Symmetry and Its Breaking in Quantum Field Theory

293 Pages·2007·72.218 MB·English

by  T. Fujita

SYMMET
D ITS BREAKING
N Q!!ANTUM
IELD TH EOK
PHYSICS RESEARCH AND TECHNOLOGY
SYMMETRY AND ITS BREAKING
IN QUANTUM FIELD THEORY
T
AKEHISA FUJITA
Nova Science Publishers, Inc.
Sew York
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Library of Congress Cataloging-in-Publication Data
Fujita, T.
Symmetry and its breaking in quantum field theol)' I Takehisa Fujita.
p. cm.
Includes index.
ISBN 978-1-60876-106-7 (softcover)
I. Broken symmetry (Physics) 2. Quantum field theory. I. Title.
QCI74.17.S9F852010
539.7'25--dc22
2010033090
Published by Nova Science Publishers, Inc. tNeIV York
Contents
Preface X"vii
I Classical Field Theory of Fermions I
1.1. Non-relativistic Fields. . ..
1.1.1. Schriidinger Equation. . . ... ,
J .1.2. Lagrangian Density for Schrodinger Fields
1.1.3. Lagrange Equation for Schrodinger Fields . 3
1.1.4. Hamiltonian Density for Schr6dinger Fields. s
J .1.5. Hamiltonian for Scluodinger Fields 5
1. J .6. Conservation of Vector Current 6
1.2. Dirac Fields . . . . . . . . . . . . ... 7
1.2.1. Dirac Equation for Free Fermion. 7
1.2.2. Lagrangian Density for Free Dirac Fields 8
1.2.3. Lagrange Equation for Free Dirac Fields. 8
1.2.4. Plane Wave Solutions of Free Dirac Equation 9
1.2.5. Quantization in Box with Periodic Boundary Conditions 10
1.2.6. Hamiltonian Density for Free Dirac Fermion \I
1.2.7. Hamiltonian for Free Dirac Fennion \I
1.2.8. Conservation of Vector Current 12
1.3. Electron and Electromagnetic Fields 12
\.3.1. Lagrangian Density ...... . 12
1.3.2. Gauge lnvariance ....... . 13
1.3.3. Lagrange Equation for Dirac Field . 14
1.3.4. Lagrange Equation for Gauge Field 14
1.3.5. Hamiltonian Density for Fermions with Electromagnetic Field 15
1.3.6. Hamiltonian for Fermions with Electromagnetic Field 15
).4. Self-interacting Fermion Fields.. .. . ........ . 16
1.4.1. Lagrangian and Hamiltonian Densities ofNJL Model . 17
1.4.2. Lagrangian Densiry of Thirring Model . .. .. 17
1.4.3. Hamiltonian Density for Thirring Model .. . 18
1.5. Quarks with Electromagnetic and Chromomagnetic Interactions 18
1.5.1. Lagrangian Density. 18
1.5.2. EDM Interactions .... 19
VI Cont~i5
2 Symmetry and Conservation Law 21
2.1. Introduction to Transformation Pro~. 21
2.2. Lorentz Invariance .... . 22
2.2.1. Lorentz Covariance . . . 23
2.3. Time Reversallnvariance .. . . 24
2.3.1. in Quantum 24
T~invariance ~k:!-:"j";:-'S
2.3.2. T -invariance in Field Tbeof\' 25
2.3.3. T -violating Interaction .. (lmagmary \fa·:; .. Tenn) 25
2.3.4. T and P-violating Interactions ED~1) 25
2.4. Parity Transformation .. . .......... . 26
2.5. Charge Conjugation 27
2.5.1. Charge Conjugation in Maxwell Equation 27
2.5.2. Charge Conjugation in Dirac Field ..... 28
2.5.3. Charge Conjugation in Quantum Chromodynamics 28
2.6. Translational [nvariance .. .......... ....,. 29
2.6.1. Energy Momentum Tensor . .... ....... . 29
2.6.2. Hamiltonian Density from Energy Momentum Tensor 30
2.7. Global Gauge Symmetry ................... . 30
2.8. Chiral Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 31
2.8.1. Expression of Chiral Tr ansfonnation in Two Dimensions . 32
2.8.2. Mass Term . . . .. ............... . . . 32
2.8.3. Chiral Anomaly . .. . ................ . 33
2.8.4. Chiral Symmetry Breaking in Massless Thirring Model. 34
2.9. SU(3) Symmetry .... ... . 34
2.9.1. Dimension of Representation [A,J1] . 35
2.9.2. Useful Reduction Formula ... . . 36
3 Quantization of Fields 37
3.). Quantization of Free Fennion Field ..... 38
3.1.1. Creation and Annihilation Operators 38
3.1.2. Equal Time Quantization ofField .. 39
3.1.3. Quantized Hamiltonian of Free Dirac Field 40
3.1.4. Vacuum of Free Field Theory 40
3.2. Quantization of Thirring Model . .. 41
3.2.1. Vacuum ofThirring Model .. 42
3.3. Quantization of Gauge Fields in QED 43
3.4. Quantization of Schrodinger Field .. 44
3.4.1. Creation and Annihilation Operators. 45
3.4.2. Fermi Gas Model ........ . . . 45
3.5. Quantized Hamiltonian of QED and Eigenstates 46
3.5.1. Quantized Hamiltonian. 46
3.5.2. Eigenvalue Equation 46
3.5.3. Vacuum State In) .. .. 47
Contents V11
4 Goldstone Theorem and Spontaneous Symme[r~ Breaking 49
4.1. Symmetry and Its Breaking in Vacuum . . . . . .. 50
4.1.1. Symmetry in Quantum Many Body Theory 51
4.1.2. Symmetry in Field Theory 51
4.2. Goldstone Theorem ... . .. 52
4.2.1. Conservation ofChiral Charge 53
4.2.2. Symmetry of Vacuum 53
4.2.3. Commutation Relation 53
4.2.4. Momentum Zero State 54
4.2.5. Pole in S-matrix 55
4.3. New Interpretation of Goldstone Theorem 55
4.3.1. Eigenstate of Hamiltonian and Qs 55
4.3.2. Index of Symmetry Breaking .. 56
4.4. Chiral Symmetry in Quantized Thirring Model 57
4.4.1. Lagrangian Density. . . . .. 57
4.4.2. Quantized Hamiltonian . . ..... . 57
4.4.3. Chiral Transfonnation for Operators. 58
Qs .
4.4.4. Unitary Operator with Chiral Charge 58
4.4.5. Symmetric and Symmetry Broken Vacuum 58
4.5. Spontaneous Chiral Symmetry Breaking. 59
4.5.1. Exact Vacuum ofThirring Model 59
4.5.2. Condensate Operator . .. . .. 59
4.6. Symmetry Breaking in Two Dimensions 60
4.6.1. Fennian Field Theory in Two Dimensions. 60
4.6.2. Boson Field Theory in Two Dimensions. 60
4.7. Symmetry Breaking in Boson Fields 60
4.7.1. Double Well Potential .. 60
4.7.2. Change ofField Variables 61
4.7.3. Current Density of Fields .. 62
4.8. Breaking of Local Gauge Symmetry? 62
4.8.1. Higgs Mechanism .. . 62
4.8.2. Gauge Fixing ...... . 63
4.8.3. What Is Physics Behind Higgs Mechanism? . 63
5 Quantum Electrodynamics 65
5.1. General Properties of QED 65
5.1.1. QED Lagrangian Density 66
5.1.2. Local Gauge Invariance 66
5.1.3. Equation of Motion. 67
5.104. Noether Current and Conservation Law 67
5.1.5. Gauge Invariance of Interaction Lagrangian 68
5.1.6. Gauge Fixing ..... . 68
5.1.7. Gauge Choices 68
5.1.8. Gauge Dependence without opJP = 0 70
5.2. S-matrix in QED . . . . . .. 71
c
V1ll
5.2.1. Definition of S·m.::.m..."'{ 71
5.2.2. Fock Space of Free FIe:", 73
5.2.3. Electron-Elewon In!== 74
5.2.4. Feynman Rules for QED 76
5.3. Schwinger Model (Massless QED:I 77
5.3.1. QED with Massless Fermions T\\"O Dt::neruaons 77
In
5.3.2. Gauge Fixing . .. . .. 78
5.3.3. Quantized Hamiltonian of Schwinger \ fodd 78
5.3.4. Bosonization of Schwinger ~lodel 79
5.3.5. Chiral Anomaly. . . . ....... . 80
5.3.6. Regularization of Vacuum Energy ... . 82
5.3.7. Bosonized Hamiltonian of Schwinger Model 82
5.4. Quantized QEDz Hamiltonian in Trivial Vacuum 83
5.4.1. Hamiltonian and Gauge Fixing . ...... . 84
5.4.2. Field Quantization in Anti-particle Representation 84
5.4.3. Dirac Representation ofy-matrices . 84
5.4.4. Quantized Hamiltonian of QED, . 85
5.4.5. Boson Fock States .. 86
5.4.6. Boson Wave Function .. 87
5.4.7. Boson Mass. . ..... 87
5.5. Bogoliubov Transformation in QEDz . 89
5.5.1. Bogoliubov Transformation 89
5.5.2. Boson Mass in Bogoliubov Vacuum 91
5.5.3. Chiral Condensate .. 92
5.6. QED, in Ligh! Cone ..... . 93
5.6.1. Light Cone Quantization 93
6 Quantum Chromodynamics 97
6.1. Properties ofQCD with SU(N,) Colors 97
6.1.1. Lagrangian Density of QCD .. 98
6. 1.2. Infinitesimal Local Gauge Transfonnation . 98
6.1.3. Local Gauge lnvariance .. .. 99
6.1.4. Noethcr Current in QCD ....... .. 100
6.1.5. Conserved Charge of Color Octet Stale .. 100
6.1.6. Gauge Non-invariance of Interaction Lagrangian 101
6.1.7. Equations of Motion ... 101
6.1.8. Hamiltonian Density ofQCD 102
6.1.9. Hamiltonian ofQCD 103
6.2. Hamiltonian ofQCD in Two Dimensions 103
6.2.1. Gauge Fixing .......... . 103
6.2.2. Quantization of Fields .. . .. 104
6.2.3. Quantized Hamiltonian ofQCD, with SU(N,) 105
6.2.4. Bogoliubov Transformed Hamiltonian . lOS
6.2.5. Determination of Bogoliubov Angle 106
6.2.6. Fennion Condensate ..... . 107
Contents IX
6.2.7. Boson Mass. 107
6.2.8. Condensate and Boson Mass in SU(N,) 108
6.3. 't Hooft Model ...... .. . 109
6.3.1. li N, Expansion .. . 109
6.3.2. Examination of 't Hooft Model 110
6.4. Spontaneous Symmetry Breaking in QCD, III
6.5. Explicit Expression of H' ..... 112
7 Thirring Model 115
7.1. Bethe Ansatz Method for Massive Thirring Model 115
7.1.1. Free Fermion System ........ . 116
7.1.2. Bethe Ansatz State in Two Particle System 117
7.1.3. Bethe Ansatz State in N Particle System . 118
7.2. Bethe Ansatz Method for Field Theory ..... 120
7.2.1. Vacuum State of Massive Thirring Model 120
7.2.2. Excited States ..... ... . 120
7.2.3. Lowest Excited State (Boson) 121
7.2.4. Higher Excited States .... 121
7.2.5. Continuum States 121
7.3. Selhe Ansatz Method for Massless Thirring Model 122
7.3.1. Vacuum State of Massless Thirring Model. 123
7.3.2. Symmetric Vacuum State . ...... . 123
7.3.3. True Vacuum (Synunetry Broken) State 124
7.3.4. Ip- Ih State ............ . 125
7.3.5. Momenrum Distribution of Negative Energy States 126
7.4. Bosonization of Thirring Model 126
7.4.1. Massless Thirring Model. 128
7.4.2. Massive Thirring Model . 129
7.4.3. Physics of Zero Mode .. 130
7.5. Massive Thirring vs Sine-Gordon Models 131
7.5.1. Sine-Gordon Field Theory Model 131
7.5.2. Correlation Functions .... . 13 I
7.5.3. Correspondence.. . ... . . 133
7.6. Bogoliubov Method for Thirring Model 133
7.6.1. Massless Thirring Model . . . . 133
7.6.2. Bogoliubov Transformation .. 134
7.6.3. Bogoliubov Transformed Hamiltonian. 134
7.6.4. Eigenvalue Equation for Boson . 135
7.6.5. Solution of Separable Interactions . 136
7.6.6. Boson Spectrum .. ..... 137
7.6.7. Axial Vector Current Conservation. 137
7.6.8. Fermion Condensate .. 138
7.6.9. Massive Thirring Model 138
7.6.10. NJL Model 139
c-'-
x
8 Lattice Field Theory 141
8.1. General Remark on Discreti.za.:::h.-""'C 141
8.1.1. Equal Spacing ... 142
8.1.2. Continuum Limit .. 142
8.2. Bethe Ansatz Method in Hei5e'."... .~... '.!-X..!- 143
8.2.1. Exchange Operator P 144
8.2.2. Heisenberg XXZ for O:c.e ~b:-.... I:. 5:z:e 144
&.2.3. Heisenberg XXZ for Tv.-o ~h:- Su:~ 145
8.2.4. Heisenberg XXZ for m '\1.:!gn\1D S~:~S 147
8.3. Equivalence between Heisenberg XYZ and .\tassiw
Thirring Models . . . . . . ..... 147
8.3.1. Jordan-Wigner Transformation ...... . 148
8.3.2. Continuum Limit . . . . . .. ...... . 149
8.3.3. Heisenberg XXZ and '\1assless Thtrring .\Iodels . 150
8.4. Gauge Fields on Lartice ..... 151
8.4.1. Discretization of Space . 151
8.4.2. Wilson's Action .... . 151
8.4.3. Wilson Loop ..... . 153
8.4.4. Critical Review on Wilson's Results 154
8.4.5. Problems in Wilson's Action 155
8.4.6. Confinement of Quarks ... ... . 157
9 Quantum Gravity 159
9.1. Problems of General Relativity . 159
9.1.1. Field Equation of Gravity 160
9.1.2. Principle of Equivalence 160
9.1.3. General Relativity .... 161
9.2. Lagrangian Density for Gravity .. 162
9.2.1. Lagrangian Density for QED. 162
9.2.2. Lagrangian Density for QED Plus Gravity . 162
9.2.3. Dirac Equation with Gravitational Interactions 163
9.2.4. Total Hamiltonian for QED Plus Gravity 163
9.3. Static-dominance Ansatz for Gravity . . 163
9.4. Quantization of Gravitational Field ...... . 164
9.4.1. No Quantization of Gravitational Field 165
9.4.2. Quantization Procedure .. 165
9.4.3. Graviton .......... . 165
9.5. Interaction of PIlOt on with Gravity .. 166
9.6. Renonnalization Scheme for Gravity. 168
9.6.1. Self·EnergyofGraviton ... 169
9.6.2. Fennion Self· Energy from Gravity. 169
9.6.3. Vertex: Correction from Gravity .. 169
9.6.4. Renonnalization Procedure ..... 170
9.7. Gravitational Interaction of Photon with Matter 170
9.7.1. Photon-Gravity Scattering Process . . . 171
Contents
XI
9.8. Cosmology .......... ... . 171
9.8.1. Cosmic Fireball Formation . . 171
9.8.2. Relics of Preceding Universe. 172
9.8.3. Remarks ..... 172
9.9. Time Shifts of Mercury and Earth Motions. 173
9.9.1. Non-relativistic Gravitational Potential 173
9.9.2. Time Shifts of Mercury, GPS Satellite and Earth 174
9.9.3. Mercury Perihelion Shift . 175
9.9.4. GPS Satellite Advance Shift ........ . 175
9.9.5. Time Shift of Earth Rotation - Leap Second 176
9.9.6. Observahles from General Relativity ... . 176
9.9.7. Prediction from General Relativity .. .. . 176
9.9.8. Summary of Comparisons between Calculations and Data 177
9.9.9. Intuitive Picture of Time Shifts. 178
9.9.10. Leap Second Dating 178
A Introduction to Field Theory 181
A.1. Natural Units .................. . 181
A2. Hermite Conjugate and Complex Conjugate .. . 1 3
A.3. Scalar and Vector Products (Three Dimensions) : 1 4
AA. Scalar Product (Four Dimensions) 184
A.4.1. Metric Tensor . . . . . . . . 185
a
A.S. Four Dimensional Derivatives p .. 185
A.S.l. P and Differential Operator 185
A.S.2. Laplacian and d' Alembertian Operators 185
A6. y-Matrices . ............ . 186
A.6.1. Pauli Matrices . .... . .. 186
A.6.2. Representation ory-matrices . 186
A.6.3. Useful Relations ofy-Matrices 187
A.7. Transformation of State and Operator 187
A.8. Fermion Current . 188
A.9. Trace in Physics . ...... . 189
A.9.1. Definition . ..... . 189
A.9.2. Trace in Quantum Mechanics 189
A.9.3. Trace in SU(N) ..... 189
p ..
A.9.4. Trace ofy-Matrices and 190
A.1 O. Lagrange Equation . . . . . . . . . . 190
A.lO.t. Lagrange Equation in Classical Mechanics 190
A.IO.2. Hamiltonian in Classical Mechanics 191
A.1O.3. Lagrange Equation for Fields. 191
All. Noether Current . . . . . . . 192
A.11.1. Global Gauge Symmetry 192
A.II.2. Chir.1 Symmetry .. 193
A.12. Hamillonian Density ..... . 194
A.12.1. Hamiltonian Density from Energy Momentum Tensor 194