2π)4f ′′(kµkµ) =

1

16π2.

With this result, we reach the same answer (1.39) again.

1.3. Some Related Issues 11

1.2.4 Fujikawa’s Uncertainty Principle

In last subsection, we find that, the anomalous result we have got is independent of the shapeof the regulator function. However, as we have put it, this is only a weak independence. Infact, we can modified the regulator in a stronger way. For instance, we may use f(∂2/M2)

instead of f(/D2/M2) as the regulator. And this time, we find that the chirally-anomalous result

disappears! Instead, the gauge symmetry breaks down, since it’s /D rather than /∂ that can keepgauge invariance.

Thus there naturally arises a question: Can we find a regulator which can keep both gaugesymmetry and chiral symmetry simultaneously? The answer is a definite no, provided by Fu-jikawa’s uncertainty principle.

The so called “uncertainty principle” generally refers to the Heisenberg’s one, which statesthat, it is impossible to measure the spatial coordinate and the momentum of an object simultane-ously. In quantum mechanics, this fact can be explained as a consequence of the non-commutablenature of the coordinate operator and the momentum operator.

Very similarly, Fujikawa indicates that, the path integral measure of the chiral SU(N) theorycan not be defined such that the gauge symmetry and the chiral symmetry are maintained simul-taneously. This can also be explained as a result of the non-commutable nature of two operators,namely, γ5 and the Dirac operator /D.

1.3 Some Related Issues

1.3.1 Anomalous Ward Identities

Recall that when deriving Ward identities, we assume that the path integral measure is in-variant under the symmetry transformations. This assumption simply fails when anomalies enter.It’s quite obvious that the anomaly contributes to the original Ward identities with a new term,arising from the nontrivial Jacobian when we transform the integral measure.

Assume the integral measure transforms as

Dϕ → exp[ ∫

d4x ϵa(x)Aa(x)]Dϕ, (1.43)


under a symmetry transformation parameterized by ϵa(x), then we can simply rewrite (1.13) as

0 =

∫Dϕ[ ∫

d4xϵa(x)Aa(x)]O1(y1) · · ·On(yn)eiS[ϕ]

=

∫Dϕ δ

[O1(y1) · · ·On(yn)eiS[ϕ]

]=

∫Dϕ[δO1(y1)O2(y2) · · ·On(yn) + · · ·

+O1(y1) · · ·On−1(yn−1)δOn(yn)]eiS[ϕ]

+ i

∫DϕO1(y1) · · ·On(yn)eiS[ϕ]δS[ϕ]. (1.44)

Then repeating the derivation, we can easily find the so called “anomalous” Ward identity:

∂xµ⟨jaµ(x)O1(y1) · · ·On(yn)

⟩=⟨∆a(x)O1 · · ·On

⟩− i⟨δaO1(y1)O2 · · ·On

⟩δ(x− y1)

− · · · − i⟨O1 · · ·On−1δ

aOn(yn)⟩δ(x− yn)− i⟨Aa(x)O1(y1) · · ·On(yn)⟩. (1.45)

The simplest example of anomalous Ward identity we could find is

∂λ⟨j5λa⟩ = −1

16π2ϵµνρσFαµνF

βρσ trT

a tr (τατβ). (1.46)

1.3.2 Triangle Diagrams (Outline)

• Calculating ⟨j5⟩ in momentum space;

• Cancelation of linear divergent diagrams gives out a finite anomalous result. This is calledABJ anomaly (Adler, Bell, and Jackiw).

• Adler-Bardeen theorem: ABJ result is exact up to all orders in perturbation theory.

1.3.3 A Phenomenological Application

The theory of chiral anomalies finds its great application in the theory of strong interactions.Today we are all know that the correct way to describe strong interaction is QCD. The kernelstructure of QCD is a Yang-Mills theory with SU(3) as its gauge group, the corresponding gaugefields are called gluons. In addition, there are 6 flavors of quarks, each of these quarks lies in thefundamental representation of the SU(3) group.

Among the six quards u, d, s, c, b and t, u quark and d quark are much lighter than the others.Thus in low energy theory, we may only consider these two quarks, then the correspondingLagrangian can be written as

Lquarks = ui /Du+ di /Du−muuu−mddd. (1.47)

1.3. Some Related Issues 13

Where we have suppressed the color indices of quarks. The covariant derivatives are:

Dµu =(∂µ − igGaµta − i 2

3 eAµ)u;

Dµd =(∂µ − igGaµta + i 1

3 eAµ)d.

In which g and e are gauge couplings of color and electric type respectively. In this section, wewill recover the conventional normalization of the gauge fields.

Since mu and md are small, we may simply drop them off, in this case the theory has theglobal SU(2)V ×SU(2)A ×U(1)V ×U(1)A symmetry, in the absence of electromagnetic fieldAµ. We denote the quark doublet (u, d) by q, then the four kinds of conserved current can bewritten as:

jµ = qγµq; j5µ = qγµγ5q; jµa = qγµτaq; j5µa = qγµγ5τaq. (1.48)

In which, jµ is the baryon number current, jµa is isospin current. These are well known sym-metry. The other two currents, j5µ and j5µa, have no significant physical meanings. In fact, wehave never find symmetry SU(2)A × U(1)A in experiments. Thus they must get lost for somereason.

There are two possibilities: spontaneous broken, and anomalies. In the following, we willshow that, SU(2)A symmetry is spontaneously broken, while U(1)A is broke by an anomaly.Or, more definitely, SU(2)A symmetry does not affected by anomalies, while U(1)A does.

From the discussion of the last section, we know that nonzero anomalous results arise fromnonvanishing gauge fields to which the quarks couple. There are two kinds of gauge fields,namely gluon fields and photon fields. We study these two cases separately.

First let us consider the effect of nonvanishing gluon fields. The anomalous Ward identityassociated with the SU(2)A and U(1)A currents are

∂λ⟨j5λa⟩ = −g2


βρσ tr (T

a) tr (τατβ), (1.49)

and

∂λ⟨j5λ⟩ = −g2


βρσ tr (1) tr (τ

ατβ), (1.50)

respectively.We see that, since tr (T a) = 0 and tr (1) = 2, thus j5µa is not affected by anomalies due to

gluon fields, but j5µ does receive anomaly.An explanation of appearance of 3 pions, as well as U(1) problem.Next let us focus on the potential anomalies contributed by photon field Aµ. For j5µa, note

that u quark has electric charge + 23e while d has − 1

3e. Thus, we can define the charge matrix Qas

Q =

(+2/3 0

0 −1/3

). (1.51)

14 BIBLIOGRAPHY

Then the anomalous Ward identity can be written as

∂λ⟨j5λa⟩ = −Nce

2

16π2ϵµνρσFµνFρσ tr (T

aQ2), (1.52)

Where the coefficient Nc is the number of color species. Remember that T a is just the familiarPauli matrices, thus only tr (T 3Q2) = 1

6 does not vanish. The corresponding Ward identity is

∂λ⟨j5λ3⟩ = −Nc3

e2

32π2ϵµνρσFµνFρσ. (1.53)

Since j5λ3 current annihilates π0 meson, thus this identity indicates that π0 can decay into 2photon, while π± can not, if there are no other contributions.

It is worth mentioning that, this process is one of the very few examples which show thereare 3 colors, by comparison between theoretical calculations and experimental measurements.

Bibliography

[1] K. Fujikawa, Phys. Rev. D 21, 2848 (1980);

[2] M. E. Peskin & D. V. Schroeder, An Introduction to Quantum Field Theory, Westview,1995;

[3] S. Weinberg, The Quantum Theory of Fields, Vol.II, Modern Applications, Cambridge,1998;

[4] R. A. Bertlmann, Anomalies in Quantum Field Theory, Oxford, 2000.

CHAPTER 2A Brief Introduction to Loop Quantum Gravity

and Its Applications

Biao Huang

Department of Physics, Beijing Normal University, Beijing 100875

(May 2, 2010)

2.1 Canonical Formulation of Classical General Relativity

2.1.1 Traditional Formalism

The Hilbert-Einstein action for general relativity is

S =1

16πG

∫d4xL =

1

16πG

∫d4x√|det g|R. (2.1)

In order to perform the canonical quantization procedure, we need to cast it into a canonical form.Recall that the Hamiltonian mechanics is ”born with the definition of time”, we need to performa 3+1 decomposition of the spacetime manifold as illustrated. The time parameter of a class ofhypersurface Σt satisfies ta∇at = 1 and ta is decomposed as

ta = Nna +Na (2.2)

where the vector na is orthogonal to Σt and Na lies in it.Next we can carry out the Legendre transformation and express the 4-D metric gab and cur-

vature scalar R with 3-D quantities, namely the spacial curvature 3R, spacial metric qab and itscanonically conjugate momentum

pab ≡ ∂L

∂qab=√q(Kab − tr(K)qab). (2.3)

15

16 Chapter 2. A Brief Introduction to Loop Quantum Gravity and Its Applications by Biao Huang

. ......................................................................... ...................................... ........................................ ...........................................

................................................................................

.............

..................................................

Σt1

. ..........................................................................

....................................... ........................................ .......................................... ............................................ .............................................. ................................................. Σt2

. ............................................... .................................................. ..................................................... ........................................................ ............................................................................................................................

M

6

-

Nna

Na

ta

Figure 2.1: 3+1 decompositon of the spacetime manifold.

Here Kab = hca∇cnb is the extrinsic curvature tensor of the 3-D space. Then the Hamiltonian ofthe gravitation field is

H =

∫d3√q(NC +NaCa), (2.4)

where

C = −3R+ q−1(tr(p2)− 1

2tr(p)2) (2.5)

Ca = −23∇b(q−1/2pab) (2.6)

are called the Hamiltonian constraint and Diffeomorphism constraint respectively. Note thatdifferent from common dynamics, the Hamiltonian given in (2.4) is actually vanishing becausethe arbitrarily chosen N and Na serve as Lagrangian multipliers. Thus, the dynamics of generalrelativity is represented by constraints which impose the symmetry requirements of the theory.In fact, we have

C = −2Gabnanb, (2.7)

Ca = −2Gabnb, (2.8)

where Gab = Rab − 12gabR is the Einstein tensor. Thus, the vanishing of constraints (2.5) and

(2.6) indeed reproduce the vacuum Einstein field equations.

2.1.2 Ashtekar Variables

It seems that we are ready to quantize the general relativity with the Hamiltonian (2.4) andthe conjugate variables qab and pab. The Hilbert space of states |ψ > would be the solution to thefamous Wheeler-Dewitt equation

H|ψ >= 0. (2.9)

However, researches stumble at this point: the Hilbert space for the metric operators is rarelyknown, and the expressions of constraints are too complex to analyze. Change came at the

2.1. Canonical Formulation of Classical General Relativity 17

1980s and 1990s when gauge field theories (GFT) were highly mature and knot theories wereproposed. These two factors gave motivations to the development of connection dynamics ofgeneral relativity which facilitated the use of tools developed in GFT.

In the connection dynamics, one abandons the idea of analyzing geometry with componentsof metrics. Instead, the triad formulation, which has long been established, is adopted.

To adopt tools in GFT, we firstly need to recover the Poincare invariance which is obvi-ously violated in traditional analysis. Thus, we adopt the tetrad eI = eIµdx

µ and cotetradeI = eµI

(∂∂xµ

)satisfying

gµν = ηIJeIµeJν . (2.10)

where ηIJ is the Minkowski metric with signature (−1,+1,+1,+1). This means

gµνdxµdxν = ηIJe

IeJ (2.11)

and any vector v = vµ(∂∂xµ

)=(vµeIµ

)eI = vIeI can be evaluated by the Minkowski rather

than the gµν metric. (Note that ηIJ does not necessarily imply a flat spacetime, because generallyeI fails to serve as coordinate basis vectors.) In this new frame, the Poincare invariance is thenpreserved by treating the indices I, J, ... = 0, 1, 2, 3 as the ”internal indices” like indices ofisospins. Correspondingly, µ, ν, ... = 0, 1, 2, 3 are regarded as spacial indices. Note that differentfrom GFT, the ”internal indices” would still be related to ”external indices” through matrices eIµby eI = eIµdx

µ, as illustrated in Fig. 2.2.

.

.............................

...............................

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..................................

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.........

................

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.........

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.......

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.

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.

..................................................

...................................................

....................................................

.

..................................................

...................................................

....................................................

.

..................................................

...................................................

....................................................

eI

eJ

dxµ

dxν

M

Figure 2.2: Treating triad indices I, J.... as internal indices (like isospins), and µ, ν..... as external(spatial) ones. Unlike that in GFT, these two indices are still related by eI = eIµdx

µ.

In this frame, I give some standard results of the connection dynamics of general relativity inthe following. The action (2.1) is rewritten as

S =1

32πG

∫MϵIJKLe

I ∧ eJ ∧ ΩKL − 1

16πGγ

∫MeI ∧ eJ ∧ ΩIJ , (2.12)

where

Ω = dω + ω ∧ ω (2.13)


is the symplectic 2-form for the Hamiltonian system, and γ is a free Barbero-Immirzi parameterwhich can be fixed by the calculation of the black hole entropy. The first term is the usual Hilbert-Einstein action, while the second term correspond to a canonical transformation leading to theAshtekar connection introduced below. In the connection dynamics of general relativity, ω is theconnection treated independently but can be totally determined by e through

de+ ω ∧ e = 0 (2.14)

This is the Cartan structure equation. Then the connection ωIJ = ωIJµ dxµ could give birth totwo so(3)-valued 1-form (representing the spin connection and the external curvature):

Γia =1

2qαa q

iIϵIJKLnJω

KLα (2.15)

Kia = qiIq

αaω

IJα nJ (2.16)

which together give the Ashtekar connection variable

Aia = Γia + γKia. (2.17)

Its conjugate momentum

Eai =1

2ϵijkϵ

abcejbekc = |det(e)|eai (2.18)

is exactly the densitiezed triad. The connection defines a covariant derivative Da and a curvatureas

Davi = ∂avi − ϵijkAjavk, (2.19)

F iab = ∂[aAi[b + ϵijkA

jaA

kb (2.20)

respectively. Their Poisson bracket isAia(x), E

bj (y)

= 8πγGδbaδ

ijδ(x− y) (2.21)

Note that a, b... are abstract indices, i, j... = 1, 2, 3 stand for internal ”spacial” indices. By now,we have obtained the Ashtekar variables A and E for further quantizations. In this new frame,the action

S =1

8πG

∫dt

∫Md3x

(1

8πGγEai LtA

ia −H

)=

1

8πG

∫dt

∫d3x

(1

8πGγ− [λiGi +NaCa +NC]

)(2.22)

gives three constraints, namely the Gauss constraint for SO(3) gauge invariance, the diffeomor-phism constraint for spacial properties, and the Hamiltonian constraint admitting the evolution

2.2. LQG Kinematical Hilbert Space 19

characters. Their explicit forms are

CG[λi] =

1

8πGγ

∫Md3xλiDaE

ai (2.23)

Cdiff[Na] =

∫Σ

d3xNa

(1

8πGγEbiF

iab −AiaGi

)(2.24)

H[N ] =1

16πG

∫Σ

d3xN

det(e)

(ϵijkF

iabE

ajEbk − 2(1 + γ2)Ki[aK

jb]E

ai E

bj

)= − 1

16πGγ2

∫Σ

d3xN

det(e)ϵijkF

iabE

ajEbk (2.25)

Analogy to GFT:Aia correspond to gauge potential whileEai serves as the field strength. Notealso that Aia, E

ai bear the ”internal indices” and therefore are Lie algebra of the SO(3) group.

Thus, due to the isomorphism between algebras su(2) and so(3), in Loop Quantum Gravitywe analyze properties of the compact SU(2) group and its algebra. This imply the use of spinnetwork developed by Roger Penrose as states in the Hilbert space. Also, it explains why strictLQG is a theory for 4-D spacetime, although toy models in E3 or E4 are widely explored inspinfoam models in an effort to provide inspiration for further researches.

2.2 LQG Kinematical Hilbert Space

One spirit of LQG is to adopt as few assumptions as possible, so there are only two basicprinciples for LQG, coming from general relativity and quantum mechanics respectively. One isthat the gravitation field and the spacetime are the same entity eI ; the other one is the probabilityexplanation of the microscopic physical system, involving states representing the system andoperators standing for the physical observables.

Common field theories assume a background (t, x1, x2, x3) where integrations are to per-formed, namely as

∫d4x. This is convenient, however, ultraviolet divergence also comes when

analyzing the gravitational field as a result of the infinite degrees of freedom. We will see belowthat when we analyze the field ”in itself” and impose the diffeomorphism requirement on statevectors, the spacetime is discrete in the Planckian scale and there is countable degrees of freedomin existence.

2.2.1 Cylindrical Functions and The Inner Product

Consider a manifold M. In general relativity, we are chiefly concerned with transport of avector space from one point A to that of another B, as illustrated in Fig 2.3. Note that when themanifold is not flat, vectors transported along different curves γ1, γ2 will change its directionwhen returning to its origin point. All such maps of vector transformations form a group, namedthe holonomy group, which is a subgroup of SO(n).


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A

B

XXy

PPi

6

HHY

@I

γ1

γ2

M

Figure 2.3: Holonomy, the parallel transportation of the vector space on the manifold

The connection A(τ) = Aia(τ)τidxa is the Lie algebra element of su(2) which can give the

element in the holonomy group as

U(A, γ) = P exp

∫γ

A. (2.26)

Here P is the path-ordering action which arrange the sequence of the integration along a path γ.

The philosophical foundation of general relativity is that spacetime exist as a relation amongone another, rather than as an absolute entity. Thus, it is natural to propose the Hilbert space ofLQG as functions of holonomies representing the relations of vector spaces of different points onthe manifold. Explicitly, we consider the so-called cylindrical functions

ΨΓ,f [A] = f (U(A, γ1), ..., U(A, γL)) , ∀Γ, f (2.27)

constituting the space S, where Γ = (γ1, ..., γL) is called a graph, and the orientation of eachpath in Γ is called a coloring.

Since state functions are defined on a compact Lie group rather than on R3, we can applythe Haar measure which demands the finiteness of integration throughout the group manifold,namely ∫

G

1da = 1. (2.28)

With this measure, the scalar product of the Hilbert space can be defined as

< ΨΓ,f |ΨΓ,g >=

∫Hol

dU1...dULf(U1...UL)g(U1...UL) (2.29)

The scalar product defined in this way is invariant under SU(2) and diffeomorphism transforma-tion. Then the completion (in Cauchy sequence) of S gives the kinematical Hilbert space K. Atthis point, the spacetime manifold is decomposed and replaced by all possible graphs Γ.


2.2.2 SU(2) Gauge Invariance: Spin Networks

A(τ) is an element in so(3) and therefore can be regarded as an element of su(2). Theholonomy group is therefore a subgroup of SU(2). Then Peter-Weyl theorem states that theirreducible representations of SU(2) consist the basis of these cylindrical functions in the Hilbertspace. Therefore, we can express any cylindrical functions in terms of a combination of j =

1/2, 1, 3/2, .... representations of the SU(2) group, with matrices denoted by

R(j)α

β(U) =< U |j, α, β > . (2.30)

Correspondingly, the graph Γ can be redrawn with non-intersecting paths γ denoted by halfintegers j, as painted in Fig. 2.4. These paths are called links of a spin network state.

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|S >

Figure 2.4: The intuitive picture for a spin network, constituting of links jl and nodes in

The connection A changes under a gauge transformation λ : Σ→ SU(2) as

A → Aλ = λAλ−1 + λdλ−1 (2.31)

Accordingly, the holonomy transforms in a simpler way

U [A, γ]→ U [Aλ, γ] = λfU [A, γ]λ−1i (2.32)

where λi, λf denote the SU(2) group element on the initial and final point of a link γl respec-tively. Then cylindrical functions transforms as

Uλ|Γ, jl, αl, βl > = R(j1)α1

α′1(λ−1f1

)R(j1)β1

β′1(λ−1i1

)...

R(jL)αL

α′L(λ−1fL

)R(j1)βL

β′L(λ−1iL

)|Γ, jl, α′l, β

′l > (2.33)

It is obvious that under gauge transformations, the connection among those links will changedue to the action of λi and λf . To obtain a gauge invariant state, we associate a tensor calledintertwiner vin to each node (connecting points) of the spin network, and vin satisfies

vβ1...βL

in α1...αL= R

(j)α′1

α1 R(j) β1

β′1...R(j)αL

αLR

(j) βL

β′L

vβ′1...β

′L

in α′1...α

′L. (2.34)


For the spin network illustrated in Fig 2.4, the formula is

|S >=∑

αl,βl,l=1,2,3

vβ1β2β3

i1 α1α2α3vβ1β2β3

i2 α1α2α3|Γ, jl, αl, βl > (2.35)

Generalization to more complicated situation consisting more nodes can be similarly defined.The state functional is

ΨS [A] =< A|S >=

(⊗l

R(jl)(H[A,Γ])

)·

(⊗n

in

). (2.36)

The indices of intertwiners contract with that of the cylindrical functions, rendering the wholeentity invariant under gauge transformations (2.33). ∗ Then |S > labeled by (Γ, jl, in) so ob-tained is the gauge invariant spin network states, which serve as the orthonormal basis of thegauge invariant kinematical Hilbert space K0.

Finite combinations of |S > give the space S0 which is subspace of S and is dense in K0. Itsdual space of S is denoted as S ′0, where the diffeomorphism invariant states will be defined in thenext subsection.

2.2.3 Diffeomorphism Invariance: s-Knot States

The most important lesson of general relativity is the requirement of diffeomorphism on phys-ical theories, which eliminates the redundancy of introducing both the notion of spacetime andthe gravitation field. Recall that we have introduced the spin network states to describe the grav-itation field; but different paths belonging to the same equivalent class under diffeomorphismtransformations will still label different spin network states. Thus, we are to erase such redun-dancy in this subsection.

The diffeomorphism map on the background manifold ϕ ∈ Diff∗ : Σ → Σ acts on theconnection through its pull back

A→ ϕ∗A. (2.37)

Then the holonomy transforms as

U [A, γ]→ U [ϕ∗A, γ] = U [A, ϕ−1γ] (2.38)

and the states in S0 transforms as

UϕΨ[A,Γ] = Ψ[(ϕ∗)−1A,Γ

]= Ψ

[A,ϕ−1Γ

](2.39)

where Uϕ is the representation of the diffeomorphism group. Here for clarity I write explicitlythe dependence of Ψ on the graph Γ. Clearly, states in S0 will change under the diffeomorphismtransformation.

∗More strictly speaking, it is the specific property of intertwiners that determines the gauge invariance of |S >. Butat present I am not clear about those mathematical details.


The action of the diffeomorphism transformation on S ′0 is defined as

(UϕΦ)(Ψ) = Φ(Uϕ−1Ψ) (2.40)

The diffeomorphism invariant kinematical Hilbert space Kdiff is therefore defined in S ′0 as

Kdiff = Φ ∈ S ′0|Φ(UϕΨ) = Φ(Ψ),∀Ψ ∈ S0 . (2.41)

These states can be found by the projection operator Pdiff : S0 → S ′0 defined by

(PdiffΨ)(Ψ′) =∑

Ψ′′=UϕΨ

< Ψ′′,Ψ′ >, ∀Ψ,Ψ′ ∈ S0. (2.42)

Note that the diffeomorphism transformation constitutes a group asUϕ1Uϕ2 = Uϕ, where ϕ1, ϕ2, ϕ ∈diff∗. Thus, < Ψ′′, UϕΨ

′ >=< Uϕ−1Ψ′′,Ψ′ > and the action Uϕ−1 is canceled by the summa-tion over representations of diff∗. The summation is always finite because on the one hand, Uϕchanges graphs or/and the colorings and such actions are discrete; on the other hand, S0 are finitecombinations of spin network states with finite graphs and ways of colorings.

Finally, with the definition of scalar product on S ′0 as

< PdiffΨS , PdiffΨS′ >Kdiff≡ (PdiffΨS)(ΨS′) =< Ψ|Pdiff|Ψ′ >, (2.43)

we arrive at the kinematical Hilbert space with both SU(2) and diffeomorphism invariance.Please note at this point what has changed in our physical picture. Originally we have a

manifold M and spin networks defined on it. Note that with graphs Γ,Γ′ not in an equivalentclass under diffeomorphism transformation, < Ψ|Pdiff|Ψ′ >= 0 according to (2.42) and (2.43).If Γ = ϕΓ′, ΦS and ΦS′ differ by colorings (the path-ordering P for the holonomy of eachlink) only. Therefore, ΦS ∈ S ′0 does not need to ”live on” a background manifold anymore.The difference among states with the same graphs and colorings but with graphs ”living” on

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s

s

s

Figure 2.5: The diffeomorphism transformation drags the spin network out of the backgroundmanifold M , rendering a spin knot state |s >= |K, c >

different points of the spacetime is identified. In mathematics, an equivalent class of graphs


without colorings is called a ”knot”. Thus, the kinematical Hilbert space for LQG is specified bya spin-knot state |s >= |K, c > where K labels knots and c, colorings.

Remark: Note that in a more mathematically rigorous way, the gauge and diffeomorphism invariance constraints are

imposed on K by solving the kernel of the corresponding constraint operators (2.23) and (2.24). That is, replacing (2.44)

and (2.47) into these two classical constraints, we find states satisfying CGΨΓ,f [A] = 0 and CdiffΨΓ,f [A] = 0 as

states satisfying these constraints. The process is purely mathematical, so in the last section I follow the book ”Quantum

Gravity” by Carlo Rovelli and just give an intuitive illustration. For more details see the book ”Modern Canonical General

Relativity” by Thomas Thiemann.

2.3 Operators

2.3.1 Area and Volume Operators On The Kinematic Space

The canonical variables are the configuration variable Aia and its momentum Eai , with whichall physical operators can be constructed. As in common quantum mechanics, the former one isconsidered as a multiplicative action and the later one is associated with a functional derivativeδ/δAia. Since both of these two variables involve the ”internal indices” i, these could only beoperators on K, variant under SU(2) transformations.

First note that cylindrical functions are combinations of holonomies, and therefore a multi-plication of Aia will generally send these functions out of K. Thus, we apply instead U [A, γ] asthe multiplicative operator with its action on K defined as

(U [A, γ]Ψ) [A] = U [A]Ψ[A] (2.44)

This is to ”smear” A along a path γ, just like that in ordinary QFT we smear operators among the3-D space.

Then we deal with Ei(S), which is closely associated with area and volume according to theclassical definition:

A =

∫S

√naEai nbE

bi d

2σi (2.45)

V =

∫Rd3x√|detE| (2.46)

The quantum operator is defined on K as:

Ei(S) = −i~∫Sdσ na(σ)

δ

δAia(σ). (2.47)

Due to its dimension, it is smeared among a 2-D surface S. Acting it on a holonomy

U [A, γ] = P exp

∫γ

A = P exp

∫γ

Aadγa = exp

∫γ

Aaγa(s)ds, (2.48)

2.3. Operators 25

and provided that γ intersects the surface for only once, we have

Ei(S)U [A, γ] = −i~∫Sdσna(σ)

∫γ

dsγa(s)δ3(γ(s), σ)× [U [A, γ1]τiU [A, γ2]]

= −i~[∫

S

∫γ

dsdσδ3(γn(s), σ)

]× [U [A, γ1]τiU [A, γ2]] (2.49)

The first rectangular bracket is the intersection number, as one intersection between γ and S willresult in 1 and no intersection, 0. Therefore,

Ei(S)U [A, γ] = ±i~U [A, γ1]τiU [A, γ2]. (2.50)

Next we consider operators on K0. A contraction of the internal indices in E2(S) = EiEi

will result in a gauge invariant operator. Considering also the classical definition (2.45), we definethe area operator A as

A(S) = limN→∞

∑N

√E2(S), (2.51)

where N denotes the discretization of the surface according to intersections, as illustrated in Fig.2.6. Note that τi = i

2σi where σi is the Pauli matrix, and that EiEi ∼ τiτi is the Casimir

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in

r rr

P1P2 P3

Figure 2.6: The decomposition of the surface S according to intersections

operator of the SU(2) group, we have the discrete spectrum of the area operator as

A|S >= ~∑P

√jP (jP + 1)|S > (2.52)

In natural units, the spectrum would be A = 8πG~c−3∑P

√jP (jP + 1). When j = 0, the link

is related to a trivial SU(2) representation, then the two nodes linked by it can be identified andthe link can be eliminated, resulting in no area at all. Then the smallest excitation of area isAj= 1

2∼ 10−66cm2. Therefore, LQG predict that in the Planck scale, there is a fundamental unit

of area and no area smaller than it can be observed.The volume operator involve taking the determinant of Eai , and therefore does not have the

simple spectrum as that ofAia. The determinant in the classical volume quantity can be expressedas

V (R) =∫Rd3x√|detE| =

∫Rd3x

√1

3!|ϵabcϵijkEaiEbjEcj | (2.53)


Then we need to regularize this operator. Note that the guage invariant ”(three-hand) loop oper-ator” will give

T abc(x, r, s, t) =1

3!ϵijkR

(1)ilγxr

Eal (r)×R(1)jmγxs

Ebm(s)×R(1)knγxt

Ecn(t)

−−−−−−→classically lim

r,s,t→x2ϵijkE

ai(x)EbjEck(x)

= 2ϵabc detE(x). (2.54)

Then we have, for a closed surface S surrounding an infinitesimal coordinate volume ϵ3 at xI ,

WI ≡ 1

16ϵ63!

∫Sd2σ

∫Sd2σ

∫Sd2σ′′|na(σ)nb(σ′)nc(σ

′′)T abc(x, σ, σ′, σ′′)|

−−−−−−→classically

1

8ϵ63!|det(E(xI)|

∫∂RI

d2σ

∫∂RI

d2τ

∫∂RI

d2ρ|na(σ)nb(τ)nc(ρ)ϵabc|

= |detE(xI)| (2.55)

This gives the volume of the whole bulkR classically

V(R) = limϵ→0

∑Iϵ

ϵ3√WIϵ (2.56)

Thus, we can obtain a regularized quantum volume operator from (2.56), using (2.54) and (2.55)without taking the classical continuous limit.

Its action on spin network states is a little complicated, and the spectrum is not as explicitas that of the area operator. But we can observe easily on which part of the spin network willthe volume operator act. Note from (2.56) that the action of V on spin network states must beinvariant when taking the coordinate volume ϵ3 → 0. Thus, we can infer that it will only act onnodes, namely, intertwiners of the spin network states. The triple integration involved in (2.55)also implies that only nodes with at least triple links will contribute to the integration. Detailedcalculation also give a discrete spectrum of the volume operator.

In conclusion, we can see how spin network, in accompany with area and volume operators,paint the physical picture in LQG. Nodes stand for volumes, and links represent the adjacent areaamong volumes, as illustrated in Fig. 2.7.

Remark: What is the difference between a quantum field theory (QFT) on a background spacetime and a background-

independent one? Here is a simple example. Consider the ground state from which the whole Hilbert space could be

constructed by ladder operators. In common QFT, the ground state is usually chosen as the Minkowski vacuum |0M >,

and the Fock space would be (a+)n|0M >. In LQG, however, one can easily find that the Minkowski vacuum

is a coherent state consisting considerable numbers of s-knot states, so that it will contain infinite spatial volume and

that it will make the connection Aia, and therefore the covariant derivative Da a trivial one. In fact, the state |0M >

is still not found at present. The ground state for LQG is actually the ”empty” or so-called ”covariant vacuum” state

|∅ >: Ψ[A] =< A|∅ >= 1. Therefore, AΨ[A] = 0, implying NO SPACE at all. Then the Hilbert space results from

the action of holonomy operators serving as ladder operators as |K, c >= U [A, γ]|∅ >. Thus, the quantum excitation is

the excitation of a grain of space (gravitation field) itself, rather than a quantum excitation on a background spacetime.

2.3. Operators 27

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Figure 2.7: The physical picture of LQG, where discrete volumes are represented by intertwinersat nodes, and their adjacent areas are represented by links connecting those nodes.

2.3.2 Dynamics: The Hamiltonian Constraint and Spinfoam Models

First we rewrite the expression (2.25) in the form most convenient to be quantized. Using thealgebra (2.21) and the expression (2.53), we can compute

V, Aia(x) = (8πGγ)3Ebj (x)E

ck(x)ϵabcϵ

ijk

2 det(e). (2.57)

Thus, the Hamiltonian constraint can be rewritten as

H[N ] = − 1

16πGγ2

∫Σ

d3xNϵijkF

iabE

ajEbk

det(e)∼∫N tr(F ∧ V, A) = 0, (2.58)

where in the last step coefficients are neglected. Now we need to regularize this operator beforequantizing it. Note that the holonomy (2.26) has the following expansion with respective todifferent types of paths. Fix a point x and two vectors from it, namely u and v as illustrated inFig. 2.8. Then for a path γx,u starting from x with length ϵ along u, we have

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*

QQQs

~u

~v

s

s

sγx,l ε

ε .

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..αx,uv

l

l′

l′′

H @= a

Figure 2.8: The action of a Hamiltonian operator on a spin network state.

U(A, γx,u) = 1 + ϵuaAa(x) +O(ϵ2). (2.59)


Similarly, for a triangle loop αx,uv with vertices as u and v, we have

U(A,αx,uv) = 1 +1

2ϵ2uavbFab(x) +O(ϵ3). (2.60)

Thus, denoting U(A, γ) ≡ hγ , we have

H = limϵ→0

∑m

ϵ3Nmϵijktr

(1

ϵ3hγ−1

xm,ukhαxm ,uiujV(Rm), hγxm ,uk

), (2.61)

where the integration over the 3d coordinate space is partitioned into regions Rm with volumesϵ3. Then the corresponding quantum operator is obtained through the following observations.

• Since holonomies are already well defined quantum operators, we can quantize the Hamil-tonian by just replacing the Poisson bracket with a quantum commutator.

• Notice also that the presence of the volume operator, the Hamiltonian only act on volumescontaining nodes.

• Finally, we can conveniently take vectors (u1, u2, u3), which initially denote arbitrary par-titions of the volume, to be these along links (l, l′, l′′) from nodes n, as illustrated in Fig.2.8.

Thus,

H|S⟩ ≡ limϵ→0

Hϵ|S⟩

= − i~limϵ→0

∑n∈S

Nn∑l,l′,l′′

ϵijktr(hγ−1

xn,ukhαxn ,uiuj [V(Rn), hγxn ,uk

])|S⟩, (2.62)

where the sum is taken over all nodes n. We know that when acting on spin networks, holonomyoperators serve as creation operators producing new links. Therefore, the action of the Hamilto-nian operator on a node will associate three additional links along (l, l′, l′′) from the node, andconnect two of them forming a loop through the action of hαxn ,uiuj . This process is illustratedin Fig. 2.8.

One delicate question is whether the limit in (2.62) exists. Actually, for general spin networkstates, this sum will diverge as what happened in common quantum field theory: a regularizedoperator can be defined but a divergence develops when removing the regulator ϵ. However,there does exist a limit for (2.62) when it is acting on the diffeomorphism invariant subclass ofstates. Intuitively speaking, this is because when shrinking ϵ→ 0, we contract the loop α in Fig.2.8. Then for diffeomorphism invariant states this action only brings about states in the sameequivalent class and therefore does not change them.

How about matter fields? In LQG, the distinction between the gravitation field and the matterfield is large conventional, which means similar steps can lead to the quantization of matter

2.3. Operators 29

fields as well. Generally, matter fields can be classified into Yang-Mills field, fermions, andscalar (Higgs) field (if any). Yang-Mills fields are represented by symmetry groups of GYM =

SU(3)× SU(2)× SU(1); therefore, we can use connection A of the new group G = GLQG ×GYM = SU(2)×GYM and perform the treatments discussed above to complete the extension.For fermions, we assume that they only sit on nodes of the new spin network.† Introduce thedensity of Grassman variable as ξ ≡ η

√detE where η is the Grassman variable. Then the new

cylindrical functions are defined as

ΨΓ,f [A,ψ] = f(U(A, γ1)...U(A, γL), ξ(x1), ...ξ(xN )) (2.63)

The situation of the scalar field is less natural than that of the Yang-Mills and fermion fields. It iscomplicated by the non-compactness of the Hilbert space of ϕ(x). One way out of this dilemmais to exponentiate it to be U(x) = exp(ϕ(x)), then U(x) takes values in GYM . Then the totalcylindrical function will also involve variables of eϕ(x1)...eϕ(xN ), and the inner product of thetotal cylindrical functions are defined as

(ΨΓ,f ,ΨΓ,g) ≡∫dULQG+YM

l

∫dξn

∫dU

′Higgsn

× f(U1, ..., UL; ξ1, ...ξN ;U ′1, ..., U

′N )

× g(U1, ..., UL; ξ1, ...ξN ;U ′1, ..., U

′N ) (2.64)

The Hamiltonian can also split into

H = HEinstein +HYM +HDirac +HHiggs (2.65)

where each terms can be found in Carlo Rovelli’s book ”Quantum Gravity” page 290.The general spectrum of the hamiltonian can hardly be obtained, as that in common quantum

mechanics. The transition amplitude W (s, s′) ≡ ⟨s|P |s′⟩ (P is the projection on the solutionspace ofHΨ = 0) is almost impossibly available in the canonical approach, and researchers haveturned to the path-integral approach for break through. In this formulation, transition amplitudeis calculated by inserting the complete basis 1 =

∑s|S⟩⟨S|:

W (S, S′) = limN→∞

⟨S|e−Ht|S′⟩K

= limN→∞

∑S1,...,SN

⟨S|e−dt∫d3xH(x)|SN ⟩K...⟨S1|e−dt

∫d3xH(x)|S′⟩K

=∑σ

A(σ) (2.66)

where e−dt∫d3xH(x) ∼ 1−dt

∫d3xH(x), andA(σ) = ΠνAν(σ),Aν(σ) = ⟨S|e−dt

∫d3xH(x)|SN ⟩K...

†This is the common point of view towards space in western society before Newton: the space is an extension of theexistence of matter.


⟨S1|e−dt∫d3xH(x)|S′⟩K. The index ν denotes the amplitude associated with each vertex. (The

vertex is the point where the Hamiltonian creates a new triangle.) The series σ = (S, SN , ..., S1, S′)

is called a spinfoam denoting the history of spin networks. Various spinfoam models, involvinganalysis in different dimensions and specific cases, is rapidly developing at present. Thus, I leavethis topic open and close this section here.

2.4 Applications

2.4.1 Black Hole Thermodynamics

Historically, black hole thermodynamics was proposed by Bekenstein in 1960s. By arguingthat an isolated system containing a black hole may decrease its entropy by losing its heat intothe black hole, Bekenstein supposed that the black hole itself should contain entropy. Noticingthat the area of a black hole cannot decrease in classical general relativity, resembling the non-decrease behavior of the entropy in an isolated system, Bekenstein associated the area of a blackhole to its entropy. Then by dimensional arguments, Bekenstein formulated

SBH = akB~G

A (2.67)

This formula received poor attention from the physics community in the beginning, due tothe following criticism. In general relativity, the global energy of a Schwarzschild black hole wasM =

√A/16πG2. Then from thermodynamics, given (2.67), the temperature of the black hole

would beT−1 =

dS

dM⇒ T =

~32akBπGM

.

Thus, the prediction of a black hole radiating like a blackbody with temperature T seriouslyviolated the concept of black hole at that moment. However, Hawking came up with similarresults from the strictly deduced QFT in curved spacetime:

T =~

8πkBGM(2.68)

SBH =kBA

4π~G(2.69)

This result initiated the research area of unifying gravitation physics and thermodynamics, andsince then the subscription BH stands for ”Bekenstein-Hawking”, instead of ”Black Hole”.

Now we deduce this result in the framework of LQG. Notice that when considering the statis-tics of a system, we need to distinguish the macroscopic statistical description and the micro-scopic dynamics of the same system. In the case of the Schwarzschild black hole, the macro-scopic behavior should be described by general relativity, which serve as the average over micro-scopic states in LQG describing microscopic dynamics of the discrete geometry.

2.4. Applications 31

Then we consider the statistical ensemble of the Schwarzschild black hole. First, we clarifythat the system under consideration constitute the black hole and its asymptotically flat externalspacetime serving as the heat source. The observer of this system locates statically with respectto the Schwarzschild coordinate outside the horizon. Second, we are considering the entropy ofthe black hole, which correlates to the heat exchange between the external spacetime and theblack hole. From M =

√A/16πG2, we know that the energy of a black hole can be entirely

determined by its coordinate area of its surface, namely, its horizon. Thus, we can count thegeometric microstates on the horizon so as to determine the macroscopic entropy of the blackhole. Therefore, according to Boltzmann’s theorem,

SBH = kB lnN(A) (2.70)

where N(A) denotes the number of possible microstates on the horizon.

Such a selection of statistical ensemble may receive two criticisms. One is that why do weconsider only the microscopic states on the horizon, rather than that within the whole blackhole. The answer is that different internal states of a black hole cannot be distinguished by anoutside observer. Consider the Kruskal extension of the Schwarzschild solution where there isanother parallel universe inside the black hole containing billions of galaxies like that of ourown. There is, however, no chance for the ”internal universe” to affect our one and the horizonobserved externally, because given the mass and the boundary condition (asymptotic flatness) ofthe spacetime, the Schwarzschild solution is uniquely determined. The second objection is that ifsuch consideration is true, then we can calculate the entropy of any system by counting only themicrostates on its surface, which is, obviously, wrong. The argument against this criticism is thatnot all surfaces would be related to the entropy of the system. Consider a system surrounded bya piece of paper, then it is clear that energy can both flow inside and outside through the paper.Thus, changing the energy distribution inside the system will surely affect the heat exchangebetween the system and the outside source. Thus, different internal states should be taken intoconsideration in this case. It is only when considering a black hole that we can treat differentinternal states as irrelevant to our observation.

Recall that LQG predicts a discrete spectrum of the area as A = 8πγ~G√j(j + 1). The

smallest element of the area is A0 = 4πγ~G√3 with j = 1/2. Thus, the largest number of

elements on the horizon is

n =A

A0=

A

4πγ~G√3. (2.71)

For each element, there are 2 states possible for the SU(2) representation of j = 1/2, namelym = ±1/2. Thus, the number of states on the horizon is N = 2n = 2A/4πγ~G

√3, and the

entropy is

SBH = kB lnN(A) =AkB ln 2

4πγ~G√3=

1

γ

ln 2

4π√3

kB~G

A. (2.72)


Comparing it with Hawking’s result (2.69), we can choose the Barbero-Immirzi parameter γ =ln 2π√3≈ 0.06 to make these two results coincide.

2.4.2 Loop Quantum Cosmology (LQC)

The most popular application of LQG is to the field of cosmology, where isotropy and thespatial homogeneity greatly reduce the system’s degrees of freedom, facilitating the analysisespecially in the dynamical level.

In usual quantum mechanics, symmetry reduction is done in the quantum level. For in-stance, when treating the hydrogen model, we first quantize the Newtonian dynamics, then weperform symmetry reduction and obtain the quantum dynamics of the system. In the case ofLQG, however, it is extremely difficult to perform symmetry reduction due to the complicatedform of constraints. Thus, around 2000 M. Bojowald firstly reduce the general relativity to theFriedmann-Robertson-Walker metric for cosmology, then perform the similar quantization pro-gram as that for full general relativity. This approach brought about significant success in resolv-ing the Big-Bang singularity and therefore have attracted intensive research on it. The full LQCrequires steps introduced in all above sections with respect to the FRW metric

ds2 = −dt2 + a2(t)

(dr2

1− kr2+ r2dθ2 + r2 sin2 θdϕ2

), (2.73)

and I do not intend to introduce it here. Instead, I illustrate its physical picture through theeffective equations initially introduced by M. Bojowald to analyze semiclassical problems ina more convenient way. This approach tackles problems in the classical formulation with theHamiltonian modified by results obtained in the quantum theory. Surprisingly, it turned out thatmajor characters obtained in full LQC are all preserved in the effective analysis, including theresolution of Big-Bang singularity.

With the FRW metric (2.73), the canonical variables are reduced to

c = γa/2, |p| = a2 (2.74)

through

Aia = cV−1/30

0ωia, Eai = pV−2/30

√0q 0eai , (2.75)

both of which are independent of spatial coordinates x. Because integrations over the wholespace usually diverge, we introduce the finite cubical fiducial volume V0 to represent the wholespace. (We are legitimate to do so due to the spatial homogeneity.) Then the triad 0eai can beconveniently chosen to be along the three edges of the cubic. Thus, the corresponding classicalPoisson bracket is

c, p = 8πGγ

3, (2.76)


and the Hamiltonian constraint is

H = Hgrav +Hϕ =

(− 3

4πGγ2c2√|p|)+

(p2ϕ

2|p |3/2

)= 0, (2.77)

where a potential term a3U(ϕ) can be added to the matter part Hϕ. Usually, the matter is cho-sen to be a scalar field ϕ with canonically conjugate momentum pϕ and the Poisson bracketϕ, pϕ = 1. The introduction of the matter field serves also to the purpose of describing thephysics without the need of a specially chosen time variable. Specifically, in most cases, ϕ willevolve monotonically and therefore can serve as the ”internal time” of the Universe. Other phys-ical quantities then ”evolve” with respect to it.

The full LQC brings about two major corrections to the classical Hamiltonian constraint,dubbed inverse-triad correction and holonomy correction respectively. The first one corrects the|p|3/2 factor in the matter Hamiltonian. Because the value of a, and therefore p, can take 0 in theclassical case, their inverse-power quantities does not have well-defined quantum counterparts.The solution is to express the inverse term by the Poisson bracket as

a−1 = aV −2/3 =16πG

γV 2/3c, V = 3

4πGγc, V 1/3. (2.78)

By this mean, the inverse a term can have a well-defined quantum correspondence. Define theinverse triad operator as d = ˆa−3, then LQC gives its spectrum as

dj,n =

12(j(j + 1)(2j + 1)γl2P )−1

j∑k=−j

k√V 1

2(|n+ 2k| − 1)

6

, (2.79)

where lP =√G~/c = 1.62 × 10−35m is the Planck length. The quantization ambiguity arise

due to different choice of SU(2) representations j. The spectrum of the volume V = a3 is givenby

V 12 (n−1) = (γl2P /6)

2/3√(n− 1)n(n+ 1), n = 0, 1, ... (2.80)

The asymptotic behavior of the spectrum is

n << 2j :

(12

7

)6

V −112 (n−1)

(n

2j

)15/2

, (2.81)

n >> 2j : V −112 (n−1)

(2.82)

The spectrum of the scale factor is a =√

16γl

2Pn. Therefore, we see that for large volumes,

d ∼ V −1 ≈ a−3. For small volumes, however, d ∼ a−3a15 = a12. With this correction, theeffective Hamiltonian constraint

Heff = − 3

4πGγ2

(γa

2

)2

a+dj(a)p

2ϕ

2= 0 (2.83)


gives the Hubble parameter

H2 ≡(a

a

)2

=16πG

3a−3Hϕ =

16πG

3

(1

2a−3dj(a)p

2ϕ

). (2.84)

Then it is clear that for a tiny a, there is a H2 ∼ a9 period where the Universe undergoes asignificant acceleration, and therefore the Big Bang singularity is resolve to be a Big Bounce, asillustrated in Fig. 2.9.

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a

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Figure 2.9: The illustration of the resolution of Big Bang singularity predicted by LQC. Two(symmetric) classical trajectory of the scale factor a run into zero at ϕ = 0. The quantum effectwhen a approaches the Planckian scale gets strong enough to revert the situation. Then twoclassical trajectories are connected by the quantum Big Bounce.

Another correction comes from that in full LQC, the configuration variable should be theholonomy of the connection c, rather than c itself. The resulting correction is that

c→ sin(µc)

µ, (2.85)

and the effective Hamiltonian becomes

Heff = − 3

4πG

sin2(µc)

µ2

√|p|+Hϕ = 0. (2.86)

This gives the Hubble parameter as

H2 =8πG

3ρ

(1− ρ

ρcrit

)(2.87)

where ρcrit = 3/(8πGγ2µ2a2) is an evolving (positive) quantity. Now that H2 ≥ 0, we have0 ≤ ρ ≤ ρcrit. This result sets an upper bound on the matter density, and prevents it from beingdivergent. From (2.87), we have, when ρ = ρcrit,

a|ρ=ρcrit = 4πGaρcrit(1 + w), (2.88)

wherew = p/ρ is the state parameter for the energy contents the Universe. ‡ For common energyforms, w ≥ −1. Therefore, given that the energy content of the Universe is not too weird, the

‡For matter content w = 0, and radiations w = 1/3. Energy conditions, including weak, strong and dominant energyconditions, uniformly set bound on w ≥ −1.


Universe should be accelerating when the energy density approaches its upper bound. This alsogives the Big Bounce picture which resolve the classical Big Bang singularity.

CHAPTER 3Anomaly, Topology and Renormalization

Group

Zhong-Zhi Xianyu


(May 9, 2010)

This is the note for the second half of my seminar talks on anomalies. At first, We explain very briefly

the topological nature of the chiral anomaly. Then we investigate the theory of renormalization group

(RG) from the viewpoint of anomalies. Basics of conformal transformations are introduced as necessary

background knowledge. Then it is shown that the breaking of the scale invariance after quantization (scale

anomaly) directly leads to the concept of RG. In particular, the famous Callan-Symanzik equation, which

serves as a quantitative description of RG, is simply the anomalous Ward identity associated with scale

anomaly. The QED beta function is also calculated at one-loop level from an evaluation of the scale anomaly.

3.1 A Brief Review of Chiral Anomaly

The appearance of anomalies can be interpreted as the nontrivial behavior of the path integralmeasure under some symmetry transformations. In what follows we will assume that, under acertain class of transformations parameterized by ϵa, various quantities transform as follows:

• The field ϕ(x):

δϕ = iϵa(x)δaϕ.

• The action S[ϕ]:

δS = −∫

d4x[∂µj

aµ −∆a(x)]ϵa(x) = 0.

37

38 Chapter 3. Anomaly, Topology and Renormalization Group by Zhong-Zhi Xianyu

• The path integral measureDϕ:

δ[Dϕ]=

∫d4x ϵa(x)Aa(x).

• Any operator (as a function of ϕ):

δOi(yi) = ϵa(yi)δaOi(yi) = δaOi(yi)

∫ddx ϵa(x)δ(x− yi).

Then we have the Ward identities, which is a directly consequence of the invariance of pathintegral under a reparametrization of fields. It is given by:


⟩=⟨∆a(x)O1 · · ·On

⟩− i⟨δaO1(y1)O2 · · ·On

⟩δ(x− y1)

− · · · − i⟨O1 · · ·On−1δ

aOn(yn)⟩δ(x− yn)− i⟨Aa(x)O1(y1) · · ·On(yn)⟩. (3.1)

Fujikawa evaluated the nontrivial transformation Jacobian of the path integral measure, andgot the following result:

A(x) = − i

16π2ϵµνρσ tr (FµνFρσ). (3.2)

All these results are studied in detail in the last lecture. Here we only list some key steps:

• Decompose the spinors in to the sum of eigenstates of Dirac operator /D:

ψ(x) =∑n

anφn(x),

ψ(x) =∑m

φ†m(x)bm.

(3.3)

• Express the transformation Jacobian of the path integral measure in terms of φns:

DψDψ ⇒ Det e−i2ϵγ5DψDψ

≃ exp[− 2i

∫d4x ϵTr (φ†

mγ5φn)]DψDψ. (3.4)

• Using gauge invariant regularization method, we get

exp[− 2i

∫d4x ϵTr (φ†

mγ5φn)]

= exp[− 2i

∫d4xϵ

∑n

φ†n(x)γ5φn(x)

]= exp

[− i

16π2

∫d4xϵϵµνρσ tr (FµνFρσ)

]. (3.5)

3.2. Anomaly and the Index Theorem 39

3.2 Anomaly and the Index Theorem

Now we will explain the relation between Fujikawa’s result and the Atiyah-Singer indextheorem. Our treatment will be very very brief.

Zero Modes First let us point out two facts about the eigenstates of Dirac operator.

• γ5φn is also an eigenstate of /D, with eigenvalue −λn. This can be easily seen from theanti-commutator /D, γ5 = 0.

• γ5φn is orthogonal to φn if λn = 0.Proof. Notice that /D†

= /D, we have

(γ5φn, /Dφn) = λn(γ5φn, φn).

On the other hand,

(γ5φn, /Dφn) = −(γ5 /Dφn, φn) = −λn(γ5φn, φn).

Thus (γ5φn, φn) = 0 when λn = 0. QED.

However, when λn = 0, both φn and γ5φn correspond to the same eigenvalue 0. They are calledzero modes in physics literature. In this case, we may construct chiral zero modes by linearcombining φn and γ5φn:

φ0n± = P±φ

0n, with P± = 1

2 (1± γ5). (3.6)

Obviously we have /Dφ0n± = 0. But here φ0

n±’s are also eigenstates of γ5:

γ5φ0n± = ±φ0

n±. (3.7)

Now we look back at the Jacobian:

J [ϵ] = exp[− 2i

∫d4xϵ(x)

∑n

φ†nγ5φn(x)

].

If we set ϵ(x) to be a constant, then it’s quite easy to find that only the integration over zeromodes survives. In this case we have:∫

d4x∑n

φ†nγ5φn(x) =

∫d4x

∑n

φ0†n γ5φ

0n(x)

=∑n

∫d4xφ0†

n+γ5φ0n+(x) +

∑n

∫d4xφ0†

n−γ5φ0n−(x)

=∑n

∫d4xφ0†

n+φ0n+(x)−

∑n

∫d4xφ0†

n−φ0n−(x)

= n+ − n−. (3.8)


Here n+ and n− are number of positive and negative chirality zero modes respectively. Thecombination n+ − n− is called the analytic index of the Dirac operator. As its name suggests, itis an analytic property of the operator itself.

On the other hand, recall the Fujikawa’s result:

J [ϵ] = exp[− iϵ

16π2

∫d4xϵµνρσ tr (FµνFρσ)

], (3.9)

The integral on the right hand side is obviously a topological quantity, since it is defined by anintegration of some differential form over the whole manifold, and will vanish when the manifoldis topologically trivial. Thus it describe some global property of the manifold on which the Diracoperator is defined. In fact, the quantity

q = − 1

16π2

∫d4xϵµνρσ tr (FµνFρσ) (3.10)

is called Pontrjagin index of the manifold.Now our earlier work shows that

n+ − n− = q. (3.11)

That is, the local property of an differential operator is related with the global topological propertyof the manifold over which the operator is defined. This is just one example of the celebratedAtiyah-Singer index theorem, which can be stated very roughly as

analytic index = topological index. (3.12)

3.3 Conformal Transformation

It’s well known that Poincare symmetry is a fundamental symmetry of the flat 3 + 1 space-time. It can be regarded as the set of all space-time coordinate transformations that leave thesquare of inteval ds2 invariant. (Here the interval of the space-time is defined by means ofMinkowskian metric.) It can be shown that any space-time coordinate transformation leavingds2 invariant is a function of space-time coordinates up to linear order. That is, any Poincaretransformation can be parameterized as

xµ → x′µ = Λµνxν + aµ. (3.13)

Where Λµν and aµ are all space-time independent quantities.It seems that any theory consistent with experiments at sufficient low energy must respect

the Poincare symmetry, since the speed of light c0 in vacuum never changes. But the constrainton a Poincare transformations is a little bit too strong: the constant nature of c0 only implies

3.3. Conformal Transformation 41

that ds2 = 0 is independent of coordinate transformations. It says nothing about ds2 when it isnonzero. Thus we are tempted to ask, what will happen if we drop off the requirement that ds2

should always be invariant, but only keep the invariance of ds2 when it equals 0?The answer is that, we will obtain a larger group of space-time transformations than Poincare

group. This larger group is called the conformal group and its elements are called conformaltransformations.

Let us investigate this in details.

3.3.1 Definition

• A conformal transformation of coordinate x is defined to be an invertible mapping x→ x′,that leaves the metric tensor invariant up to a scale:

g′µν(x′) = Λ(x)gµν(x)

• All the conformal transformations form the conformal group.

• Obviously, the Poincare group is a subgroup of the conformal group, which corresponds toa special case Λ(x) = 1.

Now we study infinitesimal conformal transformations.

• Generally, we write down a infinitesimal coordinate transformation as:

xµ → x′µ = xµ + ϵµ(x)

• The metric changes as:

gµν → g′µν = gµν −(∂µϵν + ∂νϵµ

)• Then the conformal condition requires that:

∂µϵν + ∂νϵµ = f(x)gµν

To find the general form of infinitesimal conformal transformations, we will do some simplecalculations in the following.

• Firstly, f(x) can be obtained by taking the trace:

f(x) =2

d∂µϵ

µ

where we have assumed the metric is the Euclidean one in d dimensions.


• Now acting ∂λ on:∂µϵν + ∂νϵµ = f(x)gµν

We get:∂λ∂µϵν + ∂λ∂νϵµ = ∂λf(x)gµν

• Permuting the indices:∂λ∂µϵν + ∂λ∂νϵµ = ∂λf(x)gµν

∂µ∂νϵλ + ∂µ∂λϵν = ∂µf(x)gνλ

∂ν∂λϵµ + ∂ν∂µϵλ = ∂νf(x)gλµ

• (2)+(3)−(1):2∂µ∂νϵλ = gνλ∂µf + gλµ∂νf − gµν∂λf

• Contracting with gµν :2∂2ϵλ = (2− d)∂λf

• Acting ∂ν on: 2∂2ϵµ = (2− d)∂µf :

∂2∂νϵµ = (2− d)∂µ∂νf

• Acting ∂2 on: ∂µϵν + ∂νϵµ = gµνf :

∂2∂µϵν + ∂2∂νϵµ = gµν∂2f

• Comparing these two, we get:

(2− d)∂µ∂νf = gµν∂2f

• Contracting with gµν , we get the final result:

(d− 1)∂2f = 0

• Case 1: d = 1. No constrains on f . Any coordinate transformation is conformal!

• Case 2: d = 2. We will study this case in great detail later.

• Case 3: d ≥ 3. The above constrains imply that the conformal transformations are at mostlinear:

f(x) = A+Bµxµ A,Bµ constant.

Note that f = 2d∂µϵ

µ, thus ϵ(x) is at most quadratic:

ϵ = aµ + bµνxν + cµνλx

νxλ cµνλ = cµλν

3.3. Conformal Transformation 43

We then examine each constrains on ϵµ order by order.

• No constrains on aµ. This parameter corresponds to the translations.

• Applying ∂µϵν + ∂νϵµ = gµνf = 2dgµν∂λϵ

λ, on linear term, we find:

bµν + bνµ =2

dgµνb

λλ

Thus bµν may be separated into an antisymmetric part plus a pure trace:

bµν = αgµν +mµν , mµν = −mµν

Where α represents dilation, and mµν represent rotations.

• The constrains on quadratic term imply:

cµνλ = gµλbν + gµνbρ − gνρbµ bµ ≡1

dcσσµ

The corresponding infinitesimal transformation is:

x′µ = xµ + 2(x · b)xµ − bµx2

This is called a special conformal transformation (SCT).

The finite form of conformal transformations:

translation x′µ = xµ + aµ

dilation x′µ = αxµ

rotation x′µ =Mµνx

ν

SCT x′µ =xµ − bµx2

1− 2b · x+ b2x2

• The SCT can also be expressed as:

x′µ

x′2 =xµ

x2− bµ

That is, inversion→translation→inversion.

3.3.2 Local Structure of Conformal Group

The generators of conformal transformations:

translation Pµ = −i∂µ


dilation D = −ixµ∂µ

rotation Lµν = i(xµ∂ν − xν∂µ)

SCT Kµ = −i(2xµxν∂ν − x2∂µ)

The nonzero commutators:

[D,Pµ] = iPµ

[D,Kµ] = − iKµ

[Kµ, Pν ] = 2i(gµνD − Lµν)

[Kρ, Lµν ] = i(gρµKν − gρνKµ)

[Pρ, Lµν ] = i(gρµPν − gρνPµ)

[Lµν , Lρσ] = i(gνρLµσ + gµσLνρ − gµρLνσ − gνσLµρ)

These relations in fact define the conformal algebra. The conformal algebra

• To figure out the structure of the conformal algebra, we redefine the generators, as:

Jµν =Lµν J−1,µ =1

2(Pµ −Kµ)

J−1,0 =D J0,µ =1

2(Pµ +Kµ)

Then we have Jab = −Jba, a, b ∈ −1, 0, 1, · · · , d, and the commutator becomes:

[Jab, Jcd] = i(ηadJbc + ηbcJad − ηacJbd − ηbdJac)

where the diagonal metric ηab = diag(−1, 1, 1, · · · , 1).

• Now it’s easy to find that the conformal algebra in d-dimensional Euclidean space isSO(d+ 1, 1), which has 1

2 (d+ 2)(d+ 1) parameters.

3.3.3 Conformal Invariance in Classical Field Theory

Representations of the Conformal Group in d Dimensions

• The representations of the conformal group can be constructed in the same way as forPoincare group. That is, we first consider a subgroup (namely Lorentz group) that leavesx = 0 invariant.

• Thus we first remove translation generators from the conformal algebra, and let the rest ofgenerators have the representations as:

D → ∆, Lµν → Sµν , Kµ → κµ

3.4. Scale Anomaly and Renormalization Group 45

• Obviously, these representation matrices should obey the same commutation relations inabsence of Pµ:

[∆, Sµν ] =0

[∆, κµ] = − iκµ[κµ, κν ] =0

[κρ, Sµν ] = i(gρµκν − gρνκµ)

[Sµν , Sρσ] = i(gνρSµσ + gµσSνρ − gµρSνσ − gνσSµρ)

• The vanishing of the first commutator indicates that ∆ commutes with all representationsof rotations, thus by Schur’s lemma, ∆ must be a multiple of the identity.

• We then denote ∆ = −i∆, where ∆ is a real number, called the scaling dimension of thefield.

• Then from the second commutator, we find that all κµ’s must vanish.

• An example we will encounter frequently, is the so called quasi-primary field, defined as aspinless field ϕ, that transforms like:

ϕ(x)→ ϕ′(x′) =∣∣∣ ∂x′

∂x

∣∣∣−∆/d

ϕ(x)

3.4 Scale Anomaly and Renormalization Group

3.4.1 Scaling the Phi-Four Model

Now we focus on the scale transformation only, and take the phi-four model as an example.In this case, the scalar field ϕ(x) transforms under the scale transformation x→ x′ = eαx as

ϕ(x)→ ϕ′(x) = e−αdϕ(e−αx), (3.14)

or, in the infinitesimal form:δϕ(x) = [d+ xµ∂µ]ϕ(x). (3.15)

Where d is the scale dimension of ϕ. It is easy to check that the action of massless phi-four modelwill be invariant under scale transformation if we take d = 1. Since we have

∂µϕ→ α∂µ(dϕ+ xν∂νϕ) = α(1 + d+ xν∂ν)∂µϕ, (3.16a)

(∂µϕ)2 → 2α(∂µϕ)(1 + d+ xν∂ν)∂µϕ =

[2(1 + d) + xν∂ν

](∂µϕ)

2, (3.16b)

m2ϕ2 →m2[2d+ xν∂ν

]ϕ(x), (3.16c)

λϕ4 → λ[4d+ xν∂ν

]ϕ(x). (3.16d)


Thus, if we set d = 1, then

L → (4 + xν∂ν)[12 (∂µϕ)

2 − 14!λϕ

4]+ (2 + xν∂ν)

[− 1

2 m2ϕ2]. (3.17)

Performing the integration by parts, we get

δL = m2ϕ2(x). (3.18)

That is, only the mass term breaks the scale invariance. If we set m = 0, the symmetry will beexact.

The Nother current associated with scale transformation is:

jµ = xµTµν . (3.19)

Where Tµν is the energy-momentum tensor. The current jµ satisfies the conservation equation:

∂µjµ = ∆(x) = m2ϕ2(x). (3.20)

Thus the mass term can be viewed as the source of the scale current.

3.4.2 Ward Identities in Momentum Space

Now recall the ward identities derived before:


⟩=⟨∆a(x)O1 · · ·On

⟩− i⟨δaO1(y1)O2 · · ·On

⟩δ(x− y1)

− · · · − i⟨O1 · · ·On−1δ

aOn(yn)⟩δ(x− yn)− i⟨Aa(x)O1(y1) · · ·On(yn)⟩.

Integrating this over xµ, we get:

0 =

∫d4x

⟨∆a(x)O1 · · ·On

⟩− i⟨(δaO1)O2 · · ·On

⟩− · · · − i

⟨O1 · · ·On−1(δ

aOn)⟩− i∫

d4x ⟨Aa(x)O1(y1) · · ·On(yn)⟩.

Now we take Oi(yi) = ϕ(yi), then δOi(yi) =[d+ yµi ∂µi

]ϕ(yi), and ∆ = m2ϕ2.

In order to compare this identity with the familiar relations about the Green functions, we’dbetter rewrite it in the momentum space. Therefore we define the following Green functions inmomentum space:

G∆(0, p1, · · · pn−1)(2π)4δ(4)

(∑i

pi)

≡∫

d4xd4y1 · · ·d4yn⟨∆(x)O1(y1) · · ·On(yn)

⟩ei

∑i pi·yi (3.21)

3.4. Scale Anomaly and Renormalization Group 47

G(p1, · · · pn−1)(2π)4δ(4)

(∑i

pi)

≡∫

d4y1 · · ·d4yn⟨O1(y1) · · ·On(yn)

⟩ei

∑i pi·yi (3.22)

If the scale invariance of the theory is not affected by quantization process, then the anomalousterm vanishes. In particular we have A = 0. We will temporarily assume this and see whether itis consistent with the theory of renormalization.

Keep this assumption in mind, we can now write down the Ward identity in momentum space,as

G∆(0, p1, · · · pn−1) = i[4 + n(d− 4)−

n−1∑i=1

pi∂

∂pi

]G(p1 · · · pn−1). (3.23)

This can be further expressed in terms of vertex function Γ(p1, · · · pn−1) etc. The only differenceis the dimension. Simple counting can clarify this, and lead to the result:

Γ∆(0, p1, · · · pn−1) = i[4− nd−

n−1∑i=1

pi∂

∂pi

]Γ(p1 · · · pn−1). (3.24)

Since the only dimensional quantity in our model is m, thus by dimensional analysis, the vertexfunction must have the following form:

Γ(p1, · · · , pn−1) = m4−nf(p1/m, · · · , pn−1/m), (3.25)

where f is a dimensionless function. Then it’s not difficult to find that

[4− n−

n−1∑i=1

pi∂

∂pi

]Γ(p1 · · · pn−1) = m

∂

∂mΓ(p1 · · · pn−1). (3.26)

Then we have: [m

∂

∂m+ n(1− d)

]= −iΓ∆(0, p1, · · · , pn−1). (3.27)

At sufficient high energy level, as can be proved, the right hand side can be neglected. Thenwe simply get

m∂

∂mΓ = 0. (3.28)

That is to say, Γ does not depend on m logarithmically. However this is not the case. Since thefamiliar 1-loop result of the phi-four model does depend on m logarithmically. For instance,

Γ(4) ∼ −iλ− iCλ2[log

s

m2+ log

t

m2+ log

u

m2

]. (3.29)

This problem can be solved by properly adjust the value of d away from 1. It is hoped that if dis properly chosen, all will be OK. This indeed works for Γ(4), but fails in a more complicatedcase, e.g., Γ(4)/(Γ(2))2.


Callan solved the problem by noticing that, the operator

m∂

∂m+ n(1− d)

is not the most general form. He suggested to enlarge the operator as

m∂

∂m+ nγ + β

∂

∂λ. (3.30)

Then all will be fine. This is just the origin of the Callan-Symanzik equation. Let us write itdown explicitly: [

m∂

∂m+ nγ + β

∂

∂λ

]Γ = −iΓ∆. (3.31)

Where the new operator β(x) describes the nontrivial dependence of the coupling constant onthe scale m.

3.5 QED beta function from scale anomaly

A less trivial example other than phi-four model is QED. We will show how the beta functionof QED can be got from the evaluation of the scale anomaly.

The Lagrangian of QED reads

LQED = − 14e2F

2 + ψ(i /D−m)ψ. (3.32)

When m = 0, the theory is classically scale invariant. Once again, the Nother current is given byxµTµν , and its divergence is simply Tµµ, the trace of the energy-momentum tensor. At classicallevel, Tµµ = mψψ. It vanishes indeed when m = 0.

When quantum corrections enter, the coupling parameter e changes with scale α, and the rateof this change is described by beta function,

δe = αβ(e). (3.33)

Then the trace Tµ does not vanish any more, but acquires a contribution from

δL ≡ ∂L∂α

=∂e

∂α

∂L∂e

= β(e)∂L∂e

. (3.34)

Since ∂L/∂e = 12e3F

2, thus we have

Tµµ =β(e)

2e3F 2. (3.35)

Then, an evaluation of the beta function β(e) is equivalent to calculating the following trace:

⟨Tµµ⟩ =β(e)

e3

∫d4k

(2π)4Aµ

[k2gµν − kµkν

]Aν . (3.36)

BIBLIOGRAPHY 49

Bibliography

[1] K. Fujikawa, Phys. Rev. D 21, 2848 (1980);

[2] M. E. Peskin & D. V. Schroeder, An Introduction to Quantum Field Theory, Westview, 1995;

[3] S. Weinberg, The Quantum Theory of Fields, Vol.II, Modern Applications, Cambridge, 1998;

[4] R. A. Bertlmann, Anomalies in Quantum Field Theory, Oxford, 2000.

50 BIBLIOGRAPHY

CHAPTER 4Peierls Transition in 1-Dimensional Lattice:

Three Approaches

Long Zhang


(May 16, 2010)

4.1 Introduction

Peierls transition is a specific phenomenon in one-dimensional lattices. In 1955, Sir Rudolf Ernst Peierlsproved the theorem[5], that a one-dimensional equally-spaced chain with one electron per ion is unstabletowards dimerization and metal-insulator transition. A generalization states that a one-dimensional latticewith any number of electrons is unstable towards charge density redistribution and becoming insulating.Though not taken seriously and never published formally, this result was mentioned in a brief remark inhis book[4], and was soon well-known and inspired many theoretical and experimental work to realize thetransition. In 1970’s, experimentalists observed the evidence of Peierls transition in an organic system TTF-TCNQ with X-ray diffraction[2] and neutron scattering[3], undoubtedly confirming the theoretical analysis.

First we present a phenomenological analysis based on the MFA analysis. The one-dimensional chaincan be modeled by a simple microscopic Hamiltonian, due to Su, Schrieffer and Heeger[8],

H = −tN∑n=1

(1 + un)[c†nσcn+1,σ + h.c.] +

N∑n=1

ks2(un+1 − un)2, (4.1)

in which cnσ is the electron annihilation operator with spin σ at the nth site, and un is the coordinate of thenth ion. The second term is nothing but the elastic energy of lattice distortion, while the first term mimics theelectron hopping amplitude modulated by the lattice distortion. When ions come nearer, Coulomb attractiondue to the excess positive charge draws electrons close. When the lattice possesses a static dimerization asshown in Fig. 4.1 (b), electrons accumulate around the ions, modulating the charge density to form astatic charge-density wave (CDW) order, and an energy gap opens at the Fermi surface (Fig. 4.1 (d)). Pre-

51

52 Chapter 4. Peierls Transition in 1-Dimensional Lattice: Three Approaches by Long Zhang

Figure 4.1: Charge density modulation and Peierls transition. (a) 1-D Equally-spaced chain withlattice constant a; (b) Spontaneously dimerized 1-D chain, with lattice constant doubled. (c)Band structure corresponding to (a), with half-filled electrons in metallic state; (d) In Peierlstransition, a gap opened at the Fermi surface, leading to an insulating phase. Green lines indicatethe occupied states.

assuming a staggered distortion un = (−1)nu and solving the spectrum of Eq. 4.1∗, the elastic energy costis proportional to u2, and the electron energy gain is ∼ u2 log u. Therefore, the total energy minimizes at anonzero u, and exhibits a nonuniform lattice form and charge density modulation.

Why Peierls transition happens in one-dimension, while not in two-, or three-dimensions? This isrelated to the special Fermi surface topology in one-dimension, i.e., the Fermi surface always consists ofdiscrete points in Brillouin zone, and is perfectly nested by a momentum vector ±2kF . In quasiparticlelanguage, the electron excitations near one Fermi point are scattered by phonons with momentum ±2kF tothe other Fermi point. The scattering process is of large joint density of states and thus strong enough tomake electrons form a Bragg stand-wave, namely, the charge density wave.

In the above analysis, the transition is formulated in a mean-field approximation (MFA). A wave-likeorder parameter of ion positions is assumed in advance, and the saddle point is searched without consideringthe fluctuation effect. Since in one-dimension, quantum fluctuations severely affect physical properties andmay cause the pre-assumed order to totally breakdown†, we may doubt cautiously on this issue. On the otherhand, renormalization group (RG) analysis usually turns to be more reliable, especially in low dimensions.In this paper, we try to apply RG analysis to Peierls transition, and verify the MFA results explicitly.

4.2 Effective Lagrangian and Feynman Rules

∗A careful treatment of the Hamiltonian is found as an exercise in [1], Sec. 2.4.†E.g., 1-D Ising model against ferromagnetic ordering, and Luttinger liquid against CDW formation[7]

4.2. Effective Lagrangian and Feynman Rules 53

Let’s first consider a general field theoretical model for Peierls transition. For simplicity, we consideronly the most relevant D.o.F., i.e., the low energy electron excitation near the Fermi surface and the phononswith momenta near 2kF . As we know, one-dimensional electron excitations near the Fermi surface islinearly dispersive ω = vF k, in which k = p∓ kF is the displacement from the Fermi surface, and can berepresented as massless Dirac fermions. Phonons can be formulated as a simple scalar field. The Lagrangianis as follows‡,

L = L0 + L2 + Lint, (4.2)

L0 =∑σ

[iψ†Rσ

(∂

∂t+ vF

∂

∂x

)ψRσ + iψ†

Lσ

(∂

∂t− vF

∂

∂x

)ψLσ

]+

1

2[(∂tϕ)

2 − v2(∂xϕ)2],

(4.3)

L2 = −mu1

∑σ

(ψ†LσψLσ + ψ†

RσψRσ)−1

2µϕ2, (4.4)

Lint = −igϕ∑σ

(ψ†LσψRσ − ψ

σRσψLσ). (4.5)

The L0 term represents the kinetic terms of fermions and phonons, and Lint is the electron-phonon inter-acting term, a simple Yukawa coupling. The second term, which is absent in the bare Lagrangian, is thelowest order perturbation, in chemical potential forms, that may arise in quantum fluctuations.

In preparation for a perturbative RG analysis, Feynman rules of L are presented in Fig. 4.2.

k L= i

ω+kvF−iϵθ(k) +i

ω+kvF+iϵθ(−k),

k R= i

ω−kvF+iϵθ(k) +i

ω−kvF−iϵθ(−k),

k ± 2kF= 1

ω2−v2(k±2kF )2+iϵ , = −iµ,

L L=

R R= −iµ1,

RL ←−−

2kF

= −g, LR −−→

2kF

= g,

Figure 4.2: Feynman rules for Peierls model.

‡This model is taken from [9].


4.3 Renormalization Group Analysis

Now we apply RG analysis to Peierls model. Following the standard steps learned from [7], we succes-sively integrate out a momentum shell Λ/s < |k| < Λ, s = edt each time in the original Lagrangian, thenrescale the momentum, energy and fields to keep the free theory invariant, and observe how the couplingparameters flow in a parameter space. we first note that the free theory L0 is a fixed point after integrationand the following rescaling operations,

k′ = sk, ω′ = sω,

ψ′(k′, ω′) = s−3/2ψ(k, ω),

ϕ′(k′, ω′) = s−2ϕ(k, ω).

At tree level, we find all coupling constants flow relevantly,

µ′1 = sµ1, µ′ = s2µ, g′ = sg. (4.6)

But, to break the symmetry (µ < 0) or not to break (µ > 0), that is the question. To get the answer, wemust go to one-loop level to see the RG flow of coupling constants.

For convenience, we define the “dressed” propagators,

≡ + + + · · ·

=i

ω ± kvF − µ1 ± iϵ,

≡ + + + · · ·

=i

ω2 − v2(k ± 2kF )2 − µ+ ϵ.

These are just the usual propagators with mass terms indeed.Up to one-loop level, we calculate the following corrections to coupling constants,

L L=

L L+

L k R

−2kF − k−−−−−−→

L+

L k R

2kF + k←−−−−−

L,

2kF 2kF=

2kF 2kF+

2kF

R k

2kF

L k

,

L

R

−−−→−2kF

=L

R

−−−→−2kF

4.4. Effective Potential 55

Integrating the inner line momenta over the momentum shells explicitly, and using substitutions, dµ =

µ′ − µ et al, we get the RG flow equations,

dµ1

dt= µ

(1− g2

π

1√v2(1 + 2kF /Λ)2 + µ

1

(√v2(1 + 2kF /Λ)2 + µ+ vF )2 − µ2

1

), (4.7)

dµ

dt= 2µ− g2

πvF, (4.8)

dg

dt= g. (4.9)

Note that, the Eq. 4.7 results in permanent zero chemical potential for fermions, only if the free theorydoes not include a nonzero µ1. It is a general result for fermion excitations, as long as a particle-holesymmetry is present in the theory, as in this case. Eqs. 4.8 and 4.9 can be easily solved, bringing in twointegral constant C1 and C2,

g = C1et, (4.10)

µ = − t

πvFC2

1e2t + C2e

2t, (4.11)

µ = − g2

πvFlog

g

C1+C2

C21

g. (4.12)

The corresponding RG flow graph is shown in Fig. 4.3. The most prominent phenomenon is that theone-loop order perturbation totally reverses the direction of RG flow for µ > 0.§ The mass square µ ofscalar field flows to negative infinity steadily, no matter where it locates in the free theory, as long as theelectron-phonon coupling g = 0. It reminds us of the Landau’s phase transition theory, suggesting aninstability towards nonzero ϕ expectation value and breaking of ϕ↔ −ϕ symmetry. In this sense, the MFAresult is confirmed by RG analysis.

4.4 Effective Potential

Another field theoretical approach is the effective potential method¶. A background field is first as-sumed and by integrating out fluctuating fields around a saddle point in the partition function, we rigorouslycalculate the effective potential felt by the background field. Whether the system possesses a spontaneoussymmetry breaking (SSB) can be found by analyze the effective potential. This method is similar to MFAat first glance, it is nevertheless better than the latter because it takes the quantum fluctuations into account.

In order to investigate whether the phonon field possesses a SSB towards a staggered distortion, thephonon field is assumed to be a background and fluctuations ϕ(x) = ϕ0 cos(2kFx) + ϕ(x). The fermion

§In view of this nonperturbative result, one may wonder whether higher order perturbations will also strongly alter theRG flow. In this model, a specific feature arises that the theory is “super-renormalizable” in (1+1)-D, and higher orderperturbations only result in constants and do not show up in RG flows. In this sense, we can safely say the RG equations4.7-4.9 are EXACT and capture the main physical properties of the system.

¶A heuristic introduction to effective potential method can be found in [6], Part II.


Figure 4.3: RG flow graph for Peierls transition.

fields and the fluctuating phonon field in the partition function

Z =

∫Dϕ∏σ

Dψ†RσDψRσDψ

†LσDψLσ

× ei∫d2x

12((∂tϕ)

2−v2(∂xϕ)2−µϕ2)+∑σ(ψ

†Lσψ

†Rσ

)

i(∂t − vF ∂x) −igϕigϕ i(∂t + vF ∂x)

ψLσψRσ

(4.13)

are to be integrated out, leaving the effective potential of ϕ0. Neglecting second order coupling, e.g.,coupling between ψ(k) and ψ(k′) mediated by a phonon of momentum (k − k′), the fluctuating phononfield and the static one, the fermion fields that are not of opposite momenta, are both separated. Thus theeffective potential felt by the static background phonon field is simplified to be Veff = VT + VF , in whichVT comes from the chemical potential term and the kinetic term (spatial variation term), and VF from thefluctuating fermion fields.

ST ≡∫d2xVT (ϕ) =

1

2

∫d2xv2(∂xϕ0 cos(2kFx))

2 + µϕ20 cos

2(2kFx) =

∫d2x

(v2k2F +

1

4µ

)ϕ20.

(4.14)

SF (ϕ0) ≡∫d2xVF = 2i

∫d2xTr log

(i(∂t − vF ∂x) −igϕ(x)

igϕ(x) i(∂t + vF ∂x)

)

= 2i

∫d2x

∫d2k

(2π)2log det

(ω + vF k −igϕ0/2

igϕ0/2 ω − vF k

)

= 2i

∫d2x

∫d2k

(2π)2log(ω2 − v2F k2 − g2ϕ2

0/4),

(4.15)

adding an unimportant constant, Eq. 4.15 can be Wick-rotated to Euclidean space (k0(ω) = ik2vF ) and

4.5. Conclusion 57

then regularized by a momentum cut-off Λ,

VF = 2i

∫d2k

4π2log

(ω2 − v2F k2 − g2ϕ20/4)

ω2 − v2F k2

= −2∫d2kE4π2

logk2E + g2ϕ2

0/4v2F

k2E· vF

= −vF2π

∫ Λ2

0

dk2E logk2E + g2ϕ2

0/4v2F

k2E

= −vF2π

[−Λ2 log Λ2 − 1

4g2ϕ20

v2Flog

(1

4g2ϕ2

0/v2F

)+

(Λ2 +

1

4g2ϕ2

0/v2F

)log

(Λ2 +

1

4g2ϕ2

0/v2F

)]=

(vF8π

g2

v2Flog

g2

4v2F+vF8π

g2

v2Flog ϕ2

0 −g2

4v2Flog Λ2 − vF

8π

g2

v2F

)ϕ20 +O

(ϕ20

Λ2

).

(4.16)

We then pose a renormalization condition,

d2Veffdϕ2

0

|ϕ0=M = 2v2k2F +1

2µ(M),

in which M is the renormalization scale and µ(M) is a “physical” chemical potential. The effective poten-tial Veff (ϕ0) is renormalized to be

Veff (ϕ0) =

[v2k2F +

1

4µ(M)− 3vF

8π

g2

v2F+vF8π

g2

v2Flog

(ϕ20

M2

)]ϕ20. (4.17)

The most remarkable feature of the effective potential Eq. 4.17 is that it possesses the same dependenceon ϕ0 with the MFA result, the ϕ2

0 log ϕ20 term dominates at small ϕ0, leading to instability to forming

staggered lattice distortion. This can be used to fit the experiments.Before closing, another remark is put that RG flow equations can also be derived from the effective

potential approach, corresponding to the dependence of coefficients in Veff on the renormalization scaleM . For detailed derivation method, see [6]. The results are identical to that in Wilson’s method.

4.5 Conclusion

In conclusion, we have calculated the RG flow equations and the effective potential of the Peierls modeland firmly verified the MFA results, that one-dimensional equally-spaced half-filled lattice is unstable to-wards a charge density modulation and metal-insulator transition. RG analysis is usually more powerfuland reliable, and proved to be useful to study phases and phase transition issues.

Bibliography

[1] Altland A and Simons B. Condensed Matter Field Theory. Cambridge University Press, 2006.

58 BIBLIOGRAPHY

[2] Denoyer F, Comes F and Garito A F et al. X-Ray Diffuse Scattering Evidence for a Phase Transitionin Tetrathiafulvalene Tetracyanoquinodimethane (TTF-TCNQ). Phys. Rev. Lett. 35: 445 (1975).

[3] Mook H A and Watson C R, Jr. Neutron Inelastic Scattering Study of Tetrathiafulvalene Tetra-cyanoquinodimethane (TTF-TCNQ). Phys. Rev. Lett. 36: 801 (1976).

[4] Peierls R E, Sir. Quantum Theory of Solids. Oxford University Press, 1955.

[5] Peierls R E, Sir. More Surprises in Theoretical Physics. Princeton University Press, 1991.

[6] Peskin M E and Schroeder D V. An Introduction to Quantum Field Theory. Westview Press, 1995.

[7] Shankar R. Renormalization-group approach to interacting fermions. Rev. Mod. Phys. 66: 129 (1994).

[8] Su W P, Schrieffer J R and Heeger A J. Solitons in Polyacetylene. Phys. Rev. Lett. 42: 1698 (1979).

[9] Zee A. Quantum Field Theory in a Nutshell. Princeton University Press, 2003.

CHAPTER 5Spin Connection and Local Lorentz

Transformation

Xiao Xiao

School of Physics, Peking University, Beijing 100871

(May 22, 2010)

We introduce a new concept named spin-connection, which is currently widely used in quantum fieldtheory, general relativity and string theory.

5.1 Why spin connection

In order to define a spinor theory in curved spacetime, the crucial point is to define a covariant derivativein order to take into account the non-trivial effect of translation.

∂µψ → Dµψ (5.1)

We should add an extra term after∂µψ, the form of the term is highly restricted:

1. We expect the term to contain Γµνσ , because we expect the spin–which is classically a pseudo vector,has similar property under translation as a vector.

2. We expect the term contains γ matrices, because the non-trivial transformations of spinors must beperformed with Γ matrices.

3. The term has a contra-variant index.

We try first:

Dµψ = ∂µψ + ξΓµνσ[γν , γσ]ψ (5.2)

59

60 Chapter 5. Spin Connection and Local Lorentz Transformation by Xiao Xiao

However, since the torsion-free property of the spacetime, the two lower indexes of Γµνσ are symmetric,while the commutator of Γ matrices are antisymmetric in the same indexes, the term obviously vanishes.

We try again:

Dµψ = ∂µψ + ξΓµνσγν , γσψ = ∂µψ + 2ξgνσΓµνσψ

We see that the anti-commutator of the matrices gives the metric itself and it is trivial in the internalspace, we have to discard this ansatz.

Similarly:

Dµψ = ∂µψ + ξΓµνσγνγσψ

should not be considered as well since the symmetry of the lower indexes of Γµνσ and the non-commutativityof the matrices are not consistent.

Therefore, using affine connectionΓµνσ we fail to construct a covariant derivative of spinor field, dueto the inconsistency of the commutation property of γ matrices and the torsion-free property of space-time.However, we can find another definition of connection which circumvents this discrepancy, it is spinconnection, we will return here later.

Another point, spin connection provides an easier way to calculate Riemann tensor,as we know, calcu-lating Riemann tensor in traditional way is a demanding job: one should calculate all connection coefficientsand then all components of Riemann tensor, later we will see such a process is simplified in the frameworkof tetrad.

What’s more, an important conceptual point about spin connection is that, it provides a framework inwhich general relativity can be expressed explicitly as a gauge theory, just as Yang-Mills theory. If one askedwhether gravitation field is a gauge field, a comparison may be made before a confirming answer, actuallygeneral relativists call the invariance of Einstein theory under general coordinate transformation gauge in-variance. But this cannot remove a little suspicion: after all general coordinate transformation is differentfrom local gauge invariance, the coordinate transformation cannot be regard as totally local because of therequirement of continuity, and the transformation of the coordinate basis is not orthonormal(or Lorentzian,in relativity), the tetrad description will remove this disparity and we see clearly a correspondence.

5.2 Formalism

(1)Tetrad, vielbein Many legs: einsbein, zweibein, dreibein, vierbein... vielbein! —local Lorentz framein which the metric tensor gives a Minkowski matrix. The local Lorentz frames can ”rotate independently”,i.e. the vierbeins at different spacetime points can experience different ”Local Lorentz Transformation”, wewill see this correspond to local gauge transformation.

The vielbein, or tetrad, defined as, ea, which are Lorentz frames:

g(ea, eb) = ηab

Recall that if we define a coordinate system on the spacetime manifold, we have the tangent vectors ofthe coordinate lines as the coordinate base vectors: eµ, they are not necessarily orthonormal (be Lorentz

5.2. Formalism 61

frame)g(eµ, eν) = gµν the coordinate basis gives another measurement of the metric tensor.Definition 1:

eµ = e aµ ea

eaµ is the projection of the coordinate basis on tetrad vectors.Definition 2:eµa which satisfies:e aµ e

νa = δνµ

e aµ e

µb = δab

Two defining conditions required to define the properties of the Latin and Greek indices.Inference 1:eµbeµ = eµbe

aµ ea = eb

i.e. eµa is the projection of tetrad vectors on coordinate basis.Inference 2:gµν = g(eµ, eν) = ηabe

aµ e

bν

Inference 3:gµν = ηabe

aµ e

bν

⇒ gµνeνc = ηabe

aµ e

bνeνc = ηace

aµ

⇒ eσc = gσµgµνeνc = gσµηace

aµ

We see it is deducted that the mixed tensor eσc has the usual index uppering and lowering rule withrespect to ηab and gµν

Having made tetrad well-defined, we go to co-tetrad.Definition 3:θaeb = δab

This defines the co-tetrad.Inference 4:Recall that θµeν = δµν

Therefore we have:⇒ θaeµbeµ = δab ⇒ θaeµbeµθ

ν = δab θν ⇒ θaeνbδ

ba = θν = θaeνa

eµb is the projection of co-coordinate basis on co-tetradSimilarly we haveInference 4:θa = e a

µ θµ

e aµ is the projection of co-tetrad on co-coordinate basis

Any vector can be expressed either in coordinate basis or in tetrad.V = V aea = V µeµ

We can deduce:V a = e a

µ Vµ

This is also valid for tensors:V ab = e a

µ Vµb = eνbV

aν = e a

µ eνbV

µν

(2) Covariant derivative:


Next we construct the covariant derivative with respect to tetrad.Unlike the coordinate basis, which transforms according to general coordinate transformation(GCT), or

actively—diffeomorphism, tetrad transforms according to local Lorentz transformation(LLT), it is nothingbut a local gauge transformation, in which the tetrad defined on different points transform independently,but keep the orthonormality.

Just like in the case of gauge theory, we require the covariant derivative to be covariant under LLT:DµX

a → D′µX

′a= ΛDµX

a

D′µ = ΛDµΛ

−1

We need an additional term in order to compensate the rotation of tetrad, it is just the introduction ofspin-connection:

Definition 1:DµX

a = ∂µXa + (ωµ)

abX

b

We need here a clarification of the covariant derivative of tetrad from the covariant derivative of diffeo-morphism.

Because ofeµ = e a

µ ea

Xa = e aµ X

µ

We have,∇µXa = ∂µXa

Xa is the component of a vector on tetrad.Inference 1:DµXa = ∂µXa − (ωµ)

baXb

It can be proved by requiring XaXa transforms as a scalar(invariant) under LLT.Inference 2:By applying ηab on both sides of definition 2, and consider definition 1, we have:(ωµ)ab = −(ωµ)baThis antisymmetry would play a vital role in our construction of a spinor theory in curved spacetime.Next we introduce the concept of tetrad postulate, it is a correspondence between tetrad covariant deriva-

tive and diffeomorphism covariant derivative. When the tetrad covariant derivative acts on a vector withgreek index, we require the result is of the same structure of a diffeomorphism covariant derivative actingon the same vector.

Definition 2:DµX

ν = ∇µXν

The reason why I say it is a definition is that it actually defines the action of tetrad covariant derivativeon Greek indexes.

Inference 3:Dµe

aν = ∂µe

aν − Γλµνe

aλ + (ωµ)

abe

bν

Pf:DµX

a = Dµ(eaν X

ν) = (Dµeaν )Xν + e a

ν (DµXν)

DµXa = ∂µX

a + (ωµ)abX

b

⇒ (Dµeaν )Xν + e a

ν (DµXν) = ∂µX

a + (ωµ)abX

b = ∂µ(eaν X

ν) + (ωµ)abX

b


ν (DµXν) = (∂µe

aν )Xν + e a

ν (∂µXν) + (ωµ)

abX

b


ν (∇µXν) = (∂µeaν )Xν + e a


abX

b

5.3. Spinor field in curved spacetime and the local transformation property of spin connection 63


ν (∂µXν + ΓνµλX

λ) = (∂µeaν )Xν + e a


abX

b


ν ΓνµλXλ = (∂µe

aν )Xν + (ωµ)

abX

b

⇒ (Dµeaν )Xν = (∂µe

aν)X

ν − e aλ ΓλµνX

ν + (ωµ)abe

bν X

v

⇒ Dµeaν = ∂µe

aν − e a

λ Γλµν + (ωµ)abe

bν

We further note that∇X can be expressed by both coordinate basis and tetrad in the same structure:

∇X = (∇µXν)eµ ⊗ eν = (∇µXa)eµ ⊗ eaThis leads to a definite correspondence between tetrad connection and affine connection:

Inference 4:Γνµλ = eνa(∂µe

aλ ) + eνae

bλ (ωµ)

ab

Pf:

∇X = (∇µXa)eµ ⊗ ea= (∂µX

a + (ωµ)abX

b)eµ ⊗ ea= ∂µ(e a

ν Xν) + (ωµ)

abe

bλ X

λeµ ⊗ eσaeσ= eσa(∂µe a

ν )Xν + e aν ∂µX

ν + (ωµ)abe

bλ X

λeµ ⊗ eσ= ∂µXν + eνa(∂µe

aλ )Xλ + eνae

bλ (ωµ)

abX

λeµ ⊗ eλ⇒ ∇µXν = ∂µX

ν + ΓνµλXλ = ∂µX

ν + eνa(∂µeaλ )Xλ + eνae

bλ (ωµ)

abX

λ

⇒ Γνµλ = eνa(∂µeaλ ) + eνae

bλ (ωµ)

ab

Using inference 3 and 4 we can deduce:

Inference 5:Dµe

aν = 0

This result is mostly seen as ”tetrad posutulate”, we see it’s the consequence of two requirements oftetrad covariant derivative: the action on Greek index is equivalent to diffeomorphism covariant derivative,and the action on Greek index and Latin index have identical structure.

5.3 Spinor field in curved spacetime and the local transformationproperty of spin connection

Now we have been equipped with an anti-symmetry connection, (ωµ)ab,

we can easily overcome the difficulty that we met in the beginning of this note.

Define:

Dµψ = ∂µψ + ξ(ωµ)ab[γa, γb]

where ξ is a constant.

The only sacrifice we make is changing the Greek indices of the Gamma matrices to Latin indices.

We may further the discussion of the transformation property of the spin connection.

We go back to

ωµ → Λ(∂µ + ωµ)Λ−1

Where the local Lorentz transformation:


Λ = Λ(x) = exp− i2JabΘab(x) ≃ 1− i

2JabΘab(x)

Then plug in the infinitesimal form and consider the spinor case,we get:

ωµ →i

2Sab∂µΘab(x) +

i

2[Sab, ωµ]

Sab = i4[γa, γb]

Because ωµ is antisymmetric in its two Latin indices, so are the generators of the local Lorentz group,therefore the connection can be expressed as the linear combination of the generators:

ωµ = − i2ωµabS

ab

ωµ →1

4Sab∂µΘab +

1

4[Sab, Scd]ωµcdΘab

We can prove that the definition of derivative:

Dµψ = (∂µ −i

2ωµabS

ab)ψ = (∂µ +1

8ωµab[γ

a, γb])ψ

is covariant under LLT.

5.4 Covariant exterior derivative, torsion and curvature

To define the covariant exterior derivative there are three requirements:*antisymmetry of the indices*covariant of the Latin indices under LLT*covariant of the Greek indices under GCTWe have the definition:Definition 1:(DX)aµν = DµX

aν −DνXa

µ = (∂µXaν − ΓλµνX

aλ + ωaµbX

aν )− (∂νX

aµ − ΓλνµX

aλ + ωaνbX

aµ)

= ∂µXaν − ∂νXa

µ + ωaµbXaν − ωaνbXa

µ

= (dXa)µν + (ω ∧Xa)µν

We define torsion and curvature tensors below, the equivalence of the two to the ones defined withrespect to diffeomorphism can be verified.

Definition 2:T a = dea + ωab ∧ eb

Definition 3:Rab = dωab + ωac ∧ ωcbInference 1:dT a + ωab ∧ T b = Rab ∧ eb

Pf:T a = dea + ωab ∧ eb

⇒ dT a = d(ωab ∧ eb) = dωab ∧ eb − ωab ∧ deb

Rab = dωab + ωac ∧ ωcb⇒ Rab ∧ eb = dωab ∧ eb + (ωac ∧ ωcb) ∧ eb

dT a = −ωab ∧ deb +Rab ∧ eb − (ωab ∧ ωbc) ∧ ec

= −ωab ∧ deb − ωab ∧ (ωbc ∧ ec) +Rab ∧ eb

= −ωab ∧ T b +Rab ∧ eb

5.5. Calculating the curvature tensor: 65

dT a = −ωab ∧ T b +Rab ∧ eb

Inference 2:dRab = Rac ∧ ωcb − ωac ∧RcbPf:Rab = dωab + ωac ∧ ωcb⇒dRab = d(ωac ∧ ωcb) = dωac ∧ ωcb − ωac ∧ dωcb= (Rac − ωad ∧ ωdc) ∧ ωcb − ωac ∧ (Rcb − ωcd ∧ ωdb)= Rac ∧ ωcb − ωad ∧ ωdc ∧ ωcb − ωac ∧Rcb + ωac ∧ ωcd ∧ ωdb= Rac ∧ ωcb − ωac ∧Rcb

dRab = Rac ∧ ωcb − ωac ∧RcbNotice not to confuse Rac with Ricci tensor, Rac correspond to the whole Riemann tensor and the

correspondence is:Rρσµν = eρae

bσ R

abµν

5.5 Calculating the curvature tensor:

Tetrad provide us a convenient way of calculating the Riemann tensor, given the definition of torsionand curvature, all we need is to solve torsion-free condition and calculate curvature tensor.

torsion-free condition:dea + ωab ∧ eb = 0

We need to solve this forωab,Since ωab is antisymmetric, we have in consequence:*We take a flat expanding universe:ds2 = −dt2 + a2(t)δijdx

idxj

First, we choose a co-tetrad, a orthonormal co-vierbein:e0 = dt, ei = adxi

We can deduce from the antisymmetry of ω:ω0

0 = η00ω00 + η0iωi0 = 0

ω0j = η00ω0j + η0iωij = ω0j

ωj0 = ηj0ω00 + ηjiωi0 = −ωj0⇒ ω0

j = ωj0

ωij = ηi0ω0j + ηikωkj = −ωijωji = ηj0ω0i + ηjkωki = −ωji⇒ ωij = −ωjiNext step is to compute the left hand side of the torsion-free condition:de0 = 0


dei = da ∧ dxi = ·adt ∧ dxi

And the right hand side:

ω0b ∧ eb = ω0

j ∧ ej = aω0j ∧ dxj

ωib ∧ eb = ωi0 ∧ e0 + ωij ∧ ej = ωi0 ∧ dt+ aωij ∧ dxj

Then we have the torsion-free condition as equations of connection coefficient:

ω0j ∧ dxj = 0

ωi0 ∧ dt+ aωij ∧ dxj = −·adt ∧ dxi

The first equation can be solved by observation, there are two possibilities, the first one:

ω0j = 0

But then,

⇒ ωij = −·aδijdt

violating the antisymmetry of connection coefficient.

The only possibility is the quantities on the both sides of the wedge parallel:

ω0j =

·adxj

Then we have:

ωij = 0

Having the connection efficient in hand,we can compute the curvature.

Rab = dωab + ωac ∧ ωcb dωi0 = dω0i = d(

·adxj) = d

·a ∧ dxj = ··

adt ∧ dxj

dωij = 0

ω0c ∧ ωc0 = ω0

0 ∧ ω00 + ω0

j ∧ ωj0 = 0

ωic ∧ ωc0 = ωi0 ∧ ω00 + ωij ∧ ωj0 = 0

ωic ∧ ωcj = ωi0 ∧ ω0j + ωik ∧ ωkj = (

·a)2dxi ∧ dxj

R00 = 0

R0j =

··adt ∧ dxj

Ri0 =··adt ∧ dxi

Ri j = (·a)2dxi ∧ dxj

It is important not to confuseRabwith Ricci tensor, it is equivalent to Riemann tensor, and every com-ponent in the sense of Latin index is actually a two-form in the sense of Greek index, we can see this clearlyfrom the above results.

Riemann tensor can be obtained by trading two Latin indexes to Greek indexes using vierbein compo-nents.

Rρσµν = eρaebσR

abµν

eaµ =

1 0 0 0

0 a 0 0

0 0 a 0

0 0 0 a

BIBLIOGRAPHY 67

eνb =

1 0 0 0

0 a−1 0 0

0 0 a−1 0

0 0 0 a−1

And we have:(dxα ∧ dxβ)µν = δαµδ

βν − δαν δβµ

Then it is easily obtained that:R0

j0l = a··aδjl

Ri0k0 = −··a

aδik

Ri jkl = (·a)2(δikδjl − δilδjk)

Through the simple example we see that the spin-connection coefficient, which are equivalent to Christof-fel coefficient, can be obtained by solving torsion-free condition, and the condition can be expressed bysimple wedge products, and the expression of curvature in the term of spin-connection is much simpler.

Bibliography

[1] Mathematics wiki: differential geometry, spin connection.

[2] Sean Carroll, spacetime And Geometry

[3] Ramond, ⟨Field Theory: A Modern Primer⟩

[4] C.B.Liang ⟨Introduction of Differential Geometry And General Relativity⟩

68 BIBLIOGRAPHY

CHAPTER 6An Introduction to Dark Matter

Lan-Chun Lu


(May 30, 2010)

6.1 Cosmology evidence for the existence of DM

Dark matter particle physics is one of the leading subject today .At present a number of related obser-vations with gravitational effects indicate the existence of dark matter .

It is because of these evidences ,nowadays we all believe that dark matter is an important part of theuniverse .And we are trying to find the signal of DM particles in different experiments.

6.1.1 Asymptotic flatness of the rotation curve for galaxies

The main ingredient of the gas in galaxy is Hydrogen ,of which the Doppler shift can be observed to getthe speed of its rotation .The centripetal force of its rotation is gravity ,by Newton’s Law we can computethe speed of rotate as follows

v2rotr

=GM(r)

r2(6.1)

which means

vrot =

√GM(r)

r(6.2)

where M(r) is the total mass in sphere with radius r ,assume the radius of the galaxy is r0, on thesurrounding of the galaxy the observable matter is very rare .If there is no dark matter .M(r) would beconstant when r > r0then the rotate speed of the galaxy would be

vrot ∝ r−12 (6.3)

while the experimental observation get another result.

69

70 Chapter 6. An Introduction to Dark Matter by Lan-Chun Lu

Figure 6.1: Rotation curve of a spiral galaxy: The prediction given by DM halo is close to theexperiment.

6.1.2 X Ray observation of the cluster hot gas

The gas within a cluster is under the thermal equilibrium ,of which the temperature satisfy

GM(r)

r2= − kBT

µmH[d log ρ

dr+d log T

dr] (6.4)

We can measure the X-Ray radiated by the gas in a cluster to get its temperature , then we can use the formulaabove to compute the mass of the cluster .On the other hand ,we can measure the mass of observable matterin this cluster by other observation methods .

The result for one typical cluster is :

M/Mvisible = 20 (6.5)

which indicate the existence of Dark Matter.

6.1.3 The observation of gravitational lensing

Gravitational lensing refers to the deviation of light while in the gravitational field of matters .Usuallythe lens is a special cluster while the source can be different .In the case of Weak gravitational lensing thesource is another cluster while in Sunyaev-Zeldovich deviation the source is the cosmological microwavebackground .

Through this observation we can get the total mass of the lens which is the cluster we concern about.As mentioned before .We can get the visible mass of this cluster by other means .

Here is a picture of gravitational lensing captured by T.Verdugo and his group .

6.2. The Standard Model 71

Figure 6.2: SL2S J02140-0535Gravitational lensing: We can see three image of G1, G2, G3

6.1.4 Anisotropy of Cosmology Microwave Background

WMAP has got very precise results of the Cosmological Microwave Background ,we can fit manyparameters in cosmology by data of the anisotropic of CMB .The relative abundant given by ΛCMD modelis:

ΩMh2 = 0.127+0.007

−0.013 (6.6)

ΩBh2 = 0.0223+0.0007

−0.0009 (6.7)

Which means the DM relative density is about 5 times of the visible matter.

6.2 There is no dark matter candidate within the SM

As all the evidences above and others ,people believe that DM does exist .But all the particles we havetoday in our Standard Model cannot be dark matter because of the property of DM which are:

1. Stability :As there are dark matter today ,the life time of dark matter particle has to be greater thanthe age of our universe (1016s).

2. Massive:23% of the critical density and should be non-relative particle .

3. Chargeless: No weak and electromegnetic interaction .

4. Substructure :Dark halo surrounding galaxies. Part of the DM form compact halo objects.

The Standard Model is a gauge theory with gauge group SU(3)c × SU(2)L × U(1)Y , of which thelagrangian is

L = Lgauge + Lscalar + Lfermion (6.8)

The gauge part is Y ang −Mills theory

Lgauge = −1

4W aµνW

aµν − 1

4BµνB

µν ≡ LKgauge + LIgauge (6.9)


The scalar part predicts the existence of Higgs boson. Higgs boson is a two dimension representation ofSU(2), and one dimension representation of U(1). The gauge boson get mass by the vacuum expectationof Higgs boson.

Lscalar = (DµΦ)†(DµΦ)− V (Φ) (6.10)

Fermion couple to scalar through Yukawa coupling .

LF = LLiD/lLLLi + lRiD/lRlRi − fli(LLiΦlRi + lRiΦ†Lli)

+QLiD/qLQLi + qRiD/qRqRi

−[fuijQLiΦuRj + fdijQLiΦdRj + h.c.] (6.11)

There are no DM candidates in SM which give us a hint that new physics is necessary.

6.3 Models that contain Dark Matter candidates

People have built up many models which contain DM candidates extended of the SM. Among themthere are supersymmetry in which the lightest superpartner(LSP) is DM candidate, extradimension in whichthe lightest Kaluza-Klein particle is DM candidate, axion model and mirror type matter and so on.

6.3.1 Supersymmetry (MSSM)

The motivation of supersymmetry is to solve the hierarchy problem in the SM. The basic idea of SUSYis combine the spacetime symmetry and the intrinsic symmetry together, which in other words is to extendthe symmetry from Poincare symmetry to SuperPoincare symmetry. Another hint that SUSY is a natureresult of String theory. The SUSY predicts partners of SM particles which is called superpartner.

In SUSY we introduce the SUSY operator Q which turns a bosonic state into fermionic state, and viceversa. This operator must be an anticommutative operator.

Q|Boson⟩ = |Fermion⟩

Q|Fermion⟩ = |Boson⟩

As Q and Q† are both fermionic, they both carry spin angularmomentum 12

. SUSY operators Q andQ† satisfy commutation relation as follows:

Q,Q† = Pµ

Q,Q = Q†, Q† = 0

[Pµ, Q] = [Pµ, Q†] = 0

(6.12)

The particles in a specific SUSY model which is called Minimal Supersymmetry(MSSM) are:With these particles states, we can write down the general Lagrangian of MSSM, we have to notice

that SUSY is a broken symmetry as the superpartners is heavier than its SM partner. Now come back tothe problem we concern about here, the DM candidate is lightest one in all of the superpartners(LSP). Theresult is that it is a mixing of gaugino and Higgsino which we call it neutrino.

6.3. Models for Dark Matter 73

Table 6.1: particles in scalar multiplets in MSSMNames spin-0 spin1/2 SU(3),SU(2),U(1)Q (uL, dL) (uL, dL) (3, 2, 1/6)

u uR u†R (3, 1,−2/3)d dR d†R (3, 1, 1/3)

L (v, eL) (v, eL) (1, 2,−1/2)e eR e†R (1, 1, 1)

Hu (H+u ,H

0u) (H+

u , H0u) (1, 2, 1/2)

Hd (H0d ,H

−d ) (H0

d , H−d ) (1, 2,−1/2)

Table 6.2: particles in gauge multiplets in MSSMNames spin1/2 spin1 SU(3),SU(2),U(1)gluon g g (8, 1, 0)

W (W±, W 0) (W±,W 0) (1, 3, 0)

B B0 B0 (1, 1, 0)

6.3.2 Mirror Type Matter

The motivation of Mirror theory is to keep Parity conservative. As we know that T.D.Lee and C.N.Yangwon Nobel prize for their work of Breaking Parity symmetry. Foot suggest that if we have a kind of mirrorparticles which have the opposite chirality comparing to its corresponding SM particle, then we can keepParity conserved and give the DM candidate at the same as time.The mirror transform P have the following form.

x→ −x, t→ t

Gµ → G′µ,W

µ →W ′µ, B

µ → B′µ

liL → γ0l′iR, eiR → γ0e

′iL, qiL → γ0q

′iR

uiR → γ0u′iL, diR → γ0d

′iL

Then we can write down the general Lagrangian of mirror theory.

L = LSM (eiL, qiL,Wµ, . . .) + LSM (e′iL, q

′iL,W

′µ, . . .)

+Lmix (6.13)

Generally the mixing term between these two part has the following form.

Lmix = λ′ϕ′†ϕ′ϕ†ϕ+ϵ

2FµνF ′

µν (6.14)

Our goal is the DM candidate in this theory, well, in this theory all the stable mirror particles are DMparticles if we have ϵ very small.


6.4 Experimental detection of Dark Matter

We have three kinds of Experimental methods according to way in which the reaction goes. Let’s noteχ as DM particles and ψ as SM particles.

1. Direct detection: χψ −→ χψ

2. Indirect detection: χχ −→ ψψ

3. Accelerator detection:ψψ −→ χχ

In the direct detection we assume that the DM particles are around us everywhere, so it can hit the detectorand recoil. While in the indirect detection we test the anomaly of some specific cosmic ray to see whetherit can be interpret by DM annihilation. As in accelerator detection, we create DM particles on accelerator.

6.4.1 Direct detection

Here is a list of some of the direct detection experiment around the world.

Figure 6.3: Direct detections: Direct detections in different places.

• We are observing collisions between dark matter particle and the ordinary target atom in our labframe. what we measure is the recoil energy of the target atoms, remember that our detection willhave the threshold energy E(R), all the recoiling event of recoiling energy under this threshold isinvisible.

• Our lab frame is moving together with the earth in the galaxy rest frame. The velocity distribution ofdark matter particle is different according to halo models. Typically we will choose this one: that thevelocity distribution is a Maxwellian distribution in the galaxy frame, and the density of dark matteris ρh = 0.3GeV/cm3.

• The differential cross section of this collision of DM and target comes from different DM model!

Here is a summary of present limits of direct detection. Two different signatures are proposed. One is

6.4. Detection of DM 75

Figure 6.4: Present limits of direct detections: Nowadays CDMS and XENON give the strongestbounds.

the annual modulation which is confirmed by DAMA ,as the earth goes around the sun. The second signa-ture is Diurnal modulation.

Some notes about the direct detection :As we mentioned before , the interaction rate will have a annual modulation.

R(vE) = R(v⊙) +

(∂R

∂vE

)v⊙

∆vE cosω(t− t0) (6.15)

The differential interaction rate is given by

dR

dER= NTnA′

∫dσ

dER

fA′(v, vE)

k|v|d3v (6.16)

For mirror type dark matter, the collision is like a Rutherfold scattering, of which the differential cross-section is :

dσ

dER=

λ

E2R|v|2

(6.17)

where λ = 2πϵ2Z2Z′2α2

mAF 2A(qrA)F

2A′(qrA′), F is the form factor. So we get the differential interaction

rate for mirror dark matter direct detection.dR

dER= NTnA′

λ

E2R

∫ ∞

vmin(ER)

fA′(v, vE)

k|v| d3v

≡ NTnA′λ

E2R

I (6.18)

I can be calculated as the following way:

I =

∫ ∞

vmin(ER)

fA′(v, vE)

k|v| d3v

=2π

π3/2v30[A′]

∫ ∞

vmin(ER)

dv exp [−v2 + v2Ev20[A′]

]v

∫ π

0

exp (−2vEv cos θ

v20[A′]

) sin θdθ

=2

π1/2v30[A′]

∫ ∞

vmin(ER)

dv exp [−v2 + v2Ev20[A′]

]v20[A′]

2vE

[exp (

2vEv

v20[A′]

)− exp (−2vEvv20[A′]

)

]

=1

2yv0[A′][erf(x+ y)− erf(x− y)]


where the definition of x and y are x ≡ vmin(ER)v0[A′]

y ≡ vEv0[A′]

6.4.2 Indirect detection

Figure 6.5: Indirect detection: Indirect detections in different places. Most of them are satelliteor balloon experiments.

6.4.3 Accelerator detection

There is no strong constraint given by accelerator experiment today. We hope to find something on theLHC.

Figure 6.6: Accelerator detection: Dark Matter Particle Spectroscopy at the LHC: GeneralizingMT2 to Asymmetric Event Topologies

BIBLIOGRAPHY 77

Bibliography

[1] D. Clowe, M. Bradac, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones and D. Zaritsky,Astrophys. J. 648, L109 (2006) [arXiv:astro-ph/0608407].

78 BIBLIOGRAPHY

CHAPTER 7Quantum Geometric Tensor (Fubini-Study

Metric) in Simple Quantum System: AnIntroduction

Ran Cheng

Department of Physics, University of Texas, Austin, TX 78712, USA

(June 6, 2010)

Geometric Quantum Mechanics is a novel and prospecting approach motivated by the belief that our

world is ultimately geometrical. At the heart of that is a quantity called Quantum Geometric Tensor (or

Fubini-Study metric), which is a complex tensor with the real part serving as the Riemannian metric that

measures the ‘quantum distance’, and the imaginary part being the Berry curvature. Following a physical

introduction of the basic formalism, we illustrate its physical significance in both the adiabatic and non-

adiabatic systems.

7.1 Introduction

The most intriguing feature of modern physics is the introduction of geometrical conceptsdescribing fundamental principles of the nature [1]. One one hand, the gravity emerges as thelocal space-time symmetry, where comparison between nearby local frames naturally gives riseto the concept of Christofle connection; On the other hand, electroweak and strong interactionsare unified by Yang-Mills theory, which identifies the gauge interactions as local symmetries ofinternal degrees of freedom. Again, comparison between nearby frames of the internal spaces(e.g., for SU(2), the three isospin axies) introduces the gauge connection. In electromagnetictheory, this reduces to the Weyl’s principle and the gauge connection is the usual four-potential

79

80 Chapter 7. Quantum Geometric Tensor... by Ran Cheng

Aµ. The nature resumes all the observed interactions by simply obeying space-time and gaugesymmetries.

In early 1980’s, people discovered that gauge fields not only appear in fundamental forcesbetween elemetrary particles, they also emerge in simple quantum systems under certain con-strains [2, 3, 4, 5]. For example, when a spin-1/2 electron adiabatically follows a smoothlyvarying magnetization texture, an effective gauge field now known as Berry curvature affects themotion of the electron. Following the same line, people further found that this interesting gaugestructure is just the Holonomy effect on the phase bundle of the wave function [6]. Specifically,when we identify states differe only by a local phase factor (since physical observables are blindto the phase), the Hilbert spaceH reduces to the Projected Hilbert space PH and quantum statesbeomce ‘Rays’. On this particular space, people were able to construct a geometric reformula-tion of the usual Schrodinger quantum mechanics [6, 7, 8], where covariant derivative is enabledby the emergent gauge potential. Studies along this path is often named geometric quantummechanics [9].

What is more significant in geometric quantum mechanics is the emergent metric structure inaddition to the gauge structure mentioned above. Historically, the discovery of this metric struc-ture preceeds the intensive study on the emergent gauge fields in the attempt to define ‘quantumdistance’ (or interval) between different states [10]. A remarkable feature of the quantum dis-tance comes from the fact that quantum states are denoted by complex functions, which rendersthe metric a complex tensor known as the Quantum Geometric Tensor (QGT). Thanks to the Her-mitean property of the inner product in quantum mechanics, the real and imaginary parts of thiscomplex tensor play quite separate roles. The former is symmetric and serves as the Riemanniantensor fullfilling the function of measuring quantum distance, while the latter is antisymmetricand is identified with the emergent gauge field in projected Hilbert space. There is a simple wayto see this in advance: when taking the inner product between two quantum states we need twoinformation, the overlap and the relative phase. The symmetric part of QGT measures the former,and the antisymmetric part gives the flux density of the latter.

If we represent a state |ψ⟩ =∑k Zk|ek⟩ by the complex vector Z := [Z1, Z2, · · · , Zn]

where |ek⟩ is a set of orthonormal basis, the QGT is nothing but the Fubini-Study metric on theCPn manifold [11]. To avoid confusion in terminology, we would prefer QGT in the follow-ing discussions. The QGT has enjoying renewed interests in the study of quantum statisticalmechanics [12], quantum transports in solids [13], quantum phase transitions [14], topologicalinsulators [15], etc.. In the following sections, we are going to give a more physical introductionto the basic formalism of QGT and its implications in both adiabatic and non-adiabatic systems.

7.2 Formalism - A Physicist’s way

7.2. Formalism - A Physicist’s way 81

Let us consider a family of parameter-dependent Hamiltonian [10, 15, 16, 17] H(λ) re-quiring a smooth dependence on a set of parameters λ = (λ1, λ2, ...) ∈M, which consists of thebase manifold of the quantum system. The Hamiltonian acts on the parameterized Hilbert spaceH(λ), the eigen-energies and eigen-states are denoted by En(λ) and |ϕn(λ)⟩ respectively. Thesystem state |ψ(λ)⟩ is a linear combination of |ϕn(λ)⟩ at each point inM.

Upon infinitesimal variation of the parameter dλ, we define the quantum distance:

ds2 = ||ψ(λ+ dλ)− ψ(λ)||2 = ⟨δψ|δψ⟩ = ⟨∂µψ|∂νψ⟩dλµdλν

= (γµν + iσµν)dλµdλν (7.1)

where in the last line we have decomposed the complex tensor ⟨∂µψ|∂νψ⟩ by its real and imagi-nary parts. Since the inner product of any two states is Hermitean, we know that γµν + iσµν =

γνµ − iσνµ, which indicates the symmetric properties of the two tensors:

γµν = γνµ

σµν = −σνµ (7.2)

so that σµνdλµdλν vanishes due to the antisymmetry of σµν and symmetry of dλµdλν , thus thequantum distance reduces to ds2 = ⟨δψ|δψ⟩ = γµνdλ

µdλν .However, a careful look reminds us that γµν thus defined is NOT gauge invariant which

disqualifies this tensor as the appropriate metric on measuring the quantum distance. Specifically,when we take |ψ′(λ)⟩ = expiα(λ) |ψ(λ)⟩ and define ⟨∂µψ′|∂νψ′⟩ = γ′µν + iσ′

µν , a simplecalculation shows that:

γ′µν = γµν − βµ∂να− βν∂µα+ ∂µα∂να

σ′µν = σµν (7.3)

where βµ(λ) = i⟨ψ(λ)|∂µψ(λ)⟩ is the Berry connection [2], which is purely real due to thenormalization of the quantum state ⟨ψ(λ)|ψ(λ)⟩ = 1. It is obvious that upon the above gaugetransformation, the Berry connection changes as β′

µ = βµ+∂µα. This provides us with a solutionto redefine a gauge invariant metric:

gµν(λ) := γµν(λ)− βµ(λ)βν(λ) (7.4)

changes from the second term on the right hand side cancels the changes from the first part undera gauge transformation, so that g′µν(λ) = gµν(λ). We can understand this relation through aphysicist’s way, γµν measures the distance of ‘bare states’ in Hilbert spaceH, while gµν measuresthe distance of ‘Rays’ in Projected Hilbert space PH = H/U(1). But physical observable relateto Hermitean operators acting on ‘Rays’, not the ‘bare states’, required by the principle of gauge


invariance. Therefore, we are safe to discard γµν and focus on the gauge invariant metric gµνin PH in the following discussions. For mathematical unambiguity and simplicity, we furtherdefine the ‘Quantum Geometric Tensor’ (QGT), which is the Fubini-Study metric on quantumRays, as:

Qµν(λ) := ⟨∂µψ(λ)|∂νψ(λ)⟩ − ⟨∂µψ(λ)|ψ(λ)⟩⟨ψ(λ)|∂νψ(λ)⟩ (7.5)

the previously defined quantities then relate to QGT by,

gµν = Re Qµν ; σµν = Im Qµν (7.6)

To see more explicitly that the gauge invariant tensor gµν indeed plays the role of the metric,we now turn to a different approach which comes from the perturbation theory. Take the innerproduct of the state |ψ(λ)⟩ with |ψ(λ+ dλ)⟩ up to second order in dλ so that,

⟨ψ(λ)|ψ(λ+ dλ)⟩ = 1 + iβµ(λ)dλµ +

1

2⟨ψ(λ)|∂µ∂νψ(λ)⟩dλµdλν (7.7)

Since ⟨ψ|∂µψ⟩ ∈ ℑ (pure imaginary), we know that ⟨∂µψ|∂νψ⟩ + ⟨ψ|∂µ∂νψ⟩ ∈ ℑ so thatRe⟨ψ|∂µ∂νψ⟩ = −Re⟨∂µψ|∂νψ⟩ = −γµν , thus we obtain from Eq. (7.7) the gauge invariantresult:

|⟨ψ(λ)|ψ(λ+ dλ)⟩| = 1− 1

2(γµν(λ)− βµ(λ)βν(λ))dλµdλν

= 1− 1

2gµν(λ)dλ

µdλν (7.8)

for two quantum states labeled by λI and λF , the quantum distance between them is thereforeexpressed as the integration over the metric:

|⟨ψ(λF )|ψ(λI)⟩| = 1− 1

2

∫ λF

λI

gµν(λ)dλµdλν (7.9)

the last term in this equation is the length of the geodesic curve marked by the metric gµν andwe call it ‘geodesic quantum distance’. It is worthy of noticing that the inner product of any twostates should within the range of [0, 1], by which we regard the QGT as a metric measuring thegeodesic distance of points lying on the Bloch sphere. Specifically, if we define |⟨ψ|χ⟩| = cos2 θ2 ,then dθ = 2ds = 2

√|gµνdλµdλν |.

7.3 Case One – Adiabatic System

If a quantum system is confined on a single energy level, where transitions to other levels arenegligible due to large energy gaps separating this particular level, the QGT can be defined with

7.3. Case One – Adiabatic System 83

the instantaneous eigenstate of that level. It losses no generality to consider the ground state ofthe system as an example where the energy is denoted by E0, and the corresponding eigenstateis labeled by |ϕ0(λ)⟩. We assume a sufficiently large energy gap ‘protecting’ the ground statethus transitions to any of the excited states are ignored. In condensed matter physics, peopleoften interpret such a situation as the absence of Goldstone modes and identify the elementaryexcitations as ‘massive’. This is a very common case in the study of Fractional Quantum Hallliquid, High Tc superconductivity, and the recent progress on quantum phase transitions [14].However, it is still far from clear whether there exists an intrinsic relationship between the realand imaginary parts of the QGT near the critical point of the transition [17].

The instantaneous ground eigenstate of the system is defined as H(λ)|ϕ0(λ)⟩ = E0(λ)|ϕ0(λ)⟩,the adiabaticity guarantees the confinement of the system on the subspaceHE0(λ) of the Hilbertspace. First we assume the ground state is non-degenerate, take the partial derivative ∂µ on bothsides of the above relation and consider the orthonormal condition ⟨ϕn(λ)|ϕ0(λ)⟩ = δn0, wearrive at the Feynman-Helleman equations:

⟨ϕn|ϕ0⟩ =⟨ϕn|∂µH|ϕ0⟩E0 − En

if n = 0

⟨ϕ0|∂µH|ϕ0⟩ = ∂µE0 (7.10)

Then the QGT can be defined on the ground state as:

Qµν = ⟨∂µϕ0|(1− |ϕ0⟩⟨ϕ0|)|∂νϕ0⟩

=∑n=0

⟨∂µϕ0|ϕn⟩⟨ϕn|∂νϕ0⟩

=∑n=0

⟨ϕ0|∂µH|ϕn⟩⟨ϕn|∂νH|ϕ0⟩(E0 − En)2

(7.11)

its real part gµν = ReQµν is the Rieman metric introduced in the former section which relatesdirectly to the ‘Fidelity Susceptibility’ in the study of quantum phase transition [14], and itsimaginary part σµν = Im Qµν only differs the Berry curvature [2, 3] by a factor of −2 because:

Fµν = ∂[µ,βν] = i⟨∂[µϕ0|∂ν]ϕ0⟩ = i(Qµν −Qνµ) = −2ImQµν = −2σµν (7.12)

Therefore, we finally arrive at the relation:

Qµν = gµν −i

2Fµν (7.13)

It seems that Eq. (7.11) provides no better way to calculate the QGT than the simple algorithmof taking partial derivatives on the eigenstates |ϕn(λ)⟩. However, in a real quantum system, theHamiltonian is usually too complicated to be analytically solved and people have to resort tonumerical solutions of the eigenstates. But the phase relations between two neighboring sets of


solutions |ϕn(λi)⟩ and |ϕn(λi+1)⟩ are completely random in the computer programm, thus itlosses sense to take partial derivative |∂µϕn(λ)⟩ as it will never give a definite value. Eq. (7.11)removes this phase ambiguity by transforming the partial derivative on the eigenstates to theHamiltonian, which is always well defined. Moreover, the form is explicitly gauge invariant andthe term (E0−En)2 in the denominator implies the singular behavior of the QGT near degeneratepoints.

If the ground state is degenerate H(λ)|ϕ0i(λ)⟩ = E0(λ)|ϕ0i(λ)⟩ where i labels the secondquantum number, the QGT becomes a matrix with a non-Abelian transformation property [5, 16,17]. By generalizing the Feyman-Helleman relations Eq. (7.10) to degenerate ground states, weobtaine the modified expression:

[Qµν ]ij =∑

n =0,k(n)

⟨ϕ0i|∂µH|ϕnk⟩⟨ϕnk|∂νH|ϕ0j⟩(E0 − En)2

(7.14)

where k(n) labels the possible degeneracy of the n-th level. But we won’t go into the physicalapplications of the non-Abelian QGT here, ambitious readers are highly recommended to readRef. [16, 17].

A typical example of the adiabatic case is the spin- 12 particle subject to a magnetic field withconstant amplitude and slowly time-varying orientation. When the frequency of the variation ofthe magnetic field is much smaller than the zeeman energy of the spin, we may assume the adia-batic condition that the spin always follows the instantaneous direction of the magnetic field. TheHamiltonian reads H = µσ · B where µ is the Gyromagnetic ratio constant, it is straightforwardto solve the two eigenstates:

|+⟩ =(e−iϕ/2 cos

θ

2, eiϕ/2 sin

θ

2

)T, |−⟩ =

(−e−iϕ/2 sin θ

2, eiϕ/2 cos

θ

2

)T(7.15)

where θ and ϕ are the two spherical angles labeling the orientation of the magnetic field. UsingEq. (7.11), we obtain the Riemannian metric and the Berry curvature (~ = 1 for simplicity):

gµν =

(1 0

0 sin2 θ

), Fµν =

1

2

(0 − sin θ

sin θ 0

)(7.16)

where µ and ν run between θ and ϕ. The Riemannian tensor is just the metric on S2 which isplayed by the Bloch shpere of the spin wave function; the Berry curvature is a symplectic formon S2 and if we transform it to the usual magnetic field Br = 1

2ϵrµν

r2 sin θFµν = 12r2 , it turns out to

be the magnetic field originates from a monopole lying at the origin with magnetic charge 1/2.

7.4 Case Two – non-Adiabatic System

7.4. Case Two – non-Adiabatic System 85

Now, we release the adiabatic condition and turn to the most general case of an arbitraryquantum process governed by an arbitrary Hamiltonian H(t), it may be non-linear, non-periodic,no parametrical dependence on λ, and may in general time-dependent.

This time, we are going to register the evolution of the state by the time t rather than theparameter λ. Expand |ψ(t+ dt)⟩ to second order in dt,

|ψ(t+ dt)⟩ = |ψ(t)⟩+ d

dt|ψ(t)⟩dt+ 1

2

d2

dt2|ψ(t)⟩dt2 + · · · (7.17)

and by the Full Schrodinger equation we know that,

d

dt|ψ(t)⟩ = − i

~H(t)|ψ(t)⟩, d2

dt2|ψ(t)⟩ = − i

~dH(t)

dt|ψ(t)⟩ − 1

~2H(t)2|ψ(t)⟩ (7.18)

Before checking the inner product of neighboring states, we should note an important quantity– the energy uncertainty (or energy fluctuation) defined as (∆E)2 = ⟨ψ|H2|ψ⟩ − ⟨ψ|H|ψ⟩2. Ifwe go back to the adiabatic case, this quantity would be zero where the system has a determinedenergy; but for a general process, it is non-trivial and may also be a function of time ∆E =

∆E(t). In view of all the above relations, some manipulations lead us to the following equation:

|⟨ψ(t)|ψ(t+ dt)⟩| = 1− 1

2

(∆E)2

~2dt2 +O(dt4) (7.19)

Remember we have defined the quantum distance between two arbitrary states by an ‘angle’ θ onthe Bloch sphere as |⟨ψ|χ⟩| = cos2 θ2 , then we immediately obtain from Eq. (7.19) an interestingrelation:

dθ

dt=

2|∆E|~

or θ = 2

∫|∆E|~

dt ∈ [0, π] (7.20)

The term dθdt has an obvious physical interpretation – the ‘quantum velocity’, i.e., the evolution

rate of a quantum state. Eq. (7.20) relates the quantum velocity to the energy uncertainty ofthat system, the larger the fluctuation in energy the faster the quantum evolution. It is the en-ergy fluctuation that drives the quantum evolution of a system. This was a crucial discovery infundamental quantum theory two decades ago now known as the famous ‘Anandan-Aharonovtheorem’ [6].

What is the relationship between the Anandan-Aharonov theorem and the QGT? Let us re-trieve the parameter λ but this time it is not necessarily an adiabatic parameter. We have seen thegeodesic quantum distance dθ = 2

√|gµνdλµdλν | in the previous sections, taking into account

Eq. (7.20) we would obtain,

|∆E| = ~√|gµν λµλν | (dot denotes time derivative) (7.21)

86 BIBLIOGRAPHY

this relation gives us another way to justify how well the adiabaticity is hold: The slower theparameter varies with time, the smaller the energy uncertainty, i.e., the system is maintained in asingle energy level.

Strictly speaking, when a system is far from the adiabatic region, the QGT defined on theoverall state |ψ⟩ is totally different from that defined on a particular eigenstate. Among thephysics community, people perfer to call the former the Fubini-Study metric and the latter simplyQuantum Geometric Tensor. Of course, this is just a matter of terminology, and the two tend to beequivalent at the adiabatic limit. We have not attempted to distinguish them in this literature. Aspecific illustration of this subtle difference is provided in Ref. [17] on a similar spin-1/2 modeldiscussed above but the variation of the magnetic field there is non-adiabatic. The QGT there isthe same as the adiabatic case while the FSM , though has the same form, depends on a differentset of spherical angles.

7.5 Summary

In this article, we introduced the concept of Quantum Geometric Tensor, which is the Fubini-Study metric furnishing the phase bundle of a quantum system. It gives rise to the geodesicquantum distance defined on the Bloch sphere of a quantum state measured by the Riemannianmetric, which consists of the real part of this tensor. In a general quantum evolution, the met-ric form also determines the rate of change of the system thus defining the quantum velocity.Meanwhile, the imaginary part of this tensor which is antisymmetric, plays the role of the Berrycurvature, the integral of which gives the gauge invariant geometric phase of the wave function.

Acknowledgements - The author is grateful for Prof. Qian Niu, X. Li, and Y.-Z. You forhelpful discussions. Special thank is also given to the Seminar of Quantum Field Thoery held byTsinghua University, which invited me to give a talk on this topic on summer 2010.

Bibliography

[1] K. Huang, Quarks, Leptons & Gauge Fields, 2nd Ed., World Scientific Pub. (1992).

[2] M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984);B. Simon, Phys. Rev. Lett. 51, 2167 (1983).

[3] F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984);J. Moody, A. Shapere, and F. Wilczek, ibid. 56, 893 (1986);

BIBLIOGRAPHY 87

R. Jackiw, ibid. 56, 2779 (1986).

[4] Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987);D. Banerjee, Fortschr. Phys. 44, 323-370 (1996).

[5] R. R.Aldinger, A. Bohm, and M. Loewe, Found. Phys. Lett. 4, 217 (1991).

[6] J. Anandan and Y, Aharonov, Phys. Rev. Lett. 65, 1697 (1990);J. Anandan, Found. Phys. 21, 1265 (1991).

[7] D. Minic and C.-H. Tze, Phys. Rev. D 68, 061501 (2003) and the reference therein.

[8] For relativisitc quantum mechanic, the geometrical reformulation is still absent today. Thisis partly because the concept of particles is replaced by the field quanta, and quantum fieldtheory already plays the role of geometrical theory.

[9] D. C. Brody and L. P. Hughston, J. Geom. Phys. 38, 19 (2001);Aalok, Int. J. Theor. Phys. 46, 3216 (2007).

[10] J. P. Provost and G. Vallee, Comm. Math. Phys. 76, 289 (1980).

[11] A. Parks, J. Phys. A: Math. Gen. 39,601 (2006).

[12] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994);

[13] N. Marzari and D. Vanderbilt. Phys. Rev. B 56, 12847 (1997).

[14] L. C. Venuti and P. Zanardi, Phys. Rev. Lett. 99, 095701 (2007).

[15] S. Matsuura and S. Ryu, submitted to Phys. Rev. B (2010).

[16] Y.-Q. Ma et al. Phys. Rev. B 81, 245129 (2010).

[17] R. Cheng and Q. Niu, unpublished, 2010.

88 BIBLIOGRAPHY

CHAPTER 8An Introduction to Linearized Gravity

Zhong-Zhi Xianyu


(June 13, 2010)

General relativity is a highly nonlinear theory, in the sense that it contains an infinite numbers of in-

teraction types. However, when the gravity is not too strong or the energy scale is not too high, general

relativity can be well approximated by a linearized theory. In this case, we can study a number of properties

of gravity without a knowledge of the whole theory of quantum gravity. As we will see, this study is not

only phenomenologically important but also theoretically interesting.

8.1 Relativity at Linear Order

8.1.1 Expanding the Hilbert-Einstein Action around Flat Background

Our starting point is general relativity. This theory can be summarized in the following action:

S =1

16πG

∫d4x√−g(R− 2Λ) + Smatter. (8.1)

The first term is the well-known Hilbert-Einstein action, which is the pure gravitational part ofthe theory. The second term is the action for matter field, whose explicit form will be given inthe following. Note that we have add a cosmological term with cosmological constant Λ in theaction. According to the experimental data at present, Λ is nonzero but extremely small.

As is well known, the curvature scalar R is a complicated combination of the metric gµν .

89

90 Chapter 8. An Introduction to Linearized Gravity by Zhong-Zhi Xianyu

That is:

R = gµνRµν ; (8.2)

Rκν = Rλκλν ; (8.3)

Rλκµν = Γλκµ,ν + ΓλνσΓσκµ − (µ↔ ν); (8.4)

Γλµν = 12 g

λα(gαν,µ + gαµ,ν − gµν,α). (8.5)

Also taking account of the determinant√−g, we find that, at the lowest order (quadratic order),

the Hilbert-Einstein Lagrangian takes the form of, very roughly, (∂gµν)2. This is exactly in theform of a kinetic term. With all higher order terms in mind also, we may say, the Hilbert-Einsteinaction describes the propagation of the metric, together with all of its self-interactions.

To be more precise, we will separate a background value from the whole metric gµν . Weassume the background is flat so that Lorentz symmetry is manifest. That is,

gµν = ηµν + κhµν . (8.6)

Here we insert a coefficient κ before the fluctuation field hµν . It has two implications. Firstly,κ is a parameter of dimension [mass]−1, thus the fluctuation field hµν will be dimension-1.Secondly, since κ is typically an extremely tiny quantity, thus we can use it to expand the theoryperturbatively.

The expansion is conceptually very clear but technically very tedious. We only list the resultof the expansion of Hilbert-Einstein Lagrangian, up to linear order (linear order means quadraticorder in fields here):

Lh = 14

[h∂2h− hµν∂2hµν + 2hµλ∂

µ∂νhνλ − 2h∂µ∂νhµν

]+O(κ). (8.7)

Don’t forget we also have the cosmological terms:

LΛ = − Λ

κh− Λ

4(h2 − 2hµνh

µν) +O(κ). (8.8)

You can derive it yourself to gain more confidence on this result. It’s worthy noting that we arefree to doing integration by parts during the derivation, since we are only concerned with localproperties.

8.1.2 Free Spin-2 Field

The linearized Lagrangian listed above is precisely in the form of the kinetic terms for a spin-2 field hµν . In fact, you can easily convince yourself that the four terms in Lh is the only possibleterms which are correct in dimension and Lorentz covariance. At the same time, the symmetry ofgeneral covariance is also succeeded by this Lagrangian. This fact can be seen as follows. Firstly,we know how the metric transforms under an general coordinate transformations:

g′µν(x′) =

∂xρ

∂x′µ∂xσ

∂x′νgρσ(x). (8.9)

8.1. Relativity at Linear Order 91

Then its easy to find that, when this transformation is an infinitesimal one

x′µ(x) = xµ + ϵµ, (8.10)

the metric transforms as

δgµν =− ϵλ∂λgµν − gλν∂µϵλ − gµλ∂νϵλ, (8.11)

δϕ =− ϵλ∂λϕ. (8.12)

From this we can derive the transformation property of the perturbed metric hµν as

δhµν = − 1κ (∂µϵν + ∂νϵµ)− (ϵλ∂λhµν + hλν∂µϵ

λ + hµλ∂νϵλ). (8.13)

It is convenient to rescale the parameter ϵλ as ϵλ/κ such that it becomes dimensionless. Then thetransformations should be modified to

δhµν =− ∂µϵν − ∂νϵµ − κ(ϵλ∂λhµν + hλν∂µϵλ + hµλ∂νϵ

λ). (8.14)

At linear order, we have simply

δhµν =− ∂µϵν − ∂νϵµ (8.15)

Then it can be easily checked that Lh indeed respects the symmetry generated by this transfor-mation.

Quantization It’s a standard process in field theory to quantize the linearized Lagrangian andfind a propagator the quantized field, namely the graviton. However, this is not an easy job. Infact, it is complicated enough to quantize the spin-one photon, not to mention the spin-2 graviton.

To warm up let’s review the path-integral quantization procedure of the spin-1 photon verybriefly. It is well known that the kinetic term of the photon

L = − 14 FµνF

µν = 12 Aµ(η

µν∂2 − ∂µ∂ν)Aν . (8.16)

is not invertible due to the existence of gauge symmetry. That is to say, two photon fields linkedby a gauge transformation:

Aµ → Aµ + ∂µϵ (8.17)

is physically equivalent. But in the naive path integral:

Z =

∫DA exp[iS], (8.18)

the summation goes over all fields, including an infinite number of physically equivalent fieldconfigurations. In other words, the degrees of freedom are over counted.


To cure this problem, we insert a gauge condition into the path-integral

Z =

∫DAδ(F [A]) exp[iS]. (8.19)

It can be shown that with proper choice of weighting factor, the insertion of the gauge conditionin the path-integral is equivalent to adding such a term into the Lagrangian:

LGF = 12αF

2[A]. (8.20)

When this is done, we can indeed derive a propagator for photon Aµ with the new Lagrangian.For instance, if we choose

F [A] = ∂µAµ, (8.21)

then the propagator for photon in momentum space reads:

Gµν(k) =−i

k2 + iϵ

[ηµν − (1− α) kµkν

k2

]. (8.22)

Now we turn to graviton. The gauge condition in this case can be chosen as

Fµ[h] = ∂νhµν − c∂µh. (8.23)

Here c is an arbitrary constant. When c = 1/2 the gauge is called “harmonic gauge”, whichappears frequently in literature. Since in this gauge, the propagator of graviton finds its simplestform:

Gµναβ(k) =i

k2

[ηµαηνβ + ηµβηνα − ηµνηαβ

− (1− α) kµkαηνβ + kνkαηµβ + kµkβηνα + kνkβηµαk2

]. (8.24)

When α = 1, it reduces to

Gµναβ(k) =i

k2


]. (8.25)

In general c, the propagator becomes more complicated:

Gµναβ(k) =i

k2


− (1− α) kµkαηνβ + kνkαηµβ + kµkβηνα + kνkβηµαk2

− 2c−1c−1

kµkνηαβ + kαkβηµνk2

− 1−4c2−(3−8c+4c2)α(c−1)2

kµkνkαkβk4

]. (8.26)

It is notable that when c = 1, the expression above diverges. This is a signal that the gaugedegrees of freedom are not fixed completely in this case.

8.1. Relativity at Linear Order 93

8.1.3 Recovery of Some Classical Results

Now that we have got an propagator for graviton, we can calculate some scattering processwith graviton exchange. We are especially interested in three cases, as discussed in the following.

Case 1: Newton’s law Let’s rediscover the New’s law of gravitation now by means of quantumfield theory. But to do this, we must know how gravity couples to matter. That is, we should knowsome information about the matter action Smatter.

However, we need not know a lot, a bit of information is enough. Recall the definition of theenergy-momentum tensor:

Tµν = − 2√−g

δSmatter

δgµν, (8.27)

then its easy to see that gravity couples to matter through the energy momentum tensor.On the other hand, we should also know how classical quantities are related to quantities in

quantum field theory. A simply analogy could be made from a study of path integral:

Z = ⟨0|e−iHT |0⟩ = e−iET , (8.28)

where E is can be regarded as classical energy. Also recall Z = eiW , we get, very roughly

W = −ET. (8.29)

Then to calculate the classical energy, it is amount to evaluate the connected-path integral W .With this knowledge in mind, let’s consider two bulks of matter interacting with each other

through gravity. Then, to lowest order, we have

W = − 12

∫d4k

(2π)4GT ∗µν

1 Gµναβ(k)Tαβ2 . (8.30)

In the nonrelativistic case, T 00i = miδ

(3)(x− xi) dominates. Then in momentum space,

T 00i (k) =

∫d4x eik·xmiδ

(3)(x− xi) = me−ik·xi

∫dx0eik

0x0i . (8.31)

W =− 12

∫d4k

(2π)4GT ∗µν

1

i

k2[ηµαηνβ + ηµβηνα − ηµνηαβ

]Tαβ2

≃− 12

∫d4k

(2π)4GT 00

1

i(1 + 1− 1)

k2T 002

=− 12 G

∫dx0dy0eik

0(x0−y0)∫

d4k

(2π)4m1m2e

−ik·(x2−x1)

k2

=− 12 G

∫dx0

∫d3k

(2π)3m1m2e

−ik·(x2−x1)

k2(8.32)


Working out this integral, we get

W = −∫

dx0Gm1m2

4πr, (8.33)

where∫dx0 is simply the time interval T , thus we conclude that

E ∝ m1m2

r(8.34)

8.1.4 Pauli-Fierz Mass Term

8.1.5 DVZ Discontinuity

8.1.6 Vainshtein Bound

rV =[ GMsun

m4G

]1/5(8.35)

8.2 Geometric Effective Potential

8.2.1 Traditional Effective Potential Method

In classical mechanics, the action S[ϕ], as a functional on virous kinds of fields ϕi, lies in thevery kernel of the theory. The true physical configuration (ϕphys)i can be obtained by extremizingthe action S[ϕ]:

δS[ϕ]

δϕi

∣∣∣ϕi=(ϕphys)i

= 0. (8.36)

It is this property that make the action S[ϕ] be of great importance in a classical theory.However, in quantum theory, this advantage of action functional disappears, due to quantum

fluctuation. Because of quantum corrections, the true physical configuration does not coincidewith the classical one. Thus, we are eager to find a quantum counterpart of the action functionalS[ϕ]. The result is called effective action.

Now let’s give a precise definition for the effective action. We begin with the path integralZ[J ],

Z[J ] = ⟨0|0⟩J = eiW [J] =

∫Dϕ exp

[iS[ϕ] + iJiϕi

]. (8.37)

Note that Z[J ] is a functional of the external source Ji. In terms of this functional we can expressthe vacuum expectation value (VEV) of ϕi with the presence of Ji, as

⟨ϕi⟩ ≡⟨0|ϕi|0⟩J⟨0|0⟩J

= − i

Z[J ]

δZ[J ]

δJi=

δW [J ]

δJi. (8.38)

8.2. Geometric Effective Potential 95

Now, if the VEV ⟨ϕi⟩ = ϕi is given as a known quantity, the external source Ji can also betreated as a functional of ϕi, the dependence of Ji on ϕi can be read from the relation

δW [J ]

δJi

∣∣∣Ji=Ji

= ϕi. (8.39)

With this relation in mind, we can define the effective action Γ[ϕ], as the Legendre transformationof W [J ]:

Γ[ϕ] =W [J ]− Jiϕi. (8.40)

Note that Γ[ϕ] is a functional of the VEV ϕi. Since ϕi is treated as given, we will also call itexternal field in the following.

Γ[ϕ] is indeed the quantum counterpart of the classical action S[ϕ], in the sense that it has thefollowing desired property

δΓ[ϕ]

δϕi= −Ji. (8.41)

When all external sources J(x) are turned off, the relation goes back to the classical one.

Now we derive a perturbative expression for effective action. For convenience, we definethe quantity ϕcli, called classical field, to be the solution of classical equations of motion withexternal source Ji present:

δ

δϕiS[ϕ]

∣∣∣ϕi=ϕcli

= −Ji. (8.42)

Then we can Taylor-expand S[ϕi] around ϕcli:

S[ϕcl + ϕ] = S[ϕcl] + ϕiδ

δϕiS[ϕ]

∣∣∣ϕ=ϕcl

+ 12 ϕiϕj

δ2

δϕiδϕjS[ϕ]

∣∣∣ϕ=ϕcl

+ I[ϕcl;ϕ], (8.43)

where I[ϕcl, ϕ] represents terms of higher order in ϕ. If we define the propagator D as:

iD−1ij =

δ2

δϕiδϕjS[ϕ]

∣∣∣ϕ=ϕcl

, (8.44)

and also recall (8.42), then we arrived at:

S[ϕcl + ϕ] = S[ϕcl]− Jiϕi + 12 ϕi

(iD−1

ij

)ϕj + I[ϕcl, ϕ]. (8.45)


Thus:

Z[ϕ] =

∫Dϕ exp

[iS[ϕcl + ϕ] + iJi(ϕcli + ϕi)

]=exp

[iS[ϕcl] + iJiϕcli

] ∫Dϕ exp

(i2ϕi(iD−1

ij

)ϕj + I[ϕcl + ϕ]

)=exp

(iS[ϕcl] + iJiϕcli

)Det−1/2(iD−1

ij )

×∫Dϕ exp

[i2ϕi(iD−1

ij

)ϕj + I[ϕcl + ϕ]

]∫Dϕ exp

[i2ϕi(iD−1

ij

)ϕj]

≡ exp(iS[ϕcl] + iJiϕcli

)Det−1/2(iD−1

ij )Z2[J ], (8.46)

and:W [J ] = S[ϕcl] + Jiϕcli +

i2 log Det (iD−1

ij )− i logZ2[J ]. (8.47)

Then the effective action Γ[ϕ] can be obtained by Legendre transforming W [J ]. To performthis, we define the difference of the classical field ϕcli and the expectation value ϕi to be ϕ1i :

ϕcli = ϕi + ϕ1i . (8.48)

This difference arises due to quantum corrections, thus ϕ1i , as a functional of ϕi, is of order ~.With this in mind, we can represented the effective potential by expanding it in ϕ1i , which isequivalent to expanding in ~.

Γ[ϕ] =W [J ]− Jiϕi= S[ϕcl] + Ji(ϕcli − ϕi) + i

2 log Det (iD−1ij )− i logZ2[J ]

= S[ϕ+ ϕ1] + Jiϕ1i +

i2 log Det (iD−1

ij )− i logZ2[J ]

= S[ϕ] + ϕ1iδS[ϕ]

δϕ

∣∣∣ϕ=ϕ

+ 12 ϕ

1iϕ

1j

δ2S[ϕ]

δϕiδj

∣∣∣ϕ=ϕ

+ i2 log Det (iD−1

ij )− i logZ2[J ] + Jiϕ1i

= S[ϕ] + i2 log Det (iD−1

ij ) +O(~2). (8.49)

8.2.2 Gauge-Dependence Problem of Coleman-Weinberg Model

Coleman-Weinberg model is simply a scalar QED theory with the scalar being massless. Herewe do not want to be so restrictive and allow the scalar to have a mass m. Then, the classicalLagrangian of the model is given by

L = − 14 FµνF

µν + 12 (Dµϕ)

a(Dµϕ)a − 12 m

2ϕaϕa − 14!λ(ϕ

aϕa)2, (8.50)

with the gauge field strength Fµν being

Fµν = ∂µAν − ∂νAµ, (8.51)


and the covariant derivative Dµ defined by:

(Dµϕ)a = ∂µϕ

a − eAµϵabϕb. (8.52)

Where the scalar index a = 1, 2, and ϵab is the antisymmetric tensor with ϵ12 = 1. We notethat the scalar must be complex, or equivalently multi components, in order to couple to theelectromagnetic field.

Coleman-Weinberg model is a U(1) gauge theory. The gauge transformation is given by

Aµ → Aµ + ∂µα, ϕa → ϕa + eαϵabϕb. (8.53)

It is easy to check that the action is really invariant under such a transformation.The traditional effective potential of the Coleman-Weinberg model up to 1-loop level is given

byΓ[ϕ] = S[ϕ] + i

2 log Det [S,ij ]back., (8.54)

With the choice of Lorentz gauge,

LGF = − 12ξ (∂µA

µ)2. (8.55)

the effective potential can be evaluated, the result is

V (ϕ) =1

4!ϕ4 +

1

2

∫d4kE(2π)4

[log(1 +

m2 + 12λϕ

2

k2E

)+ 3 log

(1 +

e2ϕ2

k2E

)+ log

(1 +

m2 + 16λϕ

2

k2E+ξ(m2 + 1

6λϕ2)e2ϕ2

k4E

)]. (8.56)

8.2.3 Geometric Effective Potential

The whole classical mechanics can be described by the action functional. This functional is ascalar under the general spact-time coordinate transformation. A good treatment on this methodcan be found in

Classical action also respects a much large symmetry – it is invariant under fields reparametriza-tion. Thus we may also say, action functional is a scalar in field space. This fact can be repre-sented by

S′[ϕ′] = S[ϕ]. (8.57)

Where ϕ′ = ϕ′(ϕ) is an arbitrary field reparametrization.The field reparametrization invariance of the action merely reflects the equivalence between

descriptions of the same theory by different variables. This is almost trivial in classical theory.However, the issue becomes more subtle in quantum theory. The reason is, a quantum theory

is general obtained by quantizing the corresponding classical theory. Since, in canonical descrip-tion, we need to choose a set of canonical variables and use them to build up a set of commutation


relations. Therefore, even with the same classical theory, different choices of canonical variableswill in general lead to different quantum theory. In other words, the quantized theory loses thefield reparametrization invariance.

Indeed, the traditional effective action Γ[ϕ] defined above does not respect the field reparametriza-tion invariance. We may say it is not a scalar in field space. This fact can be understood bynoticing that

eiΓ[ϕ] = exp(iW [J ]− iJiϕi

)=

∫Dϕ exp

[iS[ϕ]− i

δΓ[ϕ]

δϕi(ϕi − ϕi)

]. (8.58)

In the last line of this equality, there appears the difference of two point in field space, namelyϕi − ϕi. It is just this difference that break the field reparametrization invariance of the effectiveaction, since it does not transform as a vector in field space in general.

In fact, this is not a problem by itself. Since, a quantum theory contains more informationthan its classical counterpart, thus different quantum theory may share the same classical limit.Or equivalently, a single classical theory can lead to different quantum theory.

But, in a theory with gauge symmetry, the gauge transformation can also be regarded asfield reparametrization. Thus the disappearance of field reparametrization invariance in quantumtheory may imply (though not necessarily) the break down of gauge symmetry. As a result, theeffective action Γ[ϕ] will be gauge dependent.

One may argue that this is also not a problem. Since the effective action is not a physicallymeasurable quantity, thus it is allowed to have some dependence on gauge degrees of freedom.What we demand is that the true physical result, such as S matrix element, must be gauge inde-pendent.

Indeed, it has been proved that the S matrix elements derived from the traditional effectiveaction is gauge independent, though the effective action itself is not.

However, one may be also interested in some quantities other than S matrix. For instance, thebeta function. Since the beta function reflects the running behavior of physical coupling constantwith the energy scale, it should also be gauge independent. But, it is doubtful whether the betafunction derived from the knowledge of traditional effective action is gauge independent.

Unfortunately, this is actually the case. As we will illustrate in the following, the traditionaleffective potential method does lead to a gauge dependent beta function.

Hence, it is reasonable to introduce a effective action respecting the field reparametrizationinvariance. This is what we will do in the next section.

From the argument above, it is clear that the non-scalar behavior of the traditional effectiveaction Γ[ϕ] is caused by the non-vector nature of the difference (ϕi−ϕi) in field space in general.∗

∗From now on, we will introduce some geometrical elements on the fields space, thus it’s necessary to distinguish theupper indices and lower indices here and in what follows.


This is easy to understand. Since, the (tangent) vector lying on a space (manifold) is an elementof the tangent space of the original space. Thus, if the space manifold itself is a flat linear space,then we can identify this space with its tangent space at each point. In this case, the different oftwo points in manifold can be naturally identified as a tangent vector. However, once the spacemanifold is curved, we cannot make the identification between the space and its tangent spaceany more. Then the difference of two points fails to be a vector in this case. In other words, it isnot covariant. Thus, one way to solve this problem is to replace the difference of two points by acovariant quantity. This can be conveniently down as follows.

Suppose we already have a metricGij defined in field space. Then we can talk about the con-cept in field space such as connection, the length, the geodesic, etc. In particular, the difference(ϕi − ϕ⋆i) (where ϕ⋆ is an arbitrary point in field space) can be replaced in this way: Firstly, de-fined the length of the geodesic linking these two points be L[ϕ⋆, ϕ], then we can define anotherquantity σ[ϕ⋆, ϕ] as

σ[ϕ⋆, ϕ] =12 L

2[ϕ⋆, ϕ]. (8.59)

Then we may replace (ϕi − ϕ⋆i) by −σi[ϕ⋆, ϕ], with σi[ϕ⋆, ϕ] defined by

σi[ϕ⋆, ϕ] = Gijσ,j [ϕ⋆, ϕ], (8.60)

where the functional derivative acts on the first argument of σ[ϕ⋆, ϕ]. † Then the combinationJiσ

i[ϕ⋆, ϕ] is automatically covariant, if Ji transforms as a vector at point ϕ⋆i and is independentof ϕi.

The theory with gauge symmetry owns a lot of special features that needs a distinct treatment.This is mainly because, in gauge theory, the fields are parameterized by two different types ofvariables: the physical one and the gauge one. The former labels the physical degrees of freedom,while the latter one is merely redundant (thus unphysical) degrees of freedom. Thus we must bevery careful when dealing with the gauge degrees of freedom. In this section, we outline the mainpoints of Vilkovisky-DeWitt method when applied to a gauge theory.

Generally, in a gauge theory, there exists a set of continuous gauge transformations acting onfields ϕi, whose infinitesimal form can be written as

δϕi = ϕiϵ − ϕi = Kiα[ϕ]δϵ

α. (8.61)

Where the transformation is parameterized by infinitesimal quantity ϵα, and Kiα can be viewed

as the generator of the transformation. Here and in what follows we use Greek indices α, β, · · ·denote the gauge degrees of freedom.

Then, the statement that classical action S[ϕ] should be gauge invariance can be representedby

S,i[ϕ]Kiα[ϕ] = 0. (8.62)

†From now on, we use a comma to denote the functional derivative with respect to the field label by the index on theright side of the comma.


Orbit Decomposition of Field Space

As was mentioned above, the gauge degrees of freedom need a distinct treatment. Now let’smake this point more explicit.

Firstly, we say that two point in the field space ϕi1 and ϕi2 are gauge equivalence, if thereexists a gauge transformation that transforms ϕi1 into ϕi2. We denote this relation by ϕi1 ∼ ϕi2.Then, we can define the gauge equivalence class, as

[ϕi] = φi; φi ∼ ϕi. (8.63)

The gauge equivalence class is also called “gauge orbit” in the physical literature. If the wholefield space is denoted by F , then the quotient space F/ ∼ is called the orbit space. Each pointin this quotient space is an orbit, which serves as a unique label for physical configurations.

Note that each orbit itself is also a space spanned by gauge degrees of freedom. We denotethis space by G . Then we may say (F/ ∼) × G is, at least locally, homomorphic to F . Thuswe have locally decompose the whole field space F into the direct sum of a physical orbit spaceF/ ∼ and a gauge space G .

This decomposition may be applied to various kinds of quantities defined on the whole fieldspace. The key element of this process is the projection operator P ij , which is defined such thatit satisfies the following properties:

P ijKjα = 0; (8.64a)

KiαGijP

jk = 0; (8.64b)

P ijPjk = P ik. (8.64c)

If we define the induced metric γαβ on the gauge space G as

γαβ = KiαGijK

jβ , (8.65)

and its inverse γαβ , as

γαβγβγ = δαγ , (8.66)

then it can be shown that the following is a proper choice for P ij , that is, it satisfies all propertiesof P ij :

P ij = δij −Kiαγ

αβKkβgkj . (8.67)

Now, consider an arbitrary infinitesimal displacement δϕi in field space. This displacementconsists of displacement on two direction: one in the gauge space and the other perpendicular togauge space. By definition, the former is just a gauge transformation, which can be written as

δ∥ϕi = Ki

α[ϕ]δϵα. (8.68)


The latter one, which is a displacement in orbit space, can be conveniently expressed with thehelp with P ij :

δ⊥ϕi = P ijδϕ

j . (8.69)

Then the whole displacement has the following decomposed form:

δϕi = δ∥ϕi + δ⊥ϕ

i = P ijδϕi +Ki

α[ϕ]δϵα. (8.70)

Similarly, the metric Gij of the whole field space can also be decomposed similarly. Themetric in the gauge space γαβ has been defined above, while the metric in the orbit spaceG⊥

ij canbe defined by

G⊥ij = P kiP

ljGkl. (8.71)

Then the line element two form in the field space can be written as

ds2 = G⊥ijdϕ

idϕj + γαβdϵαdϵβ . (8.72)

Gauge Condition

Until now, we were dealing with the whole field space. But it is usually necessary to pickout one representative field from each orbit. The rule that determined which field to be picked iscalled gauge condition.

In general, the gauge condition Fα[ϕ] is a functional in the gauge space. It satisfies such aproperty: the equation

Fα[ϕ] = 0 (8.73)

has one and only one solution in each orbit. Locally, this is equivalent to saying that Fα is anormal parametrization of the gauge space. More technically, if we define

Qαβ = Fαi Kiβ , (8.74)

then the statement above is also equivalent to the condition detQαβ = 0. Any gauge conditionsatisfying this constrain is an locally acceptable choice. Globally, however, this constrain is notenough to guarantee uniqueness of the solution. In fact, the non-uniqueness of the solution isreally existed in Yang-Mills theory, which is called Gribov ambiguity in literature. But this is nota problem at present, since we are only concerned with local properties (which is always the casein perturbation theory).

Then, it is quite reasonable to use the gauge condition Fα as a coordinate system in gaugespace. Correspondingly, we may choose another coordinate system ξA in the orbit space. Herewe use the capital Latin indices to denote degrees of freedom in orbit space.

102 BIBLIOGRAPHY

Field Space Connection in Gauge Theories

Now let’s focus on the connection in the field space when gauge degrees of freedom is present.Both the classical action and the geometrical effective action are gauge independent quanti-

ties. As a functional defined in the field space, being gauge independent means it only dependson the coordinates in orbit space, and does not depend on the coordinates in gauge space.

On the other hand, the choice of connection becomes more flexible than in the non-gaugetheory. Since in this case, we does not need the whole metric Gij , but only the metric G⊥

ij inorbit space, to be parallel transported. If we denote the connection coefficients defined in thisway by Γkij , then the parallel transporting of the orbit metric means

DiG⊥jk ≡ G⊥

jk,i − Γlikg⊥jl − ΓlijG

⊥kl = 0. (8.75)

Simple algebraic calculations lead to

ΓlijG⊥lk = 1

2

(G⊥jk,i +G⊥

ik,j −G⊥ij,k

). (8.76)

According to the standard derivation of Christoffel connection, one may want to get Γlij by multi-plying both sides of this equation by the inverse metric. But this does not work here, sinceG⊥

ij , asa projected metric in the orbit space, does not invertible as a tensor in the whole field space. Thisis the hint that the connection Γkij does not take the simple Christoffel form in a gauge theory. Itturns out that the correct connection is given by (see [3] for further explanation)

Γkij = Γkij − γαβKαiKkβ;j − γαβKαjK

kβ;i

+ 12 γ

αγγβδKαiKβj(KlγK

kδ;l +Kk

δKlγ,l) +Kk

αAαij . (8.77)

8.2.4 Applications in Gravitational Theories

See, for instance, Refs. [4]

Bibliography

[1] M. J. G. Veltman, “Quantum Theory of Gravitation”, in Methods in Field Theory (LesHouches Summer School Proceedings), edited by R. Balian and J. Zinn-Justin, World Sci-entific, 1981.

[2] A. Zee, Quantum Field Theory in a nutshell, Princeton University Press, 2003.

[3] L. Parker and D. J. Toms, Quantum Field Theory in Curved Spacetime: Quantized Fieldsand Gravity, Cambridge University Press, 2009.

[4] D. J. Toms, Phys. Rev. Lett. 101, 131301 (2008); Nature, 468, 56 (2010).

CHAPTER 9On Chern-Simons Gauge Theory

Xiao Xiao

School of Physics, Peking University, Beijing 100871

(June 30, 2010)

In the chiral anomaly of the Abelian and non-Abelian gauge theory in even dimensional spacetime, it is

a curious fact that the anomalous breaking of chiral symmetry introduces a new current, whose charge yields

a Chern-Simons term defined on odd dimensional spacetime. This motivates us to probe some interesting

properties of Chern-Simons gauge theory and its relations to chiral anomaly.

9.1 Chiral Anomaly

Remember the spirit of chiral anomaly that the classical chiral symmetry is broken underradiative correction if gauge symmetry is required to be preserved in the quantum level. Cor-respondingly, the conservation of axial current is violated by a term which is quadratic in fieldstrength:

∂µJ5µ =

e2

(4π)2εµνρσFµνFρσ (9.1)

WhereFµν = ∂µAν − ∂νAµ (9.2)

Even more interesting is the fact that the conservation-violating term can be written as the diver-gence of a new current:

∂µJ5µ =

e2

(4π)2εµνρσ(∂µAν − ∂νAµ)(∂ρAσ − ∂σAρ)

= 4e2

(4π)2εµνρσ∂µAν∂ρAσ =

e2

4π2εµνρσ∂µ(Aν∂ρAσ) (9.3)

103

104 Chapter 9. On Chern-Simons Gauge Theory by Xiao Xiao

Thus we can construct a new current which is conserved in the quantum level:

∂µJ′µ = ∂µ(J

5µ − e2

4π2εµνρσAν∂ρAσ) = 0 (9.4)

Where

JµCS = − e2

4π2εµνρσAν∂ρAσ (9.5)

If we further integrate one of the component of the current in the other three dimension of space-time,we will find the emergence of an action defined in 2+1 dimensional spacetime:∫

d3−→x J3CS = − e2

4π2

∫d3−→x ενρσAν∂ρAσ = SCS (9.6)

Noticing the new antisymmetric tensor is defined in the 2+1 dimensional spacetime, we rec-ognize the action is actually Chern-Simons action which is used to describe the quantum Hallsystem in 2 spatial dimensions.

This interesting fact is not limited to Ablian anomaly, in the non-Abelian case:

∂µJ5µ =

e2

4π2εµνρσ(∂µA

aν∂ρA

aσ + gfabcAbµA

cν∂ρA

aσ)

=e2

4π2εµνρσ∂µ(A

aν∂ρA

aσ +

1

3gfabcAbρA

cνA

aσ) (9.7)

Such current can also be defined in chiral anomalies in any gauge theory defined in evendimensional spacetime.

Actually, the induced action is proportional to a topological quantity called ”Chern-Simonssecondary characteristic class” and the term representing anomalous breaking of chiral symmetryis called Pontryagin density.

9.2 Topologically Massive Theory And Higgs Mechanism

Chern-Simons term, reflecting the topological properties of gauge field, can also contribute amass to a gauge field which is not coupled to a broken scalar field, thus may be an independentresource of mass of the gauge field.

The Lagrangian of Maxwell-Chern-simons theory is:

LMCS = − 1

4e2FµνFµν +

κ

2εµνρAµ∂νAρ (9.8)

Its equation of motion is:

∂µFµν +

κe2

2εναβFαβ = 0 (9.9)

9.2. Topologically Massive Theory And Higgs Mechanism 105

In order to see the mass clearly, we introduce the dual form of the field strength tensor:

Fµν = εµνρFρ (9.10)

And the equation is:∂µ∂µ + (κe2)2F ν = 0 (9.11)

And Bianchi identity implies:∂µF

µ = 0 (9.12)

Introducing gauge-fixing term

Lgf = − 1

2ξe2(∂µA

µ)2 (9.13)

One can deduce the propagator of the gauge field:

Dµν = e2(p2gµν − pµpν − iκe2εµνρpρ

p2(p2 − κ2e4)+ ξ

pµpν(p2)2

)(9.14)

The propagator has a non zero pole indicating a non zero mass.Next we look at the effect of Chern-Simons term on the traditional Higgs mechanism, the

whole Lagrangian is:

L = − 1

4e2FµνFµν +

κ

2εµνρAµ∂νAρ + (Dµϕ)

∗Dµϕ− V (ϕ) (9.15)

With a non-zero vacuum scalar field expectation value, then quantized in R(ξ)gauge, the propa-gator is:

Dµν =e2(p2 −m2

H)

(p2 −m2+)(p

2 −m2−)

[gµν −pµpν

p2 − ξm2H

− iκe2 εµνρpρ

p2 −m2H

+ e2ξpµpν(p

2 − κ2e4 −m2H)

(p2 −m2+)(p

2 −m2−)(p

2 − ξm2H)

] (9.16)

With two poles corresponding the total effects of SSB and topological mass,where

m2H = 2e2v2 (9.17)

And

m2± = m2

H +(κe2)2

2± κe2

2

√κ2e4 + 4m2

H (9.18)

In the limit of e → ∞ the theory become a pure Chern-Simons gauge field coupled to scalarfield, and as we have m+ →∞ and m− → 2v2

κ , the propagator tends to:

Dµν =1

p2 − ( 2v2

κ )2[2v2

κgµν −

1

2v2pµpν +

i

κεµνρp

ρ] (9.19)


9.3 Non-Abelian Chern-Simons Theory

More interested is non-Abelian Chern-Simons theory, whose Lagrangian is:

LCS = κεµνρtr(Aµ∂νAρ +2

3AµAνAρ) (9.20)

Under a gauge transformation:Aµ → g(Aµ + ∂µ)g

−1 (9.21)

The Lagrangian changes by a surface term and a bulk term

LCS → LCS − κεµνρ∂µtr(∂νgg−1Aρ)−κ

3εµνρtr(g∂µg

−1g∂µg−1g∂µg

−1) (9.22)

The quantity

w(g) =1

24π2εµνρtr(g∂µg

−1g∂µg−1g∂µg

−1) (9.23)

is winding number and its integration in the spacetime yields an integer, therefore under gaugetransformation the action changes by a quantity which is proportional to an integer.

SCS → SCS − 8π2κN (9.24)

For the invariance of the generating functional which defines the quantum theory, the change ofthe action must be integer times 2π, therefore it is required by gauge invariance of the theorythat:

4πκ = integer (9.25)

In the arguments above we simply ignored the surface term by requiring the integration on theboundary of spacetime vanishes, however, there are actually circumstances in which the require-ment is not fulfilled. In the Abelian case, examine a variation of gauge field:

δ(

∫d3xεµνρAµ∂νAρ) = 2

∫d3xεµνρδAµ∂νAρ +

∫d3xεµνρ∂ν(AµδAρ) (9.26)

When the variation is a infinitesimal gauge transformation

δAµ = ∂µλ (9.27)

The variation is purely a surface term

δ(

∫d3xεµνρAµ∂νAρ) =

∫d3xεµνρ∂µ(λ∂νAρ) (9.28)

In a spacetime D × R in which the space D is a disk and its boundary is a circle, then theintegral of the total divergence yields:

9.4. Induced Chern-Simons Terms 107

∫d3xεµνρ∂µ(λ∂νAρ) =

∫S1×R

λ(∂0Aθ − ∂θA0) =

∫S1×R

λEθ (9.29)

The current is conserved inside the disk, but not on the boundary.

Jµ = κεµνρ∂νAρ (9.30)

we can see a radial current causes the creation and elimination of particles on the boundary.

Jr = κEθ (9.31)

This could be seen as anomaly of a fermion theory defined on 1+1 dimensional spacetime withthe boundary circle as the spatial dimension.

dQ

dt=

n

2πE (9.32)

9.4 Induced Chern-Simons Terms

Another interesting point about 2+1 dimensional gauge theory is that, Chern-Simons termcan be induced by radiative correction and corrected by radiative process a discrete value, whichis quite startling.

First we compute the effective action in a traditional QED model in 2+1 dimensional space-time

Seff [A,m] = Nf log det(i/∂ + /A+m) (9.33)

The action can be computed perturbatively

Seff [A,m] = Nf tr log(i/∂+m)+Nf tr(1

i/∂ +mA)+

Nf2tr(

1

i/∂ +m/A

1

i/∂ +m/A)+· · · (9.34)

The term we are interested is the quadratic one, it is possible for Chern-Simons term to appearonly in this term

S[2]eff [A,m] =

Nf2

∫d3p

(2π)3[Aµ(−p)ΓµνAν(p)] (9.35)

Where

Γµν =

∫d3k

(2π)3tr[γµ

/p+ /k −m(p+ k)2 +m2

γν/k −mk2 +m2

] (9.36)

In 3-dim spacetimetr(γµγνγρ) = −2εµνρ (9.37)

Therefore we can extract the term proportional to Levi-Civita tensor

Γµνodd(p,m) = εµνρpρΠodd(p2,m) (9.38)


Πodd(p2,m) = 2m

∫d3k

(2π)31

[(p+ k)2 +m2][k2 +m2]=

1

2π

m

|p|arcsin(

|p|√p2 + 4m2

)

(9.39)In the p→ 0 and m→∞ limit:

Γµνodd(p,m) ∼ 1

4π

m

|m|εµνρpρ +O(

p2

m2) (9.40)

Inserting this into the effective action we obtain:

SCSeff [A,m] = −iNf2

1

4π

m

|m|

∫d3xεµνρAµ∂νAρ (9.41)

The Chern-Simons term emerges in the loop computation.The term arises from the large mass limit of the massive fermion theory,such limit appears

just in the process of regularization when there are no bare mass term in the Lagrangian, inPauli-Villars regularization:

Sregeff [A,m = 0] = Seff [A,m]− limM→∞

Seff [A,M ] (9.42)

Such a regularization scheme preserves gauge invariance, but parity conservation is violated bythe mass term introduced to regularize the integrals, such violation of parity is manifest in theemergence of Chern-Simons term.

In non-Abelian theory the Chern-Simons effective action

SCSeff [A,m] = −iNf2

1

4π

m

|m|

∫d3xεµνρtr(Aµ∂νAρ +

2

3AµAνAρ) (9.43)

can also be generated.*Coleman and Hill proved that in an Abelian gauge theory, the Chern-Simons term is only

corrected in one-loop order, any higher order correction vanishes.In a pure gauge theory with bare Chern-Simons term, radiative correction can give the coef-

ficient a discrete correction.4πκren = 4πκbare +N (9.44)

In Eucilidean space

LCSYM = −1

2tr(FµνF

µν)− imεµνρtr(Aµ∂νAρ +2

3AµAνAρ) (9.45)

Fµν = ∂µAν − ∂νAµ + e[Aµ, Aν ] (9.46)

The parameter is required to be4πm

e2= integer (9.47)

9.4. Induced Chern-Simons Terms 109

The bare propagator is

∆bareµν (p) =

1

p2 +m2(δµν −

pµpνp2−mεµνρ

pρ

p2) + ξ

pµpν(p2)2

(9.48)

∆−1µν = (∆bare

µν )−1 +Πµν (9.49)

The contribution of self-energy can be decomposed as

Πµν(p) = (δµνp2 − pµpν)Πeven(p2) +mεµνρp

ρΠodd(p2) (9.50)

The renormalized propagator is

∆µν(p) =1

Z(p2)[p2 +m2ren(p

2)](δµν −

pµpνp2−mren(p

2)εµνρpρ

p2) + ξ

pµpν(p2)2

(9.51)

Where the renormalized mass is

mren(p2) =

Zm(p2)

Z(p2)m (9.52)

AndZ(p2) = 1 + Πeven(p

2) (9.53)

Zm(p2) = 1 + Πodd(p2) (9.54)

mren = mren(0) =Zm(0)

Z(0)m (9.55)

The charge is also renormalized

e2ren =e2

Z(0)(Z(0))2(9.56)

After computation we can obtain that

Zm(0) = 1 +7

12πNe2

m(9.57)

Z(0) = 1− 1

6πNe2

m(9.58)

Where N is the dimension of the gauge group

(m

e2)ren = (

m

e2)Zm(0)Z(0)

2= (

m

e2)1 + (

7

12π− 1

3π)N

e2

m = m

e2+N

4π(9.59)

Finally we get the renormalized coupling constant

4πκren = 4πκbare +N (9.60)


In a non-Abelian gauge theory whose gauge group is completely broken by vacuum, the integerrenormalization is broken to a function that is not necessarily an integer.

4πκren = 4πκbare + f(mhiggs

mCS) (9.61)

However, if the non-Abelian gauge group is broken to a non-Abelian subgroup, say SU(3) toSU(2),the integer renormalization is still robust.

9.5 Gravity With Topological Mass

The dynamics of gravity can be altered dramatically when topological mass term is included.(a)The Einstein theory acquires a spin-2, propagating massive degree of freedom when the topo-logical mass term is present(b)The topological term has third time derivative, but the propagation is also causal.(c)The topological mass contribution is the three dimensional analog of Weyl tensor.

The Einstein theory is a symmetric tensor gauge theory, the excitations are decribed by thetransverse traceless part of the spatial components, which has 1

2d(d− 3) degrees of freedom. In3-dim theory, the dynamic is trivial.

Rαβγσ = ∂γΓαβσ − ∂σΓαβγ + ΓαγµΓ

µβσ − ΓασµΓ

µβγ (9.62)

In 3 dimensional spacetime, Cαβγσ vanishes. And the Ricci tensor has all the components ofRiemann tensor.

Rαβγσ = gαγRβσ + gβσRαγ − gασRβγ − gβγRασ (9.63)

WhereRαβ = Rαβ −

1

4gαβR (9.64)

Another way to see this is to establish the equivalence between Riemann tensor and Einsteintensor.

R αβγσ = −εαβµεγσνGνµ (9.65)

WhereGµν = Rµν − 1

2Rgµν (9.66)

The equivalence of the Riemann tensor and Einstein tensor states that in the empty spacewhere there are no matter fields, the spacetime is flat, no gravitational waves and gravitons.

In 3-dim Einstein theory, a non vanishing Weyl tensor can be constructed.

Cµν =1√gεµαβ∇αRνβ (9.67)

9.5. Gravity With Topological Mass 111

Cµν satisfies a Bianchi identity, i.e. it is covariantly conserved, therefore it is a functionalderivative of a geometrical invariant, the invariant is Chern-Simons characteristic, obtained fromHirzebruch-Pontryagin density:

∗RR =1

2εµναβRµνρσR

ρσαβ = ∂µX

µ (9.68)

Integrate X3 we can obtain the characteristic class:

ICS = −1

4

∫X3 = −1

4

∫dxεµνα[Rµνabω

abα +

2

3ωcµbω

aνcω

bαa] (9.69)

In order to express the Pontryagin density of gravity in a total divergence, we apply vielbeindescription of gravity. Where

Rµνab = ∂µωνab + ωcµaωνcb − (µ←→ ν) (9.70)

The total Lagrangian is

I =1

κ2

∫dx√gR+

1

κ2µICS (9.71)

The equation of motion is

Gµν +1

µCµν = 0 (9.72)

Written as a second order equation, we have

(∇α∇α + µ2)Rµν = −gµνRαβRαβ + 3RαµRαν (9.73)

To see the degree of freedom of graviton, in the linearized gravity, we have

ILE = −1

2

∫dxhµνG

µνL (9.74)

Where

GµνL = RµνL −1

2ηµνRL (9.75)

and

RµνL =1

2(−hµν + ∂µ∂αh

αν + ∂ν∂αhαµ − ∂µ∂νh) (9.76)

Linearized mass term is1

µILCS =

1

2µ

∫dxεµαβG

ανL ∂µhβν (9.77)

The functional derivative with respect to gravitational field is

− 1

µ

δILCSδhµν

=1

µCµνL (9.78)


This is consistent with former arguments.And the Lagrangian contains two parts: linearized Ein-stein term and linearized Chern-Simons term.

IL = ILE +1

µILCS (9.79)

We decompose the field into components

hµν = (h00 ≡ N,h0i = N i, hij) (9.80)

It can be decomposed into transver-trace part and other parts

hij = (δij + ∂i∂j)φ− ∂i∂jχ+ (∂iξjT + ∂jξiT ) (9.81)

−→N =

−→N T +∇NL (9.82)

∇ ·−→N T = 0 = ∇ ·

−→ξ T (9.83)

Plugging in the decomposition in the action

IL = −1

2

∫dxφφ+ λφ+ σ2 +

1

µσλ (9.84)

whereλ = ∇2(N + 2

·NL) +

··χ−φ (9.85)

and

σ = εij∂j(NiT +

·ξi

T ) (9.86)

Eliminating the constriant, we get

I = −1

2

∫dxφ(+ µ2)φ (9.87)

which describes a single propagating degree of freedom

9.6 APPENDIX

9.6.1 DERIVATION OF (10)

Plug the definition of dual form into the equation:

εµνρ∂µFρ +

κe2

2εναβεαβσF

σ = εµνρ∂µFρ +

κe2

22δνσF

σ = 0 (9.88)

BIBLIOGRAPHY 113

Contract the equation with a antisymmetric tensor and differential operator and use the equationagain we obtain

−ελσνεµρν∂λ∂µF ρ − (κe2)2Fσ = 0 (9.89)

We know the result of contracting the indexes of the antisymmetric tensors

(gλµgσρ − gλρgσµ)∂λ∂µFρ + (κe2)2Fσ = 0 (9.90)

This is indeed∂µ∂µF

σ − ∂σ(∂µFµ) + (κe2)2Fσ = 0 (9.91)

Because we have Bianchi identity∂µF

µ = 0 (9.92)

Then we get a Klein-Gordon like massive equation

[∂µ∂µ + (κe2)2]Fσ = 0 (9.93)

Bibliography

[1] GV.Dunne, Aspect of Chern-Simons Theory arxiv:hep-th/9902115v1

[2] S.Deser,R.Jackiw,S.Templeton,Topologically Massive Gauge Theories, Annals of Physics281,409-449 (2000)

[3] RD,Pisarski,Sumathi,Rao Topologically Massive Chromodynamics in the PerturbativeRegime, Phys.Rev.D, 32,8(1985)

proceedings of the seminar on modern quantum field theory - … · 2017. 1. 16. · the seminar was...

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