Fundamental Problems in Quantum Field Theory. 9781608057542, 1608057542

Table of contents :
Cover
Title
EUL
Contents
Foreword
Preface
Chapter 01
Chapter 02
Chapter 03
Chapter 04
Chapter 05
Chapter 06
Chapter 07
Appedix
Bibliography
Index

Citation preview

Fundamental Problems in Quantum Field Theory

Authored By

Takehisa Fujita and Naohiro Kanda Department of Physics Faculty of Science and Technology Nihon University Kanda-Surugadai Tokyo 101-0062 Japan

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CONTENTS F oreword

i

P ref ace

ii

CHAPTERS 1

Maxwell and Dirac Equations . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Maxwell Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Vector Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Static Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Free Vector Field and Its Quantization . . . . . . . . . . . . . . . . 1.2.4 Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Field Energy of Photon . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Static Field Energy per Time . . . . . . . . . . . . . . . . . . . . . 1.2.7 Oscillator of Electromagnetic Wave . . . . . . . . . . . . . . . . . 1.3 Dirac Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Free Field Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Quantization of Dirac Fields . . . . . . . . . . . . . . . . . . . . . 1.3.3 Quantization in Box with Periodic Boundary Conditions . . . . . . 1.3.4 Hamiltonian Density for Free Dirac Fermion . . . . . . . . . . . . 1.3.5 Fermion Current and its Conservation Law . . . . . . . . . . . . . 1.3.6 Dirac Equation for Coulomb Potential . . . . . . . . . . . . . . . . 1.3.7 Dirac Equation for Coulomb and Gravity Potential . . . . . . . . .

3 3 4 5 5 6 7 7 8 8 9 10 13 13 14 15 15 17

2

S-Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Time Dependent Perturbation Theory and T-matrix . . . . . . . . . . . . 2.2.1 Non-static Perturbation Expansion . . . . . . . . . . . . . . . . . . 2.2.2 T-matrix in Non-relativistic Potential Scattering . . . . . . . . . . 2.2.3 Born Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Separable Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Interaction Picture and Definition of S-matrix . . . . . . . . . . . . . . . 2.3.1 Interaction Picture . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20 20 21 21 22 24 24 25 26 26

3

2.4 Photon Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Free Wave of Photon . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Feynman Propagator of Photon . . . . . . . . . . . . . . . . . . . 2.4.3 Calculation of h0|T {Aµ (x1 )Aν (x2 )}|0i . . . . . . . . . . . . . . . 2.4.4 Summation of Polarization States . . . . . . . . . . . . . . . . . . 2.4.5 Coulomb Propagator . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Correct Propagator of Photon . . . . . . . . . . . . . . . . . . . . 2.5 Feynman Propagator vs. Correct Propagator . . . . . . . . . . . . . . . . 2.5.1 Scattering of Two Fermions . . . . . . . . . . . . . . . . . . . . . 2.5.2 Loop Diagrams (Fermion Self-energy) . . . . . . . . . . . . . . . .

27 27 28 29 30 30 31 31 31 33

Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Lagrangian Density in QED . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 QED Lagrangian Density . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Local Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Noether Current and Conservation Law . . . . . . . . . . . . . . . 3.2.5 Gauge Invariance of Interaction Lagrangian . . . . . . . . . . . . . 3.2.6 Gauge Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.7 Gauge Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.8 Quantization of Gauge Fields . . . . . . . . . . . . . . . . . . . . 3.3 Renormalization Scheme in QED: Photon . . . . . . . . . . . . . . . . . . 3.3.1 Self-energy of Photon . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Photon Self-energy Contribution . . . . . . . . . . . . . . . . . . . 3.3.3 Gauge Conditions of Πµν (k) . . . . . . . . . . . . . . . . . . . . 3.3.4 Physical Processes Involving Vacuum Polarizations . . . . . . . . 3.3.5 Triangle Diagrams with Two Photons . . . . . . . . . . . . . . . . 3.3.6 Specialty of Photon Propagations . . . . . . . . . . . . . . . . . . 3.4 Renormalization Scheme in QED: Fermions . . . . . . . . . . . . . . . . 3.4.1 Vertex Correction and Fermion Self-energy . . . . . . . . . . . . . 3.4.2 Ward Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Photon-Photon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Feynman Amplitude of Photon-Photon Scattering . . . . . . . . . 3.5.2 Logarithmic Divergence . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Definition of Polarization Vector . . . . . . . . . . . . . . . . . . . 3.5.4 Calculation of Ma at Low Energy . . . . . . . . . . . . . . . . . . 3.5.5 Total Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 36 36 37 37 38 38 39 39 41 42 43 44 46 47 47 48 50 51 52 53 53 54 54 55 58

4

3.5.6 Cross Section of Photon-Photon Scattering . . . . . . . . . . . . . 3.6 Proposal to Measure Photon-Photon Scattering . . . . . . . . . . . . . . . 3.6.1 Possible Experiments of Photon-Photon Elastic Scattering . . . . . 3.6.2 Comparison with e+ + e− → e+ + e− Scattering . . . . . . . . . . 3.6.3 Comparison with γ + γ → e+ + e− Scattering . . . . . . . . . . . 3.6.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Chiral Anomaly: Unphysical Equation . . . . . . . . . . . . . . . . . . . . 3.7.1 π 0 → γ + γ process . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Standard Procedure of Chiral Anomaly Equation . . . . . . . . . . 3.7.3 Z 0 → γ + γ process . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 No Chiral Anomaly in Schwinger Model . . . . . . . . . . . . . . . . . . . 3.8.1 Chiral Charge of Schwinger Vacuum . . . . . . . . . . . . . . . . 3.8.2 Exact Value of Chiral Charge in Schwinger Vacuum . . . . . . . . 3.8.3 Summary of Chiral Anomaly Problem . . . . . . . . . . . . . . . .

58 60 61 62 62 62 63 63 65 66 69 69 70 71

Quantum Chromodynamics and Related Topics . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Properties of QCD with SU (Nc ) Colors . . . . . . . . . . . . . . . . . . . 4.2.1 Lagrangian Density of QCD . . . . . . . . . . . . . . . . . . . . . 4.2.2 Infinitesimal Local Gauge Transformation . . . . . . . . . . . . . . 4.2.3 Local Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Noether Current in QCD . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Conserved Charge of Color Octet State . . . . . . . . . . . . . . . 4.2.6 Gauge Non-invariance of Interaction Lagrangian . . . . . . . . . . 4.2.7 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.8 Hamiltonian Density of QCD . . . . . . . . . . . . . . . . . . . . 4.2.9 Hamiltonian of QCD . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Nuclear Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 One Boson Exchange Potential . . . . . . . . . . . . . . . . . . . . 4.3.2 Two Pion Exchange Process . . . . . . . . . . . . . . . . . . . . . 4.3.3 Double Counting Problem . . . . . . . . . . . . . . . . . . . . . . 4.4 Physical Observables in QCD . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Perturbation Theory or Exact Hamiltonian . . . . . . . . . . . . . 4.4.2 Electric Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Magnetic Moments of Nucleons . . . . . . . . . . . . . . . . . . . 4.4.4 Cross Section Ratio of σe+ e− →hadrons and σe+ e− →µ+ µ− . . . . . .

72 72 73 73 74 74 75 76 76 77 77 78 78 78 79 81 83 83 83 84 85

5

Weak Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Critical Review of Weinberg-Salam Model . . . . . . . . . . . . . . . . . . 5.2.1 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . 5.2.2 Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Gauge Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Gauge Freedom and Number of Independent Equations . . . . . . 5.2.5 Unitary Gauge Fixing . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6 Non-abelian Gauge Field . . . . . . . . . . . . . . . . . . . . . . . 5.2.7 Summary of Higgs Mechanism . . . . . . . . . . . . . . . . . . . 5.3 Theory of Conserved Vector Current . . . . . . . . . . . . . . . . . . . . . 5.3.1 Lagrangian Density of CVC Theory . . . . . . . . . . . . . . . . . 5.3.2 Renormalizability of CVC Theory . . . . . . . . . . . . . . . . . . 5.3.3 Renormalizability of Non-Abelian Gauge Theory . . . . . . . . . . 5.4 Lagrangian Density of Weak Interactions . . . . . . . . . . . . . . . . . . 5.4.1 Massive Vector Field Theory . . . . . . . . . . . . . . . . . . . . . 5.5 Propagator of Massive Vector Boson . . . . . . . . . . . . . . . . . . . . . 5.5.1 Lorentz Conditions of kµ ²µ = 0 . . . . . . . . . . . . . . . . . . 5.5.2 Right Propagator of Massive Vector Boson . . . . . . . . . . . . . 5.5.3 Renormalization Scheme of Massive Vector Fields . . . . . . . . . 5.6 Vertex Corrections by Weak Vector Bosons . . . . . . . . . . . . . . . . . 5.6.1 No Divergences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Electron g − 2 by Z 0 Boson . . . . . . . . . . . . . . . . . . . . . 5.6.3 Muon g − 2 by Z 0 Boson . . . . . . . . . . . . . . . . . . . . . .

86 86 87 88 89 90 91 91 92 92 92 93 93 93 94 94 94 95 96 97 97 97 98 98

6

Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Field Equation of Gravity . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Principle of Equivalence . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Lagrangian Density with Gravitational Interactions . . . . . . . . . . . . 6.2.1 Lagrangian Density for QED and Gravity . . . . . . . . . . . . . . 6.2.2 Dirac Equation with Gravitational Interactions . . . . . . . . . . . 6.2.3 Total Hamiltonian for QED and Gravity . . . . . . . . . . . . . . . 6.2.4 Static-dominance Ansatz for Gravity . . . . . . . . . . . . . . . . . 6.2.5 Quantization of Gravitational Field . . . . . . . . . . . . . . . . . 6.3 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Cosmic Fireball Formation . . . . . . . . . . . . . . . . . . . . . .

99 99 100 100 101 102 102 102 103 103 104 105 105

6.3.2 Relics of Preceding Universe . . . . . . . . . . . . . . . . . . . . . 6.3.3 Mugen-universe . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Shifts of Mercury and Earth Motions . . . . . . . . . . . . . . . . 6.4.1 Non-relativistic Gravitational Potential . . . . . . . . . . . . . . . 6.4.2 Time Shifts of Mercury, GPS Satellite and Earth . . . . . . . . . 6.4.3 Mercury Perihelion Shifts . . . . . . . . . . . . . . . . . . . . . . 6.4.4 GPS Satellite Advance Shifts . . . . . . . . . . . . . . . . . . . . . 6.4.5 Time Shifts of Earth Rotation − Leap Second . . . . . . . . . . . 6.4.6 Observables from General Relativity . . . . . . . . . . . . . . . . . 6.4.7 Prediction from General Relativity . . . . . . . . . . . . . . . . . . 6.4.8 Summary of Comparisons between Calculations and Data . . . . 6.4.9 Intuitive Picture of Time Shifts . . . . . . . . . . . . . . . . . . . . 6.4.10 Leap Second Dating . . . . . . . . . . . . . . . . . . . . . . . . . Time Shifts of Comets . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Additional Potential by Hamilton Equation . . . . . . . . . . 6.6.1 Bound State Case . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Scattering State Case . . . . . . . . . . . . . . . . . . . . . . . . . Photon-Photon Interaction via Gravity . . . . . . . . . . . . . . . . . . .

105 106 107 107 107 109 109 110 110 111 111 112 113 113 115 116 118 119

Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 EDM of Neutron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Neutron EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 CP Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Neutron EDM in One Loop Calculations . . . . . . . . . . . . . . 7.2.4 T-violation and Neutron EDM . . . . . . . . . . . . . . . . . . . . 7.2.5 Neutron EDM in Higher Loop Calculations . . . . . . . . . . . . 7.2.6 Origin of Neutron EDM . . . . . . . . . . . . . . . . . . . . . . . 7.2.7 P-Violating Electromagnetic Vertex for Proton . . . . . . . . . . 7.3 Lamb Shifts in Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Quantization of Coulomb Fields . . . . . . . . . . . . . . . . . . . 7.3.2 Uehling Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Self-energy of Electron . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Mass Renormalization and New Hamiltonian . . . . . . . . . . . 7.3.5 Energy Shifts of 2s 1 State in Hydrogen Atom . . . . . . . . . . . 2 7.3.6 Problems of Bethe’s Treatment . . . . . . . . . . . . . . . . . . . . 7.3.7 Relativistic Treatment of Lamb Shifts . . . . . . . . . . . . . . . . 7.3.8 Higher Order Center of Mass Corrections in Hydrogen Atom . .

121 121 122 123 123 123 124 124 125 125 126 127 127 128 129 129 130 130 131

6.4

6.5 6.6

6.7 7

7.4 Lamb Shifts in Muonic Hydrogen . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Vacuum Polarization and Uehling Potential . . . . . . . . . . . . 7.4.2 Lamb Shifts in Muonic Hydrogen: Theory and Experiment . . . 7.4.3 Center of Mass Effects on Lamb Shifts . . . . . . . . . . . . . . . 7.4.4 Degeneracy of 2s 1 and 2p 1 in FW-Hamiltonian . . . . . . . . . . . 2 2 7.4.5 Higher Order Center of Mass Corrections in Muonic Hydrogen . . 7.4.6 Comparison between Theory and Experiment . . . . . . . . . . . . 7.4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Lamb Shifts in Muonium . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Higher Order Center of Mass Corrections in Muonium . . . . . . . 7.5.2 Disagreement of Bethe’s Calculation with Experiment . . . . . . . 7.6 Further Corrections in QED . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Infra-Red Singularities . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Correction Terms in Coulomb Interaction . . . . . . . . . . . . . .

131 131 132 133 133 134 134 135 135 135 136 137 137 138

Appendix A Regularization A.1 Cutoff Momentum Regularization . . . . . . . . . . . . . . . . . . . . . . A.2 Pauli-Villars Regularization . . . . . . . . . . . . . . . . . . . . . . . . . A.3 ζ−Function Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Dimensional Regularization . . . . . . . . . . . . . . . . . . . . . . . . .

139 139 140 140

Appendix B Gauge Conditions B.1 Vacuum Polarization Tensor . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Vacuum Polarization Tensor for Axial Vector Coupling . . . . . . . . . B.3 Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 Decay of π 0 → 2γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.5 Decay of Vector Boson Z 0 into 2γ . . . . . . . . . . . . . . . . . . . . . B.6 Decay of Scalar Field Φ into 2γ . . . . . . . . . . . . . . . . . . . . . . . B.7 Photon-Photon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . B.8 Gauge Condition and Current Conservation . . . . . . . . . . . . . . . . B.9 Summary of Gauge Conditions . . . . . . . . . . . . . . . . . . . . . . . .

141 142 142 143 144 144 145 145 146

Appendix C Lorentz Conditions C.1 Gauge Field of Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 C.2 Massive Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Appendix D Basic Notations in Field Theory D.1 Natural Units and Constants . . . . . . . . . . . . . . . . . . . . . . . . . 150

D.2 D.3 D.4 D.5

Hermite Conjugate and Complex Conjugate . . . . . . . . . . . . . . . . Scalar and Vector Products (Three Dimensions) . . . . . . . . . . . . . Scalar Product (Four Dimensions) . . . . . . . . . . . . . . . . . . . . . Four Dimensional Derivatives ∂µ . . . . . . . . . . . . . . . . . . . . . D.5.1 pˆµ and Differential Operator . . . . . . . . . . . . . . . . . . . . . D.5.2 Laplacian and d’Alembertian Operators . . . . . . . . . . . . . . D.6 γ-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.6.1 Pauli Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.6.2 Representation of γ-Matrix . . . . . . . . . . . . . . . . . . . . . . D.6.3 Useful Relations of γ-Matrix . . . . . . . . . . . . . . . . . . . . D.7 Transformation of State and Operator . . . . . . . . . . . . . . . . . . . D.8 Fermion Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.9 Trace in Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.9.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.9.2 Trace in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . D.9.3 Trace in SU (N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . D.9.4 Trace of γ-Matrices and p /. . . . . . . . . . . . . . . . . . . . . D.10 Lagrange Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.10.1 Lagrange Equation in Classical Mechanics . . . . . . . . . . . . D.10.2 Lagrange Equation for Fields . . . . . . . . . . . . . . . . . . . . D.11 Noether Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.11.1 Global Gauge Symmetry . . . . . . . . . . . . . . . . . . . . . . . D.11.2 Chiral Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . D.12 Hamiltonian Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.12.1 Hamiltonian Density from Energy Momentum Tensor . . . . . D.12.2 Hamiltonian Density for Free Dirac Fields . . . . . . . . . . . . D.12.3 Role of Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . D.13 Variational Principle in Hamiltonian . . . . . . . . . . . . . . . . . . . D.13.1 Schr¨ odinger Field . . . . . . . . . . . . . . . . . . . . . . . . . . . D.13.2 Dirac Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151 152 153 153 154 154 154 154 155 155 156 156 157 157 157 158 158 159 159 160 161 161 162 162 163 164 164 165 165 166 167 173

i

 

 

Foreword The inherent purpose of a textbook is to teach the reader the basics of a topic, but ideally, also to inspire a learner to seek further knowledge. Memorizing the contents of textbooks, along with analytical thinking, is without argument an important component of higher education. For readers with some knowledge of physics, this book will make the reader think beyond the boundaries of current knowledge, about what physics is and may suggest a different view from what they have learnt from other textbooks, ”What are the differences?” ”Why?” Such serious contemplation is the first step in an intrinsic approach to physics. To discover something truly new, it is important not only to enrich one’s own knowledge but keep an open mind and pursue answers to the Why’s, Why not’s, and discover the How’s. Sometimes such innovative thinkers may appear as if they are slow learners. They think and contemplate for a long time. It is, however, from among such people who engage in deep thought that new ideas in Science are born. Pierre Curie was a poetic physicist. A so-called dropout, refused by schools, he spent his childhood immersed in Nature yet he became a rare physicist who formulated principles on symmetry, piezo-electricity and magnetism - approaching the essence of Nature. Curie also discovered radium and found an application for radioactivity. I believe that this is a book that will challenge the reader, and it is my hope, the reader, in being challenged, will be inspired to seek new answers in physics. Hideaki Koizumi Fellow and Corporate Officer Hitachi, Ltd.

ii

 

 

Preface Quantum field theory has been a central subject in physics research for a long time. This is basically because the fundamental physics law is essentially written in terms of quantum field theory terminology. The Newton equation is the exception in this respect since it is not the field theory equation, but it is the equation for the coordinate of the particle object. The Newton equation can be derived from the Schr¨odinger equation in terms of the Ehrenfest theorem, and thus it cannot be a fundamental equation of motion. Therefore, the general relativity that aimed at generalizing the Newton equation to a relativistic equation is not a fundamental equation of motion, either. In this respect, the physical world is described in terms of field theory terminology, and physically interesting objects must be always the field Ψ which depends on space and time. This presents a physical state of the corresponding object in nature, and the basic equation of motion can determine the behavior of the field. This world is described by fields of photon Aµ , leptons Ψ` and quarks Ψf,c which are all quantized. In addition, there are a gravitational field G and the weak vector bosons W µ , Z µ which are well included into the Lagrangian density. The fields of photon, leptons, quarks and weak vector bosons should be quantized, but the quantization of the gravitational field is not yet clear from the experimental point of view since there is no discovery of the graviton until now. In this sense, it is most likely true that the gravitational field G should not be quantized. The Lagrangian density that governs the equation of motion for all the fields with four interactions (QED, QCD, weak and gravitational interactions) can be uniquely written, and at the present stage, there is no experiment which is in contradiction to theoretical predictions of the above fields. The quantum field theory has an infinite number of freedom once its field is quantized, and therefore the theory cannot be solved exactly. This indicates that we should rely on the perturbation theory when we wish to calculate any physical observables. The evaluation of the perturbation theory is well established in terms of the S-matrix theory which is essentially the same as the non-static perturbation theory in non-relativistic quantum mechanics. In the course of the evaluations of the physical observables, some of the Feynman diagrams contain the infinity in the momentum integral. The treatment of the infinity is developed in terms of the renormalization scheme in QED. The basic strategy is that the infinity in the evaluation of the physical observables should be renormalized into the wave function since its infinity in the physical observables is just the same as that of the self-energy con-

iii

calculated by the renormalization scheme are   tribution. At present, physical observables   consistent with the experiment. However, any physical observables like the vertex corrections should be finite if the theoretical framework is sound, and in this sense, we still believe that they should not have any logarithmic divergences if we can treat them properly with correct propagators. Therefore, it is most probable that the renormalization scheme should meet a major modification in near future. Here, we should notice that science is only to understand nature, in contrast to engineering which may be connected to the invention of human technology. Therefore, science is always faced to a difficulty and, in some sense, to a fear that some of the research areas should fail to keep highest activities after this research area is completely understood. In this respect, the field theory should survive at any time of research in science since it presents the fundamental technique to understand nature whatever one wishes to study. In this textbook, we clarify the fundamental part of basic physics law which can be well understood by now. The most important of all is to understand physics in depth, which is very difficult indeed. To remember the text book knowledge is not as important as one would have thought at the beginning of his physics study. Once one can understand physics in depth, then one can apply the physics law to understanding many interesting phenomena in nature, which should be basically complicated many body problems. In the last chapter, we discuss some problems which are not understood very well at the present stage of the renormalization scheme. Some of the open problems should be solved by experimental observations, and some are solved by modifying the theoretical considerations. The motivation of writing this textbook is initiated by Asma Ahmed who repeatedly pushed one of the authors (TF) who was reluctant to preparing a new textbook which may well displease quite a few physicists with vested rights. As a result, we concentrated on writing this book from intensive discussions and hard works with our collaborators to achieve deeper but simpler understanding of the quantum field theory than ever. We should be grateful to all of our collaborators, in particular, R. Abe, H. Kato, H. Kubo, Y. Munakata, S. Obata, S. Oshima, T. Sakamoto and T. Tsuda for their great contri-i butions to this book.

Takehisa Fujita and Naohiro Kanda Department of Physics Faculty of Science and Technology Nihon University Kanda-Surugadai Tokyo 101-0062 Japan E-mails: [email protected] and [email protected]

Send Orders of Reprints at [email protected] Fundamental Problems in Quantum Field Theory, 2013, 3-

3

CHAPTER 1

Maxwell and Dirac Equations Abstract: This chapter discusses the basic equations in quantum field theory. First, we clarify some important properties of Maxwell equation so that the main part of the electromagnetisms can be easily understood. Then, we present some useful properties of the Dirac equation and its free wave solution. These two equations are the basic ingredients in understanding quantum field theory. We also give the exact energy spectrum of Dirac equation with Coulomb plus gravity potential in hydrogenlike atom Keywords: Maxwell equation, Dirac equation, photon, oscillator of electromagnetic wave, free Dirac equation, energy eigenvalue in Coulomb and gravity.

1.1

Introduction

Science is to study and understand nature, and it is always fascinating even though it is quite difficult. The physics research is intended to clarify the fundamental law of physical world. At the present stage, the dynamics of electrons and nuclei is well described by the Dirac and Maxwell equations. In addition to the electromagnetic interactions, we have now the gravitational and weak interactions which are included into the same Lagrangian density that describes the field equations of the Dirac and Maxwell fields. The Dirac equation now contains the gravitational potential in the mass term, and the weak decay processes can be just calculated in the same manner as the standard treatment of quantum field theory after the field quantization. The success of the Dirac equation is explained in the field theory textbooks, and therefore there is no need to add anything further to the standard description. However, the real examination of the Dirac equation is only done basically for one body problem and free case, and as long as the limited range of applications of the Dirac equation are concerned, it is perfectly successful. This does not mean that the Dirac equation is all correct for everything in nature. This is clear since we cannot solve even two body problems for the Dirac equation in an exact fashion. It should be interesting to note that the full relativistic Takehisa Fujita and Naohiro Kanda All rights reserved - © 2013 Bentham Science Publishers

4 Fundamental Problems in Quantum Field Theory

Fujita and Kanda

treatment of the positronium has its intrinsic difficulty and up to the present stage, there is no solid method to solve the spectrum of the positronium in a correct manner. Nevertheless the property of the matter fields is determined by the Dirac equation if we can luckily solve the many body problems. The dynamics of fermions becomes very complicated since the motion of charged particles can generate electromagnetic fields which, in turn, should affect on the motion of fermions. Here, we intend to clarify the basic physics law as clearly as possible, and the difficulty of the many body nature should be treated in future. The most important of all should be that the fundamental four interactions (electromagnetic, strong, weak and gravitational interactions) can be well described in terms of the Lagrangian density, and therefore all the physical law should be written by the Lagrange equations which are common to four fundamental interactions.

1.2

Maxwell Equation

The most fundamental equation in physics is the Maxwell equation. This equation is discovered by extracting physical law from experiments, and therefore the equation is basically related to describing nature itself. The Maxwell equation is written for the electric field E and magnetic field B as ∇ · E = eρ,

(Gauss law)

(1.1a)

∇ · B = 0,

(No magnetic monopole)

(1.1b)

∇×E+

∂B = 0, ∂t

(Faraday law)

(1.1c)

∂E = ej, (Ampere − Maxwell law) (1.1d) ∂t where ρ and j denote the charge and current densities, respectively, and we explicitly write the charge e. Here, the charge density means the density of fermions which should be later denoted as ρ = ψ † ψ for one fermion state, and therefore it does not include the charge e. This is because the charge e denotes the strength of the electromagnetic interaction with fermions, and the charge of electron, for example, should be considered as a quantum number of electron state, which is −1. Therefore, if there exist n electrons in the small area V , then the charge Q of the area V becomes Q = −en, and the charge is measured in units of e. The behavior of the charge density ρ and the current density j should be understood by solving the equations of motion for fermions. In this sense, it is important to realize that the Maxwell equation cannot tell us anything about the charge and current densities. In reality, the behavior of the charge and current density in the metal is very complicated, and it is mostly impossible to produce and understand the physics of the charge and current ∇×B−

Maxwell and Dirac Equations

Fundamental Problems in Quantum Field Theory 5

density in the metal in a proper manner. This is, of course, related to the fact that many body problems cannot be solved even for the non-relativistic equations of motion. It may be important to note that the Maxwell equation does not contain any ~ even though it is a field theory equation of motion. However, if one considers the energy of photon, then one should introduce the ~ to express the photon energy like ~ω. In this respect, one may say that the free photon is the result of the quantization of the vector field, and the classical field equation which is derived for the vector field in the absence of the matter fields does not prove the existence of photon. It only says that the wave equation for the vector field A indicates that it should behave like a free massless particle. In this sense, the Maxwell equation itself does not know about the quantization of fields, and the basic theoretical reason why one should quantize the fields is one of the most important problems left for readers as a home work. There must be some fundamental principle to understand the field quantization in connection with the electromagnetic field. On the other hand, the quantization of the Dirac field should be originated from the negative energy states which should require the field quantization with the anti-commutation relation for the creation and annihilation operators within the theoretical framework.

1.2.1

Vector Potential

In order to describe the Maxwell equation in a different way, one normally introduces the vector potential (A0 , A) as ∂A , B = ∇ × A. (1.2) ∂t In this case, the Faraday law (∇ × E = − ∂∂tB ) and no magnetic monopole (∇ · B = 0) can be automatically satisfied. In this case, the Maxwell equation can be written in terms of the vector potential (A0 , A) as E = −∇A0 −

∇2 A0 = −eρ, (Poisson equation) (1.3a) µ 2 ¶ ∂ ∂ − ∇2 A + ∇A0 = ej, with ∇ · A = 0. (1.3b) 2 ∂t ∂t In this expression, we take the Coulomb gauge fixing since this is simple and best.

1.2.2

Static Fields

If the field does not depend on time, then the electric field E can be written as E = −∇A0 because ∂∂tA = 0. By making use of the identity equation for the δ−function, ∇2

1 = −4πδ(r − r 0 ) |r − r 0 |

we can obtain the solution for the Poisson equation as Z ρ(r 0 ) 3 0 e d r A0 (r) = 4π |r − r 0 |

(1.4)

(1.5)

6 Fundamental Problems in Quantum Field Theory

Fujita and Kanda

and thus we obtain the electric field by

Z ρ(r 0 )(r − r 0 ) 3 0 e d r. E(r) = 4π |r − r 0 |3 On the other hand, the Ampere law becomes

(1.6)

∇2 A = −ej which can be easily solved, and we can obtain the solution of the above equation as Z j(r 0 ) 3 0 e d r. (1.7) A(r) = 4π |r − r 0 | In this case, the magnetic field B = ∇ × A can be expressed as Z Jdr 0 × (r − r 0 ) e , with Jdr 0 ≡ j(r 0 )d3 r0 B(r) = (1.8) 4π |r − r 0 |3 which is Biot-Savart law.

1.2.3

Free Vector Field and Its Quantization

When there exist neither charge nor current densities, that is, the vacuum state, then the Maxwell equation becomes ¶ µ 2 ∂ 2 − ∇ A(t, r) = 0 (1.9) ∂t2 which is the wave equation. However, it is clear that the vector field is a real field, and therefore there is no free field solution at the present condition for the vector field. More explicitly, the solution of the free field should be an eigenstate of the momentum operator p = −i∇. This means the solution of the vector field with its momentum k should have the following shape 1 1 A(t, r) = √ eik·r−iωt , or √ e−ik·r+iωt V V which are, however, complex functions. Therefore, we should have another condition on the vector field if we wish to have a free field solution, corresponding to a photon state. This is indeed connected to the quantization of the vector field and we write 2 h i XX 1 ˆ √ A(x) = ²k,λ c†k,λ e−iωk t+ik·r + ck,λ eiωk t−ik·r (1.10) k λ=1 2V ωk where ck,λ , c†k,λ denote the creation and annihilation operators, and ωk = |k|. Here, ²k,λ denotes the polarization vector which should satisfy the following condition from the Coulomb gauge fixing k · ²k,λ = 0 (1.11) which is the most reasonable gauge fixing condition, and up to now, it does not give rise to any problems concerning the evaluation of all the physical observables in quantum electrodynamics.

Maxwell and Dirac Equations

Fundamental Problems in Quantum Field Theory 7

Commutation Relations Since the gauge fields are bosons, the quantization procedure must be done in the commutation relations, instead of anti-commutation relations. Therefore, the quantization can be done by requiring that ck,λ , c†k,λ should satisfy the following commutation relations [ck,λ , c†k0 ,λ0 ] = δk,k0 δλ,λ0

(1.12)

and all other commutation relations vanish.

1.2.4

Photon

For this quantized vector field, we can define one-photon state with (k, λ), and it can be written as 1 ˆ Ak,λ (x) = hk, λ|A(x)|0i =√ ²k,λ eik·r−iωk t (1.13) 2V ωk which is indeed the eigenstate of the momentum operator pˆ = −i∇. Here, we see that photon is the result of the field quantization. In this respect, photon cannot survive in the classical field theory of the Maxwell equation even though the wave equation suggests that there must be some wave that can propagate like a free particle. Indeed, eq.(1.9) indicates that there should be a free wave solution. However, the vector potential A itself is a real field, and therefore it cannot behave like a free particle which should be a complex function (eik·r ). In this respect, the existence of photon should be understood only after the vector field A is quantized. After the field quantization, the energy of photon is measured in units of ~, that is, Ephoton = ~ω.

(1.14)

The fact that the Maxwell equation does not contain any ~ may be a good reason why it could not lead us to the concept of the first quantization even though it is indeed a field theory equation.

1.2.5

Field Energy of Photon

The energy of the gauge field can be calculated from the energy momentum tensor T µν of the electromagnetic fields and it becomes Z H0 =

1 T 00 d3 r = 2

Z "µ

∂A ∂t

#

¶2 + (∇ × A)

2

3

d r=

X

k,λ

¶ µ 1 † . ωk ck,λ ck,λ + 2

(1.15) This represents the energy of photons, and it is written in terms of the field quantized expression.

8 Fundamental Problems in Quantum Field Theory

  1.2.6

Fujita and Kanda

Static Field Energy per   Time

The energy increase per second can be written as Z W0 = e j · Ed3 r. This equation can be rewritten by making use of the Maxwell equation as ¶ Z µ Z 1 1 d 2 2 3 |B| + |E| d r − ∇ · Sd3 r W0 = − dt 2 2

(1.16)

(1.17)

where the Poynting vector S is defined as S = E × B.

(1.18)

This first term in this equation corresponds to the normal field energy increase of the static fields E and B. The second term is the energy flow from the Poynting vector, but we should note that the energy should flow into the inner part of the system and should be accumulated into the condenser thorough the Poynting vector. But it never flows out into the air. That means that the emission of photons has nothing to do with the Poynting vector. This is, of course, clear since the emission of photon should be only possible through electrons (fermions) as we see below.

1.2.7

Oscillator of Electromagnetic Wave

Photon can be emitted from the oscillator when the electromagnetic field is oscillating. A question is as to how it can emit photons. Now the electromagnetic interaction HI with electrons can be written as Z HI = −e j · Ad3 r (1.19) and thus we should start from this expression. The interaction energy increase per time can be written as ¸ Z · ∂j dHI ∂A 3 = −e W ≡ ·A+j· d r (1.20) dt ∂t ∂t where we consider the case without A0 term, and thus the electric field can be written as E=− Thus, W becomes

Z

∂A . ∂t

(1.21)

Z ∂j · Ad3 r + e j · Ed3 r. (1.22) ∂t From the above equation, we see that the second term is just W0 , and thus there is no need to discuss it further. Therefore, defining the first term by W1 , we obtain ¾ Z ½ Z ∂ † ∂j e 3 ˆ W1 = −e · Ad r = − (ψ pψ) · Ad3 r (1.23) ∂t m ∂t W = −e

Maxwell and Dirac Equations

Fundamental Problems in Quantum Field Theory

9

  where we take the non-relativistic   current j as j=

1 † ˆ ψ pψ, m

with pˆ = −i∇.

(1.24)

Since the Zeeman Hamiltonian HZ is written as HZ = −

e σ · B0 2m

(1.25)

we can evaluate the current variation with respect to time as · ¸ ∂j 1 ∂ψ † ∂ψ e ˆ + ψ † pˆ ∇B0 (r) = pψ = ∂t m ∂t ∂t 2m2

(1.26)

where we assume that the B0 is in the z−direction B0 = B0 ez . Thus, we find e2 W1 = − 2 2m

Z (∇B0 (r)) · Ad3 r

(1.27)

where we note that A is associated with current electrons while B0 is an external magnetic field. This is the basic mechanism for the production of the electromagnetic waves (low energy photons) through the oscillators. This clearly shows that the electromagnetic wave can be produced only when there are, at least, two coils where one coil should produce the change of the magnetic field which can affect on electrons in another coil. In most of the textbooks in electromagnetism, the description of the photon emission is insufficient, and we should be very careful for the photon emission processes.

1.3

Dirac Equations

The fundamental equation for fermions is the Dirac equation which can describe the energy spectrum of the hydrogen atom to a very high accuracy. The Dirac equation can naturally describe the spin part of the wave function and this is essentially connected to the relativistic wave equations. In addition to the spin degree of freedom, the Dirac equation contains the negative energy states which are quite new to the non-relativistic wave equations. The existence of the negative energy states requires the Pauli principle which enables us to build the vacuum state, and it should be defined as the state in which all the negative energy states are filled. In this case, this vacuum state becomes stable since no particle can be decayed into the vacuum state due to the Pauli principle. It should be noted that the Pauli principle can be derived if we ask the quantization of the Dirac field in terms of the anti-commutation relations. In this respect, the quantization of the Dirac field is essential because of the Pauli principle, and the field quantization is basically necessary within the theoretical framework.

10 Fundamental Problems in Quantum Field Theory

  1.3.1

Fujita and Kanda

Free Field Solutions  

The Dirac equation for free fermion with its mass m is written as µ ¶ ∂ i + i∇ · α − mβ ψ(r, t) = 0 ∂t where ψ has four components

  ψ1 ψ2   ψ= ψ3  . ψ4

(1.28)

(1.29)

α and β denote the Dirac matrices and can be explicitly written in the Dirac representation as µ ¶ µ ¶ 0 σ 1 0 α= , β= σ 0 0 −1 where σ denotes the Pauli matrix. The derivation of the Dirac equation and its application to hydrogen atom can be found in the standard textbooks. One can learn from the procedure of deriving the Dirac equation that the number of components of the electron fields is important, and it is properly obtained in the Dirac equation. That is, among the four components of the field ψ, two degrees of freedom should correspond to the positive and negative energy solutions and another two degrees should correspond to the spin with s = 21 . It is also important to note that the factorization procedure indicates that the four component spinor is the minimum number of fields which can take into account the negative energy degree of freedom in a proper way. Eq.(1.28) can be rewritten in terms of the wave function components by multiplying β from the left hand side (i∂µ γ µ − m)ij ψj = 0

for i = 1, 2, 3, 4

(1.30)

where the repeated indices of j indicate the summation of j = 1, 2, 3, 4. Here, gamma matrices γ µ = (γ0 , γ) ≡ (β, βα) are introduced, and the repeated indices of Greek letters µ indicate the summation of µ = 0, 1, 2, 3. The expression of eq.(1.30) is called covariant since its Lorentz invariance is manifest. It is indeed written in terms of the Lorentz scalars, but, of course there is no deep physical meaning in covariance. Lagrangian Density for Free Dirac Fields The Lagrangian density for free Dirac fermions can be constructed as ¯ µ γ µ − m)ψ L = ψi† [γ0 (i∂µ γ µ − m)]ij ψj = ψ(i∂

(1.31)

Maxwell and Dirac Equations

Fundamental Problems in Quantum Field Theory 11

  where ψ¯ is defined as

 

ψ¯ ≡ ψ † γ0 .

This Lagrangian density is just constructed so as to reproduce the Dirac equation of (1.30) from the Lagrange equation. It should be important to realize that the Lagrangian density of eq.(1.31) is invariant under the Lorentz transformation since it is a Lorentz scalar. This is clear since the Lagrangian density should not depend on the system one chooses. Lagrange Equation for Free Dirac Fields The Lagrange equation for ψi† is given as ∂µ

∂L ∂(∂µ ψi† )

≡

∂ ∂ ∂L ∂L ∂L + = ∂t ∂(∂0 ψi† ) ∂xk ∂( ∂ψi† ) ∂ψi† ∂xk

(1.32)

and one can easily calculate the following equations ∂L ∂ = 0, ∂t ∂(∂0 ψi† ) ∂L ∂ψi†

∂ ∂L =0 ∂xk ∂( ∂ψi† ) ∂xk

= [γ0 (i∂µ γ µ − m)]ij ψj

and thus, this leads to the following equation [γ0 (i∂µ γ µ − m)]ij ψj = 0

(1.33)

which is just the free Dirac equation. Here, it should be noted that the ψi and ψi† are independent functional variables, and the functional derivative with respect to ψi or ψi† gives the same equation of motion. Plane Wave Solutions of Free Dirac Equation The free Dirac equation of eq.(1.33) can be solved exactly, and it has plane wave solutions. A simple way to solve eq.(1.33) can be shown as follows. First, one writes the wave function ψ in the following shape µ ¶ 1 ϕ √ e−iEt+ip·r ψs (r, t) = (1.34) φ V where ϕ and φ are two component spinors µ ¶ n1 ϕ= , n2

φ=

µ ¶ n3 . n4

(1.35)

12 Fundamental Problems in Quantum Field Theory

Fujita and Kanda

  In this case, eq.(1.33) becomes   µ ¶µ ¶ ϕ m−E σ·p =0 φ σ · p −m − E which leads to

E 2 = m2 + p2 .

(1.36) (1.37)

This equation has the following two solutions. Positive Energy Solution (Ep =

p p2 + m2 )

In this case, the wave function becomes 1 (s) ψs(+) (r, t) = √ up e−iEp t+ip·r V s ¶ µ χs Ep + m 1 (s) up = ·p χ , with s = ± 2Ep 2 Ep +m s

(1.38a) (1.38b)

where χs denotes the spin wave function and is written as µ ¶ µ ¶ 1 0 , . χ1 = χ− 1 = 2 2 0 1 p Negative Energy Solution (Ep = − p2 + m2 ) In this case, the wave function becomes 1 (s) ψs(−) (r, t) = √ vp e−iEp t+ip·r V s à ! ·p χs |Ep | + m − |Ep |+m (s) . vp = 2|Ep | χs

(1.39a) (1.39b)

Some Properties of Spinor (s)

(s)

The spinor wave function up and vp are normalized according to (s)† (s)

(1.40a)

(s)† (s)

(1.40b)

up up = 1 vp vp = 1. Further, they satisfy the following equations when the spin is summed over 2 X

(s) (s)

up u ¯p =

s=1 2 X s=1

(s) (s)

vp v¯p =

pµ γ µ + m 2Ep

(1.41a)

pµ γ µ + m . 2Ep

(1.41b)

Maxwell and Dirac Equations

  1.3.2

Fundamental Problems in Quantum Field Theory 13

Quantization of Dirac   Fields

Here, we discuss the quantization of free Dirac fields and write the free Dirac field as ´ X 1 ³ (s) (s) (s) (s) √ an un eipn ·r−iEn t + bn vn eipn ·r+iEn t , (1.42) ψ(r, t) = 3 n,s L (s)

(s)

where un and vn denote the spinor part of the plane wave solutions as given in eqs.(1.38). Here, the basic method to quantize the fields is to require that the annihilation and creation (s0 ) (s0 ) (s) (s) operators an and a† n0 for positive energy states and bn and b† n0 for negative energy states become operators which should satisfy the anti-commutation relations. Anti-commutation Relations The creation and annihilation operators for positive and negative energy states should satisfy the following anti-commutation relations, © (s) † (s0 ) ª © (s) † (s0 ) ª an , a n0 = δs,s0 δn,n0 , bn , b n0 = δs,s0 δn,n0 . (1.43a) All the other cases of the anti-commutations vanish, for examples, © (s) (s0 ) ª © (s) (s0 ) ª © (s) (s0 ) ª an , an0 = 0, bn , bn0 = 0, an , bn0 = 0.

1.3.3

(1.43b)

Quantization in Box with Periodic Boundary Conditions

In field theory, one often puts the theory into the box with its volume V = L3 and requires that the wave function should satisfy the periodic boundary conditions (PBC). This is mainly because the free field solutions are taken as the basis states, and in this case, one can only calculate physical observables if one works in the box. It is clear that the free field can be defined well only if it is confined in the box. Since the wave function ψs (r, t) for a free particle in the box should be proportional to µ ¶ 1 ϕ √ e−iEt+ip·r ψs (r, t) ' φ V the PBC equations become eipx x = eipx (x+L) ,

eipy y = eipy (y+L) ,

eipz z = eipz (z+L) .

Therefore, one obtains the constraints on the momentum pk as px =

2π nx , L

py =

2π ny , L

pz =

2π nz , L

nk = 0, ±1, ±2, · · · .

In this case, the number of states N in the large L limit becomes Z X X L3 d3 p N= =2 3 (2π) n ,n ,n s x

y

z

(1.44)

(1.45)

14 Fundamental Problems in Quantum Field Theory

Fujita and Kanda

  where a factor of two comes from   the spin degree of freedom. At this point, we should make a comment on the validity of the periodic boundary conditions. After we solve the Schr¨odinger equation, we should impose some boundary conditions on the wave function. Normally, one puts the condition that the wave function should vanish at infinity when solving the bound state problems, and this can determine the energy eigenvalues of the Hamiltonian. On the other hand, the plane wave solutions as given in eq.(1.42) cannot satisfy this type of boundary condition that the wave function should be zero at infinity. Nevertheless, we want to confine the waves within the box, and the only possible boundary condition is the periodic boundary conditions. Up to now, there are no physical observables which are in contradiction with this condition of PBC. The important requirement is that any physical observables should not depend on the box length L if it is sufficiently large, which is called the thermodynamic limit.

1.3.4

Hamiltonian Density for Free Dirac Fermion

The Hamiltonian density for free fermion can be constructed from the energy momentum tensor T µν ! Ã X ∂L ∂L ∂ ν ψi† − Lg µν . ∂ ν ψi + (1.46) T µν ≡ ∂(∂µ ψi ) ∂(∂µ ψi† ) i

Hamiltonian Density from Energy Momentum Tensor Now, one defines the Hamiltonian density H as ! Ã X ∂L ∂L ∂ ψ † − L. ∂0 ψi + H ≡ T 00 = † 0 i ∂(∂0 ψi ) ∂(∂ ψ ) 0 i i

(1.47)

Since the Lagrangian density of free fermion is given in eq.(1.31) and is rewritten as L = iψi† ∂0 ψi + ψi† [iγ0 γ · ∇ − mγ0 ]ij ψj .

(1.48)

In this case, the Hamiltonian density becomes H = T 00 = ψ¯i [−iγ · ∇ + m]ij ψj = ψ¯ [−iγ · ∇ + m] ψ.

(1.49)

Hamiltonian for Free Dirac Fermion The Hamiltonian for free fermion fields is obtained by integrating the Hamiltonian density over all space Z Z 3 H = Hd r = ψ¯ [−iγ · ∇ + m] ψd3 r. (1.50) As we discussed in the Schr¨odinger field, the Hamiltonian itself cannot give us many information on the dynamics. One can learn some properties of the system described by the Hamiltonian, but one cannot obtain any dynamical information of the system from the

Maxwell and Dirac Equations

Fundamental Problems in Quantum Field Theory 15

  Hamiltonian. In order to calculate   the dynamics of the system in the classical field theory model, one has to solve the equation of motions which are obtained from the Lagrange equations for fields. When one wishes to consider the quantum effects of the fields or, in other words, creations of particles and anti-particles, then one should quantize the fields. In this case, the Hamiltonian becomes an operator. Therefore, one has to prepare the Fock states on which the Hamiltonian can operate. Most of the difficulties of the field theory models should be to find the correct vacuum of the interacting system. In four dimensional field theory models, only the free field theory can be solved exactly, and therefore we are all based on the perturbation theory to obtain physical observables.

1.3.5

Fermion Current and its Conservation Law

Dirac equation has a very important equation of current conservation. This is, in fact, related to the global gauge symmetry which should be always satisfied in Dirac as well as Schr¨odinger equations. If the Lagrangian density should have the following shape L = F (ψ † ψ) then it is invariant under the global gauge transformation of ψ 0 = eiα ψ. In this case, if one defines the Noether current j µ as ¸ · ∂L ∂L † ψ ψ− j µ ≡ −i ∂(∂µ ψ) ∂(∂µ ψ † )

(1.51)

then one has the conservation of current ∂µ j µ = 0.

(1.52)

For Dirac fields, one can obtain as a conserved current ¯ µψ j µ = ψγ while the conserved current j µ = (ρ, j) for the Schr¨odinger field is written as ´ 1 ³ † ψ ∇ψ − (∇ψ † )ψ ρ = ψ † ψ, j= 2im

1.3.6

(1.53)

(1.54)

Dirac Equation for Coulomb Potential

For a hydrogen-like atomic system, one can write the Dirac equation as ¶ µ ∂ Ze2 ψ(r, t) = 0 i + i∇ · α − mβ + ∂t r

(1.55)

16 Fundamental Problems in Quantum Field Theory

  where ψ has four components  

Fujita and Kanda

  ψ1 ψ2   ψ= ψ3  . ψ4

(1.56)

α and β denote the Dirac matrices and can be explicitly written in the Dirac representation as µ ¶ µ ¶ 0 σ 1 0 α= , β= σ 0 0 −1 where σ denotes the Pauli matrix. In this case, one can easily prove that the quantities that can commute with the Dirac Hamiltonian must be J and K as defined below J = L + s,

K = β(2s · L + 1)

where L and s are defined as ˆ L = r × p,

1 s= 2

(1.57)

µ ¶ σ 0 . 0 σ

Therefore, the energy eigenvalue of the Dirac field can be specified by the quantum numbers of J, Jz , K. Energy Eigenvalue with Coulomb in Hydrogen-like Atom The energy eigenvalue of the Dirac equation can be obtained for the hydrogen-like atomic system. The Dirac equation can be written as ¶ µ Ze2 ψ(r, t) = Eψ(r, t) −i∇ · α + me β − (1.58) r where me denotes the electron mass. This can be solved exactly, and the energy eigenvalue is given as  En,j

 = me 1 −

1 2

(Zα)2  q h i ¢ ¡ n2 + 2 n − (j + 21 ) (j + 21 )2 − (Zα)2 − (j + 21 )

(1.59)

1 . The quantum number n runs as where α denotes the fine structure constant with α = 137 n = 1, 2, . . . . The energy En,j can be expanded up to the order α4 as à ! ¡ ¢ 3 n me (Zα)2 me (Zα)4 6 − − + O (Zα) . (1.60) En,j − me = − 1 2n2 2n4 4 j+2

The first term in the energy eigenvalue is the familiar energy spectrum of the hydrogen-like atom in the non-relativistic quantum mechanics.

Maxwell and Dirac Equations

 

Fundamental Problems in Quantum Field Theory 17

It should be noted that this  result is mathematically exact, but the Dirac equation eq.(1.58) itself is simply obtained within one body problem, and it is, of course, an approximation. A question may arise as to how much the reduction of the one body problem can be justified. Namely, the hydrogen atom should be, at least, a two body problem since it involves electron and proton in the hydrogen atom. In fact, the relativistic two body Dirac equation cannot be solved or cannot be reduced to one body problem in a proper manner. The Dirac equation of eq.(1.58) is indeed one body equation, but the mass m should be replaced by the reduced mass, and this is indeed a very artificial procedure. In reality, it may well be even more complicated than the tow body problems, and once the fields are quantized, then the hydrogen atom should become many body problems. This means that one electron state could be mixed up by the two electron-one positron states in the electron wave function after the field quantization. At present, however, there is no reliable calculation with this additional configuration, and therefore we do not know how large these contributions to the energy should be for the hydrogen atom.

1.3.7

Dirac Equation for Coulomb and Gravity Potential

Even when one considers the hydrogen-like atom, there is a gravitational interaction between electron and proton. Here, we write a full Dirac equation in the hydrogen-like atom when the gravitational interaction is included ¶ ¸ · µ Gme Mp Z Ze2 β− Ψ = EΨ (1.61) −i∇ · α + me − r r where Mp and G denote the proton mass and the gravitational constant, respectively. The gravity is too weak to make any influence on the spectrum in the hydrogen-like atom, but theoretically it should be important that all the interactions in the hydrogen-like atom are now included in the Dirac equation. Energy Eigenvalue with Coulomb and Gravity in Hydrogen-like Atom The equation (1.61) can be solved exactly, and we obtain p −Z 2 αc0 + (γ + nr ) (Zα)2 − (Zc0 )2 + (γ + nr )2 E = me (Zα)2 + (γ + nr )2 where

p c0 ≡ GMp me , γ ≡ κ2 − (Zα)2 + (Zc0 )2

and ³ 1´ κ≡∓ j+ 2

for

j = l + 21 j = l − 21 .

When we solve the equation, we see that the allowed region of Z is changed as p 1 + 1 + 2(GMp me )2 . Z < 2α

(1.62)

18 Fundamental Problems in Quantum Field Theory

Fujita and Kanda

  Therefore, the energy eigenvalue  is rewritten as p ¸ · −Z 2 αc0 + (γ + nr ) (Zα)2 − (Zc0 )2 + (γ + nr )2 Enr , j = me (Zα)2 + (γ + nr )2 where

(1.63)

ZαE + Zc0 me nr = −γ + λ p 2 2 λ = me − E ½

and nr =

0, 1, 2, · · · 1, 2, 3, · · ·

κ 0.

nr is related to the principal quantum number n as nr + |κ| = n (n = 1, 2, 3, · · · ). If we expand it in terms of Zc0 (= ZGMp me ), then the energy eigenvalue becomes " Enr , j ' me 1 +

−³

Ã

!2 #− 1 2

Zα

q nr + (j + 21 )2 − (Zα)2

Z 2 α · GMp m2e + O((GMp Zme )2 ). q ´2 nr + (j + 21 )2 − (Zα)2 + (Zα)2

(1.64)

The frst term is just the well known Dirac’s eigenvalue of hydrogen like atom. The second term corresponds to the correction of gravitational effects. As an example, we consider 1s 1 2 state in hydrogen atom (Z = 1), and the correction of gravitational effects becomes Egr. = −

αGMp m2e p = −αGm2e Mp ' −1.2 × 10−38 eV (0 + 1 − α2 )2 + α2

(1.65)

which is too small to observe, but it is finite. In addition, we obtain the eigenfunction [1] #1 " ¾ 3 2 f ±(2λ) 2 (me ± E)Γ(2γ + nr + 1) ³ ´ (2λr)γ−1 e−λr = Γ(2γ + 1) 4m (Zαme +Zc0 E) Zαme +Zc0 E − κ n ! g e

(µ ×

Zαme + Zc0 E −κ λ

λ

¶

λ

r

)

1 F1 (−nr , 2γ + 1; 2λr) ∓ nr1 F1 (1 − nr , 2γ + 1; 2λr)

(1.66)

where 1 F1 (α, β; z) denotes the hypergeometric function. Also, f and g are radial wave functions, and therefore, the total wave function becomes à ! f (r)Ωj,l,m (θ, φ) Ψ(r, θ, φ) = (1.67) 1+l−l0 (−) 2 g(r)Ωj,l0 ,m (θ, φ)

Maxwell and Dirac Equations

Fundamental Problems in Quantum Field Theory 19

  where l=j±

1 2

l0 = 2j − l ´ ³ 0 σ · r Ωj,l,m (θ, φ) Ωj,l0 ,m (θ, φ) = il−l r and



q Ωl+ 1 ,l,m (θ, φ) = 2

1 j+m m− 2 Yl (θ, φ) 2j  q 1 j−m m+ 2 Y (θ, φ) l 2j

  q 1 j−m+1 m− 2 (θ, φ) − 2j+2 Yl . Ωl− 1 ,l,m (θ, φ) =  q 1 m+ j+m+1 2 2 (θ, φ) 2j+2 Yl Classical Limits As we see in the later chapter, the gravitational force becomes important when we discuss the motion of the planets in the Newton equation. When we make the non-relativistic reduction of the Dirac Hamiltonian, then we find H=

GMp 2 e2 Gme Mp p2 − − + p . 2me r r 2me r

(1.68)

Now, we make the classical limit of the Hamiltonian and obtain a new potential for the Newton equation with an additional gravitational potential 1 e2 Gme Mp + V (r) = − − r r 2me c2

µ

Gme Mp r

¶2 .

(1.69)

If the new potential is applied to the motion of the planets, then this additional gravitational potential turns out to be responsible for the description of the observed advance shifts of the Mercury perihelion, the GPS satellite motion and the earth rotation around the sun.

Conflict of Interest The author(s) confirm that this FKDSWHU content has no conflicts interest. Acknowledgements: Declared none.

Send Orders of Reprints at [email protected] 20

Fundamental Problems in Quantum Field Theory, 2013, 20-34

CHAPTER 2  

 

S-Matrix Theory Abstract:

In this chapter, we discuss the S-matrix theory in quantum field theory. Here, we first treat the non-relativistic scattering theory and its relation to the Tmatrix. In particular, we discuss the scattering problem in terms of the LippmannSchwinger equation. Then we discuss the S-matrix theory in quantum field theory. This is based on the perturbation theory and we present the example of the S-matrix evaluation. In particular, we discuss some basic problems in the Feynman propagator of photon and show a possible physical difference between Feynman and correct propagators of photon. Keywords: S-matrix, T-matrix, scattering amplitude, Lippmann-Schwinger equation, time-dependent perturbation theory, Born approximation, cross section, Feynman and correct propagator of photon.

2.1

Introduction

All of the evaluation in quantum field theory in four dimensions should be based on the perturbation theory. This is simply due to the fact that there is no case in which the quantum field theory models in four dimensions can be solved exactly, except a free field theory. In this sense, we should understand how we can obtain physical information from experiments. In most of the cases, the experimental information can be obtained by scattering processes in which the incident particles collide with targets. In this reaction process, people measure outgoing particles at fixed solid angles. Mostly they extract the differential cross section as the important information on the structure of the targets or on the interaction between incident particles and target matters. Here, we describe the basic theoretical framework in the S-matrix theory. The treatment should be first based on the non-relativistic quantum mechanics, and then we discuss the S-matrix expansion in the quantum field theory. It should be noted here that the difference of the theoretical frameworks between non-relativistic quantum mechanics and relativistic field theory is not very large as will be seen below. In Takehisa Fujita and Naohiro Kanda All rights reserved - © 2013 Bentham Science Publishers

S-Matrix Theory

Fundamental Problems in Quantum Field Theory 21

  particular, the time development  of the system is most important, and this property is just the same between relativistic and non-relativistic wave equations. The only but basic difference is the kinematics in the scattering process, and this is, in general, not very significant.

2.2

Time Dependent Perturbation Theory and T-matrix

In four dimensional field theory models, we should rely on the perturbation theory. This is, of course, due to the fact that there is no model field theory which can be solved exactly, except the free field theory. In this case, the strategy of the theoretical calculation is always based on the perturbation theory in which all the physical observables can be described in terms of the free field terminology, that is, electron and photon in the case of quantum electrodynamics.

2.2.1

Non-static Perturbation Expansion

Now we consider the system in which the total Hamiltonian can be written as the sum of H0 and HI H = H0 + HI (2.1) where it is assumed that all the solutions of the Hamiltonian H0 are known and also HI is relatively small. In this case, the eigenfunctions and eigenvalues of the Hamiltonian H0 can be written as H0 un (r) = En un (r), n = 1, 2, · · · . (2.2) The eigenfunction un (r) should satisfy the following orthogonalty and completeness conditions X |un ihun | = 1. (2.3) hun |um i = δnm , n

The state vector Ψ(t, r) can be expanded in terms of the wave function un (r) as Ψ(t, r) =

X

an (t)e−iEn t un (r).

(2.4)

n

Now we insert this state vector into the Schr¨odinger equation i

∂Ψ(t, r) = (H0 + HI )Ψ(t, r) ∂t

and we obtain ¸ X · dan (t) X + En an (t) e−iEn t un (r) = i (En + HI )an (t)e−iEn t un (r). dt n n

(2.5)

(2.6)

22 Fundamental Problems in Quantum Field Theory

Fujita and Kanda

†

  By operating uk (r) from the left  and integrating over the three dimensional space, we obtain X dak (t) = −i ei(Ek −En )t huk |HI |un ian (t). dt n

(2.7)

This integral equation can be formally solved, and we obtain ak (t) = ak (0) − i

XZ n

0

t

0

dt0 ei(Ek −En )t huk |HI |un ian (t0 )

(2.8)

which is just the equation for the S-matrix evaluation. This is, of course, the same as the relativistic treatment which has the same time dependence as the Schr¨odinger equation.

2.2.2

T-matrix in Non-relativistic Potential Scattering

Here we consider the scattering process in the potential model where the Hamiltonian is given as 1 pˆ2 =− ∇2 . (2.9) H = H0 + V, with H0 = 2m 2m The main aim is to calculate the scattering T-matrix in the potential scattering process. For the scattering problem, it is easier if we start from the Lippmann-Schwinger equation ψ =ϕ+

1 Vψ E − H0 + iε

(2.10)

where +iε is introduced because of the boundary condition that we take out only the outgoing wave. This is very important, and the Lippmann-Schwinger equation has already the proper boundary condition in itself. Here, ϕ denotes the state vector which can satisfy the following free wave equation (E − H0 )ϕ = 0 (2.11)

k2 and we normally take the plane wave solution with the incident energy of E = 2m ϕ(r) = eik·r

(2.12)

where the normalization constant √1V is set to unity. Here, it may be worthwhile noting and making a comment on this normalization constant. Since we set the normalization constant to unity, the wave function does not have a proper dimension, and thus the T-matrix as well. However, this is all right since all the physical observables in scattering processes should be given as the ratio between outgoing flux divided by the incident flux. Now the Lippmann-Schwinger equation can be rewritten more explicitly Z ik·r ψk (r) = e + d3 r0 G(r, r 0 )V (r 0 )ψk (r 0 ) (2.13)

S-Matrix Theory

Fundamental Problems in Quantum Field Theory 23

  where the Green’s function G(r,  r 0 ) is defined as Z m 1 1 d3 p ip·(r−r0 ) 0 0 =− e eik|r−r | . G(r, r ) = 0| p2 k2 (2π)3 2π |r − r 2m − 2m + iε

(2.14)

At large r compared to the potential range, we find G(r, r 0 ) ' −

m ikr −ik0 ·r0 e e 2πr

with

k0 ≡ k rˆ.

(2.15)

Since the scattering amplitude f (k0 , k) is defined by the following equation ψk (r) = eik·r + f (k0 , k)

eikr r

(2.16)

the f (k0 , k) can be given as m f (k , k) = − 2π

Z

0

d3 r0 e−ik ·r V (r 0 )ψk (r 0 ). 0

0

(2.17)

In this case, the probability P of finding particle at the area of r2 dΩ should be P dΩ =

jr r2 dΩ . jin

(2.18)

Since the current density in the non-relativistic case is given in eq.(1.54), we find jin =

k , m

jr =

Therefore, we obtain the differential cross section

k|f (k0 , k)|2 . mr2 dσ dΩ

(2.19)

= P as

dσ = |f (k0 , k)|2 . dΩ

(2.20)

Here, we define the scattering T-matrix by hk0 |T |ki ≡ hk0 |V |ψk i

(2.21a)

which is related to the scattering amplitude as f (k0 , k) = −

m 0 hk |T |ki. 2π

(2.21b)

This T-matrix can, of course, satisfy the Lippmann-Schwinger type equation which can be written as Z 3 00 1 d k 0 0 hk0 |V |k00 ihk00 | |k00 ihk00 |T |ki. (2.22) hk |T |ki = hk |V |ki + 3 (2π) E − H0 + iε It may also be important to note that the scattering amplitude as well as the T-matrix are not yet determined completely, and if one can solve the above equation exactly, then one can determine all the information on the scattering T-matrix.

24 Fundamental Problems in Quantum Field Theory

  2.2.3

Fujita and Kanda

Born Approximation 

In the normal case of the potential scattering, one cannot easily solve the LippmannSchwinger equation, and therefore it is important to make a reasonably good approximation. The best known approximation method is the Born approximation in which we obtain the 0 T-matrix by replacing the exact state vector ψk (r 0 ) by the plane wave eik·r . In this case, the scattering amplitude f (k0 , k) becomes Z m 0 0 0 d3 r0 ei(k−k )·r V (r 0 ). (2.23) fB (k , k) = − 2π Correspondingly, we can obtain the T-matrix with the Born approximation as hk0 |T |kiB = hk0 |V |ki.

(2.24)

Rutherford Scattering If the potential is V (r) = αr (Coulomb scattering), then the scattering amplitude can be easily calculated in the Born approximation, and one finds Z 2mα m 0 0 α 0 d3 r0 ei(k−k )·r 0 = − . (2.25) fB (k , k) = − 2π r |k − k0 |2 Therefore the differential cross section of the Rutherford scattering becomes α2 dσ . = |fB (k0 , k)|2 = dΩ 4m2 v 4 sin4 2θ

(2.26)

Even though we make the approximation, it is well-known that the Rutherford cross section obtained by the Born approximation is found to be almost the same as the exact result by solving the Lippmann-Schwinger equation in a proper way. This result can be compared with the Mott scattering formula which is obtained by calculating the relativistic T-matrix µ ¶ α2 dσ 2 2 θ 1 − v sin = (2.27) dΩ 2 4(m2 + k 2 )v 4 sin4 2θ where v is given as v =

2.2.4

√ k m2 +k2

'

k m

at the non-relativistic limit.

Separable Interaction

The Lippmann-Schwinger equation for the T-matrix cannot normally be solved analytically. However, there is a special potential which can be solved analytically, and this is the separable interaction. The separable interaction is assumed to have the following shape hk0 |V |ki = λg(k0 )g(k)

(2.28)

S-Matrix Theory

Fundamental Problems in Quantum Field Theory 25

  where g(k) and λ denote some  function and the interaction strength, respectively. Here, this interaction is a highly non-local, and therefore, this is a toy model. In this case, the T-matrix equation becomes Z 3 00 d k 0 0 0 g(k00 )GE (k00 )hk00 |T |ki (2.29) hk |T |ki = λg(k )g(k) + λg(k ) (2π)3 where

1 |k00 i. (2.30) E − H0 + iε This equation can be easily solved by assuming the following shape for the T-matrix GE (k00 ) = hk00 |

hk0 |T |ki = β(E)g(k0 )g(k).

(2.31)

In this case, the equation for the T-matrix can be written as Z 3 00 d k 0 0 0 [g(k00 )]2 GE (k00 ) β(E)g(k )g(k) = λg(k )g(k) + λβ(E)g(k )g(k) (2π)3 which can be solved for β(E) and we find β(E) = where F (E) is defined as

Z F (E) ≡

λ 1 − λF (E)

d3 k 00 [g(k00 )]2 GE (k00 ). (2π)3

(2.32)

(2.33)

Therefore, the T-matrix is completely determined as hk0 |T |ki =

λ g(k0 )g(k). 1 − λF (E)

(2.34)

In this way, one sees that the T-matrix equation for the separable interaction can be solved exactly. This is quite important since we can understand the basic structure of the T-matrix, even though the separable interaction is not realistic.

2.3

Interaction Picture and Definition of S-matrix

In the non-static perturbation theory, we obtain the integral equation for the amplitude ak (t) as XZ t 0 ak (t) = ak (0) − i dt0 ei(Ek −En )t huk |HI |un ian (t0 ). n

0

Here, we want to derive the same type of the equation as above in terms of the interaction picture. The main difference should be that, in the interaction picture approach, we do not make any expectation values of the interaction Hamiltonian with the state vectors, and therefore the integration over the space is not yet done. In addition, the Hamiltonian should be taken as operators by introducing the Fock space.

26 Fundamental Problems in Quantum Field Theory

  2.3.1

Fujita and Kanda

Interaction Picture  

Now, we introduce ΨI as

ΨI = eiH0 t Ψ.

(2.35)

It should be noted that the new state vector ΨI is different from the original one only because of the time rotation. In this case, the Schr¨odinger equation becomes i where HI (t) is defined as

2.3.2

∂ΨI (t, r) = HI (t)ΨI (t, r) ∂t

(2.36)

HI (t) = eiH0 t HI e−iH0 t .

(2.37)

S-matrix

Now we make the same procedure as above by operating u†k (r) from the left and integrating over the space. If we define ak (t) by ak (t) ≡ huk |ΨI i then we obtain exactly the same equation for ak (t) as eq.(2.7) X dak (t) = −i ei(Ek −En )t huk |HI |un ian (t). dt n Here, we take a slightly different approach by still keeping the Hamiltonian as operators. Instead, we introduce the matrix U (t, t0 ) as ΨI (t) = U (t, t0 )ΨI (t0 )

(2.38)

and thus we obtain the following equation i

∂U (t, t0 ) = HI (t)U (t, t0 ). ∂t

This equation can be easily solved in terms of iteration method, and we find Z t Z t Z t1 U (t, t0 ) = 1 − i dt1 HI (t1 ) + (−i)2 dt1 dt2 HI (t1 )HI (t2 ) + · · · . t0

t0

(2.39)

(2.40)

t0

However, this is not a very convenient shape when we define the S-matrix, and thus we introduce the T-product as   HI (t1 )HI (t2 ) (t1 > t2 ) T {HI (t1 )HI (t2 )} = (2.41)  HI (t2 )HI (t1 ) (t1 < t2 ).

S-Matrix Theory

Fundamental Problems in Quantum Field Theory 27

U (t, t0 ) as   Therefore, we can rewrite the matrix   Z tZ t Z t 1 dt1 dt2 T {HI (t1 )HI (t2 )} + · · · . U (t, t0 ) = 1 − i dt1 HI (t1 ) + (−i)2 2 t0 t0 t0 Now we can define the S-matrix as Z ∞Z ∞ Z ∞ 1 2 dt1 dt2 T {HI (t1 )HI (t2 )} + · · · S ≡1−i dt1 HI (t1 ) + (−i) 2 −∞ −∞ −∞ which can be symbolically written as ½ µ Z ¶¾ 4 S = T exp −i d xHI

(2.42)

Z with

HI (t) =

d3 rHI .

We note that the S-matrix is still an operator since the Hamiltonian should be an operator after the field quantization.

2.4

Photon Propagator

When one R calculates the S-matrix elements in the process of the electromagnetic interaction H 0 = e jµ Aµ d3 x in the second order perturbation theory, then one has to evaluate the propagator of photon. This is written as h0|T {Aµ (x1 )Aν (x2 )}|0i

(2.43)

where Aµ (x) is given as Aµ (x) =

XX

k

λ

h i 1 √ ²µk,λ c†k,λ e−iωk t+ik·r + ck,λ eiωk t−ik·r . 2V ωk

(2.44)

This should be a solution of the following equation of motion for the gauge field ∂µ (∂ µ Aν − ∂ ν Aµ ) = 0.

(2.45)

In this case, a question may arise as to how we can calculate the propagator of photon since the photon field A has one redundant degree of freedom. As we discuss below, there is some problem for determining the propagator of photon.

2.4.1

Free Wave of Photon

Now, the S-matrix evaluation should start from the free wave equation of motion. This means that we should not put the δ−function in the right hand side of the equation of motion for photon. Indeed, if we start from the free wave equation of motion, then we should insert

28 Fundamental Problems in Quantum Field Theory

Fujita and Kanda

  the solution of Aµ (x) in eq.(2.44)   into eq.(2.45) and obtain the following equation for the µ polarization vector ²k,λ as k 2 ²µ − (kν ²ν )k µ = 0. (2.46) This equation can be written in terms of the matrix equation for the polarization vector ²µ as 3 X {k 2 g µν − k µ k ν }²ν = 0 ν=0

where we write the summation explicitly. In order that the ²µ should have a non-zero solution, the determinant of the matrix should vanish, namely, det{k 2 g µν − k µ k ν } = 0.

(2.47)

Thus one finds the solution of this equation k2 = 0 which is a proper dispersion relation of photon. Then, we insert it into eq.(2.46) and obtain kµ ² µ = 0

(2.48)

and this is the solution for the polarization vector. This is the same equation as the Lorentz gauge fixing and thus the Lorentz gauge is not a proper gauge fixing even though people often use it. At this point, we should require the gauge fixing condition, and for example, we choose the Coulomb gauge fixing of ∇ · A = 0. In this case, we obtain a condition for the polarization vector as k · ² = 0, ²0 = 0. This gauge fixing is most natural, and we should take this condition which can guarantee the number of freedom of photon, which is two.

2.4.2

Feynman Propagator of Photon

Before going to the evaluation of the photon propagator in detail, we should make a comment on the Feynman propagator of photon [2]. The propagator of photon which is known as the Feynman propagator can be written as DFµν (k) = −

g µν . k 2 − iε

(2.49)

This is a standard photon propagator which can be found in most of the field theory textbooks. However, it is also well-known that this propagator cannot satisfy the condition of the polarization summation in a correct way. This is clear since it cannot satisfy the following equation kν 6= 0 kµ DFµν (k) = − 2 k − iε

S-Matrix Theory

Fundamental Problems in Quantum Field Theory 29

  where the left hand side should  be zero due to the Lorentz condition. Further, it cannot satisfy the Coulomb gauge condition, and therefore whatever they invent, there is no way to claim that the Feynman propagator is a right one. However, as will be seen below, Feynman propagator can reproduce the same T-matrix of the fermion-fermion scattering as the one calculated from the correct propagator as long as the scattering particles are on the mass shell. Since the agreement of the T-matrices evaluated from the two propagators is entirely based on the free Dirac equation of the fermions involved in the scattering process, the Feynman propagator cannot be applied for physical processes involving fermions which are not free or off the mass shell.

2.4.3

Calculation of h0|T {Aµ (x1 )Aν (x2 )}|0i

Here, we should evaluate the denominator of the propagator h0|T {Aµ (x1 )Aν (x2 )}|0i explicitly in order to avoid any confusions. First, we insert the free solution of the vector potential in eq.(2.44), and find h0|T {Aµ (x1 )Aν (x2 )}|0i = n³ h0|T

c†k,λ e−ikx1 + ck,λ eikx1

XX

k,λ k0 ,λ0

1 p ²µk,λ ²νk0 ,λ0 × 2 4V ωk ωk0

´³ ´o 0 0 c†k0 ,λ0 e−ik x2 + ck0 ,λ0 eik x2 |0i

(2.50)

which can be calculated to be µ

ν

h0|T {A (x1 )A (x2 )}|0i =

2 Z X λ=1

´ d3 k 1 µ ν ³ ikx −ikx ² ² e θ(t) + e θ(−t) (2.51) (2π)3 2ωk k,λ k,λ

where we define x = x1 − x2 ,

θ(t) = 1 for t > 0,

θ(t) = 0 for t < 0.

By noting the following complex plane integrations  iω t ie k  Z ∞  2ωk eik0 t dk0 = 2 2  −∞ (2π) k0 − k − iε  ie−iωk t 2ωk

for t > 0 (2.52) for t < 0

we can rewrite h0|T {Aµ (x1 )Aν (x2 )}|0i as Z µ

ν

h0|T {A (x1 )A (x2 )}|0i = −i

2

d4 k eik(x1 −x2 ) X µ ν × ²k,λ ²k,λ . (2π)4 k 2 − iε

(2.53)

λ=1

This is just the propagator of photon. Now, the problem comes up when we evaluate the summation of the polarization vector.

30 Fundamental Problems in Quantum Field Theory

  2.4.4

Fujita and Kanda

Summation of Polarization States  

Up to now, we have presented the expression of the propagator evaluation of photon without making any comments on the field quantization, Now, we should quantize only the vector field A which depends on time. The Coulomb field A0 is already solved from the constraint equation, and thus it cannot appear in the S-matrix expansion. Therefore, we have the condition that ²0 = 0. In addition, we should respect the Coulomb gauge condition k · ²k,λ = 0. Now, we are ready to construct the numerator of the propagator of photon, and we find 2 X λ=1

a

b

²k,λ ²k,λ

¶ µ ka kb ab = δ − 2 k

(2.54)

which is the only possible solution for the summation of the polarization vector. Note that this can satisfy the condition of k · ²k,λ = 0, because the left hand side of eq.(2.54) multiplied by k a becomes 2 X ²ak,λ ²bk,λ = 0 ka λ=1

while the right hand side can be calculated as ¶ µ k2 k b ka kb a ab = kb − 2 = 0 k δ − 2 k k and thus eq.(2.54) can satisfy all the conditions we have for the polarization vectors. Therefore, the propagator of photon Dab becomes µ ¶ ka kb 1 ab ab δ − 2 . (2.55) D (k) = 2 k − iε k

2.4.5

Coulomb Propagator

The Coulomb part is solved exactly since it does not depend on time. Namely the equation of motion for the Coulomb part is a constraint equation which has nothing to do with the quantization of field. Note that the field quantization should always involve the time dependence of fields. Now, the equation of motion for the A0 part can be written as ¯ 0 ψ ≡ −ej 0 (x) ∇2 A0 = −eψγ

(2.56)

which is a constraint equation. However, the right hand side is made of fermion fields, and the quantization of the fermion fields is already done. It should be noted that the Coulomb case is calculated from the first order perturbation theory since it arises from Z Z 1 0 0 3 (∇A0 )2 d3 r. HC = e j (t, r)A (r)d r − 2

S-Matrix Theory

Fundamental Problems in Quantum Field Theory 31

between two Dirac fields j10 and j20 becomes   In this case, the interaction Hamiltonian   Z 0 j1 (x1 )j20 (x2 ) 3 3 e2 d r1 d r2 HC = 8π |r1 − r2 | which can be rewritten in terms of the momentum representation as Z ˜0 ˜0 j1 (q)j2 (−q) d3 q 2 . HC = e q2 (2π)3

(2.57)

On the other hand, the propagator of photon should be calculated from the S-matrix expansion in the second order perturbation theory. Therefore, the Coulomb propagator is completely different from the photon propagator which is calculated from the S-matrix expansion. However, the Coulomb field interaction should be always considered for the scattering process since fermions are already quantized.

2.4.6

Correct Propagator of Photon

The correct propagator of photon is given in eq.(2.55), but if we consider the scattering process such as electron-electron scattering in which the scattering particles are all on the mass shell, then we should add the Coulomb scattering in which the Coulomb propagator is employed. Therefore, the total propagators of photon together with the Coulomb scattering become  Coul A0 − part (k) = k12   D (2.58) ³ ´  ab − ka kb  Dab (k) = 2 1 δ A − part. 2 k −iε k

2.5

Feynman Propagator vs. Correct Propagator

Here, we discuss the equivalence and/or difference between the T-matrices which are calculated from Feynman and correct propagators, The discussion of the equivalence between them is usually found in old field theory textbooks [3, 4]. However, this equivalence proof is valid only if the propagators appear in the scattering processes with free fermions. Therefore, if there is a loop involved such as the fermion self-energy, then the expected equivalence cannot be valid any more. Later in this section, we discuss some physical effects which may arise from the T-matrix difference between the Feynman and the correct propagators.

2.5.1

Scattering of Two Fermions

As an example, we present the scattering T-matrices between two fermions in which one fermion with its four momentum p1 scatters with another fermion with its four momentum p2 , and after the scattering, we find two fermions with their momenta of p0 1 and p0 2 . The four momentum transfer is defined as q = p1 − p0 1 = p0 2 − p2 .

32 Fundamental Problems in Quantum Field Theory

  (a) Feynman Propagator

Fujita and Kanda

 

In the case of Feynman propagator as given in eq.(2.49), the T-matrix can be written in a straight forward way as T (F ) = −

¤ e2 £ 0 0 u ¯(p 1 )γ u(p1 )¯ ¯(p0 1 )γu(p1 ) · u ¯(p0 2 )γu(p2 ) . u(p0 2 )γ 0 u(p2 ) − u 2 q

(2.59)

(b) Correct Propagator Now, we evaluate the T-matrix with the correct propagator of photon which is given by eq.(2.58). First, the T-matrix from the Coulomb part can be written as T (C) =

e2 u ¯(p0 1 )γ 0 u(p1 )¯ u(p0 2 )γ 0 u(p2 ). q2

(2.60)

On the other hand, the T-matrix from the vector field A becomes · ¸ e2 1 0 0 (A) 0 0 ¯(p 1 )γu(p1 )¯ ¯(p 2 )γ · qu(p2 ) . T = 2 u u(p 2 )γu(p2 ) − u ¯(p 1 )γ · qu(p1 ) 2 u q q (2.61) Now, we make use of the free Dirac equations for two fermions at the initial and final states (p/ 1 − m1 )u(p1 ) = 0, u ¯(p0 1 )(p/ 0 1 − m1 ) = 0, (p/ 2 − m2 )u(p2 ) = 0, u ¯(p0 2 )(p/ 0 2 − m2 ) = 0 and thus we can rewrite u ¯(p0 1 )γ · qu(p1 ) = u ¯(p0 1 )γ 0 u(p1 )q10 , u ¯(p0 2 )γ · qu(p2 ) = −¯ u(p0 2 )γ 0 u(p2 )q20 where q10 = E1 − E1 0 and q20 = E2 − E2 0 . Therefore, T (A) becomes · ¸ e2 q10 q20 0 0 0 0 (A) 0 0 ¯(p 1 )γu(p1 ) · u ¯(p 2 )γu(p2 ) + u ¯(p 2 )γ u(p2 ) . (2.62) T = 2 u ¯(p 1 )γ u(p1 ) 2 u q q Note that one may be tempted to assume that q10 = −q20 = q 0 at this point. However, the energy conservation can be used only at the final stage of the calculation, and therefore, the evaluation of the T-matrix should be done without using the energy conservation. It should be noted that the on-shell scattering processes like the fermion-fermion scattering must conserve the energy, and therefore one can employ the equation q10 = −q20 when one calculates the cross section. Now, it is easy to check that the sum of T (C) and T (A) becomes T (C) + T (A) = −

¤ e2 £ 0 0 0 0 0 0 u ¯ (p )γ u(p )¯ u (p )γ u(p ) − u ¯ (p )γu(p ) · u ¯ (p )γu(p ) 1 2 1 2 1 2 1 2 q2

e2 (q 0 q 0 + q10 q20 ) u ¯(p0 1 )γ 0 u(p1 )¯ u(p0 2 )γ 0 u(p2 ). (2.63) q2q2 As can be seen, the T-matrix calculated from the correct propagator has an extra-term which is not found in the T-matrix evaluated from the Feynman propagator. Therefore, there exists a clear difference between the two T-matrices in the fermion-fermion scattering case. +

S-Matrix Theory

Fundamental Problems in Quantum Field Theory 33

  Energy Conservation in T-matrix   Now, if one uses the energy conservation of q10 = −q20 = q 0 , then the Feynman propagator can reproduce the right T-matrix for the fermion-fermion scattering cross section as one can find the equivalence proof in old textbooks [3, 4]. Indeed, the on-shell scattering case is justified because the energy conservation is taken into account for the whole system. This should be one of the strong reasons why people accepted the Feynman propagator.

2.5.2

Loop Diagrams (Fermion Self-energy)

As one sees from the comparison between the Feynman and correct propagators, the use of free Dirac equations play a very important role. Therefore, it is most likely that the two propagators should give the very big difference for the fermion self-energy type diagrams in which intermediate fermions do not satisfy the free Dirac equations. (a) Feynman Propagator Using the Feynman propagator, the self-energy of fermion can be easily written as Z Σ

(F )

2

(p) = −ie

1 1 e2 d4 k µ γ γ = ln µ (2π)4 p/ − k/ − m + iε k 2 − iε 8π 2

µ

Λ m

¶ (−p/+4m)+· · · (2.64)

which is just the self-energy contribution normally found in the textbooks. (b) Correct Propagator The self-energy of fermion with the correct propagator has never been calculated up to now, but we should evaluate it since it is very important to examine whether this selfenergy contribution can agree with the normal self-energy contribution with the Feynman propagator. First, the Coulomb part does not contribute to the fermion self-energy because of the equal time operations, and thus we should only calculate the contribution from the vector potential part which can be written as Z Σ(A) (p) = ie2

³ d4 k (2π)4

γa

1 γb p/ − k/ − m + iε

a b

δ ab − kkk2 k 2 − iε

´ .

(2.65)

What we have to calculate and see is whether the Σ(A) (p) should be the same as Σ(F ) (p) or not. From the calculations, one sees that it does not agree with the one calculated from the Feynman propagator. In this respect, there is no reason any more that one can employ the Feynman propagator for the calculation that involves the photon propagation unless all fermions are on the mass shell.

34 Fundamental Problems in Quantum Field Theory

  Conflict of Interest The author(s) confirm that this chapter content has no conflicts interest. Acknowledgements: Declared none.

Fujita and Kanda

Send Orders of Reprints at [email protected] Fundamental Problems in Quantum Field Theory, 2013, 35-71

35

CHAPTER 3

Quantum Electrodynamics Abstract:

Here, we present the calculations of the self-energy of photon and electron. First the renormalization scheme of photon is discussed in connection with the triangle diagrams which have no divergence at all, and therefore the self-energy of photon is not related to any physical observables. Then, we discuss the renormalization scheme of fermion self-energy term which is still connected with the vertex correction in QED. Also, we discuss briefly the physics of Lamb shifts which should be treated in detail in chapter 7. Then, we present the calculation of the photonphoton scattering cross section and some possible experiments on the photon-photon cross section. Finally, the problem of the chiral anomaly equation is discussed, and we see that there are neither anomaly equation nor the violation of the axial current conservation in physical world. Keywords: renormalization scheme, self-energy of photon and fermion, local gauge invariance, Lamb shifts, photon-photon scattering, chiral anomaly equation.

3.1

Introduction

The basic starting point of the physics equation is, of course, the Maxwell equation. The introduction of the displacement current density term jD = ∂∂tE should be the most important development in the history of physics. Since the Maxwell equation is basically derived from the experimental observation, it always holds true for any physical situations. The Maxwell equation has neither non-relativistic limits nor classical limits, in contrast to the Newton equation which is obtained as the classical limit of the Schr¨odinger equations. Therefore, it is valid at any physical occasions. However, the field should be quantized when we want to treat the transition of one state to the other in atoms. In this case, photons can be created from the vacuum state by changing of the fermion states, which is not found in the non-relativistic quantum mechanics. In this chapter, we first describe the general properties of QED and discuss the basic Takehisa Fujita and Naohiro Kanda All rights reserved - © 2013 Bentham Science Publishers

36 Fundamental Problems in Quantum Field Theory

Fujita and Kanda

ingredients to understand QED. Then, we explain briefly the quantization of fields which are necessary for the description of the physical observables. The quantization of the vector field is required from the atomic transitions in which photons are created from the vacuum state. On the other hand, the quantization of the Dirac field is required in its own construction, that is, the Dirac equation contains negative energy solutions as a physical observable. Therefore, we have to construct the physical vacuum state in fermion field theory and require the field quantization which should impose the anti-commutation relations on the creation and annihilation operators. This should guarantee the Pauli principle which enables to construct the physical vacuum state in a Dirac field theory in which the negative energy states are all to be filled.

3.2

Lagrangian Density in QED

Here, we repeat the discussion of the fundamental QED properties which are important and helpful for the understanding of physical phenomena in QED. The basic ingredients are vector fields (gauge fields) and Dirac fermion fields which are coupled with each other.

3.2.1

QED Lagrangian Density

The Lagrangian density of QED with massive fermions ψ with its mass m can be written as ¯ µ Dµ − m)ψ − 1 Fµν F µν L = ψ(iγ 4

(3.1)

where Dµ = ∂ µ + ieAµ ,

F µν = ∂ µ Aν − ∂ ν Aµ .

Here, Aµ and F µν denote the gauge field and the field strength, respectively. Mass Scale or Cut-off Λ It should be noted that the coupling constant e in four dimensional QED is dimensionless, and it is normally denoted as e which is called charge. Therefore, the charge e is the strength of the electromagnetic interaction which happens to be universal for the electromagnetic interactions. The charge state is specified by the quantum number of fermions, and electron, for example, has a charge quantum number of (−1) while proton has (+1). The Lagrangian density of QED in four dimensions has no scale parameter with dimensions if the mass of fermion is set to zero, m = 0. In nature, electron has a finite mass, and therefore physical observables in QED are all measured by the electron mass m. For examples, the energy eigenvalue En and Bohr radius aB of the hydrogen-like atom with its potential Ze2 V (r) = − r

Quantum Electrodynamics

Fundamental Problems in Quantum Field Theory 37

in non-relativistic quantum mechanics are given as En = −

mZ 2 e4 , 2n2

aB =

1 . mZe2

(3.2)

Here, as mentioned above, the charge e is a dimensionless constant and n runs as n = 1, 2, · · · . This suggests that all of the physical observables should be measured by the electron mass m as long as we are working with the Lagrangian density with electromagnetic field and electron field as the ingredients.

3.2.2

Local Gauge Invariance

The Lagrangian density of QED is invariant under the local gauge transformation A0 = Aµ + ∂ µ χ

µ

(3.3a)

ψ 0 = e−ieχ ψ.

(3.3b)

Since the following equations can be easily proved µ

D0 ψ 0 = e−ieχ Dµ ψ,

F0

µν

= F µν

(3.3c)

the gauge invariance of the Lagrangian density of eq.(3.1) is guaranteed.

3.2.3

Equation of Motion

The equation of motion for the gauge field can be obtained from the Lagrange equation for the vector field Aν ∂L ∂L = ∂µ . (3.4) ∂(∂µ Aν ) ∂Aν This leads to the following Maxwell equation ∂µ F µν = ej ν

(3.5)

where the fermion vector current j µ is given as ¯ µ ψ. j µ = ψγ From the Lagrange equation for ψ, ∂µ

∂L ∂L = ∂(∂µ ψ) ∂ψ

one obtains the Dirac equation (i∂µ γ µ − eAµ γ µ − m)ψ = 0.

(3.6)

38 Fundamental Problems in Quantum Field Theory

3.2.4

Fujita and Kanda

Noether Current and Conservation Law

The Lagrangian density is obviously invariant under the global gauge transformation ψ 0 = eiθ ψ

(3.7)

where θ is a real constant. In this case, there is a conserved current associated with the global gauge invariance. First, one makes the infinitesimal global gauge transformation as ψ 0 = (1 + iθ)ψ = ψ + δψ

(3.8)

where θ is assumed to be infinitesimally small. Also, δψ is introduced as δψ = iθψ.

(3.9)

In this case, one obtains δL ≡ L(ψ 0 , ∂µ ψ 0 ) − L(ψ, ∂µ ψ) =

∂L ∂L δ(∂µ ψ) = 0. δψ + ∂ψ ∂(∂µ ψ)

By making use of the equation of motion for ψ, one obtains the conservation law ¶ ¸ ¸ · ·µ ∂L ∂L ∂L ψ+ (∂µ ψ) = iθ∂µ ψ = 0. δL = iθ ∂µ ∂(∂µ ψ) ∂(∂µ ψ) ∂(∂µ ψ)

(3.10)

(3.11)

Since one can calculate

∂L ¯ µ ψ = −ej µ ψ = −eψγ ∂(∂µ ψ) this leads to the conservation of the fermion vector current j µ ∂µ j µ = 0.

(3.12)

(3.13)

From eq.(3.11), one notices that the Noether current does not depend on the interaction terms. This is because the interaction terms should not depend on the field derivative of ∂µ ψ in any Lagrangian densities we discuss in this textbook.

3.2.5

Gauge Invariance of Interaction Lagrangian

The interaction Lagrangian density itself LI = −ejµ Aµ

(3.14)

is not gauge invariant at first sight, and therefore if one wishes to make any perturbation calculations, one should check it in advance that the gauge dependent part should not cause any troubles in the perturbative estimation with the interaction Lagrangian density of LI . Now, the interaction Lagrangian density can be rewritten with the gauge transformation as LI = −ejµ (Aµ + ∂ µ χ) = −ejµ Aµ − e∂ µ (jµ χ) + e(∂ µ jµ )χ.

(3.15)

The second term of the last equation is a total divergence and hence does not contribute to any physical observables in perturbation theory, and the last term vanishes to zero as long as the vector current conservation ( ∂µ j µ = 0 ) holds. Therefore, one sees that the gauge dependent parts do not cause any contributions to the perturbative calculation under the condition that the vector current conservation of fermions should be respected.

Quantum Electrodynamics

3.2.6

Fundamental Problems in Quantum Field Theory 39

Gauge Fixing

The total Hamiltonian is gauge invariant, and therefore one should fix the gauge since the gauge field Aµ has a redundancy as variables. There are many ways to fix the gauge, and of course there should not be any differences for the observables one calculates from different gauge fixings. The most popular gauge fixing must be a Coulomb gauge ∇ · A = 0.

(3.16)

This has some advantage in that the time component of the gauge field A0 is not a dynamical variable any more and becomes just a simple constraint which can be easily solved by employing equations of motion. Also, one may take the temporal gauge A0 = 0.

(3.17)

In this case, one can recover the Coulomb interactions if one calculates the interactions properly. Any physical observables like the energy spectrum should not depend on the choice of the gauge fixing if it is properly chosen. However, the gauge fixing in the perturbation theory should be treated carefully, and we will discuss it later more in detail.

3.2.7

Gauge Choices

The physical observables must be independent from gauge choices. In particular, the Coulomb interaction should be derived also by the temporal gauge A0 = 0. Here, it is shown how one can obtain the Coulomb interaction when one takes the temporal gauge. In particular, it is also shown that the conservation of the fermion vector current ( ∂µ j µ = 0 ) plays an important role and indeed without the current conservation, the different choices of the gauge fixing, the Coulomb gauge, the temporal gauge and the axial gauge, give different results on the Coulomb interactions. Temporal Gauge ( A0 = 0 ) We start from the following Hamiltonian "µ #) ¶2 Z ( ∂A 1 0 2 2 0 d3 r. H= ψ¯ (−iγ · ∇ + m) ψ − ej · A + ej0 A + − (∇A ) + B 2 ∂t (3.18) Now, the A0 = 0 gauge is taken, and therefore the equation of motion for the gauge field becomes ∂A ∇· = −ej 0 . (3.19) ∂t Here, there is still a gauge freedom left. Namely, the Hamiltonian is invariant under the following transformation A → A + ∇χ(r),

ψ → eieχ(r) ψ

(3.20)

40 Fundamental Problems in Quantum Field Theory

Fujita and Kanda

where χ(r) depends only on the coordinate r. Now, we can write the vector field A as A = AT + ∇ξ,

with ∇ · AT = 0.

(3.21)

In this case, the equation of motion for the gauge field becomes ∇2 φ0 = −ej0 ,

with φ0 ≡

∂ξ . ∂t

(3.22)

¡ ¢2 Therefore, 21 ∂∂tA term in the Hamiltonian can be written with the transverse electric field ET as defined by ∂AT ET = − . ∂t Therefore, we have ¶ Z µ Z Z Z ∂A 2 3 1 1 1 2 3 3 ET d r + ET · ∇φ0 d r + (∇φ0 )2 d3 r. d r= (3.23) 2 ∂t 2 2 The second term in the right hand side vanishes since ∇ · ET = 0 holds, and the third term is just the same as the Coulomb interaction. Therefore, the Hamiltonian with the temporal gauge becomes just the same as that obtained by the Coulomb gauge fixing Z © ª ψ¯ (−iγ · ∇ + m) ψ − ej · A d3 r H= +

e2 8π

Z

j0 (r 0 )j0 (r)d3 rd3 r0 1 + |r 0 − r| 2

Z

£

¤ ET2 + B 2 d3 r.

(3.24)

Axial Gauge ( A3 = 0 ) The axial gauge fixing is also employed where one has the gauge condition A3 = 0. In this case, the vector potential has only the transverse component and therefore one can define the transverse electric field ∂A ET = − ∂t where ∇ · ET = 0 holds. In this case, the equation of motion for the gauge field becomes ∇2 A0 = −ej 0

Quantum Electrodynamics

Fundamental Problems in Quantum Field Theory 41

and therefore A0 can be solved just in the same way as the Coulomb gauge case. In addition, ¡ ¢2 the 21 ∂∂tA term can be written in terms of ET as µ ¶ 1 ∂A 2 1 = ET 2 . 2 ∂t 2 Therefore, the Hamiltonian becomes just the same as eq.(3.24).

3.2.8

Quantization of Gauge Fields

Now we should discuss the quantization of the gauge field A. However, we should be careful for the number of the degree of freedom of the gauge fields since there is a gauge fixing condition. For example, if one takes the Coulomb gauge. ∇·A=0

(3.25)

then, the gauge field A should have two degrees of freedom. In this case, the gauge field A can be expanded in terms of the free field solutions A(x) =

h i 1 √ ²k,λ c†k,λ e−ikx + ck,λ eikx k λ=1 2V ωk

2 XX

(3.26)

where ωk = |k|. The polarization vector ²k,λ should satisfy the following relations ²k,λ · k = 0,

²k,λ · ²k,λ0 = δλ,λ0

(3.27)

since the gauge field A should satisfy eq.(3.25). Commutation Relations Since the gauge fields are bosons, the quantization procedure must be done in the commutation relations, instead of anti-commutation relations. Therefore, the quantization can be done by requiring that ck,λ , c†k,λ should satisfy the following commutation relations [ck,λ , c†k0 ,λ0 ] = δk,k0 δλ,λ0 (3.28) and all other commutation relations vanish. ˆ em of the electromagnetic fields is written in terms of In this case, the Hamiltonian H the creation and annihilation operators as ¶ µ 2 XX 1 ˆ em = . (3.29) H ωk c†k,λ ck,λ + 2 k λ=1 From eq.(3.29), one sees that there are two degrees of freedom for the quantized gauge fields. Since the gauge field A has always a gauge freedom, it may be the best to quantize the gauge field A in terms of the creation and annihilation operators ck,λ , c†k,λ after the gauge fixing is done.

42 Fundamental Problems in Quantum Field Theory

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Zero Point Energy Eq.(3.29) contains a zero point energy. That is, the vacuum state where there is no electromagnetic field present has an infinite energy Evac = 2 ×

X 1X ωk = |k| → ∞. 2 k

However, there is nothing serious since the vacuum state cannot be observed. Therefore, one should measure the energy of excited states from the vacuum, and thus ∆Eem

¶ µ 2 XX 1 † − Evac = = ωk ck,λ ck,λ + ωk c†k,λ ck,λ 2 k λ=1 k λ=1 2 XX

must be physical observables.

3.3

Renormalization Scheme in QED: Photon

In this section, we present a critical review of the renormalization scheme of photon in quantum electrodynamics. For a long time, people accepted the renormalization scheme in which the self-energy contributions of photon and fermion are considered to be related to physical observables once the infinity in the self-energy diagrams are properly renormalized into the wave function. This procedure looks always odd at the beginning when we study the theoretical frame wok of QED, but somehow we are all persuaded after some time of the study. The most important of all is that the renormalization scheme can predict and reproduce the observed values of electron g − 2. Here, we explain the essential point of the renormalization scheme of photon, and we show that the self-energy of photon is not needed in the renormalization procedure. This is basically because there is no relevant physical process which can make use of the renormalized wave function of the photon self-energy, in contrast to the fermion self-energy case which is applied to the vertex corrections. The important point is that the vertex correction corresponds to the Feynman diagram in which the external electromagnetic field couples to the intermediate fermion state in the fermion self-energy diagram. On the other hand, the triangle diagrams correspond to the Feynman diagram in which the external vertex of Γ couples to the intermediate fermion or anti-fermion states in the photon self-energy diagram. Both of the procedures in the renormalization scheme are quite similar to each other, but the vertex correction has the logarithmic divergence which should be absorbed into the renormalized wave function of fermions while the triangle diagrams have no divergences at all, and thus there is no need of the renormalization procedure for the photon self-energy case as long as we aim at producing physical observables. In what follows, we should clarify the origin of the basic mistake of the renormalization of photon self-energy. We show that it arises from gauge conditions in the various stages of

Quantum Electrodynamics

Fundamental Problems in Quantum Field Theory 43

the divergent contributions. Once one sees that this gauge condition is unphysical, then all of the renormalization procedures of photon become just simple, namely, there is no need of renormalization of photon self-energy.

3.3.1

Self-energy of Photon

In the renormalization procedure of QED, one first considers the vacuum polarization which is the contribution of the self-energy diagram of photon ¸ · Z 1 1 d4 p ν µ µν 2 Tr γ γ . (3.30) Π (k) = ie (2π)4 p/ − m p/ − k/ − m This integral obviously gives rise to the quadratic divergence (Λ2 term). However, when one considers the counter term of the Lagrangian density which should cancel this quadratic divergence term, then the counter Lagrangian density violates the gauge invariance since it should correspond to the mass term in the gauge field Lagrangian density. Therefore, one has to normally erase it by hand, and in the cutoff procedure of the renormalization scheme, one subtracts the quadratic divergence term such that one can keep the gauge invariance of the Lagrangian density. Here, we should notice that the largest part of the vacuum polarization contributions is discarded, and this indicates that there must be something which is not fully understandable in the renormalization procedure [2, 5, 6]. Physically, it should be acceptable to throw away the Λ2 term since this infinite term should not be connected to any physical observables. Nevertheless we should think it over why the unphysical infinity appears in the self-energy diagram of photon. On the other hand, the quadratic divergence term disappears in the treatment of the dimensional regularization scheme. Here, we clarify why the quadratic divergence term does not appear in the dimensional regularization treatment. That is, the treatment of the dimensional regularization employs the mathematical formula which is not valid for the evaluation of the momentum integral in the vacuum polarization. Therefore, the fact that there is no quadratic divergence term in the dimensional regularization is simply because one makes a mistake by applying the invalid mathematical formula to the momentum integral. This is somewhat surprising, but now one sees that the quadratic divergence is still there in the dimensional regularization, and this strongly indicates that we should reexamine the effect of the photon self-energy diagram from the beginning. Momentum Integral with Cutoff Λ This procedure of the photon self-energy is well explained in the textbook of Bjorken and Drell, and therefore we describe here the simplest way of calculating the momentum integral. The type of integral one has to calculate can be summarized as Z Λ2 Z 1 1 2 4 = iπ wdw (n ≥ 3) (3.31) d p 2 n (p − s + iε) (w − s + iε)n 0 where we define w = p2 .

44 Fundamental Problems in Quantum Field Theory

3.3.2

Fujita and Kanda

Photon Self-energy Contribution

The photon self-energy contribution Πµν (k) in eq.(3.30) can be easily evaluated as · ¸ Z 1 Z 2pµ pν − gµν p2 + sgµν − 2z(1 − z)(kµ kν − k 2 gµν ) 4ie2 4 dz d p Πµν (k) = (2π)4 0 (p2 − s + iε)2 · ¸ Z 1 Z Λ2 w(w − 2s)gµν + 4z(1 − z)w(kµ kν − k 2 gµν ) α dz dw (3.32) = 2π 0 (w − s + iε)2 0 where s is defined as s = m2 − z(1 − z)k 2 . This can be calculated to be (2) Πµν (k) = Π(1) µν (k) + Πµν (k)

where

µ ¶ k2 2 2 Λ +m − gµν (3.33a) 6 · µ 2 ¶ ¶¸ µ Z 1 Λ α k2 2 (2) (kµ kν − k gµν ) ln −6 dzz(1 − z) ln 1 − 2 z(1 − z) . Πµν (k) = 3π m2 e m 0 (3.33b) (1) Here, the Πµν (k) term corresponds to the quadratic divergence term and this should be (2) discarded by hand since it violates the gauge condition. The Πµν (k) term can keep the gauge invariance, and therefore one can renormalize it into the new Lagrangian density, and this is a story of the renormalization scheme. Π(1) µν (k)

α = 2π

Finite Term in Photon Self-energy Diagram After the renormalization, one finds a finite term which should affect on the propagator change in the process involving the exchange of the transverse photon A. The propagator 1 should be replaced by q2 · ¶¸ µ Z 1 2α 1 q 2 z(1 − z) 1 ⇒ 1 dzz(1 (3.34) + − z) ln 1 − q2 q2 π 0 m2 where q 2 should become q 2 ≈ −q 2 for small q 2 . It should be important to note that the correction term arising from the finite contribution of the photon self-energy should affect only on the renormalization of the vector field A. Since the Coulomb propagator is not affected by the renormalization procedure of the transverse photon (vector field A), one should not calculate its effect on the Lamb shifts. Dimensional Regularization In the evaluation of the momentum integral, people often employ the dimensional regularization [7, 8] where the integral is replaced as Z Z dD p d4 p 4−D → λ (3.35) (2π)4 (2π)D

Quantum Electrodynamics

Fundamental Problems in Quantum Field Theory 45

where λ is introduced as a parameter which has a mass dimension in order to compensate the unbalance of the momentum integral dimension. This is the integral in the Euclidean space, but D is taken to be D = 4 − ² where ² is an infinitesimally small number. In this case, the photon self-energy Πµν (k) can be calculated to be Z 4−D 2

Πµν (k) = iλ

e

¸ · 1 1 dD p Tr γµ γν (2π)D p/ − m p/ − k/ − m

· ¸ 2 α 2 (kµ kν − gµν k ) + finite term = 3π ²

(3.36)

where the finite term is just the same as eq.(3.33b). One sees that the quadratic divergence (1) term (Πµν (k)) is missing. This is surprising since the quadratic divergence term is the leading order contribution in the momentum integral, and whatever one invents in the integral, there is no way to erase it unless one makes a mistake. Indeed, in the treatment of the dimensional regularization, people employ the mathematical formula which is invalid for the integral in eq.(3.36). That is, the integral formula for D = 4 − ² Z dD p

1 D gµν pµ pν n+1 Γ(n − 2 D − 1) 2 (−1) (for n ≥ 4) (3.37) = iπ 1 2 n (p − s + iε) 2Γ(n) sn− 2 D−1

is only valid for n ≥ 4. For n = 3, the integral should have the logarithmic divergence, and this is nicely avoided by the replacement of D = 4 − ². However, the n = 2 case must have the quadratic divergence and the mathematical formula of eq.(3.37) is absolutely meaningless. Reconsideration of Photon Self-energy Diagram Now, one sees that the disappearance of the quadratic divergence term in the evaluation of Πµν (k) in the dimensional regularization is not due to the mathematical trick, but simply (1) due to a simple-minded mistake. In this respect, it is just accidental that the Πµν (k) term in the dimensional regularization vanishes to zero. The most important point is that one should (1) obtain the proper expression of the Πµν (k) term when one makes ² → 0 in the calculation of the dimensional regularization. The requirement that the original result should be recovered at the limit of ² → 0 is crucial and should be satisfied in any mathematics [9]. The evaluation of the vacuum polarization contribution gives rise to the quadratic divergence and, as long as this should exist, there is no way to renormalize it into the standard renormalization scheme [10]. In fact, Pauli and Villars proposed [11] that the quadratic divergence term should be evaded by the requirement that the calculated result should be gauge invariant when renormalizing it into the Lagrangian density. This requirement of the gauge invariance should be based on the following relation for the vacuum polarization tensor Πµν (k) as kµ Πµν (k) = 0. (3.38)

46 Fundamental Problems in Quantum Field Theory

3.3.3

Fujita and Kanda

Gauge Conditions of Πµν (k)

However, eq.(3.38) does not hold, and it is indeed a spurious equation even though it has been employed as the gauge condition. As seen above, the vacuum polarization tensor is given in eqs.(3.33), and this is obviously inconsistent with eq.(3.38). For a long time, people believe that the Πµν (k) should satisfy the relation of eq.(3.38). Now, we present the proof found in the textbook of Bjorken and Drell [10], and show that the proof of eq.(3.38) is a simple mathematical mistake. Therefore, this gauge condition is spurious, and the relation has no physical foundation at all. The standard method of the proof starts by rewriting the kµ Πµν (k) as ¶ ¸ ·µ Z 1 d4 p 1 µν 2 Tr − γν . (3.39) kµ Π (k) = ie (2π)4 p/ − k/ − m + iε p/ − m + iε In the first term, the integration variable should be replaced as q =p−k and thus one can prove that ¸ Z ¸¾ ½Z · · d4 p d4 q 1 1 ν ν µν 2 Tr γ − Tr γ = 0. kµ Π (k) = ie (2π)4 q/ − m + iε (2π)4 p/ − m + iε (3.40) At a glance, this proof looks plausible. However, one can easily notice that the replacement of the integration variable is only meaningful when the integral is finite. In order to clarify the mathematical mistake in eq.(3.40), we present a typical example which shows that one cannot make a replacement of the integration variable when the integral is infinity. Let us now evaluate the following integral Z ∞ ¡ ¢ (x − a)2 − x2 dx. (3.41) Q= −∞

If we replace the integration variable in the first term as x0 = x − a, then we can rewrite eq.(3.41) as Z ∞³ ´ 2 Q= x0 dx0 − x2 dx = 0. (3.42) −∞

However, if we calculate it properly, then we find Z ∞ Z ∞ ¡ ¢ ¡ 2 ¢ 2 2 Q= (x − a) − x dx = a − 2ax dx = a2 × ∞ −∞

(3.43)

−∞

which disagrees with eq.(3.42). If one wishes to properly calculate eq.(3.41) by replacing the integration variable, then one should do as follows ·Z Λ−a ¸ Z Λ Z Λ ¡ ¢ 2 2 02 0 2 Q = lim (x − a) − x dx = lim x dx − x dx = lim 2a2 Λ. Λ→∞ −Λ

Λ→∞

−Λ−a

−Λ

Λ→∞

(3.44)

Quantum Electrodynamics

Fundamental Problems in Quantum Field Theory 47

It is clear by now that the replacement of the integration variable in the infinite integral should not be made, and this is just the mistake which has been accepted as the gauge condition of the Πµν (k) in terms of eq.(3.38). Therefore, one sees that the requirement of the gauge condition of the vacuum polarization is unphysical.

3.3.4

Physical Processes Involving Vacuum Polarizations

In nature, there are a number of Feynman diagrams which involve the vacuum polarization. The best known physical process must be the π 0 decay into two photons, π 0 → γ + γ. This process of the Feynman diagrams can be well calculated in terms of the nucleon and anti-nucleon pair creation where these fermions couple to photon [12]. In this calculation, one knows that the loop integral gives a finite result since the apparent logarithmic divergence vanishes to zero due to the kinematical cancellation. Also, the physical process of photon-photon scattering involves the box diagrams where electrons and positrons are created from the vacuum state. As is well known, the apparent logarithmic divergence of this box diagrams vanishes again due to the kinematical cancellation, and the evaluation of the Feynman diagrams gives a finite number. This is clear since all of the perturbative calculations employ the free fermion basis states which always satisfy the current conservation of ∂µ j µ = 0. In these processes, one does not have any additional “gauge conditions” in the evaluation of the Feynman diagrams. In this respect, if the process is physical, then the corresponding Feynman diagrams should be finite without any further constraints of the gauge invariance. Renormalization of Photon Self-energy As we saw above, the divergent terms of the vacuum polarization tensor Πµν (k) can be written as µ 2 ¶ α µ ν Λ α µν 2 2 µν µν g Λ + (k k −k g ) ln + finite term. (3.45) Π (k) = 2π 3π m2 e Now it is obvious that the self-energy of photon itself is not a physical observable. Further, the important point is that the vacuum polarization diagrams are never used for the renormalization scheme of evaluating physical observables in the triangle diagrams, in contrast to the fermion self-energy case.

3.3.5

Triangle Diagrams with Two Photons

In analogy with the vertex correction, we should consider the triangle diagrams which can be viewed as an external vertex Γ coupled to the photon self-energy diagram. The vertex Γ which couples to fermion or anti-fermion can be written in the following functions, Γ = 1 (scalar) , γ 5 (pseudoscalar) , γ µ γ 5 (axialvector) , γ µ (vector).

(3.46)

48 Fundamental Problems in Quantum Field Theory

Fujita and Kanda

However, the calculated results of these T-matrices of the triangle diagrams involving two photons have no divergences and the physical processes with the vacuum polarization diagrams are all finite as shown in Appendix B. This means that there is no necessity of considering the photon self-energy into the renormalization scheme. It is surprising that this fact is indeed overlooked by experts. We should note here that the calculations of the triangle diagrams are not so easy, but the calculations in detail (12 pages explanation) can be found in the textbook of Nishijima [12]. Therefore, one can then convince oneself that all of the triangle diagrams with any vertices coupled to fermions should not have any divergences at all.

3.3.6

Specialty of Photon Propagations

As we understand by now, the only serious divergence we have in the calculation of physical observables is concerned with the vertex corrections due to the propagation of photon, since there is no divergence in the vertex corrections due to the propagation of the massive vector boson as discussed in section 6. It turns out that this logarithmic divergence arises from the choice of the propagator of photon, which is the Feynman propagator DFµν (k) = −

g µν . k 2 − iε

(3.47)

This is, of course, the standard propagator of photon, and at the present day nobody would doubt the validity of this photon propagator. However, as we discuss in chapter 2, this Feynman propagator is not a correct one, and in fact, the self-energy type diagram is not properly calculated. This means that it is only justified for the scattering processes in which there is no loop involved. Theoretically, this Feynman propagator cannot satisfy the Lorentz condition of kµ ²µ = 0 since the numerator of the propagator is obtained because the following equation is assumed 4 X ²µ (λ, k)²ν (λ, k) = −g µν . (3.48) λ=1

It is obvious that this equation is inconsistent with kµ ²µ = 0. Now, one may consider the following propagator µ ν g µν − k kk2 (3.49) Dµν (k) = − 2 k − iε which can satisfy the Lorentz condition of kµ ²µ = 0 since the numerator of Dµν (k) should come from the following equation 4 X λ=1

¶ µ kµ kν µν . ² (λ, k)² (λ, k) = − g − 2 k µ

ν

(3.50)

However, this cannot satisfy the Coulomb gauge fixing condition of k ·² = 0, and therefore, we cannot employ the above propagator. In this respect, the correct photon propagator

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Fundamental Problems in Quantum Field Theory 49

should be the following one as discussed in detail in chapter 2  Coul A0 − part (k) = k12   D ³ ´  ab − ka kb  Dab (k) = 2 1 δ A − part. 2 k −iε k

(3.51)

Even though it is not very easy to handle, one has to employ a proper propagator of photon unless one can prove that the use of the Feynman propagator can be justified for all the physical processes one considers. As we show in chapter 2, the Feynman propagator can be used for the processes in which there is no fermion loop involved such as a fermion-fermion scattering with one photon exchange. Also, we note again that the Coulomb field A0 should not be quantized, but the propagator appear in the scattering process since fermion fields are always quantized. Infra-Red Singularity Here, we should note that the vertex corrections by the photon propagation contain the infrared singularity. Up to the present stage, we have neglected this infra-red singularity since it is consistent with experiments. However, this does not mean that we have understood the problem theoretically. At present, we believe that the infra-red singularity is unphys1 in ical since it is originated from the expression of the vector potential in terms of √2ω k eq.(3.26). This shows that the ωk = 0 part should not be included (k 6= 0), and this should give rise to the infra-red singularity when evaluating Feynman diagrams. On the other hand, the renormalization scheme of the massive vector boson propagations does not have any divergences at all in the vertex corrections, and therefore there is no need of the renormalization scheme. In this sense, the field theory of the massive vector boson is well understood. This strongly suggests that the difficulty in the vertex correction in QED should be related to the gauge freedom. In this respect, a common belief that only the gauge field theory is renormalizable is just meaningless. On the contrary, the QED is the most difficult field theory as long as the treatment of the logarithmic divergence in the vertex correction is concerned, which is basically due to the redundant degree of freedom of the vector potential A.

50 Fundamental Problems in Quantum Field Theory

3.4

Fujita and Kanda

Renormalization Scheme in QED: Fermions

The self-energy of fermion is considered to be related to physical observables after its renormalization into the wave function. The main purpose of the renormalization of fermion self-energy is to understand and calculate the magnetic moment of electron as the vertex correction which has a logarithmic divergence. Therefore, we present the standard procedure of the renormalization scheme of fermions. The fermion self-energy Σ(p) is obtained from the corresponding Feynman diagram as Z 1 1 d4 k 2 γµ γµ 2 (3.52) Σ(p) = −ie 4 (2π) p/ − k/ − m k where Λ denotes the cutoff momentum. This can be calculated to be µ ¶ Λ e2 (−p/ + 4m) + finite terms. Σ(p) = 2 ln 8π m

(3.53)

Therefore, the Lagrangian density of the free fermion part ¯ / − m)ψ LF = ψ(p

(3.54)

should be modified, up to one loop contributions, by the counter term δLF µ ¶ ¸ · 2 Λ e ln (−p/ + 4m) ψ. δLF = ψ¯ 8π 2 m In this case, the total Lagrangian density of fermion becomes L0 F = LF + δLF = ψ¯ [(1 + B)p/ − (1 + A)m] ψ + finite terms where A=−

e2 ln 2π 2

µ

Λ m

¶ ,

B=−

e2 ln 8π 2

µ

Λ m

(3.55)

¶ .

Now, one defines Z2 and δm as e2 Z2 ≡ 1 + B = 1 − 2 ln 8π µ ¶ Λ 3e2 m ln . δm ≡ 2 8π m

µ

Λ m

¶ (3.56a) (3.56b)

Here, one can introduce the wave function renormalization and the bare mass m0 p √ ψb ≡ Z2 ψ = 1 + Bψ (3.57a) µ ¶¶ µ µ ¶¶ µ Λ Λ e2 e2 −1 ' m − δm m0 ≡ Z2 m(1 + A) = m 1 + 2 ln 1 − 2 ln 8π m 2π m (3.57b)

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Fundamental Problems in Quantum Field Theory 51

where one should always keep up to order of e2 . In this case, one can rewrite L0 F as L0 F = ψ¯b (p/ − m0 )ψb + finite terms

(3.58)

which has just the same shape as the original one, and thus it is renormalizable. However the self-energy of fermions itself is not physical observables, and therefore it is not at all serious even if it has the logarithmic divergence. The basic problem should appear when there is a logarithmic divergence in the evaluation of the physical observables. The most important example is the vertex correction due to the photon propagation. If it has a logarithmic divergence, then it should be renormalized into the wave function.

3.4.1

Vertex Correction and Fermion Self-energy

Now we explain the renormalization procedure of the fermion self-energy case which can be directly related to the vertex correction. The self-energy of fermion Σ(p) is written as µ ¶ Λ e2 (−p/ + 4m). Σ(p) = 2 ln (3.59) 8π m It is, of course, clear that this contribution alone is not a physical observable. However, if one calculates the vertex correction which is indeed a physical process, then one realizes that one must make use of the logarithmic divergence of the fermion self-energy contribution such that the logarithmic divergence of the vertex correction can be completely canceled out by the wave function renormalization. In fact, the total Lagrangian density of free fermion together with the fermion self-energy part can be written as µ ¶¸ · 2 ¯ /ψ = ψ¯r p/ψr ¯ /ψ − e ln Λ ψp (3.60) LF = ψp 8π 2 m where ψr is defined as

s ψr =

e2 1 − 2 ln 8π

µ

¶ Λ ψ m

(3.61)

where we only write the p/ term since the mass term is not relevant in the present discussion. Now we can calculate the vertex correction Λµ (p0 , p) as µ ¶ Z 1 1 γν e2 Λ d4 k ν µ γ γ ' ln γµ. Λµ (p0 , p) = −ie2 (2π)4 p/ − k/ − m k 2 8π 2 m p/0 − k/ − m (3.62) Therefore, the total interaction Lagrangian density LI can be written as µ ¶ Λ e3 µ¯ ¯ µ ψ = −eAµ ψ¯r γµ ψr Aµ ψγ (3.63) LI = −eA ψγµ ψ + 2 ln 8π m where the logarithmic divergent part can be completely absorbed into the renormalized wave function.

52 Fundamental Problems in Quantum Field Theory

3.4.2

Fujita and Kanda

Ward Identity

Here, we should make a comment on the Ward identity [13]. This relation starts from the following equation 1 ∂ ∂ 1 = (p/ − k/ + m) ∂pµ p/ − k/ − m + iε (p − k)2 − m2 + iε ∂pµ +(p/ − k/ + m)

1 1 ∂ 1 = γµ 2 2 ∂pµ (p − k) − m + iε p/ − k/ − m + iε p/ − k/ − m + iε

(3.64)

which is always valid as an operator equation, and the Ward relation is written as Λµ (p, p) =

∂Σ(p) . ∂pµ

(3.65)

Since the self-energy of fermion is defined in eq.(3.52), the Ward relation corresponds to taking the first term in eq.(3.64) before the momentum integrations. The second term of eq.(3.64) should contribute to the vertex corrections, though it depends on the shape of the integrand. Here, one should be careful for the validity of eq.(3.65) since one normally makes use of the following free dispersion relation of p2 = m2 . In the evaluation of the selfenergy of fermion Σ(p), we replace the p2 term by m2 in the denominator of the fermion self-energy calculations. In this case, there is no guarantee that eq.(3.65) holds true since one has neglected the second term in eq.(3.64) to calculate the self-energy of fermion Σ(p) before the differentiation with respect to pµ . Therefore, one should carefully evaluate the vertex corrections without referring to the Ward identity. In fact, the validity of eq.(3.65) can be proved for the logarithmic divergence in the evaluation of the vertex corrections from the photon propagation. However, we believe that it is simply accidental because, this is not valid any more for the vertex corrections from the massive vector boson propagation. In this respect, we see that identity equations are sometimes useful for checking mathematical formula, but we should be very careful for applying the identity equation to physical processes.

Quantum Electrodynamics

3.5

Fundamental Problems in Quantum Field Theory 53

Photon-Photon Scattering

Here, we give an example of the S-matrix calculation, and choose the S-matrix evaluation of photon-photon scattering. The evaluation of the S-matrix should start from the following Hamiltonian density Z 0 3 ¯ ˆ ˆ H = e ψ(x)γψ(x) · A(x)d r. (3.66) The corresponding Feynman diagrams of the photon-photon scattering are shown in Fig. .

(a): Ma

(b): Mb

(c): Mc

Figure 1: Feynman diagrams of the photon-photon scattering.

3.5.1

Feynman Amplitude of Photon-Photon Scattering

The Feynman amplitude of the photon-photon scattering can be written as [14] 0 0 ¯ aµνλσ + M ¯ µνλσ + M ¯ cµνλσ )²rµ (k)²sν (l)²s0 (l0 )²rσ0 (k 0 ) M rr ss = 2(Ma + Mb + Mc ) = 2(M λ b (3.67) where we note that the Feynman amplitude is not affected by the direction of the loop momentum [12]. The amplitude Ma can be explicitly written as Z (ie)4 0 0 d4 q²rµ (k)²rσ (k 0 )²sλ (l0 )²sν (l)× Ma = − 4 (2π)

Tr[γ µ (q/ − l/ − k/ + m)γ σ (q/ − l/0 + m)γ λ (q/ + m)γ ν (q/ − l/ + m)] . [(q − l − k)2 − m2 + iε][(q − l0 )2 − m2 + iε][q 2 − m2 + iε][(q − l)2 − m2 + iε] (3.68) Here, it should be important to note that the total Feynman amplitude of the photon-photon scattering does not have any logarithmic divergence, even though the amplitude Ma alone has the logarithmic divergence. This is quite important in that the physical processes should not have any divergences, and indeed the photon-photon scattering is just the case. This

54 Fundamental Problems in Quantum Field Theory

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strongly suggests that the evaluation of the Feynman amplitude of the photon-photon scattering process must be directly connected to the real physical process which should be observed by experiments.

3.5.2

Logarithmic Divergence

The amplitude Ma has the logarithmic divergence, and this can be seen when we check the large q behavior [1]. In this case, we find from eq.(3.68) Z 4 (ie)4 0 0 d4 q {g µν g λσ + g µσ g νλ − 2g νσ g µλ }²rµ ²rν ²sλ ²rσ (3.69) Ma ∼ − 4 2 2 (2π) 3(q ) which has obviously the logarithmic divergence. However, if we add all of the amplitude together, then we find from eq.(3.67) Z 1 e4 M ∼ − 4 d4 q 2 2 {g µν g λσ + g µσ g νλ − 2g νσ g µλ + g µν g σλ + g µλ g νσ − 2g νλ g µσ 6π (q ) 0

0

+g σν g λµ + g σµ g νλ − 2g νµ g σλ }²rµ ²rν ²sλ ²rσ = 0.

(3.70)

This means that the total amplitude has no divergence at all because of the cancellation, and it is indeed finite. Therefore, we do not have to employ any specific regularization scheme, and thus the evaluation is very reliable.

3.5.3

Definition of Polarization Vector

Here, we take the polarization vector as defined by Lifshitz [1] (1)

(1)

(1)

(1)

²1 = ²2 = ²3 = ²4 = (2)

²4 =

k × k0 |k × k0 |

(2)

²1 =

1¡ (1) ¢ (2) k × ²1 = −²2 ω

1¡ 0 (1) ¢ (2) k × ²4 = −²3 . ω

(3.71)

(i)

en denotes the polarization vector of photon. Here, we take the Coulomb gauge fixing ∇ · A = 0. Now, each photon has the following momenta (a) initial state photon 1 : k µ = (ω, k)

photon 2 : lµ = (ω, l)

(b) final state photon 3 : l0µ = (ω, l0 )

photon 4 : k 0µ = (ω, k0 ).

ω = |k| = |l| = |k0 | = |l0 | Noting that there is no rest system, we find l = −k,

l0 = −k0 .

Quantum Electrodynamics

3.5.4

Fundamental Problems in Quantum Field Theory 55

Calculation of Ma at Low Energy

Now, we carry out the calculation of Ma as an example ¯ aµνλσ ²rµ (k)²sν (l)²s0 (l0 )²rσ0 (k 0 ) Ma = M λ

(3.72)

¯ aµνλσ can be written as where M Z

4

¯ aµνλσ = − (ie) M (2π)4

d4 q×

Tr[γ µ (q/ − l/ − k/ + m)γ σ (q/ − l/0 + m)γ λ (q/ + m)γ ν (q/ − l/ + m)] . [(q − l − k)2 − m2 + iε][(q − l0 )2 − m2 + iε][q 2 − m2 + iε][(q − l)2 − m2 + iε] (3.73) By making use of the Feynman parameter [15], we find Z Z 1 Z z1 Z z2 (ie)4 4 µνλσ ¯ d q 3! dz1 dz2 dz3 × Ma =− (2π)4 0 0 0 Tr[γ µ (q/ − l/ − k/ + m)γ σ (q/ − l/0 + m)γ λ (q/ + m)γ ν (q/ − l/ + m)] [q 2 − 2q.A + B + iε]4 where

(3.74)

Aµ ≡ lµ z1 + (l0 − l)µ z2 + k 0µ z3 B ≡ 2l.kz3 − m2 .

By introducing t = q − A, we find 4

¯ µνλσ = − (ie) 3! M a (2π)4

Z 0

Z

1

dz1

Z

z1

0

dz2

z2

0

dz3 ×

Z

Tr[γ µ (/t + A / + m)γ ν (/t + A / − l/ + m)] / − l/ − k/ + m)γ σ (/t + A / − l/0 + m)γ λ (/t + A . 2 2 4 [t − A + B + iε] (3.75) Here, for simplicity, we define d4 t

b/ = A / − l/0 ,

a/ = A / − l/ − k, / and thus

4

¯ aµνλσ = − (ie) 3! M (2π)4 Tr[γ µ (/t

+ a/ +

m)γ σ (/t

Z 0

Z

1

dz1

c/ = A, / Z

z1

0

dz2

0

d/ = A / − l/ Z

z2

dz3

m)γ λ (/t

d4 t×

+ b/ + + c/ + m)γ ν (/t + d/ + m)] . [t2 − A2 + B + iε]4

In the numerator, we find Tr[odd-numbers of γ-matrices] = 0

(3.76)

56 Fundamental Problems in Quantum Field Theory

and by noting

Fujita and Kanda

Z d4 t t|µ tν{z · · · tλ} f (t2 ) = 0 odd

we find

Tr[γ µ (/t + a/ + m)γ σ (/t + b/ + m)γ λ (/t + c/ + m)γ ν (/t + d/ + m)] = t2 F µνλσ + Gµνλσ + Tr[γ µ t/ γ σ t/ γ λ t/ γ ν t/ ]

where

(3.77)

F µνλσ ≡ −4m2 {g µσ g λν + g µν g σλ − 2g µλ g σν } 1 − Tr[{γ µ γ σ γ λ c/ γ ν d/ + γ µ γ λ b/ γ σ γ ν d/ + γ µ γ σ b/ γ λ c/ γ ν 2 +(γ µ aγ / σ )(γ λ γ ν d/ + γ ν c/ γ λ + b/ γ λ γ ν )}] µνλσ

G

µ

σ

λ

ν

≡ Tr[γ (a/ + m)γ (/b + m)γ (/c + m)γ (d/ + m)].

(3.78a) (3.78b)

Now, we find 4

¯ aµνλσ = − (ie) 3! M (2π)4

Z

Z

1

dz1

0

0

Z

z1

dz2

0

Z

z2

dz3

d4 t

t2 F µνλσ + Gµνλσ + Tr[γ µ t/ γ σ t/ γ λ t/ γ ν t/ ] (t2 − A2 + B + iε)4

and we can carry out the integration and obtain [15] ½ µνλσ Z Z z1 Z z2 Gµνλσ (ie)4 1 2 2F µνλσ ¯ dz dz dz iπ + Ma =− 1 2 3 (2π)4 0 B − A2 (B − A2 )2 0 0 ¯ ¶¾ µ ¯ ¯ Λ2 ¯ 11 µσ λν µν λσ σν µλ ¯ ¯ − +8(g g + g g − 2g g ) ln¯ B − A2 ¯ 6

(3.79)

ω ¿ 1 and expand where Λ denotes the cutoff momentum. Here, we consider the case of m ω the integration in terms of m powers. This approximation is quite reasonable as we show it later [16]. Noting that the m never appears in k, l, k 0 , l0 in F µνλσ , Gµνλσ , but only ω appears. Therefore, we can rewrite F µνλσ , Gµνλσ and obtain

where

1 F µνλσ = −4m2 Qµνλσ − ω 2 Rµνλσ 2

(3.80a)

Gµνλσ = ω 4 S µνλσ + m2 ω 2 Rµνλσ + m4 T µνλσ

(3.80b)

Qµνλσ ≡ g µσ g λν + g µν g σλ − 2g µλ g σν Rµνλσ ≡

(3.81a)

1 Tr[{γ µ γ σ γ λ c/ γ ν d/ + γ µ γ λ b/ γ σ γ ν d/ + γ µ γ σ b/ γ λ c/ γ ν ω2

S µνλσ ≡

+(γ µ aγ / σ )(γ λ γ ν d/ + γ ν c/ γ λ + b/ γ λ γ ν )}]

(3.81b)

1 Tr[γ µ aγ / σ b/ γ λ c/ γ ν d] / ω4

(3.81c)

Quantum Electrodynamics

Fundamental Problems in Quantum Field Theory 57

T µνλσ ≡ Tr[γ µ γ  σ γ λ γ ν ].

 

(3.81d)

Here, Qµνλσ , Rµνλσ , S µνλσ , T µνλσ are all dimensionless. In this case, we can easily expand Qµνλσ , Rµνλσ , S µνλσ , T µνλσ . Therefore, we can write ¾ ½ ω2 B − A = −m 1 − 2 Ua m 2

2

i h θ θ Ua = 4 z2 (z2 − z1 − z3 ) sin2 − z3 z1 cos2 + z3 2 2 and we have 4

¯ aµνλσ = − (ie) iπ 2 M (2π)4 ½

Z

Z

1

dz1

0

Z

z1

0

dz2

z2

0

dz3 ×

¶¾ µ ¯ 2¯ ¯Λ ¯ Gµνλσ 2F µνλσ 11 2 µνλσ ¯ ¯ + + 8Q ln¯ 2 ¯ − ln |1 − ξ Ua | − B − A2 (B − A2 )2 m 6

(3.82)

ω . where we define ξ ≡ m Here, we omit the index of µ, ν, λ, σ in µνλσ µνλσ µνλσ µνλσ Q , R , S , T . Thus, we find 4

¯ aµνλσ = − (ie) iπ 2 M (2π)4

Z

Z

1

dz1

0

Z

z1

dz2

0

½

z2

dz3

0

8Q + ξ 2 R ξ 4 S + ξ 2 R + T + 1 − ξ2U (1 − ξ 2 U )2

¶¾ µ ¯ 2¯ ¯Λ ¯ 11 . +8Q ln¯¯ 2 ¯¯ − ln |1 − ξ 2 U | − m 6 Expanding in terms of ξ =

ω m,

(3.83)

we find

½ ¶ µ ¯ 2¯ ¾ ¯ Λ ¯ 11 ¯ ¯ dz1 dz2 dz3 8Q + 8Q ln¯ 2 ¯ − + T + ··· . m 6 0 0 0 (3.84) ¡ ¯ Λ2 ¯ 11 ¢ ¯ ¯ The coefficient of Q at the lowest order can be written as 8 + 8 ln m2 − 6 , which does not depend on the shape of Ma , Mb , Mc . Therefore, we find 4

¯ aµνλσ = − (ie) iπ 2 M (2π)4

Z

Z

1

4

(ie) ¯ µνλσ ∼ iπ 2 M =− a (2π)4

Z 0

Z

z1

Z

1

dz1

0

z2

Z

z1

dz2

0

z2

dz3 T = −

(ie)4 2 1 iπ T (2π)4 6

(3.85)

and we can write it explicitly as 4 ¯ aµνλσ = − (ie) iπ 2 1 T µνλσ . M (2π)4 6

(3.86)

58 Fundamental Problems in Quantum Field Theory

  3.5.5

Total Amplitude

Fujita and Kanda

 

¯ aµνλσ , M ¯ µνλσ and M ¯ cµνλσ at ω ¿ 1. Thus, we In this way, we can obtain the shape of M b m can write for Ma , Mb , Mc as ½ ¾ (ie)4 2 1 0 0 µσ λν µν σλ µλ σν iπ 4(g g + g g − g g ) ²rµ ²sν ²sλ ²rσ (3.87a) Ma = − (2π)4 6 ½ ¾ (ie)4 2 1 0 0 µλ σν µν σλ µσ λν iπ 4(g g + g g − g g ) ²rµ ²sν ²sλ ²rσ Mb = − (3.87b) 4 (2π) 6 ½ ¾ (ie)4 2 1 0 0 µσ λν σν µλ σλ µν iπ 4(g g + g g − g g ) ²rµ ²sν ²sλ ²rσ . (3.87c) Mc = − 4 (2π) 6 In this case, the total amplitude can be calculated as ½ (ie)4 2 1 1 M = Ma + Mb + Mc = − iπ 4 g µσ g λν + g µν g σλ − g µλ g σν 2 (2π)4 6 ¾ 0 0 σλ µν µλ σν µν σλ µσ λν µσ λν σν µλ ²rµ ²sν ²sλ ²rσ +g g + g g − g g + g g + g g − g g =−

´ (ie)4 2 1 ³ µσ λν 0 0 µν σλ µλ σν iπ 4 g g + g g + g g ²rµ ²sν ²sλ ²rσ . (2π)4 6

Therefore, M becomes M = −i

3.5.6

´ 4 2 ³ µσ λν 0 0 α g g + g µν g σλ + g µλ g σν ²rµ ²sν ²sλ ²rσ . 3

(3.88)

Cross Section of Photon-Photon Scattering

Now, the photon-photon scattering cross section can be written as [1] 1 1 dσ = |M |2 2 dΩ 64π (2ω)2

(3.89)

where M is given as M = −i

´ 4 2 ³ µσ λν 0 0 α g g + g µν g σλ + g µλ g σν ²rµ ²sν ²sλ ²rσ . 3

(3.90)

Now, we sum up the final states of the photon polarization state and make average of the initial polarization states 1 X 0 0 |M |2 = |M rr ss |2 4 0 0 rr ss

where the non-vanishing term is written as 1 n 1111 2 |M | + |M 2222 |2 + |M 1122 |2 + |M 2211 |2 |M |2 = 4

Quantum Electrodynamics

Fundamental Problems in Quantum Field Theory 59

o   |M 2112 |2 + |M 1212 |2 + |M 2121 |2 . +|M 1221 |2 +

 

(3.91)

Since we know that M 1122 = M 2211 ,

M 1212 = M 2121

M 1221 = M 2112 ,

we obtain o 1 X 1 n 1111 2 0 0 |M | + |M 2222 |2 + 2|M 1122 |2 + 2|M 1221 |2 + 2|M 1212 |2 . |M rr ss |2 = 4 0 0 4 rr ss (3.92) Further, we find 4 M 1111 = −i α2 · 3 3 4 M 2222 = −i α2 (1 + 2 cos2 θ) 3 4 M 1122 = −i α2 cos θ 3 4 M 1221 = i α2 cos θ 3 4 M 1212 = i α2 · 1 3 and thus the amplitude squared with the polarization sum becomes ª 16 © 1X |M |2 = α2 3 + 2 cos2 θ + cos4 θ . 4 9 Therefore, for

ω m

(3.93)

¿ 1, we find the differential cross section ¢ 1 α4 ¡ dσ = 3 + 2 cos2 θ + cos4 θ . 2 2 dΩ (6π) (2ω)

This is the photon-photon scattering cross section for

ω m

¿ 1 case.

(3.94)

60 Fundamental Problems in Quantum Field Theory

  3.6

Fujita and Kanda

Photon-Photon Scattering Proposal to Measure  

We discuss a possibility to measure the photon-photon scattering cross section at low energy in a theoretical standpoint. The cross section of photon-photon scattering at low energy can be given as above α4 dσ ' (3 + 2 cos2 θ + cos4 θ) dΩ (12π)2 ω 2 with ω the energy of photon. The magnitude of the cross section at ω ' 1 eV should be 1037 times larger than the prediction of Heisenberg and Euler by the classical picture of field theory. However, due to a difficulty of the initial condition of photon-photon reaction process, we propose to first measure γ + γ → e+ + e− reaction at a few MeV before measuring γ + γ → γ + γ elastic scattering. Historically, the photon-photon cross section was calculated by Heisenberg and Euler in 1936 [17, 18] and it is written as [1] 139α4 ³ ω ´6 dσ ' (3 + cos2 θ)2 dΩ (180π)2 m2 m

(with ω ¿ m).

(3.95)

This result was confirmed by Karplus and Neuman [19] with the modern field theory calculation. Since then, it is believed that this photon-photon cross section is the correct one, even though the quantum evaluation of the Feynman diagrams gives the cross section of eq.(3.94). However, Karplus and Neuman made use of some additional but unphysical conditions (gauge conditions). In addition, the effective Lagrangian method of Heisenberg and Euler gave the incorrect result of the vacuum polarization effects which disagree with the observation that photon is always massless [20]. Therefore, eq.(3.95) is not correct at all. Up to now, the measurements of the photon-photon scattering cross section have been made by Moulin et al. [22, 21], but they found no evidence of the photon-photon scattering. However the measurements with the sufficient accuracy must be very difficult since photon cannot be at rest but always at the speed of light. Most of the scattering experiments are the collision experiment of the incident particle with some fixed targets which are basically taken to be at rest. In this respect, the photon-photon scattering must be quite new to the conventional experiments, and therefore this experiment must be one of the most important experiments in particle physics, which is left almost untouched until now. In this sense, the main difficulty must be connected to the initial condition of the scattering experiments in which one should control the time of photon-photon collision and the focusing of the photon-photon beams. In addition, the photon-photon scattering is the reaction process arising from the particle nature of photon, in contrast to the wave nature of photon such as diffraction or interference phenomena. Initial Photon-Photon Beams Before going to discuss the elastic photon-photon scattering, we should first clarify what should be the system we can make any experiments on photon-photon scattering. This is,

Quantum Electrodynamics

Fundamental Problems in Quantum Field Theory 61

  of course, related to the fact that  we cannot specify the special system for photon since it cannot be at rest. In other words, photon has no reference frame, and thus we should first understand in which system we can carry out the experiments. In this respect, we believe that the γ + γ → e+ + e− experiment should be first used as the monitor of the initial state condition check of the photon-photon scattering experiment. The reaction cross section of γ + γ → e+ + e− is the same order of magnitude as the Compton scattering when the incident photon energy is larger than a few MeV. Therefore, it is crucial that the experimental setup should be able to reproduce the cross section of γ + γ → e+ + e− process. Even though the photon-photon cross section should be smaller than the γ + γ → e+ + e− by several order of magnitudes in a few MeV region, it should be possible to measure the photon-photon elastic cross section once the problem of the initial condition is resolved. We believe that the measurement of two photons should not be very difficult indeed, even though a very small number.

3.6.1

Possible Experiments of Photon-Photon Elastic Scattering

If the photon-photon cross section of eq.(3.95) were correct, then there was no chance to observe it in terms of the laser-laser scattering experiment since the energy of the laser is mostly lower than a few tens of eV. However, the photon-photon scattering experiment at high energy must have a serious difficulty since the experiment should be done as the headon collision in the center of mass system of two photons, and the control of the high energy photon flux must be non-trivially difficult. On the other hand, the situation is completely different if one should observe the photonphoton cross section of eq.(3.94). In this case, we should consider the laser-laser scattering where the energy of the laser is around ω = 1 eV or lower. If the measurement is carried out at θ = 90 degree, then the cross section of eq.(3.94) at ω ' 1 eV becomes 3α4 dσ ' ' 2.3 × 10−21 cm2 ' 2.3 × 106 mb/st dΩ (12π)2 ω 2

(3.96)

which should be well detectable. On the other hand, the cross section of eq.(3.95) at ω ' 1 eV becomes 3 × 417α4 ³ ω ´6 dσ ' ' 9.3 × 10−67 cm2 ' 9.3 × 10−40 mb/st (3.97a) dΩ (180π)2 m2 m which is extremely small, and it is impossible to detect in any of the experiments. We note that the above cross section of eq.(3.95) becomes larger for larger photon energy. In fact, the cross section at ω ' 1 MeV becomes 417α4 ³ ω ´6 dσ ' ' 9.3 × 10−31 cm2 ' 9.3 × 10−4 mb/st. (3.97b) dΩ (180π)2 m2 m However, in this energy region, the low energy approximation is not well satisfied even though, we believe, the order of magnitude estimation must be correct. Unfortunately, there is no such high energy laser available at present.

62 Fundamental Problems in Quantum Field Theory

  3.6.2

Fujita and Kanda

Comparison with e+  + e− → e+ + e− Scattering

In terms of the reaction process, the photon-photon scattering must be similar to the e+ e− elastic scattering. The cross section of the e+ e− elastic scattering process at high energy limit is given as · ¸ α2 1 + cos4 (θ/2) 1 + cos2 θ 2 cos4 (θ/2) dσ ' − + . (3.98) dΩ 8E 2 2 sin4 (θ/2) sin2 (θ/2) The typical number of this cross section can be seen from experiment at E = 17 GeV dσ ' 1.6 × 10−7 mb/st. In order to find a naive and θ = 90 degree, and it becomes dΩ estimation of the cross section at around E ∼ 2 MeV, we extrapolate the energy dependence dσ ∼ 12 mb/st. On the other hand, the photon-photon cross of eq.(3.98), and thus we find dΩ dσ ∼ 6 × 10−7 mb/st which is section of the present estimation at ω ∼ 2 MeV becomes dΩ + − smaller than the e e elastic scattering cross section by seven orders of magnitude. This naive estimation of the photon-photon cross section indicates that the cross section becomes quite large at low energy. However, in comparison with the head-on collisions between the e+ e− elastic scattering cross section and the photon-photon cross section at 1 MeV incident energy, the photon-photon cross section is smaller than the e+ e− elastic scattering cross section by several orders of magnitude.

3.6.3

Comparison with γ + γ → e+ + e− Scattering

If the energy of photon is larger than a few MeV, then we have to consider the scattering process in which the photon-photon scattering can produce the electron positron pair, that is, γ + γ → e+ + e− . This cross section is the same order as the e+ e− elastic scattering cross section, and therefore, at higher energy than a few MeV, the photon-photon scattering process must be dominated by the γ + γ → e+ + e− cross section. In this respect, the γ + γ → e+ + e− should be used as the monitor of the reaction process before carrying out the photon-photon elastic scattering. This is clear since the main difficulty of the photon-photon scattering should be concerned with the initial conditions of photon-photon reaction, and therefore one should examine the validity of the reaction process first by carrying out the γ + γ → e+ + e− experiment. It should be noted that the photon-photon elastic cross section must be smaller than the γ + γ → e+ + e− reaction cross section by several orders of magnitudes. However, we believe it should be observed as long as we can judge from the magnitude of the cross section.

3.6.4

Discussions

The cross section we discuss is related to the probability of the scattering process when two photons collide. The basic difficulty of this scattering problem is indeed related to the fact that this scattering process is only possible for the head-on collision. Namely, the initial condition of the scattering process must be most difficult when setting up the photon-photon scattering experiment.

Quantum Electrodynamics

  3.7

Fundamental Problems in Quantum Field Theory 63

Equation Chiral Anomaly: Unphysical  

Here, we discuss the problem of the chiral anomaly equation and clarify that the anomaly equation is not connected to any physical observables. Before going to the discussion of the chiral anomaly equation, we first confirm that the π 0 → 2γ process has no divergence, and therefore there should not be any anomalous behaviors in the theoretical evaluation of this reaction process.

3.7.1

π 0 → γ + γ process

Now, we consider the reaction process of π 0 → 2γ where the interaction Lagrangian density between fermion ψ and pion ϕ with its coupling constant gπ can be written as ¯ 5 ψϕ L1 0 = igπ ψγ where the isospin indices are suppressed. In this case, the T-matix of the corresponding Feynman diagrams for the π 0 → 2γ reaction can be written as · Z d4 p 1 2 Tr (γ²1 ) (γ²2 ) Tπ0 →2γ = e gπ 4 (2π) p/ − M + iε ¸ 1 1 5 γ + (1 ↔ 2) (3.99) × p/ − k/2 − M + iε p/ + k/1 − M + iε where k1 (k2 ) and ²µ1 (²ν2 ) denote the four momentum and the polarization vector of two photons, respectively. Also, k1 and k2 satisfy the following relations k12 = 0,

k22 = 0,

2k1 · k2 = m2π .

Here, M and mπ denote the nucleon and pion masses, respectively. Now, we can rewrite eq.(3.99) to evaluate the Trace parts as Z Aµν ²µ1 ²ν2 d4 p Tπ0 →2γ = 2e2 gπ (2π)4 (p2 − M 2 )((p − k2 )2 − M 2 )((p + k1 )2 − M 2 ) where Aµν is defined as Aµν ≡ Tr[γµ (p/ + M )γν (p/ − k/2 + M )γ 5 (p/ + k/1 + M )]. Linear Divergence Term Now the linear divergence term should correspond to the term which is proportional to p3 , and thus we can show 5 A(3) µν = Tr[p/γµ p/γν p/γ ] = 0 which is due to the property of the Trace with γ 5 matrix.

64 Fundamental Problems in Quantum Field Theory

Fujita and Kanda

  Logarithmic Divergence Term   Next, we should evaluate the p2 terms which can be written as 5 5 5 A(2) µν = Tr[γµ p/γν p/γ ] + Tr[p/γµ γν p/γ ] + Tr[p/γµ p/γν γ ] = 0

and these terms also vanish to zero by the Trace evaluation. Here we have made use of the following identity Tr[γµ p/ γν p/γ 5 ] = −4iεµρνσ pρ pσ = 0. Therefore, the T-matrix of the π 0 → 2γ process has neither linear nor logarithmic divergences, and this is proved at the level of the Trace evaluation before the momentum integrations. Finite Term Now, we can easily evaluate this momentum integral, and the result becomes Tπ0 →2γ '

e2 gπ εµναβ k1α k2β ²µ1 ²ν2 . 4π 2 M

(3.100)

As one sees, there is no divergence in this T-matrix calculation, and this is because the apparent linear and logarithmic divergences can be completely canceled out due to the Trace evaluation. In this respect, the corresponding T-matrix is finite and thus there is no chiral anomaly in this Feynman diagrams. This is, of course, well known, and the calculation of the T-matrix is explained quite in detail in the textbook of Nishijima in 1969 [12]. Decay Width of π 0 → 2γ In this case, we can calculate the decay width Γπ0 →2γ as Γπ0 →2γ

1 = 8mπ |p1 ||p2 |(2π)2

Z δ(mπ − |p1 | − |p2 |)δ(p1 + p2 )|U |2 d3 p1 d3 p2 (3.101)

where |U |2 is given as |U |2 =

1 X |Tπ0 →2γ |2 2

(3.102)

λ1 ,λ2

where λ1 and λ2 denote the polarization state of two photons. The summation of the polarization state of two photons can be carried out by making use of the Coulomb gauge fixing which gives the polarization sum as given in eq.(2.54) ´  ³ a b  2 for µ 6= 0, ν 6= 0  δ ab − kkk2 X ²∗ µk,λ ²νk,λ = (3.103)   λ=1 0 for µ, ν = 0.

Quantum Electrodynamics

Fundamental Problems in Quantum Field Theory 65

  After some calculations, we obtain   the decay width Γπ0 →2γ as Γπ0 →2γ '

α2 gπ2 m3π ' 7.4 eV 16π 2 4π M 2

(3.104)

which can be compared with the observed value [23] Γexp = 7.8 eV. π 0 →2γ As seen above, the calculation can well reproduce the observed data of the life time of the 2 gπ ' 8 which should be slightly smaller than π 0 → 2γ decay. Here, we take the value of 4π the one determined from the nucleon-nucleon scattering experiments. This is clear since the π 0 → 2γ process should naturally include the effect of the nucleon form factor, in contrast to the value of the πN N coupling constant obtained from the nucleon-nucleon scattering experiments. In the case of N N scattering, the nucleon form factors are introduced to accommodate the finite size effect of nucleons in the scattering process. Here, it should be important to note that the calculation of the decay width with the Coulomb gauge fixing is quite involved. On the other hand, the choice of the polarization sum of two photons X µ ²∗ k,λ ²νk,λ = −g µν (3.105) λ

can also reproduce the correct decay width as given in eq.(3.104), even though the calculation with this choice of the polarization sum is much easier than the case with the correct expression of the polarization sum. However, the expression of Peq.(3.105) cannot be justified for ν = µ = 0 case. This is clear since the left hand side λ |²0k,λ |2 is always positive definite while the right hand side is negative. In this respect, the employment of eq.(3.105) is accidentally justified because of the special property of the amplitude in eq.(3.100).

3.7.2

Standard Procedure of Chiral Anomaly Equation

Now, we briefly review the procedure how people obtained the chiral anomaly equation, and clarify where the anomaly equation came up in the process [24, 25]. The starting point of the chiral anomaly equation is the Feynman diagram which involves the triangle diagrams with three vertex interactions of ieγ µ , ieγ µ and igz γ µ γ5 . Here, we can write the Lagrangian density of the weak Z 0 boson Z µ and the fermion ψ with its coupling constant gz as ¯ µ γ5 ψZ µ L2 0 = gz ψγ which is due to the standard electroweak interactions [15]. In this case, the corresponding T-matrix for the axial vector coupling can be written as · Z 1 d4 p 2 Tr (γ²1 ) (γ²2 ) TAV C ' e gz (2π)4 p/ − M + iε ¸ 1 1 γ5 (γ²v ) + (1 ↔ 2) × (3.106) p/ − k/2 − M + iε p/ + k/1 − M + iε

66 Fundamental Problems in Quantum Field Theory

Fujita and Kanda

µ   where ²v denotes the polarization  vector of the Z 0 boson. In this case, there seems to be an apparent linear divergence in eq.(3.106), and therefore people worried about the renormalization procedure. Below is the description of the field theory textbook in order to derive the chiral anomaly equation in four dimensions [24, 25]. First, they define the amplitude Mµνρ in eq.(3.106) as

TAV C = ²µ1 ²ν2 ²ρv Mµνρ . In this case, they say that the amplitude Mµνρ has an ambiguity which can be expressed in terms of an arbitrary momentum a, because they thought that there must be a linear divergence in the T-matrix. Using a mathematical identity, they found the following equation Mµνρ (a) = Mµνρ (0) −

β µνρσ ε (k1 − k2 )σ 8π 2

where a is chosen as a = αk1 + (α − β)k2 . The second term in the right hand side corresponds to the surface term which becomes finite. Here, β can be determined so that the gauge condition can be satisfied, and they obtained the chiral anomaly equation. ∂µ j5µ =

e2 ρσµν ε Fρσ Fµν . 16π 2

This is a story how people found the chiral anomaly equation. In reality, as we discuss below, the triangle diagrams with the axial vector current coupling do not have any divergences (neither linear nor logarithmic). Therefore, there is no need of the regularization, and thus the anomaly equation is indeed spurious.

3.7.3

Z 0 → γ + γ process

Here, we discuss the physical processes which involve the axial vector coupling with vector bosons. In fact, if we include the weak interactions, then the triangle diagrams with the axial vector coupling should be connected to the physical observables. Namely, there is a possible decay process of a weak vector boson into two photons, that is, Z 0 → 2γ. However, this decay process is forbidden due to the Landau-Yang theorem [26, 27]. The physical reason why the decay rate of Z 0 → 2γ process vanishes to zero must be due to the symmetry arguments. The two photon state can make 1+ state of Z 0 boson due to the vector product, which is anti-symmetric. However, the two photon state must be symmetric due to its bosonic nature, and therefore the decay of Z 0 → 2γ process is forbidden as a physical process even with the parity non-conserving interaction. Nevertheless, we should carry out the T-matrix evaluation of the Feynman diagrams corresponding to the Z 0 decay into two photons, and we show that the triangle diagrams with the axial vector coupling have neither linear nor logarithmic divergences. This is proved without any regularizations since it is evaluated before the momentum integrations, and the total amplitude of Z 0 → 2γ decay process is indeed finite [28].

Quantum Electrodynamics

Fundamental Problems in Quantum Field Theory 67

T-matrix Evaluation Here, we briefly explain the T-matrix evaluation, and the corresponding T-matrix for the Z 0 boson decaying into two photons is just given in eq.(3.106) · Z 1 d4 p 2 Tr (γ²1 ) (γ²2 ) TZ 0 →2γ ' e gz 4 (2π) p/ − M + iε ¸ 1 1 γ5 (γ²v ) + (1 ↔ 2). (3.107) p/ − k/2 − M + iε p/ + k/1 − M + iε Linear Divergence First, we show that the linear divergence should vanish to zero because of the following equation Z Λ kµ =0 lim d4 k 2 Λ→∞ −Λ (k − s0 )2 which is just the same integration of the linear divergence appearing in the fermion selfenergy diagrams. In fact, the fermion self-energy Σ(p) can be written as µ ¶ Z Λ k/ e2 d4 k 2 (−p/ + 4m) − 2ie Σ(p) = 2 ln 4 2 8π m (2π) (k − s0 )2 where the last integral term is, of course, set to zero, and the first term just corresponds to the fermion self-energy contribution. In reality, however, the apparent linear divergence vanishes to zero before the momentum integration. This can be easily proved since the corresponding Trace evaluation of the T-matrix becomes Tr[p/γµ p/γν p/γρ γ5 ]²µ1 ²ν2 ²ρv + Tr[p/γµ p/γν p/γρ γ5 ]²µ2 ²ν1 ²ρv = 0

(3.108)

where we have made use of the following identity equation Tr[p/γµ p/γν p/γρ γ5 ] = −Tr[p/γν p/γµ p/γρ γ5 ]. Therefore, the linear divergence disappears in eq.(3.107) before carrying out the momentum integration. Logarithmic Divergence Term The p2 term of the numerator in eq.(3.107) contains the apparent logarithmic divergence. However, we show that the logarithmic divergence term vanishes to zero exactly [28]. First, we calculate the Trace of the γ− matrices, and find the following shape for the logarithmic (0) divergence term TZ 0 →2γ as Z (0) TZ 0 →2γ

2

' gz e

Z

1

dx 0

Z

x

dy 0

F (p, x, y) d4 p (2π)4 (p2 − s0 + i²)3

(3.109)

68 Fundamental Problems in Quantum Field Theory

Fujita and Kanda

  where s0 is written as s0 ' m2t   with mt the top quark mass of mt ' 172 GeV/c2 , and F (p, x, y) is defined as F (p, x, y) ≡ Tr[p/γµ p/γν a/γρ γ5 ] + Tr[p/γµ b/γν p/γρ γ5 ] + Tr[c/γµ p/γν p/γρ γ5 ] where a, b, c are given as a = −k1 (1 − x) − k2 (1 − y), b = −k1 (1 − x) + k2 y, c = −k1 x + k2 y. After some tedious but straight forward calculation, one obtains (0)

TZ 0 →2γ = 0

(3.110)

and therefore there is no need of the renormalization since the triangle diagrams are indeed all finite. Finite Term and Decay Width Here, we present the calculated decay width of the Z 0 → 2γ process where only the intermediate top quark state is considered since its contribution is the largest among all the other fermions. Therefore, the T-matrix for the Z 0 → 2γ process can be written as Z 1 Z x Z A(x, y) d4 p 2 (3.111) TZ 0 →2γ = gz e dx dy 4 2 (2π) (p − s0 + i²)3 0 0 where A(x, y) is given as A(x, y) = −4im2t (x + 1 − y)εµνρα ²µ1 ²ν2 ²ρv (k1α − k2α ).

(3.112)

The integrations of the momentum p and parameters x, y can be carried out in a straight forward way, and the T-matrix becomes µ ¶2 2e gz εµνρα ²µ1 ²ν2 ²ρv (k1α − k2α ). TZ 0 →2γ = − 2 (3.113) 6π 3 Now, we can prove that this should vanish to zero by choosing the system where Z 0 boson should be at rest. In this case, we can take the polarization vector ²ρv as ²ρv = (0, ²v )

(3.114a)

which can satisfy the Lorentz condition of kµ ²µv = 0. On the other hand, we can also choose the photon polarization vectors ²µ1 and ²ν1 with the Coulomb gauge fixing as ²µ1 = (0, ²1 ), ²ν1 = (0, ²2 )

with

k · ²1 = 0, k · ²2 = 0.

(3.114b)

Further, we see that the (k1α − k2α ) should be expressed as k1α − k2α = (0, 2k).

(3.114c)

Quantum Electrodynamics

Fundamental Problems in Quantum Field Theory 69

  Therefore, we can easily prove   by now that the T-matrix should be exactly zero due to the anti-symmetric nature of εµνρα where the non-zero part of the T-matrix should always satisfy the condition that µ, ν, ρ, α should be different from each other. Here, we should make a comment on the branching ratio of ΓZ 0 →2γ /Γ, and the present experimental upper limit shows [29] ¡ ¢ ΓZ 0 →2γ /Γ exp < 5.2 × 10−5 which is consistent with zero decay rate. Therefore, the theoretical prediction of the branching ratio is indeed consistent with experiments.

3.8

No Chiral Anomaly in Schwinger Model

In the Schwinger model, the chiral anomaly property is well evaluated since all the equations can be obtained analytically [30] . Here, we first review the procedure of obtaining the chiral anomaly equation in the Schwinger model [31, 32]. Then, we show that the exact value of the chiral charge Q5 can be calculated without any regularization, and we find Q5 = 1. This value does not agree with the regularized chiral charge, and therefore, the chiral anomaly equation is the artifact of the regularization as we see it below.

3.8.1

Chiral Charge of Schwinger Vacuum

The Schwinger model is the two dimensional QED with massless fermions and its Lagrangian density can be given as ¯ µ (∂ µ + igAµ )ψ − L = ψiγ

1 Fµν F µν 4

(3.115)

where F µν = ∂ µ Aν − ∂ ν Aµ . After the field quantization, one can calculate the charge and the chiral charge of the vacuum state in the Schwinger model. In this case, the charge of the vacuum state becomes infinity since one counts the number of all the negative energy particles. In order to obtain the finite number of the charge, one can employ the ζ function regularization which can be done in accordance with the large gauge transformation. Therefore, the regularized charge and chiral charge become NL X

Q=

eλ(k+

LgA1 ) 2π

Q5 =

k=−∞

eλ(−k−

LgA1 ) 2π

=

k=NR

k=−∞ NL X

∞ X

+

e

λ(k+

LgA1 ) 2π

−

∞ X k=NR

eλ(−k−

LgA1 ) 2π

2 +NL +1−NR +O(λ) λ

= NL +NR +

LgA1 π

(3.116a)

(3.116b)

where A1 denotes the vector field which only depends on time, and λ is an infinitesimally small number. The NL and NR denote integers which characterize the vacuum state for the

70 Fundamental Problems in Quantum Field Theory

Fujita and Kanda

the physical vacuum must have zero charge, we can   left and right mover fermions. Since   set NL + 1 = NR . Therefore, we can write the chiral charge as [32] Q5 = 1 +

LgA1 π

(3.117)

where we set NR = 0. Now, the important point is that A1 should depend on time and the chiral charge of the Schwinger vacuum is time-dependent, and thus it is not conserved any more. Therefore, eq.(3.117) leads to the chiral anomaly equation of ∂µ j5µ =

3.8.2

g ²µν F µν . 2π

Exact Value of Chiral Charge in Schwinger Vacuum

In order to explain the exact value of the chiral charge in the Schwinger mode, we should first start from the quantized Hamiltonian of the Schwinger mode where the fermion field is quantized as µ ¶ 2π 1 X an √ ei L nx (3.118) ψ(x) = b n L n where an and bn denote the annihilation operators of the left and right mover fermions. In this case, the Hamiltonian of the Schwinger model becomes µ ¶ ¶ X µ 2π L ˙ 2 X 2π † n + gA1 an an + H = A1 + − n − gA1 b†n bn 2 L L n n +

g2L X 1 ˜ j0 (p)j˜0 (−p) 8π 2 p2

(3.119)

p6=0

where we take the Coulomb gauge fixing of ∂1 A1 = 0. The current j˜0 (p) denotes the momentum representation of the fermion currents j0 (x), and can be written i Xh † j˜0 (p) = ak+p ak + b†k+p bk . (3.120a) k

Also, the axial vector current in momentum representation is written as i Xh † j˜5 (p) = ak+p ak − b†k+p bk .

(3.120b)

k

In this case, the chiral charge of the vacuum state |vaci which is filled with negative energy particles can be easily calculated as   (−Λ) Λ X X Q5 = hvac|j˜5 (0)|vaci = lim  1− 1 = 1 (3.121) Λ→∞

k=0

k=1

Quantum Electrodynamics

Fundamental Problems in Quantum Field Theory 71

  where we have made no regularization, and this is just the exact result. On the other hand, as we show above, the regularized chiral charge is written     (−∞) ∞ X X LgA1 LgA1 LgA1 eλ(−k− 2π )  = 1 + Q5 = lim  eλ(k+ 2π ) − λ→0 π k=0

k=1

which is different from the exact result. Now we can clearly see that the regularization induces something unphysical. Therefore, the axial vector current conservation is always valid, and there is no violation at all. Further, the regularization cannot change the conservation law in quantum field theory.

3.8.3

Summary of Chiral Anomaly Problem

The chiral anomaly problem is one of the most serious theoretical syndromes, and it means that they are mathematically correct, but physically incorrect. The regularization is a mathematical tool, and the procedure of its application to physics is mathematically correct, but the chiral anomaly equations are physically incorrect since they are discovered when the regularization method is applied to the systems which have no divergence as a physical process. It is very unfortunate that there are too many examples of ”mathematically correct, but physically incorrect” such as general relativity, field theory path integral and so on [31].

Conflict of Interest The author(s) confirm that this FKDSWHU content has no conflicts interest. Acknowledgements: Declared none.

Send Orders of Reprints at [email protected] 72

Fundamental Problems in Quantum Field Theory, 2013, 72-85

CHAPTER 4

Quantum Chromodynamics and Related Topics Abstract:

Quantum chromodynamics is the theoretical frame work in which one can treat the physics of the strong interactions. This is the non-abelian gauge field theory, and it cannot be solved in the perturbation theory since the free Lagrangian densities of quarks and gluons are not gauge invariant. In the perturbation theory, we describe all the physical observables in terms of the properties of quarks and gluons, and if they are not related to physical observables, then there is no point of employing the perturbation theory. Here, we also discuss the nucleon-nucleon interactions based on the meson exchange processes and present the calculation of the two pion exchange potential. Further, we discuss some physical observables in connection with quarks, that is, the magnetic moments of nucleons in terms of quark model and the total cross section ratio between σe+ e− →hadrons and σe+ e− →µ+ µ− . These physical quantities are related to the quark degrees of freedom. Keywords: Hamiltonian of QCD, color charge, nuclear force, two pion exchange, quark model prediction of nucleon magnetic moments.

4.1

Introduction

Physics of the strong interactions is described by quantum chromodynamics (QCD), and this is by now well established. Many experimental observations support that the number of the color must be three, and interactions between quarks should be mediated by gluons which are gauge bosons with colors. In addition to colors, quarks have six flavors of up, down, strange, charm, bottom and top. However, it is extremely difficult to solve QCD in a non-perturbative fashion and obtain any reasonable spectrum of hadrons from QCD since quantum field theory has infinite degrees of freedom. At the present stage, one should make some kind of approximations Takehisa Fujita and Naohiro Kanda All rights reserved - © 2013 Bentham Science Publishers

Quantum Chromodynamics and Related Topics

Fundamental Problems in Quantum Field Theory 73

in order to obtain physical observables. The perturbative treatment is the only possible method to calculate physical observables. However, there is a serious problem in the unperturbed QCD Hamiltonian since there are no free quark and gluon states in physical space, and indeed the unperturbed Fock space is gauge dependent. In addition, we present an inherent problem connected to the gauge non-invariance of the unperturbed and interaction Lagrangian densities due to the non-conservation of the quark color current. Therefore there is a basic difficulty of carrying out the perturbative expansion. Here, we clarify what are the physical observables in QCD since some of known quantities are not gauge invariant and thus they cannot be observed.

4.2

Properties of QCD with SU (Nc ) Colors

In this section, we explain some fundamental properties of QCD which are important for the understanding of the difficulties in QCD.

4.2.1

Lagrangian Density of QCD

The Lagrangian density of QCD for quark fields ψ with SU (Nc ) colors is described as [33] ¯ µ ∂µ − gγ µ Aµ − m)ψ − 1 Tr{Gµν Gµν } L = ψ(iγ 2

(4.1)

where Gµν is written as Gµν = ∂µ Aν − ∂ν Aµ + ig[Aµ , Aν ]

(4.2)

Nc2 −1

Aµ =

Aaµ T a

≡

X

Aaµ T a

(4.3)

a=1

where Aaµ denotes the gluon fields and T a corresponds to the generator of SU (Nc ) group and satisfies the following commutation relations [T a , T b ] = iC abc T c

(4.4)

where C abc denotes the structure constant of the group generators. For SU (2) case, the structure constant C abc becomes just the anti-symmetric symbol ²abc . In eq.(4.1), Tr { } means the trace of the group generators of SU (Nc ), and the generators T a are normalized according to 1 (4.5) Tr{T a T b } = δ ab . 2 Therefore, the last term of eq.(4.1) can be rewritten as 1 1 Tr{Gµν Gµν } = Gaµν Ga,µν 2 4

 

74 Fundamental Problems in Quantum Field Theory

  where Gaµν is described as

Fujita and Kanda

  Gaµν = ∂µ Aaν − ∂ν Aaµ − gC abc Abµ Acν .

(4.6)

m denotes the fermion mass, and at the massless limit, the Lagrangian density has a chiral symmetry.

4.2.2

Infinitesimal Local Gauge Transformation

QCD Lagrangian density is invariant under the following infinitesimal local gauge transformation ψ 0 = (1 − igχ)ψ = (1 − igT a χa )ψ, with χ = T a χa (4.7) A0 µ = Aµ + ig[Aµ , χ] + ∂µ χ

or

a

A0 µ = Aaµ − gC abc Abµ χc + ∂µ χa

(4.8a) (4.8b)

where χ is infinitesimally small. By defining the covariant derivative Dµ by Dµ = ∂µ + igT a Aaµ one can see D0 µ ψ 0 = [∂µ + igT a (Aaµ − gC abc Abµ χc + ∂µ χa )](1 − igT a χa )ψ = (1 − igT a χa )Dµ ψ.

(4.9)

Therefore, one can prove that ¯ µ Dµ ψ ψ¯0 iγ µ D0 µ ψ 0 = ψiγ

(4.10)

G0 µν = (1 − igT a χa )Gµν (1 + igT a χa )

(4.11)

and one obtains µν

Tr{G0 µν G0 } = Tr{(1 − igT a χa )Gµν Gµν (1 + igT a χa )} = Tr{Gµν Gµν }.

(4.12)

Therefore, one sees that the Lagrangian density of eq.(4.1) is invariant under the infinitesimal local gauge transformation.

4.2.3

Local Gauge Invariance

Now, the local gauge transformation with finite χ is defined as i A0 µ = U (χ)Aµ U † (χ) − U (χ)∂µ U † (χ) g ψ 0 = U (χ)ψ

 

(4.13) (4.14)

Quantum Chromodynamics and Related Topics

Fundamental Problems in Quantum Field Theory 75

  where U (χ) is described in terms  of χ as U (χ) = e−igχ .

(4.15)

Here, one can easily prove the following equations ¯ µ Dµ ψ ψ¯0 iγ µ D0 µ ψ 0 = ψiγ

(4.16)

G0 µν = U (χ)Gµν U (χ)−1

(4.17)

and by making use of the following identity na X

a a,µν G0 µν G0

0

= 2Tr{G µν G

0 µν

µν

} = 2Tr{Gµν G } =

a=1

na X

Gaµν Ga,µν

(4.18)

a=1

the gauge invariance of the Lagrangian density is easily seen.

4.2.4

Noether Current in QCD

The QCD Lagrangian density is invariant under the following infinitesimal global gauge transformation ψ 0 = (1 − igT a θa )ψ (4.19a) a

A0 ν = Aaν − gC abc Abν θc

(4.19b)

where θa is an infinitesimally small constant. In this case, one finds a

a

δL = L(ψ 0 , ∂µ ψ 0 , A0 ν , ∂µ A0 ν ) − L(ψ, ∂µ ψ, Aaν , ∂µ Aaν ) = 0.

(4.20)

By making use of the equations of motion, one obtains £ ¯ µ T a ψ + iψγ ¯ µ T a ∂µ ψ) δL = −ig(i∂µ ψγ

i −g(∂µ Gµν,c C bca Abν + Gµν,c C bca ∂µ Abν ) θa = 0.

Therefore, one easily finds that ³ ´ ¯ µ T a ψ + C abc Gµν,b Ac = 0. ∂µ ψγ ν

(4.21)

(4.22)

This means that the Noether current

is indeed conserved. That is,

I µ,a ≡ j µ,a + C abc Gµν,b Acν

(4.23)

∂µ I µ,a = 0

(4.24)

where the quark color current jµa is defined as ¯ µ T a ψ. jµa = ψγ Thus, the quark color current alone cannot be conserved, and therefore there is no conservation of the quark color charge. This is consistent with the fact that the color current of quarks is not a gauge invariant quantity.

 

76 Fundamental Problems in Quantum Field Theory

  4.2.5

Fujita and Kanda

Conserved Charge of  Color Octet State

¿From eqs.(4.22) and (4.23), one sees that the color octet vector current of one quark and one gluon state I µ,a is conserved. Since ∂µ I µ,a is a gauge invariant quantity, one can integrate it over all space Z ∂µ I

d d r= dt

µ,a 3

Z

Z I0a d3 r

dQaI ∇·I d r = + dt a 3

+

Z I a · dS =

dQaI =0 dt

where the color charge QaI is defined as Z QaI =

I 0,a d3 r.

(4.25)

Therefore, the color charge QaI is indeed a conserved quantity, and there may be some chance that the color charge QaI becomes a physical observable.

4.2.6

Gauge Non-invariance of Interaction Lagrangian

The interaction Lagrangian density of QCD that involves quark currents is written as LI = −gjµa Aµ,a .

(4.26)

Now, the interaction Lagrangian density LI is not gauge invariant, and therefore if one wishes to make any perturbation calculations involving the quark color currents, then one should check it in advance whether one can make the gauge invariant quark-quark interactions. The interaction Lagrangian density is transformed into a new shape under the infinitesimal local gauge transformation LI = −gjµa (Aµ,a + ∂ µ χa )

(4.27)

where the second term is a gauge dependent term. In the same way as QED case, one can rewrite the second term by making use of the conserved current as −gjµa ∂ µ χa = −g∂ µ (jµa χa ) + gC abc χa ∂ µ Gbµν Aν,c .

(4.28)

The first term is a total derivative and thus does not contribute. However, there is no way to erase the second term which depends on χa . Therefore, one sees that one cannot make any simple-minded perturbative calculations of quark-quark interactions in QCD, contrary to the QED case where the electron-electron interaction is well defined and calculated. This means that there is a difficulty of defining any potential between quarks, and this is of course consistent with the picture that the color charge of quarks are gauge dependent and is not a conserved quantity.

 

Quantum Chromodynamics and Related Topics

  4.2.7

Fundamental Problems in Quantum Field Theory 77

Equation of Motion  

The Lagrange equations of motion now become (iγ µ ∂µ − gγ µ Aµ − m0 )ψ = 0 ³ ´ ∂µ Gµν,a = gI ν,a = g j ν,a + C abc Gνρ,b Acρ .

(4.29) (4.30)

One can see that the equation of motion for the gauge fields has gauge field source terms in addition to the quark color current. Even though the equation of motion looks similar to that of QED, physics of QCD must be very different from the QED case. Now, one can introduce the color electric field E a and the color magnetic field B a by µ ¶ ∂A a a E =− − ∇Aa0 − gC abc Ab Ac0 (4.31a) ∂t 1 (4.31b) B a = ∇ × Aa + gC abc Ab × Ac . 2 It should be noted that the fields E a and B a themselves are not gauge invariant, contrary to the QED case. Now, eq.(4.30) can be rewritten in terms of E a and B a as ∇ · E a = gja0 − gC abc Ab · E c (4.32a) ³ ´ a ∂E ∇ × Ba − = gj a − gC abc Ab0 E c + Ab × B c . (4.32b) ∂t From eq.(4.30), one sees that the current I ν,a is a conserved quantity, ∂µ I µ,a = 0. In order to solve the dynamics of QCD, it should be inevitable to take into account the conservation of this current I ν,a . A question is, of course, as to in which way one should consider this effect of the current conservation in QCD dynamics, and this is still an open question.

4.2.8

Hamiltonian Density of QCD

Now, one can construct the Hamiltonian density of QCD just in the same way as the QED case. The Hamiltonian density H can be defined by the energy momentum tensor T µν as ! ! Ã Ã X ∂L X ∂L ∂L ˙ a† a a 00 ˙ A˙ − L. ψi + (4.33) H≡T = a ψi + ˙a k ∂ A ∂ ψ˙ i ∂ ψ˙ ia† k i k In this way, the Hamiltonian density of QCD is written as ∂Aa H = ψ¯ (−iγ · ∇ + m0 ) ψ + gj0a Aa0 − gj a · Aa − E a · ∂t 1 − (E a · E a − B a · B a ) . 2 By employing the equation of motion [eq.(4.30)], one obtains 1 H = ψ¯ [−iγ · ∇ + m0 ] ψ − gj a · Aa + (E a · E a + B a · B a ) 2 which is, of course, gauge invariant.

 

(4.34)

(4.35)

78 Fundamental Problems in Quantum Field Theory

  4.2.9

Fujita and Kanda

Hamiltonian of QCD 

The Hamiltonian can be obtained by integrating the Hamiltonian density over all space ¾ ½ Z 1 a a a a 3 a a (4.36) H = d r ψ¯ [−iγ · ∇ + m0 ] ψ − gj · A + [E · E + B · B ] . 2 In order to calculate the spectrum emerged from this Hamiltonian, one should quantize the fields Aaµ by making use of the equations of motion (iγ µ ∂µ − gγ µ Aµ − m0 )ψ = 0 ³ ´ ∂µ Gµν,a = g j ν,a + C abc Gνρ,b Acρ .

(4.37) (4.38)

After obtaining the quantized Hamiltonian, one should prepare Fock spaces and then evaluate the Hamiltonian to obtain the mass of hadrons, assuming finite quark masses. This is a difficult task, but it should be done in future by diagonalizing the total Hamiltonian in some way or the other.

4.3

Nuclear Force

Since, at present, there is no way to evaluate the strong interactions in terms of QCD, we should carry out the calculation of the nuclear interactions in terms of meson exchange processes. This is the only reliable and reasonable method to evaluate the nuclear force in a proper manner. The nuclear interaction should be mediated by the exchange processes of the observed bosons, such as pions.

4.3.1

One Boson Exchange Potential

The structure of the nucleus can be described once the nucleon-nucleon interactions are properly known. Indeed there are already sufficiently large number of works available for the determination of the nucleon-nucleon potential [34, 35, 36, 37]. The most popular nuclear interaction may be obtained by one boson exchange potential (OBEP) [38, 39, 4] where exchanged bosons are taken from experimental observations. In this case, the masses and the coupling constants of the exchanged bosons are determined from various methods, partly experimentally and partly theoretically. The discussions of the determination of these parameters may have some ambiguities, but one can see that the basic part of the nuclear force can be well understood until now. However, there is one important problem which still remains unsolved. This is related to the medium attraction of the nucleon-nucleon potential, and it is normally simulated by the effective scalar meson exchange process. Until now, however, people have discovered no massive scalar meson in nature and, therefore, the artificial introduction of the scalar meson

 

Quantum Chromodynamics and Related Topics

Fundamental Problems in Quantum Field Theory 79

  is indeed a theoretical defect of the   one boson exchange model. This is indeed a homework problem for many years of nuclear physics research. However, this important problem is left unsolved for a long time since many of the nuclear theorists moved to the quark model calculations of the nucleon-nucleon interaction. By now, it becomes clear that the evaluation of the QCD based model has an intrinsic difficulty due to the gauge dependence of the quark color charge [31] and this strongly suggests that the meson exchange approach is indeed a right direction of the nuclear force calculations.

4.3.2

Two Pion Exchange Process

In addition to the one boson exchange processes, one should consider the two pion exchange diagrams in order to obtain a proper nucleon-nucleon interaction. There are, of course, many calculations of the nucleon-nucleon interaction due to the two pion exchange processes [40, 41, 42], and this may indeed give rise to the medium range attraction even though until now there is no clear cut evaluation which can isolate the nuclear force contribution to the medium range attraction. Here, we present a careful calculation of the two pion exchange processes. The important point is that the fourth order process involving the four γ5 interactions is not suppressed at all, in contrast to the one pion exchange diagram where the γ5 coupling is indeed supπ pressed by the factor of m M with M denoting the nucleon mass. This is basically due to the parity mismatch and corresponds to the mixture of the small and the large components of the Dirac spinors. Therefore, it should be very important to calculate the two pion exchange process properly in order to understand the medium attraction of the nucleon-nucleon interaction. Now, the evaluation of the two pion exchange Feynman diagram is done in a straight forward way [10, 43], and we find the corresponding T-matrix as Z

1 1 d4 k (1) (1) iγ iγ (2π)4 5 k 2 − m2π + iε (p1 − k)µ γµ(1) − M + iε 5

T = igπ4 (τ1 · τ2 )2 (2)

×iγ5

(q −

k)2

1 1 (2) iγ 2 (2) − mπ + iε (p2 + k)µ γµ − M + iε 5

(4.39)

where p1 (p0 1 ) and p2 (p0 2 ) denote the initial (final) four momenta of the two nucleons, and q is the four momentum transfer which is defined as q = p1 − p0 1 . Here, we have ignored the crossed diagram which is much smaller than eq.(4.39). By noting (1)

(2)

(γ5 )2 = 1, (γ5 )2 = 1, γ5 γ µ = −γ µ γ5 we can rewrite eq.(4.39) as Z T =

 

igπ4 (τ1

· τ2 )

2

(1)

1 1 (p1 − k)µ γµ − M d4 k (2π)4 k 2 − m2π (q − k)2 − m2π (p1 − k)2 − M 2

80 Fundamental Problems in Quantum Field Theory

 

 

Fujita and Kanda (2)

(p2 + k)µ γµ − M . × (p2 + k)2 − M 2

(4.40)

Now, we introduce the Feynman parameters x, y, z as 1 =6 abcd

Z

Z

1

Z

x

dx

y

dy

0

dz

0

0

1 . [a + (b − a)x + (c − b)y + (d − c)z]4

Further, we assume that the nucleons at the initial state are on the mass shell (p/1 − M )u(p1 ) = 0,

(p/2 − M )u(p2 ) = 0

and therefore we also find u ¯(p0 1 )q µ γµ u(p1 ) = 0. In addition, we take the non-relativistic limit for the nucleon motion and thus obtain Z T '

−6igπ4 (τ1

· τ2 )

2

Z

1

dx

Z

x

dy

0

0

Z

y

dz 0

d4 k 41 k 2 + M 2 (2z − y)2 (2π)4 (k 2 − s)4

(4.41)

where s is defined as ¡ ¢ s = q 2 (y − x)2 − y + x + M 2 (2z − y)2 + m2π (1 − y). The momentum integration of k can be easily carried out, and we find ¸ Z 1 Z x Z y · 1 2M 2 (2z − y)2 gπ4 2 dz (τ1 · τ2 ) dx dy − . T '− 32π 2 s s2 0 0 0

(4.42)

(4.43)

This three dimensional integration of x, y, z can be done only numerically, and the calculated result can be well fit by the following shape [44] T ' −(τ1 · τ2 )2

A gπ4 × 2 2 32π q + m2s

(4.44)

where A and ms are found to be A ' 0.57,

ms ' 4.7mπ ' 650 MeV.

Here, we replace the four momentum transfer of q 2 as q 2 = q02 − q 2 ' −q 2 since we may use the static approximation to a good accuracy ¶2 µq q 2 0 2 2 2 M + p1 − M + p 1 ' (q0 ) = 2

 

1 2 (p2 − p0 1 )2 ¿ q 2 . 4M 2 1

(4.45)

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Fundamental Problems in Quantum Field Theory 81 2 gπ 4π gs2

  If we take the value of the πN N  coupling constant as T ' −(τ1 · τ2 )2

' 8, then we find

q 2 + m2s

(4.46)

where

1 g4 gs2 ' × π 2 × 0.57 ' 1.45 (4.47) 4π 4π 32π which are consistent with the values determined from the nucleon-nucleon scattering experiments. It should be important to note that the present calculation suggests that the T = 0 channel of the nucleon-nucleon interaction is very strong in comparison with the T = 1 case. This means that the proton-neutron interaction is much stronger than the interactions between identical particles. .

4.3.3

Double Counting Problem

In general, the evaluation of the two boson exchange potential should be carefully done due to the double counting problem. This is clear since the solution of the Schr¨odinger equation with the one boson exchange potential should contain the repeat of the one boson exchange process in some way or the other. Ladder Diagrams In order to understand the double counting problem, we should first start from the Lippmann-Schwinger equation for the T-matrix, and the T-matrix equation for the nucleonnucleon scattering case can be written as 1 T (4.48) T = VN N + VN N E − H0 + iε where VN N and H0 denote the nucleon-nucleon potential and the two nucleon Hamiltonian in the free state, respectively. Suppose this VN N should be one pion exchange potential Vπ , and we insert it into eq.(4.48) and expand it into the ladder type contributions 1 Vπ + · · · . T = Vπ + Vπ (4.49) E − H0 + iε Here it is claimed that the second term should correspond to the contributions from the two pion exchange potential. Indeed, it indicates that some part of the two pion exchange process should be taken into account in this T-matrix equation. However, this is not necessarily correct for the pion exchange process since the one pion exchange potential is suppressed a great deal due to the γ5 interaction which picks up the product of the large and small components of the Dirac wave function. On the other hand, the second order ladder calculation can take into account only the large components of the Dirac wave functions. This is clear since the Lippmann-Schwinger equation is solved only for the non-relativistic wave function. In addition, the OPE potential is obtained already by making the approximation of the non-relativistic reduction, and thus the two pion exchange process is completely different from the second order ladder contribution of OPE potential.

 

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  One Pion Exchange Potential   Here, we see that the T-matrix of the one pion exchange process is written as T (OP E) = −gπ2 u ¯(p01 )τ1 γ 5 u(p1 )

q2

1 u ¯(p02 )τ2 γ 5 u(p2 ) − m2π + iε

(4.50)

and after some static and non-relativistic approximations, we obtain the OPE potential · µ 2 ¶ ¸ mπ mπ 1 m2π f 2 (τ1 · τ2 ) e−mπ r S12 + + 2 + σ1 · σ2 (4.51) Vπ (r) = 4π m2π r 3 r r 3 where S12 ≡ 3(σ1 · rˆ)(σ2 · rˆ) − σ1 · σ2 ,

f≡

mπ gπ . 2Mp

mπ . The most important point is that the OPE poHere, one finds the suppression factor of 2M p tential can be obtained only after one takes the expectation value of the γ 5 matrix with free Dirac wave functions, and the suppression factor comes from this point of the expectation value u ¯(p)γ 5 u(p) which is proportional to the product of the large and small components of the Dirac wave function.

Two Pion Exchange Potential On the other hand, the two pion exchange diagram does not have any such suppressions because one considers all the intermediate states which pick up the states strongly coupled to the γ 5 vertex with the initial nucleon state. We can write it more explicitly [44] Z T

(T P EP )

=

igπ4 (τ1

2

· τ2 )

1 1 d4 k (1) (1) u ¯(p01 ) 2 γ5 γ5 u(p1 ) 4 2 (2π) k − m + iε p/ 1 − k/ − M + iε (2)

ׯ u(p02 )γ5

1 1 (2) γ u(p2 ). (4.52) p/ 2 + k/ − M + iε 5 (q − k)2 − m2 + iε

(1)

Here, one can see that the two γ5 s appear between the spinors u ¯(p01 ) and u(p1 ) and thus there is no suppression. In addition, one sees that the ladder contribution of eq.(4.49) can only take into account the intermediate states which are always described in terms of the non-relativistic wave functions in the Lippmann-Schwinger equation, and the OPEP is obtained only after one has taken the expectation value of the γ 5 with the free Dirac states. In summary, the second order ladder type calculation of the small potential like one pion exchange potential cannot give rise to such a large contribution of the potential obtained in eq.(4.46), and therefore, there is no sizable double counting problem in this calculation, apart from the very small part of the two pion exchange potential, which should be a corπ 2 rection of the order of ( m Mp ) to the two pion exchange potential.

 

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Fundamental Problems in Quantum Field Theory 83

in QCD Physical Observables  

We believe that the QCD should be a right theoretical frame work which can describe the physics of strong interactions, even though the constituents of QCD, namely, quarks and gluons do not have free states by construction. In this case, it is clear that we cannot describe physical observables in terms of the quark and gluon terminology. Therefore, it should be important to clarify what are, in fact, the physical observables in connection with QCD.

4.4.1

Perturbation Theory or Exact Hamiltonian

In QCD, the free quark and free gluon Lagrangian densities are gauge dependent as we saw in the previous sections. Therefore, there is no way to carry out the perturbation expansion, and thus the S-matrix theory cannot be developed. Color Charge It is easy to prove that the color charges of quarks should be gauge dependent, and therefore there is no way to observe free quark states. This is, of course, consistent with experiment since no free quark state is observed up to the present stage. This is just the representation of the quark confinement. Total Hamiltonian As we see in the previous sections, the total Hamiltonian is indeed gauge independent, and therefore it should be a physical observable. In fact, it is observed as the mass of baryons or mesons. In this respect, a physically interesting quantity should be the total Hamiltonian which should be diagonalized in some way or the other. This is the only physical quantity which can be directly calculated in QCD.

4.4.2

Electric Charge

We also understand that the electric charge of quarks should be gauge independent, and therefore it should be related to physical observables even though quarks should be confined inside hadrons. In fact, all the experimental evidence that QCD must be a right theory should be based on the observations due to the electric probe. For example, the QCD jet phenomena can be understood as the physical processes in which e+ + e− collisions can create the virtual q − q¯ state which should decay into many hadrons. The jet indicates that the total momentum of hadrons in one jet must be equivalent to the momentum of quark or anti-quark. However, we cannot make any reliable estimations of the hadron formations how the quark can decay into many hadrons. This is, of course, clear because we cannot be based on the perturbation theory.

 

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Magnetic Moments of   Nucleons

When the quark model is introduced, people calculate the magnetic moments of proton and neutron. They use the SU(6) wave functions of baryons in which the spin parts are also included into the SU(3) flavor wave functions. In particular, they apply this wave functions of proton and neutron to calculate the magnetic moments of nucleons. The SU(6) wave functions of proton and neutron in terms of quark model can be written as 1 h ↑ ↓ ↑ 2u1 d2 u3 + 2u↑1 u↑2 d↓3 + 2d↓1 u↑2 u↑3 |P ↑i = √ 18 i −u↑1 d↑2 u↓3 − u↓1 d↑2 u↑3 − u↑1 u↓2 d↑3 − u↓1 u↑2 d↑3 − d↑1 u↓2 u↑3 − d↑1 u↑2 u↓3 (4.53a) 1 h ↑ ↓ ↑ 2d1 u2 d3 + 2d↑1 d↑2 u↓3 + 2u↓1 d↑2 d↑3 |N ↑i = √ 18 i −d↑1 u↑2 d↓3 − d↓1 u↑2 d↑3 − d↑1 d↓2 u↑3 − d↓1 d↑2 u↑3 − u↑1 d↓2 d↑3 − u↑1 d↑2 d↓3 (4.53b) where u↑i (d↓i ) denotes the u−quark (d−quark) spin up (down) wave functions for i−th quark state, respectively. Now, the magnetic moment operator of quarks inside nucleons should be written as µz = µ0

3 X

ei σz(i)

(4.54)

i=1

where ei denotes the electric charge of quarks, and it is written 2 eu = e, 3

1 ed = − e. 3

(4.55)

Further, µ0 indicates some scale parameter which cannot be easily calculated in the quark model. The only thing we can say is that it should have the dimension of length like a radius of nucleon. Now the magnetic moments of proton and neutron can be calculated in a straight forward way, and we find  P3 (i)   µp = hP |µ0 i=1 ei σz |P i = µ0 (4.56)   µ = hN |µ P3 e σ (i) |N i = − 2 µ . n 0 i=1 i z 3 0 µ

Therefore, the magnetic moments ratio µnp between proton and neutron can be written as µ ¶ µp = −1.50 (4.57a) µn th which should be compared with the observed value µ ¶ µp = −1.46. µn exp

 

(4.57b)

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Fundamental Problems in Quantum Field Theory 85

  The agreement between theory and experiment of the nuclear magnetic moments ratio is almost perfect, and there is a good reason for the excellent agreement. This is connected to the nature of the magnetic moment operator which does not depend on the radial distribution of quarks inside hadron. Therefore, the magnetic moments of nucleon should be determined from the kinematics of quark spins. Further, this agreement of the nucleon magnetic moments between the quark model prediction and experiment strongly suggests that the charge of the quarks should be described properly by eq.(4.55) as long as the charges of u and d quarks are concerned. In this sense, the naive quark model is indeed consistent with the experimental observations.

4.4.4

Cross Section Ratio of σe+ e− →hadrons and σe+ e− →µ+ µ−

There is an interesting physical observable which involves quarks, and this is the total cross section ratio between σe+ e− →hadrons and σe+ e− →µ+ µ− . The total cross section of σe+ e− →hadrons can be considered to be e+ + e− → q + q¯ → hadrons and therefore, the ratio R between the total cross sections of σe+ e− →hadrons and σe+ e− →µ+ µ− can be written if the incident energy of Ee+ e− is sufficiently large enough to produce the top quarks R=

3 σe+ e− →hadrons = 2 (e2u + e2d + e2s + e2c + e2b + e2t ) σe+ e− →µ+ µ− e

(4.58)

where a factor of 3 comes from the color degrees of freedom of quarks. The comparison between theory and experiment confirms that the color degrees of freedom of quarks must be 3. This is quite important since the color degrees of freedom are introduced mainly because there exist no free quarks and no free gluons in nature, and therefore their confinement should be exact, which is indeed supported by the non-abelian nature of QCD, that is, the color charge is gauge dependent.

Conflict of Interest The author(s) confirm that this FKDSWHU content has no conflicts interest. Acknowledgements: Declared none.

 

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Fundamental Problems in Quantum Field Theory, 2013, 86-98

CHAPTER 5

Weak Interactions Abstract: In this chapter, we first present a brief review of the weak interaction theory. In particular, we discuss why the conserved vector current model had to be modified to a new theory. After that, we clarify the physics of the spontaneous symmetry breaking and then discuss the intrinsic problem of the Higgs mechanism in the Weinberg-Salam model. In addition, we present the calculation of the vertex correction due to the weak vector bosons and show that there is no logarithmic divergence in this vertex corrections. Therefore, there is no need of the renormalization procedure in the weak interaction models with massive vector bosons. Keywords: CVC theory, vertex corrections of weak vector boson, Weinberg-Salam model, spontaneous symmetry breaking, Higgs mechanism, right propagator of massive vector boson, Lorentz condition.

5.1

Introduction

The physics of weak interactions started from the Fermi model of the four fermion interaction Hamiltonian. This model pointed to the essentially correct physics picture of the weak decay processes. However, the four fermion interaction model has a quadratic divergence in the second order perturbation calculations even with a very small coupling constant. Therefore, the model cannot be accepted for a correct theory unless one makes some modifications, even though this model is applied to physical processes with the first order perturbation theory and has made a great success. At the same time, there were several strong experimental evidences that the four fermion interaction model should be mediated by very heavy bosons, and indeed, the experimental discovery of the weak vector bosons (W ± , Z 0 ) was followed. In the mean time, Weinberg and Salam proposed a weak interaction model which is based on the SU (2) ⊗ U (1) nonabelian gauge theory. The reason why they employ the gauge theory is simply because they believed that the gauge theory should be renormalizable, though without any foundations. However, the problem is that this standard model has two serious mistakes. The first Takehisa Fujita and Naohiro Kanda All rights reserved - © 2013 Bentham Science Publishers

Weak Interactions

Fundamental Problems in Quantum Field Theory 87

  one is related to the non-abelian  nature of gauge fields in the model Lagrangian density. As we discuss in the previous chapter, the charges of the non-abelian gauge fields are gauge dependent, and therefore they are not physical observables at all. This means that these gauge fields cannot become free particles unless one makes mistakes somewhere within the theoretical framework. The second mistake in the standard model is connected to their treatment of the Higgs mechanism. There, the local gauge invariance is broken by hand in order to give a finite mass to the gauge field at the Lagrangian density level, and this is a wrong procedure. This is mainly based on the fact that the symmetry breaking physics is completely misunderstood at the time of the construction of the theory, and indeed the symmetry breaking physics is only concerned with the property of the interacting vacuum state, and it cannot induce any change of the gauge field properties in the Hamiltonian since the symmetry breaking has nothing to do with the field operators. In reality, the chiral symmetry is never broken spontaneously, as we see below. In this chapter, we review what is the basic problem of the standard model of the weak interactions. In short, the problem of the Weinberg-Salam model is concerned with the symmetry nature which should be kept at any time in the Lagrangian density, even though the state (here the vacuum state of the interacting field theory model) can find the symmetry property which is different from the one found in the free field theory model. There is nothing surprising since the true vacuum of the interacting Hamiltonian may well have a non-vanishing charge associated with the symmetry of the Hamiltonian while the free field theory model may have zero charge of the symmetry group. On the other hand, the Weinberg-Salam model had to break the symmetry itself at the Lagrangian density level because it started from the local gauge theory whose fields must be always massless, and this is more than a serious defect of the model Hamiltonian, but it is physically a wrong procedure. This clearly indicates that, instead of the Weinberg-Salam model, one should find a new model Hamiltonian with three massive vector bosons from the beginning, and it turns out that this is indeed renormalizable. In fact, there is no logarithmic divergence in the calculations of any physical observables in the new model, and thus one does not have to worry about the renormalization procedure.

5.2

Critical Review of Weinberg-Salam Model

The Weinberg-Salam model has basically two important ingredients. The first one is concerned with the fermion and vector field coupling that leads to the four fermion interaction model in the second order perturbative calculations. This is a very reasonable assumption, and indeed one sees that the model can reproduce almost all of the experimental observations. The second part is the Higgs mechanism which has, in fact, a serious problem in connection with the unitary gauge fixing. In this mechanism, the condition of φ = φ† is imposed on the Higgs fields. However, this does not correspond to a proper gauge fixing. Instead, this is simply a procedure for giving a finite but very large mass to a gauge field by breaking the local gauge invariance by hand. This suggests that the starting Lagrangian

 

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  density of the weak interactions  should be reconsidered, and indeed we should start from the three massive vector boson fields from the beginning. The massive vector fields should couple to the fermion currents as the initial ingredients. Here, it is shown that the new renormalization scheme with massive vector bosons has no intrinsic problem, and the massive vector boson fields do not give rise to any divergences for physical observables and therefore we do not need any renormalization procedure.

5.2.1

Spontaneous Symmetry Breaking

Before going to the discussion of the Higgs mechanism, we should clarify the physics of the spontaneous symmetry breaking. The whole idea of the symmetry breaking has been critically examined in the recent textbook [31, 45, 46], and the physics of the spontaneous symmetry breaking is, by now, well understood in terms of the standard knowledge of quantum field theory. In particular, if one wishes to understand the vacuum state in a field theory model of fermions, then one has to understand the structure of the negative energy states of the corresponding field theory model. The terminology of the spontaneous symmetry breaking is misleading, and one should say that it is incorrectly used. It does not express the right physics of the symmetry breaking [47, 48, 49]. This is simply because the breaking of the symmetry cannot, of course, occur in the Hamiltonian of isolated system [50, 51]. If the symmetry breaking is concerned with the comparison of the vacuum states between the free field theory and the interacting field theory models, then we see that the chiral charge associated with the chiral symmetry transformation in the interacting vacuum state may well have a finite but different charge from the vacuum state of the free field theory which indeed has a zero chiral charge. For the total Hamiltonian H = H0 + HI , we have the vacuum state |vaciexact which is an eigenstate of H, and the vacuum state may well have the eigenvalue of the chiral charge ˆ 5 as [31] operator Q ˆ eiαQ5 |vaciexact = einα |vaciexact where n is ±1 for the Thirring model. On the other hand, the free vacuum state |vacif ree which is an eigenstate of H0 should have ˆ

eiαQ5 |vacif ree = |vacif ree . Here, one can see that there is nothing special in this symmetry arguments. The most important of all is that there is no symmetry breaking in the Hamiltonian of H. Only the exact vacuum state has a finite chiral charge, in contrast to the zero chiral charge of the free vacuum state. On the other hand, some people completely misunderstood this physics of symmetry breaking and thought that the vacuum state of the interacting Hamiltonian itself broke the chiral symmetry [49]. This should arise from the two kinds of misunderstanding in their calculations. The first point is that they made use of the approximation scheme of Bogoliubov transformation, and this approximation method happens to induce a deceptive term which looks like a mass term though its mass is infinite [31]. The second misunderstanding

 

Weak Interactions

Fundamental Problems in Quantum Field Theory 89

  is concerned with the concept of  the cutoff momentum, and in fact, their result of the mass term is expressed by the cutoff momentum Λ which should be set to infinity at the end of the calculation. In this respect, it is clear that one cannot discuss its physics by rewriting the Lagrangian density into a new shape. As one knows, the property of the vacuum state should be determined from the eigenstate of the total Hamiltonian in the corresponding field theory model. In summary, the symmetry of the Hamiltonian can never be spontaneously broken, and the eigenstate of the Hamiltonian should keep the symmetry property, unless the symmetry breaking terms should be added to the Hamiltonian by hand. As we discuss below, the physics of the Higgs mechanism has nothing to do with the property of the vacuum state, and therefore it is not related to the symmetry breaking physics at all [52].

5.2.2

Higgs Mechanism

As we show below, the whole procedure of the Higgs mechanism cannot be justified at all. This is mainly connected to the misunderstanding of the gauge fixing where one degree of freedom of the gauge fields must be reduced in order to solve the equations of motion of the gauge fields. Therefore, one cannot insert the condition of the gauge fixing into the Lagrangian density. This is clear since the Lagrangian density only plays a role for producing the equation of motions. Indeed, the Lagrangian density itself is not directly a physical observable, and the Hamiltonian constructed from the Lagrangian density is most important after the fields are quantized. For the field quantization, one has to make use of the gauge fixing condition which can determine the gauge field Aµ together with the equation of motions. This means that only the final Hamiltonian density is relevant to the description of physical observables, and thus the success of the Glashow-Weinberg-Salam model [53, 54, 55] is entirely due to the final version of the weak Hamiltonian which is not at all the gauge field theory but is a model field theory of the massive vector fields which couple to the fermion currents. The success of the standard model is, of course, due to the fact that it can be reduced to the theory of conserved vector current (CVC). In this respect, it is very important to examine the renormalizability of the final version of the weak Hamiltonian. Here, we show that the renormalizability of the model field theory can be indeed justified. This is basically due to the fact that there is no divergence in the vertex corrections of fermions due to the massive vector boson propagations once we employ a proper propagator of the massive vector bosons. Here, we briefly review how we can obtain the new propagator of the massive vector boson, and the correct shape of the propagator of the massive vector bosons should be given as [56] µ ν

g µν − k kk2 . D (k) = − 2 k − M 2 − iε µν

(5.0)

This shape is determined by solving the equations of motion for the massive vector bosons. As long as we employ the above propagator, we find that the anomalous magnetic moment of electron due to weak Z 0 bosons does not have any divergences and it is indeed very

 

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  small number which is consistent   with experiment. Thus, one can see that the physical observables with the massive vector boson propagations are all finite and that there are neither conceptual nor technical problems in the renormalization scheme of the massive vector bosons interacting with fermions. Namely, there is no need of the wave function renormalization.

5.2.3

Gauge Fixing

Now we discuss the basic problem of the Higgs mechanism [46]. The Lagrangian density of the Higgs mechanism is given as ¢2 1 1 ¡ 1 L = (Dµ φ)† (Dµ φ) − u0 |φ|2 − λ2 − Fµν F µν 2 4 4

(5.1)

where Dµ = ∂ µ + igAµ ,

F µν = ∂ µ Aν − ∂ ν Aµ .

(5.2)

Here, we only consider the U(1) case since it is sufficient for the present discussions. The above Lagrangian density is indeed gauge invariant, and in this respect, the scalar field may interact with gauge fields in eq.(5.1). However, it should be noted that there is no experimental indication that the fundamental scalar field can interact with any gauge fields in terms of the Lagrangian density of eq.(5.1). In this sense, this is only a toy model. Now, the equations of motion for the scalar field φ become ¡ ¢ ∂µ (∂ µ + igAµ )φ = −u0 φ |φ|2 − λ2 − igAµ (∂ µ + igAµ )φ

(5.3)

¡ ¢ ∂µ (∂ µ − igAµ )φ† = −u0 φ† |φ|2 − λ2 + igAµ (∂ µ − igAµ )φ† .

(5.4)

On the other hand, the equation of motion for the gauge field Aµ can be written as

where

∂µ F µν = gJ ν

(5.5)

o 1 n J µ = i φ† (∂ µ + igAµ )φ − φ(∂ µ − igAµ )φ† . 2

(5.6)

One can also check that the current J µ is conserved, that is ∂µ J µ = 0.

(5.7)

This Lagrangian density of eq.(5.1) has been employed for the discussion of the Higgs mechanism.

 

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  5.2.4

Fundamental Problems in Quantum Field Theory 91

Gauge Freedom and  Number of Independent Equations

Now, we should count the number of the degrees of freedom and the number of equations. For the scalar field, we have two independent functions φ and φ† . Concerning the gauge fields Aµ , we have four since there are A0 , A1 , A2 , A3 fields. Thus, the number of the independent fields is six. On the other hand, the number of equation is five since the equation for the scalar fields is two and the number of the gauge fields is three. This number of three can be easily understood, even though it looks that the independent number of equations in eq.(5.5) is four, but due to the current conservation the number of the independent equations becomes three. This means that the number of the independent functions is six while the number of equations is five, and they are not equal. This is the gauge freedom, and therefore in order to solve the equations of motion, one has to put an additional condition for the gauge field Aµ like the Coulomb gauge which means ∇ · A = 0. In this respect, the gauge fixing is simply to reduce the redundant functional variable of the gauge field Aµ to solve the equations of motion, and nothing more than that.

5.2.5

Unitary Gauge Fixing

In the Higgs mechanism, the central role is played by the gauge fixing of the unitary gauge. The unitary gauge means that one takes φ = φ† .

(5.8)

This is the constraint on the scalar field φ even though there is no gauge freedom in this respect. For the scalar field, the phase can be changed, but this does not mean that one can erase one degree of freedom. One should transform the scalar field in the gauge transformation as φ0 = e−igχ φ but one must keep the number of degree of freedom after the gauge transformation. Whatever one fixes the gauge χ, one cannot change the shape of the scalar field φ since it is a functional variable and must be determined from the equations of motion. The gauge freedom is, of course, found in the vector potential Aµ as we discussed above. In this sense, one sees that the unitary gauge fixing is a simple mistake [57]. The basic reason why people overlooked this simple-minded mistake must be due to their obscure presentation of the Higgs mechanism. Also, it should be related to the fact that, at the time of presenting the Higgs mechanism, the spontaneous symmetry breaking physics was not understood properly since the vacuum of the corresponding field theory was far beyond the proper understanding. Indeed, the Goldstone boson after the spontaneous symmetry breaking was taken to be almost a mysterious object since there was no experiment which suggests any existence of the Goldstone boson. Instead, a wrong theory prevailed among physicists. Therefore, they could assume a very unphysical procedure of the Higgs mechanism and people pretended that they could understand it all.

 

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Non-abelian Gauge Field  

Now, one should be careful for the renormalizability of the non-abelian gauge field theory. As one can easily convince oneself, the non-abelian gauge theory has an intrinsic problem of the perturbation theory [58]. This is connected to the fact that the color charge in the non-abelian gauge field depends on the gauge transformation, and therefore it cannot be physical observables. This means that the free gauge field which has a color charge is gauge dependent, and thus one cannot develop the perturbation theory in a normal way. In QCD, this is exhibited as the experimental fact that both free quarks and free gluons are not observed in nature. The absence of free fields is a kinematical constraint and thus it is beyond any dynamics. Therefore, one cannot discuss the renormalizability of the nonabelian gauge field theory models due to the lack of the perturbation scheme in this model field theory [31, 58]. Therefore, the problem of the renormalizability in the non-abelian field theory model is a meaningless subject since the perturbation theory is not defined in this model field theory.

5.2.7

Summary of Higgs Mechanism

The intrinsic problem of the Higgs mechanism is discussed in terms of the gauge fixing condition. This is also related to the misunderstanding of the spontaneous symmetry breaking physics. Here, we have shown that the Higgs mechanism cannot be justified since the gauge invariance of the Lagrangian density is violated by hand. However, we believe that the final version of the weak Hamiltonian should be correct, and therefore we should discuss the renormalization scheme of the massive vector bosons in detail. As we discuss above, the basic reason why the standard model Hamiltonian becomes a reasonable model is due to the fact that they make mistakes twice and thus it gets back to the right Hamiltonian which can describe the nature. The first mistake is related to the non-abelian character of the gauge field theory model while the second mistake is concerned with the breaking of the local gauge invariance in terms of Higgs mechanism, and it is, of course, an incorrect treatment. Therefore, if we remove the Higgs fields and the non-abelian nature of the massive vector bosons from the Weinberg-Salam model, then the final Hamiltonian of the standard model should be physically acceptable.

5.3

Theory of Conserved Vector Current

It should be important to construct the Lagrangian density which can describe the weak interaction processes. The basic starting point is, of course, the conserved vector current (CVC) theory which can describe most of the observed weak decay processes quite well. This CVC theory should be derived from the second order perturbation theory by exchanging the weak vector bosons between corresponding fermions.

 

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  5.3.1

Fundamental Problems in Quantum Field Theory 93

Lagrangian Density of   CVC Theory

The theory of the weak interactions is developed in terms of the four fermion interaction model [59] by Fermi and, after some time, Feynman and Gell-Mann extended it to the conserved vector current (CVC) theory, which is quite successful for describing experiments [60, 61, 62, 63]. The Lagrangian density of the CVC theory can be written as GF L = − √ Jµ† J µ + h.c. 2 where GF denotes the weak coupling constant GF ' 1.2 × 10−5 M12 . Also J µ is composed p of the leptonic and hadronic currents and is written as J µ = j`µ + jhµ where both of the currents can be expressed as j`µ = ψ¯νe γ µ (1 − γ 5 )ψe + ψ¯νµ γ µ (1 − γ 5 )ψµ + · · · jhµ = cos θψ¯u γ µ (1 − γ 5 )ψd + sin θψ¯u γ µ (1 − γ 5 )ψs + · · · . It should be important to note that the current-current interaction model can describe many experimental data to a very high accuracy, and this is, indeed, a well-known fact before the discovery of the weak vector bosons of W ± , Z 0 .

5.3.2

Renormalizability of CVC Theory

However, this model Hamiltonian of CVC theory should have a serious problem related to the divergence in the second order perturbation theory. Since the coupling constant GF is very small compared to the fine structure constant, one can expect that the second order perturbation must be reliable. On the contrary, however, the second order calculation has a quadratic divergence since the coupling constant GF has the dimension of the inverse square of the energy. Therefore, it is clear that this theoretical framework should have an intrinsic problem of the divergence, and thus it should be very important to construct a theory which should not have any divergence.

5.3.3

Renormalizability of Non-Abelian Gauge Theory

Now, in order to construct a theory which is renormalizable, it was believed that the gauge field theory should be renormalizable at the time when people discovered the CVC theory. Therefore, it is natural that the non-abelian gauge theory of SU (2) ⊗ U (1) was proposed by Weinberg-Salam. However, one sees by now that the non-abelian gauge field has a charge associated with its gauge group, but the charge is not a physical observable since it is gauge dependent. Therefore, there is no way to develop any perturbation theory in this non-abelian gauge field theory. This means that the non-abelian gauge theory has an intrinsic problem before going to the renormalization scheme. [64]

 

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Fujita and Kanda

Lagrangian Density   of Weak Interactions

Even though the Higgs mechanism itself has an intrinsic problem, the final Hamiltonian density may well be physically meaningful. This is clear since, from this Hamiltonian density, one can construct the CVC theory which describes the experimental observables quite well.

5.4.1

Massive Vector Field Theory

In this respect, we may write the simplest Lagrangian density for two flavor leptons which couple to the SU(2) vector fields Wµa ¯ ` (i∂µ γ µ − m)Ψ` − gJµa W µ,a + 1 M 2 Wµa W µ,a − 1 Gaµν Gµν,a L=Ψ 2 4

(5.9)

where M denotes the mass of the vector boson. Here, we do not write the hadronic part, for simplicity. The lepton wave function Ψ` has two components µ ¶ ψe Ψ` = . (5.10) ψν Correspondingly, the mass matrix can be written as µ ¶ me 0 m= . 0 mν

(5.11)

The fermion current Jµa and the field strength Gaµν are defined as ¯ ` γµ (1 − γ5 )τ a Ψ` , Jµa = Ψ

Gaµν = ∂µ Wνa − ∂ν Wµa .

(5.12)

This Lagrangian density is almost the same as the standard model Lagrangian density, apart from the Higgs fields and the abelien nature. In fact, there is no experiment in weak process which cannot be described by the Lagrangian density of eq.(5.9). The only thing which, people thought, may be a defect in the above Lagrangian density is concerned with the renormalization of the theory. As we see below, the problem of the renormalization is completely solved by employing the right propagator of the massive vector bosons. This means that we find that there is no logarithmic divergence in the evaluation of the vertex corrections due to the propagations of the massive vector bosons. Therefore, we do not need any renormalization procedure since all the physical observables are calculated to be finite.

5.5

Propagator of Massive Vector Boson

Here, we briefly review the derivation of the new propagator of the massive vector boson which has recently been evaluated properly in terms of the polarization vector [56]. The

 

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Fundamental Problems in Quantum Field Theory 95

is found to be the one given as   correct shape of the boson propagator   µ ν

g µν − k kk2 . D (k) = − 2 k − M 2 − iε µν

(5.13)

This is quite important since this does not generate any quadratic divergences in the selfenergy diagrams of fermions any more while the old propagator in the textbooks µ ν

µν Dold (k) = −

g µν − kMk2 k 2 − M 2 − iε

gives rise to the quadratic divergence [31, 15]. This old propagator is obtained by making use of the Green’s function method. However, the summation of the polarization vectors cannot be connected to the Green’s function as we discuss below, and thus the employment of the old propagator is incorrect if one should treat the physical processes which involve the loop integral.

5.5.1

Lorentz Conditions of kµ ²µ = 0

Here, we briefly explain how we can obtain eq.(5.13). The free Lagrangian density for the vector field Z µ with its mass M is written as 1 1 LZ = − Gµν Gµν + M 2 Zµ Z µ 4 2 with Gµν = ∂ µ Z ν − ∂ ν Z µ . In this case, the equation of motion becomes ∂µ (∂ µ Z ν − ∂ ν Z µ ) + M 2 Z ν = 0.

(5.14)

Since the free massive vector boson field should have the following shape of the solution Z µ (x) =

i h 1 √ ²µk,λ ck,λ eikx + ck† ,λ e−ikx k λ=1 2V ωk

3 XX

(5.15)

Here, we can insert this solution into eq.(5.14) and obtain the following equation for the polarization vector ²µ (k 2 − M 2 )²µ − (kν ²ν )k µ = 0. (5.16) The condition that there should exist a non-zero solution for the ²µ requires that the determinant of the matrix should be zero, namely det{(k 2 − M 2 )g µν − k µ k ν } = 0.

(5.17)

This equation can be easily solved, and we find the following equation k2 − M 2 = 0

 

(5.18)

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of eq.(5.17). Therefore we insert this solution into   which is the only physical solution   eq.(5.16) and obtain the equation for the polarization vector ²µ kµ ² µ = 0

(5.19)

which should always hold. Here, we should note that this process of determining the condition on the wave function of ²µ is just the same as solving the free Dirac equation. Obviously this is the most important process of determining the wave functions in quantum mechanics, and surprisingly, this has been missing in the treatment of determining not only the massive vector boson propagator but also the photon propagator as well. Also, one can notice that the condition of eq.(5.19) is just the same as the Lorentz gauge fixing condition in quantum electrodynamics (QED), and this is often employed as the gauge fixing. However, one sees by now that the Lorentz condition itself can be obtained from the equation of motion, and therefore it is more fundamental than the gauge fixing, even though the theory of massive bosons has no gauge freedom. This indicates that the Lorentz gauge fixing in QED should not be a proper gauge fixing procedure since the Lorentz gauge fixing cannot give a further constraint on the polarization vector in the perturbation theory of QED. In addition, the number of degrees of freedom for the gauge fields can be understood properly since photon must have the two degrees of freedom due to the two constraint equations (the Lorentz condition and the gauge fixing condition).

5.5.2

Right Propagator of Massive Vector Boson

Now, we can evaluate the propagator of the massive vector field in the S-matrix expression. The second order perturbation of the S-matrix for the bosonic part can be written in terms of the T-product of the boson fields and it becomes 3 Z X d4 k µ ν eik(x1 −x2 ) µ ν ² . (5.20) h0|T {Z (x1 )Z (x2 )}|0i = i ² (2π)4 k,λ k,λ k 2 − M 2 − iε λ=1

After over the polarization states, we find the following shape for P3 theµ summation ν λ=1 ²k,λ ²k,λ as ¶ µ 3 X kµ kν µ ν µν ²k,λ ²k,λ = − g − 2 k

(5.21)

λ=1

which satisfies the Lorentz invariance and the condition of the polarization vector kµ ²µ = 0. One sees that this is the only possible solution. From eq.(5.21), one finds that the right propagator of the massive vector boson should be the one given in eq.(5.13) µ ν

Dµν (k) = −

g µν − k kk2 . k 2 − M 2 − iε

Here it may be important to note that the polarization vector ²µk,λ should depend only on the four momentum k µ , and it cannot depend on the boson mass at this expression. Later, one may replace the k 2 term by M 2 in case the vector boson is found at the external line. But in the propagator, the replacement of the k 2 term by M 2 is forbidden.

 

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  5.5.3

Fundamental Problems in Quantum Field Theory 97

Renormalization Scheme of Massive Vector Fields  

In 1970’s, people found that some experiments indicate there might be heavy vector bosons exchanged between leptons and baryons in the weak processes. Therefore, people wanted to start from the massive vector bosons. However, it was somehow believed among educated physicists that only gauge field theories must be renormalizable. We do not know where this belief came from. In fact, there is no strong reason that only the gauge field theory is renormalizable. On the contrary, we know by now that only QED may well have a strange divergence in the vertex corrections.

5.6

Vertex Corrections by Weak Vector Bosons

Now we can calculate the vertex correction Λρ (p0 , p) of electromagnetic interaction due to the Z 0 boson. The Lagrangian density for the Z 0 boson which couples to the electron field ψe should be written as 1 1 LZ 0 = − Gµν Gµν + M 2 Zµ Z µ − gz ψ¯e γµ (1 − γ5 )ψe Z µ 4 2

(5.22)

where the free Lagrangian density part of electron is not written here for simplicity. This vertex correction is a physical process which can be directly related to the physical observables, and therefore we should be concerned with its divergences. The vertex correction Λρ (p0 , p) can be written by evaluating the corresponding Feynman diagrams as [56] Ã ! µ ν Z g µν − k kk2 1 1 d4 k ρ 0 2 γρ γµ γ 5 0 γν γ 5 Λ (p , p) = −igz e 4 2 2 (2π) k − M − iε p/ − k/ − me p/ − k/ − me (5.23) where only the term corresponding to the γ 5 γµ is written for simplicity.

5.6.1

No Divergences

First, we show that the apparent logarithmic divergent terms in eq.(5.23) vanish to zero, and this can be easily proved since we can find ³ ´ ρ k/γ µ − k/k/γ ρ k/k/ Z 1 Z γ k /γ 4 µ 2 d k k 2xdx =0 Λρ (p, p) = −iegz2 (5.24) (2π)4 0 (k 2 − s − iε)3 where s = M 2 (1 − x) + m2e x2 . Therefore, there is no logarithmic divergence for the vertex correction from the weak massive vector boson propagations. This is very important in that the physical processes do not have any divergences when we make use of the proper propagator of the massive vector boson.

 

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  5.6.2

Fujita and Kanda

Electron g − 2 by Z 0 Boson

The finite part of the vertex correction due to the Z 0 boson can be easily calculated and, therefore, the electron g − 2 should be modified by the weak interaction to 7αz ³ me ´2 g−2 ' ' 2 × 10−14 2 12π M

(5.25)

where

gz2 ' 2.73 × 10−3 . 4π This is a very small effect, and therefore, it is consistent with the electron g − 2 experiment. We should note that, if we employed the standard propagator of the massive vector boson as given in the field theory textbooks [15], then we would have obtained a very large effect on the electron g − 2, even if we had successfully treated the problem of the quadratic and logarithmic divergences in some way or the other, by renormalizing them into the fermion self-energy contributions. This strongly suggests from the point of view of the renormalization scheme that the propagator of the massive vector field should be the one given by eq.(5.13). αz =

5.6.3

Muon g − 2 by Z 0 Boson

Here, we should also give a calculated value of the muon g − 2 due to the Z 0 boson since it is just the same formula as eq.(5.24) except the mass of lepton. The result becomes ¶ µ 7αz ³ mµ ´2 g−2 ' ' 8.6 × 10−10 (5.26) 2 12π M µ which is much larger than the electron case. This is, however, still too small to be observed by the muon g − 2 experiments at the present stage.

Conflict of Interest The author(s) confirm that this FKDSWHU content has no conflicts interest. Acknowledgements: Declared none.

 

Send Orders of Reprints at [email protected] Fundamental Problems in Quantum Field Theory, 2013, 99-120

99

CHAPTER 6

Gravity Abstract: In this chapter, we present the new model of the quantum field theory of gravitation which is, by now, properly included into the Lagrangian density of quantum electrodynamics. In this model of QED plus gravity, Dirac fields couple to the electromagnetic field Aµ as well as the gravitational field G. The gravity appears ¯ with the coupling constant of g, thus keeping the in the mass term as m(1 + gG)ψψ local gauge invariance of the total Lagrangian of QED plus gravity. Here, after a brief review of the new gravity model, we present the discussion of applying the new gravity model to the time shifts of various kinds of planet motions. Keywords: gravity, Dirac equation for gravity, time shifts of Mercury, time shifts of earth motion, time shitts of GPS, general relativity, time shifts of comets.

6.1

Introduction

The motion of the earth is governed by the gravitational force between the earth and the sun, and the Newton equation is written as m¨ r = −GmM

r r3

where G, m and M denote the gravitational constant, the mass of the earth and the mass of the sun, respectively. This is the classical mechanics which works quite well. The gravitational potential that appears in the Newton equation is experimentally determined. However, the theoretical derivation of the gravity cannot be achieved in any of the equations such as Newton equation or Maxwell equation. Einstein presented the equation of general relativity which should be some analogous equations to the Maxwell equation in the sense that the gravitational field should be determined by the equation of general relativity. However, since he employed the principle of equivalence which has nothing to do with real nature, the general relativity became an equation that determines the metric tensor. This does not mean that one can determine the gravitational interaction, and indeed, the correct Takehisa Fujita and Naohiro Kanda All rights reserved - © 2013 Bentham Science Publishers

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direction in physics has been completely lost for a long time. Therefore, a theoretical frame work should be found which can determine the gravitational interaction with fermions in a proper manner. Here, we should first discuss the fundamental problems in the theory of general relativity. Basically, there are two serious physical problems in the general relativity, the lack of field equation under the gravity and the assumption of the principle of equivalence. In addition, the general relativity is not a field theory but a classical mechanics, and this is the most important defect of the general relativity. This is related to the observation that space and time are simply parameters in field theory, and they cannot be a target of the physics research.

6.1.1

Field Equation of Gravity

When one wishes to write the Dirac equation for a particle under the gravitational interaction, then one faces to the difficulty. Since the Dirac equation for a hydrogen-like atom can be written as ¶ µ Ze2 Ψ = EΨ (6.1) −i∇ · α + mβ − r one may include the gravitational potential in the Dirac equation either in the zeroth component in the interaction term like the Coulomb case or in the mass term. This problem is solved completely, and by now, we know the following Dirac equation for the Coulomb potential together with the gravitational potential [31] ·

6.1.2

µ ¶ ¸ GmM Z Ze2 + m− β Ψ = EΨ. −i∇ · α − r r

(6.2)

Principle of Equivalence

The theory of general relativity is entirely based on the principle of equivalence. Namely, Einstein started from the Gedanken experiment that physics of the two systems (a system under the uniform external gravity and a system that moves with a constant acceleration) must be the same. This looks plausible from the experience on the earth. However, one can easily convince oneself that the system that moves with a constant acceleration cannot be defined properly since there is no such an isolated system (space and time) in a physical world. The basic problem is that the assumption of the principle of equivalence is concerned with the two systems which specify space and time, not just the numbers in connection with the acceleration of a particle. Note that the acceleration of a particle is indeed connected to the gravitational acceleration, z¨ = −g, but this is, of course, just the Newton equation. Therefore, the principle of equivalence inevitably leads Einstein to the space deformation. It is clear that physics must be the same between two inertia systems, and any assumption which contradicts this basic principle cannot be justified at all.

 

Gravity

Fundamental Problems in Quantum Field Theory 101

Frame or Coordinate Transformation Besides, this problem can be viewed differently in terms of Lagrangian. For the system under the uniform external gravity, one can write the corresponding Lagrangian. On the other hand, there is no way to construct any Lagrangian for the system that moves with a constant acceleration. One can define a Lagrangian for a particle that moves with a constant acceleration, but one cannot write the system (or space and time) that moves with a constant acceleration. Physics in one inertia frame must be equivalent to that of another inertia frame, and this requirement is very severe. It is not only a coordinate change of space and time with Lorentz transformation, but also physical observables must be the same between two systems. In this respect, the principle of equivalence violates this important condition, and therefore, it is very hard to accept the assumption of the principle of equivalence even with the most modest physical intuition.

6.1.3

General Relativity

Einstein generalized the Poisson type equation for gravity ∇2 φg = 4πGρ

(6.3)

to the tensor equations which should have some similarities with the Maxwell equation. Therefore, he had to find some tensor quantity like the field strength F µν of the electromagnetic field, and the metric tensor g µν is chosen as the basic tensor field since he started from the principle of equivalence. Thus, the general relativity is the equation for the metric tensor g µν which, he believed, should be connected to the gravitational field φg . By assuming that 2φg (6.4) g 00 ' 1 + 2 c together with T 00 ' ρ (6.5) with the energy momentum tensor of T µν , he arrived at the equation of the general relativity 1 (6.6) Rµν − g µν R = 8πGT µν 2 where Rµν denotes the Ricci tensor which can be described in terms of the metric tensor g µν . However, the physical meaning of the g µν is unclear, and that is the basic problem of the general relativity. Consistency with Special Relativity Before going to the discussion of the Lagrangian density of the gravity, it should be important to clarify the origin of the coordinate xµ in the metric tensor g µν (x) from where it is measured. From Einstein’s equation, it is clear that the origin of the coordinate should be

 

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found in the matter field center. Therefore, one can see that the metric tensor g µν should be in contradiction with the principle of special relativity since its space and time become different from the space and time of the other inertia system. This peculiar behavior of the metric tensor g µν is just the result of the principle of equivalence which is not consistent with the principle of special relativity.

6.2

Lagrangian Density with Gravitational Interactions

The gravitational force is important in the Newton equation since it can properly describe the motion of the planets around the sun. However, the Newton equation can be derived from the Schr¨odinger equation by making the classical limit. Further, this Schr¨odinger equation can be obtained from the Dirac equation, and therefore it is crucial that the gravitational interaction should be included in the Dirac equation. This indicates that we should construct the Lagrangian density which can take into account the gravitational interactions. In this chapter, we present a model Lagrangian density which can describe electrons interacting with the electromagnetic field Aµ as well as the gravitational field G.

6.2.1

Lagrangian Density for QED and Gravity

Now, we propose to write the Lagrangian density for electrons interacting with the electromagnetic field as well as the gravitational field G [65] ¯ µ ∂µ ψ − eψγ ¯ µ Aµ ψ − m(1 + gG)ψψ ¯ − 1 Fµν F µν + 1 ∂µ G ∂ µ G L = iψγ 4 2

(6.7)

where the gravitational field G is assumed to be a massless scalar field and the field strength F µν is defined as F µν = ∂ µ Aν − ∂ ν Aµ . (6.8) It is easy to prove that the new Lagrangian density is invariant under the local gauge transformation Aµ → Aµ + ∂ µ χ, ψ → e−ieχ ψ. (6.9) This is, of course, quite important since the introduction of the gravitational field does not change the most important local gauge symmetry of the electromagnetic interaction.

6.2.2

Dirac Equation with Gravitational Interactions

Now, one can easily obtain the Dirac equation for electrons from the new Lagrangian density iγ µ ∂µ ψ − eγ µ Aµ ψ − m(1 + gG)ψ = 0. (6.10) Also, one can write the equation of motion of gravitational field ¯ ∂µ ∂ µ G = −mg ψψ.

 

(6.11)

Gravity

Fundamental Problems in Quantum Field Theory 103

The symmetry property of the new Lagrangian density can be easily examined, and one can confirm that it has a right symmetry property under the time reversal transformation, parity transformation and the charge conjugation.

6.2.3

Total Hamiltonian for QED and Gravity

The Hamiltonian can be constructed from the Lagrangian density Z Z © ª 3 j0 (r 0 )j0 (r)d3 rd3 r0 e2 ¯ H= ψ (−iγ · ∇ + m(1 + gG)) ψ − ej · A d r + 8π |r 0 − r| # # ¶ Z "µ Z "µ ¶2 ∂A 2 ∂G 1 1 + + (∇ × A)2 d3 r + + (∇G)2 d3 r (6.12) 2 ∂t 2 ∂t ¯ µ ψ. In this expression of the Hamiltonian, the gravitational where j µ is defined as j µ = ψγ energy is still written without making use of the equation of motion. In the next section, we will treat the gravitational energy and rewrite it into an expression which should enable us to easily understand the structure of gravitational force between fermions.

6.2.4

Static-dominance Ansatz for Gravity

In eq.(6.8), the gravitational field G is introduced as a real scalar field, and therefore it cannot be a physical observable as a classical field [66]. In this case, it may be reasonable to assume that the gravitational field G can be written as the sum of the static and timedependent terms and that the static part should carry the information of diagonal term in the external source term. Thus, the gravitational field G is assumed to be written as ¯ G = G0 (r) + G(x)

(6.13)

where G0 (r) does not depend on time. This ansatz is only a sufficient condition, and its validity cannot be verified mathematically, but it can be examined experimentally. The ¯ equations of motion for G0 (r) and G(x) become ∇2 G0 = mgρg

(6.14)

¯ [non−diagonal] + (ψψ) ¯ [diagonal rest] } ¯ ∂µ ∂ µ G(x) = −mg{(ψψ)

(6.15)

where ρg is defined as

¯ [diagonal] ρg ≡ (ψψ)

¯ [diagonal] denotes the diagonal part of the ψψ ¯ while (ψψ) ¯ [non−diagonal] term where (ψψ) denotes a non-diagonal part. In this case, we can solve eq.(6.14) exactly and find a solution Z ρg (r 0 ) 3 0 mg d r. (6.16) G0 (r) = − 4π |r 0 − r|

 

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As long as the solution can satisfy the equation of motion of eq.(6.14), it is physically sufficient. The solution of eq.(6.16) is quite important for the gravitational interaction since this is practically a dominant gravitational force in nature. Here, we assume that the diagonal ¯ [diagonal] is mostly time independent, and in this case, the static gravitational term of (ψψ) S can be written as energy which we call HG Z S HG

= mg

1 ρg G0 d r + 2 3

Z

m2 G (∇G0 ) d r = − 2 2 3

Z

ρg (r 0 )ρg (r) 3 3 0 d rd r |r 0 − r|

(6.17)

where the gravitational constant G is related to the coupling constant g as G=

g2 . 4π

(6.18)

Eq.(6.17) is just the gravitational interaction energy for the matter fields, and one sees that the gravitational interaction between electrons is always attractive. This is clear since the gravitational field is assumed to be a massless scalar. It may also be important to note that S of eq.(6.17) is obtained without making use of the perturbation theory, and it is the HG indeed exact, apart from the static ansatz of the field G0 (r).

6.2.5

Quantization of Gravitational Field

In quantum field theory, we should consider the quantization of fields. For fermion fields, we should quantize the Dirac field by the anti-commutation relations of fermion operators. This is required from the Pauli principle, that is, a fermion can occupy only one quantum state. In order to accommodate this experimental fact, we should always quantize the fermion fields with the anti-commutation relations. On the other hand, for gauge fields, we must quantize the vector field in terms of the commutation relation which is required from the experimental observation that one photon is emitted by the electron transition between 2p and 1s states in hydrogen atoms. That is, a photon is created from the vacuum of the electromagnetic field, and therefore the field quantization is an absolutely necessary procedure. However, it is not very clear whether the gravitational field G should be quantized or not. No Quantization of Gravitational Field At present, we should take a standpoint that the gravitational field G should not be quantized since there is no requirement from experiments. In this sense, the gravitational field G should remain to be a classical field. In this case, we do not have to worry about the renormalization of the graviton propagator, and we obtain the gravitational interaction between fermions which is always attractive, and this is consistent with the experimental requirement.

 

Gravity

6.3

Fundamental Problems in Quantum Field Theory 105

Cosmology

What should be a possible picture of our universe in the new quantum theory of gravity ? By now we have sufficient knowledge concerning the cosmology how the present universe is created and what should be its fate near future. Below is a simple picture one can easily draw, even though it is almost a story. In order to make it into physics, hard works may be required, though it must be a doable task. Here, we should define a hierarchy of the mugen-universe (infinite number of universes) with very rough numbers 1057 × protons ⇒ star

:

1012 × stars ⇒ galaxy

1012 × galaxies ⇒ universe

6.3.1

:

:

∞ × universe ⇒ mugen − universe.

Cosmic Fireball Formation

Since the gravity is always attractive, it is clear that all of the galaxies should eventually get together. A question may arise as to in which way these galaxies would collapse into a Cosmic Fireball. It is most likely true that, after the end of the expansion of the present universe, a few galaxies should coalesce into a larger galaxy, and this coalescence should take place repeatedly until two or three giant clusters of galaxies should be formed. Finally, these giant clusters would eventually collide into a Cosmic Fireball which should be quite similar to the initial stage of the big bang. After the Cosmic Fireball is created, it should rapidly expand, and during the expansion, light nuclei should be created. In this picture, galaxies should be naturally formed since the expansion after the explosion should not be very uniform. This is in contrast to the big bang cosmology in which the galaxy formation must be quite difficult since the big bang should be extremely uniform. In this respect, the universe should repeatedly make the same formation of galaxies. The universe should have existed from the infinite time of past, and it should make the galaxy formation and collision in the infinite time of future. Here, it should be noted that the concept of the infinite time of past or future is beyond the understanding of human being. Also, the mugen-universe should have the infinite space, but again the infinite space should not be a target of physics research.

6.3.2

Relics of Preceding Universe

According to the present picture of the universe, there may well be some relics of the preceding universe before the Cosmic Fireball. Large Scale Structure of Universe In the present universe, there is a large scale structure of the universe among cluster of galaxies such as the Great Attractor. This should be related to the remnants of the Cosmic Fireball formation when the preceding universe got together into the Cosmic Fireball.

 

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Photon Baryon Ratio Another possible relic must be the large number of photons compared to the number of baryons in the present universe. This photon-baryon ratio may well be understood in terms of the relic of photons in the preceding universe since photon has some interactions with strong gravitational fields [31], and therefore some of photons may be trapped during the Cosmic Fireball formation. On the other hand, neutrino should not be trapped due to the lack of the interactions with baryons. Therefore, the number of neutrinos must be much smaller than the number of photons in our universe.

6.3.3

Mugen-universe

The mugen-universe must have an infinite number of universes, and the present universe should be only one of them. This is based on the assumption that our universe should have existed from the infinite time of past. Dilemma of One Universe Scenario If the present universe is the only universe in the whole world, then our universe must have already lost its whole energy into the space outside the universe. This is clear since the present universe must have been radiating away a finite portion of the gravitational energy by photons and neutrinos through the Cosmic Fireball explosions. The infinite time indicates that a small but finite energy loss must become infinite even for our universe, and therefore, by now, nothing should have been left for our universe. Mugen-universe Scenario If the whole world has an infinite number of universes (mugen-universe), then the dilemma of the energy loss by radiations can be nicely avoided. This is because the energy loss can be compensated by the energy gain of photons and neutrinos from the rest of the mugenuniverse. In this case, however, a question may arise as to how the present universe can be stable against the collapse of the mugen-universe. This puzzle can be solved in the following way. Suppose our universe is sitting at the center of the infinite space of −∞ < x < ∞ in which one dimensional space must be sufficient for the present discussion. Now, in the left hand side, there are infinitely many universes which attract the present universe. However, infinitely many universes in the right hand side should also attract our universe in the same way, and therefore our universe must be stable against the collapse since both sides of the universes are attracting our universe each other. This infinite space can be replaced by a circle with a radius R which should be set to infinity, and in this case, our universe sitting at one point of the circle should feel the same attractions from universes in the left and right hand sides because they are just the same, at least, mathematically, because of the circle. In reality, our universe is in three dimension, and thus it should be even more stable than one dimension. This is, of course, only a story and not physics, but a possible inconsistency within the theoretical framework should be removed.

 

Gravity

6.4

Fundamental Problems in Quantum Field Theory 107

Time Shifts of Mercury and Earth Motions

The new gravity model is applied to the description of the observed advance shifts of the Mercury perihelion, the earth rotation and the GPS satellite motion. First, we should obtain the gravitational potential which can be calculated from the non-relativistic reduction of the Dirac equation in terms of the Foldy-Wouthuysen transformation. Then, we should make the classical limit of the Hamiltonian so that we can obtain the classical potential for the gravity.

6.4.1

Non-relativistic Gravitational Potential

The Hamiltonian of the Dirac equation in the gravitational field can be written as ¶ µ GmM β H = −i∇ · α + m − r

(6.19)

where M denotes the mass of the gravity center. This Hamiltonian can be easily reduced to the non-relativistic equation of motion by making use of the Foldy-Wouthuysen transformation [10]. Here, we only write the result in terms of the Hamiltonian H H =m+

GmM 1 GmM 2 1 GM m p2 − + p − (s · L) 2 2m r 2m r 2m2 r3

(6.20)

where the last term denotes the spin-orbit force, but we do not consider it here. Now, we make the classical limit to derive the Newton equation. In this case, it is safe to assume the factorization ansatz for the third term, that is, À ¿ À ¿ 1 GmM ­ 2 ® 1 GmM 2 p = p . (6.21) 2m2 r 2m2 r By making use of the Virial theorem for the gravitational potential À ¿ 2À ¿ GmM p = m r we obtain the new gravitational potential for the Newton equation µ ¶ 1 GmM 2 GmM + V (r) = − r 2mc2 r

(6.22)

(6.23)

where we explicitly write the light velocity c in the last term of the equation.

6.4.2

Time Shifts of Mercury, GPS Satellite and Earth

The Newton equation with the new gravitational potential can be written as m¨ r=−

 

`2 G2 M 2 m GmM + + . r2 mr3 c2 r3

(6.24)

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Therefore, we can introduce a new angular momentum L as G 2 M 2 m2 . c2

L2 ≡ `2 +

(6.25)

Further, we define the angular velocity ω and radius R by ` , mR2

ω≡

R≡

`2 3

GM m2 (1 − ε2 ) 4

(6.26a)

where ε denotes the eccentricity. Correspondingly, we can define a new angular velocity Ω associated with ω as G2 M 2 Ω2 ≡ ω 2 + 2 4 = ω 2 (1 + η) (6.26b) c R where η is defined as η=

G2 M 2 . c2 R4 ω 2

(6.27)

The equation (6.24) can be immediately solved, and one finds the solution of the orbit r=

A ¡ ¢ 1 + ε cos L` ϕ

(6.28)

where A and ε are given as L2 , A= GM m2

s 1+

ε=

2L2 E . m(GmM )2

Physical observables can be obtained by integrating ϕ˙ = ` m

Z

Z

T

2π

dt = 0

Z 2

2

r dϕ = A 0

0

2π

` mr2

(6.29)

over the period T

1 ¡ ¡ ¢¢2 dϕ. 1 + ε cos L` ϕ

(6.30)

This can be easily calculated to be ωT = 2π(1 + 2η) (1 − εη) ' 2π{1 + (2 − ε)η}

(6.31)

where ε is assumed to be small. Therefore, the new gravity potential gives rise to the advance shift of the time shift, and it can be written as ¶ µ ∆T ' (2 − ε)η. (6.32) T th This is a physical observable which indeed can be compared to experiment.

 

Gravity

  6.4.3

Fundamental Problems in Quantum Field Theory 109

Mercury Perihelion Shifts  

The Mercury perihelion advance shift ∆θ is well known to be [67] ∆θ ' 43 00 per 100 year. Since Mercury has the 0.24 year period, it can amount to the shift ratio δθ ¶ µ ∆T ' 8.0 × 10−8 . δθobs ≡ T obs

(6.33)

(6.34)

The theoretical calculation of the new gravity model shows G2 M 2 ' 2.65 × 10−8 c2 R4 ω 2 where the following values are used for the Mercury case η=

(6.35)

R = 5.73 × 1010 m, M = 1.989 × 1030 kg, ω = 8.30 × 10−7 . Therefore, the theoretical shift ratio δθth becomes ¶ µ ∆T ' 4.8 × 10−8 δθth ≡ T th

(6.36)

which should be compared to the observed value in eq.(6.34). As can be seen, this agreement is indeed remarkable since there is no free parameter in the theoretical calculation [68].

6.4.4

GPS Satellite Advance Shifts

Many GPS satellites which are orbiting around the earth should be influenced rather heavily by the new gravitational potential. The GPS satellite advance shift can be estimated just in the same way as above, and we obtain G2 M 2 ' 1.69 × 10−10 c2 R4 ω 2 where we employ the following values for the GPS satellite [69, 70] η=

R = 2.6561 × 107 m, M = 5.974 × 1024 kg, ω = 1.4544 × 10−4

(6.37)

(6.38)

since the satellite circulates twice per day. Therefore, the advance shift of the GPS satellite becomes ¶ µ ∆T ' 3.4 × 10−10 . (6.39) T th This should be compared to the observed value of ¶ µ ∆T ' 4.5 × 10−10 . (6.40) T obs As seen from the comparison between the calculation and the observed value, the new gravity theory can indeed achieve a remarkable agreement with experiment.

 

110 Fundamental Problems in Quantum Field Theory

  6.4.5

Fujita and Kanda

Time Shifts of Earth  Rotation − Leap Second

Here, we calculate the time shift of the earth rotation around the sun [71]. First, we evaluate the value of η G2 M 2 (6.41) η = 2 4 2 ' 0.992 × 10−8 c R ω where we employ the following values for R, M and ω R = 1.496 × 1011 m, M = 1.989 × 1030 kg, ω = 1.991 × 10−7 .

(6.42)

In this case, we find the time shift for one year (∆T )th ' 0.621 s/year

(6.43)

where ε = 0.0167 is taken. In fact, people have been making corrections for the leap second, and according to the data, they made the first leap second correction in June of 1972. After that, they have made the leap second corrections from December 1972 to December 2008. The total corrections amount to 23 seconds for 36.5 years since we should start from June 1972. This corresponds to the time shift per year (∆T )obs ' 0.63 ± 0.02 s/year

(6.44)

where the errors are supposed to come from one year shift of the observation. This agrees surprisingly well with the theoretical time shift of the earth. Indeed, this confirms that the new gravity model is a correct theory of gravitation.

6.4.6

Observables from General Relativity

Now, we discuss the calculated results by the general relativity [72, 67]. For the Mercury perihelion shift, the result is quite well known, and it can be written in terms of the angular shift. In fact, the angular variable ϕ is modified by the general relativity to cos ϕ −→ cos(1 − γ)ϕ

(6.45)

where γ is found to be γ=

3G2 M 2 . c2 R4 ω 2

(6.46)

Now people thought that this change of the shift in the angular variable could explain the observed Mercury perihelion shift. However, as shown below, this effect vanishes to zero in the case of ε = 0, that is, for the circular orbit. This is, of course, unphysical in that the effect of the general relativity is valid only for the elliptic orbit case. In Newton dynamics, the angular momentum ` is the only quantity which can be affected from the external effects like the general relativity or the additional potential.

 

Gravity

  6.4.7

Fundamental Problems in Quantum Field Theory 111

Relativity Prediction from General  

Now, we should calculate the physical observables as to how the general relativity can induce the perihelion shift. In this case, one finds that the change appears in eq.(6.28) as r=

A A ¡L ¢ ⇒ r = 1 + ε cos ((1 − γ)ϕ) 1 + ε cos ` ϕ

(6.47)

and thus the physical observable becomes ωT ' 2π(1 + 2εγ). Therefore, the advance shift of the Mercury perihelion becomes ¶ µ ∆T ' 3.3 × 10−8 δθth ≡ T th

(6.48)

(6.49)

which is a factor of 2.5 smaller than the observed value of the Mercury perihelion shift. It should be noted that the predicted shift in eq.(6.47) is indeed the advance shift of the Mercury perihelion as given in eq.(6.48). In addition, the GPS satellite shift predicted by the general relativity becomes ¶ µ ∆T ' 0.10 × 10−10 (6.50) T th which is very small. This is because the GPS satellite motion has almost the circular orbit around the earth. Further, the time shift of the earth rotation around the sun predicted by the general relativity becomes (∆T )th ' 0.031 s/year. (6.51) This shows that it is much too small compared to the observed time shift of the earth rotation around the sun. In reality, if the angular momentum is affected from the external potential as given in eq.(6.25), then not only the angular variable but also A in eq.(6.29) should be changed, and therefore as the total effects of the physical observables in the general relativity, eq.(6.48) is modified to ωT ' 2π{1 − 2(2 − ε)γ} (6.52) which is, unfortunately, a retreat shift since ε is smaller than unity.

6.4.8

Summary of Comparisons between Calculations and Data

We summarize the calculated results of the Mercury perihelion shift, GPS satellite advance shift and Leap Second corrections due to the new gravity model as well as the general relativity. Here, the observed data are compared with the predictions of the model calculations in Table 1.

 

112 Fundamental Problems in Quantum Field Theory

 

 

Observed data New Gravity General Relativity

Fujita and Kanda

Table 1

Mercury (∆T /T ) 8.0 × 10−8 4.8 × 10−8 3.3 × 10−8

GPS (∆T /T ) 4.5 × 10−10 3.4 × 10−10 0.10 × 10−10

Leap Second ∆T 0.63 ± 0.02 s/year 0.62 s/year 0.031 s/year

Table 1 shows the calculated results of the Mercury perihelion shift, GPS satellite advance shift and Leap Second corrections together with the observed data. The New Gravity shows the prediction of the new gravity model calculations which are discussed here. The General Relativity is the calculation in which we only consider the angular shift following Einstein. From this table, one sees that the general relativity cannot describe the observed data before employing the prediction of the physical observables from eq.(6.52).

6.4.9

Intuitive Picture of Time Shifts

It may be interesting to note that the velocity of the Mercury or the earth around the sun is one of the fastest objects we can observe as a classical motion. This velocity v is around v ∼ 1.0 × 10−4 c, which leads to the correction of the relativistic effects in physical observables as ³ v ´2 ∼ 1.0 × 10−8 c which is just the same magnitude as the values observed in the Mercury perihelion shift (∆T /T ∼ 5 × 10−8 ) and the leap second corrections (∆T /T ∼ 2 × 10−8 ). Therefore, it should not be surprising at all that the new additional gravitational potential which is obtained as the relativistic effects of the gravity potential in Dirac equation can account for the advance shifts of the planets orbiting around the sun. In this sense, the physical effect of the earth rotation velocity on the perihelion shift can be compared to the Michelson-Morley experiment. The interesting point is that the Michelson-Morley experiment is essentially to examine the kinematical effect of the relativity that the light velocity is not influenced by the earth rotation velocity, even though the classical mechanics indicates it should be affected. On the other hand, the leap second correction is the relativistic effect of the dynamical motion of the earth rotation, and it is a deviation from the Newton mechanics. Both of the observed facts can be understood by the relativistic effects of the earth motion around the sun, and in fact, the Michelson-Morley experiment proves that the light velocity is independent of the speed of the earth rotation, which leads to the concept of the special relativity, while the perihelion shift of the planets confirms the existence of the new additional gravity potential which is derived from the non-relativistic reduction of the Dirac equation with the gravitational potential.

 

Gravity

  6.4.10

Fundamental Problems in Quantum Field Theory 113

Leap Second Dating 

Since we know quite accurately the time shift of the earth rotation around the sun by now, we may apply this time shift to the dating of some archaeological objects such as pyramids or Stonehenge. For example, the time shift of 1000 years amounts to 10.3 minutes, and some of the archaeological objects may well possess a special part of the building which can be pointed to the sun at the equinox. In this case, one may be able to find out the date when this object was constructed. This new dating procedure is basically useful for the stone-made archaeological objects in contrast to the dating of the wooden buildings which can be determined from the Carbon dating. It should be noted that the new dating method has an important assumption that there should be no major earthquake in the region of the archaeological objects. It should be worthwhile noting that one should be careful for the Leap Second Dating method in the realistic application. This is clear since the earth is also rotating in its own axis when it is rotating around the sun. Therefore, the advance time shift of the earth rotation around the sun should correspond to the retreat time shift of the earth’s own rotation if one measures it at one fixed point of the earth surface.

6.5

Time Shifts of Comets

Here, we discuss the time shifts of the comets whose orbits may not be a simple ellipse. Now the total energy of the comet can be written as α 1 ³ α ´2 1 (6.53) E = mv 2 − + 2 r 2mc2 r where α ≡ GM m. If we use the virial theorem, then we find µ ¶ 1 DαE 1DαE 1− E=− 2 r mc2 r where

D1E r

=

³1´ r

av

1 = T

Z

T

0

(6.54)

1 1 dt = . r a

Here, a denotes the long axis of the ellipse. Therefore, its energy becomes E = E0 (1 − γ¯ )

(6.55)

where E0 and γ¯ are defined as γ¯ ≡

 

1 D α E GM = 2 mc2 r c a

(6.56a)

114 Fundamental Problems in Quantum Field Theory

 

 

E0 ≡ −

Now, we introduce η η≡

Fujita and Kanda

1DαE . 2 r

(6.56b)

³ GM ´³ m ´2 c

At the first order approximation, we find

(6.57)

`

` m

q q 2πab 2π 2 ` = = a 1 − ε20 = GM a(1 − ε20 ) m T T

(6.58)

where a, b denote the long and short axis lengths. Thus, η becomes η≡

GM 1 c2 a 1 − ε20

(6.59)

and ε is written r 2EL2 2E0 `2 = 1+ (1 − γ)(1 + η). ε= 1+ 2 GM m GM m2 r

Now, we define

r ε0 ≡ 1 + δ ≡η−γ =

and in the case of

δ ε0

(6.60)

2E0 `2 GM m2

GM ε20 c2 a 1 − ε20

¿ 1, we find ε as

where D≡

ε ' ε0 + D

(6.61)

E0 `2 1 δ. ε0 (GM m)2 m

(6.62)

Now, the period T becomes · ¾2 ¸2 ½Z 2π `2 m 1 2 p dϕ T = (1 + η) ` GM m2 (1 + (ε0 + D) cos 1 + ηϕ 0 µ · ¶ ¾ ¸2 ½ 3π π `2 m 2π ' η . 3 + 1 + ` GM m2 (1 + ε20 )2 (1 − ε2 ) 2 (1 − ε2 ) 2 0

If we define T0 as

0

· ¸2 `2 m 2π T0 = 3 2 ` GM m (1 − ε2 ) 2 0

 

(6.63)

(6.64)

Gravity

Fundamental Problems in Quantum Field Theory 115

of the Newton result. Now the deviation from the   then, this corresponds to the period   Newton period T0 ∆T ≡ T − T0 can be obtained as

` π ∆T = m c2

½

¾ 1 . 1 + (1 + ε20 )2 (1 − ε20 ) 2 3

(6.65)

This expression is valid for all the values of the ε0 , and therefore we obtain  2π2 a2 ¡ ¢ 2 2 − ε0 , for ε0 ¿ 1    T0 c2 µ ∆T ' q ¶  2 a2  2π 1  T c2 3 + 2 d2 , for ε0 = 1 − d, d ¿ 1. 0

(6.66)

Here, we present some calculations of the period ∆T for several cases. Table 2 ε0 0.967 0.847 0.634 0.820 0.258 0.0167

Halley Encke Pons-Winnecke Tuttle Whipple Earth

6.6

a (AU ) 17.8 2.22 3.43 5.70 4.17 1.0

T0 (year) 75.3 3.3 6.37 13.6 8.50 1.0

`/m 4.79 × 1015 3.53 × 1015 6.38 × 1015 6.09 × 1015 8.78 × 1015 4.46 × 1015

∆T (s/period) 2.0 0.73 0.95 1.2 1.1 0.62

Effects of Additional Potential by Hamilton Equation

In this section, we carry out the extensive calculation of the Newton equation with the additional potential. Here, instead of employing the Virial theorem, we solve the Hamilton equation without making any approximations. The Hamiltonian we start is written as H=

α 1 α 2 p2 − + p . 2m r 2m2 c2 r

(6.67)

Now, we make use of the Hamilton equations and obtain r˙ =

1 α p + 2 2 p, m m c r

where p is defined as p= ³

 

p˙ = − mr˙

1+

1 α mc2 r

α 1 α 2 r+ p r 3 r 2m2 c2 r3

´=

mr˙ . A(r)

(6.68)

116 Fundamental Problems in Quantum Field Theory

  Also, we have

  A(r) ≡ 1 +

Fujita and Kanda

1 α ≡ 1 + γ(r) mc2 r

and thus the Lagrangian becomes L=

α 1 mr˙ 2 + . 2 A(r) r

(6.69)

Therefore, we can find the equation of motion as m¨ r=−

˙ α 1 1 α r˙ 2 α (r · r) α ˙ r − γ(r)r + r− r. 3 3 2 3 2 3 r r 2(1 + γ(r)) c r 1 + γ(r) c r

(6.70)

The energy and angular momentum are conserved quantity, which can be written as d nm 1 2 αo r˙ − =0 dt 2 A r o dn1 r × mr˙ = 0. dt A By denoting the energy E, we find the equation of motion in polar coordinates as r ³ r2 α ´ `2 2 dr = − 2A . (6.71) 2Am E + dϕ `A r r This can be easily solved, and we obtain r=

p p 1 + e cos( 1 + 2τ ϕ)

where α2 τ ≡ 2 2, c `

6.6.1

`2 (1 + 2τ ) ´, ³ p= E mα 1 − mc 2

v u 2E`2 (1 + 2τ ) e ≡u ´2 . ³ t1 + E 2 mα 1 − mc2

(6.72)

(6.73)

Bound State Case

Here, we consider the case in which an orbiting body is in the bound state and evaluate the period of the orbit. First, the angular momentum ` = A1 mr2 ϕ˙ is a conserved quantity and thus we find Z T Z 2π Z 2π 1 ` 2 dt = r2 (1 − γ(r))dϕ. (6.74) r dϕ ' m 0 1 + γ(r) 0 0 Therefore, we obtain Z 2π nZ 2π o 1 1 ` α 2 p p T =p . − m (1 + e cos( 1 + 2τ ϕ))2 mc2 p 0 1 + e cos( 1 + 2τ ϕ) 0

 

Gravity

Fundamental Problems in Quantum Field Theory 117

  After some calculations, we can  write e as s 1 + 2τ 2E0 `20 1 − γ¯ ' e0 + D e= 1+ 2 2 mα (1 + γ¯ ) (1 + 21 γ¯ )2 where

r e0 ≡ 1 +

2E0 `20 , mα2

D≡

(6.75)

1 2E0 `20 (2τ − 4¯ γ ). e0 mα2

Also, p can be written as µ ¶ 1 5 `20 1 + 2τ ' p0 1 + 2τ − γ¯ p= mα (1 + γ¯ )2 1 + 21 γ¯ 2

(6.76)

where

`20 . mα Since the period T0 in the Newton equation is written as p0 ≡

T0 ≡

2π m 2 p0 `0 (1 − e2 ) 23 0

we can define the deviation ∆T ≡ T − T0 and thus obtain ½ ¾ 1 1 ` π + 2 p . ∆T = m c2 1 − e20 (1 + e0 )2

(6.77)

This can be expressed for the two limiting cases of e0 ¿ 1 and e0 ∼ 1 as  2π2 a2 ¡ ¢ for e0 ¿ 1  2 4 1 − e0 ,  T c 0  µ q ¶ ∆T '  2 a2  d  2π T0 c2 2 + 2 , for e0 = 1 − d, d ¿ 1.

(6.78)

Here, we present some of the calculated results Table 3

Halley Encke Pons-Winnecke Tuttle Whipple Earth where we use

 

e0 0.967 0.847 0.634 0.820 0.258 0.0167

a (AU ) 17.8 2.22 3.43 5.70 4.17 1.0

T0 (year) 75.3 3.3 6.37 13.6 8.50 1.0

`0 /m 4.79 × 1015 3.53 × 1015 6.38 × 1015 6.09 × 1015 8.78 × 1015 4.46 × 1015

q 2π 2 `0 = a 1 − e20 . m T0

∆T (s/period) 1.4 0.54 0.74 0.87 1.02 0.61

118 Fundamental Problems in Quantum Field Theory

  6.6.2

Fujita and Kanda

Scattering State Case 

Next, we consider the cases in which the orbiting body should be found in the scattering state. First we start from the equation of motion `A 2

r dϕ = q 2mA(E + αr ) −

`

2

r dr ' q `2 2 A 2m(1 − γ)(E + αr ) − r2

`2 r2

dr

(6.79)

and we carry out the integration Z ϕ0 '

`

∞

2

r q rmin 2m(1 − γ)(E + α ) − r

where rmin ≡ r(ϕ = 0) =

p 1+e

and τ ≡

α2 c2 `2

1 dr = p (π − 2ψ0 ) `2 1 + 2τ

(6.80)

r2

are introduced. In this case, ψ0 becomes

sin2 ψ0 = −

1−e . 2e

Therefore, the scattering angle θ becomes 1 (π − 2ψ0 ). θ = π − 2p 1 + 2τ

(6.81)

Now, the eccentricity is described in terms of the energy E and angular momentum ` which are defined at infinity as v ¡ u 2¢ u 2E∞ `2∞ + 2α c2 t (6.82) e= 1+ ¢ . ¡ E∞ 2 mα2 1 − mc 2 This can be expanded as e ' eN +

1 ´ E∞ ³ e + = eN + δ N mc2 eN

where

r eN ≡ 1 +

and δ≡

(6.83)

2`2∞ E∞ mα2

1 ´ E∞ ³ e + . N mc2 eN

Thus, ψ0 becomes r −1

ψ0 = sin

 

s r 1 e−1 −1 eN − 1 −1 eN + δ − 1 q δ. (6.84) = sin ' sin + 2e 2(eN + δ) 2eN 2eN e2N − 1

Gravity

Fundamental Problems in Quantum Field Theory 119

  Now we write

 

where

ψ0 = ψ0N + ∆ r

ψ0N

−1

≡ sin

eN − 1 , 2eN

∆≡

(6.85) 1 q δ. 2eN e2N − 1

(6.86)

Since eN ≥ 1, thus δ and ∆ should always satisfy the following conditions δ ≥ 0, ∆ ≥ 0.

(6.87)

Here we write the scattering angle θ as θ = θ0 + δθ

(6.88)

where θ0 ≡ π − 2(π − 2ψ0N ), And τ can be written as τ=

δθ ≡ 4∆ + 2τ (π − 2ψ0N ).

(6.89)

α2 . c2 `2∞

(6.90)

From eqs. (6.86) and (6.89), we see that δθ ≥ 0.

(6.91)

This calculated result shows that the scattering angle in planets should be larger than the Newton equation. In addition, we may expect that the fly-by phenomena for the satellite motion, for example, should deviate from the Newton equation as well.

6.7

Photon-Photon Interaction via Gravity

The interaction between two photons can take place via vacuum polarization of fermions as discussed in chapter 3. Here, we show that photon can interact with photon via gravity through the vacuum polarization of fermions. In this case, the intermediate Coulomb interaction should be replaced by the gravitational interaction, and otherwise the corresponding Feynman diagram is just the same as the photon case. However, the process must be extremely small since it involves the gravity which is roughly speaking, smaller than photon by 10−36 or so. In this respect, the evaluation of the photon-photon interaction via gravity is only for academic interests. Therefore, we give here only the calculated result of the cross section between photons via gravity 1 1 dσ = dΩ 4 (2ω)6

 

µ

Gαm2 m02 π2

¶2

1 2 4 4 (1 + cos θ + 2 cos θ) sin θ

(6.92)

120 Fundamental Problems in Quantum Field Theory

Fujita and Kanda

  which is calculated as the sixth order perturbation theory. As can be easily seen, this cross section should be extremely small, but it is indeed finite.

Conflict of Interest The author(s) confirm that this FKDSWHU content has no conflicts interest. Acknowledgements: Declared none.

 

Send Orders of Reprints at [email protected] Fundamental Problems in Quantum Field Theory, 2013, 121-138

121

CHAPTER 7

Open Problems Abstract:

In this chapter, we discuss some of the problems which are not yet fully understood in the calculations of quantum field theory. First, we discuss the neutron EDM in terms of the vertex corrections from the W ± bosons. Then, we present the calculation of the weak charge which may arise from the parity violating interaction. Also, we discuss the Lamb shifts in muonic hydrogen as well as in muonium, and present the calculation of the center of mass corrections in these exotic atoms. It is shown that the effect of the center of mass corrections destroys the close agreement between theory and experiment in muonic hydrogen. In addition, the effect in muonium is so large that it destroys the excellent agreement between Bethe’s calculation and experiment of the Lamb shifts, and this may well affect on the understanding of the fundamental mechanism of the Lamb shifts. Keywords: open problem, neutron EDM, Lamb shifts, muonic hydrogen, muonium, vacuum polarization potential, center of mass corrections, infra-red singularity, CP transformation, Bethe’s calculation of Lamb shifts.

7.1

Introduction

The most important theory in physics is quantum field theory which can describe the basic behavior of nature in the microscopic world. We believe that the basic law of physics that governs nature is, by now, understood well, and there is no mystery in physical world any more. However, there are some phenomena which should be still beyond our understanding of the present knowledge of the quantum field theory. The violation of time reversal invariance is one of the example which cannot be found in the fundamental law of physics. It should be quite important to know whether the interaction which is not found in the present physics law may exist in nature or not. Among physical observables which are well calculated and understood, there is one observed quantity which is still far from a complete understanding. This is the Lamb shift Takehisa Fujita and Naohiro Kanda All rights reserved - © 2013 Bentham Science Publishers

122 Fundamental Problems in Quantum Field Theory

Fujita and Kanda

energy, and the interpretation of the Lamb shifts in terms of the electron self-energy effects has a difficulty of the logarithmic divergence, unless one makes a fancy manipulation of connecting the infra-red and ultraviolet divergences. But this is physically out of the question. There are some technical open problems which are to be understood. In particular, the infra-red singularity which often appears in the treatment of the photon propagation causes some problems, and a question is as to whether the singularity can be removed or we can leave it out since it is not related to physical observables.

7.2

EDM of Neutron

The examination of the time reversal invariance is one of the most important subject in modern physics. Yet, the understanding of the T-violation is still far from satisfactory in the theoretical point of view [73, 74, 75]. The most stringent experimental test must come from the neutron electric dipole moments (EDM) whose existence should present a direct evidence of the T-violation term [76, 77]. The origin of the T-violation interaction may come from the CP-violation term, but the connection of the T-violation term with the CP violation is still not yet established. Here, we calculate the neutron EDM in terms of the vertex correction due to the weak ± W vector bosons. It turns out that the neutron EDM should be exactly zero due to the hermiticity of the CP violating weak interactions. Therefore, the measurement of EDM should not be related to the CP violation, but it should be rather due to a new type of the interaction violating the time reversal invariance which should have some different origin from the CP violation mechanism. In addition, we calculate the P-violating electromagnetic vertex for protons due to the weak interactions, and we see that it is directly related to the CKM matrix elements of Vdt . Therefore, the measurement of the parity violating part of the electromagnetic interaction with protons can give a good test of the CKM matrix elements, and this should be important since the CKM matrix elements can be examined by the different physical processes from the weak interactions. In this evaluation of the vertex corrections for the neutron EDM, we employ a proper propagator of the massive vector boson of eq.(5.13) µ ν

g µν − k kk2 . D (k) = − 2 k − M 2 − iε µν

The important point is that this does not give rise to the logarithmic divergence in the vertex corrections, and thus we obtain some finite numbers for the electron g − 2 and for the neutron EDM. It turns out that the effect of Z 0 boson on the electron g − 2 is very small (δg ∼ 10−14 ), which is consistent with experiment. On the other hand, the neutron EDM due to the vertex corrections of the weak vector bosons turns out to be exactly zero, and the CP violating phase of the CKM matrix elements do not give rise to a finite EDM of neutron.

 

Open Problems

7.2.1

Fundamental Problems in Quantum Field Theory 123

Neutron EDM

The neutron EDM (electric dipole moments) can be calculated in terms of the vertex corrections due to the W ± boson. Here, we only consider the top quark state since it can generate the most important contribution to the neutron EDM. The interaction Lagrangian density is written as [15] © ª LI = gw Vdt ψ¯t γ µ (1 − γ 5 )ψd Wµ+ + Vtd ψ¯d γ µ (1 − γ 5 )ψt Wµ− (7.1) where Vdt denotes the CKM matrix element in the standard parameterization [29] in which Vdt should contain the complex phase.

7.2.2

CP Transformation

Before going to the evaluation of the neutron EDM, we first show how the Lagrangian density of eq.(7.1) behaves according to the basic symmetry transformation of the CP and the time reversal T. Under the CP transformation, the Lagrangian density of eq.(7.1) becomes © ª (7.2) LI = gw Vdt ψ¯d γ µ (1 − γ 5 )ψt Wµ− + Vtd ψ¯t γ µ (1 − γ 5 )ψd Wµ+ while, under the T-transformation, it becomes © ª LI = gw Vdt∗ ψ¯d γ µ (1 − γ 5 )ψt Wµ− + Vtd∗ ψ¯t γ µ (1 − γ 5 )ψd Wµ+ .

(7.3)

Therefore, under the CPT, we should obtain the same Lagrangian density as the original one as given in eq.(7.1). Thus, we find Vdt = Vtd∗ .

(7.4)

The equation of (7.4) shows that the Lagrangian density of the weak interactions is hermitian. In this respect, the CP violation is just due to the complex phase of the coupling constant in the weak interaction [78]. This is quite different from the parity violation case ¯ µ γ 5 ψWµ violates the parity invariance at the operator level as in which the term ψγ ¯ µ γ 5 ψWµ → −ψγ ¯ µ γ 5 ψWµ P : ψγ

(7.5)

which gives rise to the new additional terms in weak interactions.

7.2.3

Neutron EDM in One Loop Calculations

Here, we review the one loop calculation of the neutron EDM [79, 80]. The calculation can be carried out in a straightforward way, and Shabakin shows the neutron EDM should be proportional to ¸ · 1 1 1 1 0 0 Γqq − Γq q dn ∝ p/ + k/ − m p/ − m0 p/ + k/ − m0 p/ − m where Γq0 q s are the regularized quark electromagnetic transition moments which should satisfy Γq0 q = Γqq0 up to one loop order. Thus, the neutron EDM should vanish at the limit of k → 0.

 

124 Fundamental Problems in Quantum Field Theory

7.2.4

Fujita and Kanda

T-violation and Neutron EDM

Now, we carry out the calculation of the neutron EDM explicitly in terms of the vertex correction arising from the W boson propagation. Here, the d quark in neutron emits a W boson and changes into the top quark state. This top quark makes an electromagnetic interaction with an external photon before it gets back to the d quark state by absorbing the W boson. The T-matrix of this process can be written in terms of the vertex correction as 4e 2 Vdt Vtd Λ (p , p) = −i gw 3 ρ

Z

0

d4 k (2π)4

Ã

µ ν

g µν − k kk2 k 2 − M 2 − iε

! γµ γ 5

1 1 γρ γν p/0 − k/ − mt p/ − k/ − mt

+(exchange diagram).

(7.6)

This can be easily calculated to be eαw |Vdt |2 Λ (p , p) ' π ρ

0

µ

m2t s¯

¶ γργ5

(7.7)

2

where αw = g4πw ' 4.3 × 10−3 and s¯ = 43 m2t . This is obviously a real number, and thus there is no imaginary term. This means that there is no neutron EDM from the one loop calculation of the CP violating weak interactions. This is, however, clear from the beginning since we start from the hermitian operators of the Hamiltonian density, and therefore the result must be, for sure, hermitian.

7.2.5

Neutron EDM in Higher Loop Calculations

The vertex corrections of two and higher loops can be calculated just in the same way as one loop calculation. However, without carrying out any explicit evaluations, we can prove that the neutron EDM must be zero because of the hermiticity of the original Hamiltonian density of weak interactions. As one sees, the Lagrangian density of the weak interactions including the CP violating terms is hermitian, and therefore the calculated result of the vertex corrections must be also hermitian. This means that the vertex correction terms can be reduced to the following final expression Λρ (p0 , p) = Cγ ρ γ 5 + · · ·

(7.8)

where C must be a real number due to the condition that the calculated result must be hermitian. This is basically due to the fact that the T-violation is directly related to the hermiticity as far as the constant term is concerned. Therefore, there is no chance that the vertex correction has any imaginary part and thus the CP violation phase cannot be related to the neutron EDM [81, 79, 80].

 

Open Problems

7.2.6

Fundamental Problems in Quantum Field Theory 125

Origin of Neutron EDM

The measurement of the neutron EDM is important in its own right [76, 77]. We see that the neutron EDM cannot be caused by the CP violating CKM phase. However, this does not mean that the neutron EDM should vanish to zero. It should be still quite important to observe the neutron EDM in order to find the T-violating interaction, and this time it should be an interaction which violates the T-invariance at the operator level. The T-violating interaction should have a well-known shape, and it should be given as i ¯ 5 µν Hedm = − df ψσ µν γ ψF 2

(7.9)

where df denotes the EDM of the corresponding fermion. The only point that should be considered more carefully is concerned with the dimension of the coupling constant df . The df has a dimension of length and therefore it may cause some troubles for the renormalization procedure, if necessary. However, it is most likely true that there should be no need of the renormalization procedure in connection with eq.(7.9) as far as any physical observables are concerned. At the same time, it seems to be natural that the T-violating interaction is specified by some dimensional constant. This is because a breaking of the fundamental symmetry like T-violation should be determined by some breaking of dimensional property. In this respect, it should be very interesting to understand whether the T-violation can take place in nature or not.

7.2.7

P-Violating Electromagnetic Vertex for Proton

The vertex correction due to the weak interactions can induce the parity violating electromagnetic interaction for protons, and we can now determine the strength quite accurately in terms of the CKM matrix elements. Parity Violating Charge ep The vertex corrections for proton becomes Λρ (p0 , p) =

eαw |Vdt |2 γ ρ γ 5 = ep γ ρ γ 5 2π

(7.10)

where the parity violating charge ep is defined as ep =

eαw |Vdt |2 ' 4.6 × 10−8 e 2π

(7.11)

where |Vdt | = 8.5 × 10−3 is used. It should be noted that the effective interaction here is only for the transverse part of the vector potential A, and therefore the Coulomb interaction is not affected at all. This is, of course, due to the fact that the electromagnetic field Aµ has only two degrees of freedom. Thus, the transverse part of the vector field A should be quantized and can generate the effect of the P-violating interaction.

 

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Physical Observables of P-Violating Electromagnetic Vertex The equation of (7.11) indicates that the proton current with the parity violation in the nonrelativistic expression can be written as jp = ep σ.

(7.12)

In this case, the interaction Hamiltonian of protons under the magnetic fields can be described as 1 (7.13) Hpv ' ep r × σ · B 2 where B denotes the uniform magnetic field. At the present stage, we do not know what kind of experiments should be best to study the effect of ep . In any case, the measurement of the parity violating electromagnetic interaction should become a direct check of the CKM matrix elements of Vdt .

7.3

Lamb Shifts in Hydrogen Atom

The Lamb shift energy in hydrogen atom is a symbol of the success in the QED renormalization scheme, and indeed the renormalization effect of the electron self-energy is considered to be responsible for the small deviation of the 2s 1 energy from the prediction of the Dirac 2 equation. In the theoretical calculation of the Lamb shift energy, however, there is a serious ambiguity which arises from the cutoff momentum Λ since the calculation is only possible for the non-relativistic treatment, at least, up to the present stage. In the non-relativistic evaluation, the Lamb shift energy has a logarithmic divergence, and Bethe first took the cutoff momentum Λ as electron mass, that is, Λ = me [82]. However, this has, of course, no physically plausible reason. In his original paper, he stated that ”the relativity theory would provide a natural cutoff for the momentum and a relativistic calculation to establish the limit Λ is in progress”. However, since then, there has been no progress in carrying out the relativistic calculation [10, 83], and the reason of its difficulty of the relativistic calculation is discussed and clarified here. In connection with the Lamb shift energy, people discuss the modification of the Coulomb propagator since it only affects on the 2s 1 state. The Coulomb propagator mod2 ification is taken from the renormalization scheme of the vacuum polarization and is considered to be µ ¶ 1 α q2 1 ⇒ 2 1− (7.14) q2 q 15π m2e which is discussed in the textbook of Bjorken and Drell [10]. This is indeed connected to the Uehling potential as we discuss below more in detail. However, the Coulomb field A0 should not be quantized since it is a time independent field [31]. Therefore, there should not be any change of the Coulomb propagator and the modified propagator discussed in Bjorken and Drell is not correct at all. The Coulomb propagator always stays in the same shape since it is evaluated exactly.

 

Open Problems

  7.3.1

Fundamental Problems in Quantum Field Theory 127

Fields Quantization of Coulomb  

Before going to the discussion of the Lamb shift energy, we should first clarify the quantization of the electromagnetic field Aµ . After taking the Coulomb gauge fixing of ∇ · A = 0, the field equation ∂µ F µν = ej ν can be written for the A0 field as ∇2 A0 = −eje0

(7.15)

where je0 denotes the current density of electron. This is a constraint equation and therefore the A0 field can be solved and written in terms of the electron current je0 as Z je0 (r 0 ) 3 0 e d r. (7.16) A0 (r) = 4π |r − r 0 | This means that there is no way to quantize the A0 field, even though, in the literatures [10], this field is often quantized in the same way as the vector field A. The important point is that fields should be quantized only when they are time dependent. The creation and annihilation operators depend, of course, on time since they occur at some fixed point of time in the reaction processes. In terms of the Hamiltonian, the Coulomb interaction HC can be written as [31, 12] Z 0 0 0 jp (r )je (r)d3 rd3 r0 e2 (7.17) HC = − 4π |r 0 − r| where jp0 denotes the proton current density. This expression is independent of the gauge choice and it clearly states that the Coulomb interaction is not influenced by the higher order corrections since eq. (7.17) is exact. It may be important to make a comment on the quantization of the Coulomb field. Feynman quantized the Coulomb field and therefore obtained the Feynman propagator of µν photon as Dµν = − gk2 . However, this propagator happens to be justified only for leptonlepton scattering [3, 4]. But this is accidental because the scattering particles are on the mass shell. In this respect, one should employ the right propagator of photon without the Coulomb propagator, and therefore, there is no further effect on the Coulomb force.

7.3.2

Uehling Potential

Here, we should critically review the Uehling potential which is essentially the same as the finite term of the vacuum polarization in the Coulomb potential [84, 18, 85]. Uehling obtained the induced charge distribution due to the creation of electron and positron pairs in the vacuum as α δρ(r) = − (7.18) ∇2 ρ(r) 15πm2e which can generate the Uehling potential. However, when the fermions interact with the Coulomb field A0 , there is no physical process which can create the fermion pairs. As an intuitive picture, one may say that the static Coulomb field cannot make any polarizations in the vacuum since the pair creations are physical processes which involve time dependent interactions, and they are only given by the vector field A.

 

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  Classical Picture of Polarization   This classical picture of the fermion pair creations (Uehling potential) should come from the misunderstanding of the structure of the vacuum state. In the medium of solid state physics, the polarization can take place when there is an electric field present. In this case, the electric field can indeed induce the electric dipole moments in the medium, and this corresponds to the change of the charge density. However, this is a physical process which can happen in the real space (configuration space). On the other hand, the fermion pair creation in the vacuum in field theory is completely different in that the negative energy states are all filled in momentum space, and the time independent field of A0 which is only a function of coordinates cannot induce any changes on the vacuum state. Therefore, unless some time dependent field is present in the reaction process, the pair creation of fermions cannot take place in physical processes.

7.3.3

Self-energy of Electron

Here, we briefly review the calculation of the Lamb shift energy by Bethe’s method. We start from the non-relativistic Hamiltonian for electron in hydrogen atom with the electromagnetic interaction e2 e p2 − − p·A (7.19) H= 2m0 r m0 where the A2 term is ignored in the Hamiltonian. In this calculation, the electromagnetic field A should be quantized A(x) =

i h 1 √ ²k,λ c†k,λ e−ikx + ck,λ eikx k λ=1 2V ωk

2 XX

(7.20)

where c†k,λ and ck,λ denote the creation and annihilation operators which satisfy the following commutation relations [ck,λ , c†k0 ,λ0 ] = δk,k0 δλ,λ0 (7.21) and all other commutation relations vanish. Now, the second order perturbation energy due to the electromagnetic interaction for a free electron state can be written as X X X µ e ¶2 1 |hp0 |²k,λ · p|pi|2 δE = − m0 2V ωk Ep0 + k − Ep λ k p0

(7.22)

where |pi and |p0 i denote the free electron state with its momentum. Since the photon energy (ωk = k) is much larger than the energy difference of the electron states (Ep0 −Ep ), one obtains µ ¶ e 2 2 1 p (7.23) δE = − 2 Λ 6π m0

 

Open Problems

Fundamental Problems in Quantum Field Theory 129

  where Λ is the cutoff momentum   of photon. This divergence is proportional to the cutoff Λ which is not the logarithmic divergence. However, this is essentially due to the nonrelativistic treatment, and if one carries out the relativistic calculation of quantum field theory, then the divergence becomes logarithmic.

7.3.4

Mass Renormalization and New Hamiltonian

Defining the effective mass δm as e2 Λ 3π 2

(7.24)

p2 p2 p2 − δm ' 2m0 2m20 2(m0 + δm)

(7.25)

δm = the free energy of electron can be written as EF =

where one should keep only the term up to order of e2 because of the perturbative expansion. Now, one defines the renormalized (physical) electron mass me by me = m0 + δm

(7.26)

and rewrites the Hamiltonian H in terms of the renormalized electron mass m H=

e2 p2 p2 e − + δm − p · A. 2 2me r 2me me

(7.27)

p2 δm ) corresponds to the counter term which cancels out the second Here, the third term ( 2m 2 e order perturbation energy.

7.3.5

Energy Shifts of 2s 1 State in Hydrogen Atom 2

Now, we consider hydrogen atom, and its Hamiltonian can be written as H0 =

e2 p2 − 2mr r

(7.28)

where mr denotes the reduced mass of the electron and proton system. Using eq.(7.27), one can calculate the first and the second order perturbation energies due to the electromagnetic interaction for the 2s1/2 state in hydrogen atom ∆E2s1/2

1 = 2Λ 6π

µ

e me

¶2

h2s1/2 |p2 |2s1/2 i

X X X µ e ¶2 1 |hn, `|²k,λ · p|2s1/2 i|2 − me 2V ωk En,` + k − E2s1/2 λ k n,`

 

(7.29)

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  where the first term comes from  the counter term. This Lamb shift energy for the 2s1/2 state can be rewritten as µ ¶ Z Λ En,` − E2s1/2 e 2X 1 2 |hn, `|p|2s1/2 i| . (7.30) ∆E2s1/2 = 2 dk 6π me En,` + k − E2s1/2 0 n,`

After some calculations, we obtain ∆E2s1/2

1 m3r α5 ln = 6π m2e

à < En,`

Λ > −E2s1/2

! (7.31)

where we have neglected the (n, `) dependence in the denominator when summing up (n, `), and < En,` > is defined as some average value of the excitation energies with respect to the 2s1/2 state. For the cutoff Λ, people normally take Λ ' me , but there is no special reason why one should take the Λ as electron mass.

7.3.6

Problems of Bethe’s Treatment

As one sees, the calculated result of the Lamb shift energy depends on the cutoff Λ, which is not satisfactory at all. In most of the textbooks [4], they explain that the logarithmic divergence can be removed by the unphysical method by connecting the ultra-violet divergence to the infra-red singularity. But this is a meaningless procedure. The important point is that we should understand the origin of the value of the cutoff Λ which may be understood in the relativistic treatment of the Lamb shifts, and if we cannot understand it even with the relativistic treatment of the Lamb shift theory, then the physical origin of the Lamb shift energy itself should not be originated from the renormalization scheme. This point will be discussed later in this section.

7.3.7

Relativistic Treatment of Lamb Shifts

It is believed that the correct scenario of the relativistic treatment of Lamb shifts must be as follows. In the non-relativistic treatment, the mass counter term is linear divergent. However, if one treats it relativistically, the divergence is logarithmic. This reason of the one rank down of the divergence is originated from the fact that the relativistic treatment considers the negative energy states which in fact reduce the divergence rank due to the cancellation. Now, we consider the renormalization effect in hydrogen atom, and if we calculate the Lamb shift energy in the non-relativistic treatment, then it has the logarithmic divergence as we saw above, and this is the one rank down of the divergence. This is due to the fact that the evaluation of the Lamb shift energy is based on the cancellation between the counter term and the perturbation energy in hydrogen atom. In the same way, if one can calculate the Lamb shift energy relativistically, then one should obtain the one rank down of the divergence, and this means that it should be finite. This is a story which has been believed up to now. Unfortunately, one cannot carry out the relativistic calculation of the Lamb shift energy. There are basically two different kinds of difficulties in carrying

 

Open Problems

Fundamental Problems in Quantum Field Theory 131

  out the relativistic calculations  of the Lamb shift energy. The first one is connected to the relativistic bound state problem of two particles (proton and electron), and there is no reliable treatment of the relativistic two body bound state calculations. The second one is more serious than the difficulty of the relativistic two body problem, and the relativistic treatment of the Lamb shifts requires the right treatment of the negative energy states of the bound state particles, and this is far beyond the present understanding of the relativistic field theory. The reliable and clear understanding of the negative energy states are only for the free particle states.

7.3.8

Higher Order Center of Mass Corrections in Hydrogen Atom

Here, we should make a comment on the effect of higher order center of mass corrections which should be discussed later in detail. The calculation on the Lamb shift energy in hydrogen atom can be carried out, and the calculated result becomes CM (Hydrogen Atom) = 0.067 × 10−6 eV ∆E(2s 1 −2p 1 ) 2

2

while the experimental value is exp (Hydrogen Atom) = 4.371 × 10−6 eV. ∆E(2s 1 −2p 1 ) 2

2

Therefore, the higher order center of mass correction term is too small to reproduce the Lamb shift experiment in hydrogen atom.

7.4

Lamb Shifts in Muonic Hydrogen

In order to understand the basic mechanism of the Lamb shift energy, it should be quite important to discuss the energy splitting in muonic hydrogen. In this case, the muon orbit is much closer to proton than the electron orbit in hydrogen atom, and there may well appear some important physics in the energy splitting of 2s 1 and 2p 1 states. 2

7.4.1

2

Vacuum Polarization and Uehling Potential

In chapter 3, we discuss the renormalization scheme in QED and show that there is no need of the renormalization of the photon self-energy contribution. In fact, there is no ambiguity in the theoretical treatment of the vacuum polarization effect. However, it is always important that any theory should be examined by experiment in some way or the other as carefully as possible. The best candidate of the examination of the vacuum polarization effect must be that of the Uehling potential which is equivalent to the potential which is obtained after the renormalization of the self-energy of photon. In the normal treatment of the vacuum polarization diagram, people assume that some physical effects can arise from the finite term of the vacuum polarization diagram. After

 

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  the renormalization, one finds a  finite term which should affect the propagator change in the process involving the exchange of the transverse photon A. Even though it is not the 2 effect for the Coulomb propagator, people assume that the Coulomb propagator qe2 should be replaced by [10] · ¶¸ µ Z e2 2α 1 q 2 z(1 − z) e2 ⇒ 1 dzz(1 . (7.32) + − z) ln 1 − q2 q2 π 0 m2 This corresponds to the additional Coulomb potential V (r) as · Z ¶¸ µ Z d3 q iq·r e2 1 q 2 z(1 − z) 2α e dzz(1 − z) ln 1 + V (r) = − π (2π)3 q2 0 m2

(7.33)

where q 2 is replaced by the three momentum vector (−q 2 ) which is sufficiently reliable approximation as long as we discuss the hydrogen-like atom. This potential is equivalent to the Uehling potential.

7.4.2

Lamb Shifts in Muonic Hydrogen: Theory and Experiment

The Lamb shift energy in muonic hydrogen should be quite different from the normal hydrogen atom if one believes the vacuum polarization potential. The splitting between 2s 1 2 and 2p 1 should be quite large for the muonic hydrogen since the electron mass may appear 2 in eq.(7.33). In this case, the effect of eq.(7.33) become very large for the 2s 1 state en2 ergy which is attractive. The energy splitting between 2s 1 and 2p 1 in muonic hydrogen is 2 2 calculated as [83] VP = −205.0 meV. (7.34) ∆E(2s 1 −2p 1 ) 2

2

On the other hand, one sees that the proton finite size effect can amount to FS = 3.8 meV ∆E(2s 1 −2p 1 ) 2

(7.35)

2

which is indeed a sizable and measurable effect in muonic hydrogen. Experimental Observation of Lamb Shifts in Muonic Hydrogen The Lamb shifts in muonic hydrogen is measured by Pohl et al. [86] as exp = −206.3 meV. ∆E(2s 1 −2p 3 ) 2

(7.36)

2

Here, we should note that this experimental measurement is assumed to be a difference between 2s 1 and 2p 3 states, and, in this respect, it is not a Lamb shift energy which 2 2 is normally stated for the shifts in 2s 1 and 2p 1 states. This observed energy difference 2 2 exp is compared to the theoretical calculation with QED corrections in which the ∆E(2s 1 −2p 3 ) 2

 

2

Open Problems

Fundamental Problems in Quantum Field Theory 133

  most important contribution is the   vacuum polarization with electron mass included. ¿From the comparison of the measurements with the theoretical value, they extract the proton finite size effect on the energy difference, and they found that their observed energy of the finite size effect on the energy splitting ∆E(2s 1 −2p 3 ) cannot be consistent with the proton finite 2

2

size effect which is based on the observed proton radius as determined from the electron scattering experiment. The discrepancy of theory and experiment is −0.3 meV, which is sufficiently accurate and reliable both from theoretical and experimental point of view.

7.4.3

Center of Mass Effects on Lamb Shifts

In hydrogen atom, one should always consider the center of mass correction since we should not forget that electron is bound by proton. In fact, the hydrogen atom is a two body problem in the Dirac equation, but unfortunately, we do not know in which way we can reduce the two body problem to one body problem. In particular, the situation becomes much more pronounced when one considers the muonic hydrogen. In this case, the mass of muon is 105.66 MeV which is larger than one tenth of proton mass (938.28 MeV). This clearly indicates that the higher order center of mass effect in muonic hydrogen must be crucially important. Nevertheless we do not know how we can solve the Dirac equation for the two body system, and therefore, it is very important to develop a reliable method to evaluate the center of mass corrections for the 2s 1 and 2p 1 states. 2

7.4.4

2

Degeneracy of 2s 1 and 2p 1 in FW-Hamiltonian 2

2

Here, we start from the higher order Hamiltonian of the non-relativistic expansion in the Foldy-Wouthuysen transformation for lepton and proton H = m + Mp +

1 e P2 + (p − eA)2 + eA0 − σ·B 2Mp 2m 2m

(7.37a)

e e p4 − σ·E×p− ∇·E (7.37b) 3 2 8m 4m 8m2 where H 0 term denotes the higher order effects which are relevant to the present discussion. Now, we define the reduced mass µ as µ ¶ mMp m −1 =m 1+ (7.38) µ= m + Mp Mp H0 = −

and rewrite all the Hamiltonian in terms of µ as HN R =

e2 e p4 e e 1 (p − eA)2 − − σ · B − 3 − 2 σ · E × p − 2 ∇ · E. (7.39) 2µ r 2µ 8µ 4µ 8µ

It is important to examine that this Hamiltonian should indeed give the degenerate energy spectrum of 2s 1 and 2p 1 , and in fact, one can easily reproduce the degeneracy of the two 2 2 states.

 

134 Fundamental Problems in Quantum Field Theory

  7.4.5

Fujita and Kanda

Higher Order Center  of Mass Corrections in Muonic Hydrogen

m . This is Now, there should be left some important terms which must be of the order of M p 0 the correction term which should come from the H Hamiltonian when replacing m by µ, and we can write it as [87] ¶ µ ¶ µ ¶ µ 2m e 2m e 3m p4 0 + σ·E×p+ ∇·E (7.40) H CM = 3 2 Mp 8µ Mp 4µ Mp 8µ2

which corresponds to the higher order corrections for the kinetic energy, spin-orbit and the Darwin terms arising from the center of mass replacement. As can be seen, this Hamiltonian from the center of mass corrections should be repulsive. Now, we can carry out the calculations of the above Hamiltonian in muonic hydrogen, and the calculated results for the 2s 1 and 2p 3 states become 2

2

 CM   ∆E2s =

23 m µα4 64 Mp 2

  ∆E CM = 2p

13 m µα4 192 Mp 2

for 2s 1 2

(7.41)

for 2p 3 . 2

Therefore, the higher order center of mass correction effect on the Lamb shifts in muonic hydrogen amounts to CM = 4.4 meV (7.42) ∆E(2s 1 −2p 3 ) 2

2

which is even larger than the finite size correction. Here, we should give the expectation value for the atomic states in general, and the effect of center of mass corrections can be written as ½ ¾ 1 9 2 1 m µ(Zα)4 CM 0 + − . ∆En,l,m,j = hn, l, m, s|HCM |n, l, m, si = 3 n Mp 2 4n j + 21 l + 21 (7.43)

7.4.6

Comparison between Theory and Experiment

The calculated energy difference between 2s 1 and 2p 3 states due to the QED correction 2 2 with the vacuum polarization becomes [83] VP = −210.4 meV ∆E(2s 1 −2p 3 ) 2

(7.44)

2

and, therefore, the total corrections of the vacuum polarization, proton finite size and center of mass effect become V P +F S+CM = −(210.4 − 3.8 − 4.4) meV = −202.2 meV ∆E(2s 1 −2p 3 ) 2

2

which should be compared with the observed value exp = −206.3 ∆E(2s 1 −2p 3 ) 2

 

2

meV.

(7.45)

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Fundamental Problems in Quantum Field Theory 135

theory and experiment is now completely destroyed,   As one sees, the agreement between   and the change of the proton size cannot explain the experimental result any more. At present, there is no way to interpret the experimental result of the Lamb shift energy in muonic hydrogen, and most likely, both theory and experiment of the Lamb shifts should be carefully reconsidered and studied in future as one of the most important themes in modern physics research. In this respect, it should be crucially important that one should carry out reliable observations of the energy spectrum in muonic hydrogen.

7.4.7

Summary

Here, we have presented a new calculation of the higher order center of mass effects on the Lamb shifts in muonic hydrogen. This new effect destroys the close agreement of the Lamb shift energies between theory and experiment, which has been so far believed. Therefore, the puzzle becomes more fundamental than ever. This suggests that the energy from vacuum polarization potential should be unrealistic. Theoretically, there are two reasons that the vacuum polarization energy should not exist. The first reason is that the renormalization scheme of self-energy of photon is not needed any more as clarified in [56]. Secondly, the Coulomb interaction is not time dependent, and it is solved exactly. In fact, the Feynman µν propagator of photon DFµν = − gk2 which includes the quantization of Coulomb field happens to be justified only for lepton-lepton scattering [3, 4]. But this is accidental, and one should employ the right propagator of photon without the Coulomb propagator. Therefore, there is no further effect on the Coulomb force, and this means that there is no Uehling potential. In this respect, a possible reason of the discrepancy between theory and experiment in the Lamb shift energy in muonic hydrogen should be mainly because the energy from the vacuum polarization is unphysical. However, it is also clear that more careful experimental measurements should be necessary.

7.5

Lamb Shifts in Muonium

Now, we apply the effects of the center of mass corrections on the Lamb shift energy for muonium. The center of mass effect in muonium should be quite important since the mass ratio between electron and muon is much larger than that of hydrogen atom.

7.5.1

Higher Order Center of Mass Corrections in Muonium

Now, the H 0 Hamiltonian of the higher order center of mass correction is obtained in the previous section as µ H 0 CM =

 

3m Mµ

¶

p4 + 8µ3

µ

2m Mµ

¶

e σ·E×p+ 4µ2

µ

2m Mµ

¶

e ∇·E 8µ2

(7, 46)

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  which corresponds to the higher  order corrections for the kinetic energy, spin-orbit and the Darwin terms arising from the center of mass replacement. As can be seen, this Hamiltonian from the center of mass corrections should be repulsive. Now, we can carry out the calculations of the above Hamiltonian in muonium, and the calculated results for the 2s 1 and 2p 1 states become 2

2

 CM   ∆E2s =

23 m µα4 64 Mµ 2

  ∆E CM = 2p

37 m µα4 192 Mµ 2

for 2s 1 2

(7.47)

for 2p 1 . 2

Therefore, the higher order center of mass correction effect on the Lamb shifts in muonium amounts to CM ' 0.58 × 10−6 eV. (7.48) ∆E2s 1 −2p 1 2

7.5.2

2

Disagreement of Bethe’s Calculation with Experiment

As one knows, it is believed that the observed Lamb shift energy is reproduced by Bethe’s calculation which is based on the renormalization procedure of the fermion self-energy diagrams. In this case, however, the calculation has the logarithmic divergence as is wellknown, but this divergence is removed in terms of the very artificial manipulation which is not physically understandable at all. In any case, one can find the very ”precise Lamb shift energy” as · ¸ mα5 m 11 1 1 log − + (7.49) + ∆E2s 1 −2p 1 = 6π 2 < EI − En=2,`=0 >av 24 5 2 2 2 which is given in most of the field theory textbooks [4]. In muonium, this value becomes Bethe = 4.35 × 10−6 eV ∆E2s 1 −2p 1 2

(7.50)

2

where the reduced mass of muonium is used. On the other hand, the experimental value of the Lamb shifts in muonium becomes [88, 89, 90] exp = (4.31 ± 0.09) × 10−6 eV ∆E2s 1 −2p 1 2

(7.51)

2

which agrees very well with the prediction of Bethe’s calculation. However, we know by now that there exists the contribution to the Lamb shifts energy from the higher order center of mass correction which is always there and an inevitable effect. This is given in eq.(7.48), and therefore we should add the Lamb shift energy to the Bethe’s result. In this case, the total Lamb shift energy of muonium becomes th ' 4.93 × 10−6 eV ∆E2s 1 −2p 1 2

 

2

(7.52)

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Fundamental Problems in Quantum Field Theory 137

of the theoretical prediction with experiment of   which completely destroys the agreement   Lamb shift energy. This is very serious, and it is clear that we should reconsider the basic mechanism of the Lamb shift energy from the beginning. At least, Bethe’s calculation is now inconsistent with experiment, and this strongly suggests that the resolution of the degeneracy in the 2s 1 and 2p 1 states in hydrogen atom may not necessarily be due to the 2 2 mass renormalization procedure of the self-energy of fermions.

7.6

Further Corrections in QED

Here, we discuss some of the effects which arise from the correction terms in QED. We note that they do not affect on the physical observables as a result, but it should be important that we should understand them in depth.

7.6.1

Infra-Red Singularities

The photon propagator should contain the infra-red singularity. In fact, the solution of the photon state A is written as ˆ A(x) =

h i 1 √ ²k,λ c†k,λ e−iωk t+ik·r + ck,λ eiωk t−ik·r k λ=1 2V ωk

2 XX

(7.53)

where one sees that it has a singularity at ωk = 0 in this equation. Since there is no photon with zero momentum, it is obvious we should exclude the k = 0 part in the summation. This clearly shows that any infra-red singularities must be unphysical, and thus we should not worry about the problems arising from the k = 0 point. Momentum Summation However, the treatment of this part is not so easy. If we do not make the summation into the integral, then the original singularity can be well avoided since it is just the k = 0 part only. In this case, however, we do now know what we can do for the thermodynamic limit, that is, we should make the volume infinity. Momentum Integral In reality, we should make the summation into the momentum integral as Z d3 k 1 X ⇒ V (2π)3 k

(7.54)

and the singularity at k = 0 becomes obscure. Nevertheless the divergence which come from the infra-red origin should not be serious at all, and it may well be that we should

 

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  introduce a infra-red mass λ which can be kept finite throughout the calculation. However, as one can see now, there is no case in which the infra-red mass λ is connected to physical observables. If one connects it to the physical observables like the infinity of Lamb shifts, then it is simply a mistake.

7.6.2

Correction Terms in Coulomb Interaction

The evaluation of the T-matrix in the fermion-fermion scattering shows that there arises a correction term as an additional Coulomb interaction which is written as δT (C) = −

e2 q10 q20 u ¯(p0 1 )γ 0 u(p1 )¯ u(p0 2 )γ 0 u(p2 ). q4

(7.55)

In the non-relativistic limit, the additional T-matrix can be written as δT (C) ' −

e2 (q 2 + 2q · p)2 4m1 m2 q 4

(7.56)

which is evaluated in the center of mass system, and thus find q = p1 − p01 = p02 − p2 ,

p = p2 = −p1 .

(7.57)

Here, we should take into account the energy conservation in the scattering process, and therefore we have q q q q p21 + m21 + p22 + m22 = p01 2 + m21 + p02 2 + m22 (7.58) which becomes in the non-relativistic limit q 2 + 2q · p = 0.

(7.59)

This indicates that there is no effect from the additional term of the Coulomb interaction on physical observables as long as the scattering particles are on the mass shell.

Conflict of Interest The author(s) confirm that this FKDSWHU content has no conflicts interest. Acknowledgements: Declared none.

 

Fundamental Problems in Quantum Field Theory, 2013, 139-166

139

Appendix A

Regularization It should be worthwhile clarifying what the regularization means in physics. Mathematically, most of the regularizations are clear, except the dimensional regularization which has made crucial mistakes in making use of mathematical formula.

A.1 Cutoff Momentum Regularization The simplest and most reliable regularization method is known in terms of the cutoff Λ in which the integral of the momentum p can be set to Z

Z

∞

F (p)dp → lim

Λ→∞ 0

0

Λ

F (p)dp

(A.1)

where Λ is called the cutoff momentum. This has a good physical meaning since the integral over the momentum corresponds to the summation of all the possible states in the Fock space of the field theory one considers. Therefore, the introduction of the cutoff momentum means that the maximum number of the states in the field theory model is now fixed to N = 2π L Λ with L the box length. In this sense, if the cutoff momentum Λ is much larger than any scales in the model field theory, then one can reliably obtain the calculated results under the condition that the physical observables should not depend on the Λ.

A.2 Pauli-Villars Regularization Now, another popular regularization must be the Pauli-Villars regularization [11]. This is rather simple and it makes the divergent integral to the convergent integral in the following way Z

1 =⇒ d p 2 p − m2 + iε 4

Z d4 p

1 Λ2 ' Λ2 log(Λ/m) p2 − m2 + iε p2 + Λ2

Takehisa Fujita and Naohiro Kanda All rights reserved - © 2013 Bentham Science Publishers

(A.2)

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which is indeed convergent. However, if we make the Λ to infinity, then we can get back to the infinity as the original integral (l.h.s. of eq.(A.2)) indicates. Therefore, there is no point to employ the Pauli-Villars regularization.

A.3

ζ−Function Regularization

The third example can be the ζ−function regularization [32], and in this case, the summation can be replaced in the following way ∞ X n=0

1 →

∞ X n=0

e−λn =

1 1 − e−λ

1 λ→0 λ

→ lim

(A.3)

where the original infinity is certainly kept in terms of λ. In this respect, the apparent infinity can be expressed in terms of some finite numbers and the original infinity can be recovered when the parameter is set to zero or infinity depending on the regularization. Mathematically, the regularizations we discuss here can satisfy the important condition that the original divergence can be recovered by setting the parameters to zero or infinity.

A.4 Dimensional Regularization Finally, we discuss the dimensional regularization which is, however, quite different from other examples [7, 8]. It cannot satisfy this most important mathematical condition that the original infinity should be recovered when we set the parameter to zero or infinity. In the dimensional regularization, the parameter is ε since they replace the integral dimension from 4 to D = 4 − ε. In this case, one uses the following integral formula Z dD p

1 D g µν pµ pν n+1 Γ(n− 2 D−1) 2 (−1) = iπ 1 (p2 −s+iε)n 2Γ(n) sn− 2 D−1

(for n ≥ 4).

(A.4)

The important point is that eq.(A.4) is only valid for n ≥ 4, and this is the very strict condition. In fact, if one applies eq.(A.4) to the calculation of the photon self-energy diagram (n = 2), then one cannot recover the quadratic divergence in the dimensional regularization even when one sets the value of the parameter ε to infinitesimally small. What does this means ? It indicates that the dimensional regularization must be mathematically incorrect for the quadratic and higher divergent evaluations. For the case of the logarithmic divergence, the dimensional regularization can give a correct result, though the divergence level is somewhat different from the normal regularizations. In this respect, the dimensional regularization is a useless regularization method.

Appendix

Fundamental Problems in Quantum Field Theory 141

Appendix B

Gauge Conditions In connection with the chiral anomaly problem, the gauge condition is closely related to the derivation of the chiral anomaly equation. Therefore, we should clarify the situation of the gauge invariance in QED since there is a serious misunderstanding among some of the educated physicists concerning the gauge invariance of the calculated amplitudes which involve the external photon lines. Their argument is as follows. The polarization vector ²µ is gauge dependent and therefore the calculated results must be kept invariant under the transformation of ²µ → ²µ + ck µ . However, this condition is unphysical since we already fixed a gauge (for example, Coulomb gauge fixing of k·² = 0) before the field quantization. The gauge invariance of the S-matrix evaluation is guaranteed as far as the fermion current is conserved, which is always satisfied in the perturbation calculation. Here, we present several examples of the gauge conditions whether the calculated Feynman amplitudes can satisfy the gauge conditions or not.

B.1 Vacuum Polarization Tensor The best example can be found in the vacuum polarization tensor Πµν . People believe that the following gauge condition should be satisfied [10] kµ Πµν = 0.

(B.1)

However, this is a wrong equation, and in fact, we can easily calculate Πµν as [20, 4] ¸ µ ¶ · Z 1 α k2 1 d4 p ν 2 2 µ µν 2 Tr γ γ = Λ +m − g µν Π (k) = ie (2π)4 p/ − m + iε p/ − k/ − m + iε 2π 6 · µ 2 ¶ ¶¸ µ Z 1 Λ α µ ν k2 2 µν −6 dzz(1 − z) ln 1 − 2 z(1 − z) (B.2) + (k k − k g ) ln 3π m2 e m 0 where Λ denotes the cutoff momentum. There is no way that the first term of the right hand side can satisfy the gauge condition of eq.(B.1). Namely we find kµ Πµν 6= 0

(B.3)

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and therefore, eq.(B.1) cannot be satisfied by the vacuum polarization tensor. People believed eq.(B.1) should hold true, basically because of the mathematical mistake due to the wrong replacement of the integration variables in the infinite integrals [31, 20]. Since then, the gauge conditions are imposed by hand on the amplitudes which have some external photon lines. However, it should be noted that the same type of the calculations of the vacuum polarization was done a long time ago by Heisenberg and Euler [84, 18] and they obtained the similar results as above including the quadratic divergence and logarithmic divergence as well. Even though their argument of the polarization process is not in the right direction in terms of the renormalization scheme [5, 6], their calculations themselves are indeed correct.

B.2 Vacuum Polarization Tensor for Axial Vector Coupling Here, we should check whether the gauge condition can be satisfied for other reaction processes or not. First, we should present an example of the vacuum polarization tensor which is induced by the axial vector current which couples to fermions as ¯ µ γ 5 ψAµ . L0 = g 0 ψγ P In this case, the vacuum polarization tensor for the axial vector current can be written as Z 0 Πµν AV (k) = g

g0 2 = 2 8π

2

¸ · 1 d4 p 1 ν 5 µ 5 Tr γ γ γ γ (2π)4 p/ − m + iε p/ − k/ − m + iε

µ ¶ · µ 2 ¶ k2 g0 2 Λ 2 2 µν µ ν 2 µν Λ −m − g + (k k − k g ) ln 2 6 12π m2 e ¶¸ µ Z 1 k2 −6 dzz(1 − z) ln 1 − 2 z(1 − z) . m 0

(B.4)

This clearly shows that the axial vector current conservation is not related to the gauge condition since we have kµ Πµν (B.5) AV (k) 6= 0. On the other hand, the Compton scattering case is different and it can satisfy the gauge condition since there is no fermion loop in this calculation.

B.3 Compton Scattering The Feynman amplitude of the Compton scattering can be written as ¸ · 1 1 ν µ µ µν 2 0 ν γ u(p). γ +γ 0 M = −ie u ¯(p ) γ p/ + k/ − m + iε p/ − k/ − m + iε

(B.6)

Appendix

Fundamental Problems in Quantum Field Theory 143

Therefore, we can check µν

kµ M

¸ 1 1 ν γ u(p). k/ + k/ 0 = −ie u ¯(p ) γ p/ + k/ − m + iε p/ − k/ − m + iε 2

·

0

ν

(B.7)

Now, using some identities k/ = p/ + k/ − m − (p/ − m),

k/ = −(p/0 − k/ − m) + (p/0 − m)

and the free Dirac equations of (p/ − m)u(p) = 0, we can easily prove

u ¯(p0 )(p/0 − m) = 0

kµ Mµν = 0.

(B.8) (B.9)

This is, of course, clear since the Compton scattering does not contain a loop diagram, and therefore the gauge condition, ²µ → ²µ + ck µ just corresponds to the conservation of the fermion current. This can be easily seen since the initial and final fermion in the Compton scattering can satisfy the free Dirac equation. On the other hand, if the Feynman diagrams involve fermion loops, then the gauge condition does not correspond to the fermion current conservation since the free Dirac equation cannot be used.

B.4 Decay of π 0 → 2γ Among the Feynman diagrams that contain the fermion loop, the decay of the π 0 → 2γ can satisfy the gauge condition. Now, the T-matrix of π 0 → 2γ can be evaluated to be ¸ · Z 1 1 1 d4 p 5 2 Tr (γ²1 ) (γ²2 ) γ Tπ0 →2γ ' gπ e (2π)4 p/ − M + iε p/ − k/2 − M + iε p/ + k/1 − M + iε e2 gπ εµνρσ k1ρ k2σ ²µ1 ²ν2 . 4π 2 M Defining the amplitude Mµν as Tπ0 →2γ = Mµν ²µ1 ²ν2 , we can prove '

(B.10)

e2 gπ εµνρσ k1µ k1ρ k2σ = 0 (B.11) 4π 2 M which is due to the anti-symmetric character of the εµνρσ tensor. This property is basically due to the γ 5 interaction which generates the anti-symmetric nature of the invariant amplitude. In this respect, it is very special that the π 0 → 2γ decay process satisfies the gauge condition, and it is not due to the nature of the electromagnetic interactions. This pion and nucleon interaction is, in fact, beyond QED, and it indeed involves the strong interaction. Since the strong interaction satisfies the parity invariance, the Feynman diagram of the decay process keeps the anti-symmetric nature, and thus the amplitude satisfies eq.(B.11), and this is, of course, accidental from the point of view of the gauge condition. This point can be clearly seen if we examine the reaction process of the scalar meson decay into two photons since the scalar interaction has the symmetric nature. k1µ Mµν =

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B.5 Decay of Vector Boson Z 0 into 2γ The T-matrix for the Z 0 → 2γ decay process is given as [28] TZ 0 →2γ

gz =− 2 6π

µ

2e 3

¶2

(k1α − k2α )εµνρα ²µ1 ²ν2 ²ρv .

(B.12)

Therefore we can define the amplitude Mµνρ as TZ 0 →2γ = Mµνρ ²µ1 ²ν1 ²ρV , we can now prove µ ¶2 2e gz k1µ k2α εµνρα 6= 0. k1µ Mµνρ = 2 (B.13) 6π 3 Therefore, the gauge condition is not satisfied in the case of of Z 0 → 2γ decay process. This is, of course, clear since the γ µ γ 5 interaction has a symmetric nature and therefore it is just opposite to the γ 5 interaction.

B.6 Decay of Scalar Field Φ into 2γ Now, we consider the interaction Lagrangian density where the scalar field Φ couples to fermions as ¯ L0 = g0 ψψΦ where g0 is the coupling constant. In this case, the T-matrix of Φ → 2γ can be evaluated as Z TΦ→2γ ' e2 g0

¸ · 1 1 d4 p 1 Tr (γ · ² ) · ² ) (γ 2 1 (2π)4 p/ − M + iε p/ + k/1 − M + iε p/ − k/2 − M + iε

' e2 g0 M ²1 · ²2

(B.14)

where M denotes the nucleon mass. Defining the amplitude Mµν as TΦ→2γ = Mµν ²µ1 ²2 ν , we can now prove k1µ Mµν = e2 g0 M k1µ gµν 6= 0.

(B.15)

Therefore, the gauge condition is not satisfied in the case of of Φ → 2γ decay process. This is, again, easy to understand since the scalar interaction has a symmetric nature and therefore it is opposite to the γ 5 interaction. It should be noted that there is no scalar meson in nature which decays into two photons. However, the similar type of the Feynman diagram becomes important when we consider the photon-gravity interaction. In fact, photon can interact with the gravitational field via loop diagrams which are essentially the same as the T-matrix given in eq.(B.14) [31]. In this respect, the T-matrix of eq.(B.14) can be considered to be a physical process.

Appendix

Fundamental Problems in Quantum Field Theory 145

B.7 Photon-Photon Scattering The T-matrix of the box diagrams in the photon-photon scattering can be written as · Z 1 d4 p 1 4 Tr (γ · ²1 ) (γ · ²3 ) Tγ−γ ' e (2π)4 p/ − m p/ − k/ 3 − m ¸ 1 1 (γ · ²2 ) (B.16) ×(γ · ²4 ) p/ − k/ 1 − m p/ − k/ 1 − k/ 2 − m where the energy of photon can be written as ω = |k1 | = |k2 | = |k3 | = |k4 | at the center of mass system of two photons. The leading behavior of the finite terms in this T-matrix can be easily evaluated under the condition of m À ω and we write it in terms of Mµνλσ which is defined Tγ−γ = ²µ ²ν ²λ ²σ Mµνλσ as · ³ ω ´4 ¸ ³ ω ´2 µνλσ 4 + c2 (g µσ g λν + g µν g σλ + g µλ g σν ) M ' e 1 + c1 (B.17) m m where c1 and c2 denote some numerical constants. Therefore, it is clear that the gauge conditions do not hold kµ Mµνλσ 6= 0. (B.18) As we explain in detail in chapter 3, the apparent divergences can be completely cancelled out due to the kinematical cancellation by adding up three independent Feynman diagrams together, and the disappearance of the divergences is not due to the regularization [14].

B.8 Gauge Condition and Current Conservation As we saw above, the serious mistake must be concerned with the confusion between the gauge conditions and the current conservation. Somehow, people believed that the gauge condition should be directly connected to the current conservation [25]. Or in other words, the gauge condition of the Feynman amplitude (we denote it as T = ²µ ²ν Mµν ) for some reaction process kµ Mµν = 0 (B.19) should be identical to the current conservation of ∂µ j µ = 0.

(B.20)

This is, of course, a wrong statement, and the current conservation must hold true for any occasions while the gauge condition of eq.(B.19) is in some cases satisfied and in other cases not, depending on the reaction processes as we saw above. Basically, the current conservation cannot manifestly be traced in the Feynman amplitude of Mµν unless there are only external fermion lines which can satisfy the free Dirac equation, and consequently can be related to the current conservation.

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In this sense, the condition of the reaction amplitude T = ²αv ²µ ²ν Mαµν involving the axial vector vertex of γ µ γ 5 with its polarization vector ²αv q α Mαµν 6= 0

(B.21)

does not mean that the axial current conservation is violated. We have shown that the gauge condition of ²µ → ²µ + ck µ does not have any physical meaning, but the replacement of ²µv → ²µv + c0 k µ is even worse than the gauge condition since the axial vector coupling has nothing to do with the gauge theory. In this respect, the whole business of the chiral anomaly equation is just the castle in the air.

B.9 Summary of Gauge Conditions To summarize, we see that the Compton scattering and π 0 → 2γ decay process can satisfy the gauge condition, while other examples of the photon self-energy, the vacuum polarization for the axial vector current, the Z 0 → 2γ decay process, photon-photon scattering diagrams and Φ → 2γ decay process do not satisfy the gauge condition, and this is mainly because they have a fermion loop. It is by now clear that the gauge condition of ²µ → ²µ + ck µ is physically a meaningless procedure. This is basically due to the fact that the Lorentz condition of ²µ k µ = 0 is obtained from the equation of motion as explained in Appendix C. Therefore, this constraint equation has nothing to do with the gauge fixing condition, and thus the requirement of ²µ → ²µ + ck µ is physically a wrong procedure.

Appendix

Fundamental Problems in Quantum Field Theory 147

Appendix C

Lorentz Conditions Here, we clarify that the Lorentz condition of kµ ²µ = 0 should be obtained from the equation of motion, and therefore it is more fundamental than the requirement of the gauge fixing condition in QED. For the massive vector bosons, the Lorentz condition plays a fundamental role for determining the polarization sum of the vector boson.

C.1 Gauge Field of Photon We write the Lagrangian density for the free gauge field as 1 Lem = − Fµν F µν 4

(C.1)

with F µν = ∂ µ Aν − ∂ ν Aµ . In this case, the equation of motion becomes ∂µ (∂ µ Aν − ∂ ν Aµ ) = 0.

(C.2)

Since the free photon field should have the following solution Aµ (x) =

2 i XX ²µ (k, λ) h † −ikx √ ck,λ e + ck,λ eikx k λ=1 2V ωk

(C.3)

we can insert this solution into eq.(C.2) and obtain the following equation for ²µ (k, λ) k 2 ²µ − (kν ²ν )k µ = 0.

(C.4)

This equation can be written in terms of the matrix equation for the polarization vector ²µ 3 X ν=0

{k 2 g µν − k µ k ν }²ν = 0

(C.5)

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where we write the summation explicitly. In order that the ²µ should have a non-zero solution, the determinant of the matrix should vanish to zero det{k 2 g µν − k µ k ν } = 0.

(C.6)

Now it is easy to prove that k 2 = 0 is the only physical solution of eq.(C.6) since one finds det{−k µ k ν } = 0. Therefore, putting the solution of k 2 = 0 into eq.(C.4), we obtain kµ ² µ = 0

(C.7)

which becomes the solution for the polarization vector. Here, we should note that this process of determining the condition on the wave function of ²µ is just the same as solving the free Dirac equation. Obviously this is the most important process of determining the wave functions in quantum mechanics, and surprisingly, this has been missing in the treatment of determining not only the massive vector boson propagator but also the photon propagator as well. This constraint equation of eq.(C.7) is obtained from the equation of motion, even though it is just the same equation as Lorentz gauge fixing condition. As can be seen by now, the gauge fixing condition is still left for use. In fact, if we take the Coulomb gauge fixing of ∇ · A = 0, then we find k · ² = 0 which leads to the condition of ²0 = 0. Therefore, we now see that the photon field has only two degrees of freedom which can be naturally obtained from the equation of motion and the gauge fixing condition. In addition, one realizes that the Lorentz gauge fixing is not allowed in the free field gauge theory since the same equation of the Lorentz gauge fixing is already obtained from the equation of motion. Namely, it cannot give a further constraint on the polarization vector. In this respect, one sees that the Coulomb gauge fixing gives a proper condition on the polarization vector.

C.2 Massive Vector Fields The massive vector field can be treated just in the same manner as above. We first write the free Lagrangian density for the vector boson field Z µ with its mass M 1 1 LW = − Gµν Gµν + M 2 Zµ Z µ 4 2

(C.8)

with Gµν = ∂ µ Z ν − ∂ ν Z µ . In this case, the equation of motion becomes ∂µ (∂ µ Z ν − ∂ ν Z µ ) + M 2 Z ν = 0.

(C.9)

Since the free massive boson field should have the following shape of the solution Z µ (x) =

3 i XX ²µ (k, λ) h √ ck,λ eikx + c†k,λ e−ikx k λ=1 2V ωk

(C.10)

Appendix

Fundamental Problems in Quantum Field Theory 149

we can insert this solution into eq.(C.9) and obtain the following equation for the polarization vector ²µ (k 2 − M 2 )²µ − (kν ²ν )k µ = 0. (C.11) In the same way as above, we can prove that k2 − M 2 = 0 should hold, and this is the only physical solution of eq.(C.11). Therefore we obtain the following equation for the polarization vector ²µ kµ ² µ = 0

(C.12)

which should always hold. This is just the same equation as Lorentz gauge fixing condition in QED. However, there is no gauge freedom for the massive vector boson, and therefore the degrees of freedom of the polarization vector ²µ for the massive vector boson is three, in contrast to the gauge field. Now, we understand that the massive vector field should have a spin of s = 1 which has indeed three components as we saw above. In this sense, the photon field is special in that it has a spin of s = 1 with only two degrees of freedom. This should be directly related to the massless nature of photon which is required from the gauge invariance of the Lagrangian density of the vector field.

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Appendix D

Basic Notations in Field Theory In field theory, one often employs special notations which are by now commonly used. In this Appendix, we explain some of the notations which are particularly useful in field theory calculations.

D.1 Natural Units and Constants Here, we employ the natural units because of its simplicity c = 1, ~ = 1.

(D.1.1)

If one wishes to get the right dimensions out, one should use ~c = 197.33 MeV · fm. For example, pion mass is mπ ' 140 MeV/c2 . Its Compton wave length is ~c 197 MeV · fm 1 = = ' 1.4 fm. mπ mπ c2 140 MeV e2 e2 1 e2 = = = . Fine structure constant: α = e2 = ~c 4π 4π~c 137.036  Electron mass : me = 0.511 MeV/c2    Muon mass : mµ = 105.66 MeV/c2   Some constants:   Proton mass : Mp = 938.28 MeV/c2    1 = 0.529 × 10−8 cm Bohr radius : a0 = me e2

(D.1.2)

Appendix

Fundamental Problems in Quantum Field Theory 151 1 Mp2

Gravitational constant:

G = 5.906 × 10−39

Weak coupling Constant:

GF = 1.166 × 10−5 (GeV)−2

  Magnetic moments :  

Weak bosons :

Electron : µe = 1.00115965219 Muon :

µµ = 1.001165920

e~ 2me c e~ 2mµ c

  W ± − boson : MW = 80.4 GeV/c2 ,

αW ' 4.3 × 10−3

 Z 0 − boson : M = 91.2 GeV/c2 , z

αZ ' 2.73 × 10−3

D.2 Hermite Conjugate and Complex Conjugate For a complex c-number A A = a + bi (a, b : real).

(D.2.1)

Its complex conjugate A∗ is defined as A∗ = a − bi.

(D.2.2)

Matrix A If A is a matrix, one defines the hermite conjugate A† (A† )ij = A∗ji .

(D.2.3)

Differential Operator Aˆ If Aˆ is a differential operator, then the hermite conjugate can be defined only when the Hilbert space and its scalar product are defined. For example, suppose Aˆ is written as ∂ . Aˆ = i ∂x In this case, its hermite conjugate Aˆ† becomes µ ¶T ∂ ∂ † ˆ A = −i =i = Aˆ ∂x ∂x

(D.2.4)

(D.2.5)

which means Aˆ is Hermitian. This can be easily seen in a concrete fashion since ¶ Z∞ Z∞ µ ∂ ∂ † † ˆ ˆ hψ|Aψi = ψ (x)i (D.2.6) ψ(x) dx = −i ψ (x) ψ(x) dx = hAψ|ψi, ∂x ∂x −∞

−∞

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where ψ(±∞) = 0 is assumed. The complex conjugate of Aˆ is simply ∂ ˆ 6 A. = Aˆ∗ = −i ∂x

(D.2.7)

Field ψ If the ψ(x) is a c-number field, then the hermite conjugate ψ † (x) is just the same as the complex conjugate ψ ∗ (x). However, when the field ψ(x) is quantized, then one should always take the hermite conjugate ψ † (x). When one takes the complex conjugate of the field as ψ ∗ (x), one may examine the time reversal invariance.

D.3 Scalar and Vector Products (Three Dimensions) : Scalar Product For two vectors in three dimensions r = (x, y, z) ≡ (x1 , x2 , x3 ),

p = (px , py , pz ) ≡ (p1 , p2 , p3 )

(D.3.1)

the scalar product is defined r·p=

3 X

xk pk ≡ xk pk ,

(D.3.2)

k=1

where, in the last step, we omit the summation notation if the index k is repeated twice. Vector Product The vector product is defined as r × p ≡ (x2 p3 − x3 p2 , x3 p1 − x1 p3 , x1 p2 − x2 p1 ).

(D.3.3)

This can be rewritten in terms of components, (r × p)i = ²ijk xj pk ,

(D.3.4)

where ²ijk denotes anti-symmetric symbol with ²123 = ²231 = ²312 = 1,

²132 = ²213 = ²321 = −1,

otherwise = 0.

Appendix

Fundamental Problems in Quantum Field Theory 153

D.4 Scalar Product (Four Dimensions) For two vectors in four dimensions, xµ ≡ (t, x, y, z) = (x0 , r),

pµ ≡ (E, px , py , pz ) = (p0 , p)

(D.4.1)

the scalar product is defined x · p ≡ Et − r · p = x0 p0 − xk pk .

(D.4.2)

This can be also written as xµ pµ ≡ x0 p0 + x1 p1 + x2 p2 + x3 p3 = Et − r · p = x · p,

(D.4.3)

where xµ and pµ are defined as xµ ≡ (x0 , −r),

pµ ≡ (p0 , −p).

(D.4.4)

Here, the repeated indices of the Greek letters mean the four dimensional summation µ = 0, 1, 2, 3. The repeated indices of the roman letters always denote the three dimensional summation throughout the text. Metric Tensor It is sometimes convenient to introduce the metric tensor g µν which has the following properties   1 0 0 0 0 −1 0 0  g µν = gµν =  (D.4.5) 0 0 −1 0  . 0 0 0 −1 In this case, the scalar product can be rewritten as x · p = xµ pν gµν = Et − r · p.

(D.4.6)

D.5 Four Dimensional Derivatives ∂µ The derivative ∂µ is introduced for convenience µ ¶ µ ¶ µ ¶ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂µ ≡ = , , , = , , , = ,∇ , ∂xµ ∂x0 ∂x1 ∂x2 ∂x3 ∂t ∂x ∂y ∂z ∂t

(D.5.1)

where the lower index has the positive space part. Therefore, the derivative ∂ µ becomes µ ¶ µ ¶ ∂ ∂ ∂ ∂ ∂ ∂ µ ∂ ≡ = ,− ,− ,− = , −∇ . (D.5.2) ∂xµ ∂t ∂x ∂y ∂z ∂t

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D.5.1 pˆµ and Differential Operator Since the operator pˆµ becomes a differential operator as µ ¶ ∂ µ ˆ ˆ = i , −i∇ = i∂ µ pˆ = (E, p) ∂t the negative sign, therefore, appears in the space part. For example, if one defines the current j µ in four dimension as j µ = (ρ, j), then the current conservation is written as ∂µ j µ =

∂ρ 1 + ∇ · j = pˆµ j µ = 0. ∂t i

(D.5.3)

D.5.2 Laplacian and d’Alembertian Operators The Laplacian and d’Alembertian operators, ∆ and 2 are defined as ∂2 ∂2 ∂2 + + , ∂x2 ∂y 2 ∂z 2 ∂2 2 ≡ ∂µ ∂ µ = 2 − ∆. ∂t

∆≡∇·∇=

D.6 γ-Matrix Here, we present explicit expressions of the γ-matrices in two and four dimensions. Before presenting the representation of the γ-matrices, we first give the explicit representation of Pauli matrices.

D.6.1 Pauli Matrix Pauli matrices are given as µ ¶ µ ¶ 0 1 0 −i σx = σ1 = , σy = σ2 = , 1 0 i 0

µ ¶ 1 0 σz = σ3 = . 0 −1

Below we write some properties of the Pauli matrices. Hermiticity σ1† = σ1 ,

σ2† = σ2 ,

σ3† = σ3 .

Complex Conjugate σ1∗ = σ1 ,

σ2∗ = −σ2 ,

σ3∗ = σ3 .

(D.6.1)

Appendix

Fundamental Problems in Quantum Field Theory 155

  Transposed

  σ1T = σ1 ,

σ2T = −σ2 ,

σ3T = σ3 (σkT = σk∗ ).

Useful Relations σi σj = δij + i²ijk σk ,

(D.6.2)

[σi , σj ] = 2i²ijk σk .

(D.6.3)

D.6.2 Representation of γ-matrix (a) Two dimensional representations of γ-matrices µ ¶ µ ¶ µ 1 0 0 1 0 0 1 5 0 1 Dirac : γ = , γ = , γ =γ γ = 0 −1 −1 0 1 µ ¶ µ ¶ µ 0 1 0 −1 1 0 1 5 0 1 Chiral : γ = , γ = , γ =γ γ = 1 0 1 0 0

¶ 1 , 0 ¶ 0 . −1

(b) Four dimensional representations of gamma matrices µ ¶ µ ¶ 1 0 0 σ 0 Dirac : γ = β = , γ= , 0 −1 −σ 0 µ ¶ µ ¶ 0 1 0 σ 5 0 1 2 3 γ = iγ γ γ γ = , α= , 1 0 σ 0 µ ¶ µ ¶ 0 1 0 −σ 0 Chiral : γ = β = , γ= , 1 0 σ 0 µ ¶ µ ¶ 1 0 σ 0 γ 5 = iγ 0 γ 1 γ 2 γ 3 = , α= . 0 −1 0 −σ µ ¶ µ ¶ 0 0 1 0 where 0 ≡ , 1≡ . 0 0 0 1

D.6.3 Useful Relations of γ-Matrix Here, we summarize some useful relations of the γ-matrices. Anti-commutation relations {γ µ , γ ν } = 2g µν , {γ 5 , γ ν } = 0.

(D.6.4)

Hermiticity 㵆 = γ0 γµ γ0 (γ0† = γ0 , γk† = −γk ),

γ5† = γ5 .

(D.6.5)

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γ0∗ = γ 0 , γ1∗ = γ1 , γ2∗ = −γ2 , γ3∗ = γ3 ,

γ5∗ = γ5 .

(D.6.6)

Transposed γµT = γ 0 㵆 γ 0 , γ5T = γ5 .

(D.6.7)

D.7 Transformation of State and Operator When one transforms a quantum state |ψi by a unitary transformation U which satisfies U †U = 1 one writes the transformed state as |ψ 0 i = U |ψi.

(D.7.1)

The unitarity is important since the norm must be conserved, that is, hψ 0 |ψ 0 i = hψ|U † U |ψi = 1. In this case, an arbitrary operator O is transformed as O0 = U OU −1 .

(D.7.2)

This can be obtained since the expectation value of the operator O must be the same between two systems, that is, hψ|O|ψi = hψ 0 |O0 |ψ 0 i. (D.7.3) Since hψ 0 |O0 |ψ 0 i = hψ|U † O0 U |ψi = hψ|O|ψi one finds U † O0 U = O which is just eq.(D.7.2).

D.8 Fermion Current We summarize the fermion currents and their properties of the Lorentz transformation. We also give their nonrelativistic expressions since the basic behaviors must be kept in the

Appendix

Fundamental Problems in Quantum Field Theory 157

  nonrelativistic expressions. Here,  the approximate expressions are obtained by making use of the plane wave solutions for the Dirac wave function.  ¯ '1 Scalar : ψψ   ¯ 5 ψ ' ·p  Pseudoscalar : ψγ m   ´ ³ (D.8.1) Fermion currents :   Vector : ¯ µ ψ ' 1, p ψγ  m  ³σ · p ´  ¯ µγ5ψ ' ,σ Axialvector : ψγ m Therefore, under the parity Pˆ and time reversal Tˆ transformation, the currents behave  0 0 ¯ ψ¯ ψ = ψ¯Pˆ −1 Pˆ ψ = ψψ   ψ¯0 γ5 ψ 0 = ψ¯Pˆ −1 γ5 Pˆ ψ = −ψγ ¯ 5ψ  (D.8.2) Parity Pˆ :   ψ¯0 γk ψ 0 = ψ¯Pˆ −1 γk Pˆ ψ = −ψγ ¯ kψ  ¯ k γ5 ψ ψ¯0 γk γ5 ψ 0 = ψ¯Pˆ −1 γk γ5 Pˆ ψ = ψγ     Time Reversal Tˆ :   

¯ ψ¯0 ψ 0 = ψ¯Tˆ−1 Tˆψ = ψψ ¯ 5ψ ψ¯0 γ5 ψ 0 = ψ¯Tˆ−1 γ5 Tˆψ = ψγ ¯ kψ ψ¯0 γk ψ 0 = ψ¯Tˆ−1 γk Tˆψ = −ψγ

(D.8.3)

¯ k γ5 ψ ψ¯0 γk γ5 ψ 0 = ψ¯Tˆ−1 γk γ5 Tˆψ = −ψγ

D.9 Trace in Physics D.9.1 Definition The trace of N × N matrix A is defined as Tr[A] =

N X

Aii .

(D.9.1)

i=1

It is easy to prove Tr[AB] = Tr[BA].

(D.9.2)

D.9.2 Trace in Quantum Mechanics The trace of the Hamiltonian H becomes Tr[H] = Tr[U HU −1 ] =

X

En ,

(D.9.3)

n=1

where U is a unitary operator, and En denotes the energy eigenvalue of the Hamiltonian.

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Trace in SU (N )

Fujita and Kanda

 

In SU (N ), the element U a can be described in terms of the generator T a U a = eiαT

a

(D.9.4)

where the generator must be hermitian and traceless since ¡ ¢ ¡ ¢ detU a = exp Tr [ln U a ] = exp iα Tr [T a ] = 1

(D.9.5a)

Tr [T a ] = 0.

(D.9.5b)

The generators of SU (N ) group satisfy the following commutation relations [T a , T b ] = iC abc T c ,

(D.9.6)

where C abc denotes a structure constant. The generators are normalized such that Tr [T a T b ] =

1 ab δ . 2

(D.9.7)

D.9.4 Trace of γ-Matrices and p/ Trace of γ-matrices : Tr [1] = 4, Tr [γµ ] = 0, Tr [γ5 ] = 0. Symbol p/ :

(D.9.8)

p/ ≡ pµ γ µ

Useful Relations: γµ p/γ µ = −2p/

(D.9.9)

p/q/ = p · q − iσµν pµ q ν

(D.9.10)

Tr [p/q/] = 4p · q

(D.9.11)

Tr [γ5 p/q/] = 0 (D.9.12) n o Tr [p/1 p/2 p/3 p/4 ] = 4 (p1 · p2 )(p3 · p4 ) − (p1 · p3 )(p2 · p4 ) + (p1 · p4 )(p2 · p3 ) (D.9.13) Tr [γ 5 p/1 p/2 p/3 p/4 ] = −4iεαβγδ pα1 pβ2 pγ3 pδ4

(D.9.14)

5

Tr [γ γµ1 γµ2 γµ3 γµ4 γµ5 γµ6 ] = −4i [gµ1 µ2 εµ3 µ4 µ5 µ6 − gµ1 µ3 εµ2 µ4 µ5 µ6 +gµ2 µ3 εµ1 µ4 µ5 µ6 + gµ4 µ5 εµ1 µ2 µ3 µ6 − gµ4 µ6 εµ1 µ2 µ3 µ5 + gµ5 µ6 εµ1 µ2 µ3 µ4 ]

(D.9.15)

Appendix

 

Fundamental Problems in Quantum Field Theory 159

¯ µ ¯ ¯δ 0 δ µ 0 δ µ 0 δ µ 0 ¯ ν α β ¯ ¯ µ ¯δ ν 0 δ ν 0 δ ν 0 δ ν 0 ¯ ¯ ν α β ¯ εµναβ εµ0 ν 0 α0 β 0 = − ¯ αµ α α α ¯ δ δ δ δ ¯ µ0 ν0 α0 β0 ¯ ¯ β ¯ ¯δ µ0 δ βν 0 δ βα0 δ ββ 0 ¯ ¯ ¯ ¯δ ν 0 δ ν 0 δ ν 0 ¯ ¯ ν α β ¯ ¯ ¯ εµναβ εµν 0 α0 β 0 = − ¯δ αν 0 δ αα0 δ αβ 0 ¯ ¯ ¯ β ¯ δ ν 0 δ β α0 δ β β 0 ¯ ¯ ¯ ¯δ α 0 δ α 0 ¯ ¯ β ¯ εµναβ εµνα0 β 0 = −2 ¯ βα ¯ ¯δ α0 δ ββ 0 ¯

 

(D.9.16)

(D.9.17)

(D.9.18)

εµναβ εµναβ 0 = −6δ ββ 0

(D.9.19)

εµναβ εµναβ = −24

(D.9.20)

D.10 Lagrange Equation In classical field theory, the equation of motion is most important, and it is derived from the Lagrange equation. Therefore, we review briefly how we can obtain the equation of motion from the Lagrangian density.

D.10.1 Lagrange Equation in Classical Mechanics Before going to the field theory treatment, we first discuss the Lagrange equation (Newton equation) in classical mechanics. In order to obtain the Lagrange equation by the variational principle in classical mechanics, one starts from the action S as defined Z S = L(q, q) ˙ dt, (D.10.1) where the Lagrangian L(q, q) ˙ depends on the general coordinate q and its velocity q. ˙ At the time of deriving equation of motion by the variational principle, q and q˙ are independent as the function of t. This is clear since, in the action S, the functional dependence of q(t) is unknown and therefore one cannot make any derivative of q(t) with respect to time t. Once the equation of motion is established, then one can obtain q˙ by time differentiation of q(t) which is a solution of the equation of motion. The Lagrange equation can be obtained by requiring that the action S should be a minimum with respect to the variation of q and q. ˙ ¶ µ Z Z ∂L ∂L δ q˙ dt δq + δS = δL(q, q) ˙ dt = ∂q ∂ q˙ ¶ Z µ ∂L d ∂L δq dt = 0, = (D.10.2) − ∂q dt ∂ q˙

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Thus one obtains the Lagrange equation   where the surface terms should vanish.   d ∂L ∂L = 0. − ∂q dt ∂ q˙

(D.10.3)

Hamiltonian in Classical Mechanics The Lagrangian must be invariant under the infinitesimal time displacement ² of q(t) as q(t + ²) → q(t) + q², ˙ q(t ˙ + ²) → q(t) ˙ + q¨² + q˙

d² . dt

(D.10.4)

Therefore, one finds δL(q, q) ˙ = L(q(t + ²), q(t ˙ + ²)) − L(q, q) ˙ =

∂L ∂L ∂L d² q¨² + q˙ = 0. (D.10.5) q² ˙ + ∂q ∂ q˙ ∂ q˙ dt

Since the surface term vanishes, one obtains µ ¶¸ · µ ¶¸ · d ∂L d ∂L ∂L ∂L q¨ − q˙ ² = L− q˙ ² = 0 δL(q, q) ˙ = q˙ + ∂q ∂ q˙ dt ∂ q˙ dt ∂ q˙

(D.10.6)

where the term in bracket is a conserved quantity, and thus the Hamiltonian H is defined as H≡

∂L q˙ − L. ∂ q˙

(D.10.7)

D.10.2 Lagrange Equation for Fields The Lagrange equation for fields can be obtained almost in the same way as the particle case. For fields, we should start from the Lagrangian density L and the action is written as ¶ Z µ ∂ψ ˙ d3 r dt, S = L ψ, ψ, (D.10.8) ∂xk ∂ψ where ψ(x), ∂ψ ∂t and ∂xk are independent functional variables. Hereafter, we use the nota˙ tion of ψ(x) ≡ ∂ψ ∂t . The Lagrange equation can be obtained by requiring that the action S should be a minimum with respect to the variation of ψ, ψ˙ and ∂ψ , ∂xk

µ µ ¶ ¶! Z Z à ∂L ∂ψ ∂ψ ∂L ∂L 3 ˙ ˙ δψ + δS = δL ψ, ψ, δ d r dt = δψ + d3 r dt ∂ψ ∂xk ∂ψ ∂x ∂ ψ˙ ∂( ∂xk ) k à ! Z ∂ ∂L ∂ ∂L ∂L − = (D.10.9) δψ d3 r dt = 0, − ˙ ∂ψ ∂t ∂ ψ ∂xk ∂( ∂ψ ) ∂xk

where the surface terms are assumed to vanish. Therefore, one obtains ∂ ∂L ∂ ∂L ∂L + , = ∂ψ ∂t ∂ ψ˙ ∂xk ∂( ∂ψ ) ∂xk

(D.10.10)

Appendix

Fundamental Problems in Quantum Field Theory 161

covariant way as   which can be expressed in the relativistic   ¶ µ ∂L ∂L . = ∂µ ∂ψ ∂(∂µ ψ)

(D.10.11)

D.11 Noether Current If the Lagrangian density is invariant under the transformation of the field with a continuous variable, then there is always a conserved current associated with this symmetry. This is called Noether current and can be derived from the invariance of the Lagrangian density and the Lagrange equation.

D.11.1 Global Gauge Symmetry The Lagrangian density which is discussed in this textbook should have the following functional dependence in general © ª ¯ µ ∂ µ ψ − mψψ ¯ + LI ψψ, ¯ ψγ ¯ 5 ψ, ψγ ¯ µψ L = iψγ which is obviously invariant under the global gauge transformation ψ 0 = eiα ψ,

†

ψ 0 = e−iα ψ † ,

(D.11.1)

where α ia a real constant. Therefore, the Noether current is conserved in this system. To derive the Noether current conservation for the global gauge transformation, one can consider the infinitesimal global transformation, that is, |α| ¿ 1 ψ 0 = ψ + δψ, †

ψ 0 = ψ † + δψ † ,

δψ = iαψ.

(D.11.2a)

δψ † = −iαψ † .

(D.11.2b)

Invariance of Lagrangian Density Now, it is easy to find †

†

δL = L(ψ 0 , ψ 0 , ∂µ ψ 0 , ∂µ ψ 0 ) − L(ψ, ψ † , ∂µ ψ, ∂µ ψ † ) = 0

(D.11.3a)

which becomes ³ ´ ∂L † ∂L ∂L ∂L † δ ∂ ψ δ (∂µ ψ) + δψ δψ + + µ ∂ψ ∂(∂µ ψ) ∂ψ † ∂(∂µ ψ † ) ¶ ¸ ¶ µ ·µ ∂L ∂L ∂L ∂L † † ψ − ∂µ ψ ψ+ ∂µ ψ − ∂µ = iα ∂µ ∂(∂µ ψ) ∂(∂µ ψ) ∂(∂µ ψ † ) ∂(∂µ ψ † ) ¸ · ∂L ∂L † ψ− ψ =0 = iα∂µ (D.11.3b) ∂(∂µ ψ) ∂(∂µ ψ † )

δL =

where the equation of motion for ψ is employed.

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Therefore, one defines the current j µ as ·

∂L ∂L ψ† ψ− j ≡ −i ∂(∂µ ψ) ∂(∂µ ψ † )

¸

µ

(D.11.4)

and one has the current conservation ∂µ j µ = 0.

(D.11.5)

For Dirac fields, one finds the conserved current ¯ µ ψ. j µ = ψγ

(D.11.6)

D.11.2 Chiral Symmetry When the Lagrangian density is invariant under the chiral transformation, ψ 0 = eiαγ5 ψ

(D.11.7)

then there is another Noether current. Here, δψ as defined in eq.(D.11.2) becomes δψ = iαγ5 ψ.

(D.11.8)

Therefore, a corresponding conserved current for massless Dirac fields becomes j5µ = −i

∂L ¯ µ γ5 ψ γ5 ψ = ψγ ∂(∂µ ψ)

(D.11.9)

and we have ∂µ j5µ = 0.

(D.11.10)

The conservation of the axial vector current holds for massless field theory models.

D.12 Hamiltonian Density The Hamiltonian density H is constructed from the Lagrangian density L. If the Lagrangian density is invariant under the translation aµ , then there is a conserved quantity which is the energy momentum tensor T µν . The Hamiltonian density is constructed from the energy momentum tensor of T 00 .

Appendix

  D.12.1

Fundamental Problems in Quantum Field Theory 163

Hamiltonian Density   from Energy Momentum Tensor

´ ³ ∂ψi . If one considers the following Now, the Lagrangian density is given as L ψi , ∂0 ψi , ∂x k infinitesimal translation aµ of the field ψi and ψi† ψi0 = ψi + δψi ,

δψi = (∂ν ψi )aν ,

0

ψi† = ψi† + δψi† ,

δψi† = (∂ν ψi† )aν ,

then the Lagrangian density should be invariant δL ≡ L(ψi0 , ∂µ ψi0 ) − L(ψi , ∂µ ψi ) # " X ∂L ∂L ∂L ∂L † † δ(∂µ ψi ) = 0. (D.12.1) δ(∂µ ψi ) + δψi + δψi + = ∂ψi ∂(∂µ ψi ) ∂ψi† ∂(∂µ ψi† ) i Making use of the Lagrange equation, one obtains µ ¶¸ X · ∂L ∂L ∂L (∂µ ∂ν ψi ) − ∂µ ∂ν ψi aν δL = (∂ν ψi ) + ∂ψi ∂(∂µ ψi ) ∂(∂µ ψi ) i

+

X i

"

∂L

(∂ν ψi† ) +

∂ψi†

" = ∂µ Lg µν −

X

Ã

i

Ã

∂L ∂(∂µ ψi† )

(∂µ ∂ν ψi† ) − ∂µ

∂L ∂L ∂ ν ψi† ∂ ν ψi + † ∂(∂µ ψi ) ∂(∂µ ψi )

∂L ∂(∂µ ψi† ) !#

!# ∂ν ψi†

aν = 0.

aν

(D.12.2)

Energy Momentum Tensor T µν Therefore, if one defines the energy momentum tensor T µν by ! Ã X ∂L ∂L ∂ ν ψi† − Lg µν ∂ ν ψi + T µν ≡ ∂(∂µ ψi ) ∂(∂µ ψi† )

(D.12.3)

i

then, T µν is a conserved quantity, that is ∂µ T µν = 0. This leads to the definition of the Hamiltonian density H in terms of T 00 ! Ã X ∂L ∂L 0 † 0 00 ∂ ψi − L. ∂ ψi + H≡T = ∂(∂0 ψi ) ∂(∂0 ψi† ) i

(D.12.4)

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Fujita and Kanda

Hamiltonian Density   for Free Dirac Fields

For a free Dirac field with its mass m, the Lagrangian density becomes £ ¤ L = ψi† ψ˙ i + ψi† iγ 0 γ · ∇ − mγ 0 ij ψj .

(D.12.5)

Therefore, we find the Hamiltonian density as H = T 00 = ψ¯i [−iγk ∂k +m]ij ψj = ψ¯ [−iγ · ∇+m] ψ.

(D.12.6)

Hamiltonian for Free Dirac Fields The Hamiltonian H is obtained by integrating the Hamiltonian density over all space Z Z 3 H = H d r = ψ¯ [−iγ · ∇ + m] ψ d3 r. (D.12.7) In classical field theory, this Hamiltonian is not an operator but is just the field energy itself. However, this field energy cannot be evaluated unless one knows the shape of the field ψ(x) itself. Therefore, one should determine the shape of the field ψ(x) by the equation of motion in the classical field theory.

D.12.3 Role of Hamiltonian The classical field Hamiltonian itself is not useful. This is similar to the classical mechanics case in which one has to derive the Hamilton equations in order to calculate physical properties of the system, and the Hamilton equations are equivalent to the Lagrange equations in classical mechanics. Classical Field Theory In classical field theory, the situation is just the same as the classical mechanics case. If one stays in the classical field theory, then one should derive the field equation from the Hamiltonian by the functional variational principle. Quantized Field Theory The Hamiltonian of the field theory becomes important when the fields are quantized. In this case, the Hamiltonian becomes an operator, and thus one has to solve the eigenvalue ˆ problem for the quantized Hamiltonian H ˆ H|Ψi = E|Ψi,

(D.12.8)

where |Ψi is called Fock state and should be written in terms of the creation and annihilation operators of fermion and anti-fermion. The space spanned by the Fock states is called Fock space. In normal circumstances of the field theory models such as QED and QCD, it is

Appendix

Fundamental Problems in Quantum Field Theory 165

  practically impossible to find the  eigenstate of the quantized Hamiltonian. The difficulty of the quantized field theory comes mainly from two reasons. Firstly, one has to construct the vacuum state which is composed of infinite many negative energy particles interacting with each other. The vacuum state should be the eigenstate of the Hamiltonian ˆ H|Ωi = EΩ |Ωi, where EΩ denotes the energy of the vacuum and it is in general infinity with the negative sign. The vacuum state |Ωi is composed of infinitely many negative energy particles |Ωi =

Y

p,s

(s)

b† p |0ii,

where |0ii denotes the null vacuum state. In the realistic calculations, the number of the negative energy particles must be set to a finite value, and this should be reasonable since physical observables should not depend on the deep negative energy particles.

D.13 Variational Principle in Hamiltonian Now, one can derive the equation of motion by requiring that the Hamiltonian should be minimized with respect to the functional variation of the state ψ(r).

D.13.1 Schr¨odinger Field When one minimizes the Hamiltonian ¸ Z · 1 † 2 † ψ ∇ ψ + ψ U ψ d3 r H= − 2m

(D.13.1)

with respect to ψ(r), then one can obtain the static Schr¨odinger equation. Functional Derivative First, one defines the functional derivative for an arbitrary function ψi (r) by δψi (r 0 ) = δij δ(r − r 0 ). δψj (r)

(D.13.2)

This is the most important equation for the functional derivative, and once one accepts this definition of the functional derivative, then one can evaluate the functional variation just in the same way as normal derivative of the function ψi (r).

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  Functional Variation of Hamiltonian For the condition on ψ(r), one requires that it should be normalized according to Z ψ † (r)ψ(r) d3 r = 1. (D.13.3) In order to minimize the Hamiltonian with the above condition, one can make use of the Lagrange multiplier and make a functional derivative of the following quantity with respect to ψ † (r) ¸ Z · 1 † 0 02 ψ (r )∇ ψ(r 0 ) + ψ † (r 0 )U ψ(r 0 ) d3 r0 H[ψ] = − 2m µZ ¶ † 0 0 3 0 −E ψ (r )ψ(r ) d r − 1 , (D.13.4) where E denotes a Lagrange multiplier and just a constant. In this case, one obtains · ¸ Z 1 δH[ψ] 0 2 0 0 0 0 ∇ ψ(r ) + U ψ(r ) − Eψ(r ) d3 r0 = 0. (D.13.5) = δ(r − r ) − δψ † (r) 2m Therefore, one finds

1 ∇2 ψ(r) + U ψ(r) = Eψ(r) 2m which is just the static Schr¨odinger equation. −

(D.13.6)

D.13.2 Dirac Field The Dirac equation for free field can be obtained by the variational principle of the Hamiltonian eq.(D.12.7). Below, we derive the static Dirac equation in a concrete fashion by the functional variation of the Hamiltonian. Functional Variation of Hamiltonian For the condition on ψi (r), one requires that it should be normalized according to Z ψi† ψi (r) d3 r = 1. (D.13.7) Now, the Hamiltonian should be minimized with the condition of eq.(D.13.7) Z £ ¤ H[ψi ] = ψi† (r) −i(γ 0 γ · ∇)ij + m(γ 0 )ij ψj (r) d3 r µZ −E

¶ ψi† (r)ψi (r) d3 r

−1 ,

(D.13.8)

where E is just a constant of the Lagrange multiplier. By minimizing the Hamiltonian with respect to ψi† (r), one obtains (−iα · ∇ + mβ) ψ(r) − Eψ(r) = 0 which is just the static Dirac equation for free field.

(D.13.9)

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Fundamental Problems in Quantum Field Theory, 2013, 173-176

 

173

 

Index γ matrix, 154 γ + γ → e+ + e− experiment, 62 π 0 decay, 143 g − 2 with Z 0 boson, 98 SU (2) ⊗ U (1) gauge theory, 93 SU (Nc ) colors, 73 Z 0 decay, 144 additional gravity potential, 107, 115 Ampere-Maxwell, 4 annihilation operator, 6, 41, 128 anomaly, 64 anomaly equation, 65 anomaly equation in four dimensions, 66 anomaly equation in two dimensions, 70 anomaly in Schwinger model, 69 anti-commutation relation, 9, 13, 155 axial gauge fixing, 40 Bethe’s Lamb shifts, 130, 136 Biot-Savart law, 6 Bohr radius, 37 Born approximation, 24 box diagrams, 145 charge of symmetry group, 87 chiral anomaly equation, 63 chiral charge, 69, 88 chiral symmetry, 88, 162 CKM matrix elements, 123 color charge, 76, 79, 83, 92 color current, 75 color degree of freedom, 85 color electric field, 77 color magnetic field, 77 color octet, 76 comets, 113 commutation relation, 7, 41 complex conjugate, 151, 154

Compton scattering, 142 conservation law, 38 conservation of axial vector current, 71 conserved color current, 76 conserved current in QCD, 76 conserved vector current model, 92 correct propagator, 29 correct propagator of photon, 31, 32 Cosmic Fireball, 105 cosmology, 105 Coulomb gauge fixing, 5, 28, 39, 49, 54,91 127, 141 Coulomb interaction, 39, 138 Coulomb propagator, 30, 31 counter term, 129 coupling constant, 36, 81, 93, 99, 104 123, 144 CP transformation, 123 CPT theorem, 123 creation operator, 6, 41, 128 cross section, 24, 58, 60, 85 cutoff Λ, 43 CVC theory, 93 d’Alembertian operators, 154 Darwin term, 134 decay width, 68 decay width of π 0 → 2γ, 64 differential cross section, 23, 24, 59 dimensional regularization, 44, 140 Dirac equation, 9, 11, 37, 173 Dirac equation for Coulomb, 15 Dirac equation for gravity, 17, 102 Dirac field, 10, 164, 166 Dirac representation of gamma matrix, 10 double counting problem, 81

Takehisa Fujita and Naohiro Kanda All rights reserved - © 2013 Bentham Science Publishers

174 Fundamental Problems in Quantum Field Theory

earth rotation, 107 EDM of neutron, 122 EDM origin, 125 eigenfunction, 21 eigenstate, 6, 88, 165 electric charge, 83 electric dipole moments, 123 electromagnetic wave, 8 electron g − 2, 98 electron self-energy, 128 energy momentum tensor, 7, 14, 101, 163 equation for gauge field, 27, 37, 40 equation of motion for gravity, 102 equation of motion in QCD, 77 exact energy for Coulomb and gravity, 17 exact energy for Coulomb potential, 16 exact Hamiltonian, 83 exact solution of T-matrix, 25 exact vacuum, 88 Faraday law, 4 Fermi model, 86 fermion current, 15, 37, 156 fermion quantization, 31 Feynman propagator of photon, 28, 32 field equation of gravity, 100 field strength, 36, 94, 102 fine structure constant, 150 Foldy-Wouthuysen transformation, 107, 133 free Dirac field, 10, 11 free field of photon, 6, 27, 41, 137, 147 free massive vector field, 95, 148 free wave of photon, 27 functional derivative, 165, 166 gamma matrix, 16 gauge choice, 39 gauge condition, 46, 141 gauge field, 37, 40 gauge fixing, 39 gauge invariance, 74 gauge invariant interaction, 38 Gauss law, 4

Fujita and Kanda

Gedanken experiment, 100 general relativity, 101, 110 generator, 158 generator of SU(N), 73 global gauge symmetry, 161 global gauge transformation, 15, 38 gluon, 76 gluon field, 73 GPS satellite, 107, 109 gravitational constant, 99, 104, 151 gravitational interaction, 99 gravitational potential, 107 gravity, 99 Green function, 23 Hamilton equations, 115 Hamiltonian density, 14, 162, 164 hermite conjugate, 151 Higgs mechanism, 89, 90 higher loop calculations, 124 higher order CM correction, 131, 134 hydrogen atom, 9, 16, 19, 100, 129 infinitesimal transformation, 38, 74.76 infra-red singularity, 49, 137 interaction picture, 25 ladder diagrams, 81 Lagrange equation, 11, 37, 159, 160 Lagrangian density, 10, 161 Lagrangian density of gravity, 102 Lagrangian density of QCD, 73 Lagrangian density of QED, 36 Lamb shifts in hydrogen, 126 Lamb shifts of CM effect, 133 Lamb shifts of muonic hydrogen, 132 Index 175 Lamb shifts of muonium, 135, 136 Lamb shifts without ., 136 Laplacian operator, 154 large scale structure, 105 leap second, 110 leap second dating, 113 linear divergence, 67

Index

Fundamental Problems in Quantum Field Theory 175

Lippmann-Schwinger equation, 22 local gauge invariance, 37 local gauge transformation, 74 logarithmic divergence, 68 loop diagram, 33 Lorentz conditions, 28, 95, 147 Lorentz gauge fixing, 28, 96, 149 magnetic moments of nucleons, 84 mass renormalization, 129 mass scale, 36 massive vector fields, 94 massless scalar field, 102 Maxwell equation, 4, 37 Mercury perihelion, 107, 109 metric tensor, 101 Michelson-Morley experiment, 112 Mott cross section, 24 mugen-universe, 105 mugen-universe scenario, 106 muonic hydrogen, 131 muonium, 135 natural units, 150 negative energy solution, 12 new angular momentum, 108 new gravitational potential, 107 Newton equation with new gravity, 107 no divergence in triangle diagrams, 47 no magnetic monopole, 4 Noether current, 15, 38, 161 Noether current in QCD, 75 Noether theorem, 38 non-abelian gauge fields, 76, 92 non-abelian gauge theory, 73, 93 non-local interaction, 25 non-static perturbation, 21 nuclear force, 78 one one one one one

boson exchange potential, 78 loop calculation of EDM, 123 photon state, 7 pion exchange potential, 82 universe scenario, 106

P-violating charge, 125 parity violation, 125 Pauli matrix, 154 Pauli-Villars regularization, 139 periodic boundary condition, 13 photon baryon ratio, 106 photon emission, 8 photon field, 7 photon propagator, 27 photon self-energy, 47 photon-photon via gravity, 119 photon-photon beam, 60 photon-photon cross section, 58 photon-photon experiment, 61 photon-photon scattering, 47, 53 plane wave solution, 11 Poisson equation, 5 Poisson equation for gravity, 101 polarization states, 30 positive energy solution, 12 potential scattering, 22 principle of equivalence, 100 QCD, 73, 78 QED, 36, 43, 50 QED and gravity, 102, 103 quadratic divergence, 43, 45 quantization in box, 13 quantization of Coulomb field, 127 quantization of Dirac field, 13 quantization of gauge field, 41 quantization of gravity field, 104 quantization procedure, 7 quantum number, 4, 16, 36 quark field, 73 quark model, 84 redundant variable of vector field, 39 regularization, 139 regularized charge, 69 regularized chiral charge, 71 relics of preceding universe, 105 renormalization scheme, 42, 50, 93, 97

176 Fundamental Problems in Quantum Field Theory

renormalized mass, 129 Ricci tensor, 101 right propagator, 96 Rutherford scattering, 24 S-matrix definition, 25 S-matrix formula, 27 scalar field, 102 scalar meson, 81 scalar product, 152 scattering amplitude, 23 scattering T-matrix, 23 Schr¨ odinger field, 165 Schwinger model, 69 Schwinger vacuum, 70 self-energy of fermion, 50 self-energy of photon, 43 separable interaction, 24 spin wave function, 12 spin-orbit force, 107 spinor, 12 spontaneous symmetry breaking, 88 static field, 6 static field energy per time, 8 static-dominance ansatz, 103 structure constant, 73 symmetry breaking, 87, 88 T-matrix, 21, 22, 25, 32 T-matrix of π 0 → 2γ, 63 T-matrix of Z 0 decay, 67 T-matrix of NN interaction, 81 T-matrix of OPE, 82 T-matrix of two pion exchange, 79 T-matrix with Coulomb interaction, 138 T-product, 26 T-transformation, 123

Fujita and Kanda

T-violation, 124 temporal gauge fixing, 39 thermodynamic limit, 14 time dependent perturbation, 21 time shifts of earth, 107 time shifts of GPS, 107 time shifts of Mercury, 107 top quark, 68, 124 trace in physics, 157 trace of operator, 73 traceless matrix, 158 transposed, 155 triangle diagrams, 47 two pion exchange potential, 79 Uehling potential, 127, 132 unitary gauge fixing, 91 vacuum expectation of S-matrix, 29 vacuum polarization, 47, 133, 141 vacuum polarization potential, 127, 131 vacuum polarization tensor, 43, 45 variational principle, 159, 165 vector field, 7, 36 vector potential, 5, 137 vector product, 152 velocity of earth, 112 vertex correction, 51, 97, 98, 124, 125 Virial theorem, 107 Ward identity, 52 wave function renormalization, 50 weak coupling constant, 151 weak interaction, 94 weak massive vector boson, 94, 97 Weinberg-Salam model, 87 zero point energy of photon, 42