Quantum fields

with classical perturbations

(revised version)

Jan Dereziński
Dept. of Math. Methods in Phys.,
Faculty of Physics, University of Warsaw
Hoza 74, 00-682 Warszawa, Poland
email Jan.Derezinski@fuw.edu.pl
January 14, 2020

Abstract
The main purpose of these notes is a review of various models of Quan-
tum Field Theory involving quadratic Lagrangians. We discuss scalar and
vector bosons, spin 12 fermions, both neutral and charged. Beside free the-
ories, we study their interactions with classical perturbations, called, de-
pending on the context, an external linear source, mass-like term, current
or electromagnetic potential. The notes may serve as a first introduction
to QFT.

Contents
0 Introduction 7

1 Basic concepts 11
1.1 Minkowski space . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.1 Coordinates in Minkowski space . . . . . . . . . . . . . . 11
1.1.2 Causal structure . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.3 Invariant measure . . . . . . . . . . . . . . . . . . . . . . 13
1.1.4 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . 13
1.1.5 Lorentz and Poincaré groups . . . . . . . . . . . . . . . . 14
1.1.6 Double coverings of Lorentz and Poincaré groups . . . . . 15
1.1.7 Finite dimensional representations of the Lorentz group . 17
1.2 Symplectic dynamics and its quantization . . . . . . . . . . . . . 18
1.2.1 Symplectic form vs Poisson bracket in classical mechanics 18
1.3 Darboux coordinates and symplectic vector spaces . . . . . . . . 19
1.4 General concepts of quantum field theory . . . . . . . . . . . . . 19

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 1.4.1 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . 19
1.4.2 Time reversal . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.3 Relativistic quantum mechanics . . . . . . . . . . . . . . . 21
1.4.4 Haag-Kastler axioms for observable algebras . . . . . . . . 23
1.4.5 Haag-Kastler axioms for field algebras . . . . . . . . . . . 23
1.4.6 Global symmetries . . . . . . . . . . . . . . . . . . . . . . 24
1.4.7 Neutral quantum fields . . . . . . . . . . . . . . . . . . . . 25
1.4.8 Wightman axioms for neutral fields . . . . . . . . . . . . . 25
1.4.9 Relationship between Haag-Kastler and Wightman axioms 26
1.4.10 Global symmetries in the Wightman formalism . . . . . . 27
1.4.11 Charged fields . . . . . . . . . . . . . . . . . . . . . . . . 27
1.4.12 Wightman axioms for neutral and charged fields . . . . . 28
1.4.13 U (1) symmetry . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4.14 Charge conjugation . . . . . . . . . . . . . . . . . . . . . . 30
1.4.15 Parity invariance . . . . . . . . . . . . . . . . . . . . . . . 30
1.4.16 Time reversal invariance . . . . . . . . . . . . . . . . . . . 31
1.4.17 The CPT Theorem . . . . . . . . . . . . . . . . . . . . . . 31
1.4.18 The CPT Theorem in a P and T -invariant theory . . . . 32
1.4.19 N -point Wightman and Green’s functions . . . . . . . . . 33
1.5 General scattering theory . . . . . . . . . . . . . . . . . . . . . . 34
1.5.1 Time ordered exponential . . . . . . . . . . . . . . . . . . 34
1.5.2 Schrödinger and Heisenberg picture . . . . . . . . . . . . 35
1.5.3 Schrödinger and Heisenberg picture for time-dependent
Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.5.4 Time-dependent perturbations . . . . . . . . . . . . . . . 37
1.5.5 Vacuum energy . . . . . . . . . . . . . . . . . . . . . . . . 38
1.5.6 Time-ordered Green’s functions . . . . . . . . . . . . . . . 38
1.5.7 Adiabatic switching and the energy shift . . . . . . . . . . 39
1.5.8 Adiabatic switching and Green’s functions . . . . . . . . . 42
1.5.9 Adiabatic scatttering theory . . . . . . . . . . . . . . . . . 43

2 Neutral scalar bosons 45

2.1 Free neutral scalar bosons . . . . . . . . . . . . . . . . . . . . . . 45
2.1.1 Special solutions and Green’s functions . . . . . . . . . . 45
2.1.2 Space of solutions . . . . . . . . . . . . . . . . . . . . . . 48
2.1.3 Classical fields . . . . . . . . . . . . . . . . . . . . . . . . 50
2.1.4 Smeared fields . . . . . . . . . . . . . . . . . . . . . . . . 51
2.1.5 Lagrangian formalism . . . . . . . . . . . . . . . . . . . . 52
2.1.6 Stress-energy tensor . . . . . . . . . . . . . . . . . . . . . 53
2.1.7 Positive and negatve frequency space . . . . . . . . . . . . 54
2.1.8 Plane wave functionals . . . . . . . . . . . . . . . . . . . . 55
2.1.9 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.1.10 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.1.11 Quantization in terms of smeared fields . . . . . . . . . . 59
2.1.12 Quantization in terms of C ∗ -algebras . . . . . . . . . . . . 59
2.1.13 Two-point functions . . . . . . . . . . . . . . . . . . . . . 60

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 2.2 Neutral scalar bosons with a linear source . . . . . . . . . . . . . 61
2.2.1 Classical fields . . . . . . . . . . . . . . . . . . . . . . . . 61
2.2.2 Lagrangian and Hamiltonian formalism . . . . . . . . . . 62
2.2.3 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.2.4 Operator valued source . . . . . . . . . . . . . . . . . . . 64
2.2.5 Scattering operator . . . . . . . . . . . . . . . . . . . . . . 64
2.2.6 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . 66
2.2.7 Path integral formulation . . . . . . . . . . . . . . . . . . 67
2.2.8 Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . 67
2.2.9 Vacuum energy . . . . . . . . . . . . . . . . . . . . . . . . 69
2.2.10 Problems with the scattering operator . . . . . . . . . . . 69
2.2.11 Energy shift and scattering theory for a stationary source 70
2.2.12 Travelling source . . . . . . . . . . . . . . . . . . . . . . . 71
2.2.13 Scattering cross-sections . . . . . . . . . . . . . . . . . . . 73
2.2.14 Inclusive cross-section . . . . . . . . . . . . . . . . . . . . 74
2.3 Neutral scalar bosons with a mass-like perturbation . . . . . . . 75
2.3.1 Classical fields . . . . . . . . . . . . . . . . . . . . . . . . 75
2.3.2 Lagrangian and Hamiltonian formalism . . . . . . . . . . 76
2.3.3 Dynamics in the interaction picture . . . . . . . . . . . . 77
2.3.4 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.3.5 Quantum Hamiltonian . . . . . . . . . . . . . . . . . . . . 79
2.3.6 Path integral formulation . . . . . . . . . . . . . . . . . . 79
2.3.7 Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . 80
2.3.8 Vacuum energy . . . . . . . . . . . . . . . . . . . . . . . . 81
2.3.9 Pauli-Villars renormalization . . . . . . . . . . . . . . . . 82
2.3.10 Renormalization of the vacuum energy . . . . . . . . . . . 84
2.3.11 Method of dispersion relations . . . . . . . . . . . . . . . 85
2.3.12 Wick rotation . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.3.13 Dimensional renormalization . . . . . . . . . . . . . . . . 87
2.3.14 Energy shift . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3 Massive photons 89
3.1 Free massive photons . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.1.1 Space of solutions . . . . . . . . . . . . . . . . . . . . . . 90
3.1.2 Equations of motion . . . . . . . . . . . . . . . . . . . . . 90
3.1.3 Symplectic structure on the space of solutions . . . . . . . 91
3.1.4 Smeared 4-potentials . . . . . . . . . . . . . . . . . . . . . 92
3.1.5 Lagrangian formalism and stress-energy tensor . . . . . . 92
3.1.6 Diagonalization of the equations of motion . . . . . . . . 94
3.1.7 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.1.8 Positive frequency space . . . . . . . . . . . . . . . . . . . 97
3.1.9 Spin averaging . . . . . . . . . . . . . . . . . . . . . . . . 97
3.1.10 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.2 Massive photons with an external 4-current . . . . . . . . . . . . 100
3.2.1 Classical 4-potentials . . . . . . . . . . . . . . . . . . . . . 100
3.2.2 Lagrangian and Hamiltonian formalism . . . . . . . . . . 101

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 3.2.3 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.2.4 Causal propagators . . . . . . . . . . . . . . . . . . . . . . 104
3.2.5 Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . 105
3.2.6 Path integral formulation . . . . . . . . . . . . . . . . . . 106
3.2.7 Energy shift . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.3 Alternative approaches . . . . . . . . . . . . . . . . . . . . . . . . 108
3.3.1 Classical 4-potentials without the Lorentz condition . . . 108
3.3.2 The Lorentz condition . . . . . . . . . . . . . . . . . . . . 110
3.3.3 Diagonalization of the equations of motion . . . . . . . . 110
3.3.4 Positive frequency space . . . . . . . . . . . . . . . . . . . 112
3.3.5 “First quantize, then reduce” . . . . . . . . . . . . . . . . 113
3.3.6 Quantization without reduction on a positive definite Hilbert
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.3.7 The Gupta-Bleuler approach . . . . . . . . . . . . . . . . 115

4 Massless photons 116

4.1 Free massless photons . . . . . . . . . . . . . . . . . . . . . . . . 117
4.1.1 Space of solutions and the gauge invariance . . . . . . . . 117
4.1.2 Symplectic structure on the space of solutions . . . . . . . 119
4.1.3 Smeared 4-potentials . . . . . . . . . . . . . . . . . . . . . 120
4.1.4 Lagrangian formalism and the stress-energy tensor . . . . 121
4.1.5 Diagonalization of the equations of motion . . . . . . . . 122
4.1.6 Positive frequency space . . . . . . . . . . . . . . . . . . . 123
4.1.7 Spin averaging . . . . . . . . . . . . . . . . . . . . . . . . 123
4.1.8 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.1.9 Quantization in terms of C ∗ -algebras . . . . . . . . . . . . 125
4.2 Massless photons with an external 4-current . . . . . . . . . . . . 126
4.2.1 Classical fields . . . . . . . . . . . . . . . . . . . . . . . . 126
4.2.2 Lagrangian and Hamiltonian formalism . . . . . . . . . . 127
4.2.3 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.2.4 Causal propagators . . . . . . . . . . . . . . . . . . . . . . 129
4.2.5 Path integral formulation . . . . . . . . . . . . . . . . . . 130
4.2.6 The m → 0 limit . . . . . . . . . . . . . . . . . . . . . . . 130
4.2.7 Current produced by a travelling particle . . . . . . . . . 132
4.2.8 Energy shift . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.3 Alternative approaches . . . . . . . . . . . . . . . . . . . . . . . . 133
4.3.1 Manifestly Lorentz covariant formalism . . . . . . . . . . 133
4.3.2 The Lorentz condition . . . . . . . . . . . . . . . . . . . . 134
4.3.3 Positive frequency space . . . . . . . . . . . . . . . . . . . 135
4.3.4 “First quantize, then reduce” . . . . . . . . . . . . . . . . 135
4.3.5 Quantization with a subsidiary condition . . . . . . . . . 136
4.3.6 The Gupta-Bleuler approach . . . . . . . . . . . . . . . . 137

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5 Charged scalar bosons 137
5.1 Free charged scalar bosons . . . . . . . . . . . . . . . . . . . . . . 138
5.1.1 Classical fields . . . . . . . . . . . . . . . . . . . . . . . . 138
5.1.2 Smeared fields . . . . . . . . . . . . . . . . . . . . . . . . 139
5.1.3 Lagrangian formalism . . . . . . . . . . . . . . . . . . . . 140
5.1.4 Classical 4-current . . . . . . . . . . . . . . . . . . . . . . 140
5.1.5 Stress-energy tensor . . . . . . . . . . . . . . . . . . . . . 141
5.1.6 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.1.7 Positive and negative frequency subspace . . . . . . . . . 143
5.1.8 Plane wave functionals . . . . . . . . . . . . . . . . . . . . 144
5.1.9 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.1.10 Quantum 4-current . . . . . . . . . . . . . . . . . . . . . . 147
5.1.11 Quantization in terms of smeared fields . . . . . . . . . . 149
5.2 Charged scalar bosons in an external 4-potential . . . . . . . . . 149
5.2.1 Classical fields . . . . . . . . . . . . . . . . . . . . . . . . 149
5.2.2 Lagrangian and Hamiltonian formalism . . . . . . . . . . 150
5.2.3 Classical discrete symmetries . . . . . . . . . . . . . . . . 152
5.2.4 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.2.5 Quantum Hamiltonian . . . . . . . . . . . . . . . . . . . . 154
5.2.6 Quantized discrete symmetries . . . . . . . . . . . . . . . 154
5.2.7 2N -point Green’s functions . . . . . . . . . . . . . . . . . 156
5.2.8 Path integral formulation . . . . . . . . . . . . . . . . . . 157
5.2.9 Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . 157
5.2.10 Vacuum energy . . . . . . . . . . . . . . . . . . . . . . . . 159
5.2.11 Pauli-Villars renormalization . . . . . . . . . . . . . . . . 160
5.2.12 Renormalization of the vacuum energy . . . . . . . . . . . 162
5.2.13 Method of dispersion relations . . . . . . . . . . . . . . . 163
5.2.14 Dimensional renormalization . . . . . . . . . . . . . . . . 164
5.2.15 Abstract gauge covariance . . . . . . . . . . . . . . . . . . 165
5.2.16 Ward identities . . . . . . . . . . . . . . . . . . . . . . . . 165
5.2.17 Energy shift . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6 Dirac fermions 168

6.1 Free Dirac fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.1.1 Dirac spinors . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.1.2 Special solutions and Green’s functions . . . . . . . . . . 169
6.1.3 Space of solutions . . . . . . . . . . . . . . . . . . . . . . 171
6.1.4 Classical fields . . . . . . . . . . . . . . . . . . . . . . . . 172
6.1.5 Smeared fields . . . . . . . . . . . . . . . . . . . . . . . . 172
6.1.6 Diagonalization of the equations of motion . . . . . . . . 173
6.1.7 Plane wave functionals . . . . . . . . . . . . . . . . . . . . 174
6.1.8 Positive and negative frequency subspaces . . . . . . . . . 175
6.1.9 Spin averaging . . . . . . . . . . . . . . . . . . . . . . . . 175
6.1.10 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.1.11 Quantization in terms of smeared fields . . . . . . . . . . 178
6.1.12 Dirac sea quantization . . . . . . . . . . . . . . . . . . . . 178

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 6.1.13 Fermionic Hamiltonian formalism . . . . . . . . . . . . . . 179
6.1.14 Fermionic Lagrangian formalism . . . . . . . . . . . . . . 180
6.1.15 Classical 4-current . . . . . . . . . . . . . . . . . . . . . . 181
6.1.16 Quantum 4-current . . . . . . . . . . . . . . . . . . . . . . 183
6.2 Dirac fermions in an external 4-potential . . . . . . . . . . . . . . 184
6.2.1 Dirac equation in an external 4-potential . . . . . . . . . 184
6.2.2 Lagrangian and Hamiltonian formalism . . . . . . . . . . 185
6.2.3 Classical discrete symmetries . . . . . . . . . . . . . . . . 186
6.2.4 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.2.5 Quantum Hamiltonian . . . . . . . . . . . . . . . . . . . . 188
6.2.6 Quantized discrete symmetries . . . . . . . . . . . . . . . 188
6.2.7 2N -point Green’s functions . . . . . . . . . . . . . . . . . 189
6.2.8 Path integral formulation . . . . . . . . . . . . . . . . . . 191
6.2.9 Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . 191
6.2.10 Vacuum energy . . . . . . . . . . . . . . . . . . . . . . . . 192
6.2.11 Pauli-Villars renormalization . . . . . . . . . . . . . . . . 193
6.2.12 Method of dispersion relations . . . . . . . . . . . . . . . 194
6.2.13 Dimensional renormalization . . . . . . . . . . . . . . . . 194
6.2.14 Energy shift . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7 Majorana fermions 196

7.1 Free Majorana fermions . . . . . . . . . . . . . . . . . . . . . . . 196
7.1.1 Charge conjugation . . . . . . . . . . . . . . . . . . . . . . 196
7.1.2 Space of solutions . . . . . . . . . . . . . . . . . . . . . . 197
7.1.3 Smeared fields . . . . . . . . . . . . . . . . . . . . . . . . 197
7.1.4 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7.1.5 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 198
7.1.6 Quantization in terms of smeared fields . . . . . . . . . . 200
7.2 Majorana fermions with a mass-like perturbation . . . . . . . . . 200
7.2.1 Classical fields . . . . . . . . . . . . . . . . . . . . . . . . 200
7.2.2 Lagrangian and Hamiltonian formalism . . . . . . . . . . 201
7.2.3 Quantum fields . . . . . . . . . . . . . . . . . . . . . . . . 201
7.2.4 Path integral formulation . . . . . . . . . . . . . . . . . . 202
7.2.5 Vacuum energy . . . . . . . . . . . . . . . . . . . . . . . . 202
7.2.6 Renormalization of the vacuum energy . . . . . . . . . . . 202
7.2.7 Pauli-Villars renormalization of the 2nd order term . . . . 203

A Appendix 204
A.1 Second quantization . . . . . . . . . . . . . . . . . . . . . . . . . 204
A.1.1 Fock spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 204
A.1.2 Creation/annihilation operators . . . . . . . . . . . . . . . 205
A.1.3 Weyl/antisymmetric and Wick quantization . . . . . . . . 206
A.1.4 Second quantization of operators . . . . . . . . . . . . . . 208
A.1.5 Implementability of Bogoliubov translations . . . . . . . . 208
A.1.6 Implementability of Bogoliubov rotations . . . . . . . . . 209
A.1.7 Infimum of a van Hove Hamiltonian . . . . . . . . . . . . 209

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 A.1.8 Infimum of a Bogoliubov Hamiltonian . . . . . . . . . . . 210
A.2 Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
A.2.1 Identities for Feynman integrals . . . . . . . . . . . . . . . 211
A.2.2 Identities for the dimensional regularization . . . . . . . . 212
A.2.3 Operator identities . . . . . . . . . . . . . . . . . . . . . . 213
A.2.4 Coulomb and Yukawa potential . . . . . . . . . . . . . . . 213
A.2.5 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . 213
A.2.6 Dispersion relations . . . . . . . . . . . . . . . . . . . . . 214

0 Introduction
In these notes we discuss various models of Quantum Field Theory in 1+3
dimensions involving quadratic Lagrangians or, equivalently, quadratic Hamil-
tonians.
First of all, we describe basic types of free fields:
(1) neutral scalar bosons,
(2) neutral massive vector bosons (“massive photons”),
(3) neutral massless vector bosons (“massless photons”),
(4) charged scalar bosons,
(5) (charged) Dirac fermions,
(6) (neutral) Majorana fermions.
We also consider free fields perturbed by a linear or quadratic perturbation
involving a classical (c-number) function.
(1) neutral scalar bosons interacting with a linear source,
(2) neutral scalar bosons interacting with a mass-like perturbation,
(3) massive photons interacting with a classical 4-current,
(4) massless photons interacting with a classical 4-current,
(5) charged scalar bosons interacting with an electromagnetic 4-potential,
(6) Dirac fermions interacting with an electromagnetic 4-potential,
(7) Majorana fermions interacting with a mass-like perturbation.
All the above models are (or at least can be) well understood in the non-
perturbative sense. Perturbation theory is not necessary to compute their scat-
tering operators and Green’s functions, which is not the case (at least so far) of
truly interacting models.
Quantum fields interacting with classical perturbations is a topic with many
important applications to realistic physical systems. Therefore, the formalism
developed in our text is well motivated physically.
Clearly, many important issues of quantum field theory are outside of the
scope of free fields interacting with classical perturbations. However, surpris-
ingly many difficult topics can be discussed already on this level. Therefore, we

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believe that our text has pedagogical value, as a kind of an introduction to full
quantum field theory.
In our text we stress the deductive character of quantum field theory. Models
that we discuss are quite rigid and built according to strict principles. Among
these principles let us mention the Poincaré covariance, the Einstein causality
and the boundedness of the Hamiltonian from below. Some of these principles
are encoded in the Haag-Kastler and Wightman axioms. Even if these axioms
are often too restrictive, they provide useful guidelines.
The only known models for Haag-Kastler or Wightman axioms in 1+3 dimen-
sions are free theories. Their scattering theory is trivial. To obtain interesting
physical information one needs interacting theories. Unfortunately, interacting
theories are known only perturbatively.
Free theories are the quantizations of covariant 2nd order linear hyperbolic
equations on the Minkowski space. These equations can be perturbed by 0th or
1st order terms involving an arbitrary space-time functions called, depending on
the context, a classical (=external) linear source, mass-like term, 4-current or
electromagnetic 4-potential. We can consider the quantization of the perturbed
equation. Such a theory is still essentially exactly solvable, since the Hamilto-
nian is quadratic. It has no Poincaré covariance. However, it still gives rise to
a net of observable algebras satisfying the Einstein causality.
In our discussion we always start from the study of a classical theory, which
we discuss from the Hamiltonian and Lagrangian point of view. Then we dis-
cuss its quantization. Even though in all the cases we consider the Hamiltonian
is quadratic, its quantization often has various subtle points. In some cases,
especially for vector fields, there are several natural approaches to quantization,
which in the end lead to the same physical results. We try to discuss vari-
ous possible approaches. In our opinion, the existence of seemingly different
formalisms for the same physical system constitutes one of the most confusing
aspects of quantum field theory.
Classical perturbations that we consider are usually described by smooth
space-time functions that decay fast both in space and time. In particular,
their dynamics is typically described by time-dependent Hamiltonians. This is
a certain minor difficulty, which is often ignored in the literature. We discuss
how to modify the usual formalism in order to deal with this problem.
The models that we discuss illustrate many problems of interacting theories,
such as the ultraviolet problem, the infrared problem and the gauge invariance.
The ultraviolet problem means that when we try to define a theory in a
naive way some integrals are divergent for large momenta. In the context of our
paper this is never due to classical perturbations, which we always assume to be
smooth – the source of ultraviolet divergences is the behavior of propagators.
The ultraviolet problem is already visible when we consider neutral fields
with a masslike perturbation or charged fields with a classical electromagnetic
4-potential. In these systems classical dynamics exists under rather weak as-
sumptions. However there are problems with the quantum dynamics.
In some cases the quantum dynamics cannot be implemented on a Hilbert
space. This is the case of charged particles (bosons or fermions) in the presence

8
of variable spatial components of the 4-potential. On the other hand, the the
scattering operator exists under rather weak assumptions for 4-potential going
to zero in the past and future.
Even if we are able to implement the classical dynamics or the classical scat-
tering operator, we encounter another unpleasant surprise. The only quantity
that is not fixed by the classical considerations is the phase factor of the scat-
tering operator, written as e−iE/~ , where E is usually called the vacuum energy.
Computed naively, it often turns out to be divergent. In order to make this
phase factor finite it is necessary to renormalize the naive expression. This di-
vergence appears in low order vacuum energy diagrams. It was first successfully
studied by Heisenberg and Euler in the 30’s. A quantity closely related to this
phase factor is the effective action, which for a constant field was computed
exactly by Schwinger.
The infrared problem means that in the naive theory some integrals are di-
vergent for small momenta. This problem appears already in non-relativistic
quantum mechanics – in scattering theory with Coulomb forces. These forces
are long-range, which makes the usual definition of the scattering operator im-
possible [14]. Its another manifestation is the appearance of inequivalent repre-
sentations of canonical commutation relations, when we consider scattering of
photons against a classical 4-current that has a different direction in the past
and in the future [13, 15]. Thus, even in these toy non-relativistic situations
the usual scattering operator is ill-defined. Therefore, it is not surprising that
(much bigger) problems are present eg. in the full QED. One can cope with the
infrared problem by approximating massless photons with massive ones and re-
stricting computations only to inclusive cross-sections justified by an imperfect
resolution of the measuring device [57, 27, 55].
The expression gauge invariance has in the context of quantum field theory
several meanings.
(1) The most common meaning, discussed already in the context of classi-
cal electrodynamics, is the fact that if a total derivative is added to a
4-potential solving the Maxwell equation, then it still solves the Maxwell
equations. Of course, this no longer holds for the Proca equations – the
massive generalization of the Maxwell equations. Therefore, it is often
stressed that gauge invariance implies that the photons are massless.
(2) There exists another meaning of gauge invariance: we can multiply charged
fields by a space-time dependent phase factor and compensate it by chang-
ing the external potentials.
1. and 2. go together in the full QED, which is invariant with respect to
these two gauge transformations applied simultaneously.
(3) One often uses the term “gauge invariance” in yet another meaning: To
compute the scattering operator we can use various (free) photon prop-
agators. Equivalently, we have the freedom of choosing a Lagrangian in
the path integral formalism. This meaning applies both to massive and
massless photons. Some of these propagators are distinguished, such as

9
 the propagator in the Feynman or the Coulomb gauge. (Note, however,
that time-ordered N -point Green’s functions depend on the choice of the
propagator).
All these three meanings of gauge invariance can be illustrated with models
that we consider.
The paper is most of the time rigorous mathematically. In the places where
it is not, we believe that many readers can quite easily make it rigorous. We
try to make the presentation of various models parallel by applying, if possible,
coherent notation and formalism. This makes our text sometimes repetitious –
we believe that this helps the reader to understand small but often confusing
differences between distinct models.
Mathematical language that we use is most of the time elementary. Some-
times we use some mathematical concepts and facts that are, perhaps, less
commonly known, such as C ∗ -algebras, von Neumann algebras, the Schwartz
Kernel Theorem. The readers unfamiliar with them should not be discouraged
– their role in the article is minor.
Most of the material of this work has been considered in one way or another
in the literature. Let us give a brief and incomplete review of references.
On the formal level examples of quantum fields with classical perturbations
are discussed in most textbooks on quantum field theory, see eg. [26, 27, 46, 50,
55, 54, 5].
Linear hyperbolic equations is a well established domain of partial differential
equations, see eg [3].
Axioms of quantum field theory are discussed in [52, 23, 22].
A necessary and sufficient condition for the implementability of Bogoliubov
transformation was given by Shale for bosons [48] and by Shale and Stinespring
for fermions [49], see also [15]
Problems with implementability of the dynamics of charged particles in ex-
ternal potentials was apparently first noticed on a heuristic level in [45]. It was
studied rigorously by various authors. In particular, charged bosons were stud-
ied in [47, ?, 36, 37, 25, 1] and charged fermions in [40, 30, 29, 43, 12]. Rigorous
discussion of the smeared out local charge for charged fermions is contained in
[33].
The renormalization of the vacuum energy goes back to pioneering work of
[24]. In the mathematically rigorous literature it leads to the concept of a causal
phase discussed in the fermionic case in [44, 21].
The infrared problem goes back to [7, 28], see also [13].
The Gupta-Bleuler method of quantization of photon fields goes back to
[19, 6]. The C ∗ -algebraic formulation of the subsidiary condition method is
discussed in [53].
Rigorous study of vacuum energy for Dirac fermions in a stationary potential
is given in [18].
A topic that not included in these notes are anomalies in QFT, which to
a large extent can be treated in the context of external classical perturbations
[20, 32, 10]

10
 The notes also treat only dimension 1+3. Note, however, that related prob-
lems can be considered in other dimensions. Of particular importance is the
case of 1+1 dimension with a large literature, eg. [11, 34]

Acknowledgements The origins of these notes go back to a lecture course

on QED at LMU, Munich, which I prepared at the request of H.Siedentop.
The course, with some modifications, was then repated at IHP, Paris, at the
request of M.Lewin, at the Copenhagen University at the request of J.P.Solovej,
at Warsaw University, and at CRM, Montreal, at the request of V.Jaksic and
R.Seiringer. I am grateful to the colleagues who invited me to give these
courses for their support and advice. Remarks of participants of my courses,
among them of P.Majewski, E.Stockmeier, P.Pickl, D.-A.Deckert, P.Duch and
M.Duch are also appreciated. I profited also from the criticism of E.Langmann,
M.Skowron and anonymous referees.
I am grateful to the experts in quantum field theory, K.Pachucki, I.B.Bialynicki-
Birula, K.Meissner, V.Shabaev and P.Chankowski, who answered my various
questions.
Last but not least, I profited greatly from my long time collaboration with
C.Gérard.
My research was supported in part by the National Science Center (NCN)
grant No. 2011/01/B/ST1/04929.

1 Basic concepts
1.1 Minkowski space
1.1.1 Coordinates in Minkowski space
The coordinates of the Minkowski space R1,3 will be typically denoted by xµ ,
µ = 0, 1, 2, 3. By definition, the Minkowski space is the vector space R4 equipped
with the canonical pseudo-Euclidean form of signature (− + ++)
3
X
gµν xµ xν = −(x0 )2 + (xi )2 .
i=1

(Throughout these notes the velocity of light has the value 1 and we use the
Einstein summation convention). We use metric tensor [gµν ] to lower indices
and its inverse [g µν ] to raise indices:

xµ = gµν xν , xµ = g µν xν .

For a function R1,3 3 x 7→ f (x), we will sometimes use various kind of

notation for partial derivatives:

∂f (x)
= ∂xµ f (x) = ∂µ f (x) = f,µ (x).
∂xµ

11
 Writing R3 we will typically denote the spatial part of the Minkowski space
obtained by setting x0 = 0. If x ∈ R1,3 , then ~x will denote the projection of
x onto R3 . Latin letters i, j, k will sometimes denote the spatial indices of a
vector. Note that xi = xi .
ijk denotes the 3-dimensional Levi-Civita tensor (the fully antisymmetric
tensor satisfying 123 = 1).
~ x) we define its divergence and rotation in
For a vector field R3 3 ~x 7→ A(~
the standard way:
~ = ∂i Ai , (rotA)
divA ~ i = ijk ∂j Ak .

We write ∂~ A
~ as the shorthand for the tensor ∂i Aj , moreover,
~ 2 :=
X
2
∂~ A


∂i Aj .
ij
1,3
On R we have the standard Lebesgue measure denoted dx. The notation
d~x will be used for the Lebesgue measure on R3 ⊂ R1,3 .
We will often write t for x0 = −x0 . The time derivative will be often denoted
by a dot:
∂f (t) ∂f (x0 )
f˙(t) = = ∂t f (t) = = ∂0 f (x0 ) = f,0 (x0 ).
∂t ∂x0
θ(t) will denote the Heaviside function. We set |t|+ := θ(t)|t|.

1.1.2 Causal structure

A nonzero vector x ∈ R1,3 is called
timelike if xµ xµ < 0,
causal if xµ xµ ≤ 0,
lightlike if xµ xµ = 0,
spacelike if xµ xµ > 0.
A causal vector x is called
future oriented if x0 > 0,
past oriented if x0 < 0.
The set of future/past oriented causal vectors is called the future/past light
cone and denoted J ± . We set J := J + ∪ J − .
If O ⊂ R1,3 , its causal shadow is defined as J(O) := O + J. We also define
its future/past shadow J ± (O) := O + J ± .
Let Oi ⊂ R1,3 , i = 1, 2. We will write O1 × O2 iff J(O1 ) ∩ O2 = ∅, or equiv-
alently, O1 ∩ J(O2 ) = ∅. We then say that O1 and O2 are spatially separated.
A function on R1,3 is called space-compact if there exists a compact K ⊂ R1,3
such that suppf ⊂ J(K). It is called future/past space-compact if there exists a
compact K ⊂ R1,3 such that suppf ⊂ J ± (K).
∞
The set of space-compact smooth functions will be denoted Csc (R1,3 ). The
∞
set of future/past space-compact smooth functions will be denoted C±sc (R1,3 ).

12
1.1.3 Invariant measure
Let f be a function on R. The following fact (under appropriate assumptions)
is easy:
Lemma 1.1 Let δ be an approximate delta function, that is
Z
lim δ (t)φ(t)dt = φ(0).
&0

Then Z X φ(si )
lim δ (f (s))φ(s)ds = . (1.1)
&0 |f 0 (si )|
f (si )=0

This suggests the following notation:

X φ(si )
δ(f (s))φ(s)ds = . (1.2)
|f 0 (si )|
f (si )=0

Thus we have
 p   p 
δ k 0 − ~k 2 + m2 δ k 0 + ~k 2 + m2
δ(k 2 + m2 )dk = p d~k + p d~k. (1.3)
2 ~k 2 + m2 2 ~k 2 + m2
To derive (1.3) from (1.2) we fix ~k and use
d(k 2 + m2 )
= 2k 0 ,
dk 0
p
and k 2 + m2 = 0 iff k 0 = ± ~k 2 + m2 .
Now (1.3) is a measure on R1,3 invariant wrt Lorentz transformations. In
fact,
 1 1 
2πiδ(k 2 + m2 ) = lim + , (1.4)
&0 k 2 + m2 − i k 2 + m2 + i
where the rhs is obviously Lorentz invariant.

1.1.4 Fourier transform

The definition of the Fourier transform of R3 3 ~x 7→ f (~x) will be standard:
Z
~
Ff (~k) := e−ik·~x f (~x)d~x.

Often, we will drop F – the name of the variable will indicate whether we use
the position or momentum representation:
Z Z
~ −i~ 1 ~
f (k) = e k·~
x
f (~x)d~x, f (~x) = eik·~x f (~k)d~k.
(2π)3
For the time variable (typically t) we reverse the sign in the Fourier transform:
Z Z
1
iεt
f (ε) = e f (t)dt, f (t) = e−iεt f (ε)dε.
2π

13
1.1.5 Lorentz and Poincaré groups
The pseudo-Euclidean group O(1, 3) is called the full Lorentz group. Its con-
nected component of unity is denoted SO↑ (1, 3) and called the connected Lorentz
group.
The full Lorentz group contains special elements: the time reversal T and
the space inversion (the parity) P and the space-time inversion X := PT:
T(x0 , ~x) = (−x0 , ~x), P(x0 , ~x) = (x0 , −~x), Xx = −x.
It consists of four connected components
SO↑ (1, 3), T·SO↑ (1, 3), P·SO↑ (1, 3), X·SO↑ (1, 3).
O(1, 3) has three subgroups of index two: the special Lorentz group (preserving
the spacetime orientation), the orthochronous Lorentz group (preserving the
forward light cone) and the chiral Lorentz group (preserving the parity):
SO(1, 3) = SO↑ (1, 3) ∪ X·SO↑ (1, 3), (1.5)
↑ ↑ ↑
O (1, 3) = SO (1, 3) ∪ P·SO (1, 3), (1.6)
Ochir (1, 3) = SO↑ (1, 3) ∪ T·SO↑ (1, 3). (1.7)
The affine extension of the full Lorentz group R1,3 o O(1, 3) is called the full
Poincaré group. Its elements will be typically written as (y, Λ). We will often
write y instead of (y, 1l) and Λ instead of (0, Λ). It is the full symmetry group
of the Minkowski space.
Quantum field theory models are often not invariant wrt the full Poincaré
group but one of its subgroups: the connected, special, ortochronous or chiral
Poincaré group, which have the obvious definitions.
Example 1.2 Let us determine O(1, 1). We set
1 1
x+ := x + t, x+ := x − t; x= (x+ + x− ), t= (x+ − x− ).
2 2
 
a b
Now, let A = .
c d
x2 − t2 = x+ x0 = (ax+ + bx− )(cx+ + dx− )
is solved by
ad + bc = 1, ac = 0, bd = 0.
This has 4 types of solutions:
a > 0, d > 0, b = c = 0, (1.8)
a < 0, d < 0, b = c = 0, (1.9)
b > 0, c > 0, a = d = 0, (1.10)
b > 0, c > 0, a = d = 0. (1.11)
Finally, we set    
1 1 −1 a b 1 1
A= .
2 1 1 c d −1 1

14
1.1.6 Double coverings of Lorentz and Poincaré groups
The full Poincaré group or one of its subgroups discussed above is sufficient to
describe spacetime symmetries on the level of observables. On the level of the
Hilbert space one needs to replace it by one of its double coverings.
There exists a unique, up to an isomorphism, connected group Spin↑ (1, 3)
such that the following short exact sequence is true:

1l → Z2 → Spin↑ (1, 3) → SO↑ (1, 3) → 1l. (1.12)

We say that Spin↑ (1, 3) is a connected double covering of SO↑ (1, 3). The group
Spin↑ (1, 3) happens to be isomorphic to SL(2, C). The kernel of the homomor-
phism Spin↑ (1, 3) → SO↑ (1, 3) consists of 1l and −1l (in the notation inherited
from SL(2, C)).
We would like to extend (1.12) to O(1, 3). There are two natural choices
defined by adjoining the elements P̃± , T̃± that cover P , T , and demanding that
they satisfy
P̃2± = ±1l, T̃2± = ±1l, P̃± T̃± = −T̃± P̃± . (1.13)
One obtains the groups P in+ (1, 3) and P in− (1, 3), which satisfy the following
diagram with exact rows and columns commutes:

1l 1l 1l
↓ ↓ ↓
1l → Z2 → Spin↑ (1, 3) → SO↑ (1, 3) → 1l
↓ ↓ ↓
1l → Z2 → P in± (1, 3) → O(1, 3) → 1l, (1.14)
↓ ↓ ↓
1l → Z2 × Z2 → Z2 × Z2 → 1l
↓ ↓
1l 1l

Later on we will need the homomorphism θ : P in± (1, 3) → Z2 = {1, −1},

which is θ(Λ̃) = 1 for orthochronous Λ̃ and θ(Λ̃) = −1 for non-orthochronous
Λ̃. It is called the time orientation.
Clearly, R1,3 o P in± (1, 3) is a double covering of the full Poincaré group. Its
elements will be often written as (y, Λ̃) and then the corresponding element of
R1,3 o O(1, 3) will be denoted by (y, Λ).
Actually, the distinction between P in+ (1, 3) and P in− (1, 3) seems irrele-
vant for practical purposes. One can embed both in a quadruple covering of
O(1, 3), that we call P inext (1, 3). It is generated by P in− (1, 3) and i1l, where
i1l commutes with all elements of P in− (1, 3) and (i1l)2 = −1l. Then by setting
P̃+ := iP̃− and T̃+ := iT̃− , we see that P inext (1, 3) contains also P in+ (1, 3).
(Obviously, one can inerchange the roles of P in− (1, 3) and P in+ (1, 3) in the

15
above construction). We obtain a diagram

1l 1l 1l
↓ ↓ ↓
1l → Z2 → Spin↑ (1, 3) → SO↑ (1, 3) → 1l
↓ ↓ ↓
1l → Z4 → P inext (1, 3) → O(1, 3) → 1l. (1.15)
↓ ↓ ↓
1l → Z2 → Z2 × Z2 × Z2 → Z2 × Z2 → 1l
↓ ↓ ↓
1l 1l 1l

Note that if we set X̃ := P̃+ T̃+ = −P̃− T̃− , then X̃2 = −1l and P̃± , T̃± , X̃
anticommute among themselves.

Remark 1.3 As noted in [51] Sect. 3.10, there exists 8 nonisomporphic groups
that are double coverings extending (1.12), that is groups G such that can be
put in the diagram (1.14) in the place of P in± (1, 3). Indeed, we can demand
independently that the elements P̃, T̃ that cover P, T satisfy

P̃2 = ±1l, T̃2 = ±1l, P̃T̃ = ±T̃P̃ (1.16)

We have 2 · 2 · 2 = 8 possibilities. It seems, however, that the choices leading to

P in± (1, 3) are preferred.

Each of the groups (1.5), (1.6) and (1.7) has two nonisomorphic double
coverings. We always prefer those contained in P inext (1, 3).
We have
1l 1l 1l
↓ ↓ ↓
1l → Z2 → Spin↑ (1, 3) → SO↑ (1, 3) → 1l
↓ ↓ ↓
1l → Z2 → Spin(1, 3) → SO(1, 3) → 1l, (1.17)
↓ ↓ ↓
1l → Z2 → Z2 → 1l
↓ ↓
1l 1l

The group Spin(1, 3) is contained in both P in+ (1, 3) and P in− (1, 3). It is
obtained from Spin↑ (1, 3) by adjoining X̃ satisfying X̃2 = −1l. (The other
double covering of SO(1, 3), obtained by adjoining X̃ satisfying X̃2 = 1l, is not
contained in P inext (1, 3)).
We also have two double coverings of O↑ (1, 3) extending (1.12), one contained

16
in P in+ (1, 3), the other in P in− (1, 3):
1l 1l 1l
↓ ↓ ↓
1l → Z2 → Spin↑ (1, 3) → SO↑ (1, 3) → 1l
↓ ↓ ↓
1l → Z2 → P in↑± (1, 3) → O↑ (1, 3) → 1l, (1.18)
↓ ↓ ↓
1l → Z2 → Z2 → 1l
↓ ↓
1l 1l

P in↑± (1, 3) is obtained by adjoining P̃± satisfying P̃2± = ±1l.

Finally, we have two double coverings of Ochir (1, 3) extending (1.12), one
contained in P in+ (1, 3), the other in P in− (1, 3):
1l 1l 1l
↓ ↓ ↓
1l → Z2 → Spin↑ (1, 3) → SO↑ (1, 3) → 1l
↓ ↓ ↓
1l → Z2 → P inchir
± (1, 3) → O
chir
(1, 3) → 1l, (1.19)
↓ ↓ ↓
1l → Z2 → Z2 → 1l
↓ ↓
1l 1l

P inchir 2
± (1, 3) is obtained by adjoining T̃± satisfying T̃± = ±1l.

1.1.7 Finite dimensional representations of the Lorentz group

Identifying Spin↑ (1, 3) with SL(2, C), we obtain its two basic representations
on C2
SL(2, C) 3 A 7→ A,
SL(2, C) 3 A 7→ A.
Equivalence classes of finite dimensional irreducible representation of Spin↑ (1, 3)
are parametrized by a pair of nonnegative integers (j, k). The representation of
type (j, k) acts on ⊗js C2 ⊗ ⊗ks C2 ' Cj+1 ⊗ Ck+1 and is given by
SL(2, C) 3 A 7→ ⊗js A ⊗ ⊗ks A. (1.20)
(1.20) is sometimes called the representation of spin ( 2j , k2 ).
We can identify the group Spin(1, 3) with
SL(2, C) ∪ iSL(2, C),
so that the element X̃ ∈ Spin(1, 3), which covers X ∈ SO(1, 3), corresponds to
i1l. Under this identification, each representation of the type (j, k) has a unique

17
extension to Spin(1, 3) given by the formula in (1.20). Note in particular that
the representation of X̃ is
ij−k 1lCj+1 ⊗Ck+1 .

1.2 Symplectic dynamics and its quantization

1.2.1 Symplectic form vs Poisson bracket in classical mechanics
In classical mechanics states of a system are described by a phase space, typ-
ically, a manifold Y. The space C ∞ (Y) of smooth functions on Y describes
possible observables.
To define a dynamics one needs to fix a vector field V on Y and one has the
equations of motion
d

ζ(t) = V (ζ(t) . (1.21)
dt
The evolution of an observable F ∈ C ∞ (Y) is then given by

d
F (ζ) = dF (ζ)V (ζ). (1.22)
dt
The phase space Y is typically a symplectic manifold. This means, it is
equipped with a symplectic form ω, that is, a nondegenerate differential 2-form
ω satisfying dω = 0. If y ∈ Y and ζ1 , ζ2 ∈ Ty Y, then we can evaluate ω(ζ1 , ζ2 ).
Actually, it is convenient to write this as

ω(ζ1 , ζ2 ) = ζ1 · ωζ2 , (1.23)

where ω is treated as a linear map from Ty Y to its dual (Ty Y)# . Then we
can define the Poisson bracket, which is a bilinear antisymmetric map C ∞ (Y) ×
C ∞ (Y) → C ∞ (Y)
 
{F, G} := −dF · ω −1 dG = ω ω −1 (dF ), ω −1 (dG) , F, G ∈ C ∞ (Y). (1.24)

The Poisson bracket satisfies the Jacobi identity

{{F, G}, H} + {{G, H}, F } + {{H, F }, G} = 0. (1.25)

One usually assumes that the dynamics is Hamiltonian. This means that
the vector field V (ζ) is given by a function H ∈ C ∞ (Y), called a Hamiltonian,
by
V (ζ) = ω −1 dH. (1.26)
The evolution of observables is given by
d
F (ζ) = {F, H}(ζ). (1.27)
dt
By the Jacobi identity, a Hamiltonian dynamics preserves the Poisson bracket,
and hence also the symplectic form.

18
1.3 Darboux coordinates and symplectic vector spaces
The Darboux Theorem says that on any symplectic manifold locally we can
always choose coordinates, say φi , πj , i = 1, . . . , n, such that

ω = dφi ∧ dπi , (1.28)

To simplify, assume for the moment that we have  one degree of freedom.
0 1
The symplectic form is given by the matrix ω = . Clearly, −ω −1 =
  −1 0
0 1
. Noting that dF = (∂φ F, ∂π F ), dG = (∂φ G, ∂π G), and going back to
−1 0
an arbitrary number of degrees of freedom, we can write

{F, G} = ∂φi F ∂πi G − ∂πi F ∂φi G. (1.29)

This is often encoded by writing

{φi , φj } = {πi , πj } = 0, {φi , πj } = δji . (1.30)

The phase space has often the structure of a vector space and φi , πj , i =
1, . . . , n, can be chosen to be the coordinates in a basis. Then the tangent
space to Ty Y at any point y ∈ Y can be identified with Y itself and the form
ω is simply a nondegenerate antisymmetric bilinear form on Y. The Darboux
Theorem says that we can identify a symplectic manifold with a symplectic
vector space at least locally.
The space Y # of linear functionals on Y obviously is contained in C ∞ (Y). For
a linear functional on Y, its derivative is the original functional itself. Therefore,
(1.28) can be simplified and written as

ω = φi ∧ πi , (1.31)

If dim Y is finite, then Y # = ωY (because ω is nondegenerate) and (??)

determines the Poisson bracket on the whole C ∞ (Y), consistently with (1.28).
In classical field theory, the phase space is infinitely dimensional. Fortu-
nately, it is also a vector space. We will try to implement the above procedure,
ignoring its technical difficulties.

1.4 General concepts of quantum field theory

1.4.1 Quantum mechanics
Pure quantum states are described by normalized vectors in a Hilbert space.
In typical situations the dynamics is generated by a bounded from below self-
adjoint operator called the Hamiltonian. It does not affect any physical pre-
dictions if we subtract from the Hamiltonian the infimum of its spectrum. The
Hamiltonian has often a ground state. The ground state is typically nondegen-
erate.
It will be convenient to formalize these properties.

19
Definition 1.4 We will say that H, H, Ω satisfy the standard requirements of
quantum mechanics (QM) if
(1) H is a Hilbert space;
(2) H is a positive self-adjoint operator on H (called the Hamiltonian);
(3) Ω is a normalized eigenvector of H with eigenvalue 0;
(4) Ω is nondegenerate as an eigenvector of H.

1.4.2 Time reversal

If R is a unitary operator R reversing the time, that is, satisfying

Re−itH R−1 = eitH ,

then RHR−1 = −H. Therefore, if H is positive, then H = 0. Hence unitary

operators are not appropriate for the time reversal invariance.
Following Wigner, by a time reversal operator we will mean an anti-unitary
operator T satisfying
T e−itH T −1 = eitH .
We have then T HT −1 = H, which is compatible with the positivity of H.
Let us review some concepts and notation related to linear and especially
anti-linear operators, motivated by their applications as Wigner’s time reversal.
Consider the complex vector space W = Cn . Let ρ be a linear operator on W.
Then there exists a matrix [ρab ] such that
X
(ρw)a = ρab wb , (1.32)
b

where w = [wa ] ∈ W. We will call [ρab ] the matrix of ρ. Note that it is natural
to denote the operator and its matrix by the same symbol. In particular, the
matrix of the product of linear operators is simply the product of their matrices.
Let κ be an antilinear operator on W. Then there exists a matrix [κab ] such
that X
(κw)a = κab wb , (1.33)
b

where, as usual, the bar denotes the complex conjugation. We will say that
[κab ] is the matrix of κ. Unfortunately, it is dangerous to use the same letter
for an antilinear operator and its matrix, even if we will sometimes do so, as in
(1.33). Note in particular that
X
(κσ)ac = κab σ bc . (1.34)
b

Sometimes it will be convenient to denote linear transformations on W by

L1 (W) instead of the usual L(W). Then antilinear transformations will be
denoted by L−1 (W).

20
 Let G be a group equipped with a homomorphism θ : G → Z2 = {1, −1}. It
yields an obvious partition of G:

G = G1 ∪ G−1 .

We will say that G 3 g 7→ π(g) is a θ-linear representation on W if we have a

pair of maps

G1 3 g 7→ π(g) ∈ L1 (W), (1.35)

G−1 3 g 7→ π(g) ∈ L−1 (W), (1.36)

which together form a representation of G. One can write (1.35) and (1.36)
more compactly:

G3g 7→ π(g) ∈ Lθ(g) (W). (1.37)

Suppose that W is equipped with a scalar product. Sometimes it will be

convenient to denote unitary transformations on W by U1 (W) instead of the
usual U (W). Then anti-unitary transformations will be denoted by U−1 (W).
We say that G 3 g 7→ π(g) is a θ-unitary representation on W if we have a pair
of maps

G1 3 g 7→ π(g) ∈ U1 (W), (1.38)

G−1 3 g 7→ π(g) ∈ U−1 (W), (1.39)

which together form a representation of G. Again, (1.38) and (1.39) can be

written more compactly:

G3g 7→ π(g) ∈ Uθ(g) (W). (1.40)

1.4.3 Relativistic quantum mechanics

Relativistic covariance of a quantum system described by a Hilbert space H
is expressed by choosing a strongly continuous unitary representation of the
double cover of the connected Poincaré group

R1,3 o Spin↑ (1, 3) 3 (y, Λ̃) 7→ U (y, Λ̃) ∈ U (H). (1.41)

We will denote the self-adjoint generator of space-time translations by P =

(P 0 , P~ ). P 0 = H is the Hamiltonian. P~ is called the momentum. Thus
~
U ((t, ~y ), 1l) = e−itH+i~yP .

(We assume that the Planck constant ~ equals 1).

Representations of Spin↑ (1, 3) can be divided into two categories. Integer
spin representations induce a representation of SO↑ (1, 3), and half-integer rep-
resentations do not. The projections
1
1

1l + U (0, −1l) , resp. 1l − U (0, −1l)
2 2

21
project onto the spaces of representations of integer, resp. half-integer spin. We
will write
I := U (0, −1l).
Obviously, U (y, Λ̃)I = IU (y, Λ̃). Anticipating the connection of spin and statis-
tics we will call I the fermionic parity.
Denote the ∗-automorphism defined by U (y, Λ̃) by U(y,Λ̃) :

U(y,Λ̃) (A) := U (y, Λ̃)AU (y, Λ̃)∗ .

Restricted to the commutant of I

{I}0 := {A ∈ B(H) : IA = AI}

U(y,Λ̃) = U(y,−Λ̃) , and thus we obtain a representation of the Poincaré group:

R1,3 o SO↑ (1, 3) 3 (y, Λ) 7→ U(y,Λ) ∈ Aut {I}0 .

Definition 1.5 The following conditions will be called the basic requirements
of relativistic quantum mechanics (RQM):
(1) Existence of a Poincaré invariant vacuum: There exists a (normalized)
vector Ω invariant with respect to R1,3 o Spin↑ (1, 3).
(2) Spectral condition: The joint spectrum of the energy-momentum operator
is contained in the forward light cone, that is, sp(P ) ⊂ J + .
(3) Uniqueness of the vacuum: The vector Ω is unique up to a phase factor.
(4) Integer and half-integer spin states live in separate superselection sectors:
Observables are contained in {I}0 .

Note that conditions (1)-(3) imply the standard requirements of QM.

More precisely, (2) implies H ≥ 0. Conversely, the Poincaré invariance and
the boundedness from below of H implies (2).
(2) implies also that Ω is the ground state of H. (3) implies that this ground
state is unique.
Obviously, IΩ = Ω.

Remark 1.6 Sometimes the expression relativistic quantum mechanics is used

for the theory of relativistic linear hyperbolic equations, such as the Klein-
Gordon and Dirac equation. For the Klein-Gordon equation this is certainly
incorrect. This is a classical equation – in particular, it does not have a nat-
ural interpretation in terms of a unitary dynamics on a Hilbert space. In our
terminology Dirac equation is also a classical equation – its unitary dynamics is
non-physical because the Hamiltonian is unbounded from below.

22
1.4.4 Haag-Kastler axioms for observable algebras
We still need some postulates that express the idea of causality. In the mathe-
matical physics literature one can find two kinds of axioms that try to formalize
this concept: the Haag-Kastler and the Wightman axioms. Even though the
Wightman axioms were formulated earlier, it is more natural to start with the
Haag-Kastler axioms.
Definition 1.7 We keep the basic requirements of RQM.
In addition, to each open bounded set O ⊂ R1,3 we associate a von Neumann
algebra A(O) ⊂ {I}0 . We will say that the family A(O), O open in R1,3 , is a
net of observable algebras satisfying the Haag-Kastler axioms if the following
conditions hold:
(1) Isotony: O1 ⊂ O2 implies A(O1 ) ⊂ A(O2 ).
(2) Poincaré covariance: for (y, Λ̃) ∈ R1,3 o Spin↑ (1, 3), we have



U(y,Λ̃) A(O) = A (y, Λ)O .

(3) Einstein causality: Let O1 × O2 . Then

Ai ∈ A(Oi ), i = 1, 2, implies A1 A2 = A2 A1 .

Self-adjoint elements of the algebras A(O) are supposed to describe observ-

ables in O. This means that in principle an observer contained in O can perturb
the dynamics by a self-adjoint operator from A(O), and only from A(O).
Remark 1.8 One can ask why von Neumann algebras are used in the Haag-
Kastler axioms to describe sets of observables. We would like to argue that it is
a natural choice.
Suppose we weaken the Haag-Kaster axioms as follows: We replace the fam-
ily of von Neumann algebras A(O) by arbitrary sets B(O) of self-adjoint ele-
ments of B(H), and otherwise we keep the axioms unchanged. Then, if we set
A(O) := B(O)00 (which obviously contain B(O)), we obtain a family of von
Neumann algebras satisfying the usual Haag-Kastler axioms. In particular, to
see that the Einstein causality still holds, we use the following easy fact:
Let B1 , B2 , be two ∗-invariant subsets of B(H) such that
A1 ∈ B1 , A 2 ∈ B2 implies A1 A2 = A2 A1 .
Set A1 := B001 , A2 := B002 . Then
A1 ∈ A1 , A2 ∈ A2 implies A1 A2 = A2 A1 .

1.4.5 Haag-Kastler axioms for field algebras

It is often natural to consider nets of algebras containing not only observables,
but also other operators that can be useful to construct observables. They
are called field algebras and satisfy a slightly modified version of Haag-Kastler
axioms.

23
Definition 1.9 We assume the basic requirements of RQM. We say that a fam-
ily of von Neumann algebras F(O) ⊂ B(H) associated to bounded open subsets
O of R1,3 is a net of field algebras in the sense of Haag-Kastler axioms if the
following conditions hold:
(1)’ Isotony: O1 ⊂ O2 implies F(O1 ) ⊂ F(O2 ).
(2)’ Poincaré covariance: for (y, Λ̃) ∈ R1,3 o Spin↑ (1, 3), we have



U(y,Λ̃) F(O) = F (y, Λ)O .

(3)’ Twisted Einstein causality. Let O1 × O2 . Then

Ai ∈ F(Oi ), Ai = (−1)ji IAi I, i = 1, 2, implies A1 A2 = (−1)j1 j2 A2 A1 .
S 
(4)’ Cyclicity: F(O) Ω is dense in H.
O

The main reason for introducing the twisted Einstein causality is the need
to accommodate anticommuting fermionic fields. Clearly, if the net F(O), O ⊂
R1,3 satisfies the Haag-Kastler axioms for field algebras, then the net of their
fermionic even subalgebras
F0 (O) := {B ∈ F(O) : IBI = B}, O ⊂ R1,3 ,
satisfies the Haag-Kastler axioms for observable algebras.
Note that in our formulation the decomposition H = H0 ⊕ H1 given by the
operator I plays a double role.
(1) It describes the decomposition of the Hilbert space into integer and half-
integer spin representations.
(2) In the Einstein causality axiom, block-diagonal operators have the bosonic
character and block-off-diagonal operators have the fermionic character.
A priori it is not obvious that these two properties should give the same decom-
position. However, one can show that it is natural to assume from the beginning
that this is the case. This is the content the theorem about the connection of
the spin and statistics, described eg. in [52].
Setting Λ̃ = −1l in Axiom (2)’ shows that the bosonic/fermionic superselec-
tion rule is local, ie., IF(O)I = F(O) for all O.

1.4.6 Global symmetries

Field algebras can be used to describe global symmetries.
Suppose that a group G has a unitary representation on the Hilbert space
H:
G 3 g 7→ R(g) ∈ U (H)
We assume that R(g), g ∈ G, commute with U (y, Λ̃) and leave invariant Ω. This
implies that I commutes with R(g). Let Rg denote the automorphism defined
by R(g):
Rg (A) := R(g)AR(g)−1 , A ∈ B(H).

24
We define the gauge invariant subalgebras
Fgi (O) = {B ∈ F0 (O) : Rg (B) = B, g ∈ G}
or, equivalently,
Fgi (O) = F0 (O) ∩ {R(g) : g ∈ G}0 .
Then the net O 7→ Fgi (O) satisfies the Haag-Kastler axioms for observable
algebras.

1.4.7 Neutral quantum fields

In practical computations of quantum field theory the information is encoded
in quantum fields. Some of these fields are (formally) Hermitian, and then they
are called neutral fields. Some of them are not – they are usually called charged
fields. We will first consider only neutral fields. Charged fields will be discussed
later.
Neutral fields are typically denoted by R1,3 3 x 7→ φ̂a (x), where a = 1, . . . , n
enumerates the “internal degrees of freedom”, eg. the species of particles and
the value of their spin projected on a distinguished axis. Some of the fields
are bosonic, some are fermionic. They commute or anticommute for spatially
separated points, which is expressed by the commutation/anticommutation re-
lations
[φ̂a (x), φ̂b (y)]± = 0, (x − y)2 > 0.
One can try to interpret neutral quantum fields as “operator valued tempered
distributions”, which become (possibly unbounded) self-adjoint operators when
smeared out with real Schwartz test functions. We can organize the internal
degrees of freedom of neutral fields into a finite dimensional vector space V = Rn .
Thus for any f = (fa ) ∈ S(R1,3 , Rn ) we obtain a smeared out quantum field,
which is the operator
XZ
φ̂[f ] := fa (x)φ̂a (x)dx. (1.42)
a

1.4.8 Wightman axioms for neutral fields

Let us now formulate the Wightman axioms for neutral fields.
Definition 1.10 We assume that the basic requirements of RQM are satisfied.
V is a finite dimensional real vector space equipped with a representation
Spin↑ (1, 3) 3 Λ̃ 7→ σ(Λ̃) ∈ L(V). (1.43)
We have a unique decomposition V = V0 ⊕ V1 . where V0 , resp. V1 is the space
of integer spin, resp. half-integer spin.
We suppose that D is a dense subspace of H containing Ω and we have a
map
S(R1,3 , V) 3 f 7→ φ̂[f ] ∈ L(D) (1.44)
satisfying the following conditions:

25
(1) Continuity: For any Φ, Ψ ∈ D,

S(R1,3 , V) 3 f 7→ (Φ|φ̂[f ]Ψ) (1.45)

is continuous.
(2) Poincaré covariance: for (y, Λ̃) ∈ R1,3 o Spin↑ (1, 3) we have
h i
U(y,Λ̃) φ̂[f ] = φ̂ σ(Λ̃)f ◦ (y, Λ)−1 .

(3) Einstein causality: Let suppf1 × suppf2 , where fi have values in Vji , i =
1, 2. Then
φ̂[f1 ]φ̂[f2 ] = (−1)j1 j2 φ̂[f2 ]φ̂[f1 ].

(4) Cyclicity of the vacuum: Let Falg denote the algebra of polynomials gener-
ated by φ̂[f ]. Then Falg Ω is dense in H.
(5) Hermiticity: For any Φ, Ψ ∈ D,

(Φ|φ̂[f ]Ψ) = (φ̂[f ]Φ|Ψ).

In what follows a map (1.44) satisfying Axiom (1) will be called an operator
valued distribution. By saying that it is cyclic we will mean that it satisfies
Axiom (4).
Setting Λ̃ = −1l in Axiom (2), we see that f ∈ S(R1,3 , Vj ) implies

φ̂[f ] = (−1)j I φ̂[f ]I.

1.4.9 Relationship between Haag-Kastler and Wightman axioms

“Morally”, Wightman axioms are stronger than the Haag-Kastler axioms. In
fact, let Falg (O) be the algebra of polynomials in φ̂[f ] with suppf ⊂ O, which
can be treated as a ∗-subalgebra of L(D). Then the family O 7→ Falg (O) is
almost a net of field algebras and O 7→ Falg 0 (O) is almost a net of observable
algebras in the sense of the Haag-Kastler axioms. Unfortunately, elements of
Falg (O) are defined only on D and not on the whole H, and often do not extend
to bounded operators on H.
We know that the fields φ̂[f ] are Hermitian on D. Suppose they are es-
sentially self-adjoint. Then their closures are self-adjoint operators on H. We
could consider the von Neumann algebra F(O) generated by bounded functions
of φ̂[f ], suppf ⊂ O. Let F0 (O) be its fermionic even part. Then there is still no
guarantee that the net O 7→ F0 (O) satisfies the Haag-Kastler axioms: we are
not sure whether the Einstein causality holds.
To understand this, we recall that there are serious problems with com-
mutation of unbounded operators [41]. One says that two self-adjoint operators
commute (or strongly commute) if all their spectral projections commute. There
exist however examples of pairs of two self-adjoint operators A, B and a sub-
space D ⊂ DomA ∩ DomB with the following property:

26
(1) A and B preserve D and are essentially self-adjoint on D.
(2) A and B commute on D.
(3) A and B do not commute strongly.
(4) D is dense.
More about what is known about the relationship between the Haag-Kastler
and Wightman axioms the reader can find in [2], Sect. 4.9.

1.4.10 Global symmetries in the Wightman formalism

In the Wightman formalism we can encode global symmetries. Let G be a
group acting with the unitary representation U (g) and let Ug the corresponding
∗-automorphism, as described in Subsect. 1.4.6. Suppose in addition that g
acts on V such that Rg (φ̂[f ]) := φ̂[gf ], or in the unsmeared notation

X
Rg φ̂a (x) = gab φ̂b (x),
b

where g commutes with σ(Λ̃).

Rg can be interpreted as a ∗-automorphism of the polynomial algebra Falg .
We set Falg alg
gi (O) to be the subalgebra of fixed points of the action of G on F0 (O).
One could argue that this ∗-algebra should describe observables in O.
Note that what we described is a global symmetry and not a local gauge
invariance. (In the older literature sometimes the former is called the gauge in-
variance of the first kind and the latter the gauge invariance of the second kind).
Satisfactory treatment of local gauge invariance, even Abelian, in the framework
of Wightman axioms seems to be problematic. In fact, a convenient description
of gauge fields seems to require a space with an indefinite scalar product. This
goes beyond the usual Wightman axioms and poses serious technical problems
[56].
Haag-Kastler axioms seem to provide a satisfactory general framework for
quantum field theory on a flat spacetime, also for theories with local gauge in-
variance. Their weakness is the abstractness and great generality. For instance,
we do not see how to recognize that a given family of algebras satisfying Haag-
Kastler axioms corresponds to a theory with local gauge invariance. (There
exists, however, a beautiful theory developed by Doplicher-Haag-Roberts that
allows us to recognize global symmetries.)
Wightman axioms seem more concrete. However, they have flaws. As we
mentioned earlier, they seem to be incompatible with the local gauge invariance.
In any case, both Haag-Kastler and Wightman axioms are useful as guiding
principles for quantum field theory.

1.4.11 Charged fields

Sometimes, instead of Hermitian fields one uses a pair of fields R1,3 3 x 7→
ψ̂a (x), ψ̂a∗ (x), a = 1, . . . , m. We will call them charged fields. One assumes that

27
after smearing with complex test functions
XZ
ψ̂[h] := ha (x)ψ̂a (x)dx,
a
XZ
ψ̂ ∗ [h] := ha (x)ψ̂a∗ (x)dx,
a

one obtains linear operators on D Hermitian conjugate to one another.

One can organize species of charged fields into a complex space W = Cm .
Clearly, for any charged field ψa , by setting
1
φ̂a,R (x) := √ (ψ̂a (x) + ψ̂a∗ (x)),
2
1
φ̂a,I (x) := √ (ψ̂a (x) − ψ̂a∗ (x))
i 2
we obtain a pair of neutral fields. Thus introducing charged fields to the Wight-
man axioms is essentially only a notational change, which, as we will see, is
convenient for describing U (1) symmetries.
1.4.12 Wightman axioms for neutral and charged fields
The modified Wightman axioms that admits both neutral and charged fields are
very similar to the Wightman axioms for neutral fields described in Subsubsect.
1.4.8. It would be boring to state them in full detail. In fact, almost all state-
ments from the Wightman axioms for neutral fields remain a part of the new
axioms. The only exception is Axiom (4) about the cyclicity of the vacuum,
which needs to be replaced by a new one. Below we will list the additional ele-
ments that need to be added. We indicate by (...) the places where appropriate
statements from Subsubsect. 1.4.8 should be inserted.

Definition 1.11 (...) We assume that W is a finite dimensional complex vector

space equipped with a representation

Spin↑ (1, 3) 3 Λ̃ 7→ τ (Λ̃) ∈ L(W). (1.46)

We have a unique decomposition W = W0 ⊕ W1 . where W0 , resp. W1 is the

space of integer spin, resp. half-integer spin.
(...) We have maps

S(R1,3 , W) 3 h 7→ ψ̂[h], ψ̂ ∗ [h] ∈ L(D). (1.47)

(1) Continuity: (...)

S(R1,3 , W) 3 h 7→ (Φ|ψ̂[h]Ψ) (1.48)
is continuous.
(2) Poincaré covariance: (...)
h i
U(y,Λ̃) ψ̂[h] = ψ̂ τ (Λ̃)h ◦ (y, Λ)−1 .

28
(3) Einstein causality: (...) Let supph1 ×supph2 , where hi have values in Wji ,
i = 1, 2. Then (...)

φ̂[f1 ]ψ̂[h2 ] = (−1)j1 j2 ψ̂[h2 ]φ̂[f1 ],

ψ̂[h1 ]ψ̂[h2 ] = (−1)j1 j2 ψ̂[h2 ]ψ̂[h1 ],
ψ̂[h1 ]ψ̂ ∗ [h2 ] = (−1)j1 j2 ψ̂ ∗ [h2 ]ψ̂[h1 ].

(4) Cyclicity of the vacuum: Let Falg denote the algebra of polynomials in φ̂[f ],
ψ̂[h] and ψ̂ ∗ [h]. Then Falg Ω is dense in H.
(5) Hermiticity: (...)
(Φ|ψ̂[h]Ψ) = (ψ̂ ∗ [h]Φ|Ψ).

It will be convenient to reformulate the axiom about the Poincaré invariance

in terms of the unsmeared fields:
X
−1


U(y,Λ̃) φ̂a (x) = σab (Λ̃)φ̂b (Λx + y), (1.49)
b
X
−1


U(y,Λ̃) ψ̂a (x) = τab (Λ̃)ψ̂b (Λx + y). (1.50)
b

1.4.13 U (1) symmetry

Consider the group U (1) = R/2πZ. A global U (1) symmetry is usually encoded
by dividing fields into neutral φ̂ and complex ψ̂. Let Hn be the closed span of
vectors of the form

φ̂[f1 ] · · · φ̂[fk ]ψ̂ ∗ [h1 ] · · · ψ̂ ∗ [hp ]ψ̂[h01 ] · · · ψ̂[h0q ]Ω, n = p − q.

Note that the cyclicity of vacuum implies that the sum of Hn is dense in H.
Assume that Hn are mutally orthogonal, so that we have the decomposition
H = ⊕ Hn . For θ ∈ U (1) we define R(θ) := ⊕ einθ . Clearly, R(θ)Ω = Ω and
n∈Z n∈Z
U (1) 3 θ 7→ R(θ) is a unitary representation commuting with U (y, Λ̃). Let Rθ
be the corresponding ∗-automorphism:

Rθ (A) = R(θ)AR(−θ).

We then have


Rθ φ̂a (x) = φ̂a (x),
e−iθ ψ̂a (x),


Rθ ψ̂a (x) =
Rθ ψ̂a∗ (x) eiθ ψ̂a∗ (x).


=

Thus we have an example of a global symmetry, as in Subsect. 1.4.10.

29
1.4.14 Charge conjugation
Let C be a unitary operator such that CΩ = Ω. Let C be the corresponding
∗-automorphism:
C(A) := CAC −1 .
We say that it is a charge conjugation if it satisfies
X
−1


C φ̂a (x) = αab φ̂b (x), (1.51)
b
X
κ−1 ∗


C ψ̂a (x) = ab ψ̂b (x), (1.52)
b

and hence
X
C ψ̂a∗ (x) κ−1


= ab ψ̂b (x), (1.53)
b

where α ∈ L(V) and κ is an antilinear transformation on W with matrix [κab ].

We may also assume that
α4 = 1l, κ4 = 1l, (1.54)
so that C 4 = 1l.
We have
CRθ = R−θ C,
which is the reason for the name charge conjugation.
Note that C is linear, even though C acts on fields antilinearly.

1.4.15 Parity invariance

Recall that the Wightman axioms involve the connected Lorentz group Spin↑ (1, 3).
In particular, we have representations

Spin↑ (1, 3) 3 Λ̃ 7→ σ(Λ̃) ∈ L(V), (1.55)

↑
Spin (1, 3) 3 Λ̃ 7→ τ (Λ̃) ∈ L(W), (1.56)
1,3 ↑
R o Spin (1, 3) 3 (y, Λ̃) 7→ U (y, Λ̃) ∈ U (H). (1.57)

Choose + or −. Replace the group Spin↑ (1, 3) in the Wightman axioms by

P in↑± (1, 3), so that we have the representations

P in↑± (1, 3) 3 Λ̃ 7→ σ(Λ̃) ∈ L(V), (1.58)

P in↑± (1, 3) 3 Λ̃ 7→ τ (Λ̃) ∈ L(W), (1.59)
R 1,3
o P in↑± (1, 3) 3 (y, Λ̃) 7→ U (y, Λ̃) ∈ U (H). (1.60)

The resulting set of axioms will be called the Wightman axioms of a P -invariant
theory.

30
 In particular, the space inversion (parity) P̃± ∈ P in↑± (1, 3) is represented in
the Hilbert space by the unitary operator P± := U (P̃± ). P± := UP̃± denotes
the corresponding automorphism. It acts on the fields as follows:
X
−1
P± φ̂a (x0 , ~x) = (P̃± )φ̂b (x0 , −~x),


σab
b
X
0 −1
(P̃± )ψ̂b (x0 , −~x).


P± ψ̂a (x , ~x) = τab
b

We have P±2 = ±1l.

Obviously, P± is linear and P± acts on fields linearly.

1.4.16 Time reversal invariance

Choose again + or −. Let us replace the group Spin↑ (1, 3) in the Wightman
axioms by P inchir
± (1, 3). We have now representations

P inchir
± (1, 3) 3 Λ̃ 7→ σ(Λ̃) ∈ L(V), (1.61)
P inchir
± (1, 3) 3 Λ̃ 7→ τ (Λ̃) ∈ Lθ(Λ̃) (W), (1.62)
R1,3 o P inchir
± (1, 3) 3 (y, Λ̃) 7→ U (y, Λ̃) ∈ Uθ(Λ̃) (H). (1.63)

Note that we demand that (1.62) is θ-linear and (1.63) is θ-unitary. We denote
by [τab (Λ̃)] the matrix of τ . The resulting set of axioms will be called the
Wightman axioms of a T -invariant theory.
In particular, the time reversal is implemented by the anti-unitary operator
T± := U (T̃± ). T± := UT̃± denotes the corresponding automorphism. The time
reversal acts on the fields as follows
X
−1
T± φ̂a (x0 , ~x) = (T̃± )φ̂b (−x0 , ~x),


σab
b
X
0 −1
(T̃± )ψ̂b (−x0 , ~x).


T± ψ̂a (x , ~x) = τab
b

We have T±2 = ±1l.

Note that T± is antilinear, but T± acts on fields linearly.

1.4.17 The CPT Theorem

Suppose that we have a theory satisfying the Wightman axioms (without the
P and T invariance). As described in Subsubsection 1.1.7 the representations
(1.55) and (1.57) possess natural extensions

Spin(1, 3) 3 Λ̃ 7→ σ(Λ̃) ∈ L(V), (1.64)

Spin(1, 3) 3 Λ̃ 7→ τ (Λ̃) ∈ L(W). (1.65)

31
Let us stress that (1.65) is linear and not θ-linear! A deep theorem, called the
CP T Theorem, says that we can extend the representation (1.57) to a θ-unitary
representation

R1,3 o Spin(1, 3) 3 (y, Λ̃) 7→ U (y, Λ̃) ∈ Uθ(Λ̃) (H), (1.66)

such that for non-orthochronous Λ̃ ∈ Spin(1, 3) we have

X
−1


U(y,Λ̃) φ̂a (x) = σab (Λ̃)φ̂b (Λx + y), (1.67)
b
X
τ −1 ∗


U(y,Λ̃) ψ̂a (x) = ab (Λ̃)ψ̂b (Λx + y). (1.68)
b

In particular, the spacetime inversion is implemented by the anti-unitary

operator X := U (X̃). X := UX̃ denotes the corresponding automorphism. Then
X
−1


X φ̂a (x) = σab (X̃)φ̂b (−x),
b
X
τ −1 ∗


X ψ̂a (x) = ab (X̃)ψ̂b (−x).
b

Note that X is antilinear and X acts on the fields antilinearly.

1.4.18 The CPT Theorem in a P and T -invariant theory

Suppose we have a theory that satisfies the Wightman axioms with both P and
T invariance. More precisely, let us assume that we have representations

P inext (1, 3) 3 Λ̃ 7→ σ0 (Λ̃) ∈ L(V), (1.69)

P inext (1, 3) 3 Λ̃ 7→ τ0 (Λ̃) ∈ Lθ(Λ̃) (W), (1.70)
R1,3 o P inext (1, 3) 3 (y, Λ̃) 7→ U0 (y, Λ̃) ∈ Uθ(Λ̃) (H). (1.71)

Note that we demand that (1.70) is θ-linear and (1.71) is θ-unitary. We assume
(1.49) and (1.50) for P inext (1, 3). In particular, we have the antilinear operators
σ0 (X̃), τ0 (X̃), and U0 (X̃).
By the CPT Theorem we also have the representations (1.64) and (1.65)
of Spin(1, 3) satisfying (1.67) and (1.68). In particular, we have the operators
σ(X̃), τ (X̃± ), and U (X̃). Note that we put the subscript 0 in (1.69), (1.70) and
(1.71) to distinguish the representations used in Wightman axioms from the
representations obtained by the CPT Theorem.
Define

α := σ(X̃)σ0 (X̃−1 ),
κ := τ (X̃)τ0 (X̃−1 )
C := U (X̃)U0−1 (X̃−1 ).

32
Then C is unitary, and the corresponding automorphism C(A) = CAC −1 satis-
fies (1.51) and (1.52).
If P = U0 (P̃+ ) and T = U0 (T̃+ ), then U0 (X̃) = U0 (P̃+ )U0 (T̃+ ), and hence
X = CP T . This explains the name of the CPT Theorem. (Let us stress,
however, that the theorem holds also if the theory is not P and T invariant, so
that we cannot write X = CP T , as described in the previous subsection).

1.4.19 N -point Wightman and Green’s functions

For simplicity, in this subsubsection we use Wightman axioms for neutral fields.
They allow us to define a multilinear map

S(R1,3 , V) × · · · × S(R1,3 , V)
3 (fN , . . . , f1 ) 7→ (Ω|φ̂[fN ] · · · φ̂[f1 ]Ω) ∈ C, (1.72)

which is separately continuous in its arguments. By the Schwartz Kernel The-

orem [16, 41], (1.72) can be extended to a linear map
Z
1,3 N ⊗N


S (R ) , V 3 F 7→ W (xN , . . . , x1 )F (xN , . . . , x1 )dxN · · · dx1 ,

where R(1,3)N 3 (xN , . . . , x1 ) 7→ W (xN , . . . , x1 ) is a tempered distribution on

R(1,3)N with values in the space dual to V ⊗N , called the N -point Wightman
function, so that (1.72) equals
Z
W (xN , . . . , x1 )fN (x1 ) · · · f1 (x1 )dxN . . . dx1 .

From the point of view of the Wightman axioms, the collection of Wightman
functions WN , N = 0, 1, . . . , contains all the information about a given quantum
field theory. In particular,
 
φ̂[fN ] · · · φ̂[f1 ]Ω|φ̂[gM ] · · · φ̂[g1 ]Ω
Z
= W (y1 , . . . , yN , xM , . . . , x1 )

×f1 (x1 ) · · · fN (xN )gM (yM ) · · · g1 (y1 )dx1 · · · dxN dyM · · · dy1 .

This is expressed in the so called Wightman Reconstruction Theorem [52].

In practical computations Wightman functions are not often used. Much
more frequent are the so-called (time-ordered) Green’s functions. Their formal
definition is as follows:

hφ̂(xN ) · · · φ̂(x1 )i (1.73)

X    
0 0 0 0
:= sgn (σ)θ xσ(N ) − xσ(N −1) · · · θ xσ(2) − xσ(1) W (xσ(N ) , . . . , xσ(1) ),
σ∈SN

where sgna (σ) is the sign of the permutation of the fermionic elements among
N, . . . , 1.

33
 Note that we multiply a distribution with a discontinuous function in (1.73),
which strictly speaking is illegal. Disregarding this problem, Green’s functions
are covariant due to the commutativity/anticommutativity of fields at spacelike
separations.

1.5 General scattering theory

1.5.1 Time ordered exponential
We will often use the formalism of time-dependent Hamiltonians. In this sub-
section we describe the main concepts of this formalism.
Assume that I is a unitary involution. (In applications, I will be the
fermionic parity operator). We call an operator B even, resp. odd, if B = ±IBI.
Such operators will be called of pure parity.
Let t 7→ Bn (t), . . . , B1 (t) be time dependent operators of pure parity. Let
tn , . . . , t1 be pairwise distinct. We define the time-ordered product of Bn (tn ),...,
B1 (t1 ) by

T (Bn (tn ) · · · B1 (t1 )) := sgna (σ)Bσn (tσn ) · · · Bσ1 (tσ1 ),

where (σ1 , . . . , σn ) is the permutation such that tσn ≥ · · · ≥ tσ1 and sgna (σ) is
the sign of this permutation restricted to the odd elements among Bn , . . . , B1 .
Consider a family of self-adjoint operators

t 7→ H(t). (1.74)

We will assume that H(t) are even. For t+ > t− , we define the time-ordered
exponential
!
Z t+
Texp −i H(t)dt (1.75)
t−
∞
X Z Z
:= (−i)n ··· H(tn ) · · · H(t1 )dtn · · · dt1
n=0 t+ ≥tn ≥···≥t1 ≥t−
∞ Z t+ Z t+
X
n 1
= (−i) ··· T (H(tn ) · · · H(t1 )) dtn · · · dt1 .
n=0 t− t− n!

For brevity, we will write U (t+ , t− ) for (1.75) and call it the dynamics generated
by t 7→ H(t). (Of course, if H(t) are unbounded, the above definition should
be viewed only as a heuristic indication how to define the family of unitary
operators U (t+ , t− ). In general, in most of this subsection we are not very
precise about the boundedness of operators, limits, etc.)
We also set U (t− , t+ ) := U (t+ , t− )−1 .
Clearly, if H(t) = H, then U (t+ , t− ) = e−i(t+ −t− )H .

34
1.5.2 Schrödinger and Heisenberg picture
Suppose that H is a (time-independent) Hamiltonian. If we prepare a state ρ at
time 0 and measure an observable A at time t > 0, then the expectation value
of the measurement is
TrρeitH Ae−itH . (1.76)
In quantum physics two equivalent ways of expressing (1.76) are used:
(1) The Schrödinger picture: We let the state evolve ρt := e−itH ρeitH and keep
the observable constant. Then (1.76) equals Trρt A.
(2) The Heisenberg picture: We let the observable evolve At := eitH Ae−itH
and keep the state constant. Then (1.76) equals TrρAt .
(By the Schrödinger picture one also means the unitary evolution Ψt := e−itH Ψ
on H.)

1.5.3 Schrödinger and Heisenberg picture for time-dependent Hamil-

tonians
The above formalism is somewhat more complicated if the Hamiltonian is time-
dependent. Then the Hamiltonian in the Schrödinger picture and the Hamilto-
nian in the Heisenberg picture can be different.
Suppose we fix a time-dependent family of self-adjoint operators t 7→ HSp (t)
called the Hamiltonian in the Schrödinger picture. We assume that the evolution
U (t+ , t− ) is defined by HSp (t):
!
Z t+
U (t+ , t− ) = Texp −i HSp (t)dt . (1.77)
t−

and thus is the solution of the equations

d
U (t+ , t− ) = −iHSp (t+ )U (t+ , t− ), U (t, t) = 1l; (1.78)
dt+
d
or U (t+ , t− ) = U (t+ , t− )iHSp (t− ), U (t, t) = 1l. (1.79)
dt−
The evolution of a density matrix ρ in the Schrödinger picture to time t is

U (t, 0)ρU (0, t). (1.80)

Consequently the evolution of an observable A in the Heisenberg picture is then

AHp (t) := U (0, t)AU (t, 0), (1.81)

where we treat t = 0 as the reference time. Equivalently, AHp (t) is the solution
of
d
AHp (t) = i [HHp (t), AHp (t)] , (1.82)
dt
A(0) = A, (1.83)

35
where the Hamiltonian in the Heisenberg picture is defined as

t 7→ HHp (t) := U (0, t)HSp (t)U (t, 0). (1.84)

Thus a quantum dynamics is described by two time-dependent Hamiltonians:

t 7→ HSp (t) and t 7→ HHp (t). If they do not depend on time, they coincide.
The dynamics is also generated by the Hamiltonian in the Heisenberg pic-
ture, similarly to (1.78) and (1.79), but one of the times has to be the reference
time (in our case 0), and the Hamiltonian appears “on the wrong side”:

d
U (t, 0) = −U (t, 0)iHHp (t); (1.85)
dt
d
U (0, t) = iHHp (t)U (0, t). (1.86)
dt
Above we assumed that the observable A is time-independent in the Schrödinger
picture. It is sometimes useful to be more general and to consider an obsevable
t 7→ ASp (t), which is time-dependent in the Schrödinger picture. Then in the
Heisenberg picture it is defined as

AHp (t) := U (0, t)ASp (t)U (t, 0). (1.87)

Let us note that a similar distinction between two pictures exists in classical
dynamical systems. Consider a flow on Rd given by the equation
d

x(t) = v t, x(t) .
dt
For any initial condition x(0) = x0 ∈ Rd , we obtain a solution R 3 t 7→ x(t, x0 ).
Thus any time dependent observable F has two descriptions:

R × Rd 3 (t, x) 7→ F (t, x), (1.88)

d


R × R 3 (t, x0 ) 7→ F t, x(t, x0 ) . (1.89)

In fluid dynamics, (1.88) is sometimes called the Eulerian description, and (1.89)
the Lagrangian description.
In classical mechanics the phase space is described by coordinates (φ, π) ∈
Rm × Rm . The time evolution is described by the Hamilton equations


φ̇(t) = ∂π H t, φ(t), π(t) ,


π̇(t) = −∂φ H t, φ(t), π(t) .


For any initial condition φ0 , π0 ∈ Rm × Rm , we obtain a solution of the
Hamilton equations. Similarly as in the quantum case, we have two kinds of the
classical Hamiltonian:

R × Rm × Rm 3 (t, φ, π) 7→ H(t, φ, π), (1.90)

m m


R×R ×R 3 (t, φ0 , π0 ) 7→ H t, φ(t, φ0 , π0 ), π(t, φ0 , π0 ) . (1.91)

36
(1.90) is the Hamiltonian in the Eulerian description and (1.91) is the Hamilto-
nian in the Lagrangian description. The former is the analog of the Schrödinger
picture and the latter of the Heisenberg picture. If they do not depend on time,
they coincide.
We will use the classical Hamiltonian in the Lagrangian description and the
quantum Hamiltonian in the Schrödinger picture as the standard ones. Note
that for the time being we are rather fussy about puttimg the subscripts such as
Sp and Hp. In practice they are usually omitted and determined by the context.

1.5.4 Time-dependent perturbations

Our time-dependent Hamiltonians will usually have the form

HSp (t) := Hfr + λVSp (t),

where Hfr is a self-adjoint operator and R 3 t 7→ VSp (t) is a family of self-adjoint

operators.
For any operator A, treated as an observable in the Schrödinger picture, be-
sides the Heisenberg picture wrt. the full dynamics, we also have the Heisenberg
pictures wrt. the free dynamics, called sometimes the interaction picture or the
Furry picture:
Afp (t) := eitHfr Ae−itHfr . (1.92)
Equivalently, Afp (t) is the solution of

d
Afp (t) = i [Hfr , Afp (t)] ,
dt
Afp (0) = A,

We will also use the free Heisenberg picture for time-dependent observables
t 7→ ASp (t):

Afp (t) := eitHfr ASp (t)e−itHfr . (1.93)

We define the evolution in the interaction picture

UInt (t+ , t− ) := eit+ Hfr U (t+ , t− )e−it− Hfr .

Note that
!
Z t+
UInt (t+ , t− ) = Texp −i Vfp (t)dt .
t−

Thus the interaction in the free picture

Vfp (t) = eitHfr V (t)e−itHfr (1.94)

serves as the Hamiltonian for the interaction picture, therefore, (1.94) has an
alternative notation HInt (t).

37
 We define the scattering operator by

S := lim UInt (t+ , t− )

t+ ,−t− →∞
 Z ∞ 
= Texp −i HInt (t)dt . (1.95)
−∞

We also introduce the Møller operators

S− := lim U (0, −t)eitHfr = lim UInt (0, −t)

t→∞ t→∞
 Z 0 
= Texp −i HInt (t)dt , (1.96)
−∞
−itHfr
S+ := lim U (0, t)e = lim UInt (0, t)
t→∞ t→∞
 Z ∞ ∗
= Texp −i HInt (t)dt . (1.97)
0

Clearly, S = S +∗ S − .

1.5.5 Vacuum energy

Assume that Hfr and VSp (t) are even. Let Φfr be a fixed even vector with
Hfr Φfr = 0, which we will call the vacuum. (In our applications, Φfr will be al-
ways the ground state of Hfr .) Let t 7→ Aj,Sp (t), j = 1, . . . , k be time-dependent
operators of fixed parity. We can consider them in the free picture

Aj,fp (t) := eitHfr Aj,Sp (t)e−itHfr . (1.98)

The free time-ordered Green’s functions are defined as

hAk,fp (tk ) · · · A1,fp (t1 )ifp

:= Φfr |T Ak,fp (tk ) · · · A1,fp (t1 ) Φfr . (1.99)

Then we compute

lim (Φfr |U (t+ , t− )Φfr ) (1.100)

t+ →∞, t− →−∞

= (Φfr |SΦfr ) (1.101)

∞
(−iλ)n ∞
X Z Z ∞
= dsn · · · ds1 hVfp (sn ) · · · Vfp (s1 )ifp .
n=0
n! −∞ −∞

1.5.6 Time-ordered Green’s functions

We can consider also the full Heisenberg pictures, where we drop the subscript
Hp:

Aj (t) := U (0, t)Aj,Sp (t)U (t, 0). (1.102)

38
 Suppose that there exist

Φ± := lim U (0, t)Φfr . (1.103)

t→±∞

The interacting time-ordered Green’s functions are defined as

hAk (tk ) · · · A1 (t1 )i

Φ+ |T Ak (tk ) · · · A1 (t1 ) Φ− .



:= (1.104)

The following theorem is a simplified version of a result of Gell-Mann and

Low. It says that we can express interacting Green’s functions by free ones.
∞
(−iλ)n ∞
X Z Z ∞
hAk (tk ) · · · A1 (t1 )i = dsn · · · ds1 (1.105)
n=0
n! −∞ −∞

×hVfp (sn ) · · · Vfp (s1 )Ak,fp (tk ) · · · A1,fp (t1 )ifp .

Let t 7→ f (t) = (f1 (t), . . . , fn (t)) be an ntuple of functions. If Ai is even,

we simply assume that fi (t) has real values, if Ai is odd, the values of fi (t) are
(anticommuting) Grassmann numbers. The generating function is defined as
∞
(−i)N X
X Z XZ
Z(f ) = f (t1 )dt1 · · · f (tN )dtN hAiN (tN ) · · · Ai1 (t1 )i.
N! i i
N =0 1 N

Note that the generating function is the vacuum expectation value of a certain
scattering operator:
Z(f ) = (Φfr |S(f )Φfr ), (1.106)
where S(f ) is the scattering operator (1.95) with λV (t) replaced by
X
λV (t) + fi (t)Ai .
i

1.5.7 Adiabatic switching and the energy shift

In most of our notes we concentrate on time dependent perturbations that decay
sufficiently fast in the past and future. Such perturbations lead to a relatively
simple scattering theory, described in Subsubsection 1.5.4 and 1.5.6.
One would like also to consider the case of time-independent perturbations.
In fact, let V be a (time-independent) self-adjoint perturbation. In this and the
following two subsubsections we are interested in the time-independent Hamil-
tonian H := Hfr + λV .
It is often convenient to extract information about H from the time-dependent
formalism. This can be done by introducing the so called adiabatical switching
invented by Gell-Mann and Low.
In this subsubsection we describe how to compute the energy shift using
adiabatic switching. We follow quite closely [39].

39
 Let  > 0. We define V (t) := e−|t| V . We will write

H (t) := Hfr + λV (t)

for the corresponding time-dependent Hamiltonian. We also introduce the cor-

responding HInt , U (t+ , t− ), UInt (t+ , t− ), S± , S .

Proposition 1.12 We have

iλ∂λ U (t+ , t− ) (1.107)

(
H (t+ )U (t+ , t− ) − U (t+ , t− )H (t− ), 0 ≥ t+ ≥ t− ;
=
−H (t+ )U (t+ , t− ) + U (t+ , t− )H (t− ), t+ ≥ t− ≥ 0;

iλ∂λ UInt (t+ , t− ) (1.108)

(
HInt (t+ )UInt (t+ , t− ) − UInt (t+ , t− )HInt (t− ), 0 ≥ t+ ≥ t− ;
=
−HInt (t+ )UInt (t+ , t− ) + UInt (t+ , t− )HInt (t− ), t+ ≥ t− ≥ 0;

±iλ∂λ S± = HS± − S± Hfr ; (1.109)

iλ∂λ S = Hfr S + S Hfr − 2S+∗ HS− . (1.110)

Proof. Display the dependence on λ by writing U,λ (t+ , t− ). For t+ ≥ t− ≥

0 we have
U,λ (t+ , t− ) = U,λeθ (t+ + θ, t− + θ).
Hence,
d
0 = U θ (t+ + θ, t− + θ)
dθ ,λe θ=0
d d
= λ∂λ U,λ (t+ , t− ) + U,λ (t+ , t− ) + U,λ (t+ , t− )
dt+ dt−
= λ∂λ U,λ (t+ , t− ) − iH,λ (t+ )U,λ (t+ , t− ) + iU,λ (t+ , t− )H,λ (t− ).

This proves (1.107), from which the remaining identities follow. 2

Assume that Φfr is an eigenvector of Hfr with Hfr Φfr = Efr Φfr . Set
S± Φfr
Φ±
 := ,
(Φfr |S± Φfr )
(Φfr |HS± Φfr )
E± := = (Φfr |HΦ±
 ).
(Φfr |S± Φfr )
Proposition 1.13

(H − Efr ± iλ∂λ )S± Φfr = 0, (1.111)

∓iλ∂λ log(Φfr |S± Φfr ) = E± − Efr , (1.112)
H − E± ± iλ∂λ Φ±


 = 0. (1.113)

40
 Proof. Applying (1.109) to Φfr yields (1.111). We scalar multiply it with
Φfr obtaining (1.112). Combining (1.111) with (1.112) gives (1.113). 2
In the following theorem we argue that the adiabatic switching often allows
us to compute an eigenvector of H and its eigenvalue.
Theorem 1.14 (1) Assume that
there exist a nonzero lim Φ±
 . (1.114)
&0

Then there exist

±
EGL := lim E± ,
&0

Φ±
GL := lim |(Φfr |S± Φfr )|Φ±
 .
&0

(2) Supose in addition that

lim λ∂λ Φ±
 = 0. (1.115)
&0

Then
HΦ± ± ±
GL = EGL ΦGL .

±
Proof. (1) The existence of EGL is immediate. Next we note that (Φ± ±
 |Φ ) =
|(Φfr |S± Φfr )|−2 . Hence lim |(Φfr |S± Φfr )| exists. This implies the existence of
&0
Φ±
GL .
By (1.113), we have
±
(H − EGL ) lim Φ±
 = 0.
&0

This implies (2). 2

In the remaining part of this subsubsection we assume that (1.114) and
(1.115) are true.
Suppose now that for small enough λ0 > 0 and |λ| < λ0 , Hλ has a unique
nondegenerate eigenvalue close to Efr , which we denote Eλ , depending contin-
±
uously on λ, with E0 = Efr . If EGL,λ also depends continuously on λ, then we
see that EGL,λ = EGL,λ = Eλ and ΦGL,λ is proportional to Φ−
+ − +
GL,λ . Note that
−
both Φ+ GL,λ and Φ GL,λ are normalized, hence they may differ only by a phase
factor.
In what follows we simply assume that
+ − iα −
E := EGL = EGL , Φ := Φ+
GL = e ΦGL . (1.116)
is true (even if the argument given above does not apply).
Lemma 1.15 Let B be an operator. Then
(S+ Φfr |BS− Φfr )
(Φ|BΦ) = lim . (1.117)
&0 (Φfr |S Φfr )

41
Proof. The right hand side of (1.117) equals
−
(S+ Φfr |BS− Φfr ) (Φ+ −
 |BΦ ) (Φ+
GL |BΦGL )
lim = lim = .
&0 (S+ Φfr |S− Φfr ) + −
&0 (Φ |Φ ) + −
(ΦGL |ΦGL )
2
The following theorem describes the Sucher formula often used in practical
computations of the energy shift.

Theorem 1.16
iλ
E − Efr = lim ∂λ log(Φfr |S Φfr ). (1.118)
&0 2

Proof. We sandwich (1.110) with Φfr and divide with (Φfr |S Φfr ) obtaining

(S+ Φfr |HS− Φfr )

−iλ∂λ log(Φfr |S Φfr ) = 2Efr − 2 .
(S+ Φfr |S− Φfr )

The last term, by Lemma 1.15, converges to 2E. 2

Note that the right hand side of the Sucher formula may have a nonzero
imaginary part. In this case we expect that it describes a resonance close to
Efr .

1.5.8 Adiabatic switching and Green’s functions

Recall that if A is an operator then A(t) = eitH Ae−itH . To define the interacting
Green’s functions we fix a vector Φ, which is a bound state of H, and we set



hAk (tk ) · · · A1 (t1 )i := Φ|T Ak (tk ) · · · A1 (t1 ) Φ .

Suppose that H = Hfr +λV . The Gell-Mann and Low Theorem about Green’s
functions allows us to express interacting Green’s functions by the free ones:

Theorem 1.17 Suppose that (1.114), (1.115) and (1.116) are true, so that we
can apply the results of the previous subsubsection. Then
∞
1 X (−iλ)n
hAk (tk ) · · · A1 (t1 )i = lim (1.119)
&0 (Φfr |S Φfr ) n!
n=0
Z ∞ Z ∞
× dsn · · · ds1 hV (sn ) · · · V (s1 )Ak (tk ) · · · A1 (t1 )ifr ,
−∞ −∞
∞ n Z∞ Z ∞
X (−iλ)
(Φfr |S Φfr ) = lim dsn · · · ds1 (1.120)
&0
n=0
n! −∞ −∞

×hV (sn ) · · · V (s1 )ifr .

42
 Proof. (1.120) follows from (1.95) applied to U .
Let us prove (1.119). Let tk ≥ · · · ≥ t1 . Let Ai, (t) denote the operator Ai
in the Heisenberg picture for the evolution U . The left-hand side of (1.119) is

(Φ|Ak (tk ) · · · A1 (t1 )Φ)

= lim (Φ|Ak, (tk ) · · · A1, (t1 )Φ)
&0

(S+ Φfr |Ak, (tk ) · · · A1, (t1 )S− Φfr )

= lim , (1.121)
&0 (Φfr |S Φfr )
where at the last step we used Lemma 1.15.
Let h· · · i denote Green’s functions for the dynamics U . Then the numerator
of (1.121) can be written as

hAk, (tk ) · · · A1, (t1 )i . (1.122)

Applying (1.105) to (1.122) we arrive at (1.119). 2

Let us note that on the right of (1.119) only the free dynamics appears. One
can forget about the dynamics U , whose use can be treated as a trick. Some
authors consider (1.119) as a (perturbative) definition of Green’s functions, for-
getting about the auxiliary nonphysical dynamics U .

1.5.9 Adiabatic scatttering theory

In our notes we concentrate on time-dependent Hamiltonians, with perturba-
tions decaying fast as |t| → ∞. For such Hamiltonians the definitions of Møller
and scattering operators given in (1.95), (1.96) and (1.97) work well.
One would also like to consider scattering theory of time-independent Hamil-
tonians. Unfortunately, in QFT these definitions typically need to be modified.
In Quantum Mechanics the situation is much better. For (time-independent)
Schrödinger Hamiltonians H := Hfr + V (x), Hfr := −∆, (at least with short-
range potentials) there exists a very satisfactory scattering theory based on the
Møller operators
S ± := s− lim eitH e−itHfr (1.123)
t→±∞

and the scattering operator by S := S +∗ S − .

This approach almost never works in QFT, even for simple-minded models
with classical (time-independent) perturbations considered in these notes. In
particular, the limit (1.123) almost never exists. One of the reasons is the
existence of the ground state for Hfr in quantum field theory. H has essentially
always a different ground state or no ground state at all. It usually has a
different spectrum.
One can try to remedy this problem by introducing adiabatic switching
together with a renormalization of the phase, which is divergent as  & 0,
as in the following teorem:

Theorem 1.18 Suppose that (1.114), (1.115) and (1.116) are true.

43
(1) Assume also that there exist the adiabatic or Gell-Mann–Low Møller oper-
ators
± |(Φfr |S± Φfr )| ±
SGL := w− lim S , (1.124)
&0 (Φfr |S± Φfr )

and
|(Φfr |S± Φfr )| ±
w− lim λ∂λ S = 0. (1.125)
&0 (Φfr |S± Φfr )
Then
± ±
SGL (Hfr − Efr ) = (H − ReE)SGL , (1.126)
±
SGL Φfr = Φ±
GL . (1.127)

(2) Define the adiabatic or Gell-Mann–Low scattering operator

+∗ −
SGL = SGL SGL . (1.128)
Then
SGL Hfr = Hfr SGL ,
SGL Φfr = eiα Φfr .
q
Proof. Using |ff | = f |f | |f | i∂λ f
f , we obtain i∂λ f = − f Re f . Therefore, setting
f := (Φfr |S± Φfr ) we compute
|(Φfr |S± Φfr )| |(Φfr |S± Φfr )| (Φfr |(HS± − S± Hfr )Φfr )
∓iλ∂λ = − Re
(Φfr |S± Φfr ) (Φfr |S± Φfr ) (Φfr |S± Φfr )
|(Φfr |S± Φfr )|
Re E± − Efr .


= − ±
(Φfr |S Φfr )
Therefore,
|(Φfr |S± Φfr )| ±
∓iλ∂λ S
(Φfr |S± Φfr )
|(Φfr |S± Φfr )|
−HS± + S± Hfr + Re(E± − Efr )S± .


= − ±
(Φfr |S Φfr )
Using ReEfr = Efr , we obtain (1.126). (1.127) follows by definition. 2
±
If the limit in (1.125) is strong and not only weak, SGL are unitary. This
will be the case in the examples we consider in our text. In general, they do
not have to be unitary, and one needs to perform an additional wave function
renormalization, which we will not discuss.
Remark 1.19 Many textbooks use (1.123) as the starting point for a derivation
of the rules of QFT, eg. [55]. As indicated above, this is quite far from being
correct. Gell-Mann and Low invented the adiabatic switching as an attempt to
make this derivation somewhat more satisfactory. This approach often works in
QFT models with classical perturbations.

44
2 Neutral scalar bosons
In this section we consider the Klein-Gordon equation

(−2 + m2 )φ(x) = 0 (2.1)

and we quantize the space of its real solutions. We study two kinds of interac-
tions: an external linear source

(−2 + m2 )φ(x) = −j(x), (2.2)

and a mass-like perturbation

(−2 + m2 )φ(x) = −κ(x)φ(x). (2.3)

2.1 Free neutral scalar bosons

2.1.1 Special solutions and Green’s functions
Every function ζ that solves the (homogeneous) Klein-Gordon equation

(−2 + m2 )ζ(x) = 0 (2.4)

can be written as
Z
dk
ζ(x) = eikx g(k)δ(k 2 + m2 )
(2π)3
√
XZ d~k  q  0 ~2 2 ~
= p g ± ~k 2 + m2 , ~k e∓ix k +m +i~xk ,
±
3 ~
(2π) 2 k + m2 2

where g is a function on the two-sheeted hyperboloid k 2 +m2 = 0. A special role

is played by the following 3 special solutions of the homogeneous Klein-Gordon
equation.
(1) The positive frequency or Wightman, resp. negative frequency or anti-
Wightman solution:
Z
dk
D(±) (x) = ±i eikx θ(±k 0 )δ(k 2 + m2 )
(2π)3
√
d~k
Z
0 ~2 2 ~
= ±i p e∓ix k +m +i~xk
(2π)3 2 ~k 2 + m2
1
= sgnx0 δ(x2 )
4π
mθ(−x2 ) ∓sgnx0 p 2 miθ(x2 ) √
± √ H1 (m −x ) ± √ K1 (m x2 ).
8π −x2 4π 2 x2

where H1± are the Hankel functions and K1 is the MacDonald function of
the 1st order.

45
(2) The Pauli-Jordan or the commutator function:
Z
dk
D(x) = i eikx sgn(k 0 )δ(k 2 + m2 )
(2π)3
d~k
Z  q 
x~
i~ k 0 ~2 2
= p e sin x k + m
(2π)3 ~k 2 + m2
1 msgnx0 θ(−x2 ) p
= sgnx0 δ(x2 ) − √ J1 (m −x2 ),
2π 4π −x2
where J1 is the Bessel function of the 1st order. D(x) is the unique solution
of the Klein-Gordon equation satisfying

D(0, ~x) = 0, Ḋ(0, ~x) = δ(~x).

Solutions of
(−2 + m2 )ζ(x) = δ(x), (2.5)
are called Green’s functions or fundamental solutions of the Klein-Gordon equa-
tion. Formally, they can be written as
eikx
Z
dk
(k + m2 ) (2π)4
2

where a prescription how to regularize the singularity has to be given.

In particular, let us introduce the following 3 Green’s functions.
(1) The retarded, resp. advanced Green’s function:

eikx
Z
dk
D± (x) =
(k + m ∓ i0sgnk ) (2π)4
2 2 0

1 mθ(−x2 )θ(±x0 ) p
= θ(±x0 )δ(x2 ) − √ J1 (m −x2 ).
2π 4π −x 2

In the literature, D+ (x) is usually denoted Dret (x) and D− (x) is usually
denoted Dadv (x).
(2) The Feynman(-Stueckelberg) Green’s function:

eikx
Z
dk
DF (x) =
(k + m2 − i0) (2π)4
2

1 mθ(−x2 ) − p 2 miθ(x2 ) √
= δ(x2 ) − √ H1 (m −x ) + √ K1 (m x2 ).
4π 8π −x2 4π 2 x2

The special solutions and Green’s functions introduced above are often called
propagators.

Remark 2.1 Both the Feynman propagator is called causal Green’s function by
Bogoliubov-Shirkov. The Pauli-Jordan bisolution is called the causal propagator
by e.g. Fredenhagen. We will avoid these names.

46
Proposition 2.2 We have suppD± ⊂ J ± and suppD ⊂ J. The propagators
satisfy the following relations

D(x) = D(x) = −D(−x) = D(+) (x) + D(−) (x) (2.6)

+ −
= D (x) − D (x), (2.7)
(+) (−)
D(−) (x) =D (x) = −D (−x), (2.8)
+ − 0
D+ (x) = D (x) = D (−x) = θ(x )D(x), (2.9)
D− (x) = D− (x) = D+ (−x) = −θ(−x0 )D(x), (2.10)
F F 0 (+) 0 (−)
D (x) = D (−x) = θ(x )D (x) − θ(−x )D (x) (2.11)
(+) −
=D (x) + D (x) (2.12)
+ (−)
= D (x) − D (x). (2.13)
1 1
Proof. (2.6) is obvious. (2.7) follows using f −i0 − f +i0 = 2πiδ(f ).
Let us now prove that suppD+ ⊂ J + . By the Lorentz invariance it suffices
to prove that D+ is zero on the lower half-plane. We write

eikx
Z
dk
D+ (x) =
(k + m − i0sgnk 0 ) (2π)4
2 2
0 0 ~
e−ik x +ik~x dk 0 d~k
Z
= .
~k 2 + m2 − (k 0 + i0)2 (2π)4

Next we continuously deform the contour of integration, replacing k 0 by k 0 +iR,

where R ∈ [0, ∞[. We do not cross any singularities of the integrand and note
0 0
that e−ix (k +iR) goes to zero (remember that x0 < 0).
Analogously one proves suppD− ⊂ J − . By (2.7) we obtain suppD ⊂ J.
(2.9) and (2.10) also follows from (2.7) and the support properties of D± .
Let us prove (2.11).

eikx
Z
dk
DF (x) =
~k 2 + m2 − (|k 0 | + i0)2 (2π)4


0 0 ~
e−ik x +ik~x
Z
1
=   dk (2.14)
(2π)4
p p
2 ~k 2 + m2 ~k 2 + m2 − |k 0 | − i0
0 0 ~
e−ik x +ik~x
Z
dk
+ p  . (2.15)
~k 2 + m2 + |k 0 | + i0 (2π)4
p
2 ~k 2 + m2

In (2.15) we can replace i0 with −i0. Then the parts of (2.14) and (2.15) with

47
k 0 < 0 are swapped:
0 0 ~
e−ik x +ik~x
Z
dk
= p p  4
2 ~k 2 + m2 ~k 2 + m2 − k 0 − i0 (2π)
0 0 ~
e−ik x +ik~x
Z
dk
+ p p  4
,
~ 2
2 k +m 2 ~ 2 2 0
k + m + k − i0 (2π)
Z −i√~k2 +m2 x0 +i~k~x ~
0 e dk
= iθ(x ) p 3
~ 2
2 k +m 2 (2π)
√
Z i ~k2 +m2 x0 +i~k~x ~
0 e dk
+iθ(−x ) p 3
2 ~k 2 + m2 (2π)
= θ(x0 )D(+) (x) − θ(−x0 )D(−) (x).

where in the last step we integrate wrt k 0 using

0 0
e−ix k
Z
0

0
dk 0 = 2πie∓ix ε θ(±x0 ).
ε ∓ k − i0

Finally, let us prove (2.12). (2.6) and (2.7) imply D(+) = −D(−) +D+ −D− .
Inserting this into (2.11) we obtain

DF = θ(x0 ) + θ(−x0 ) D(+) − θ(−x0 )D+ + θ(−x0 )D−

= D(+) + D− .

(2.13) is proven analogously. 2

2.1.2 Space of solutions

A space-like subspace of codimension 1 will be called a Cauchy subspace.
Solutions of the Cauchy problem are uniquely parametrized by their Cauchy
data (the value and the normal derivative on a Cauchy surface). They can be
expressed by the Cauchy data with help of the Pauli-Jordan function.

Theorem 2.3 Let ς, ϑ ∈ Cc∞ (R3 ). Then there exists a unique ζ ∈ Csc
∞
(R1,3 )
that solves
(−2 + m2 )ζ = 0 (2.16)
with initial conditions ζ(0, ~x) = ς(~x), ζ̇(0, ~x) = ϑ(~x). It satisfies suppζ ⊂
J(suppς ∪ suppϑ) and is given by
Z Z
ζ(t, ~x) = Ḋ(t, ~x − ~y )ς(~y )d~y + D(t, ~x − ~y )ϑ(~y )d~y . (2.17)
R3 R3

48
 ∞
For ζ1 , ζ2 ∈ Csc (R1,3 ) we define

j µ (x) = j µ (ζ1 , ζ2 , x) := ∂ µ ζ1 (x)ζ2 (x) − ζ1 (x)∂ µ ζ2 (x). (2.18)

We easily check that

∂µ j µ (x) = (2 − m2 )ζ1 (x)ζ2 (x) − ζ1 (x)(2 − m2 )ζ2 (x),

This implies Green’s identity

Z Z
j (t+ , ~x)d~x − j 0 (t− , ~x)d~x
0
(2.19)
Z  
= (−2 + m2 )ζ1 (x)ζ2 (x) − ζ1 (x)(−2 + m2 )ζ2 (x) dx. (2.20)
t− <x0 <t+

Let YKG , resp. CYKG denote the space of real, resp. complex, space-compact
solutions of the Klein-Gordon equation. If ζ1 , ζ2 ∈ CYKG , then

∂µ j µ (x) = 0.

One says that j µ (x) is a conserved 4-current.

The flux of j µ across any Cauchy subspace S does not depend on its choice.
It defines a symplectic form on YKG
Z
ζ1 ωζ2 = j µ (ζ1 , ζ2 , x)dsµ (x)
ZS  
= −ζ̇1 (t, ~x)ζ2 (t, ~x) + ζ1 (t, ~x)ζ̇2 (t, ~x) d~x. (2.21)

Clearly, the form (2.21) is well defined also if only ζ2 ∈ YKG , and ζ1 is a
distributional solution of the Klein-Gordon equation.
The Poincaré group R1,3 o O(1, 3) acts on YKG and CYKG by

r(y,Λ) ζ(x) := ζ (y, Λ)−1 x .

r(y,Λ) are symplectic (preserve the symplectic form) for Λ ∈ O↑ (1, 3), otherwise
they are antisymplectic (change the sign in front of the symplectic form).
The Pauli-Jordan function D can be used to construct solutions of the Klein-
Gordon equation parametrized by space-time functions, which are especially
useful in the axiomatic formulation of QFT.

Proposition 2.4 (1) For any f ∈ Cc∞ (R1,3 , R), D ∗ f ∈ YKG , where
Z
D ∗ f (x) := D(x − y)f (y)dy.

(2) Every element of YKG is of this form.

(3) Z
−(D ∗ f1 )ω(D ∗ f2 ) = f1 (x)D(x − y)f2 (y)dxdy. (2.22)

49
(4) If suppf1 × suppf2 , then

(D ∗ f1 )ω(D ∗ f2 ) = 0.

Proof. Let us prove (2.22). Choose time t later than suppfi , i = 1, 2. Then
we have D ∗ fi = D+ ∗ fi . Now, using Green’s identity (2.20), we obtain

−(D ∗ f1 )ω(D ∗ f2 )
Z  
= (Ḋ+ ∗ f1 )(t, ~x)(D+ ∗ f2 )(t, ~x) − (D+ ∗ f1 )(t, ~x)(Ḋ+ ∗ f2 )(t, ~x) d~x
Z 
= − (2 − m2 )(D+ ∗ f1 )(x)(D+ ∗ f2 )(x)
x0 <t

+(D+ ∗ f1 )(x)(2 − m2 )(D+ ∗ f2 )(x) dx
Z
f1 (x)(D+ ∗ f2 )(x) − (D+ ∗ f1 )(x)f2 (x) dx


=
Z Z
f1 (x)(D+ ∗ f2 )(x) − f1 (x)(D− ∗ f2 )(x) dx =


= f1 (x)(D ∗ f2 )(x)dx.

2
The right hand side of (2.22) is sometimes called the Peierls bracket of f1
and f2 .

2.1.3 Classical fields

More precisely, we can endow the space YKG with the standard topology of
Cc∞ (R3 ) ⊕ Cc∞ (R3 ) given by the initial conditions. The space of real, resp.
# #
complex continuous functionals on YKG will be denoted by YKG , resp. by CYKG .
#
The action of T ∈ CYKG on ζ ∈ YKG will be denoted by hT |ζi, and sometimes
simply by T ζ.
If T ∈ CYKG#
, we define T ∗ ∈ CYKG
#
by

hT ∗ |ζi := hT |ζi, ζ ∈ YKG .

Note that in this context the star does not denote the Hermitian conjugation
(which in our text is the standard meaning of the star).
Let us stress that the space YKG is real, which reflects the fact that in this
section we consider neutral fields. In the section devoted to charged fields the
main role will be played by the complexification of YKG , that is WKG := CYKG .
For x ∈ R1,3 , φ(x), π(x) will denote the functionals on YKG given by

hφ(x)|ζi := ζ(x), hπ(x)|ζi := ζ̇(x).

They are called classical fields. Clearly, for any ζ ∈ YKG we have

(−2 + m2 )hφ(x)|ζi = 0.

50
Thus the equation
(−2 + m2 )φ(x) = 0 (2.23)
is a tautology.
# #−1
On YKG we have the action of the Poincaré group (y, Λ) 7→ r(y,Λ) . Note that

#−1
r(y,Λ) φ(x) = φ(Λx + y).

Clearly, φ̇(x) = π(x) and, by (2.17),

Z Z
φ(t, ~x) = Ḋ(t, ~x − ~y )φ(0, ~y )d~y + D(t, ~x − ~y )π(0, ~y )d~y . (2.24)

By (2.21), the symplectic form can be written as

Z


ζ1 ωζ2 = − hπ(t, ~x)|ζ1 ihφ(t, ~x)|ζ2 i + hφ(t, ~x)|ζ1 ihπ(t, ~x)|ζ2 i d~x,

or more simply, Z
ω= φ(t, ~x) ∧ π(t, ~x)d~x. (2.25)

The conserved 4-current can be written as

jµ (x) = φ(x) ∧ ∂µ φ(x).

By (2.25), the symplectic structure on the space YKG leads to the Poisson
bracket

{φ(t, ~x), φ(t, ~y )} = {π(t, ~x), π(t, ~y )} = 0,

{φ(t, ~x), π(t, ~y )} = δ(~x − ~y ). (2.26)

The relations (2.26) can be viewed as mnemotechnic identities that yield the
correct Poisson bracket for more regular functions, eg. the smeared out fields in
(??) or (2.29) described below. Note that formally φ(t, ~x) and π(t, ~x) generate
the algebra of all functions on YKG .
Using (2.24) we obtain

{φ(x), φ(y)} = D(x − y).

Therefore, the Pauli-Jordan solution is often called the commutator function.

2.1.4 Smeared fields

There are two basic ways to smear fields.
The first is given by the symplectic form. In fact, ω can be treated as a
#
linear map from YKG to YKG , which satisfies

−(ωζ)ρ = ζωρ.

51
 #
Therefore, each element of YKG defines an element of YKG by the pairing given
by the symplectic form. It is convenient to allow complex smearing functions
paired antilinearly. More precisely, for ζ ∈ CYKG we introduce the functional
on YKG given by
hφ((ζ))|ρi := ζωρ, ρ ∈ YKG .
A possible alternative notation for φ((ζ)) is −ωζ or ζω.
Clearly,
Z  
ζω = φ((ζ)) = −ζ̇(t, ~x)φ(t, ~x) + ζ(t, ~x)π(t, ~x) d~x. (2.27)

Note that
{φ((ζ1 )), φ((ζ2 ))} = ζ 1 ωζ 2 . (2.28)
We can also smear fields with space-time functions. For f ∈ Cc∞ (R1,3 , R),
we set Z
φ[f ] := f (x)φ(x)dx.

We have
Z Z
{φ[f1 ], φ[f2 ]} = f1 (x)D(x − y)f2 (y)dxdy. (2.29)

Proposition 2.5 Here is the relationship between the two kinds of smearing:

φ[f ] = φ(( − D ∗ f )). (2.30)

Proof. To see (2.30), write an element of YKG as D ∗ g for some g ∈

Cc∞ (R1,3 , R):
Z
hφ(( − D ∗ f ))|D ∗ gi = (D ∗ f )ω(D ∗ g) = f (x)D(x − y)g(y)dxdy
Z
= f (x)hφ(x)|D ∗ gidx = hφ[f ]|D ∗ gi.

2.1.5 Lagrangian formalism

In classical mechanics we have the Hamiltonian formalism, where the basic
object is the phase space equipped with a symplectic form, and the Lagrangian
formalism, where we start from the configuration space. In classical field theory
we can also use both formalisms.
In this context, the Hamiltonian approach is often called the on-shell for-
malism. This means that the field φ(x) acts on the space of solutions of the
equations of motion. In other words, the field φ(x) that we use in the Hamilto-
nian formalism satisfies the equation (2.23) – one says that it is on-shell.

52
 In the Lagrangian formalism one also uses a classical field, which we will
denote by φ(x), as before. But now, this field is off-shell. This means, we do
not enforce any equation on φ(x). One can interpret φ(x) as the functional on,
say, C ∞ (R1,3 ) or Csc
∞
(R1,3 ) such that hφ(x)|f i := f (x).
In QFT one uses the local Lagrangian density L(x), which is a function of
the field φ(x), ∂µ φ(x) =: φ,µ (x) and of x ∈ R1,3 . The Euler-Lagrange equation
reads then
∂L(x)
∂φ(x) L(x) − ∂µ =0 (2.31)
∂φ,µ (x)
To obtain the Klein-Gordon equation, using φ(x) in the off-shell formalism,
introduce the Lagrangian density
L(x) = − 21 ∂µ φ(x)∂ µ φ(x) − 21 m2 φ(x)2 . (2.32)
The Euler-Lagrange equation yields the Klein-Gordon equation (2.1).
When we go from the Lagrangian to Hamiltonian formalism, we enforce the
on-shell condition, that is, the Euler-Lagrange equation, and we introduce the
variable conjugate to φ(x):
∂L(x)
π(x) := = φ,0 (x) = φ̇(x).
∂φ,0 (x)
Then we express everything in terms of φ(x) and π(x).

2.1.6 Stress-energy tensor

We can also introduce the stress-energy tensor
∂L(x) ,ν
T µν (x) := − φ (x) + g µν L(x) (2.33)
∂φ,µ (x)
1
= ∂ µ φ(x)∂ ν φ(x) − g µν ∂α φ(x)∂ α φ(x) + m2 φ(x)2 .


2
It is easy to check that the stress-energy tensor is conserved for a solution
of the Klein-Gordon equation (on shell)
∂µ T µν (x) = 0.
We express the stress-energy tensor in terms of φ(x) and π(x). Its compo-
nents with the first temporal coordinate are called the Hamiltonian density and
momentum density:
1 ~

2 
H(x) := T 00 (x) = π(x)2 + ∂φ(x) + m2 φ(x)2 ,
2
P i (x) := T 0i (x) = −π(x)∂ i φ(x).
They are examples of quadratic functionals on YKG :
1 ~

2 
hH(x)|ζi = ζ̇(x)2 + ∂ζ(x) + m2 ζ(x)2 ,
2
hP (x)|ζi = −ζ̇(x)∂ i ζ(x).
i

53
 We introduce the (total) Hamiltonian and momentum:
Z Z
µ0
H := T (x)dsµ (x) = H(t, ~x)d~x, (2.34)
ZS Z
P i := T µi (x)dsµ (x) = P i (t, ~x)d~x. (2.35)
S

where S is any Cauchy subspace.

H and P~ are the generators of the time and space translations:

φ̇(x) = {φ(x), H}, π̇(x) = {π(x), H},

~ ~
∂φ(x) = −{φ(x), P~ }, ∂π(x) = −{π(x), P~ }.

The observables H, P 1 , P 2 and P 3 are in involution. (This means that the

Poisson bracket of every pair among these observables vanishes).

2.1.7 Positive and negatve frequency space

p
For ~k ∈ R3 , set ε = ε(~k) := ~k 2 + m2 .
Every ζ ∈ CYKG can be written in a unique way as

ζ = ζ (+) + ζ (−) , (2.36)

where
Z
1 ~ 0 ~
ζ (±) = ζ (±) (k) p q e±i(−ε(k)x +k~x) d~k. (2.37)
(2π)3 2ε(~k)

This gives a decomposition of the space CYKG into two subspaces

(+) (−) (−) (+)
CYKG = WKG ⊕ WKG , WKG = WKG .

We have
Z Z
(+) (+) (−) (−)
iζ1 ωζ2 = ζ1 (k)ζ2 (k)d~k − ζ1 (k)ζ2 (k)d~k (2.38)

(+)
Note that (3.67) restricted to WKG is positive definite. For g1 , g2 ∈ W (+)
we will write
(g1 |g2 ) := ig 1 ωg2 .
The Hilbert space of positive energy solutions is denoted ZKG , and is the
(+)
completion of WKG in this scalar product.
1,3 ↑
R o O (1, 3) leaves ZKG invariant.
(+)
We have a natural identification of YKG with WKG . Indeed, ζ ∈ YKG can
(+)
be projected onto ζ (+) ∈ WKG , as in (2.36)-(2.53). This identification allows us
to define a real scalar product on YKG :
(+) (+)
hζ1 |ζ2 iY := Re(ζ1 |ζ2 ).

54
We can compute explicitly this scalar product:
Z Z
hζ1 |ζ2 iY = ζ̇1 (0, ~x)D(+) (0, ~x − ~y )ζ̇2 (0, ~y )d~xd~y (2.39)
Z Z
+ ζ1 (0, ~x)(−∆~x + m2 )D(+) (0, ~x − ~y )ζ2 (0, ~y )d~xd~y .

2.1.8 Plane wave functionals

Let k := (ε(~k), ~k). k ∈ R1,3 of this form will be called on shell. For k on shell,
we define plane wave functionals a(k), a∗ (k) as functionals on the real space
YKG by
ha(k)|ζi := ζ (+) (k), ha∗ (k)|ζi := ζ (−) (k) = ha(k)|ζi (2.40)
Clearly, from hφ(x)|ζi = ζ(x), we obtain

d~k
Z
eikx a(k) + e−ikx a∗ (k) ,


φ(x) = q
(2π)3 2ε(~k)
p
q
Z d~k ε(~k)
√ eikx a(k) − e−ikx a∗ (k) .


π(x) = p
3
i (2π) 2

After setting x0 = 0, we can invert these relations:

s !
ε(~k)
Z
d~x −i~
k~x i
a(k) = p e φ(0, ~x) + q π(0, ~x) , (2.41)
(2π)3 2
2ε(~k)
s !
Z
d~x ~ ε(~k) i
a∗ (k) = p eik~x φ(0, ~x) − q π(0, ~x) . (2.42)
(2π)3 2
2ε(~k)

We have
{a(k), H} = −iε(~k)a(k), (2.43)
∗
{a (k), H} = iε(~k)a∗ (k); (2.44)
0 ∗ ∗ 0
{a(k), a(k )} = {a (k), a (k )} = 0, (2.45)
∗
{a(k), a (k )} 0
= −iδ(~k − ~k 0 ). (2.46)
a(k), a∗ (k) diagonalize simultaneously the Hamiltonian, momentum and
symplectic form:
Z
H = d~kε(~k)a∗ (k)a(k), (2.47)
Z
P~ = d~k~ka∗ (k)a(k), (2.48)
Z
iω = d~ka∗ (k) ∧ a(k). (2.49)

55
 With ζ1 , ζ2 ∈ YKG , the last identity is the shorthand for
Z  
iζ1 ωζ2 = ha(k)|ζ1 iha(k)|ζ2 i − ha(k)|ζ1 iha(k)|ζ2 i d~k.

2.1.9 Plane waves

Let k be on shell. A positive/negative frequency plane wave |k)/|k) is defined as
1 1
(x|k) = p q eikx , (x|k) = p q e−ikx . (2.50)
(2π)3 2ε(~k) (2π)3 2ε(~k)

Note that negative frequency plane waves in the neutral case play secondary
role.
Following Dirac, we denote plane waves using the “ket notation” |k) when
they appear on the right of a bilinear form. We also write (x|k) for the evaluation
of |k) at the point x ∈ R1,3 .
If a plane wave appears on the left, we employ the “bra notation”, which
implies an additional complex conjugation:
1 1
(k|x) = p q e−ikx , (k|x) = p q eikx .
(2π)3 2ε(~k) (2π)3 2ε(~k)

For k = (ε(~k), ~k), k 0 = (ε(~k 0 ), ~k 0 ),

i(k|ω|k 0 ) = i(k|ω|k 0 ) = 0,
−i(k|ω|k 0 ) = i(k|ω|k ) 0
= δ(~k − ~k 0 ). (2.51)
Thus plane waves diagonalize the symplectic form.
Plane waves are solutions of the Klein-Gordon equation. They are not space
compact, however by forming packets of plane waves we can obtain space com-
pact solutions. Indeed, (2.53) can be written as
Z
ζ (+) = ζ (+) (k)|k)d~k, (2.52)
Z
ζ (−) = ζ (−) (k)|k)d~k. (2.53)

The scalar product on W (+) can be written as

Z
(g1 |g2 ) := (k|g1 )(k|g2 )d~k. (2.54)

#
Recall that if ζ ∈ YKG , then it determines ζω, an element of YKG . We can
∗
interpret the plane wave functionals a(k), a (k) as linear functionals on the real
space YKG
a(k)ζ := i(k|ωζ, (2.55)
∗
a (k)ζ := −i(k|ωζ. (2.56)

56
In other words,

a(k) = iφ((|k)))
Z  
= i ∂t (k|0, ~x)φ(0, ~x) − (k|0, ~x)π(0, ~x) d~x,

a∗ (k) = −iφ((|k)))
Z  
= −i ∂t (k|0, ~x)φ(0, ~x) − (k|0, ~x)π(0, ~x) d~x.

Z  
φ(x) = (x|k)a(k) + (x|k)a∗ (k) d~k.

2.1.10 Quantization
Let us describe the quantization of the Klein-Gordon equation, following the
formalism of quantization of neutral bosonic systems [15]. We will use the
“hat” to denote the quantized objects.
We want to construct H, Ĥ, Ω satisfying the standard requirements of QM
(1)-(3) and a self-adjoint operator valued distribution

R1,3 3 x 7→ φ̂(x), (2.57)

˙
such that, with π̂(x) := φ̂(x),
(1) (−2 + m2 )φ̂(x) = 0,
(2) [φ̂(0, ~x), φ̂(0, ~y )] = [π̂(0, ~x), π̂(0, ~y )] = 0,
[φ̂(0, ~x), π̂(0, ~y )] = iδ(~x − ~y ).
(3) eitĤ φ̂(x0 , ~x)e−itĤ = φ̂(x0 + t, ~x).
(4) Ω is cyclic for φ̂(x).
The above problem has a solution, which is essentially unique. Indeed, let
H, Ĥ, Ω, R1,3 3 x 7→ φ̂(x), π̂(x) solve the above problem. Decorating (2.41) and
(2.42) with hats leads to the definitions of two operator valued distributions
Hermitian conjugate to one another:
s !
Z
d~x ~ ε( ~k) i
â(k) : = p e−ik~x φ̂(0, ~x) + q π̂(0, ~x) , (2.58)
(2π)3 2
2ε(~k)
s !
ε(~k)
Z
∗ d~x i~
k~x i
â (k) = p e φ̂(0, ~x) − q π̂(0, ~x) . (2.59)
(2π)3 2
2ε(~k)

57
Using (2) and (3) we obtain the quantized versions of (2.43)-(2.46):

[â(k), Ĥ] = ε(~k)â(k), (2.60)

∗
[â (k), Ĥ] = −ε(~k)â∗ (k), (2.61)
0 ∗ ∗ 0
[â(k), â(k )] = [â (k), â (k )] = 0, (2.62)
∗
[â(k), â (k )]0
= δ(~k − ~k 0 ). (2.63)

ĤΩ = 0 implies Ĥâ(k)Ω = −ε(~k)â(k)Ω. But Ĥ ≥ 0. Thus we should

assume
â(k)Ω = 0. (2.64)
By (4), Ω is cyclic for â(k) and â∗ (k). Using (2.62), (2.63) and (2.64) we see
that Ω is cyclic just for â∗ (k). In other words, H is spanned by vectors of the
form Z
Ψ(~k1 , . . . , ~kn )â∗ (k1 ) · · · â∗ (kn )Ωd~k1 · · · d~kn .

Using again (2.60), (2.63) and (2.64) we see that the scalar product of two
vectors Ψ, Ψ0 is zero if n 6= n0 , and otherwise it is
Z
(Ψ|Ψ0 ) = n! Ψ(~k1 , . . . , ~kn )Ψ0 (~k1 , . . . , ~kn )d~k1 · · · d~kn .

Therefore, H can be identified with Γs L2 (R3 ) , Ω with the Fock vacuum, â∗ (k)

with the creation operators in the “physicist’s notation”, the quantum field is
d~k
Z
eikx â(k) + e−ikx â∗ (k) .


φ̂(x) := q
(2π)3 2ε(~k)
p

The quantum Hamiltonian is a quadratic expression in fields, therefore, putting

hats on the classical formula (2.34) and (2.47) gives expressions differing by an
(infinite) constant due to the ordering problem. We choose (2.47), which gives
a well-defined operator, and also has a ground state of zero energy.
In the case of momentum, putting hats on both (2.35) and (2.48) yields the
same quatization. Thus the quantum Hamiltonian and momentum are
Z
Ĥ := â∗ (k)â(k)ε(~k)d~k,
Z
~
P̂ := â∗ (k)â(k)~kd~k.

By (2.54) we can identify L2 (R3 ) with the positive frequency Hilbert space
ZKG . Using the “mathematician’s notation” on the right we can write
â∗ (k) = â∗ |k) .


(2.65)

 Note thatthe whole R

1,3
oO↑ (1, 3) is unitarily implemented on H by U (y, Λ) :=
Γ r(y,Λ) :
ZKG
U (y, Λ)φ̂(x)U (y, Λ)∗ = φ̂ (y, Λ)x .

58
This is true even though we only required that time translations are imple-
mented.
We have
[φ̂(x), φ̂(y)] = −iD(x − y).
For f ∈ Cc∞ (R1,3 , R) set
Z
φ̂[f ] := f (x)φ̂(x)dx. (2.66)

(2.66) satisfy the Wightman axioms with D := Γfin

s (ZKG ).
For an open set O ⊂ Rd we set

A(O) := {exp(iφ̂[f ]) : f ∈ Cc∞ (O, R)}00 .

The algebras A(O) satisfy the Haag-Kastler axioms.

2.1.11 Quantization in terms of smeared fields

There exists an alternative equivalent formulation of the quantization program,
which uses smeared fields instead of point fields, which may better appeal to
some people.
Again, we want to construct H, Ĥ, Ω satisfying the standard requirements
of QM (1)-(3). Instead of (2.57) we look for a linear function

YKG 3 ζ 7→ φ̂((ζ))

with values in self-adjoint operators such that

(1)
[φ̂((ζ1 )), φ̂((ζ2 ))] = iζ1 ωζ2 . (2.67)

(2)
φ̂((r(t,~0) ζ)) = eitĤ φ̂((ζ))e−itĤ .

(3) Ω is cyclic for the algebra generated by φ̂((ζ)).

One can pass between these two versions of the quantization by
Z  
φ̂((ζ)) = −ζ̇(t, ~x)φ̂(t, ~x) + ζ(t, ~x)π̂(t, ~x) d~x. (2.68)

2.1.12 Quantization in terms of C ∗ -algebras

Let us mention yet another equivalent approach to quantization, using the lan-
guage of C ∗ -algebras.
Let CCR(YKG ) denote the (Weyl) C ∗ -algebra of the CCR over YKG . By
definition, it is generated by W (ζ), ζ ∈ YKG , such that
ζ1 ωζ2
W (ζ1 )W (ζ2 ) = e−i 2 W (ζ1 + ζ2 ), W (ζ)∗ = W (−ζ).

59
R1,3 o O↑ (1, 3) acts on CCR(YKG ) by ∗-automorphisms defined by


r̂(y,Λ) (W (ζ)) := W r(y,Λ) (ζ) .
We are looking for a cyclic representation of this algebra with the time evolution
generated by a positive Hamiltonian.
The solution is provided by the state on CCR(YKG ) defined by

 1 
ψ W (ζ) = exp − hζ|ζiY .
2
Let (Hψ , πψ , Ωψ ) be the GNS representation generated by the state ψ. Then this
representation has the required properties. Hψ can be identified with Γs (ZKG )
and the fields are related to the Weyl operators by
πψ (W (ζ)) = eiφ̂((ζ)) .

2.1.13 Two-point functions

Note the identities
(Ω|φ̂(x)φ̂(y)Ω) = D(+) (x − y), (2.69)
F
(Ω|T(φ̂(x)φ̂(y))Ω) = −iD (x − y). (2.70)
In fact,
d~kd~k 0
Z Z
0
(Ω|φ̂(x)φ̂(y)Ω) = √ √ eikx−ik y (Ω|â(k)â∗ (k 0 )Ω)
(2π)3 2ε 2ε0
d~k
Z
= eik(x−y)
(2π)3 2ε(~k)
= D(+) (x − y);
(Ω|T(φ̂(x)φ̂(y))Ω) = θ(x0 − y 0 )(Ω|φ̂(x)φ̂(y)Ω) + θ(y 0 − x0 )(Ω|φ̂(y)φ̂(x)Ω)
= −iθ(x0 − y 0 )D(+) (x − y) − iθ(y 0 − x0 )D(+) (y − x)
= −iDF (x − y).
(2.69) implies the following identities for spacetime smeared fields and Weyl
operators:
Z Z
(Ω|φ̂[f ]2 Ω) = f (x)D(+) (x − y)f (y)dxdy, (2.71)
 Z Z 
1
(Ω|eiφ̂[f ] Ω) = exp − f (x)D(+) (x − y)f (y)dxdy . (2.72)
2
Differentiating if needed (2.69) with respect time we obtain the equal time
correlation functions expressed as real symmetric kernels:
(Ω|φ̂(0, ~x)φ̂(0, ~y )Ω) = D(+) (0, ~x − ~y ), (2.73)
(Ω|φ̂(0, ~x)π̂(0, ~y )Ω) = 0, (2.74)
(Ω|π̂(0, ~x)π̂(0, ~y )Ω) = −∂t2 D(+) (0, ~x − ~y )
= (−∆~x + m2 )D(+) (0, ~x − ~y ). (2.75)

60
This yields the identities for spatially smeared fields and Weyl operators, where
the scalar product h·|·iY on YKG was introduced in (2.39):
Z Z
2
(Ω|φ̂((ζ)) Ω) = ζ̇(0, ~x)D(+) (0, ~x − ~y )ζ̇(0, ~y )d~xd~y
Z Z
ζ(0, ~x)(−∆~x + m2 )D(+) (0, ~x − ~y )ζ(0, ~y )d~xd~y

= hζ|ζiY , (2.76)
1
(Ω|eiφ̂((ζ)) Ω)


= exp − hζ|ζiY . (2.77)
2

2.2 Neutral scalar bosons with a linear source

2.2.1 Classical fields
We go back to the classical theory. The fields studied in the previous subsection
will be called free fields. We change slightly the notation: free classical fields
will be now denoted by φfr (x), πfr (x). Clearly, they satisfy

(−2 + m2 )φfr (x) = 0, (2.78)

πfr (x) = φ̇fr (x).

Fix a function
R1,3 3 x 7→ j(x) ∈ R, (2.79)
which will be called the (external) linear source. In most of this subsection we
will assume that (2.79) is Schwartz. The interacting fields satisfy the equation

(−2 + m2 )φ(x) = −j(x), (2.80)

π(x) = φ̇(x). (2.81)

We also require that the interacting fields have the same equal-time Poisson
brackets as the free fields:

{φ(t, ~x), φ(t, ~y )} = {π(t, ~x), π(t, ~y )} = 0,

{φ(t, ~x), π(t, ~y )} = δ(~x − ~y ). (2.82)

There are several, usually equivalent, ways to introduce interacting fields.

One way is to treat them as functionals on the space of solutions to the free
Klein-Gordon equation, YKG . We can demand in addition that

φ(~x) := φfr (0, ~x) = φ(0, ~x),

π(~x) := πfr (0, ~x) = π(0, ~x). (2.83)

This condition determines the field φ(x) uniquely:

φ(x) := φfr (x) (2.84)

Z
D+ (x − y)θ(y 0 ) + D− (x − y)θ(−y 0 ) j(y)dy.


+

61
 Let us mention some alternative ways to define the interacting fields φ(x).
First of all, there is nothing special about the time t = 0 in (2.83) – we can
replace it with any t = t0 . Alternatively, we can demand
lim (φfr (t, ~x) − φ(t, ~x)) = 0, lim (πfr (t, ~x) − π(t, ~x)) = 0,
t→∞ t→∞
or lim (φfr (t, ~x) − φ(t, ~x)) = 0, lim (πfr (t, ~x) − π(t, ~x)) = 0.
t→−∞ t→−∞

Another possibility is to introduce YKG (j), the space of smooth real space-
compact solutions of
(−2 + m2 )ζ(x) = −j(x), (2.85)
and define φ(x) by
hφ(x)|ζi := ζ(x), ζ ∈ YKG (j).

2.2.2 Lagrangian and Hamiltonian formalism

We can obtain the equations (2.80) as the Euler-Lagrange equations for the
Lagrangian density
L(x) = − 21 ∂µ φ(x)∂ µ φ(x) − 21 m2 φ(x)2 − j(x)φ(x). (2.86)
The conjugate variable is
∂L
π(x) := = ∂0 φ(x),
∂φ,0 (x)
just as in the free case.
The Legendre transformation leads to the Hamiltonian density
1 ~

2 
H(x) := π(x)2 + ∂φ(x) + m2 φ(x)2 + j(x)φ(x).
2
and the (time-dependent) Hamiltonian
Z
H(t) = H(t, ~x)d~x (2.87)

2 m2
Z 1 1 ~ 
= d~x π(t, ~x)2 + ∂φ(t, ~x) + φ(t, ~x)2 + j(t, ~x)φ(t, ~x) .
2 2 2
The Hamiltonian generates the dynamics:
φ̇(t, ~x) = {φ(t, ~x), H(t)}, π̇(t, ~x) = {π(t, ~x), H(t)}. (2.88)
In (2.87) and (2.88), H(t), φ(t, ~x) and π(t, ~x) should be understood as the
functions of the initial conditions at t = 0. Therefore, we use the Lagrangian
description (see Subsubsect. 1.5.2).
We can also introduce the Hamiltonian in the Eulerian description, which
is convenient for quantization. It uses the fields φ(~x) and π(~x) introduced in
(2.83):

2 m 2
Z 1 1 ~ 
HEul (t) = d~x π(~x)2 + ∂φ(~ x) + φ(~x)2 + j(t, ~x)φ(~x) .
2 2 2

62
2.2.3 Quantization
We will use the notation φ̂fr (x) for the free quantum fields studied in the previous
subsection. We are now looking for interacting quantum fields φ̂(x) satisfying
(−2 + m2 )φ̂(x) = −j(x). (2.89)
We also set
˙
π̂(x) := φ̂(x) (2.90)
and require the equal time commutation relations
[φ̂(t, ~x), φ̂(t, ~y )] = [π̂(t, ~x), π̂(t, ~y )] = 0,
[φ̂(t, ~x), π̂(t, ~y )] = iδ(~x − ~y ). (2.91)
We would like to solve (2.89) and (2.91) in terms of free fields. That means,
we are looking for φ̂(x) on the Hilbert space of the free Klein-Gordon fields,
Γs (ZKG ). We will in addition demand that the interacting and free fields at
time t = 0 coincide:
φ̂(~x) : = φ̂(0, ~x) = φ̂fr (0, ~x),
π̂(~x) : = π̂(0, ~x) = π̂fr (0, ~x). (2.92)
Clearly, the unique solution is obtained by decorating (2.84) with hats:
φ̂(x) := φ̂fr (x)
Z  
+ D+ (x − y)θ(y 0 ) + D− (x − y)θ(−y 0 ) j(y)dy. (2.93)

It can be written as
 Z 0   Z t 
φ̂(t, ~x) = Texp −i Ĥ(s)ds φ̂(0, ~x)Texp −i Ĥ(s)ds , (2.94)
t 0

where the Schrödinger picture Hamiltonian is

m2 2
Z  
1 1
Ĥ(t) := d~x : π̂ 2 (~x) + ∂i φ̂(~x)∂i φ̂(~x) + φ̂ (~x) + j(t, ~x)φ̂(~x) :. (2.95)
2 2 2
Note that Ĥ(t) is obtained from HEul (t) by the Wick quantization A.1.3, which
is expressed by decorating the fields with “hats” and putting the “double dots”.
In principle, one could replace Ĥ(t) by Ĥ(t) + C(t) for any real function
t 7→ C(t). The choice that we made satisfies
(Ω|Ĥ(t)Ω) = 0, t ∈ R. (2.96)
Condition (2.96) is quite arbitrary – the vector Ω is the ground state of the
free Hamiltonian at time zero – in particular, it depends on the choice t = 0 in
(2.92).
We also have the interaction picture Hamiltonian
Z
ĤInt (t) = j(t, ~x)φ̂fr (t, ~x)d~x. (2.97)

63
2.2.4 Operator valued source
So far we assumed that j(x) is a c-number. Most of the formalism works, at
least formally, for operator valued sources. The main additional difficulty is the
need to distinguish between the source in various pictures.
Let us start with the Schrödinger picture. Let R1,3 3 x 7→ ĵ(x) be an
operator-valued function (or distribution) that commutes with time zero fields:

[φ̂(~x), ĵ(t, ~y )] = [π̂(~x), ĵ(t, ~y )] = 0, ~x, ~y ∈ R3 , t ∈ R.

Define the Schrödinger picture Hamiltonian Ĥ(t) by (2.95), where j(x) is re-
placed by ĵ(x).
Then we define the Heisenberg picture fields φ̂(x), π̂(x), as in (2.94). We
also have the source in the Heisenberg picture
 Z 0   Z t 
ĵHP (t, ~x) := Texp −i Ĥ(s)ds ĵ(t, ~x)Texp −i Ĥ(s)ds
t 0

having the commutation relations

[φ̂(t, ~x), ĵHP (t, ~y )] = [π̂(t, ~x), ĵHP (t, ~y )] = 0.
The Klein-Gordon equation (2.89) and the relation (2.93) generalize:

(−2 + m2 )φ̂(x) = −ĵHP (x), (2.98)

φ̂(x) := φ̂fr (x)
Z  
+ D+ (x − y)θ(y 0 ) + D− (x − y)θ(−y 0 ) ĵHP (y)dy.

We can also introduce the source in the interaction picture

ĵInt (t, ~x) := eitĤfr ĵ(t, ~x)e−itĤfr ,

satisfying the commutation relations
[φ̂fr (t, ~x), ĵInt (t, ~y )] = [π̂fr (t, ~x), ĵInt (t, ~y )] = 0, ~x, ~y ∈ R3 , t ∈ R.
The interaction picture Hamiltonian is
Z
ĤInt (t) = ĵInt (t, ~x)φ̂fr (t, ~x)d~x,

which is obtained from (2.97) by replacing j(t, ~x) with ĵInt (t, ~x).

2.2.5 Scattering operator

We go back to a c-number source j(x). The interaction picture Hamiltonian
written in terms of creation and annihilation operators equals

d~k
Z  
−itε(~
k) ~k)â(k) + eitε(~k) j(t, ~k)â∗ (k) .
ĤInt (t) = q e j(t,
(2π)3 2ε(~k)
p

64
The scattering operator (1.95) can be computed exactly. On the level of creation
and annihilation operators it acts as

j ε(~k), ~k


∗ ∗ ∗
Ŝâ (k)Ŝ = â (k) + i p q , (2.99)
(2π)3 2ε(~k)
j ε(~k), ~k


∗
Ŝâ(k)Ŝ = â(k) − i p q . (2.100)
(2π)3 2ε(~k)

We have an explicit formula:

|j(k)|2
 Z 
i dk
Ŝ = exp (2.101)
2 (k 2 + m2 − i0) (2π)4
   
Z
j(ε( ~k), ~k) d~k Z
j(ε(~k), ~k) d~k
× exp −i q â∗ (k) p  exp −i q â(k) p .
(2π)3 (2π) 3
~
2ε(k) ~
2ε(k)
~ j(t,~
k)
To see this, we insert f (t, ~k) := −eitε(k) √ √ into (A.10). In particular,
(2π)3 2ε(~
k)
the exponent of (A.10) becomes:
~
ei(t2 −t1 )ε(k) θ(t1 − t2 )j(t1 , ~k)j(t2 , ~k)dt1 dt2 d~k
Z Z
−
2ε(~k) (2π)3
|j(k 0 , ~k)|2 dk 0 d~k dk 0 d~k
Z Z
= i
2ε(~k) ε(~k) − k 0 − i0 2π (2π)3

|j(k)|2 
Z Z
i 1 1  dk
= +
2 2ε(~k) (ε(~k) − k 0 − i0) (ε(~k) + k 0 − i0) (2π)4
|j(k)|2
Z Z
i dk
= .
2 ε(~k)2 − (k 0 )2 − i0 (2π)4

Note that we used j(k 0 , ~k) = j(−k 0 , −~k), ε(~k) = ε(−~k)

For distinct k1 , . . . , kn on shell, set
|kn , . . . , k1 ) := â∗ (kn ) · · · â∗ (k1 )Ω.
Matrix elements of the scattering operator between such vectors are called scat-
tering amplitudes:
 
k1+ , . . . , kn++ | Ŝ |kn−− , . . . , k1− (2.102)
+ −
|j(k)|2 (−i)n +n
 Z 
i dk
= exp
(k 2 + m2 − i0) (2π)4
p
2 (2π)3(n+ +n− )
j(ε(~k + ), ~k1+ ) j(ε(~k ++ ), ~kn++ ) j(ε(~kn−− ), ~kn−− ) j(ε(~k − ), ~k1− )
× q 1 ··· q n q ··· q 1 .
2ε(~k1+ ) 2ε(~kn++ ) 2ε(~kn−− ) 2ε(~k1− )

65
2.2.6 Green’s functions
Recall that the N -point Green’s function is defined for xN , . . . , x1 as follows:

hφ̂(xN ) · · · φ̂(x1 )i


:= Ω+ |T φ̂(xN ) · · · · · · φ̂(x1 ) Ω− , (2.103)

where
 Z 0 
Ω± := lim Texp −i Ĥ(s)ds Ω
t→±∞ t
 Z 0 
= Texp −i ĤInt (s)ds Ω.
±∞

and the fields φ̂(x) are in the Heisenberg picture:

 Z 0   Z t 
φ̂(t, ~x) = Texp − i Ĥ(s)ds φ̂(~x)Texp − i Ĥ(s)ds . (2.104)
t 0

One can organize Green’s functions in terms of the generating function:

∞ Z
(−i)N
X Z
Z(f ) = ··· hφ̂(xN ) · · · φ̂(x1 )if (xN ) · · · f (x1 )dxN · · · dx1
N!
N =0
  Z ∞ Z   
+ −
= Ω Texp −i Ĥ(t) + f (t, ~x)φ̂(~x)d~x dt Ω
−∞
  Z ∞ Z  
= Ω Texp −i ĤInt (t)dt − i f (x)φ̂fr (x)dx Ω
−∞
|j(k) + f (k)|2 dk
 Z 
i
= exp . (2.105)
2 (k 2 + m2 − i0) (2π)4

One can retrieve Green’s functions from the generating function:

∂N
hφ̂(xN ) · · · φ̂(x1 )i = iN Z(f ) . (2.106)
∂f (xN ) · · · ∂f (x1 ) f =0

The Fourier transform of Green’s function is denoted as usual by the change of

the variables:

hφ̂(kN ) · · · φ̂(k1 )i
Z Z
:= dxn · · · dx1 e−ixn kn −···−ix1 k1 hφ̂(xN ) · · · φ̂(x1 )i.

We introduce also amputated Green’s functions:

hφ̂(kn ) · · · φ̂(k1 )iamp

= (kn2 + m2 ) · · · (k12 + m2 )hφ̂(kn ) · · · φ̂(k1 )i. (2.107)

66
Amputated Green’s functions can be used to compute scattering amplitudes:
 
kn++ , . . . , k1+ | Ŝ |kn−− , . . . , k1− (2.108)

hφ̂(k1+ ) · · · φ̂(kn++ )φ̂(−kn−− ) · · · φ̂(−k1− )iamp

= q q q q ,
(2π)3(n+ +n− ) 2ε(k1+ ) · · · 2ε(kn++ ) 2ε(kn−− ) · · · 2ε(k1− )
p

where all ki± are on shell, that is ki± = ε(~ki± ), ~ki± .

Note that the Feynman rules for scattering amplitudes follow now from
the rules for the vacuum expectation value of the scattering amplitude, if we
Radd additional insertion vertices—one-legged vertices corresponding to the term
dxf (x)φ̂fr (x).

2.2.7 Path integral formulation

Recall that the generating function equals
 Z 
i
F

Z(f ) = exp j(x) + f (x) D (x − y) j(y) + f (y) dxdy
2
|j(k) + f (k)|2 dk
 Z 
i
= exp . (2.109)
2 (k 2 + m2 − i0) (2π)4
We have the following expressions for the action integral:
Z Z
1
Lfr (x)dx = − φ(x)(−2 + m2 )φ(x)dx,
2
Z Z Z
(L(x) − φ(x)f (x)) dx = Lfr (x)dx − φ(x)(j(x) + f (x))dx.

Consider heuristically the space of all (off-shell) configurations with the Lebesgue
measure Π dφ(x). Physicists like to rewrite (2.109) as
x
R R


Π dφ(x) exp i L(x) − f (x)φ(x) dx
x
Z(f ) = R R
, (2.110)
Π dφ(x) exp i Lfr (x)dx
x

which follows by basic rules of Gaussian integrals. Note that strictly speaking
(2.110) is ambiguous, since DF , the causal propagator, is only one of many
inverses (Green’s functions) of −2 + m2 . The choice of the causal propagator
is an additional convention that is not explicitly contained in the expression
(2.110).

2.2.8 Feynman rules

Perturbative expansions can be organized in terms of Feynman diagrams. The
prescriptions how to draw Feynman diagrams and to evaluate them are called
Feynman rules. We restrict ourselves to Feynman rules in the momentum space.

67
 We have 1 kind of lines and 1 kind of vertices. At each vertex just one
line ends. Vertices are denoted by solid dots. Lines have no distinguished
orientation. However, when we fix the orientation of a line, we can associate to
it a momentum k.
Diagrams for Green’s functions, in addition to internal lines have external
lines ending with insertion vertices, which will be denoted by small circles. To
compute Green’s functions we do as follows:
(1) We draw all possible Feynman diagrams. More precisely, we put N dots for
insertion vertices, labelled 1, . . . , N . We put n dots, labelled 1, . . . , n, for
interaction vertices. Then we connect them in all possible allowed ways.
The expression for the diagram is then divided by n!.
(2) To each vertex we associate the factor −ij(k), where k is the momentum
flowing towards this vertex.
(3) To each line we associate the propagator

c −i
−iDfr (k) = .
k 2 + m2 − i0

d4 k
(4) For internal lines we integrate over the variables with the measure (2π)4 .

Figure 1: Diagram for Green’s function.

Diagrams used to compute scattering amplitudes with N − incoming and N +

outgoing particles are similar to diagrams for N − +N + -point Green’s functions,
except that instead of insertion vertices we have incoming and outgoing particles.
For the incoming lines, −k are on-shell, for the outgoing lines, k is on-shell. The
rules are changed only concerning the external lines:
(i) To each incoming external line we associate √ 1 .
(2π)3 2ε(~
k)
1
(ii) To each outgoing external line we associate √ .
(2π)3 2ε(~
k)

68
 Figure 2: Diagram for a scattering amplitude.

2.2.9 Vacuum energy

Let D denote the value of the (unique) connected diagram with no external
lines. We have
|j(k)|2
Z
i dk D
log(Ω|ŜΩ) = = .
2 (k 2 + m2 − i0) (2π)4 2

Figure 3: Diagram for vacuum energy.

We can derive it diagrammatically as follows. At the order 2m there are

(2m)!
pairings. Hence
2m m!

∞
X 1 (2m)! m
(Ω|ŜΩ) = D = exp(D/2).
m=0
(2m)! 2m m!

2.2.10 Problems with the scattering operator

Ŝ can be ill defined.
First of all,

|j(k)|2 |j(k)|2
Z Z
1 dk 1 dk
Re = (2.111)
2 (k 2 + m2 − i0) (2π)4 2 (k 2 + m2 ) (2π)4

69
can be infinite. This is not a very serious problem. (2.111) is responsible only
for the phase of the scattering amplitude and does not influence scattering cross-
sections.
We can try to remedy the problem by an apropriate renormalization of the
phase. In particular, in the case of a stationary source or, more generally, a
source travelling with a constant velocity, we can use the adiabatic switching
and the Gell-Mann and Low construction to obtain a meaningful scattering
operator. We will describe this construction in the next subsubsections.
The problem with Ŝ is more serious if

|j(k)|2 |j(ε(~k), ~k)|2 d~k

Z Z
1 dk 1
Im =
2 (k + m2 − i0) (2π)4
2 2 ε(~k) (2π)3

is infinite. Then no unitary operator Ŝ satisfies the relations (2.99) and (2.100),
see Thm A.1. The scattering operator is ill defined. However, as we describe in
Subsubsect. 2.2.14, also in this situation there is a pragmatic solution – we can
define inclusive cross-sections.
Note that if k 7→ j(k) is Schwartz, then Ŝ is well-defined, even if m = 0.

2.2.11 Energy shift and scattering theory for a stationary source

Suppose now that the source does not depend on time and is given by a Schwartz
function R3 3 ~x 7→ j(~x). Then we have the time-independent Hamiltonian

2 m2
Z 
1 1 
Ĥ = : π̂(~x)2 + ∂~ φ̂(~x) + φ̂(~x)2 + j(~x)φ̂(~x) :d~x (2.112)
2 2 2
d~k
Z Z
∗ ~ ~ j(~k)a(k) + j(~k)â∗ (k) p


= â (k)â(k)ε(k)dk + q . (2.113)
(2π)3 2ε(~k)

By the method of completing the square (A.14) we compute the infinum of Ĥ:

e−m|~x−~y|
Z
1
E=− j(~x) j(~y )d~xd~y (2.114)
2 4π|~x − ~y |
|j(~k)|2 d~k
Z
1
=− . (2.115)
2 (2π)3 (~k 2 + m2 )

Obviously, the standard Møller operators Ŝ ± (1.123) are ill defined and we
need to use the Gell-Mann–Low construction. Let us replace j with j± (t, ~x) :=
θ(±x)j(~x)e−|t| . Its Fourier transform is j± (k 0 , ~k) = ∓i(k 0 ∓i)−1 j(~k), Inserting

70
this into (2.101) we obtain

±
 Z j(~k) d~k 
Ŝ = exp − q a∗ (k) p
2ε(~k)(ε(~k) ∓ i) (2π)3

Z j(~k) d~k 
× exp q a(k) p
2ε(~k)(ε(~k) ± i) (2π)3
 1Z |j(~k)|2 d~k |j(~k)|2 d~k 
Z
i
× exp − + .
2 2ε(~k)(ε(~k)2 + 2 ) (2π)3 2 2(ε(~k)2 + 2 ) (2π)3
±
Note that the phase of Ŝ± behaves as O(−1 ). In the definition of SGL we take
this phase away and put  & 0, see (1.124). We obtain

±
 Z j(~k) d~k 
ŜGL = exp − q a∗ (k) p
2ε(~k)3 (2π)3

Z j(~k) d~k 
× exp q a(k) p
2ε(~k)3 (2π)3
 1 Z |j(~k)|2 d~k 
× exp − .
2 2ε(~k)3 (2π)3
R
If m > 0 or if j(~x)d~x = 0, then Ĥ has a ground state and the operators
±
ŜGL are well defined. We have
± ±
ŜGL Ĥfr = (Ĥ − E)ŜGL .
+ −
Note a somewhat disappointing feature: ŜGL = ŜGL , and hence the scatter-
+∗ −
ing operator ŜGL := ŜGL ŜGL = 1l is trivial.
R
If m = 0 and j(~x)d~x 6= 0, then Ĥ has no ground state (even though it is
±
bounded from below) and the operators ŜGL are ill defined.

2.2.12 Travelling source

Consider now a source of a profile given by a function q ∈ Cc∞ (R3 ) travelling
with velocity ~v . That means
j(t, ~x) = q(~x − t~v ). (2.116)
We note that the Fourier transform of (2.116) in the spatial variables equals
~
j(t, ~k) = q(~k)e−it~vk .
The interaction picture Hamiltonian becomes
d~k
Z 


−it ε(~ v~ ε(~ v~
Ĥ,Int (t) = q e k)−~ k
q(~k)â(k) + eit k)−~ k
q(~k)â∗ (k) .
(2π)3 2ε(~k)
p

71
This is the interaction picture Hamiltonian for a time-independent perturbation
where in the dynamics we replace the 1-particle energy ε(~k) by ε(~k) − ~v~k.
We use the Gell-Mann–Low type adiabatic switching, so that we replace j
by
j (t, ~x) := e−|t| j(t, ~x).
We slightly generalize the Gell-Mann–Low Møller operators:

± |(Ω|Ŝ± Ω)|
ŜGL = lim Ŝ±
&0 (Ω|Ŝ± Ω)
 Z q(~k) d~k 
= exp − q a∗ (k) p
2ε(~k)(ε(~k) − ~v~k) (2π)3

Z q(~k) d~k 
× exp q a(k) p
2ε(~k)(ε(~k) − ~v~k) (2π)3
 1Z |q(~k)|2 d~k 
× exp − .
2 2ε(~k)(ε(~k) − ~v~k)2 (2π)3
Note that if |v| < 1 (if the source is slower than light) and m > 0, then
+ −
ŜGL = ŜGL are well defined unitary operators.
±
R
If m = 0 and q(~x)d~x 6= 0, then the infrared problem shows up: ŜGL are ill
defined.
It is interesting to assume that the source has a different asymptotics in the
future and in the past. For simplicity, suppose that the change occurs sharply
at time t = 0 and consider

q− (~x − t~v− ), t < 0,
j(t, ~x) =
q+ (~x − t~v+ ), t > 0.

We can introduce first the scattering operator Ŝ with the adiabatically
switched interaction. Then we can define a Gell-Mann–Low type scattering
operator by taking  & 0 and renormalizing the phase:
|(Ω|Ŝ Ω)|
lim Ŝ (2.117)
&0 (Ω|Ŝ Ω)
!
q+ (~k) q− (~k) d~k
Z
1  
∗
= exp − â (k) p
~ ~ ~ ~
q
2ε(~k) (ε(k) − ~v+ k) (ε(k) − ~v− k) (2π)3
!
q + (~k) q − (~k) d~k
Z
1  
× exp − + â(k) p
(ε(~k) − ~v+~k) (ε(~k) − ~v−~k)
q
2ε(~k) (2π)3
!
q+ (~k) q− (~k) 2 d~
Z
1 1 k
× exp − − .
2 2ε(~k) (ε(~k) − ~v+~k) (ε(~k) − ~v−~k) (2π)3
Let m = 0. Then (2.117) is ill defined if

72
 R R
(1) q+ (~x)d~x 6= q− (~x)d~x,
or
R R
(2) q+ (~x)d~x = q− (~x)d~x 6= 0 and v+ 6= v− .
(2.117) is given by (2.101) where we replace

|j(k)|2
Z
dk
(k + m − i0) (2π)4
2 2

with

|j(k)|2 |j(ε(~k), ~k)|2 d~k

Z Z
dk
Im = .
(k + m2 − i0) (2π)4
2
2ε(~k) (2π)3

Here, j(k) is the Fourier transform of the source (2.116):

Z
~ 0
j(k) = j(t, ~x)e−ik~x+ik t d~xdt

iq+ (~k) iq− (~k)

= − + .
~k~v+ − k 0 − i0 ~k~v− − k 0 + i0

If we do not like the adiabatic switching approach we can directly define the
Møller operators by removing the (possibly infinite) phase shift from (2.101).
Alternatively, we can multiply the two Møller operators:
!
~k)q− (~k)  d~k


2Im q (
Z
+∗ − i 1 +
ŜGL ŜGL = ŜGL exp − .
2 2ε(~k) (ε(~k) − ~v+~k)(ε(~k) − ~v−~k) (2π)3

2.2.13 Scattering cross-sections

We consider again an arbitrary source term j. Given on-shell momenta of in-
coming particles kn−− , . . . , k1− and outgoing particles k1+ , . . . , kn++ we can compute
the scattering cross-section for the corresponding process, or actually its density
w.r.t. the Lebesgue measure dk1+ · · · dkn++ :

σ k1+ , . . . , kn++ ; kn−− , . . . , k1−

 2
= k1+ , . . . , kn++ | Ŝ |kn−− , . . . , k1−

1  Z |j(ε(~k), ~k)|2 d~k 

= p exp −
(2π)(n+ +n− ) 2ε(~k) (2π)3
|j(ε(~k1+ ), ~k1+ )|2 |j(ε(~kn++ ), ~kn++ )|2 |j(ε(~kn−− ), ~kn−− )|2 |j(ε(~k1− ), ~k1− )|2
× · · · ··· .
2ε(~k1 )+
2ε(~kn+ )
+
2ε(~kn− )
−
2ε(~k1− )

Note that the crosssections are sometimes zero. In the next subsubsection
we describe how to cope with this problem.

73
2.2.14 Inclusive cross-section
Let δ > 0. The 1-particle Hilbert space can be split as Z = Z<δ ⊕ Z>δ corre-
sponding to the soft momenta |~k| < δ and hard momenta |~k| > δ. Clearly,

Γs (Z) ' Γ(Z<δ ) ⊗ Γ(Z>δ ), Ω ' Ω<δ ⊗ Ω>δ .

Assume first that m > 0 and the scattering operator is computed as above.
Clearly, the scattering operator and scattering cross-sections factorize:

Ŝ ' Ŝ<δ ⊗ Ŝ>δ , σ = σ<δ σ>δ .

More precisely, let

|~q1+ |, . . . , |~qm
+
q1− |, . . . , |~qm
+ |, |~
−
− | < δ. (2.118)

Then we have the soft scattering cross-sections

− −
σ<δ q1+ , . . . , qm +


+ ; qm− , . . . , q1
 2
− −
= q1+ , . . . , qm
+
+ | Ŝ<δ |qm− , . . . , q1

1  Z |j(ε(~q), ~q)|2 d~q 

= p exp −
(2π)(m+ +m− ) |~
q |<δ 2ε(~q) (2π)3
+ + − −
|j(ε(~q1+ ), ~q1+ )|2 |j(ε(~qm + ), ~
qm 2
+ )| |j(ε(~
qm − ), ~
qm − )|
2
|j(ε(~q1− ), ~q1− )|2
× ··· − ··· .
+
2ε(~q1 ) +
2ε(~qm+ ) 2ε(~qm− ) 2ε(~q1− )

Likewise, let
|~k1+ |, . . . , |~kn++ |, |~k1− |, . . . , |~kn−− | > δ. (2.119)
The corresponding hard scattering cross-section are

σ>δ k1+ , . . . , kn++ ; kn−− , . . . , k1−

 2
= k1+ , . . . , kn++ | Ŝ>δ |kn−− , . . . , k1−

1  Z |j(ε(~k), ~k)|2 d~k 

= p exp −
(2π)(n+ +n− ) |~
k|>δ 2ε(~k) (2π)3
|j(ε(~k1+ ), ~k1+ )|2 |j(ε(~kn++ ), ~kn++ )|2 |j(ε(~kn−− ), ~kn−− )|2 |j(ε(~k1− ), ~k1− )|2
× · · · · · · .
2ε(~k1+ ) 2ε(~kn++ ) 2ε(~kn−− ) 2ε(~k1− )

We have

σ>δ k1+ , . . . , kn++ ; kn−− , · · · , k1−

(2.120)
+ + − −


= σ k1 , . . . , kn+ ; kn− , · · · , k1
∞ Z
X Z
σ k1+ , . . . , kn++ , q1 , . . . , qj ; kn−− , · · · , k1− d~q1 · · · d~qj .


+ ···
j=1 |~
q1 |<δ |~
qj |<δ

74
σ>δ describes the experiment that does not measure outgoing particles of mo-
mentum less than δ and in the incoming state there are no particles of mo-
mentum less than δ. Actually, we would have obtained the same scattering
cross-section if the part of the incoming state below the momentum δ was ar-
bitrary. This is an example of an inclusive cross-section – a cross-section which
involves summing over many unobserved final states.
If m & 0, the soft scattering operator Ŝ<δ has no limit. All σ<δ go to zero.
In fact, they are proportional to
 Z |j(ε(~q), ~q)|2 d~q 
σ<δ = exp − .
|~
q |<δ 2ε(~q) (2π)3

The hard scattering operator Ŝ>δ and σ>δ have well defined limits and can have
a physical meaning.
One can imagine various experimental scenarios that lead to different inclu-
sive cross-sections. For example, imagine that our apparatus does not detect
the details of an outgoing state if the total energy of soft particles is less than
δ. This leads to the following inclusive cross-section:
app
k1+ , . . . , kn++ ; kn−− , · · · , k1− := σ k1+ , . . . , kn++ ; kn−− , · · · , k1−



σ>δ
X∞ Z
σ k1+ , . . . , kn++ , q1 , . . . , qj ; kn−− , · · · , k1− d~q1 · · · d~qj .


+
j=1 ε(~
q1 )+···+ε(~
qj )<δ

app
Note that both σ>δ and σ>δ are proportional to one another:
app
k1+ , . . . , kn++ ; kn−− , · · · , k1−


σ>δ
σ>δ k1+ , . . . , kn++ ; kn−− , · · · , k1−

∗
:= (Ω<δ |Ŝ<δ 1l[0,δ] (Ĥfr )Ŝ<δ Ω<δ ) = σ<δ (; ) (2.121)
∞
X Z
+ σ<δ (q1 , . . . , qj ; ) d~q1 · · · d~qj .
j=1 ε(~
q1 )+···+ε(~
qj )<δ

This ratio is in practice not very interesting – it contributes a common numerical

factor to all scattering cross-sections for hard particles.

2.3 Neutral scalar bosons with a mass-like perturbation

2.3.1 Classical fields
A scalar field can be also perturbed by a mass-like perturbation. Classically,
this is expressed by the equation

(−2 + m2 )φ(x) = −κ(x)φ(x), (2.122)

where R1,3 3 x 7→ κ(x) is a given function. In most of this subsection we will

assume that κ is Schwartz and m > 0. We introduce also π(x) := φ̇(x).

75
 Let us define the corresponding retarded and advanced propagators as the
unique distributional solutions of

− 2x + m2 + κ(x) D± (x, y) = δ(x − y),

(2.123)

satisfying
suppD± ⊂ {x, y : x ∈ J ± (y)}.
We also generalize the Pauli-Jordan function:

D(x, y) := D+ (x, y) − D− (x, y).

Note that
suppD ⊂ {x, y : x ∈ J(y)}.
The function D can be used to solve the initial value problem of (2.122):
Z
φ(t, ~x) = − ∂s D(t, ~x, s, ~y ) φ(0, ~y )d~y
s=0
Z
+ D(t, ~x, 0, ~y )π(0, ~y )d~y . (2.124)

We would like to interpret the classical field φ(x) satisfying (2.122) as a

functional on the space YKG coinciding with the free field at time 0, as in
(2.83). By (2.124), this allows us to express uniquely the field φ in terms of the
free field.

2.3.2 Lagrangian and Hamiltonian formalism

The Lagrangian density is
1 1
L(x) = − ∂µ φ(x)∂ µ φ(x) − (m2 + κ(x))φ(x)2 .
2 2
As in Subsubsect. 2.2.2, the variable conjugate to φ(x) is π(x). We easily obtain
the Hamiltonian density
1 2 1 ~
2 1 2
H(x) = π (x) + ∂φ(x) + (m + κ(x))φ2 (x),
2 2 2
so that the full Hamiltonian generating the dynamics is
Z
H(t) = H(t, ~x)d~x.

76
2.3.3 Dynamics in the interaction picture
The classical interaction picture Hamiltonian can be expressed in terms of plane
wave functionals:
Z
1
HInt (t) = κ(t, ~x)φ2fr (t, ~x)d~x (2.125)
2
d~k1 d~k2 κ(t, ~k1 + ~k2 )  −itε(~k1 )−itε(~k2 )
Z
1
= q q e a(−k1 )a(−k2 )
2
(2π)3 2ε(~k1 ) 2ε(~k2 )

~ ~ ~ ~
+2eitε(k1 )−itε(k2 ) a∗ (k1 )a(−k2 ) + eitε(k1 )+itε(k2 ) a∗ (k1 )a∗ (k2 ) .

Consider the equations of motion in the interaction picture:

ȧ∗t (k) = {a∗t (k), HInt (t)}

d~k1 κ(t, −~k + ~k1 )
Z
= i q q
(2π)3 2ε(~k) 2ε(~k1 )
 
~ ~ ~ ~
× e−itε(k)−itε(k1 ) at (−k1 ) + e−itε(k)+itε(k1 ) a∗t (k1 ) ,
a∗0 (k) = a∗ (k).

The solution of these equations at two times are related by a matrix of the form
" #
pt+ ,t− qt+ ,t−
(2.126)
qt+ ,t− pt+ ,t−

in the following way:

" ∗ # Z " #" ∗ #
at+ (k) pt+ ,t− (k, k1 ) qt+ ,t− (k, k1 ) at− (k1 )
~
= dk1 .
at+ (k) qt+ ,t− (k, k1 ) pt+ ,t− (k, k1 ) at− (k1 )

(2.126) has a limit as t+ , −t− → ∞, which can be called the classical scattering
operator.
One can try to solve the equations of motion by iterations. The first iteration
is often (at least in the quantum context) called the Born approximation, and
it gives the following formula for the elements of (2.126):
t+
κ(s, −~k + ~k1 )
Z
~ ~
pBorn
t+ ,t− (k, k1 ) = δ(~k − ~k1 ) + i ds q q e−isε(k)+isε(k1 ) ,
t− (2π)3 2ε(~k) 2ε(~k1 )
t+
κ(s, −~k + ~k1 )
Z
~ ~
qtBorn
+ ,t−
(k, k1 ) = i ds q q e−isε(k)−isε(k1 ) .
t− (2π)3 2ε(~k) 2ε(~k1 )

77
2.3.4 Quantization
Again, we are looking for quantum fields R1,3 7→ φ̂(x) satisfying
(−2 + m2 )φ̂(x) = −κ(x)φ̂(x), (2.127)
˙
with the conjugate field π̂(x) := φ̂(x) having the equal time commutators (2.91),
and coinciding with the free field at time 0, as in (2.92). The solution is given
by putting “hats” onto (2.124).
We would like to check whether the classical scattering operator and the clas-
sical dynamics are implementable in the Fock space for nonzero κ. By Thm A.2,
we need to check the Shale condition, that is, whether the off-diagonal elements
of (2.126) are square integrable. For simplicity, we will restrict ourselves to the
Born approximation; the higher order terms do not change the conclusion.
The verification of the Shale condition is easier for the scattering operator.
Consider
Z ∞
κ(s, −~k + ~k1 ) ~ ~
Born
q∞,−∞ (k, k1 ) = i ds q q e−isε(k)−isε(k1 ) . (2.128)
−∞ (2π)3 2ε(~k) 2ε(~k1 )
Recall that κ is a Schwartz function. Therefore, we can integrate by parts as
many times as we want:
Z ∞ ~ ~
Born n+1 ∂sn κ(s, −~k + ~k1 ) e−isε(k)−isε(k1 )
q∞,−∞ (k, k1 ) = i ds
. (2.129)
~ ~ n
q q
−∞ (2π)3 2ε(~k) 2ε(~k1 ) ε(k) + ε(k1 )

This decays in ~k and ~k1 as any inverse power, and hence is square integrable on
R3 × R3 . Therefore the classical scattering operator is implementable.
Next let us check the implementability of the dynamics, believing again that
it is sufficient to check the Born approximation. We integrate by parts once:
qtBorn
+ ,t−
(k, k1 )
~ ~ ~ ~
−κ(t+ , −~k + ~k1 )e−it+ ε(k)−it+ ε(k1 ) + κ(t− , −~k + ~k1 )e−it− ε(k)−it− ε(k1 )
= q q
(2π)3 2ε(~k) 2ε(~k1 ) ε(~k) + ε(~k1 )

~ ~
∂s κ(s, −~k + ~k1 )e−isε(k)−isε(k1 )
Z t+
+ ds q q
. (2.130)
t− (2π)3 2ε(~k) 2ε(~k1 ) ε(~k) + ε(~k1 )

Using that κ(s, ~k + ~k1 ) decays fast in the second variable, we see that (2.130)
can be estimated by
C
,
(ε(k) + ε(~k1 ))2
~
which is square integrable. Therefore, the dynamics is implementable for any
t− , t+ .
By a similar computation we check that if we freeze t0 ∈ R, the dynamics
generated by the momentary Hamiltonian HInt (t0 ) is implementable.

78
2.3.5 Quantum Hamiltonian
We may try to write the quantum Hamiltonian as
Z 
1 1
2 1 
Ĥ(t) := : π̂ 2 (~x) + ∂~ φ̂(~x) + (m2 + κ(t, ~x))φ̂2 (x) :d~x. (2.131)
2 2 2

We will see later on that the Wick-ordered expression (2.131) does not define
an operator. However we will use it to derive the Feynman rules, which unfor-
tunately will lead to divergent diagrams.
Formally (2.94) remains true if we add a time dependent constant C(t) to
(2.131). We will see that in order to define correct Hamiltonians Ĥ(t) this
constant has to be infinite. We will obtain bounded from below Hamiltonians
Ĥren (t), however the vacuum will not be contained in their form domain. There-
fore, the condition (Ω|Ĥren (t)Ω) = 0 for all t, which is equivalent to the Wick
ordering, cannot be imposed.
The interaction picture Hamiltonian is
Z
1
ĤInt (t) = κ(t, ~x):φ̂2fr (t, ~x):d~x (2.132)
2
d~k1 d~k2 κ(t, ~k1 + ~k2 )  −itε(~k1 )−itε(~k2 )
Z
1
= q q e â(−k1 )â(−k2 )
2
(2π)3 2ε(~k1 ) 2ε(~k2 )

~ ~ ~ ~
+2eitε(k1 )−itε(k2 ) â∗ (k1 )â(−k2 ) + eitε(k1 )+itε(k2 ) â∗ (k1 )â∗ (k2 ) .

As in the case of linear sources, we define the scattering operator, scattering

amplitudes, Green’s functions, amputated Green’s functions and the generating
function, see (2.102)–(2.108).

2.3.6 Path integral formulation

The generating function (and hence all the other quantities introduced above)
can be computed exactly. It is
 
−1  1  12
det − 2 + m 2
− 2 + m + κ − i0
2


Z(f ) = exp κ
−2 + m2 − i0
 
i
× exp f (−2 + m2 + κ − i0)−1 f
2

c −1

c

 21
= det 1l + κDfr exp κDfr
 
i c c −1
× exp f Dfr (1l + κDfr ) f . (2.133)
2

Here, the determinant is understood (at least formally) as the Fredholm deter-

1
minant on the space L2 (R1,3 ). The term exp κDfr
c 2
is responsible for the Wick
ordering.

79
 Similarly as in the case of (2.110), (2.133) is often expressed in terms of path
integrals as
Z  Z 


C Π dφ(x) exp i L(x) − f (x)φ(x) dx . (2.134)
x

Here, C is a normalization constant, which does not depend on f . Again, the

formula (2.134) is only symbolic, the full information is contained in (2.133).

2.3.7 Feynman rules

Figure 4: Diagram for Green’s function.

Figure 5: Diagram for scattering amplitude.

80
 Feynman rules are similar as in the case of a linear source. The difference
is that now vertices have 2-legs. The rule (2) for calculating Green’s functions
changes: for each vertex with incoming momenta k1 , k2 we insert the number
−iκ(k1 + k2 ), where k1 and k2 are the momenta of lines entering the vertex.
Another difference is that we do not allow a line to begin and end at the same
vertex – this is because we use the Wick ordered Ĥ(t).
Diagrams can be decomposed into connected components of two kinds:
1. lines ending at insertion vertices (for Green’s functions) or on-shell parti-
cles (for scattering amplitudes) with 0, 1, 2, . . . interaction vertices;
2. loops with 2, 3, . . . interaction vertices.

Note that loops with 1 interaction vertex do not appear because of the Wick
ordering.
Diagrams without loops (both for Green’s functions and scattering ampli-
tudes) are finite, because the external momenta are fixed and on interaction
vertices we have the fast decaying function κ.
Consider a loop with 4-momenta k1 , . . . , kn flowing around it. On vertices we
have the function κ, which essentially identifies ki with ki+1 . The propagators
give the power |ki |−2 . Thus we are left with 4 degrees of freedom and the
integrand that behaves as |k|−2n . This is integrable if n > 2, but divergent for
the 2-vertex loop. We will see that only the imaginary part of this diagram is
divergent.

2.3.8 Vacuum energy

The classical scattering operator is well defined. The quantum scattering oper-
ator, if computed naively (that is, using the Wick ordered Hamiltonian) is ill
defined. Its problem comes from the overall phase, which is not fixed by the
classical transformation.
One can say that this phase has no physical meaning, since it does not appear
in scattering cross-sections. However, it may be relevant for a more complete
theory. We will see that there is a natural choice of this phase, which leads
to a renormalized scattering operator Ŝren (κ). We will also see that there is a
natural renormalized Hamiltonian Ĥren (t).
The logarithm of the vacuum-to-vacuum scattering amplitude times the imag-
inary unit will be called the vacuum energy. It can be computed exactly:

E := i log(Ω|ŜΩ) = i log Z(0)

i  
= Tr log(−2+m2 −i0) − log(−2+m2 +κ−i0) + κ(−2+m2 −i0)−1
2
i  c c

= Tr − log(1 + κDfr ) + κDfr
2
∞ ∞
X (−1)n c n
X
= i Tr(κDfr ) =: En . (2.135)
n=2
2n n=2

81
 Figure 6: Vacuum energy

Here, Tr is understood (at least formally) as the usual trace of operators on

L2 (R1,3 ). En is the nth order contribution to the vacuum energy. Note that for
n = 1 there is no contribution because of the Wick ordering and for n = 2 it is
divergent.
We have En = i D n c n
2n , where Dn = (−1) Tr(κDfr ) is the value of the loop with
n

n vertices. This is a special case of a more general rule saying that to compute
log(Ω|ŜΩ) we need to sum over all connected diagrams with no external lines
divided by the symmetry factor (the order of the group of the symmetries of
the diagram). In the case of a loop with n vertices its group of symmetries is
the nth dihedral group, hence the symmetry factor is 2n.

2.3.9 Pauli-Villars renormalization

The lowest contribution to the vacuum energy is of the second order and comes
from the loop with two vertices. Formally, it can be written as
Z Z
dk dk
E2 = κ(−k)κ(k)π(k 2 ) 4
= |κ(k)|2 π(k 2 ) ,
(2π) (2π)4

where the right hand side defines the vacuum energy function π(k 2 ). Unfortu-
nately, computed naively, π(k 2 ) is logarithmically divergent.
The renormalization of a mass-like perturbation is not very difficult and can
be done in many ways. We will describe 3 methods of renormalization. All of
them will lead to the same renormalized vacuum energy function π ren (k 2 ).
We start with the Pauli-Villars method. In the context of a mass-like per-
turbation, the Pauli-Villars regularization consists in introducing an additional
fictitious field that has a (large) mass M and appears only in loops. (Thus we
ignore diagrams involving external lines of the fictitious particle). In addition,
each loop of the fictitious field has a (nonphysical) coefficient −1. We organize
our computations by setting m0 = m, C0 = 1, m1 = M , and C1 = −1. The
Pauli-Villars regularized vacuum energy function is the sum of the loop of the
physical particle and of the fictitious one:

82
 d4 q X
Z
2 1
4πM (k ) = i Ci
(2π)4 i ((q + 21 k)2 + m2i − i0)((q − 12 k)2 + m2i − i0)
Z ∞ Z ∞
d4 q
Z  
X 
2 1 2 2

= −i dα1 dα2 Ci exp −i(α1 + α2 ) q + k + mi − i(α1 − α2 )qk
(2π)4 0 0 i
4
Z ∞ Z ∞  
1 X 1 2 α1 α2 2
= − dα 1 dα 2 C i exp −i(α 1 + α2 )m i − i k
(4π)2 0 0 i
(α1 + α2 )2 α1 + α2
Z 1 Z ∞
(1 − v 2 )k 2
  
1 dρ X
= − 2
dv Ci exp −iρ m2i +
(4π) 0 0 ρ i 4
Z 1 X
1  k 2 (1 − v 2 ) 
= 2
dv Ci log m2i + − i0
(4π) 0 i
4
Z 1 X  
(1 − v 2 )k 2
 
1 2
= dv Ci log 1 + − i0 + log mi .
(4π)2 0 i
4m2i

We used the identities (A.20) and (A.22). We inserted

Z ∞
1= dρδ(ρ − α1 − α2 ), (2.136)
0

and then changed the variables as α1 = ρ (1−v) (1+v)

2 , α2 = ρ 2 , so that dα1 dα2 =
1
2 ρdvdρ. We also used the symmetry v 7→ −v to restrict the integration from
[−1, 1] to [0, 1]. At the end we use the identity (A.24).
We define the renormalized vacuum energy function as

π ren (k 2 ) := lim πM (k 2 ) − πM (0)

M →∞
 1 M2 
= lim πM (k 2 ) + log
M →∞ 4(4π)2 m2
1
k 2 (1 − v 2 )
Z
1  
= log 1 + − i0 dv. (2.137)
4(4π)2 0 4m2

Note that π ren (0) = 0.

We obtain
√
ren 2 1 1 1+θ  k2
π (k ) = log − 2 , θ= √ , 0 < k2 ;
4(4π)2 θ 1−θ k 2 + 4m2
√
1 2  −k 2
= arctan θ − 2 , θ= √ , − 4m2 < k 2 < 0;
4(4π)2 θ k + 4m2
2
√
1 1 θ+1   −k 2
= 2
log − iπ − 2 , θ= √ , k 2 < −4m2 .
4(4π) θ θ−1 −k 2 − 4m2

83
Here is the calculation. First we assume that k 2 > 0. Then we can drop i0 and
Z 1  k 2 (1 − v 2 ) 
log 1 + dv (2.138)
0 4m2
√
k2 r !
k 2 + 4m2
Z 2m
2m
=√ √ log − w dw. (2.139)
k 2 − 2m k2 4m2

Using log(w)dw = w log(w) − w, this gives the value of π ren (k 2 ) in the first
R

region. Then we use analytic continuation, remembering

√
that k 2 may have
2 k 2
negative imaginary part. As k decreases, θ := √k2 +4m2 first varies from 1 to
1+θ
0, then from 0 to −i∞, finally, from ∞ to 1. Therefore, 1−θ first varies from ∞
to 1, then goes over the lower semicircle, finally, from −1 to −∞. Next we use
1+iy
log 1−iy = 2i arctan y, and for y < 0, log(y − i0) = log |y| − iπ.

2.3.10 Renormalization of the vacuum energy

The renormalized 2nd order vacuum energy is
Z
ren dk
E2 = π ren (k)|κ(k)|2
(2π)4
Z
dk
πM (k) − πM (0) |κ(k)|2


= lim
M →∞ (2π)4
Z Z !
2 dk 2
= lim πM (k)|κ(k)| − πM (0) κ(x) dx .
M →∞ (2π)4

The full renormalized vacuum energy has a compact formula:

∞
X
E ren = E2ren + En
n=3
c 2
i  c c (κDfr )
= − Tr log(1 + κDfr ) − κDfr +
2Z 2
dk
+ |κ(k)|2 π ren (k) . (2.140)
(2π)4

We can formally write π∞ (k) := lim πM (k) (which is typically infinite).

M →∞
Note that the renormalized scattering operator Ŝren is a well defined unitary
operator and the renormalized Hamiltonian Ĥren (t) is a well defined self-adjoint
operator:
κ(x)2 dx
R
Ŝren = eiπ∞ (0) Ŝ, (2.141)
Z
Ĥren (t) = Ĥ(t) − π∞ (0) κ(t, ~x)2 d~x. (2.142)

84
 The counterterm has an infinite coefficient π∞ (0). Otherwise, it is quite well
behaved – it depends locally on the interaction, and therefore the renormaliza-
tion preserves the Einstein causality. This manifests itself in the identity

Ŝren (κ2 )Ŝren (κ1 ) = Ŝren (κ2 + κ1 ),

whenever suppκ2 is later than suppκ1 .

Formally, the correct Lagrangian density is

Lren (x) = L(x) + π∞ (0)κ(x)2 .

2.3.11 Method of dispersion relations

There exists an alternative method to renormalize and compute the vacuum
energy. We start with computing the imaginary part of π(k), which gives a
finite result:
d4 q
Z
i 1
Imπ ren (k 2 ) = Im
4 (2π)4 ((q + 21 k)2 + m2 − i0)((q − 12 k)2 + m2 − i0)
Z 1  
(1 − v 2 )k 2
 
1 2
= Im dv log 1 + − i0 + log m .
4(4π)2 0 4m2

Using log(t − i0) = log |t| − iπθ(−t), we see that the imaginary part of the
logarithm is very simple. Hence
Z 1 
ren 2 π (1 − v 2 )k 2 
Imπ (k ) = − θ − 1 − dv
4(4π)2 0 4m2
r
π
= − √ − k 2 − 4m2 .
4(4π) −k 2
2 +

We can obtain the real part by using the fact that π ren (0) = 0 and the once
subtracted dispersion relations for the variable s = −k 2 , see Thm A.4:
Z ∞  
ren 2 1 ren 1 1
Reπ (k ) = P dsImπ (−s) − . (2.143)
π 4m2 s + k2 s

2.3.12 Wick rotation

Recall that the causal propagator is defined as
1
R1,3 3 p 7→ DF (p) = .
p2 + m2 − i0
It can be interpreted as a boundary value of a holomorphic function


C\ ] − ∞, −m] ∪ [m, ∞[ × R3 3 (p0 , p~) (2.144)
1 1
7→ DF (p) = = 2 .
−(p0 )2 + p~2 + m2 p + m2

85
The physical region R1,3 of (2.144) lies at the boundary–on ]0, ∞[×R3 from
above and on ] − ∞, 0[×R3 from below:

DF (p) = lim DF (eiφ p0 , p~).

φ&0

Define the Euclidean scalar product as

hp|qiE := p0 q 0 + p~~q,

and the Euclidean propagator



C\ ] − i∞, −im] ∪ [im, i∞[ × R3 3 (p0 , p~) (2.145)
1 1
7→ DE (p0 , p~) := D(ip0 , p~) = = .
(p0 )2 + p~2 + m2 hp|pi2E + m2

Clearly, we can express the causal propagator in terms of the Euclidean propa-
gator with help of the Wick rotation:

DF (p0 , p~) = lim DE (e−iφ p0 , p~).

φ%π/2

Suppose now that a physical quantity is given by an integral

d4 q G(p2 , pq, q 2 )
Z
R1,3 3 p 7→ F (p) :=
, (2.146)
(2π)4 ap2 + 2bpq + cq 2 + m2 − i0 n


a b
where G is holomorphic and the matrix is positive definite. Then
b c
instead of F we can consider the holomorphic function


C\ ] − ∞, −m] ∪ [m, ∞[ × R3 3 (p0 , p~) (2.147)
d4 q G(p2 , pq, q 2 )
Z
7→ F (p) :=
,
(2π)4 ap2 + 2bpq + cq 2 + m2 n

where there is no need to put i0, because the denominator is automatically

invertible. The physical function (2.146) is the boundary value of (2.147):

lim F (eiφ p0 , p~).

φ&0

We can also introduce the Euclidean version of F given by

F E (p) = F E (p0 , p~) := F (ip0 , p~)

id4 q G(hp|pi2E , hp|qiE , hq|qi2E )
Z
=
,
(2π) ahp|pi2E + 2bhp|qiE + chq|qi2E + m2 n
4

where in the integral we substituted (iq 0 , ~q) for (q 0 , ~q). This substitution can be
reached from the original variables inside the holomorphy domain by the Wick

86
rotation, hence it does not affect the integral. F E is holomorphic on the domain
of (2.145). We can retrieve the physical values of F from F E by

F (p0 , p~) = lim F E (e−iφ p0 , p~).

φ%π/2

In what follows, whenever we use Euclidean functions such as F E , we will

use the Euclidean scalar product hp|qiE . We will denote this scalar product
simply by pq, since its use will be obvious from the context.

2.3.13 Dimensional renormalization

Let us renormalize the vacuum energy by yet another method – the method of
dimensional regularization. We will use the Euclidean quantities.
Let us first compute formally the 2-vertex loop:

d4 q
Z
1
4π E (k 2 ) = −
(2π)4 ((q + 21 k)2 + m2 )((q − 12 k)2 + m2

1 1 d4 q
Z Z
1
= − dv 4
2
2 −1 (2π) q + 2 k 2
2
4 + m + vqk
Z 1 Z
d4 q 1
= − dv
, (2.148)
0 (2π) q 2 + (1 − v 2 ) + m2 2
4 k 2
4

where we used the Feynman identity (A.28), replaced q + vk

2 with q, used the
1 1
R R1
symmetry v → −v to replace 2 −1 dv with 0 dv. After this preparation, we
use the dimensional regularization:

dq 4 µ4−d Ωd ∞ d−1
Z Z
is replaced by |q| d|q|, (2.149)
(2π)4 (2π)d 0

where Ωd is the “area of the unit sphere in d dimension”, see (A.30). Thus
instead of (2.148) we consider its dimensionally regularized version:
Z ∞
µ4−d Ωd 1 |q|d−1
Z
4π E,d (k 2 ) = − d
dv
2 d|q|
(2π) 0 0
k 2
q 2 + 4 (1 − v 2 ) + m2
Z 1   k2
1 2 2 2

' − dv − γ + log(µ 4π) − log (1 − v ) + m
(4π)2 0 4
1
− . (2.150)
(4π)2 (2 − d/2)

To renormalize we demand that π E,ren (0) = 0. Thus

 
π E,ren (k 2 ) = lim π E,d (k 2 ) − π E,d (0)
d→4
Z 1
1  k2 2

= dv log 1 + (1 − v ) ,
4(4π)2 0 4m2

87
which coincides with the Wick rotated result obtained by the Pauli-Villars
method. Thus the renormalization of (2.150) amounts to choosing

µ2
log = γ − log 4π, (2.151)
m2
dropping the pole term and setting d = 4.

2.3.14 Energy shift

Suppose that the perturbation does not depend on time and is given by a
Schwartz function R3 3 ~x 7→ κ(~x). The naive (Wick ordered) Hamiltonian
is Z 1 1 ~
2 1 
π̂ 2 (~x) + ∂ φ̂(~x) + m2 + κ(~x) φ̂2 (x) :d~x


Ĥ := :
2 2 2
The infimum of a quadratic Wick ordered Hamiltonian can be computed exactly
(A.19):
1 1 1 
E = Tr (−∆ + m2 + κ)1/2 − (−∆ + m2 )1/2 − (−∆ + m2 )−1/2 κ
2 2 4 !
2 2
−∆ + m + κ −∆ + m τ2
Z
dτ
= Tr − − κ
(−∆ + m2 + κ + τ 2 ) (−∆ + m2 + τ 2 ) (−∆ + m2 + τ 2 )2 2π
!
τ2 τ2 τ2
Z
dτ
= Tr 2 2
− 2 2
− 2 2 2
κ
(−∆ + m + τ ) (−∆ + m + κ + τ ) (−∆ + m + τ ) 2π
Z
1 1 dτ
= − Tr κ κτ 2
(−∆ + m2 + τ 2 )2 (−∆ + m2 + κ + τ 2 ) 2π
∞ Z n−1 dτ
X 1  1
= (−1)n−1 Tr 2 + τ 2 )2
κ 2 + τ 2)
κ τ2
n=2
(−∆ + m (−∆ + m 2π
∞
X (−1)n Z n dτ
 1 
= Tr 2 2
κ .
n=2
2n (−∆ + m + τ ) 2π

Above, we rewrote the square root by using the identities (A.35) and (A.36),
expanded the denominator in the Neumann series and at the end we used the
identity (A.37). Note that the nth term of the above expansion corresponds to
the loop with n vertices. They are all well defined except for n = 2, which needs
renormalization. We can guess that the renormalized energy shift is

d~k
Z Z
1
E ren = π ren (~k 2 )|κ(~k)|2
3
+ Tr κ (2.152)
(2π) (−∆ + m2 + τ 2 )
1 1 1 dτ
× 2 2
κ 2 2
κ 2 2
τ2 ,
(−∆ + m + τ ) (−∆ + m + κ + τ ) (−∆ + m + τ ) 2π

where we rewrote the sum of terms with n ≥ 3 in a compact form, and π ren was
introduced in (2.137).

88
 Another way to derive the expression for E ren is to use Sucher’s formula.
We introduce the adiabatically switched perturbation e−|t| κ(~x) multiplied by a
coupling constant λ, which will be put to 1 at the end. The Fourier transform
of the switching factor e−|t| is 2 2
+τ 2 Therefore,

Eren = i log(Ω|Ŝren Ω)
42 dτ d~k
Z
= λ2 π ren (−τ 2 + ~k 2 ) 2 2 2
|κ(~k)|2 + O(λ3 ).
( + τ ) (2π)4

By Sucher’s formula,
iλ
E ren = lim ∂λ log(Ω|Ŝren Ω)
&0 2
Z
43 ~
= lim λ 2
π ren (−τ 2 + ~k 2 ) |κ(~k)|2 dτ dk + O(λ3 )
&0 (2 + τ 2 )2 (2π)4
d~k
Z
= λ2 π ren (~k 2 )|κ(~k)|2 + O(λ3 ),
(2π)3
43
R
where we used (2 +τ 2 )2 dτ = 2π. Eventually, we put λ = 1 and we obtain
(2.152).

3 Massive photons
Let m > 0. In this section we discuss the quantization of the Proca equation

−∂µ F µν (x) + m2 Aν (x) = 0, (3.1)

where
F µν := ∂ µ Aν − ∂ ν Aµ . (3.2)
Beside the free equation, we will also consider the Proca equation interacting
with a given vector function J µ , called an external 4-current:

−∂µ F µν (x) + m2 Aν (x) = −J ν (x). (3.3)

We will assume that the 4-current is conserved, that is

∂ν J ν (x) = 0. (3.4)

There are several possible approaches to the Proca equation on the classical
and, especially, quantum level. In particular, one can use from the beginning the
reduced phase space, both for the classical description and quantization. This
is the approach that we will treat as the standard one. Alternative approaches
will be discussed later.

89
3.1 Free massive photons
3.1.1 Space of solutions
Let YPr , resp. CYPr denote the set of real, resp. complex smooth space-compact
solutions of the Proca equation

−∂ µ (∂µ ζν − ∂ν ζµ ) + m2 ζν (x) = 0. (3.5)

The Poincaré group R1,3 o O(1, 3) acts on YPr by

r(y,Λ) ζµ (x) := Λµν ζν (y, Λ)−1 x .

r(y,Λ) are symplectic for Λ ∈ O↑ (1, 3), otherwise they are antisymplectic.
We introduce the functionals Aµ (x) called 4-potentials. They act on ζ ∈ YPr
giving

hAµ (x)|ζi := ζµ (x).

# #−1
On YPr we have the action of the Poincaré group (y, Λ) 7→ r(y,Λ) . Note that
#−1
r(y,Λ) Aµ (x) = (Λ−1 )µν Aν (Λx + y).

We also introduce the field tensor and the electric field vector:

Fµν (x) := ∂µ Aν (x) − ∂ν Aµ (x),

Ei (x) := F0i (x) = Ȧi − ∂i A0 .

3.1.2 Equations of motion

Clearly, the free Proca equation (3.1) is satisfied. Equivalently, we have

(−2 + m2 )Aµ (x) = 0, (3.6)

ν
∂ Aν (x) = 0. (3.7)

Yet another equivalent system of equations convenient for further analysis is

~˙ = 0,
(−∆ + m2 )A0 + divA (3.8)
(−2 + m2 )A~ = 0. (3.9)

~˙ − ∂A
~ =A
Taking the divergence of the definition of the electric field E ~ 0 , then
~
using (3.8), we can express A0 in terms of E:

m2 A0 = ~
−divE. (3.10)

Thus only A~ is dynamical: A0 can be computed from E.

~ Note also that (3.7)
can be rewritten as

Ȧ0 = ~
divA. (3.11)

90
 Finally, we have the following version of the evolution equations in terms of
~ A
E, ~ with only first order derivatives:

~˙ =
A ~ − 1 ∂div
E ~ E, ~ (3.12)
m2
~˙
E = −(−∆ + m2 )A ~ A.
~ − ∂div ~ (3.13)

3.1.3 Symplectic structure on the space of solutions

It is easy to see that for ζ1 , ζ2 ∈ CYPr the following expression defines a conserved
4-current:
µ
jPr (ζ1 , ζ2 , x) (3.14)
µ
∂ν ζ1µ (x) ζ2ν (x) ∂ µ ζ2ν (x) ∂ ν ζ2µ (x) .



:= ∂ ζ1ν (x) − − ζ1ν (x) −

YPr is a symplectic space with the symplectic form

Z
µ
ζ1 ωPr ζ2 = jPr (ζ1 , ζ2 , x)dsµ (x) (3.15)
S
Z  
˙ ~ 10 (t, ~x) ζ~2 (t, ~x) + ζ~1 (t, ~x) ζ~˙2 (t, ~x) − ∂ζ
  
= − ζ~1 (t, ~x) − ∂ζ ~ 20 (t, ~x) d~x,

where S is any Cauchy surface.

Note that the more naive current

j µ (ζ1 , ζ2 , x) := ∂ µ ζ1ν (x)ζ2ν (x) − ζ1ν (x)∂ µ ζ2ν (x) (3.16)

differs from (3.14) by an exterior derivative:

∂ν ζ1µ (x)ζ2ν (x) − ζ1ν (x)ζ2µ (x) .

(3.17)

The flux of (3.17) over a Cauchy surface is zero. Hence (3.16) gives the same
symplectic form as (3.14).
The symplectic form on YPr (3.15) can be written as
Z
~ ~x) ∧ E(t,
ωPr = A(t, ~ ~x)d~x.

It leads to a Poisson bracket on functions on YPr :

{Ai (t, ~x), Aj (t, ~y )} = {Ei (t, ~x), Ej (t, ~y )} = 0,

{Ai (t, ~x), Ej (t, ~y )} = δij δ(~x − ~y ). (3.18)

We have  
∂µ ∂ν
{Aµ (x), Aν (y)} = gµν − D(x − y),
m2
where D(x − y) is the Pauli-Jordan function.

91
 Indeed, this follows after we insert (3.12), (3.10) and (3.11) into
Z 
~˙ ~y ) + Ḋ(t, ~x − ~y )A(0,

~ ~x) =
A(t, D(t, ~x − ~y )A(0, ~ ~y ) d~y ,
Z  
A0 (t, ~x) = D(t, ~x − ~y )Ȧ0 (0, ~y ) + Ḋ(t, ~x − ~y )A0 (0, ~y ) d~y ,

~ ~x).
and then we commute them with A0 (0, ~x) and A(0,

3.1.4 Smeared 4-potentials

We can use the symplectic form to pair distributions and solutions. For ζ ∈ YPr ,
the corresonding spatially smeared 4-potential is the functional on YPr given by

hA((ζ))|ρi := ζωρ.

Note that
{A((ζ1 )), A((ζ2 ))} = ζ 1 ωζ 2 .
Z  
˙
~ 0 ~ ~ ~
A((ζ)) = −(ζ(t, ~x) − ∂ζ (t, ~x))A(t, ~x) + ζ(t, ~x)E(t, ~x) d~x. (3.19)

Another way of smearing the 4-potentials is also useful. For a space-time vec-
tor valued functions f ∈ Cc∞ (R1,3 , R1,3 ) the corresponding space-time smeared
4-potential is Z
A[f ] := fµ (x)Aµ (x)dx. (3.20)

Note that A[f ] = A((ζ)), where

∂µ ∂ ν
ζµ = −D ∗ fµ + D ∗ fν .
m2
Adding to f µ a derivative ∂ µ χ for χ ∈ Cc∞ (R1,3 ) does not change (3.20).

3.1.5 Lagrangian formalism and stress-energy tensor

Consider the Lagrangian density in the off-shell formalism

1 m2
L := − Fµν F µν − Aµ Aµ .
4 2
The resulting Euler-Lagrange equations
 
∂L ∂L
= ∂µ
∂Aα ∂Aα,µ

coincide with the Proca equation.

92
 The canonical stress-energy tensor, which follows directly from the Noether
Theorem, equals

µν ∂L
Tcan = g µν L − A ,ν
∂Aα,µ α
1 m2 
= −g µν Fαβ F αβ + Aα Aα + F µα Aα,ν .
4 2
One usually prefers to replace it with the Belifante-Rosenfeld stress-energy ten-
sor. It is defined as

T µν µν
= Tcan − ∂α Σµνα
1 m2 
= −g µν Fαβ F αβ + Aα Aα + ∂α F αµ Aν + F µα F να ,
4 2
where
Σµνα = −Σανµ := F µα Aν . (3.21)
On solutions of the Euler-Lagrange equations we have
1 m2 
T µν = −g µν Fαβ F αβ + Aα Aα + m2 Aµ Aν + F µα F να ,
4 2
µν
∂µ Tcan = ∂µ T µν = 0.

Note that T µν is symmetric.

L does not contain the time derivative of A0 . Therefore, A0 is not dynamical.
The dynamical variables are Ai . Their conjugate variables are
∂L(x)
= F0i = Ei .
Ȧi (x)
The Hamiltonian and the momentum density obtained from the Belifante-
~ and E
Rosen tensor and expressed in terms of A ~ are
2
1 ~2 1 ~ 2 (x) + m A
~ 2 (x) + (rotA) ~ 2 (x),
H(x) := T 00 (x) = E (x) + (div E)
2 2m2 2
P j (x) := T 0j (x) = m2 A0 (x)Aj (x) + E i (x)F ji (x).

The total Hamiltonian and momentum obtained from both Belifante-Rosen

and canonical stress-energy tensor coincide:
Z Z
00
H := H(t, ~x)d~x = Tcan (t, ~x)d~x,
Z Z
P j := P j (t, ~x)d~x = Tcan
0j
(t, ~x)d~x.

Using (3.12) and (3.13) we check that H generates the equations of motion and
P~ the translations.
The observables H, P~ are in involution.

93
3.1.6 Diagonalization of the equations of motion
For ~k ∈ R3 , ~k 6= ~0 fix two spatial vectors ~e1 (~k), ~e2 (~k) that form an oriented
orthonormal basis of the plane orthogonal to ~k. Define
1  
~e(~k, ±1) := √ ~e1 (~k) ± i~e2 (~k) .
2
Note that
~k × ~e(~k, ±1) = ±i|~k|~e(~k, ±1),
~e(~k, σ) · ~k = 0,
ei (~k, σ)ei (~k, σ 0 ) = δσ,σ0 ,
X ki kj
ei (~k, σ)ej (~k, σ) = δij − .
~k 2
σ=±1
p
Let k ∈ R1,3 with k 0 = ε(~k) = ~k 2 + m2 . Introduce

 |~k| ε(~k)~k 
u(k, 0) := , , (3.22)
m m|~k|
 
u(k, ±1) := 0, ~e(~k, ±1) . (3.23)

Note that

uµ (k, σ)k µ = 0, (3.24)

µ 0
uµ (k, σ)u (k, σ ) = δσ,σ0 , (3.25)
X kµ kν
uµ (k, σ)uν (k, σ) = gµν + . (3.26)
σ=0,±1
m2

Set
Z
~ t (~k)
A ~ ~x)e−i~k~x p d~x ,
=
A(t,
(2π)3
Z
~ t (~k) =
E ~ ~x)e−i~k~x p d~x .
E(t,
(2π)3

We have the equations of motion

~
~˙ t (~k)
A ~ t (~k) + k ~k·E
= E ~ t (~k),
m2
~˙ t (~k)
E = −(~k 2 + m2 )A~ t (~k) + ~k ~k·A
~ t (~k),

the relations
A∗i (~k) = Ai (−~k), Ei∗ (~k) = Ei (−~k),

94
and the Poisson brackets

{A∗ti (~k), Atj (~k 0 )} = {Eti

∗ ~
(k), Etj (~k 0 )} = 0,
{A∗ (~k), Etj (~k 0 )}
ti = δij δ(~k − ~k 0 ).

Set

At (~k, ±1) := ~e(~k, ±1)·A

~ t (~k),

Et (~k, ±1) := ~e(~k, ±1)·E ~ t (~k),

m ~k ~ ~
At (~k, 0) := ·At (k),
ε(~k) |~k|
ε(~k) ~k ~ ~
Et (~k, 0) := ·Et (k).
m |~k|

We have the equations of motion

Ȧt (~k, σ) = Et (~k, σ),

Ėt (~k, σ) = −ε(~k)2 At (~k, σ),

the relations

A∗t (~k, σ) = At (−~k, −σ), Et∗ (~k, σ) = Et (−~k, −σ),

and the Poisson brackets

{A∗t (~k, σ), At (~k 0 , σ 0 )} = {Et∗ (~k, σ), Et (~k 0 , σ 0 )} = 0, (3.27)

{A∗ (~k, σ), Et (~k 0 , σ 0 )}
t = δσσ0 δ(~k − k ).
~0

We set
s
ε(~k) i
at (k, σ) := At (~k, σ) + q Et (~k, σ),
2
2ε(~k)
s
ε(~k) ∗ ~ i
a∗t (k, σ) := At (k, σ) − q Et∗ (~k, σ).
2 ~
2ε(k)

We have the equations of motion

ȧt (k, σ) = −iε(~k)at (k, σ),

a˙∗t (k, σ) = iε(~k)a∗ (k, σ).
t

We will usually write a(k, σ), a∗ (k, σ) for a0 (k, σ), a∗0 (k, σ), so that
~
at (k, σ) = e−itε(k) a(k, σ),
~
a∗t (k, σ) = eitε(k) a∗ (k, σ).

95
The direct definitions of a(k, σ), a∗ (k, σ) are
s !
ε(~k) ~
Z
d~x ~
−ik~ x ~ i ~ ~
a(k, σ) = p e ~e(k, σ)A(0, ~x) + q ~e(k, σ)E(0, ~x) ,
(2π)3 2
2ε(~k)
s !
ε(~k) ~
Z
d~x i~ i
∗
a (k, σ) = p e k~x ~ ~x) − q
~e(k, σ)A(0, ~e(~k, σ)E(0,
~ ~x) .
(2π)3 2
2ε(~k)

Their Poisson brackets are

{a(~k, σ), a(~k 0 , σ 0 )} = {a∗ (~k, σ), a∗ (~k 0 , σ 0 )} = 0,

{a(~k, σ), a∗ (~k 0 , σ 0 )} = −iδ(~k − ~k 0 )δσ,σ0 .

The 4-potentials can be written as

X Z d~k 
ikx −ikx ∗

Aµ (x) = q u µ (k, σ)e a(k, σ) + u µ (k, σ)e a (k, σ) .
(2π)3 2ε(~k)
p
σ=0,±1

We have accomplished the diagonalization of the Hamiltonian, momentum

and symplectic form:

X Z
H = d~kε(~k)a∗ (k, σ)a(k, σ),
σ=0,±1
X Z
P~ = d~k~ka∗ (k, σ)a(k, σ),
σ=0,±1
X Z
iω = a∗ (k, σ) ∧ a(k, σ)d~k.
σ=0,±1

3.1.7 Plane waves

A plane wave is defined as
1
|k, σ)µ = p q uµ (k, σ)eikx , (3.28)
(2π)3 2ε(~k)
p
with k 0 = ε(~k) = ~k 2 + m2 . We have |k, σ) = | − k, −σ) and

i(k, σ|ω|k 0 , σ 0 ) = i(k, σ|ω|k 0 , σ 0 ) = 0,

−i(k, σ|ω|k 0 , σ 0 ) = i(k, σ|ω|k , σ )0 0
= δ(~k − ~k 0 )δσ,σ0 .

96
a(k, σ) can be called plane wave functionals:

a(k, σ) = −iA((|k, σ)))

Z  


= −i ∂t (k, σ|x)i − ∂i (k, σ|x)0 Ai (0, ~x) − (k, σ|x)i Ei (0, ~x) d~x,

a∗ (k, σ) = iA((| − k, σ)))

Z  


= i ∂t (k, σ|x)i − ∂i (k, σ|x)0 Ai (0, ~x) − (k, σ|x)i Ei (0, ~x) d~x.

Note that

iha(k, σ)|ζi = (k, σ|ωζ, (3.29)

−iha(k, σ)|ζi = (k, σ|ωζ. (3.30)

3.1.8 Positive frequency space

(+)
WPr will denote the subspace of CYPr consisting of positive frequency solutions:
(+)
WPr := {g ∈ CYPr : (k, σ|ωg = 0, σ = ±, 0}.
(+)
Every g ∈ WPr can be written as

X Z d~k
gµ (x) = q eikx uµ (k, σ)ha(k, σ)|gi.
~
p
σ=0,±1 3
(2π) 2ε(k)

(+)
For g1 , g2 ∈ WPr we define the scalar product
X Z
(g1 |g2 ) := ig 1 ωg2 = ha(k, σ)|g1 iha(k, σ)|g2 id~k. (3.31)
σ=0,±1

(+)
We set ZPr to be the completion of WPr in this scalar product. R1,3 o
O↑ (1, 3) leaves ZPr invariant.
We have
ha(k, σ)|gi = (k, σ|g).
We can identify ZPr with L2 (R3 , C3 ) and rewrite (3.31) as
X Z
(g1 |g2 ) = (k, σ|g1 )(k, σ|g2 )d~k.
σ=0,±1

3.1.9 Spin averaging

For a given k ∈ R1,3 with k 2 = m2 , let M, N be vectors with

M µ kµ = N ν kν = 0.

97
The following identity allows us to average over spin and is useful in computa-
tions of scattering cross-sections:
X
M µ uµ (k, σ)uν (k, σ)N ν = M µ Nµ . (3.32)
σ=0,±1

In fact,
X kµ kν
uµ (k, σ)uν (k, σ) = gµν + .
σ=0,±1
m2

Therefore, the left hand side of (3.32) equals

(M · k)(N · k)
M µ gµν N ν + .
m2
But
k · M = k · N = 0.

3.1.10 Quantization
We want to construct (H, Ĥ, Ω) satisfying the standard requirements of QM and
a self-adjoint operator-valued distribution R1,3 3 x 7→ µ (x) such that, setting
~ ~˙
Ê = Â − ∂~ Â0 , we have
(1) −∂ µ (∂µ Âν − ∂ν µ ) + m2 Âν (x) = 0;
(2) [Âi (0, ~x), Âj (0, ~y )] = [Êi (0, ~x), Êj (0, ~y )] = 0,
[Âi (0, ~x), Êj (0, ~y )] = iδij δ(~x − ~y );
(3) eitĤ µ (x0 , ~x)e−itĤ = µ (x0 + t, ~x);
(4) Ω is cyclic for µ (x).
The above problem has an essentially unique solution, which we describe
below.
For the Hilbert space we should take the bosonic Fock space H = Γs (ZPr )
and for Ω the Fock vacuum. With ZPr ' L2 (R3 , C3 ) and k on shell we have
creation operators
â∗ (k, σ) = â∗ |k, σ) ,

written in both “physicist’s” and “mathematician’s notation” satisfying

[â(k, σ), â(k 0 , σ 0 )] = [â∗ (k, σ), â∗ (k 0 , σ 0 )] = 0,

∗ 0
[â(k, σ), â (k , σ )] 0
= δ(~k − ~k 0 )δσ,σ0 .

Therefore the quantum 4-potentials

µ (x) (3.33)

d~k
Z X  
3
= (2π)− 2 q uµ (k, σ)eikx â(k, σ) + uµ (k, σ)e−ikx â∗ (k, σ)
2ε(~k) σ=0,±1

98
satisfy the required commutation relations. The quantum Hamiltonian and mo-
mentum are
X Z
Ĥ = ε(~k)â∗ (k, σ)â(k, σ)d~k,
σ=0,±1

~ X Z
P̂ = ~kâ∗ (k, σ)â(k, σ)d~k.
σ=0,±1

1,3 ↑
 R o O (1, 3) is unitarily implemented on H by U (y, Λ) :=
 The group
Γ r(y,Λ) . We have
ZPr

U (y, Λ)µ (x)U (y, Λ)∗ = (Λ−1 )µν Âν (y, Λ)x .

Moreover,
 
∂µ ∂ν
[µ (x), Âν (y)] = −i gµν − D(x − y).
m2

Theorem 3.1 Note the identities

 
∂µ ∂ν
(Ω|µ (x)µ (y)Ω) = −i gµν − D(+) (x − y),
m2
 
∂µ ∂ν
(Ω|T(µ (x)Âν (y))Ω) = −i gµν − DF (x − y). (3.34)
m2

The second identity is valid away from x = y.

Proof. To see (3.34, we compute, using (3.26):

(Ω|µ (x)µ (y)Ω)

d~k
Z X  
3
= Ω|(2π)− 2 q uµ (k, σ)eikx â(k, σ) + uµ (k, σ)e−ikx â∗ (k, σ)
2ε(~k) σ=0,±1
!
3
Z
d ~0
k X  0 0

×(2π)− 2 q uµ (k 0 , σ 0 )eik y â(k 0 , σ 0 ) + uµ (k 0 , σ 0 )e−ik y â∗ (k 0 , σ 0 ) Ω
~ 0
2ε(k ) σ =0,±1
0

d~k ik(x−y) X
Z
=(2π)−3 e uµ (k, σ)uµ (k 0 , σ 0 )
2ε(~k) σ=0,±1
Z ~
dk ik(x−y)  kµ kν 

∂µ ∂ν

−3
=(2π) e gµν + = −i gµν − D(+) (x − y).
2ε(~k) m2 m2

99
Here is a proof of (3.34):

(Ω|T(µ (x)Âν (y))Ω)

=θ(x0 − y 0 )(Ω|µ (x)Âν (y)Ω) + θ(y 0 − x0 )(Ω|µ (y)Âν (x)Ω)
   
∂µ ∂ν ∂µ ∂ν
= − iθ(x0 − y 0 ) gµν − D (+)
(x − y) + iθ(y 0
− x 0
) gµν − D(−) (x − y)
m2 m2
 
∂µ ∂ν
= − i gµν − DF (x − y),
m2

where the last equality is clear for x0 − y 0 6= 0. However, D(+) (x − y) and

D(−) (x − y) coincide for spacelike separated x, y. Hence only x = y remains
dubious. 2
For f ∈ Cc∞ (R1,3 , R1,3 ) set
Z
Â[f ] := f µ (x)µ (x)dx.

We obtain a family that satisfies the Wightman axioms with D := Γfin

s (ZPr ).
For an open set O ⊂ Rd we set
n o
A(O) := exp(iÂ[f ]) : f ∈ Cc∞ (O, R1,3 ) .

The algebras A(O) satisfy the Haag-Kastler axioms.

3.2 Massive photons with an external 4-current

3.2.1 Classical 4-potentials
We return to the classical Proca equation. We assume that

R1,3 3 x 7→ J(x) = [J µ (x)] ∈ R1,3 (3.35)

is a given function called an external 4-current, which satisfies

∂ν J ν (x) = 0. (3.36)

In most of this subsection we will assume that (3.35) is Schwartz. In its presence
the Proca equation takes the form

−∂ µ (∂µ Aν − ∂ ν Aµ ) + m2 Aν = −J ν . (3.37)

Note that (3.37) and (3.36) imply the Lorentz condition

∂ν Aν = 0. (3.38)

We have therefore
(−2 + m2 )Aµ = −J µ . (3.39)

100
 The only dynamical variables are the spatial components, satisfying the
equation
~ = −J.
(∂02 − ∆ + m2 )A ~ (3.40)
The temporal component of (3.37) has no time derivative:

~ + m2 A0 = −J0 .
−∆A0 + ∂0 divA (3.41)

~ at the same time:

Therefore, we can compute A0 in terms of A
~ + J0 ).
A0 = −(−∆ + m2 )−1 (∂0 divA (3.42)

We can introduce
~
E(x) ~˙
= A(x) ~ 0 (x).
− ∂A (3.43)
~
We can compute A0 in terms of E:
1 ~
A0 = − (J0 + divE). (3.44)
m2
The current (3.14) is conserved also in the presence of interaction and defines
the symplectic form (3.15). Consequently, the equal time Poisson brackets are
the same as in the free case:

{Ai (t, ~x), Aj (t, ~y )} = {Ei (t, ~x), Ej (t, ~y )} = 0,

{Ai (t, ~x), Ej (t, ~y )} = δij δ(~x − ~y ). (3.45)

3.2.2 Lagrangian and Hamiltonian formalism

The Lagrangian density is

1 m2
L := − Fµν F µν − Aµ Aµ − Jµ Aµ
4 2
1 1 m2
= − ∂µ Aν ∂ µ Aν + ∂µ Aν ∂ ν Aµ − Aµ Aµ − Jµ Aµ
2 2 2
2 2
1 ~ 2 + 1 (∂A ~ 0 )2 + 1 A
~˙ 2 − A


~ 0 + m A2 − m A
~˙ ∂A ~ 2 − J~A
~ + J 0 A0 .
= − (rotA)
2 2 2 2 0 2
~
As noted before, only spatial components A(x) are dynamical. The conjugate
~
variable is E(x).
The definition of the canonical and Belifante-Rosen energy-momentum ten-
sor are defined analogously as in the free case. In particular, the canonical

101
Hamiltonian density is
∂L(x)
Hcan (x) = −L(x) + Ȧi (x)
∂ Ȧi (x)
1 ~ 2 (x) − 1 (∂A~ 0 )2 (x) + 1 A ~˙ 2 (x)


= (rotA)
2 2 2
m2 2 m2 ~ 2 ~ A(x)~
− A (x) + A (x) + J(x) − J0 (x)A0 (x)
2 0 2
1 ~ 2 (x) + 1 (E)~ 2 (x) + E(x)
~ ∂A ~ 0 (x)
= (rotA)
2 2
m2 2 m2 ~ 2 ~ A(x)~
− A0 (x) + A (x) + J(x) − J0 (x)A0 (x).
2 2
~ 0


We add to it a spatial divergence div E(x)A (x) and use (3.44) to eliminate
A0 , obtaining the usual (Belifante-Rosen) Hamiltonian density
1 ~2 1 ~ 2 (x)
H(x) := E (x) + (rotA)
2 2
2
1 ~ 2 (x) + m A~ 2 (x) + J(x)
~ A(x).
~
+ 2 J 0 − divE)
2m 2
The Hamiltonian
Z Z
H(t) = H(t, ~x)d~x = Hcan (t, ~x)d~x (3.46)

generates the equations of motion. Using the splitting of A ~ and E ~ into the
transversal and longitudinal part as in (A.41) and (A.42), we can rewrite H(t)
as
Z 1
~ tr
2 1~ ~ tr (t, ~x) + J(x)
~ A ~ tr (x)

H(t) = d~x E (t, ~x) + A x)(−∆ + m2 )A
tr (t, ~
2 2
Z 1 1 1
2
+ d~x Elg (t, ~x)2 + 2
J 0 (t, ~x) − (−∆) 2 Elg (t, ~x)
2 2m
m2 
+ Alg (t, ~x)2 . (3.47)
2
We can interpret interacting fields as functionals on YPr satisfying
~ ~x) = A
A(0, ~ fr (0, ~x), ~ ~x) = E
E(0, ~ fr (0, ~x).

3.2.3 Quantization
We are looking for operator valued distributions R1,3 3 x 7→ µ (x) satisfying
−∂µ (∂ µ Âν (x) − ∂ ν µ (x)) + m2 Âν (x) = −J ν (x),
˙
having the standard equal time commutation relations with Ê i := Âi − ∂i Â0
[Âi (0, ~x), Âj (0, ~y )] = [Êi (0, ~x), Êj (0, ~y )] = 0,
[Âi (0, ~x), Êj (0, ~y )] = iδij δ(~x − ~y ).

102
 ~ ~
We will assume that Â, Ê coincide with free fields at t = 0:

Âi (~x) := Âi (0, ~x) = Âifr (0, ~x),

i
Ê (~x) := Ê i (0, ~x) = Êfr
i
(0, ~x).

We have
 Z 0   Z t 
µ (t, ~x) := Texp −i Ĥ(s)ds µ (~x)Texp −i Ĥ(s)ds ,
t 0

where the Hamiltonian Ĥ(t), and the corresponding Hamiltonian in the inter-
action picture are
Z 1
~ 1 ~
Ĥ(t) = d~x : Ê 2 (~x) + (J 0 (t, ~x) − divÊ(~x))2
2 2m2
1 ~ m2 ~ 2 ~
~ ~x)Â(~

+ (rotÂ)2 (~x) + Â (~x) + J(t, x) :, (3.48)
Z 2 2
1 ~ ~ ~x)Â ~ 1 0 
ĤInt (t) = d~x − 2 J 0 (t, ~x)divÊfr (t, ~x) + J(t, fr (t, ~
x) + J (t, ~
x ) 2
.
m 2m2
Recall that by (3.10)
~
m2 Â0fr = divÊfr . (3.49)
Using this we express ĤInt (t) in terms of creation/annihilation operators:
Z  1 0 
ĤInt (t) = d~x Jµ (t, ~x)µfr (t, ~x) + J (t, ~
x ) 2
2m2
d~k
Z 
~
= q eitε(k) Jµ (t, ~k)uµ (k, σ)â∗ (k, σ)
(2π)3 2ε(~k)
p

 Z d~k |J 0 (t, ~k)|2

~
+e−itε(k) Jµ (t, ~k)uµ (k, σ)â(k, σ) + .
(2π)3 2m2
We can compute the scattering operator

 Z Z 
i dk µ 0 ν i 0 2
Ŝ = exp − J (k)Dµν (k)J (k) − J (x) dx
2 (2π)4 2m2
 
X Z d~k u µ (k, σ)
× exp −i p a∗ (k, σ) q J µ (ε(~k), ~k)
(2π) 3
σ=0,±1 ~
2ε(k)
 
X Z ~
dk uµ (k, σ) µ ~ ~ 
× exp −i p a(k, σ) q J (ε(k), k) , (3.50)
(2π) 3
σ=0,±1 2ε(~k)

where  
0 1 kµ kν
Dµν (k) = 2 gµν + . (3.51)
m + k 2 − i0 m2

103
 0
(The superscript 0 over Dµν will be explained later on).
For xN , . . . , x1 , the N -point Green’s function is defined as follows:
hµN (xN ) . . . µ1 (x1 )i


:= Ω+ |T µN (xN ) · · · µ1 (x1 ) Ω− .

Green functions can be organized into the generating function

X∞ Z Z
· · · hµN (xN ) · · · µ1 (x1 )i(−i)N f µN (xN ) · · · f µ1 (x1 )dxN · · · dx1
n=0
  Z ∞ Z  
µ
= Ω Texp −i ĤInt (s)ds − i f (x)µ (x)dx Ω =: Z(f ).
−∞

The amputated N -point Green’s functions are

hµN (kN ) · · · µ1 (k1 )iamp
2
:= (kN + m2 ) · · · (k12 + m2 )hµN (kN ) · · · µ1 (k1 )i.
For k1 , . . . , kN on shell, set
|kN , σN , . . . , k1 , σ1 ) := â∗ (kN , σN ) · · · â∗ (k1 , σ1 )Ω.
Amputated Green’s functions can be used to compute scattering amplitudes:
 
kn++ , σn++ , . . . , k1+ , σ1+ | Ŝ |kn−− , σn−− , . . . , k1− , σ1−

+ + − −
uµ1 (k1+ , σ1+ ) · · · uµn+ (kn++ , σn++ )uµn− (kn−− , σn−− ) · · · uµ1 (k1− , σ1− )
= q q q q
n+ +n−
(2π) 2 2ε(k1+ ) · · · 2ε(kn++ ) 2ε(kn−− ) · · · 2ε(k1− )
×hµ+ (k1+ ) · · · µ+ (kn++ )µ− (−kn−− ) · · · µ− (−k1− )iamp .
1 n+ n− 1

3.2.4 Causal propagators

The causal propagator used to compute Green’s functions and scattering  ampli- 
∂µ ∂ν
tudes that follows directly from the interaction picture Hamiltonian is gµν − m 2 Dc ,
see (3.34). If we compute scattering amplitudes, we can pass from this propa-
gator to another by adding kµ fν (k) + fµ (k)kν for an arbitrary function fµ (k).
To see this note that after adding kµ fν (k) + fµ (k)kν the contribution of each
line changes by
J µ (k) (kµ fν (k) + fµ (k)kν ) J ν (k),
which is zero, because kµ J µ (k) = 0. For scattering amplitudes, external lines
do not involve the propagator. Therefore, scattering amplitudes do not change.
Below we will list a number of useful causal propagators. (In principle,
they should be decorated by the superscript c, for causal, which we however
suppress).

104
 For any α ∈ R, we can pass to the following propagators
 
α 1 kµ kν
Dµν = gµν + (1 − α) 2 .
m2 + k 2 − i0 αk + m2

The above propagator for α = 0 was obtained in the Hamiltonian approach, see
(3.34). For α = 1 we obtain the so-called propagator in the Feynman gauge

Feyn 1
Dµν (k) = .
m2 + k 2 − i0
α = ∞ corresponds to the propagator in the Landau or Lorentz gauge:
 
Lan 1 kµ kν
Dµν = gµν − .
m2 + k 2 − i0 k2

We can introduce the propagator in the Yukawa gauge:

 
Yuk 1 Yuk Yuk 1 ki kj
D00 =− , D0j = 0, Dij = 2 δ ij − .
m2 + ~k 2 m + k 2 − i0 m2 + ~k 2
Yuk Feyn
We have Dµν = Dµν + kµ fνYuk (k) + fµYuk (k)kν , where

k0 ki
f0Yuk (k) = , fiYuk (k) = − .
(k 2 + m2 − i0)2(m2 + ~k 2 ) (k 2 + m2 − i0)2(m2 + ~k 2 )

(The propagator in the Yukawa gauge is the massive analog of the propagator
in the Coulomb gauge.)
The propagator in the temporal gauge is
 
tem tem tem 1 ki kj
D00 = 0, D0j = 0, Dij = 2 δij − .
k + m2 − i0 k02
tem Feyn
We have Dµν = Dµν + kµ fνtem (k) + fµtem (k)kν , where

1 ki
f0tem (k) = , fitem (k) = − 2 .
(m2 + k 2 − i0)2k0 (m + k 2 − i0)2k02

3.2.5 Feynman rules

Perturbation expansion can be organized with help of Feynman diagrams, which
are very similar to diagrams for neutral fields interacting with a linear source.
We have 1 kind of lines and 1 kind of vertices. At each vertex just one line ends.
To compute Green’s functions we do as follows:
(1) In the nth order we draw all possible Feynman diagrams with n vertices
and external lines.
(2) To each vertex we associate the factor −iJ µ (k).
 
0 −i kµ kν
(3) To each line we associate the propagator −iDµν (k) = m2 +k2 −i0 gµν + m2 .

105
 d4 k
(4) For internal lines we integrate over the variables with the measure (2π)4 .
To compute scattering amplitudes with N − incoming and N + outgoing par-
ticles we draw the same diagrams as for N − + N + -point Green’s functions. The
rules are changed only concerning the external lines.
(i) With each incoming external line we associate √ 1 u(k, σ).
(2π)3 2ε(~
k)
1
(ii) With each outgoing external line we associate √ u(k, σ).
(2π)3 2ε(~
k)
0
If we prefer, we can use a different causal propagator instead of Dµν . Green’s
functions change, because of external lines, however scattering amplitudes do
not.

3.2.6 Path integral formulation

We can compute exactly the generating function:

Z(f ) (3.52)
−2
 Z 
i (gµν + m k k
µ ν )
= exp J µ (k) + f µ (k) (J ν (k) + f ν (k))dk .
2 (k 2 + m2 − i0)

This can be rewritten in the path integral formalism. Recall that

Z Z 
1
Lfr (x)dx = − ∂µ Aν (x)∂ µ Aν (x) − ∂µ Aν (x)∂ ν Aµ (x)
2

+m2 Aµ (x)Aµ (x) dx
Z
1
Aµ (x) g µν (−2 + m2 ) + ∂ µ ∂ ν Aν (x)dx,


= −
2
Z Z Z
(L(x) − f µ (x)Aµ )(x)dx = Lfr (x)dx − (J µ + f µ (x))(x)Aµ (x)dx.

0 g +m−2 k k 0
Note that Dµν (k) = µνk2 +m2 −i0
µ ν
, or in the position representation Dµν =
−2 F µν 2 µ ν
(gµν − m ∂µ ∂ν )D is one of the inverses of g (−2 + m ) + ∂ ∂ . Therefore,
(3.52) is often formally rewritten as



Π Π dAµ (x) exp i L(x) − (J µ (x) + f µ (x))Aµ (x) dx
R R
µ x
Z(f ) = R R
.
Π Π dAµ (x) exp i Lfr (x)dx
µ x

•
Let Dµν be one of the propagators considered in Subsubect. 3.2.4. Let B•µν
be its inverse. We have the corresponding “free action”
Z
1
T•fr = − Aµ (x)B•µν (x − y)Aν (y)dxdy.
2

106
We define the corresponding generating function as

Z• (f ) (3.53)
 Z 
i
:= exp (J µ (k) + f µ (k)D•µν (k)(J ν (k) + f ν (k))dk
2
 Z 
i µ µ ν ν
= exp (J (x) + f (x))D•µν (x − y)(J (y) + f (y))dxdy
2


Π Π dAµ (x) exp iT•fr + i (J µ (x) + f µ (x))Aµ (x)dx
R R
µ x
= R .
Π Π dAµ (x) exp (iT•fr )
µ x

•
In general, Z• (f ) differs for various propagators Dµν , unless f satisfies the
Lorentz condition. However, all Z• (f ) can be used to compute the same scat-
tering operator.
Likewise, the Euler-Lagrange equations obtained from those various action
integrals differ from the Proca equation. However, YPr belong always to their
solutions.
If we take the Lagrangian
1
− ∂µ Aν (x)∂ µ Aν (x) + m2 Aν (x)Aν (x)
2 
+(α − 1)∂µ Aµ (x)∂ν Aν (x) , (3.54)

α
then we obtain the propagator Dµν . Indeed,

g µν (k 2 + m2 ) + (α − 1)k µ k ν
α
is the inverse of Dµν (k).
If we restrict the integration by the Lorentz condition

∂µ Aµ (x) = 0. (3.55)

and take the free Lagrangian (3.54) (they now coincide for all α), then we obtain
the propagator in the Landau/Lorentz gauge.
If we take the free Lagrangian
1
− ∂µ Ai (x)∂ µ Ai (x) + m2 Ai (x)Ai (x)
2
1
+ 2 ∂µ ∂i Ai (x)∂ µ ∂j Aj (x) + ∂i Ai (x)∂j Aj (x)
m 
−∂i A0 (x)∂i A0 (x) − m2 A0 (x)2 ,

Yuk
we obtain Dµν . Indeed,
 
ki kj
2 2
(k + m ) δij + 2 − δµ0 δ0ν (~k 2 + m2 )
m

107
 Yuk
is the inverse of Dµν (k).
If we take the free action
Z
1
∂µ Ai (x)∂ µ Ai (x) + m2 Ai (x)Ai (x) dx


−
2
Z
1 
− ∂µ ∂i Ai (x)(−2)−1 (x − y)∂ µ ∂j Aj (y)
2

+∂i Ai (x)(−2)−1 (x − y)∂j Aj (y) dxdy,
tem
(which is nonlocal and does not involve A0 ), we obtain Dµν . Indeed,
 
ki kj
(k 2 + m2 ) δij − 2
k
tem
is the inverse of Dij (k).

3.2.7 Energy shift

Suppose that the 4-current is stationary and is given by a Schwartz function
R3 3 ~x 7→ J µ (~x). Note that divJ(~ ~ x) = 0.
Using the quantum version of (3.47), we can write the Hamiltonian as
Z 1
~2 1~ ~ ~
~ x)Â

Ĥ = d~x : Êtr (~x) + Âtr (~x)(−∆ + m2 )Âtr (~x) + J(~ tr (~
x) : (3.56)
2 2

2 m 2
Z 1 1 1

+ d~x: Êlg (~x)2 + J 0
(~
x ) − (−∆) 2 Ê (~
lg x ) + Â lg (~
x )2
:. (3.57)
2 2m2 2
We check that infimum of Ĥ is
Z Z −m|~x−~y|
1 ~ x) e ~ y)
E = − d~xd~y J(~ J(~ (3.58)
2 4π|~x − ~y |
e−m|~x−~y| 0
Z Z
1
+ d~xd~y J 0 (~x) J (~y ). (3.59)
2 4π|~x − ~y |
In fact, by (A.14) the infimum of (3.56) is (3.58). Rewriting (3.57) as
m2
Z 
1
d~x 2
Êlg (~x)(−∆ + m2 )Êlg (~x) + Âlg (~x)2 (3.60)
2m 2

1 0 2 1 1
+ J (~
x ) − J0 (~
x )(−∆) 2 Ê (~
lg x )
2m2 m2
and applying (A.14), we check that that the infimum of (3.57) is (3.59).

3.3 Alternative approaches

3.3.1 Classical 4-potentials without the Lorentz condition
So far our treatment of massive photons was based on the Proca equation (3.1).
As we remember, the Proca equation is equivalent to the Klein-Gordon equation

108
for vector fields (3.6) together with the Lorentz condition (3.7). This suggests
an alternative approach to the massive photons.
In this approach one considers first the Klein-Gordon equation on functions
with values in R1,3 :
(−2 + m2 )ζµ (x) = 0. (3.61)
The space of smooth real space-compact solutions of (3.61) will be denoted by
Yvec . The following 4-current
µ
jvec (ζ1 , ζ2 , x) := ∂ µ ζ1,ν (x)ζ2ν (x) − ζ1,ν (x)∂ µ ζ2ν (x)

is conserved, that is
µ
∂µ jvec (x) = 0.
It defines in the usual way a symplectic form on Yvec
Z
µ
ζ1 ωvec ζ2 = jvec (ζ1 , ζ2 , x)dsµ (x)
S
Z  
= −ζ̇1ν (t, ~x)ζ2ν (t, ~x) + ζ1ν (t, ~x)ζ̇2ν (t, ~x) d~x,

where S is any Cauchy surface.

One introduces the 4-potentials Aµ (x) as the functionals on Yvec defined by

hAµ (x)|ζi := ζ µ (x).

We clearly have
(−2 + m2 )Aµ (x) = 0. (3.62)
We can use the Lagrangian

1 m2
L(x) := − Aµ,ν (x)Aµ,ν (x) − Aµ (x)Aµ (x).
2 2
The conjugate variables are
∂
Πµ (x) := L(x) = Ȧµ (x).
∂ Ȧµ (x)

The Poisson structure is given by the equal time brackets

{Aµ (t, ~x), Aν (t, ~y )} = {Πµ (t, ~x), Πν (t, ~y )} = 0,

{Aµ (t, ~x), Πν (t, ~y )} = gµν δ(~x − ~y ).

The stress-energy tensor is

∂L
T µν = − A ,ν + g µν L
∂Aα,µ α
1
Aα,µ Aα,ν − g µν Aα,β Aα,β + m2 Aα Aα .


=
2

109
The Hamiltonian and momentum density are

1 1 m2
H(x) = T 00 (x) = Πµ (x)Πµ (x) + Aµ,i (x)Aµ,i (x) + Aµ (x)Aµ (x),
2 2 2
P i (x) = T 0i (x) = −Πµ (x)Aµ,i (x).

As usual, we can define the Hamiltonian and momentum

Z
H = H(t, ~x)d~x, (3.63)
Z
Pj = P j (t, ~x)d~x.

The Hamiltonian (3.63) is unbounded from below.

3.3.2 The Lorentz condition

Introduce two subspaces of Yvec

YLor := {ζ ∈ Yvec : ∂µ ζ µ = 0},

Ysc := {ζ ∈ Yvec : ζ µ = ∂ µ χ, χ ∈ YKG }.

Note that Yvec = YLor ⊕ Ysc is a decomposition into symplectically orthogonal

subspaces, each preserved by the Poincaré group. If ζ ∈ Yvec , then its projection
onto Ysc is
µ 1
ζsc := 2 ∂ µ ∂ν ζ ν . (3.64)
m
Elements of YLor satisfy the Proca equation, so that we can make the iden-
tification
YLor = YPr .
On YLor the forms ωvec and ωPr coincide.
Clearly, we are back with the theory that was used in most of this section.
In particular, the Hamiltonian (3.63) restricted to YLor is now positive.

3.3.3 Diagonalization of the equations of motion

In order to diagonalize the Hamiltonian, besides the vectors u(k, σ) with σ =
0, ±1 introduced in (3.22), we will need the vectors for the scalar plane waves
1
u(k, sc) := (ε(~k), ~k).
m
Note that

uµ (k, σ)uµ (k, σ 0 ) = δσ,σ0 ,

X
uµ (k, σ)uν (k, σ) = gµν .
σ

110
 Set
Z
~ t (~k)
A = ~ ~x)e−i~k~x p d~x ,
A(t,
(2π)3
Z
~ t (~k) =
Π ~ ~x)e−i~k~x p d~x .
Π(t,
(2π)3
We have the equations of motion
Ȧt (~k) = Πt (~k),
Π̇t (~k) = −ε(~k)2 At (~k),
the relations
A∗t (~k) = At (−~k), Π∗t (~k) = Πt (−~k),
and the Poisson brackets
{A∗tµ (~k), Atν (~k 0 )} = {Π∗tµ (~k), Πtν (~k 0 )} = 0,
{A∗ (~k 0 ), Πtµ (~k)}
tν = gµν δ(~k − ~k 0 ). (3.65)
Set
At (~k, σ) := uµ (~k, σ)Aµt (~k),
Πt (~k, σ) := uµ (~k, σ)Πµt (~k).

We have the equations of motion

Ȧt (~k, σ) = Πt (~k, σ),
Π̇t (~k, σ) = −ε(~k)2 At (~k, σ),

the relations
A∗t (~k, σ) = At (−~k, −σ), Π∗t (~k, σ) = Πt (−~k, −σ),
and the Poisson brackets
{A∗t (~k, σ), At (~k 0 , σ 0 )} = {Π∗t (~k, σ), Πt (~k 0 , σ 0 )} = 0,
{A∗ (~k, σ), Πt (~k 0 , σ 0 )}
t = κσσ0 δ(~k − ~k 0 ), (3.66)
where κσ,σ0 = 1 for σ = σ 0 = ±1, 0 and κsc,sc = −1. We set
s
ε(~k) i
at (k, σ) := At (~k, σ) − q Πt (~k, σ),
2 ~
2ε(k)
s
ε(~k) ∗ ~ i
a∗t (k, σ) := At (k, σ) + q Π∗t (~k, σ).
2 ~
2ε(k)

111
 We have the equations of motion

ȧt (k, σ) = −iε(~k)at (k, σ),

a˙∗t (k, σ) = iε(~k)a∗ (k, σ),
t

and the Poisson brackets

{a(k, σ), a(k 0 , σ 0 )} = {a∗ (k, σ), a∗ (k 0 , σ 0 )} = 0,

∗ 0
{a(k, σ), a (k , σ )} 0
= −iκσ,σ0 δ(~k − ~k 0 ).

We diagonalize the Hamiltonian and momentum:

X Z Z
H = dkε(k)a (k, σ)a(k, σ) − d~kε(~k)a∗ (k, sc)a(k, sc),
~ ~ ∗

σ=0,±1
X Z Z
P~ = d~k~ka∗ (k, σ)a(k, σ) − d~k~ka∗ (k, sc)a(k, sc).
σ=0,±1

The 4-potentials can be decomposed as

XZ d~k 
ikx −ikx ∗

Aµ (x) = q uµ (k, σ)e a(k, σ) + uµ (k, σ)e a (k, σ) .
(2π)3 2ε(~k)
p
σ

Clearly, the restriction to YLor amounts to dropping all scalar components.

3.3.4 Positive frequency space

(+)
Wvec will denote the subspace of CYvec consisting of positive frequency solutions:
(+)
Wvec := {g ∈ CYPr : (k, σ|ωvec g = 0, σ = ±, 0, sc}.
(+)
Every g ∈ Wvec can be written as

d~k
X Z
gµ (x) = q eikx uµ (k, σ)ha(k, σ)|gi.
~
p
σ=0,±1,sc 3
(2π) 2ε(k)

(+)
For g1 , g2 ∈ Wvec we have a natural scalar product
X Z
(g1 |g2 ) := ig 1 ωvec g2 = ha(k, σ)|g1 iha(k, σ)|g2 id~k
σ=0,±1
Z
− ha(k, sc)|g1 iha(k, sc)|g2 id~k
Z
= g µν haµ (k)|g1 ihaν (k)|g2 id~k. (3.67)

112
Unfortunately, the above definition gives an indefinite scalar product. We can
also introduce a positive definite scalar product, which unfortunately is not
covariant:
XZ
(g1 |g2 )+ := haµ (k)|g1 ihaµ (k)|g2 id~k.
µ

(+)
The positive frequency space Wvec equipped with the scalar product (3.67) can
be completed in the norm given by (·|·)+ . It will be called Zvec . It is an example
of the so-called Krein space, which is a space with an indefinite scalar product
and has a topology given by a positive scalar product.
(+)
Using the projection (3.64), Wvec can be decomposed into the direct sum
(+) (+) (+)
of orthogonal subspaces WLor and Wsc . On WLor the scalar product (3.67)
(+)
is positive definite, on Wsc it is negative definite. Their completions will be
denoted ZLor and Zsc .
Every ζ ∈ Yvec can be uniquely written as ζ = ζ (+) + ζ (+) , where ζ (+) ∈
(+)
Wvec . This allows us to define a real scalar product on Yvec :
(+) (+)
hζ1 |ζ2 iY := Re(ζ |ζ2 ) (3.68)
Z Z1
= ζ̇1µ (0, ~x)(−i)D(+) (0, ~x − ~y )ζ̇2µ (0, ~y )d~xd~y
Z Z
+ ζ1µ (0, ~x)(−∆~x + m2 )(−i)D(+) (0, ~x − ~y )ζ2µ (0, ~y )d~xd~y .

Again, (3.68) is positive definite on YLor and negative definite on Ysc .

3.3.5 “First quantize, then reduce”

The quantization described in Subsect. 3.1 will be called “first reduce, then
quantize”. There exist alternative methods of quantization, which use the sym-
plectic space Yvec introduced in (3.61) as the basis. There are two basic ways
to implement this idea.
The first insists on using only positive definite Hilbert spaces. Unfortunately,
the Hamiltonian turns out to be unbounded from below.
In the Gupta-Bleuler approach the 4-potentials µ (x) evolve with positive
frequencies. Unfortunately, it uses an indefinite scalar product.

3.3.6 Quantization without reduction on a positive definite Hilbert

space
In this approach we use the Hilbert space

Γs (ZLor ⊕ Z sc ) (3.69)

equipped with a positive definite scalar product. More explicitly, we replace

a(k, σ) with â(k, σ) for σ = 0, ±1. We replace a(k, sc) with b̂∗ (k, sc). They

113
satisfy the standard commutation relations

[â(k, σ), â∗ (k 0 , σ 0 )] = δσ,σ0 δ(~k − ~k 0 ),

[b̂(k, sc), b̂∗ (k 0 , sc)] = δ(~k − ~k 0 ).

â(k, σ), b̂(k, sc) kill the vacuum:

â(k, σ)Ω = b̂(k, sc)Ω = 0.

The quantized 4-potentials, Hamiltonian and momentum become

X Z d~k 
ikx −ikx ∗

µ (x) = q u µ (k, σ)e â(k, σ) + u µ (k, σ)e â (k, σ)
(2π)3 2ε(~k)
p
σ=0,±1

d~k
Z  
+ p q uµ (k, sc)eikx b̂∗ (k, sc) + uµ (k, sc)e−ikx b̂(k, sc) ,
(2π)3 2ε(~k)

X Z Z
Ĥ = d~kε(~k)â∗ (k, σ)â(k, σ) − d~kε(~k)b̂∗ (k, sc)b̂(k, sc),
σ=0,±1
X Z Z
P̂ = d~k~kâ∗ (k, σ)â(k, σ) − d~k~k b̂∗ (k, sc)b̂(k, sc).
σ=0,±1

The propagator in the position representation is given by


  2 
Ω|T µ (x)Âν (y) Ω = −i gµν − 2 ∂µ ∂ν DF (x − y),
m
and in the momentum representation
 
−i kµ kν
gµν + 2 2 .
k + m2 − i0
2 m

It is an example of a propagator from the class considered in Subsubsect. 3.2.4.

Note also that
2 2
(Ω|Â((ζ)) Ω) = hζ|ζiY + h∂µ ζ µ |∂ν ζ ν iY , (3.70)
m2
which is the scalar product (3.68) corrected by a term given by the scalar product
(2.39). Note that (3.70) is positive definite.
Vectors built by applying fields satisfying the Lorentz condition to the vac-
uum will be called physical. Equivalently, physical vectors are elements of the
(+)
algebraic Fock space built on WLor . After the completion the physical space
coincides with Γs (ZLor ). Thus we obtain the same space as in the method “first
reduce, then quantize”.

114
 It will be convenient to describe this method in the C ∗ -algebraic language.
Let CCR(Yvec ) denote the (Weyl) C ∗ -algebra of the CCR over Yvec , that is, the
C ∗ -algebra generated by W (ζ), ζ ∈ Yvec , such that
ζ1 ωvec ζ2
W (ζ1 )W (ζ2 ) = e−i 2 W (ζ1 + ζ2 ), W (ζ)∗ = W (−ζ). (3.71)

We have the obvious action of R1,3 oO↑ (1, 3) on CCR(Yvec ) by ∗-automorphisms:

r̂(y,Λ) (W (ζ)) := W r(y,Λ) (ζ) .

Choose the state on CCR(Yvec ) defined by

ψ W (ζ) (3.72)
 1 1 
= exp − hζ|ζiY − 2 h∂µ ζ µ |∂ν ζ ν iY
2 m
Let (Hψ , πψ , Ωψ ) be the GNS representation generated by the state ψ. Using
(3.70) we see that Hψ can be identified with Γs (ZLor ⊕ Z sc ) and the fields are
related to the Weyl operators by

πψ (W (ζ)) = eiÂ((ζ)) .

3.3.7 The Gupta-Bleuler approach

This approach also uses the symplectic space Yvec as the basic input. It follows
almost verbatim the usual steps of quantization of the Klein-Gordon equation.
We introduce the bosonic Fock space Γs (Zvec ), which has an indefinite scalar
product and can be viewed as a Krein space.
We replace a(k, σ) by â(k, σ). The commutation relations have a wrong sign
for the scalar component:

[â(k, σ), â∗ (k 0 , σ 0 )] = κσ,σ0 δ(~k − ~k 0 ).

The annihilation operators kill the vacuum:

â(k, σ)Ω = 0.

The expressions for the Hamiltonian, momentum and 4-potentials are the same
as in the classical case:
X Z Z
Ĥ = d~kε(~k)â∗ (k, σ)â(k, σ) − d~kε(~k)â∗ (k, sc)â(k, sc),
σ=0,±1

~ X Z Z
P̂ = d~k~kâ∗ (k, σ)â(k, σ) − d~k~kâ∗ (k, sc)â(k, sc),
σ=0,±1

XZ d~k 
ikx −ikx ∗

µ (x) = q uµ (k, σ)e â(k, σ) + uµ (k, σ)e â (k, σ) .
(2π)3 2ε(~k)
p
σ

115
Note that all eigenvalues of Ĥ are positive, however its expectation values (wrt
the indefinite scalar product) can be negative. We have

(Ω|µ (x)Âν (y)Ω) = −igµν D(+) (x − y),

(Ω|T(µ (x)Âν (y))Ω) = −igµν Dc (x − y).

In particular, the 2-point Green’s function is the propagator in the Feynman

gauge. Smeared 4-potentials Â((g)) are well defined operators.
Similarly as in the previous method, vectors created by applying fields sat-
isfying the Lorentz condition to the vacuum will be called physical. Again we
(+)
obtain the algebraic Fock space built on WLor . This space is positive definite
and after the completion coincides with Γs (ZLor ). Thus the physical space is
the same as before.

4 Massless photons
In this section we discuss the quantization of the Maxwell equation

−∂µ F µν (x) = 0, (4.1)

where, as in the previous section,

F µν := ∂ µ Aν − ∂ ν Aµ .

We will also consider an external conserved 4-current, that is a vector func-

tion J ν (x) satisfying
∂ν J ν (x) = 0. (4.2)
The Maxwell equation in the presence of the current J reads

∂µ F µν (x) = J ν (x). (4.3)

Similarly as in the massive case, there are several possible approaches to the
Maxwell equation on the classical and, especially, quantum level. The approach
based from the beginning on the reduced phase space, both for the classical
description and quantization, will be treated as the standard one. The situation
is however somewhat more complicated than in the massive case, since the
Lorentz condition is not enough to fully reduce the phase space. Alternative
approaches will be discussed later.
We try to make the discussion of massive and massless photons as parallel as
possible. This is not entirely straightforward. In particular, the massless limit is
quite subtle – to describe it one needs to fix the time coordinate. The covariant
4-potential converges as m & 0 in an appropriate sense to the noncovariant
4-potential in the Coulomb gauge.

116
4.1 Free massless photons
4.1.1 Space of solutions and the gauge invariance
It is well known that the Maxwell equation

−∂µ ∂ µ ζ ν (x) − ∂ ν ζ µ (x)

= 0 (4.4)

is invariant w.r.t. the replacement of ζµ with ζµ + ∂µ χ, where χ is an arbitrary

smooth function on the space-time. In particular, there is no uniqueness of the
Cauchy problem for (4.4).
This property is called gauge invariance. It poses problems both for the
classical and quantum theory. One could avoid the problem of gauge invariance
by considering fields and not 4-potentials as basic objects. However, when one
quantizes the Maxwell equation with a classical 4-current, it is more convenient
to use 4-potentials. Therefore, we will stick to 4-potentials.
There exist several ways to cope with gauge invariance. The approach that
we will use as the standard one can be called first reduce, then quantize.
In this approach we start with the Maxwell equation in the form (4.4). The
space of smooth space compact solutions of (4.4) is denoted YMax g The space of
gauge equivalence classes of elements of YMax
g is denoted

g /{ζ ∈ YMax
YMax := YMax g : ζ = ∂χ}.

Aµ (x) denotes the functional on YMax

g given by

hAµ (x)|ζi := ζµ (x) (4.5)

called the 4-potential. Obviously, Aµ (x) is not defined on YMax .

Moreover, we introduce the functionals Fµν (x) on YMaxg , called the fields:

hFµν (x)|ζi := ∂µ ζν (x) − ∂ν ζµ (x).

They do not depend on the gauge, hence can be interpreted as functionals on

YMax . Both Aµ (x) and Fµν (x) are Lorentz covariant.
We will write Ei (x) = F0i (x).
Let us write separately the temporal and spatial Maxwell equations:

−∆A0 + divA ~˙ = 0,
~ − ∂~ Ȧ0 + ∂div
∂02 − ∆ A ~ A ~


= 0.

~ at the same time:

We can compute A0 in terms of A
~
A0 (x) = ∆−1 ∂0 divA(x). (4.6)

We can insert this into spatial equations obtaining

2A
~ tr = 0, (4.7)

117
where
~ tr
A := A ~
~ + ∂(−∆)−1 ~
divA.

Thus A~ tr can be treated as the only dynamical variables.

If ζ ∈ YMax ~
g , then divζ =: Θ is an arbitrary space-time function, which can
be used to determine ζ0 :
~
ζ0 (x) = −(−∆)−1 ∂0 divζ(x). (4.8)

The simplest choice is Θ = 0, which corresponds to the Coulomb gauge.

Coul
Thus for every ζ ∈ YMax
g , we define ζ by

ζ0Coul := 0, (4.9)
ζ~Coul := ζ~tr = ζ~ + ∂(−∆)
~ −1 ~
divζ. (4.10)

Note that

ζµCoul := ζµ + ∂µ χ, (4.11)
where χ(t, ~x) := (−∆) −1 ~ ~x).
divζ(t, (4.12)

Clearly, ζ Coul is the unique solution of the Maxwell equation gauge equivalent
to ζ satisfying
ζ0Coul = 0, divζ~Coul = 0.
Neither χ nor ζ Coul have to be space-compact. They however
R decay in space
~ ~x)d~x = 0,
directions quite fast. In fact, the Stokes theorem yields that divζ(t,
therefore χ = O(|~x|−2 ) because of (A.39). Hence

ζ Coul = O(|~x|−3 ), ∂0 ζ Coul = O(|~x|−3 ). (4.13)

We introduce the functional ACoul

µ (x) on YMax
g , called the the 4-potential in
the Coulomb gauge,

ACoul
0 (x) := 0, (4.14)
~ Coul (x) := A
A ~ tr (x) = A(x)
~ ~
+ ∂(−∆) −1 ~
divA(x). (4.15)

Note that
hACoul
µ (x)|ζi = hAµ (x)|ζ Coul i = ζµCoul (x).
ACoul (x) does not depend on the gauge, hence can be interpreted as a functional
on YMax . It is not, however, Lorentz covariant.
Thus the classical 4-potential in the Coulomb gauge ACoul (x) satisfies

ACoul
0 = 0, 2A
~ Coul = 0, ~ Coul = 0.
divA

Clearly,
~ = ∂t A
E ~ Coul , ~
divE(x) = 0. (4.16)

118
4.1.2 Symplectic structure on the space of solutions
We introduce a current defined by the same formula as in the case of the Proca
equation:

j µg (ζ1 , ζ2 , x) (4.17)
Max
µ
∂ν ζ1µ (x) ζ2ν (x) − ζ1ν (x) ∂ µ ζ2ν (x) − ∂ ν ζ2µ (x) .



:= ∂ ζ1ν (x) −

It is conserved and leads to the form defined on YMax

g , as in (3.15):

ζ1 ωMax
g ζ2 (4.18)
Z  
˙ ~ 10 (t, ~x) ζ~2 (t, ~x) + ζ~1 (t, ~x) ζ~˙2 (t, ~x) − ∂ζ
  
= − ζ~1 (t, ~x) − ∂ζ ~ 20 (t, ~x) d~x.

However, unlike in the massive case, this form is only presymplectic not sym-
plectic (it is degenerate).
(4.18) does not depend on the gauge. To see this it is enough to note that
for ζ ∈ YMax
g
j µ (∂χ, ζ) = −∂ν χ(∂ µ ζ ν − ∂ ν ζ µ ) .


(4.19)
The presymplectic form can be written as

ζ1 ωMax
g ζ2 = ζ1Coul ω g ζ2Coul (4.20)
Z  Max
˙ ˙

= −ζ~1Coul (t, ~x)ζ~2Coul (t, ~x) + ζ~1Coul (t, ~x)ζ~2Coul (t, ~x) d~x.

Note that by (4.13) the integrand of (4.20) behaves as O(|~x|−6 ), hence is inte-
grable.
Proposition 4.1 Let ζ ∈ YMax
g . We have the following equivalence:
(1) ζ ∈ KerωMax
g.

(2) ζ Coul = 0.
(3) ζ = ∂χ.

Proof. (4.11) and (4.12) imply (2)⇒(3).

The implication (3)⇒(1) follows from the gauge invariance of the form ωMax g.
Let us prove (1)⇒(2). Let ζ Coul 6= 0. Then one of the transversal vector
˙
fields R3 3 ~x 7→ ζ~tr (0, ~x), ζ~tr (0, ~x) is nonzero. Therefore we can find compactly
supported smooth transversal vector fields R3 3 ~x 7→ ~u(~x), ~v (~x) such that
Z 
˙

−~u(~x)ζ~Coul (0, ~x) + ~v (~x)ζ~Coul (0, ~x) d~x 6= 0. (4.21)

∞
There exists a unique ξ ∈ Csc (R4 , R4 ) such that
˙ ~x) = 0, ~u(~x) , ξ(0, ~x) = 0, ~v (~x) , 2ξ = 0.



ξ(0,

119
ξ clearly belongs to YMax
g and is in the Coulomb gauge. We have

Coul
ξωMax
gζ = ξωMax
gζ
Z 
~˙ ~x)ζ~Coul (0, ~x) + ξ(0,
~ ~x)ζ~˙ Coul (0, ~x) d~x,

= −ξ(0,

g. 2
which equals (4.21) and is nonzero. Hence ζ 6∈ KerωMax
Thus we can write
YMax := YMax g,
g /KerωMax

which means that YMax is obtained by the symplectic reduction of the presym-
plectic space YMax
g . Clearly, YMax is equipped with a natural symplectic form
ωMax . R1,3 o O↑ (1, 3) acts on YMax by symplectic transformations.
In what follows we will usually drop the subscript Coul from ACoul (x). This
introduces a possible ambiguity with A(x) defined in (4.5). However, when we
speak about YMax , then (4.5) is ill defined, only ACoul (x) is well defined, so we
think that the risk of confusion is small.
The symplectic structure on the space YMax can be written as
Z
ωMax = Ai (t, ~x) ∧ Ei (t, ~x)d~x, (4.22)

where we need to remember about the transversality constraint

~ ~x) = divE(t,
divA(t, ~ ~x) = 0. (4.23)
∂ ∂
δij − i∆ j is the orthogonal projector from vector fields on R3 to transversal
vector fields. Therefore, (4.22) and (4.23) lead to the following Poisson bracket

{A i (t, ~x), A j (t, ~y )} = {Ei (t, ~x), Ej (t, ~y )} = 0,

 
∂i ∂j
{A i (t, ~x), Ej (t, ~y )} = δij − δ(~x − ~y ). (4.24)
∆

From the above relations we deduce

 
∂i ∂j
{A i (x), A j (y)} = δij − D(x − y).
∆

4.1.3 Smeared 4-potentials

We can use the symplectic form to pair distributions and solutions. For ζ ∈
YMax we introduce the corresponding spatially smeared 4-potentials, which is
the functional on YMax given by

hA((ζ))|ρi := ρωMax ζ, ρ ∈ CYMax .

Note that
{A((ζ1 )), A((ζ2 ))} = ζ 1 ωMax ζ 2 ,

120
 Z  
A((ζ)) = −ζ̇µ (t, ~x)Aµ (t, ~x) + ζµ (t, ~x)E µ (t, ~x) d~x. (4.25)

Let us stress that A((ζ)) depends on ζ only modulo gauge transformations and
is Lorentz covariant.
We can also introduce space-time smeared 4-potentials in the Coulomb gauge,
which are the functionals on YMax , for f ∈ Cc∞ (R1,3 , R1,3 ) given by
Z
A[f ] := f µ (x)Aµ (x)dx. (4.26)

Note that A[f ] = A((ζ)), where

∂i ∂ j
 
ζi = −D ∗ fi − fj , ζ0 = 0.
∆

~
(4.26) is not Lorentz covariant. Adding to f µ any function of the form (ρ, ∂χ)
∞ 1,3
for ρ, χ ∈ Cc (R ) does not change (4.26), because of the Coulomb gauge.

4.1.4 Lagrangian formalism and the stress-energy tensor

The Euler-Lagrange equations for the Lagrangian density
1
L := − Fµν F µν
4
coincide with the Maxwell equation.
The canonical stress-energy tensor is

µν ∂L
Tcan = g µν L − A ,ν
∂Aα,µ α
1
= −g µν Fαβ F αβ − F µα A,να .
4
One usually replaces it with the Belifante-Rosenfeld stress-energy tensor. It is
defined as

T µν = µν
Tcan + ∂α Σµνα
1
= −g µν Fαβ F αβ + F µα F να ,
4
where
Σµνα = −Σανµ := Fµα Aν . (4.27)
On solutions of the Euler-Lagrange equations we have

∂ µ Tµν
can
= ∂ µ Tµν = 0.

In addition, Tµν is symmetric.

To pass to the Hamiltonian formalism we use the Coulomb gauge, writing
Aµ for ACoul
µ
~ = 0. The variable
. Recall that in this gauge A0 = 0 and divA

121
 ~ = 0. We express T can
conjugate to Ai is ∂Ȧi L = E i , which also satisfies divE µν
and Tµν in terms of A~ and E.
~ We introduce the Hamiltonian, momentum

1 ~2 
~ 2 (x) ,
H(x) := T 00 (x) = E (x) + (rotA)
2
P j (x) := T 0j (x) = E i (x)F ji (x)..

They yield the Hamiltonian, momentum and polarization as in (3.22) satis-

fying analogous properties.

4.1.5 Diagonalization of the equations of motion

As in the massive case, we would like to diagonalize simultaneously the Hamil-
tonian, momentum and symplectic form.p
For ~k ∈ R3 we set k = (ε, ~k), ε(~k) := ~k 2 . The vectors u(k, ±1) are defined
as in (3.23). u(k, 0) are not defined at all.
For σ = ±1, define the following functionals on YMax , called plane wave
functionals:

a(k, σ)
s
ε(~k) −i~x~k
Z 
i ~
 d~x
= e uj (k, σ)Aj (0, ~x) − q e−ik~x uj (k, σ)E j (0, ~x) p .
2 (2π)3
2ε(~k)

We have accomplished the promised diagonalization

X Z
H = d~kε(~k)a∗ (k, σ)a(k, σ),
σ=±1
X Z
P~ = d~k~ka∗ (k, σ)a(k, σ),
σ=±1
X Z
iωMax = a∗ (k, σ) ∧ a(k, σ)d~k.
σ=±1

The 4-potentials can be written as

X Z d~k 
ikx −ikx ∗

Aµ (x) = q uµ (x, σ)e a(k, σ) + uµ (x, σ)e a (k, σ) .
(2π)3 2ε(~k)
p
σ=±1

Plane waves are defined as in the massive case, with σ = ±1. We have

a(k, σ) = iA((|k, σ)))

and
X Z  
Aµ (x) = (x|k, σ)a(k, σ) + (x|k, σ)a∗ (k, σ) d~k.
σ=±1

122
4.1.6 Positive frequency space
(±)
WMax will denote the subspace of CYMax consisting of classes of solutions that
in the Coulomb gauge have positive, resp. negative frequencies.
(+)
Every g ∈ WMax can be written as

X Z d~k
g(x) = q eikx u(k, σ)ha(k, σ)|gi.
~
p
σ=±1 3
(2π) 2ε(k)

(+)
For g1 , g2 ∈ WMax we define the scalar product
X Z
(g1 |g2 ) := ig 1 ωMax g2 = ha(k, σ)|g1 iha(k, σ)|g2 id~k. (4.28)
σ=±1

(+)
The definition of WMax depends on the choice of coordinates. It is however
(+)
easy to see that the space WMax is invariant w.r.t. R1,3 o O↑ (1, 3).
(+)
We set ZMax to be the completion of WMax in this scalar product.
We have
ha(k, σ)|gi = (k, σ|g).
We can identify ZMax with L2 (R3 , C2 ) and rewrite (4.28) as
X Z
(g1 |g2 ) = (k, σ|g1 )(k, σ|g2 )d~k.
σ=±1

(+)
We can identify YMax with WMax and transport the scalar product onto
YMax , which for ζ1 , ζ2 is given by
(+) (+)
hζ1 |ζ2 iY := Re(ζ1 |ζ2 ) (4.29)
Z Z
Coul
= ζ̇1i (0, ~x)(−i)D(+) (0, ~x − ~y )ζ̇2i
Coul
(0, ~y )d~xd~y
Z Z
Coul
+ ζ1i (0, ~x)(−∆~x )(−i)D(+) (0, ~x − ~y )ζ2i Coul
(0, ~y )d~xd~y .

4.1.7 Spin averaging

Let us describe the spin averaging identities useful in computations of scattering
cross-sections. For a given k ∈ R1,3 with k 2 = 0, let M, N be vectors with

M µ kµ = N ν kν = 0.

Then we have X
M µ uµ (k, σ)uν (k, σ)N ν = M µ Nµ . (4.30)
σ=±1

123
 To see (4.30), note that

X ~kµ~kν
uµ (k, σ)uν (k, σ) = gµν + δµ0 δν0 − .
σ=±1 |~k|2

Therefore, the left hand side of (4.30) equals

~ ~k)(N
(M ~ ~k)
M µ gµν N ν + M 0 N 0 − .
|~k|2

But
~k M ~ ~k N~
M0 = , N0 = .
|~k| |~k|

4.1.8 Quantization
We would like to quantize the Maxwell equation starting from the symplectic
space YMax . We will use the 4-potentials in the Coulomb gauge (where, as usual,
we drop the superscript Coul). The quantization is similar to the Proca equation
based on YPr described in Subsubsect. 3.2.3, with Condition (1) replaced by

−2Âi (x) = 0, ∂i Âi (x) = 0, Â0 (x) = 0,

and Condition (2) replaced by

[Âi (0, ~x), Aj (0, ~y )] = [Êi (0, ~x), Êj (0, ~y )] = 0,

 
∂i ∂j
[Âi (0, ~x), Êj (0, ~y )] = i δij − δ(~x − ~y ).
∆

The above problem has a solution unique up to a unitary equivalence. We

set H := Γs (ZMax ). The creation operators will be denoted by

â∗ (k, σ) = â∗ (|k, σ)).

Ω will be the Fock vacuum. We set

d~k
Z X  
Âi (x) := q ui (k, σ)eikx â(k, σ) + ui (k, σ)e−ikx â∗ (k, σ) .
(2π)3 2ε(~k) σ=±1
p

The quantum Hamiltonian and momentumare

X Z
Ĥ := â∗ (k, σ)â(k, σ)ε(~k)d~k,
σ=±1

~ X Z
P̂ := â∗ (k, σ)â(k, σ)~kd~k.
σ=±1

124
 1,3 ↑
 The wholegroup R oO (1, 3) is unitarily implemented on H by U (y, Λ) :=
Γ r(y,Λ) . We have
ZMax
0 0
U (y, Λ)F̂µν (x)U (y, Λ)∗ = Λµµ Λνν F̂µ0 ν 0 (y, Λ)x .

Moreover,
 
∂i ∂j
[Âj (x), Âi (y)] = −i δij − D(x − y).
∆
Note the identities
 
∂i ∂j
(Ω|Âi (x)Âj (y)Ω) = −i δij − D(+) (x − y),
∆
 
∂i ∂j
(Ω|T(Âi (x)Âj (y))Ω) = −i δij − Dc (x − y).
∆
The family
Z
Cc∞ (R1,3 , R1,3 ) 3 f 7→ Â[f ] := f µ (x)µ (x)dx

with D := Γfin
s (ZMax ) does not satisfy the Wightman axioms because of two
problems: the noncausality of the commutator and the absence of the Poincaré
covariance.
If we replace µ with F̂µν , we restore the causality and the Poincaré covari-
ance.
For an open set O ⊂ R1,3 we set

A(O) := {exp(iF̂ [f ]) : f ∈ Cc∞ (O, ⊗2a R1,3 )}.

The algebras A(O) satisfy the Haag-Kastler axioms.

4.1.9 Quantization in terms of C ∗ -algebras

∗
Let CCR(YMaxg ) denote the (Weyl) C -algebra of canonical commutation rela-
tions over YMax
g . By definition, it is generated by W (ζ), ζ ∈ YMax
g , such that

ζ1 ω g ζ2
Max
W (ζ1 )W (ζ2 ) = e−i 2 W (ζ1 + ζ2 ), W (ζ)∗ = W (−ζ).

R1,3 o O↑ (1, 3) acts on CCR(YMax g ) by ∗-automorphisms defined by

r̂(y,Λ) (W (ζ)) := W r(y,Λ) (ζ) .

We are looking for a cyclic representation of this algebra with the time evolution
generated by a positive Hamiltonian.
Consider the state on CCR(YMax g ) defined for ζ ∈ YMaxg by

1

ψ W (ζ) = exp − hζ|ζiY ,
2

125
where hζ|ζiY is defined in (4.29).
Note that the state is gauge and Poincare invariant. Let (Hψ , πψ , Ωψ ) be
the GNS representation. Hψ is naturally isomorphic to Γs (ZMax ). Ωψ can
be identified with the vector Ω. πψ (W (ζ)) can be identified with eiÂ((ζ)) . In
particular, if ζ1 and ζ2 are gauge equivalent, then Â((ζ1 )) = Â((ζ2 )). However,
Â(x) in the sense of (4.5) is not well defined.

4.2 Massless photons with an external 4-current

4.2.1 Classical fields
We return to the classical Maxwell equation. We consider an external 4-current
given by function R1,3 3 x 7→ J(x) = [J µ (x)] ∈ R1,3 satisfying

∂ν J ν (x) = 0. (4.31)

In most of this subsection we assume that J is Schwartz. The Maxwell equation

reads
−∂µ ∂ µ Aν + ∂ν ∂µ Aµ = −Jν . (4.32)
Let ζ be a solution of

−∂µ ∂ µ ζν + ∂ν ∂µ ζ µ = −Jν . (4.33)

We write separately the temporal and spatial equations:

˙
−∆ζ0 + divζ~ = −J0 ,
∂02 − ∆ ζ~ − ∂~ ζ̇0 + ∂div
~ ζ~ = −J.
~

We can compute ζ0 in terms of ζ~ at the same time:

~
ζ0 (x) = ∆−1 (J0 + ∂0 divζ)(x). (4.34)

We can insert this into spatial equations, using J˙0 = divJ,

~ obtaining

2ζ~tr = J~tr , (4.35)

where

ζ~tr := ζ~ − ∂∆
~ −1 divζ,
~
J~tr ~ ~ −1
:= J − ∂∆ divJ. ~

Thus ζ~tr can be treated as the only dynamical variables. divζ~ =: Θ is an

arbitrary space-time function, which can be used to determine ζ0 .
The simplest choice is Θ = 0, which corresponds to the Coulomb gauge:

ζ0 = ∆−1 J0 ,
2ζ~ = J~tr ,
divζ~ = 0. (4.36)

126
The Coulomb gauge seems to be the most natural gauge for the Hamiltonian
approach.
Let ζ be a space compact solution of (4.33). Setting
ζµCoul := ζµ + ∂µ χ,
~ ~x), we obtain a solution of (4.36). ζ Coul is the
where χ(t, ~x) := (−∆)−1 divζ(t,
unique solution of (4.36) gauge equivalent to ζ. It does not have to be space
compact.
Thus the classical 4-potential in the Coulomb gauge ACoul (x) satisfies
ACoul
0 = −(−∆)−1 J0 ,
2A
~ Coul = J~tr ,
~ Coul
divA = 0.
~˙ ∂A
~ = A−
The electric field is E ~ 0 . It is easy to see that if we use the Coulomb
~ ~˙ Coul
gauge, then Etr = A .
Similarly as in the previous subsection, we will drop the superscript Coul in
what follows.

4.2.2 Lagrangian and Hamiltonian formalism

The Lagrangian density is
1
L := − Fµν F µν − Jµ Aµ
4
1 ~ 2 + 1 (∂A
~ 0 )2 + 1 A
~˙ 2 − A


~˙ ∂A
~ 0 − J~A
~ + J0 A0 .
= − (rotA)
2 2 2
~
The dynamic variables are A(x). The corresponding conjugate variable is
~ ~˙ ~
E(x) = A(x) − ∂A0 (x).
The canonical Hamiltonian density is

Hcan (x) = −L(x) + E ~˙

~ A(x)
1 ~ 2 (x) + 1 E
~ 2 (x) + E(x)
~ ∂A ~ 0 (x) + J(x)
~ A(x)
~
= (rotA) − J0 (x)A0 (x).
2 2
~ 0 (x) , use
~ ∂A


We add to it a spatial divergence div E(x)
~ = −J0
divE (4.37)
and express A0 in terms of J0 . We obtain the Hamiltonian density
1 ~2 1 ~ 2 (x) + J(x)
~ A(x).
~
H(x) = E (x) + (rotA)
2 2
Similarly as in the massive case, the Hamiltonian
Z Z
H(t) = H(t, ~x)d~x = Hcan (t, ~x)d~x (4.38)

127
generates the equations of motion.
The longitudinal part of A ~ does not enter in the Hamiltonian because of
(A.45). The longitudinal part of E~ is fixed by the constraint (4.37). Therefore,
we can impose the Coulomb gauge, so that only the transversal part of A ~=A ~ tr .
The corresponding conjugate variable is E ~ tr . We can rewrite the Hamiltonian
as
Z 
1 ~2 1 ~ 2 1 
H(t) = Etr (x) + (∂~ A) ~ A(x)
(x) + J(x) ~ + J0 (−∆)−1 J0 (x) d~x.
2 2 2

4.2.3 Quantization
To quantize the Maxwell equation in the presence of an external 4-current we
will use the Coulomb gauge, dropping as usual the subscript Coul.
We are looking for quantum 4-potentials R1,3 3 x 7→ µ (x) satisfying

ÂCoul
0 = −(−∆)−1 J0 ,
~ Coul
2Â = J~tr ,
~
divÂCoul = 0,

~˙
~
having the following commutation relations with E(x) = Â(x) − ∂~ Â0 (x).

[Â i (t, ~x), Â j (t, ~y )] = [Êi (t, ~x), Ej (t, ~y )] = 0,

 
∂i ∂j
[A i (t, ~x), Ej (t, ~y )] = i δij − δ(~x − ~y ).
∆

~ ~
The above conditions determine Â0 . To fix  and Ê we assume that they
coincide with their free quantum counterparts at t = 0:

~ ~ ~
Â(0, ~x) = Âfr (0, ~x) =: Â(~x),
~ ~ ~
Ê(0, ~x) = Êfr (0, ~x) =: Ê(~x).

The Schrödinger picture Hamiltonian and the corresponding interaction pic-

ture Hamiltonian are
Z 1
~ 1 ~ ~

Ĥ(t) = d~x : Ê 2 (~x) + (∂~ Â)2 (~x) + J(t, ~ ~x)Â(~ x) :
2 2
Z Z
1 1
+ d~xd~y J 0 (t, ~x) J 0 (t, ~y ),
2 4π|~x − ~y |
Z
~ ~x)Â ~
ĤInt (t) = + d~xJ(t, fr (t, ~
x)
Z Z
1 1
+ d~xd~y J 0 (t, ~x) J 0 (t, ~y ).
2 4π|~x − ~y |

128
 The scattering operator can be computed exactly:
 Z 
i dk µ Coul ν
Ŝ = exp J (k)Dµν (k)J (k)
2 (2π)4
 
X Z d~k uµ (k, σ)
× exp −i p â∗ (k, σ) q J µ (ε(~k), ~k)
(2π)3
σ=±1 ~
2ε(k)
 
X Z ~
dk uµ (k, σ) µ ~ ~ 
× exp −i p â(k, σ) q J (ε(k), k) , (4.39)
(2π)3
σ=±1 ~
2ε(k)

where the propagator in the Coulomb gauge is defined as

 
Coul 1 Coul Coul 1 ki kj
D00 = − , D0j = 0, Dij = 2 δij − .
~k 2 k − i0 ~k 2
We did not use the fact that J µ is conserved.

4.2.4 Causal propagators

If we compute scattering amplitudes, we can pass from the propagator in the
Coulomb gauge to another by adding kµ fν (k)+fµ (k)kν for an arbitrary function
fµ (k). Let us list a number of useful propagators in other gauges.
We distinguish the family of propagators
   
1 1 kµ kν
gµν + −1 .
k 2 − i0 α k2
Some of them have special names:
 
Lan 1 kµ kν
Dµν := k2 −i0 gµν − k2 Landau or Lorentz gauge,
Feyn 1
Dµν := k2 −i0 gµν Feynman gauge,
 
FY 1 k k
Dµν := k2 −i0 gµν + 2 µk2 ν Fried and Yennie gauge.

Coul Feyn
We have Dµν = Dµν + kµ fνCoul (k) + fµCoul (k)kν , where
k0 ki
f0Coul (k) = , fiCoul (k) = − .
(k 2 − i0)2~k 2 (k 2 − i0)2~k 2
The propagator in the temporal gauge
 
tem tem tem 1 ki kj
D00 = 0, D0j = 0, Dij = δij − .
k 2 − i0 k02
tem Feyn
We have Dµν = Dµν + kµ fνtem (k) + fµtem (k)kν , where
1 ki
f0tem (k) = , fitem (k) = − 2 .
(k 2 − i0)2k0 (k − i0)2k02

129
4.2.5 Path integral formulation
Let Dµν•
be one of the propagators considered in Sect. 3.2.4. Let B•µν be its
inverse. Then we can use the corresponding action to express the generating
function by path integrals, as described in Sect. 3.2.6, where this approach for
massive vector fields was considered.
α tem
The discussion of the propagators Dµν and Dµν is an obvious generalization
of the massive case.
Coul
To obtain the propagator in the Coulomb gauge Dµν , we take the La-
grangian
1 
− ∂µ Ai (x)∂ µ Ai (x) − ∂i A0 (x)∂i A0 (x) ,
2
and restrict the integration by the condition
~
divA(x) = 0.

4.2.6 The m → 0 limit

Assume that J µ is a conserved 4-current. Recall from (3.50) that the scattering
operator for a positive mass m is
 Z Z 
i dk µ c ν i 0 2
Ŝ = exp − J (k)Dµν (k)J (k) − J (x) dx
2 (2π)4 2m2
 
X Z ~
dk uµ (k, σ) µ ~ ~ 
× exp −i p a∗ (k, σ) q J (ε(k), k)
(2π) 3
σ=0,±1 2ε(~k)
 
X Z d~k uµ (k, σ)
× exp −i p a(k, σ) q J µ (ε(~k), ~k) , (4.40)
(2π) 3
σ=0,±1 ~
2ε(k)

where Dc is one of the propagators for the Proca equation and in the expressions
where we use the 3-dimensional integration d~k, the 4-momenta are on shell, that
is, k = (ε(~k), ~k). Using the propagator in the Yukawa gauge and multiplying it

130
by an obvious phase factor we can write it as
 Z 
i 0 2
Ŝ exp J (x) dx
2m2
Z  
i dk i 1 ki kj
= exp J (k) 2 gij − J i (k)
2 (2π)4 m + k 2 − i0 m2 + ~k 2
Z !
i dk 1 0 2
− |J (k)|
2 (2π)4 ~k 2 + m2
 
X Z ~
dk uµ (k, σ) µ 
× exp −i p â∗ (k, σ) q J (k)
(2π) 3
σ=0,±1 ~
2ε(k)
 
X Z
d~k u µ (k, σ)
× exp −i p â(k, σ) q J µ (k)
(2π) 3
σ=0,±1 ~
2ε(k)
= Ŝtr ⊗ Ŝlg ,

Here, the transversal scattering operator is

Z   !
i dk i 1 ki kj
Ŝtr = exp J (k) 2 gij − J j (k)
2 (2π)4 m + k 2 − i0 ~k 2
 
X Z d ~k u µ (k, σ)
× exp −i p â∗ (k, σ) q J µ (k)
(2π) 3
σ=±1 ~
2ε(k)
 
X Z dk~ uµ (k, σ) µ 
× exp −i p â(k, σ) q J (k)
(2π) 3
σ=±1 2ε(~k)
Z   !
i dk i 1 ki kj j
→ exp J (k) 2 gij − J (k)
2 (2π)4 k − i0 ~k 2
 
X Z d ~k u µ (k, σ)
× exp −i p â∗ (k, σ) q J µ (k)
(2π) 3
σ=±1 2ε(~k)
 
X Z dk~ uµ (k, σ) µ 
× exp −i p â(k, σ) q J (k)
(2π) 3
σ=±1 2ε(k)~

The longitudinal scattering operator is

131
 !!
k02
Z
i dk 0 2
1 1  1
Ŝlg = exp |J (k)| − −
2 (2π)4 (m2 + k 2 − i0) ~k 2 (m2 + ~k 2 ) (~k 2 + m2 )
 
d~k
Z
uµ (k, 0) µ 
× exp −i p â∗ (k, 0) q J (k)
(2π)3 2ε(~k)
 
Z
d ~k u µ (k, 0)
× exp −i p â(k, 0) q J µ (k)
(2π)3 2ε(k)~
!
dk  m2 |J0 (k)|2 |J0 (k)|2 
Z
i
= exp −
2 (2π)4 (m2 + k 2 − i0)~k 2 ~k 2
 
d~k mJ 0 (k) 
Z
× exp i p
 â∗ (k, 0) q
(2π)3 |~k| 2ε(~k)
 
Z
d~k mJ 0 (k)
× exp i p â(k, 0) q 
(2π)3 ~
|k| 2ε(k) ~
0 2
 
dk |J (k)|
Z
i
→ exp − .
2 (2π)4 ~k 2
Thus the renormalized massive scattering operator converges to the massless
scattering operator in the Coulomb gauge as m & 0.

4.2.7 Current produced by a travelling particle

Consider a classical particle travelling along the trajectory t 7→ ~y (t) with a
constant profile q(~x). Then its 4-current equals
 d~y (t) 
J(t, ~x) = q(~x − ~y (t)) 1, .
dt
Assume that ~y (t) = t~v ± for ±t > 0. Then
Z
~ 0
µ
J (k) = J µ (t, ~x)e−ik~x+ik t dxdt
!
i(1, ~v+ )µ i(1, ~v− )µ
= − + q(~k)
~k~v+ − k 0 − i0 ~k~v− − k 0 + i0
ipµ+ ipµ−
 
= − + q(~k),
kp+ − i0 kp− + i0
where p± = √ m
(1, ~v ± ).
v ± )2
1−(~
Consider photons of mass m ≥ 0 coupled to the 4-current J µ . Similarly as
in Subsubsect. 2.2.12, we define the scattering operator ŜGL by replacing
Z
dk µ
J (k)Dµν (k)J ν (k)
(2π)4

132
in (4.39) with Z
dk µ
Im J (k)Dµν (k)J ν (k). (4.41)
(2π)4
R
(4.41) is infrared divergent if m = 0, q(~x)d~x 6= 0 and ~v+ 6= ~v− .
We could try to justify the use of ŜGL similarly as in Subsubsect. 2.2.12,
by introducing the Gell-Mann–Low adiabatic switching. This justification is
adopted by many physicists, eg. [31]. One could criticize this approach, since
after multiplying by the switching function e−|t| the 4-current is no longer
conserved. Therefore, as indicated above, we prefer to define the scattering
operator ŜGL simply by removing the (typically infinite) phase shift.

4.2.8 Energy shift

Suppose that the 4-current is stationary and is given by a Schwartz function
~ x) = 0.
R3 3 ~x 7→ J µ (~x) with divJ(~
The Hamiltonian is given by
Z 1
~2 1 ~
2 ~ ~

Ĥ = d~x : Êtr (~x) + ∂~ Â(~x) + J(~ x)Â(~x) :
2 2
Z Z
1 1
+ d~xd~y J 0 (~x) J 0 (~y ).
2 4π|~x − ~y |

By (A.14), the infimum of Ĥ is

Z Z
1 ~ x) 1 ~ y)
E = − d~xd~y J(~ J(~
2 4π|~x − ~y |
Z Z
1 1
+ d~xd~y J 0 (~x) J 0 (~y ).
2 4π|~x − ~y |

4.3 Alternative approaches

4.3.1 Manifestly Lorentz covariant formalism
So far, our treatment of the Maxwell equation was based on the Coulomb gauge,
which depends on the choice of the temporal coordinate. One can ask whether
massless vector fields can be studied in a manifestly covariant fashion.
Let Ξ be an arbitrary space-time function. The Maxwell equation allows us
to impose a generalized Lorentz condition
∂µ Aµ = Ξ. (4.42)
The Maxwell equation together with (4.42) imply
−2Aµ = −J µ + ∂ µ Ξ. (4.43)
The function Ξ has no physical meaning. Therefore it is natural to adopt the
simplest choice Ξ = 0, that is the usual Lorentz condition, for which (4.43) reads
−2Aµ = −J µ . We will discuss this approach in what follows. For simplicity,
we will limit ourselves to free fields.

133
4.3.2 The Lorentz condition
Recall that the Proca equation is equivalent to the Klein-Gordon equation for
vector fields together with the Lorentz condition. Therefore, one can first de-
velop its theory on the symplectic space Yvec , and then reduce it to the subpace
YLor , as described before.
One can follow a similar route for the Maxwell equation. However, there is
a difference: the reduction by the Lorentz condition is insufficient, one has to
make an additional reduction.
Anyway, let us start, as described in Subsubsect. 3.3.1, by introducing the
space Yvec , the form ωvec , the subspace YLor , the 4-potentials Aµ (x), Πµ (x) :=
Ȧµ (x), where now m = 0.
In the massive case YLor was symplectic (that means, the form ωvec restricted
to YLor was nondegenerate). This is no longer true in the massless case. Instead,
the following is true.

Proposition 4.2 YLor is coisotropic. That means, if ζ is symplectically orthog-

onal to YLor , then ζ ∈ YLor .

Proof. Using −2∂µ Aµ (x) = 0 we see that, for any fixed t, we can replace

∂µ Aµ (x) = 0 (4.44)

with

0 = ∂µ Aµ (t, ~x) = (−Π0 + ∂i Ai )(t, ~x), (4.45)

µ 0 i
0 = ∂µ Π (t, ~x) = (−∆A + ∂i Ȧ )(t, ~x) (4.46)

as the defining conditions for YLor . YLor is coisotropic iff

{∂µ Aµ (t, ~x), ∂µ Aµ (t, ~y )} = 0, (4.47)

µ µ
{∂µ Π (t, ~x), ∂µ Π (t, ~y )} = 0, (4.48)
µ µ
{∂µ A (t, ~x), ∂µ Π (t, ~y )} = 0. (4.49)

It is clear that (4.47) and (4.48 are true. To see (4.49) we compute:

{∂µ Aµ (t, ~x), ∂ν Πν (t, ~y )}

= ∆~y δ(~x − ~y ) + ∂~xi ∂~yi δ(~x − ~y ) = 0.

2
YLor is a subspace of YMax
g and on YLor the forms ωMax
g and ωvec coincide.

Proposition 4.3 Any ζ ∈ YMax

g is gauge equivalent to an element of YLor .

Proof. We can find smooth functions ξ+ and ξ− such that

∂µ ζ µ = ξ+ + ξ− ,

134
ξ− is past space compact and ξ+ is future space compact. By using the advanced
and retarded Green’s functions we can solve

−2χ− = ξ− , −2χ+ = ξ+ ,

where χ− is past space compact and χ+ is future space compact. Then ζµ + ∂µ χ

belongs to YLor . 2
Therefore, the symplectically reduced YLor coincides with the symplectically
reduced YMax
g , that is, with YMax . This shows that both approaches to the
Maxwell equation are equivalent on the classical level.

4.3.3 Positive frequency space

(±)
WLor will denote the subspace of CYLor consisting of solutions that have posi-
tive, resp. negative frequencies.
(+)
For g1 , g2 ∈ WLor we define the scalar product

(g1 |g2 ) := ig 1 ωvec g2

= ig Coul
1 ωvec g2Coul . (4.50)

Note that the definition (4.50) does not depend on the choice of coordinates and
is invariant wrt. the group R1,3 o O↑ (1, 3).
The scalar product is positive semidefinite, but not strictly positive definite.
(+) (+)
Let WLor,0 be the subspace of elements WLor with a zero norm. Using Prop.
(+) (+) (+)
4.1 we see that WLor,0 consists of pure gauges. The factor space WLor /WLor,0
has a nondegenerate scalar product. Its completion is naturally isomorphic to
the space ZMax , which we constructed in Subsubsect. 4.1.6.
(+)
We have a natural identification of YLor with WLor given by the obvious
projection. For ζ ∈ YLor we will denote by ζ (+) the corresponding element
(+)
of WLor . This identification allows us to define a positive semidefinite scalar
product on YLor :
(+) (+)
hζ1 |ζ2 iY := Re(ζ |ζ2 )
Z Z1
Coul
= ζ̇1i (0, ~x)(−i)D(+) (0, ~x − ~y )ζ̇2i
Coul
(0, ~y )d~xd~y
Z Z
Coul
+ ζ1i (0, ~x)(−∆~x )(−i)D(+) (0, ~x − ~y )ζ2i
Coul
(0, ~y )d~xd~y .

4.3.4 “First quantize, then reduce”

One can try to use the symplectic space Yvec of real vector valued solutions of
the Klein-Gordon equation as the basis for quantization. In the literature, this
starting point is employed by two approaches.
The first, which we call the approach with a subsidiary condition has the
advantage that it uses only positive definite Hilbert spaces. Unfortunately, in

135
this approach there are problems with the 4-potential µ (x). Besides, the full
Hilbert space turns out to be non-separable.
In the Gupta-Bleuler approach the 4-potentials µ (x) are well defined and
covariant. Unfortunately it uses indefinite scalar product spaces.

4.3.5 Quantization with a subsidiary condition

The quantization of the Proca equation described in Subsubsec. 3.3.6 is prob-
lematic in the zero mass limit. If m = 0, we cannot use the Hilbert space (3.69)
for the quantization, since it is not well defined.
However, the C ∗ -algebraic formulation survives the m & 0 limit. In par-
ticular, the (Weyl) C ∗ -algebra of canonical commutation relations over Yvec ,
introduced in (3.71) and denoted CCR(Yvec ), is well defined also for m = 0 and
is invariant wrt the Poincaré group.
Strictly speaking, the spaces Yvec and hence the algebras CCR(Yvec ) are
different for various m. If we fix a Cauchy subspace we can identify them by
using the initial conditions.
Recall that in the massive case
2 2
(Ω|Â((ζ)) Ω) = hζ|ζiY + 2 h∂µ ζ µ |∂ν ζ ν iY . (4.51)
m
Recall that ζ ∈ YLor if ∂µ ζ µ = 0. Therefore, in the limit m & 0,

2 hζ|ζiY , ζ ∈ YLor ,
(Ω|Â((ζ)) Ω) =
+∞, ζ 6∈ YLor .
So, the following state on CCR(Yvec ) is the limit of the state (3.72) for m & 0:
(  

exp − 12 hζ|ζiY , ζ ∈ YLor ,
ψ W (ζ) =
0, ζ 6∈ YLor .
Let (Hψ , πψ , Ωψ ) denote the GNS representation for this state. We can
identify
J : Hψ → l2 (Yvec /YLor , Γs (ZMax )) . (4.52)
To describe this identification, first note that Yvec /YLor can be parametrized by
smooth space-compact functions
Ξ = ∂µ ζ µ ,
which can be called the values of the Lorentz condition. For each Ξ choose
ζΞ ∈ Yvec such that ∂µ ζΞµ = Ξ. We demand that
(

 Ω, ∂µ ζ µ = Ξ,
Jπψ W (ζΞ ) Ωψ (Ξ) =
0, ∂µ ζ µ 6= Ξ.
Then J is given by
( i


e 2 ζωvec ζΞ eiÂ((ζ−ζΞ )) Ω, ∂µ ζ µ = Ξ,
Jπψ W (ζ) Ωψ (Ξ) =
0, ∂µ ζ µ 6= Ξ.

136
Note that Hψ is non-separable – it is an uncountable direct sum of superselection
sectors corresponding to various values of the Lorentz condition. All these
superselection sectors are separable.
Special role is played by the (separable) subspace (superselection sector)
corresponding to the Lorentz condition Ξ = 0. We can choose ζΞ=0 = 0 and
thus this subspace is naturally isomorphic to Γs (ZMax ) with the fields obtained
by the usual quantization obtained by the method “first reduce, then quantize”.
Note that πψ (W (ζ)) maps between various sectors of (4.52) if ζ 6∈ YLor . The
unitary group R 3 t 7→ πψ (W (tζ)) is strongly continuous if and only if ζ ∈ YLor .
If this is the case, we can write πψ (W (ζ)) = eiÂ((ζ)) . We have Â((ζ1 )) = Â((ζ2 ))
if in addition ζ1 differs from ζ2 by a pure gauge. Â((ζ)) is ill defined if ζ 6∈ YLor .
To my knowledge, the approach that we described above, restricted to the
0th sector, was essentially one of the first approaches to the quantization of
Maxwell equation. It is typical for older presentations, eg. [27]. However,
without the language of C ∗ -algebras it is somewhat awkward to describe. One
usually says that the Lorentz condition ∂µ µ (x) = 0 is enforced on the Hilbert
space of states and constitutes a subsidiary condition.

4.3.6 The Gupta-Bleuler approach

The Gupta-Bleuler approach follows the same lines as in the massive case until
(+)
we arrive at the algebraic Fock space built on WLor . As we know, the scalar
(+) (+)
product on WLor is only semidefinite. We factor WLor by the null space of
(+)
its scalar product, obtaining WMax . We complete it, obtaining ZMax and we
take the corresponding Fock space Γs (ZMax ) – this coincides with the usual
quantization.
(+)
Equivalently, we can take the (algebraic) Fock space over WLor . It has a nat-
ural semidefinite product. We divide by its null space and take the completion.
Again, the resulting Hilbert space can be naturally identified with Γs (ZMax ).

5 Charged scalar bosons

In this section we consider again the Klein-Gordon equation

(−2 + m2 )ψ(x) = 0. (5.1)

This time we will quantize the space of its complex solutions.

The formalism used in physics to describe complex fields, and especially to
quantize them, is different from the real case, therefore we devote to it a separate
section.
The advantage of complex fields, as compared with real fields, is the possi-
bility to include an external electromagnetic 4-potential A(x) = [Aµ (x)] and to
consider the equation

− (∂µ + iAµ (x)) (∂ µ + iAµ (x)) + m2 ψ(x) = 0.

137
5.1 Free charged scalar bosons
5.1.1 Classical fields
WKG will denote the space of smooth space-compact complex solutions of the
Klein-Gordon equation
(−2 + m2 )ζ = 0. (5.2)
(In the context of neutral fields, it was denoted CYKG , because it was an aux-
iliary object, the complexification of the phase space YKG . Now it is the basic
object, the phase space itself).
Clearly, the space WKG is equipped with a complex conjugation ζ 7→ ζ and
a U (1) symmetry ζ 7→ eiθ ζ, θ ∈ R/2πZ = U (1).
If T is a real linear functional on W, then we have two kinds of natural
complex conjugations of T :

hT |ζi := hT |ζi, hT ∗ |ζi := hT |ζi. (5.3)

Both maps T 7→ T and T 7→ T ∗ are antilinear. When restricted to the real

subspace YKG ⊂ WKG , the functionals T and T ∗ coincide.
A special role is played by complex linear functionals on W. The space of
such functionals will be denoted W # . If T ∈ W # , then T ∈ W # , unlike T ∗ ,
which is antilinear.
In the neutral case a crucial role was played by the conserved 4-current
jµ (ζ1 , ζ2 ), where ζ1 , ζ2 ∈ YKG ; see (2.18). In the charged case we will use its
sesquilinear version defined on WKG :

j µ (ζ 1 , ζ2 , x) := ∂ µ ζ1 (x)ζ2 (x) − ζ1 (x)∂ µ ζ2 (x). (5.4)

If we decompose elements of WKG into their real and imaginary part ζ = ζR +iζI ,
then the real part of the 4-current splits into a part depending on ζR and on ζI :

Rej µ (ζ 1 , ζ2 , x)
= ∂ µ ζR,1 (x)ζR,2 (x) − ζR,1 (x)∂ µ ζR,2 (x)
+∂ µ ζI,1 (x)ζI,2 (x) − ζI,1 (x)∂ µ ζI,2 (x).

Thus WKG can be viewed as the direct sum of two symplectic spaces with the
form
2Reζ 1 ωζ2 = 2ζR,1 ωζR,2 + 2ζI,1 ωζI,2 .
For x ∈ R1,3 , one can introduce the fields φR (x), φI (x), πR (x), πI (x) as the
real linear functionals on WKG given by
√ √
hφR (x)|ζi := 2Reζ(x), hφI (x)|ζi := 2Imζ(x), (5.5)
√ √
hπR (x)|ζi := 2Reζ̇(x), hπI (x)|ζi := 2Imζ̇(x). (5.6)

Clearly, we have the usual equal time Poisson brackets (we write only the non-
vanishing ones):

{φR (t, ~x), πR (t, ~y )} = {φI (t, ~x), πI (t, ~y )} = δ(~x − ~y ). (5.7)

138
 In practice instead of (5.5) and (5.6) one prefers to use complex fields ψ(x), η(x) ∈
W # defined by

hψ(x)|ζi := ζ(x), hψ ∗ (x)|ζi := ζ(x),

hη(x)|ζi := ζ̇(x), hη ∗ (x)|ζi := ζ̇(x).

Clearly,
1 1
ψ(x) = √ φR (x) + iφI (x) , ψ ∗ (x) = √ φR (x) − iφI (x) ,



2 2
1 1
η(x) = √ πR (x) + iπI (x) , η ∗ (x) = √ πR (x) − iπI (x) .



2 2
Note that
Z Z
ψ(t, ~x) = Ḋ(t, ~x − ~y )ψ(0, ~y )d~y + D(t, ~x − ~y )η(0, ~y )d~y . (5.8)

The only non-vanishing equal-time Poisson brackets are

{ψ(t, ~x), η ∗ (t, ~y )} = {ψ ∗ (t, ~x), η(t, ~y )} = δ(~x − ~y ). (5.9)

Using (5.8) we obtain

{ψ(x), ψ(y)} = {ψ ∗ (x), ψ ∗ (y)} = 0,

{ψ(x), ψ ∗ (y)} = D(x − y).

5.1.2 Smeared fields

We can use the symplectic form to pair distributions and solutions. For ζ ∈ WKG
the corresponding spatially smeared fields are the functionals on WKG given by

hψ((ζ))|ρi := ζωρ,
hψ ∗ ((ζ))|ρi := ζωρ, ρ ∈ WKG .

Equivalently,
Z  
ψ((ζ)) = −ζ̇(t, ~x)ψ(t, ~x) + ζ(t, ~x)η(t, ~x) d~x,
Z  
ψ ∗ ((ζ)) = −ζ̇(t, ~x)ψ ∗ (t, ~x) + ζ(t, ~x)η ∗ (t, ~x) d~x.

Note that

{ψ((ζ1 )), ψ((ζ2 ))} = {ψ ∗ ((ζ1 )), ψ ∗ ((ζ2 ))} = 0,

∗
{ψ((ζ1 )), ψ ((ζ2 ))} = ζ 1 ωζ2 .

139
 We can also introduce space-time smeared fields. To a space-time function
f ∈ Cc∞ (R1,3 , C) we associate
Z
ψ[f ] := f (x)ψ(x)dx,
Z
ψ ∗ [f ] := f (x)ψ ∗ (x)dx.

Clearly,

{ψ[f1 ], ψ[f2 ]} = {ψ ∗ [f1 ], ψ ∗ [f2 ]} = 0,

Z Z
{ψ[f1 ], ψ ∗ [f2 ]} = f1 (x)D(x − y)f2 (y)dxdy,

ψ[f ] = −ψ((D ∗ f )), ψ ∗ [f ] = −ψ ∗ ((D ∗ f )).

5.1.3 Lagrangian formalism

In the Lagrangian formalism we use the complex off-shell fields ψ(x) and ψ ∗ (x)
as the basic variables. We introduce the Lagrangian density

L(x) = −∂µ ψ ∗ (x)∂ µ ψ(x) − m2 ψ ∗ (x)ψ(x).

The Euler-Lagrange equations

∂L ∂L
∂ψ∗ L − ∂µ ∗
= 0, ∂ψ L − ∂µ =0 (5.10)
∂ψ,µ ∂ψ,µ

yield (5.1). The variables conjugate to ψ(x) and ψ ∗ (x) are

∂L
η ∗ (x) := = ∂0 ψ ∗ (x),
∂ψ,0 (x)
∂L
η(x) := ∗ (x) = ∂0 ψ(x).
∂ψ,0

5.1.4 Classical 4-current

The Lagrangian is invariant w.r.t. the U (1) symmetry ψ 7→ e−iθ ψ. The Noether
4-current associated to this symmetry is called simply the 4-current. It is
 ∂L(x) ∂L(x) 
J µ (x) := i ψ ∗ (x) ∗
− ψ(x)
∂ψ,µ ∂ψ,µ
i ∂ µ ψ ∗ (x)ψ(x) − ψ ∗ (x)∂ µ ψ(x) .


=

It is conserved on shell and real:

∂µ J µ (x) = 0,
µ ∗
J (x) = J µ (x).

140
Up to a coefficient, it coincides with (5.4) viewed as a quadratic form:
hJ µ (x)|ζi = ij µ (ζ, ζ, x)
i ∂ µ ζ(x)ζ(x) − ζ(x)∂ µ ζ(x) .


=
The 0th component of the 4-current is called the charge density
Q(x) := J 0 (x) = i −η ∗ (x)ψ(x) + ψ ∗ (x)η(x) .

We have the relations

{Q(t, ~x), ψ(t, ~y )} = iψ(t, ~y )δ(~x − ~y ),
{Q(t, ~x), η(t, ~y )} = iη(t, ~y )δ(~x − ~y ),
{Q(t, ~x), Q(t, ~y )} = 0. (5.11)
The (total) charge Z
Q := Q(t, ~x)d~x

is conserved (does not depend on time).

For χ ∈ Cc∞ (R3 , R), let αχ denote the ∗-automorphism of the algebra of
functions on WKG defined by
αχ (ψ(0, ~x)) := e−ieχ(~x) ψ(0, ~x),
αχ (η(0, ~x)) := e−ieχ(~x) η(0, ~x). (5.12)
Obviously,
αχ (ψ ∗ (0, ~x)) = eieχ(~x) ψ ∗ (0, ~x),
αχ (η ∗ (0, ~x)) = eieχ(~x) η ∗ (0, ~x). (5.13)
(5.12) is called the gauge transformation at time t = 0 corresponding to χ. Set
Z
Q(χ) = χ(~x)Q(0, ~x)d~x. (5.14)

Q(χ) generates the one-parameter group of gauge transformations R 3 s 7→ αsχ

(5.12). In other words, for any classical observable B (a function on WKG ))

∂s αsχ (B) = αsχ (B), eQ(χ) ,
α0χ (B) = B.

5.1.5 Stress-energy tensor

The Lagrangian is invariant w.r.t. space-time translations. This leads to the
stress-energy tensor
∂L(x) ν ∂L(x)
T µν (x) := − ∂ ψ(x) − ∂ ν ψ ∗ (x) ∗ + g µν L(x)
∂ψ,µ (x) ∂ψ,µ (x)
= ∂ µ ψ ∗ (x)∂ ν ψ(x) + ∂ ν ψ ∗ (x)∂ µ ψ(x)
−g µν ∂α ψ ∗ (x)∂ α ψ(x) + m2 ψ ∗ (x)ψ(x) .

141
It is conserved on shell
∂µ T µν (x) = 0.
The components of the stress-energy tensor with the first temporal coordinate
are called the Hamiltonian density and momentum density. We express them
on-shell in terms of ψ(x), ψ ∗ (x), η(x) and η ∗ (x):

H(x) := T 00 (x) ~ ∗ (x)∂ψ(x)

= η ∗ (x)η(x) + ∂ψ ~ + m2 ψ ∗ (x)ψ(x),
i 0i
P (x) := T (x) = −η ∗ (x)∂ i ψ(x) − ∂ i ψ ∗ (x)η(x).
~
H(x) and P(x) acting on ζ ∈ WKG yield
~
hH(x)|ζi = |ζ̇(x)|2 + |∂ζ(x)| 2
+ m2 |ζ(x)|2 ,
~
hP(x)|ζi ~
= −ζ̇(x)∂ζ(x) ~
− ∂ζ(x)ζ̇(x).
We can define the Hamiltonian and momentum
Z Z
~ ~x)d~x.
H = H(t, ~x)d~x, P~ = P(t,

H and P~ are the generators of the time and space translations. The observables
H, P1 , P2 , P3 and Q are in involution.

5.1.6 Plane waves

Recall that in the neutral case the generic notation for the energy-momentum
was k. The p on-shell condition was k 2 + m2 = 0, k 0 > 0. In other words,
k 0 = ε(~k) := ~k 2 + m2 .
In the charged case, following [17], it will be convenient to use different
letters for the generic notation of the energy-momentum. It energy-momentum
p2 + m2 = 0, p0 > 0.
will be denoted generically by p with the on-shell condition p
We will also use a different letter for the energy: E(~p) := p~2 + m2 . In other
words, p = (E(~ p), p~).
In the charged case we use essentially the same plane waves as those intro-
duced in the neutral case in (2.50). There are minor differences in the notation:
the generic notation for the energy-momentum is now p and the plane waves
with a negative frequency p0 are now on the equal footing as those with a
positive frequency. To stress this we write | − p) instead of |p). Thus we have
1
(x|p) = p p eipx , (5.15)
(2π)32E(~
p)
1
(x| − p) = p p e−ipx . (5.16)
(2π)3 2E(~p)
0
Let p0 , p0 > 0. The relations (2.51) can be now written as
i(−p|ω|p0 ) = i(p|ω| − p0 ) = 0,
0 0
−i(−p|ω| − p ) = i(p|ω|p ) p − p~0 ).
= δ(~

142
5.1.7 Positive and negative frequency subspace
When we discussed neutral scalar fields we introduced positive/negative fre-
quency spaces, which in the notation used in the charged case can be defined
by
(+)
WKG := {g ∈ CYKG : (p|ωg = 0, p0 < 0},
(−) (+)
WKG := WKG = {g ∈ CYKG : (p|ωg = 0, p0 > 0}.

Every ζ ∈ WKG can be uniquely decomposed as ζ = ζ (+) + ζ (−) with ζ (±) ∈

(±)
WKG .
(+)
We equip WKG with the scalar product

(+) (+) (+) (+)

(ζ1 |ζ2 ) := iζ1 ωζ2 d~
p. (5.17)
(+) (+) (+)
We set ZKG to be the completion of WKG in this scalar product. ZKG can be
identified with L2 (R3 ) and (5.17) rewritten as
Z
(+) (+) (+) (+)
(ζ1 |ζ2 ) = (p|ζ1 )(p|ζ2 )d~
p.

(−)
Instead of WKG for quantization we will use the corresponding complex
(−)
conjugate space denoted WKG and equipped with the scalar product
(−) (−) (−) (−)
(ζ 1 |ζ 2 ) := iζ1 ωζ2 id~
p. (5.18)

(−) (−) (−)

We set ZKG to be the completion of WKG in this scalar product. ZKG can be
identified with L2 (R3 ) and (5.18) rewritten as
Z
(−) (−) (−) (−)
(ζ 1 |ζ 2 ) = (−p|ζ 1 )(−p|ζ 2 )d~
p.

(−) (+)
Note that WKG = WKG , where we use the usual (internal) complex conju-
gation in WKG . In paticular,
| − p) = |p). (5.19)
(−) (+)
Therefore in principle we could identify ZKG and ZKG . This identification will
be important for the definition of the charge conjugation. Normally, however,
(−) (+)
we treat ZKG and ZKG as two separate Hilbert spaces.
1,3 ↑ (+) (−)
R o O (1, 3) acts on ZKG and ZKG in a natural way.

143
5.1.8 Plane wave functionals
Plane wave functionals are defined as linear or antilinear functionals on the
complex space WKG , for any ζ ∈ WKG given by
ha(p)|ζi = i(p|ωζ = (p|ζ (+) ), (5.20)
∗
ha (p)|ζi = −i(p|ωζ = (p|ζ (+) ), (5.21)
hb(p)|ζi = i(−p|ωζ = (−p|ζ (−) ), (5.22)
∗
hb (p)|ζi = −i(−p|ωζ = (−p|ζ (−) ). (5.23)
Thus
Z r
E(~
p) i  d~x
a(p) = ψ(0, ~x) + p η(0, ~x) e−i~p~x p ,
2 2E(~ p) (2π)3
Z r
∗ E(~
p) ∗ i  d~x
a (p) = ψ (0, ~x) − p η ∗ (0, ~x) ei~p~x p ,
2 2E(~ p) (2π)3
Z r
E(~
p) ∗ i  d~x
b(p) = ψ (0, ~x) + p η ∗ (0, ~x) e−i~p~x p ,
2 2E(~ p) (2π)3
Z r
∗ E(~
p) i  d~x
b (p) = ψ(0, ~x) − p η(0, ~x) ei~p~x p ,
2 2E(~ p) (2π)3
The only non-vanishing Poisson bracket are
{a(p), a∗ (p0 )} = {b(p), b∗ (p0 )} = −iδ(~
p − p~0 ).
We have the following expressions for the fields:
Z
d~p
eipx a(p) + e−ipx b∗ (p) ,


ψ(x) = p p
3
(2π) 2E(~ p)
Z p
p E(~
d~ p)
√ eipx a(p) − e−ipx b∗ (p) .


η(x) = p
i (2π)3 2
We have accomplished the diagonalization of the basic observables:
Z
p) a∗ (p)a(p) + b∗ (p)b(p) ,


H = d~
pE(~
Z
~ pp~ a∗ (p)a(p) + b∗ (p)b(p) ,


P = d~
Z
p a∗ (p)a(p) − b∗ (p)b(p) .


Q = d~

In the alternative notation

a(p) = iψ((|p))),
∗
a (p) = −iψ ∗ ((|p))),
b(p) = iψ ∗ ((| − p))).
b∗ (p) = −iψ((| − p))).

144
5.1.9 Quantization
In principle, we could quantize the complex Klein-Gordon equation as a pair of
real Klein-Gordon fields. However, we will use the formalism of quantization of
charged bosonic systems [15].
We want to construct (H, Ĥ, Ω) satisfying the usual requirements of QM
(1)-(3) and an operator valued distribution

R1,3 3 x 7→ ψ̂(x) (5.24)

˙
satisfying, with η̂(x) := ψ̂(x),
(1) (−2 + m2 )ψ̂(x) = 0;
(2) the only non-vanishing 0-time commutators are

[ψ̂(0, ~x), η̂ ∗ (0, ~y )] = iδ(~x − ~y ), [ψ̂ ∗ (0, ~x), η̂(0, ~y )] = iδ(~x − ~y ); (5.25)

(3) eitĤ ψ̂(x0 , ~x)e−itĤ = ψ̂(x0 + t, ~x);

(4) Ω is cyclic for ψ̂(x), ψ̂ ∗ (x).
The above problem has an essentially unique solution, which we describe
below.
We set
(+) (−)
H := Γs (ZKG ⊕ ZKG ).
(+)
Creation/annihilation operators for the particle space ZKG ' L2 (R3 ) are de-
(−)
noted with the letter a and for the antiparticle space ZKG ' L2 (R3 ) with the
letter b. Thus, for p on the mass shell, using physicist’s notation on the left and
mathematician’s on the right, creation operators for particles/antiparticles are
written as
â∗ (p) = â∗ |p) ,


(5.26)
b̂∗ (p) = b̂∗ | − p) .


(5.27)
Ω is the Fock vacuum. The quantum field is
Z
d~
p  
ψ̂(x) := p p eipx â(p) + e−ipx b̂∗ (p) ,
(2π)3 2E(~ p)
Z p
d~
p E(~ p)  
η̂(x) := p √ eipx â(p) − e−ipx b̂∗ (p) .
i (2π)3 2
The quantum Hamiltonian, momentum and charge are
Z  
Ĥ := â∗ (p)â(p) + b̂∗ (p)b̂(p) E(~p)d~
p, (5.28)
Z 
~

P̂ := â∗ (p)â(p) + b̂∗ (p)b̂(p) p~d~
p,
Z  
Q̂ := â∗ (p)â(p) − b̂∗ (p)b̂(p) d~p.

145
 Equivalently, for any t
Z  
Ĥ = : η̂ ∗ (t, ~x)η̂(t, ~x) + ∂~ ψ̂ ∗ (t, ~x)∂~ ψ̂(t, ~x) + m2 ψ̂ ∗ (t, ~x)ψ̂(t, ~x) :d~x,
Z 
~

P̂ = : −η̂ ∗ (t, ~x)∂~ ψ̂(t, ~x) − ∂~ ψ̂ ∗ (t, ~x)η̂(t, ~x) :d~x,
Z  
Q̂ = i : −η̂ ∗ (t, ~x)ψ̂(t, ~x) + ψ̂ ∗ (t, ~x)η̂(t, ~x) :d~x.

Thus all these operators are expressed in terms of the Wick quantization of their
classical expressions.
 Note thatthe whole
 R1,3 o O↑ (1, 3) acts unitarily on H by U (y, Λ) :=
group 
Γ r(y,Λ) (+)
⊗ Γ r(y,Λ) (−)
, with
ZKG ZKG

U (y, Λ)ψ̂(x)U (y, Λ)∗ = ψ̂ (y, Λ)x .

Moreover,
[ψ̂(x), ψ̂ ∗ (y)] = −iD(x − y), [ψ̂(x), ψ̂(y)] = 0.
Note the identities

(Ω|ψ̂(x)ψ̂ ∗ (y)Ω) = −iD(+) (x − y),

(Ω|T(ψ̂(x)ψ̂ ∗ (y))Ω) = −iDF (x − y).

For f ∈ Cc∞ (R1,3 , C) we set

Z
ψ̂[f ] := f (x)ψ̂(x)dx,
Z
ψ̂ ∗ [f ] := f (x)ψ̂ ∗ (x)dx.

We obtain an operator valued distribution satisfying the Wightman axioms with

(+) (−)
D := Γfin
s (ZKG ⊕ ZKG ).
For an open set O ⊂ R1,3 the field algebra is defined as
n   o00
F(O) := exp iψ̂ ∗ [f ] + iψ̂[f ] : f ∈ Cc∞ (O, C) .

The observable algebra A(O) is the subalgebra of F(O) fixed by the automor-
phism
B 7→ eiθQ̂ Be−iθQ̂ .
The algebras F(O) and A(O) satisfy the Haag-Kastler axioms.

146
5.1.10 Quantum 4-current
Let us try to introduce the (quantum) 4-current density by

i µ ∗
Jˆµ (x) = ∂ ψ̂ (x)ψ̂(x) + ψ̂(x)∂ µ ψ̂ ∗ (x)
2 
−ψ̂ ∗ (x)∂ µ ψ̂(x) − ∂ µ ψ̂(x)ψ̂ ∗ (x) . (5.29)

In Subsubsect. 5.2.6 later on we will introduce a certain unitary operator C

called the charge conjugation satisfying CΩ = Ω, C Jˆµ (x)C −1 = −Jˆµ (x). The
existence of such C implies

(Ω|Jˆµ (x)Ω) = 0.

Therefore, (5.29) can be replaced with the following equivalent definition:

 
Jˆµ (x) = i: ∂ µ ψ̂ ∗ (x)ψ̂(x) − ψ̂ ∗ (x)∂ µ ψ̂(x) :. (5.30)

Thus Jˆµ (x) can be defined both as the Weyl quantization (5.29) and the Wick
quantization (5.30) of the corresponding quadratic classical expression.
Formally, we can check the relations

∂ µ Jˆµ (x) = 0,
Jˆµ (x)∗ = Jˆµ (x).

In particular, we have the (quantum) charge density

Q̂(x) := Jˆ0 (x) = i: −η̂ ∗ (x)ψ̂(x) + ψ̂ ∗ (x)η̂(x) :

with the relations

[Q̂(t, ~x), ψ̂(t, ~y )] = −ψ̂(t, ~y )δ(~x − ~y ),

[Q̂(t, ~x), η̂(t, ~y )] = −η̂(t, ~y )δ(~x − ~y ),
[Q̂(t, ~x), Q̂(t, ~y )] = 0. (5.31)

Similarly, as in the classical case, for χ ∈ Cc∞ (R3 , R), let αχ denote the cor-
responding gauge transformation at time t = 0 defined as the ∗-automorphism
of the algebra generated by the fields operators satisfying

αχ (ψ̂(0, ~x)) := e−iχ(~x) ψ̂(0, ~x),

αχ (η̂(0, ~x)) := e−iχ(~x) η̂(0, ~x). (5.32)

Obviously,

αχ (ψ̂ ∗ (0, ~x)) = eiχ(~x) ψ̂ ∗ (0, ~x),

αχ (η̂ ∗ (0, ~x)) = eiχ(~x) η̂ ∗ (0, ~x). (5.33)

147
 Assume that χ 6= 0. Let us check whether αχ is unitarily implementable.
On the level of annihilation operators we have
Z Z s s !
E(~
p1 ) E(~p) d~xd~
p1 i(~p1 −~p)~x−ieχ(~x)
αχ (â(p)) = + e â(p1 )
E(~p) p1 ) 2(2π)3
E(~
Z Z s s !
E(~
p1 ) E(~p) d~xd~
p1 −i(~p1 +~p)~x−ieχ(~x) ∗
+ − e b̂ (p1 ).
E(~p) p1 ) 2(2π)3
E(~

Let qχ (~
p, p~1 ) denote the integral kernel on the second line above. By the Shale
criterion (Thm A.2), we need to check whether it is square integrable. Now
s s !
E(~p1 ) E(~p)
− (5.34)
E(~
p) E(~
p1 )
p1 | − |~
(|~ p|)(|~
p1 | + |~
p|)
=
p .
E(~
p) + E(~ p1 ) E(~ p)E(~ p1 )

After integrating in ~x we obtain fast decay of qχ in p~ + p~1 , which in particular

allows us to control the term |~
p1 | − |~p|. We obtain
Z
C
p, p~1 )|2 d~
|q(~ p∼ ,
p1 )2
E(~

which is not integrable. Thus αχ is not implementable.

Formally, if we set
Z
Q̂(χ) := χ(~x)Q̂(0, ~x)d~x, (5.35)

then eieQ̂(χ) implements the gauge transformation:

αχ (B̂) = eieQ̂(χ) B̂e−ieQ̂(χ) .

But we know that αχ is not implementable. Thus for nonzero χ (5.35) cannot
be defined as a closable operator.
However, the (quantum) charge
Z
Q̂ = Q̂(t, ~x)d~x,

as we have already seen, is a well defined operator.

For further reference let us express the charge and current density in terms

148
of creation and annihilation operators:
Z Z s s !
d~p1 d~
p2 E(~ p1 ) E(~p2 )
Q̂(x) = +
2(2π)3 E(~ p2 ) E(~p1 )
 
× e−ixp1 +ixp2 â∗ (p1 )â(p2 ) − eixp1 −ixp2 b̂∗ (p2 )b̂(p1 )
Z Z s s !
d~p1 d~
p2 E(~ p1 ) E(~ p2 )
+ −
2(2π)3 E(~ p2 ) E(~ p1 )
 
× −e−ixp1 −ixp2 â∗ (p1 )b̂∗ (p2 ) + eixp1 +ixp2 b̂(p1 )â(p2 ) ,
Z Z
~ d~
p d~ p
Ĵ (x) = p1 2 (~
p1 + p~2 )
2(2π)3 E(~ p1 )E(~p2 )
 
× −e−ixp1 +ixp2 â∗ (p1 )â(p2 ) + eixp1 −ixp2 b̂∗ (p2 )b̂(p1 )
Z Z
d~
p d~p
+ p1 2 p1 − p~2 )
(~
3
2(2π) E(~ p1 )E(~p2 )
 
× −e−ixp1 −ixp2 â∗ (p1 )b̂∗ (p2 ) + eixp1 +ixp2 b̂(p1 )â(p2 ) .

5.1.11 Quantization in terms of smeared fields

An alternative equivalent formulation of the quantization program uses smeared
fields instead of point fields. Instead of (2.57) we look for an antilinear function
WKG 3 ζ 7→ ψ̂((ζ))
with values in closed operators such that
(1) [ψ̂((ζ1 )), ψ̂ ∗ ((ζ2 ))] = iζ 1 ωζ2 , [ψ̂((ζ1 )), ψ̂((ζ2 ))] = 0.
itĤ −itĤ
(2) ψ̂((r(t,~0) ζ)) = e ψ̂((ζ))e .
(3) Ω is cyclic for the algebra generated by ψ((ζ)), ψ ∗ ((ζ)).
One can pass between these two versions of the quantization by
Z  
ψ̂((ζ)) = −ζ̇(t, ~x)ψ̂(t, ~x) + ζ(t, ~x)η̂(t, ~x) d~x. (5.36)

5.2 Charged scalar bosons in an external 4-potential

5.2.1 Classical fields
Let us go back to the classical theory. Let
R1,3 3 x 7→ A(x) = [Aµ (x)] ∈ R1,3 (5.37)
be a given function called the (external electromagnetic) 4-potential. In most of
this subsection we will assume that (5.37) is Schwartz. The (complex) Klein-
Gordon equation in the external 4-potential A is
−(∂µ + ieAµ (x))(∂ µ + ieAµ (x)) + m2 ψ(x) = 0.


(5.38)

149
If ψ satisfies (5.38) and R1,3 3 x 7→ χ(x) ∈ R is smooth, then e−ieχ ψ satisfies
(5.38) with A replaced with A + ∂χ.
In this subsection, the field satisfying the Klein-Gordon equation with A = 0
will be denoted ψfr .
The retarded/advanced Green’s function is defined as the unique solution of

−(∂µ + ieAµ (x))(∂ µ + ieAµ (x)) + m2 D± (x, y) = δ(x − y)

(5.39)

satisfying
suppD± ⊂ {x, y : x ∈ J ± (y)}.
We generalize the Pauli-Jordan function:

D(x, y) := D+ (x, y) − D− (x, y).

Clearly,
suppD ⊂ {x, y : x ∈ J(y)}.
The Cauchy problem of (5.38) can be expressed with help of the function D:
Z
ψ(t, ~x) = − ∂s D(t, ~x; s, ~y ) s=0 ψ(0, ~y )d~y (5.40)
R3
Z
+ D(t, ~x; 0, ~y )ψ̇(0, ~y )d~y .
R3

We would like to introduce a field R1,3 3 x 7→ ψ(x) satisfying (5.38). As we

will see shortly, the conjugate field is

η(x) := ∂0 ψ(x) + ieA0 (x)ψ(x).

For definiteness, we will assume that ψ(x), η(x) act on WKG and at time t = 0
coincide with free fields:

ψ(0, ~x) = ψfr (0, ~x),

η(0, ~x) = ηfr (0, ~x).

This determines the field ψ uniquely:

Z
ψ(t, ~x) = − ∂s D(t, ~x; s, ~y ) s=0 ψfr (0, ~y )d~y (5.41)
3
ZR


+ D(t, ~x; 0, ~y ) ηfr (0, ~y ) − ieA0 (0, ~y )ψfr (0, ~y ) d~y .
R3

5.2.2 Lagrangian and Hamiltonian formalism

Consider the Lagrangian density

L(x) = − ∂µ − ieAµ (x) ψ ∗ (x) ∂ µ + ieAµ (x) ψ(x) − m2 ψ ∗ (x)ψ(x).

The Euler-Lagrange equations (5.10) yield (5.38).

150
 Let us introduce the variable conjugate to ψ ∗ (x) and ψ(x):

∂L
η(x) := ∗ (x) = ∂0 ψ(x) + ieA0 (x)ψ(x),
∂ψ,0
∂L
η ∗ (x) = = ∂0 ψ ∗ (x) − ieA0 (x)ψ ∗ (x).
∂ψ,0 (x)

We introduce the Hamiltonian density

∂L(x) ∂L(x) ∗
H(x) = ψ̇(x) + ψ̇ (x) − L(x)
∂ ψ̇(x) ∂ ψ̇ ∗ (x)
= η ∗ (x)η(x) + ieA0 (x) (ψ ∗ (x)η(x) − η ∗ (x)ψ(x))
+(∂i − ieAi (x))ψ ∗ (x)(∂i + ieAi (x))ψ(x) + m2 ψ ∗ (x)ψ(x)
= η ∗ (x)η(x) + ∂i ψ ∗ (x)∂i ψ(x)
+ieA0 (x) (ψ ∗ (x)η(x) − η ∗ (x)ψ(x)) − ieAi (x) (ψ ∗ (x)∂i ψ(x) − ∂i ψ ∗ (x)ψ(x))
~ 2 ψ ∗ (x)ψ(x) + m2 ψ ∗ (x)ψ(x).
+e2 A(x)

The Hamiltonian Z
H(t) = H(t, ~x)d~x

can be used to generate the dynamics

ψ̇(t, ~x) = {ψ(t, ~x), H(t)}, η̇(t, ~x) = {η(t, ~x), H(t)}.

The interaction picture Hamiltonian can be partially expressed in terms of

the free 4-current density:
Z  
HInt (t) = d~x eAµ (t, ~x)Jfrµ (t, ~x) + e2 A(t,
~ ~x)2 ψfr ∗
(t, ~x)ψfr (t, ~x)
Z 
= d~x eA0 (t, ~x)Qfr (t, ~x) (5.42)

~ ~x)J~fr (t, ~x) + e2 A(t,
+eA(t, ~ ~x)2 ψfr
∗
(t, ~x)ψfr (t, ~x)

151
 Z Z s s !
e d~p1 d~
p2 E(~
p1 ) E(~
p2 )
= +
2 (2π)3 E(~
p2 ) E(~
p1 )
 
× A0 (t, p~1 − p~2 )eitE(~p1 )−itE(~p2 ) a∗ (p1 )a(p2 ) − A0 (t, −~ p1 + p~2 )e−itE(~p1 )+itE(~p2 ) b(p1 )b∗ (p2 )
Z Z s s !
e d~p1 d~
p2 E(~ p1 ) E(~ p2 )
+ −
2 (2π)3 E(~ p2 ) E(~ p1 )
 
× A0 (t, p~1 + p~2 )eitE(~p1 )+itE(~p2 ) a∗ (p1 )b∗ (p2 ) − A0 (t, −~ p1 − p~2 )e−itE(~p1 )−itE(~p2 ) b(p1 )a(p2 )
Z Z
e d~
p d~ p
+ p 1 2 (~
p1 + p~2 )
2 3
(2π) E(~ p1 )E(~ p2 )
 
× −A(t,~ p~1 − p~2 )eitE(~p1 )−itE(~p2 ) a∗ (p1 )a(p2 ) + A(t, ~ −~ p1 + p~2 )e−itE(~p1 )+itE(~p2 ) b(p1 )b∗ (p2 )
Z Z
e d~
p d~ p
+ p 1 2 p1 − p~2 )
(~
2 3
(2π) E(~ p1 )E(~ p2 )
 
× −A(t,~ p~1 + p~2 )eitE(~p1 )+itE(~p2 ) a∗ (p1 )b∗ (p2 ) + A(t, ~ −~ p1 − p~2 )e−itE(~p1 )−itE(~p2 ) b(p1 )a(p2 )
e2
Z Z
d~p d~ p
+ p 1 2p
2 3
(2π) E(~ p1 ) E(~ p2 )

× A ~ (t, p~1 − p~2 )e
2 itE(~p 1 )−itE(~p2) ∗
a (p1 )a(p2 ) + A ~ 2 (t, −~
p1 + p~2 )e−itE(~p1 )+itE(~p2 ) b(p1 )b∗ (p2 )

+A~ 2 (t, p~1 + p~2 )eitE(~p1 )+itE(~p2 ) a∗ (p1 )b∗ (p2 ) + A
~ 2 (t, −~p1 − p~2 )e−itE(~p1 )−itE(~p2 ) b(p1 )a(p2 ) .

5.2.3 Classical discrete symmetries

If ζ solves the Klein-Gordon equation with the 4-potential A, then so does ζ
with the 4-potential −A. Thus replacing

ψ(x), ψ ∗ (x), A(x)

with ψ ∗ (x), ψ(x), −A(x)

is a symmetry of the complex Klein-Gordon equation with an external 4-potential

(5.38). It is called charge conjugation and denoted C.
Choose ξP ∈ {1, −1}. Recall that P(x0 , ~x) := (x0 , −~x) denotes the space
inversion. Replacing
~
ψ(x), ψ ∗ (x), A0 (x), A(x)

~
with ξP ψ(Px), ξP ψ ∗ (Px), A0 (Px), −A(Px)

is a symmetry of (5.38) called parity and denoted P.

Choose ξT ∈ {1, −1}. Recall that T(x0 , ~x) := (−x0 , ~x) denotes the time
reflection. Replacing
~
ψ(x), ψ ∗ (x), A0 (x), A(x)

~
with ξT ψ(Tx), ξT ψ ∗ (Tx), A0 (Tx), −A(Tx)

152
is a symmetry of (5.38) called time reversal and denoted T .
Let ξX := ξP ξT . The composition of C, P and T consists in replacing
ψ(x), ψ ∗ (x), A(x)
with ξX ψ ∗ (−x), ξX ψ(−x), −A(−x).
It is called the CPT symetry and is denoted X .
C, P, T and X commute with one another and we have the relations
C 2 = P 2 = T 2 = X 2 = id.

5.2.4 Quantization
We are looking for a quantum field satisfying
−(∂µ + ieAµ (x))(∂ µ + ieAµ (x)) + m2 ψ̂(x) = 0.


(5.43)
We set
η̂(x) := ∂0 ψ̂(x) + ieA0 (x)ψ̂(x).
We will assume that ψ̂, η̂ act on the Hilbert space of free fields
(+) (−)
Γs (ZKG ⊕ ZKG ),
and at time t = 0 we have
ψ̂(~x) := ψ̂(0, ~x) = ψ̂fr (0, ~x),
η̂(~x) := η̂(0, ~x) = η̂fr (0, ~x).
The solution is unique and is obtained by decorating (5.41) with “hats”.
We would like to ask whether the quantum fields are implemented by a
unitary dynamics. Equivalently, we want to check if the classical dynamics
generated by HInt (t) satisfies the Shale criterion (Thm A.2).
By following the discussion of Subsubsect. 2.3.4 we check that the classical
scattering operator is unitarily implementable.
The Shale criterion is satisfied for the dynamics from t− to t+ iff the spatial
part of the 4-potential is the same at the initial and final time:
~ + , ~x) = A(t
A(t ~ − , ~x), ~x ∈ R3 . (5.44)
To see this note that HInt (t) consists of three terms described in (5.42).
~ ~x)2 ψ ∗ (t, ~x)ψfr (t, ~x) is very similar to the mass-like pertur-
The term e2 A(t, fr
bation considered already in Subsubsect. 2.3.4, which did not cause problems
with the Shale criterion for the dynamics for any t+ , t− .
The same is true for the term eA0 (t, ~x)Qfr (t, ~x). Indeed, a similar term
was discussed before in the context of gauge transformations, see in particular
(5.34). Then there was a problem with the square integrability. But now we
can integrate by parts, which improves the decay.
~ ~x)J~fr (t, ~x) is problematic – it has worse behavior for large mo-
The term eA(t,
menta than the previous two terms. The integration by parts creates a boundary
term that is not square integrable unless (5.44) holds, when it vanishes.

153
5.2.5 Quantum Hamiltonian
Formally, the fields undergo a unitary dynamics given by
 Z 0   Z t 
ψ̂(t, ~x) := Texp −i Ĥ(s)ds ψ̂(0, ~x)Texp −i Ĥ(s)ds ,
t 0

where the Schrödinger picture Hamiltonian is

Z 
d~x η̂ ∗ (~x)η̂(~x) + ieA0 (t, ~x): ψ̂ ∗ (~x)η̂(~x) − η̂ ∗ (~x)ψ̂(~x) :


Ĥ(t) =

+(∂i − ieAi (t, ~x))ψ̂ ∗ (~x)(∂i + ieAi (t, ~x))ψ̂(~x)


+m2 ψ̂ ∗ (~x)ψ̂(~x) . (5.45)

Note that the above Hamiltonian is formally the Weyl quantization of its corre-
sponding classical expressions. This is perhaps not obvious the way it is written.
To see this we should note that equal time ψ̂ and ψ̂ ∗ commute, the same is true
for equal time η̂ and η̂ ∗ , finally the mixed term can be expressed by the 4-current
where the Wick and Weyl quantizations coincide, see Subsubsect. 5.1.10.
In any case, the analysis of the previous subsubsection shows that the above
Hamiltonian is often ill defined and should be understood as a formal expression,
even when we try renormalize by adding a constant C(t). We will need it to
develop perturbation expansion for the quantum scattering operator and to
compute the energy shift.
(5.45) can be compared with the free Hamiltonian without the Wick order-
ing, which differs from (5.28) by an (infinite) constant:
Z  
Ĥfr = d~x η̂ ∗ (~x)η̂(~x) + ∂i ψ̂ ∗ (~x)∂i ψ̂(~x) + m2 ψ̂ ∗ (~x)ψ̂(~x) . (5.46)

This leads to the following interaction picture Hamiltonian:

Z  
ĤInt (t) = d~x eAµ (t, ~x)Jˆfrµ (t, ~x) + e2 A(t,~ ~x)2 ψ̂fr
∗
(t, ~x)ψ̂fr (t, ~x)
Z 
= d~x eAµ (t, ~x)Jˆfrµ (t, ~x) + e2 A(t, ~x)2 ψ̂fr
∗
(t, ~x)ψ̂fr (t, ~x)

∗
+e2 A0 (t, ~x)2 ψ̂fr (t, ~x)ψ̂fr (t, ~x) . (5.47)

5.2.6 Quantized discrete symmetries

The discrete symmetries considered in Subsubsect. 5.2.3 remain true when we
decorate the fields with “hats”. Thus on the level of quantum observables the
discrete symmetries are the same as in the classical case.
A separate discussion is needed concerning the implementation of these sym-
(+) (−)
metries by unitary or antiunitary operators on the Hilbert space Γs (ZKG ⊕ZKG ).
We will discuss this for free fields, that is, for A = 0. Free fields are used to com-
pute the scattering operator for the 4-potential A, denoted by Ŝ(A). Therefore,
our analysis will lead to some identities for Ŝ(A).

154
 First consider the charge conjugation. As we have already pointed out in
(+) (−)
Subsubsect. 5.1.7, the spaces ZKG and ZKG can be naturally identified. There-
(+) (−)
fore, we can define a unitary operator on ZKG ⊕ ZKG

χ(g1 , g 2 ) := (g 2 , g1 ).

Clearly,
χ|p) = | − p), χ| − p) = |p).
We set C := Γ(χ). We have C 2 = 1l, CΩ = Ω,

C ψ̂fr (x)C −1 = ψ̂fr

∗ ∗
(x), C ψ̂fr (x)C −1 = ψ̂fr (x),

~ ~
C Q̂fr (x)C −1 = −Q̂fr (x), C Ĵfr (x)C −1 = −Ĵfr (x),
C Ŝ(A)C −1 = Ŝ(−A).
(+) (−)
Define a unitary operator on ZKG ⊕ ZKG


π(g1 , g 2 ) := ξP g1 ◦ P, ξP g2 ◦ P .

Clearly,
π|E, p~) = ξP |E, −~
p), π|−E, −~
p) = ξP |−E, p~).
We have a natural implementation of the parity P := Γ(π). It satisfies P 2 = 1l,
P Ω = Ω,

P ψ̂fr (x)P −1 = ξP ψ̂fr (Px), P ψ̂fr

∗
(x)P −1 = ξP ψ̂fr
∗
(Px),

~ ~
P Q̂fr (x)P −1 = Q̂fr (Px), P Ĵfr (x)P −1 = −Ĵfr (Px),
~ −1 = Ŝ(A0 ◦ P, −A
P Ŝ(A0 , A)P ~ ◦ P).
(+) (−)
Define the following antiunitary operator on ZKG ⊕ ZKG :


τ (g1 , g 2 ) := ξT g1 ◦ T, ξT g2 ◦ T .

Clearly,
τ |E, p~) = ξT |E, −~
p), τ |−E, −~
p) = ξT |−E, p~).
Set T := Γ(τ ). We have T 2 = 1l, T Ω = Ω,

T ψ̂fr (x)T −1 = ξ T ψ̂fr (Tx), T ψ̂fr

∗
(x)T −1 = ξT ψ̂fr
∗
(Tx),

~ ~
T Q̂fr (x)T −1 = Q̂fr (Tx), T Ĵfr (x)T −1 = −Ĵfr (Tx),
~ −1 = Ŝ(A0 ◦ T, −A
T Ŝ(A0 , A)T ~ ◦ T).

155
5.2.7 2N -point Green’s functions
For yN , . . . y1 , xN , . . . , x1 , the 2N point Green’s function are defined as follows:

ψ̂ ∗ (y1 ) · · · ψ̂ ∗ (yN )ψ̂(xN ) · · · ψ̂(x1 )

   
:= Ω+ |T ψ̂ ∗ (y1 ) · · · ψ̂ ∗ (yN )ψ̂(xN ) · · · ψ̂(x1 ) Ω− .

One can organize Green’s functions in terms of the generating function:

Z(g, g)
∞ Z
(−1)N ∗
X Z
:= ··· 2
ψ̂ (y1 ) · · · ψ̂ ∗ (yN )ψ̂(xN ) · · · ψ̂(x1 )
n=0
(N !)
×g(y1 ) · · · g(yN )g(xN ) · · · g(x1 )dy1 · · · dyN dxN · · · dx1
  Z ∞ Z Z  
∗
= Ω Texp −i ĤInt (t)dt − i g(x)ψ̂fr (x)dx − i g(x)ψ̂fr (x)dx Ω .
−∞

One can retrieve Green’s functions from the generating function:

ψ̂ ∗ (y1 ) · · · ψ̂ ∗ (yN )ψ̂(xN ) · · · ψ̂(x1 )

∂ 2N
= (−1)N Z(g, g) .
∂g(y1 ) · · · ∂g(yN )∂g(xN ) · · · ∂g(x1 ) g=g=0

We introduce also the amputated Green’s function

ψ̂ ∗ (p01 ) · · · ψ̂ ∗ (p0N )ψ̂(pN ) · · · ψ̂(p1 ) amp

(p01 )2 + m2 · · · (p0N )2 + m2 (pN )2 + m2 · · · (p1 )2 + m2





:=
× ψ̂ ∗ (p01 ) · · · ψ̂ ∗ (p0N )ψ̂(pN ) · · · ψ̂(p1 ) .

Set

|−p0n0 , . . . , −p01 , pn , . . . , p1 ) := b̂∗ (p0n0 ) · · · b̂∗ (p01 )â∗ (pn ) · · · â∗ (p1 )Ω.

One can compute scattering amplitudes from the amputated Green’s functions:
 
−0 −
−p+0
n +0 , . . . , p +
n + , . . . | Ŝ |−p n −0 , · · · , pn − , · · ·
−0 −
· · · ψ̂(p+ ∗ +0 ∗
n+ ) · · · ψ̂ (pn+0 )ψ̂(−pn−0 ) · · · ψ̂ (−pn− ) · · · amp
= q q q q ,
−0 −
p
(2π)3(n+ +n+0 +n−0 +n− ) · · · 2E(p+ n+ ) · · · 2E(p +0
n+0 ) 2E(p n−0 ) · · · 2E(p n− ) · · ·

where all p± ±0
i , pi are on shell.

156
5.2.8 Path integral formulation
Since the Hamiltonian that we consider is quadratic, we can compute exactly
the generating function in terms of the Fredholm determinant on L2 (R1,3 ):
Z(g, g) (5.48)

−1
det − 2 + m 2
− (∂µ + ieAµ (x))(∂ µ + ieAµ (x)) + m2 − i0


=

−1 
× exp ig (∂µ + ieAµ (x))(∂ µ + ieAµ (x)) + m2 − i0 g
   −1
= det 1l + −ieAµ (x)∂ µ − ie∂ µ Aµ (x) + e2 Aµ (x)Aµ (x) Dfr
c

 
c

µ µ 2 µ

c −1
× exp igDfr 1l + −ieAµ (x)∂ − ie∂ Aµ (x) + e Aµ (x)A (x) Dfr g .

Let us stress that the above formulas are based on the formal expression for
the Hamiltonian (5.47) where we used the Weyl quantization, in contrast to
the analogous formula (2.135) for the mass-like perturbation, which were Wick
ordered. The expression is to a large degree ill-defined.
Formally, (5.48) can be rewritten in terms of path integrals as
 R

Π dψ ∗ (x) Π0 dψ(y) exp i L(x) − g(x)ψ ∗ (x) − g(x)ψ(x) dx
R
y y
R R
.
Π dψ ∗ (y) Π dψ(y 0 ) exp i Lfr (x)dx
y y0

5.2.9 Feynman rules

Let us describe the Feynman rules for the charged scalar field in an external
4-potential. We have 1 kind of lines and 2 kinds of vertices. Each line has an
arrow. At each vertex two lines meet, one with an arrow pointing towards, one
with an arrow pointing away from the vertex. The 1-photon vertex is denoted
by an attached “photon line” ending with a small cross. The 2-photon vertex
has two “photon lines”, each ending with a cross. Note that the “photon lines”
are in this context only decorations of the vertices – there are no photons in
this theory. They are usually denoted by wavy, sometimes dashed lines. For
typographical reasons we use dashed lines.
To compute Green’s functions we do as follows:
(1) We draw all possible Feynman diagrams.
(2) (i) To each 1-photon vertex we associate the factor
− −
ie(p+ ν +
ν + pν )A (p − p ).

(ii) To each 2-photon vertex we associate the factor

−ie2 (Aν Aν )(p+ − p− ).

(3) To each line we associate the propagator

c −i
−iDfr (p) = .
p2 + m2 − i0

157
 d4 p
(4) We integrate over the variables of internal lines with the measure (2π)4 .
It is immediate to derive the Feynman rules for charged scalar bosons from
the path integral formula (5.48).
The derivation of the Feynman rules within the Hamiltonian formalism using
the Dyson expansion of the scattering operator is relatively complicated, since
one has to use not only the two-point functions of “configuration space fields”
ψ, ψ ∗ , but also of conjugate fields η, η ∗ [26]:
∗ c
(Ω|T(ψ̂fr (x)ψ̂fr (y))Ω) = −iDfr (x − y),
∗ c
(Ω|T(η̂fr (x)ψ̂fr (y))Ω) = −i∂x0 Dfr (x − y),
∗ c
(Ω|T(ψ̂fr (x)η̂fr (y))Ω) = −i∂y0 Dfr (x − y),
∗ c
(Ω|T(η̂fr (x)η̂fr (y))Ω) = −i∂x0 ∂y0 Dfr (x − y) − iδ(x − y).

Figure 7: Diagram for Green’s function.

To compute scattering amplitudes with N − incoming and N + outgoing par-

ticles we draw similar diagrams as for N − + N + -point Green’s functions, where
as usual the incoming lines are drawn on the right and outgoing lines on the
left. The rules are changed only concerning the external lines.
(i) With each incoming external line we associate
1
• charged boson: √ .
(2π)3 2E(~
p)
1
• charged anti-boson: √ .
p0 )
(2π)3 2E(~

(ii) With each outgoing external line we associate

1
• charged boson: √ .
(2π)3 2E(~
p)
1
• charged anti-boson: √ .
p0 )
(2π)3 2E(~

158
 Figure 8: Diagram for scattering amplitudes.

5.2.10 Vacuum energy

Formally, the vacuum energy can be computed exactly:
E := i log(Ω|ŜΩ) = i log Z(0, 0)


iTr log −2+m2 −i0 − log −(∂µ +ieAµ (x))(∂ µ +ieAµ (x))+m2 −i0


=
 
c 
= −iTr log 1l + −ieAµ (x)∂ µ −ie∂ µ Aµ (x) + e2 Aµ (x)Aµ (x) Dfr
X D`
= i . (5.49)
n`
`

Here D` is the value of the loop ` and n` is its symmetry factor. Any such a
loop is described by a cyclic sequence (α1 , . . . , αn ), where αj = 1, 2 correspond
to 1− and 2−photon vertices. The symmery factor n` is the order of the group
of the authomorphisms of this loop. The loop is oriented, hence this group is
always a subgroup of rotations. In particular, if the loop has n identical vertices,
the group is Zn and n` = n.
Actually, it is better to organize (5.49) not in terms of the number of vertices
on a loop but in terms of the order wrt e. Using the unitary charge conjugation
operator C and CΩ = Ω we obtain
(Ω|Ŝ(A)Ω) = (Ω|C Ŝ(A)C −1 Ω) = (Ω|Ŝ(−A)Ω).
Therefore, diagrams of an odd order in e vanish. This is the content of Furry’s
theorem for charged bosons. Hence (5.49) can be written as
∞
X
E= e2n En .
n=1

159
 Figure 9: Divergent diagrams for vacuum energy.

The expressions for En obtained from the Feynman rules are convergent for
n ≥ 3. E2 is logarithmically divergent, but its physically relevant gauge invariant
part is convergent. E1 is quadratically divergent and its gauge-invariant part
is logarithmically divergent. It needs an infinite renormalization, which will be
described below.

5.2.11 Pauli-Villars renormalization

The lowest nonzero loop diagrams are of the second order in e, and hence of
e2
the first order in α = 4π . There are two kinds of loops of this order: a loop
with two 1-photon vertices with symmetry factor 2 and a loop with a 2-photon
vertex with symmetry factor 1, see the Fig. 9. The sum of their contributions
has the form Z
dp
e2 E1 = Aµ (−p)Aν (p)Πµν (p). (5.50)
(2π)4
(5.50) defines the vacuum energy tensor Πµν (p).
We will first compute Πµν using the Pauli-Villars regularization. The ultra-
violet problem is more severe now than it was for the mass-like perturbation,
where a single additional fictitious particle sufficed to make the expressions well
defined. Now we need two fictitious particles:

m20 := m2 , C0 := 1,
m21 := m2 + 2Λ2 , C1 := 1,
m22 2
:= m + Λ , 2
C2 := −2.

Using
2
X 2
X
Ci = Ci m2i = 0 (5.51)
i=0 i=0

we can check that with this choice the sums used in the following computations
are integrable.
In the following formula we have a contribution of the loop with 2 single-
photon vertices and twice the contribution of the loop with a single 2-photon

160
vertex. It is convenient to write the latter as the sum of two terms, equal to
one another.

d4 q X 
Z
2 4qµ qν
2ΠµνΛ (p) = ie Ci
(q + 12 p)2 + m2i − i0 (q − 21 p)2 + m2i − i0



(2π)4 i
gµν gµν 
− −
(q + 21 p)2 + m2i − i0 (q − 12 p)2 + m2i − i0

d4 q X 4qµ qν − 2gµν (q 2 + 14 p2 + m2i )

Z
= ie2 Ci
(q + 2 p)2 + m2i − i0 (q − 21 p)2 + m2i − i0
1
4



(2π) i

∞ ∞
e2 (α1 − α2 )2
Z Z X
= − dα1 dα2 (g p2 − pµ pν )
Ci
4 µν
(4π)2
0 0 i
(α 1 + α2 )
 2
!
α1 α2 i m i
+2gµν p2 − +
(α1 + α2 )4 (α1 + α2 )3 (α1 + α2 )2
 
α1 α2 2
× exp −i(α1 + α2 )m2i − i p
α1 + α2
=: (−gµν p2 + pµ pν )2Πgi 2 gd 2
Λ (p ) + 2ΠµνΛ (p ).

We used the identity (A.23).

The gauge dependent part of the vacuum energy tensor vanishes:

−Πgd 2
µνΛ (p )
Z ∞ Z ∞
e2
 
X
2 α1 α2 2
= Ci dα1 dα2 exp −i(α1 + α2 )mi − i p
i
(4π)2 0 0 α1 + α2
α1 α2 p2 m2i
 
i
×gµν − +
(α1 + α2 )4 (α1 + α2 )3 (α1 + α2 )2
2 Z ∞ Z ∞   
X e 2 α1 α2 2
= Ci ρ∂ρ dα1 dα2 exp −iρ (α1 + α2 )mi + p
i
(4π)2 0 0 α1 + α2
igµν
×
ρ(α1 + α2 )3 ρ=1
Z ∞ Z ∞
e2
  
X
2 α1 α2 2
= Ci ρ∂ ρ dα1 dα 2 exp −i (α1 + α2 )m i + p
i
(4π)2 0 0 α1 + α2
igµν
× = 0.
(α1 + α2 )3

To compute the gauge invariant part we proceed similarly as in Subsubsec. 2.3.9,

161
 see (2.136), and we obtain
Z ∞ Z ∞
e2 X (α1 − α2 )2
Πgi
Λ (p2
) = − 2
dα 1 dα2 Ci
2(4π) 0 0 i
(α1 + α2 )4
 
α1 α2 2
× exp −i(α1 + α2 )m2i − i p
α1 + α2
2 Z 1 Z ∞
(1 − v 2 )p2
  
e dρ X
2 2
= − dv C i v exp −iρ m i +
2(4π)2 0 0 ρ i 4
1
e2 (1 − v 2 )p2
Z X  
= 2
dv Ci v 2 log m2i + − i0 .
2(4π) 0 i
4

We define

Πren (p2 ) Πgi 2 gi

:= lim Λ (p ) − ΠΛ (0) (5.52)
Λ→∞
2 Z 1
(1 − v 2 )p2
 
e 2
= dvv log 1 + − i0 .
2(4π)2 0 4m2

Using (A.27), and then analytic continuation, we obtain

Πren (p2 )
! p
e2 1 1+θ 2 2 p2
= log − − , θ= p , 0 < p2 ;
2 · 3(4π)2 θ3 1 − θ 3 θ2 p + 4m2
2

! p
e2 2 2 2 −p2
= arctan θ − − 2 , θ= p , −4m2 < p2 < 0;
2 · 3(4π)2 θ3 3 θ 2
p + 4m 2
! p
e2 1 θ+1  2 2 −p2
= log − iπ − − 2 , θ= p , p2 < −4m2 .
2 · 3(4π)2 θ3 θ−1 3 θ −p − 4m2
2

5.2.12 Renormalization of the vacuum energy

Note that the Fourier transform of the electromagnetic field is

Fµν (p) = pµ Aν (p) − pν Aµ (p). (5.53)

Hence
1
− Fµν (p)F µν (p) = −p2 |A(p)|2 + |pA(p)|2 . (5.54)
2
Thus the renormalized 1st order contribution to the vacuum energy is
Z
dp
E1ren = − Πren (p2 )Fµν (p)F µν (p). (5.55)
2(2π)4

162
 gi
We can formally write Πgi
∞ (k) := lim ΠΛ (k) (which is typically infinite).
Λ→∞
Note that the renormalized scattering operator Ŝren is a well defined unitary
operator:
gi
i
Fµν (x)F µν (x)dx
R
Ŝren = e− 2 Π∞ (0) Ŝ. (5.56)

However, there is no correctly defined renormalized Hamiltonian. Formally, the

correct Lagrangian density is obtained by replacing L(x) with
1
Lren (x) = L(x) − Πgi (0)Fµν (x)F µν (x).
2 ∞

5.2.13 Method of dispersion relations

There exists an alternative method to renormalize and compute the vacuum
energy. We start with computing just the imaginary part of the gauge invariant
vaccum energy function, which does not require a renormalization, so that we
obtain ImΠren (p2 ) from the very beginning:
Z 1
e2 (1 − v 2 )p2
 
ren 2 2
ImΠ (p ) = dvv (−π)θ −1 −
2(4π)2 0 4m2
2
e π 2
3
2 2
= − −p − 4m . (5.57)
2 · 3(4π)2 (−p2 )3/2 +

As in (2.143), using Πren (0) = 0 and Thm A.4 we obtain

1 ∞
Z  
ren 2 ren 1 1
ReΠ (p ) = dsImΠ (−s) − . (5.58)
π 4m2 s + p2 s

Note that (5.57) is nonzero only for p2 < −4m2 , and then it is negative. For
such p we can find a coordinate system with p = (p0 , ~0). Then

−gµν p2 + pµ pν = p20 (gµν + δµ0 δ0ν )

and
~ 0 , ~0)|2 .
−Fµν (p0 , ~0)F µν (p0 , ~0) = p20 |A(p (5.59)
Thus the imaginary part of (5.55) is negative (and is responsible for the decay).

163
5.2.14 Dimensional renormalization
We present an alternative computation of Πren µν based on the dimensional regu-
larization. We use the Euclidean formalism.
d4 q 
Z
4qµ qν
2ΠEµν (p) = −e 2
4 1
(2π) ((q + 2 p) + m2 )((q − 12 p)2 + m2 )
2

2gµν 
− 2
q + m2
d4 q 4qµ qν − 2gµν (q 2 + 41 p2 + m2 )
Z
= −e2
(2π)4 ((q + 21 p)2 + m2 )((q − 12 p)2 + m2 )
e2 1 d4 q 4qµ qν − 2gµν (q 2 + 14 p2 + m2 )
Z Z
= − dv 2
2 −1 (2π)4 (q 2 + p4 + m2 + vqp)2
p2
d4 q 4qµ qν − 2gµν (q 2 + 41 p2 + m2 ) + v 2 (pµ pν − gµν 2 )
Z 1 Z
= −e2 dv 2 , (5.60)
0 (2π)4 (q 2 + p4 (1 − v 2 ) + m2 )2

where we used the Feynman identity (A.28), replaced q + vp 2 withR q, used the
R1 R1 1
symmetry v → −v to remove −1 dvv and replace 12 −1 dv with 0 dv. After
this preparation, we use the dimensional regularization:

d4 q µ4−d Ωd ∞ d−1
Z Z
is replaced by |q| d|q|, (5.61)
(2π)4 (2π)d 0
Z ∞
qµ qν d4 q µ4−d Ωd
Z
is replaced by gµν |q|d+1 d|q|, (5.62)
(2π)4 d(2π)d 0

where Ωd is given by (A.30). Thus (5.60) is replaced by

1 ∞
µ4−d Ωd
Z Z
ΠE,d
µν (p) = −e
2
dv |q|d−1 d|q|
(2π)d 0 0
2

(4/d − 2)gµν q 2 − 2gµν ( 14 p2 + m2 ) + v 2 (pµ pν − gµν p2
× 2
2
q 2 + p4 (1 − v 2 ) + m2
Z 1 
e2 µ2 4π 2−d/2
= − 2
dv p2 Γ(2 − d/2)
(4π) 0 2 2
4 (1 − v ) + m
!
 p2  1   p 2
× 2gµν (1 − v 2 ) + m2 − 2gµν p2 + m2 + v 2 pµ pν − gµν
4 4 2
1
e2 µ2 4π
Z  2−d/2
= − dv 2 Γ(2 − d/2)v 2 (pµ pν − gµν p2 )
(4π)2 0 p
(1 − v 2 ) + m2
4
Z 1 
e2 2
 p2
2 2

' − dv − γ + log µ 4π − log (1 − v ) + m v 2 (pµ pν − gµν p2 )
(4π)2 0 4
e2
− (pµ pν − gµν p2 ) (5.63)
3(4π)2 (2 − d/2)

164
We can now renormalize (5.63):

ΠE,ren (p2 )(pµ pν − gµν p2 )

 
= lim ΠE,dµν (p2
) − Π E,d
µν (0)
d→4
Z 1
1 2
 p2 2

= dvv log 1 + (1 − v ) (pµ pν − gµν p2 ).
2(4π)2 0 4m2

This coincides with the Wick rotated result obtained by the Pauli-Villars method.

5.2.15 Abstract gauge covariance

Let us adopt for a moment an abstract setting. Let R 3 t 7→ Ĥ(t) be a time-
dependent Hamiltonian generating the dynamics
 Z t+ 
Û (t+ , t− ) := Texp − i Ĥ(s)ds .
t−

Let t 7→ Ŵ (t) be a family of unitary operators that have the interpretation of

time dependent gauge transformations. We will assume that Ŵ (t) converges to
identity as t → ±∞ and is generated by a time dependent family of self-adjoint
operators t 7→ R̂(t), so that
 Z t 
Ŵ (t) := Texp − i R̂(s)ds .
−∞

Then
 Z t+ 
Ŵ (t+ )Û (t+ , t− )Ŵ ∗ (t− ) = Texp − i ĤR (s)ds ,
t−

where the gauge-transformed Hamiltonian is

ĤR (t) := Ŵ (t)Ĥ(t)Ŵ ∗ (t) + R̂(t). (5.64)

5.2.16 Ward identities

Let us go back to the setting of quantized charged scalar fields. The gauge invari-
ance implies strong conditions on the scattering operator and Green’s functions.
Let Ŝ(A) denote the scattering operator for the external 4-potential A. Let
χ be a Schwartz function on R1,3 . It is easy to see that the scattering operator
is gauge-invariant:
Ŝ(A) = Ŝ(A + ∂χ). (5.65)
Differentiating this identity w.r.t. χ and setting χ = 0 we obtain the Ward(-
Takahashi) identities for the scattering operator in the position representation:

∂
∂yµ Ŝ(A) = 0.
∂Aµ (y)

165
In the momentum representation these identities read
∂
pµ Ŝ(A) = 0.
∂Aµ (p)
We will write hψ̂ ∗ (x01 ) · · · ψ̂ ∗ (x0N )ψ̂(xN ) · · · ψ̂(x1 )iA to express the depen-
dence of Green’s functions on the external 4-potential A. We have
hψ̂ ∗ (x01 ) · · · ψ̂ ∗ (x0N )ψ̂(xN ) · · · ψ̂(x1 )iA+∂χ (5.66)
0 0
∗
= hψ̂ (x01 ) · · · ψ̂ ∗ (x0N )ψ̂(xN ) · · · ψ̂(x1 )iA eieχ(x1 )+···+ieχ(xN )−ieχ(xN )−···−ieχ(x1 ) .
By differentiating with respect to χ(y) and setting χ = 0 we obtain the Ward(-
Takahashi) identities for Green’s functions in the position representation:
∂
∂yµ hψ̂ ∗ (x01 ) · · · ψ̂ ∗ (x0N )ψ̂(xN ) · · · ψ̂(x1 )iA
∂Aµ (y)
N N
!
X X
0
= e i δ(y − xj ) − i δ(y − xj ) hψ̂ ∗ (x01 ) · · · ψ̂ ∗ (x0N )ψ̂(xN ) · · · ψ̂(x1 )iA .
j=1 j=1

In the momentum representation these identities read

∂
qµ hψ̂ ∗ (p01 ) · · · ψ̂ ∗ (p0N )ψ̂(pN ) · · · ψ̂(p1 )iA
∂Aµ (q)
N
X
= hψ̂ ∗ (p01 ) · · · ψ̂ ∗ (p0j − q) · · · ψ̂ ∗ (p0N )ψ̂(pN ) · · · ψ̂(p1 )iA +
j=1
N
X
− hψ̂ ∗ (p01 ) · · · ψ̂ ∗ (p0N )ψ̂(pN ) · · · ψ̂(pj + q) · · · ψ̂(p1 )iA .
j=1

(5.65) and (5.66) are essentially obvious if we use the path integral expres-
sions. It is instructive to derive these statements also in the Hamiltonian for-
malism. This derivation is not fully rigorous, since transformations cannot be
implemented, and in general the dynamics does not have a well defined Hamil-
tonian.
Formally, we define the gauge transformation as a unitary operator
 Z 
Ŵ (χ, t) := exp −ie d~x χ(t, ~x)Q̂(~x)
 Z t Z 
= exp −ie ds d~x χ̇(s, ~x)Q̂(~x) (5.67)
−∞
 Z t Z 
= Texp −ie ds d~x χ̇(s, ~x)Q̂(~x) .
−∞

To see the second identity it is enough to note that [Q̂(~x), Q̂(~y )] = 0, hence
we can replace Texp with exp in (5.67). Clearly,
Ŵ (χ, t)ψ̂(~x)Ŵ (χ, t)∗ = eieχ(t,~x) ψ̂(~x),
Ŵ (χ, t)η̂(~x)Ŵ (χ, t)∗ = eieχ(t,~x) η̂(~x).

166
Let Ĥ(A, t) denote (5.45), that is the Schrödinger picture Hamiltonian. Let
Û (A, t+ , t− ) be the corresponding dynamics.
Z
∗
Ŵ (χ, t)Ĥ(t, A)Ŵ (χ, t) + e χ̇(t, ~x)Q̂(~x)d~x
Z 
d~x η̂ ∗ (~x)η̂(~x) − ie A0 (t, ~x) + χ̇(t, ~x) : ψ̂ ∗ (~x)η̂(~x) − η̂ ∗ (~x)ψ̂(~x) :



=

+(∂i − ieAi (t, ~x))eieχ(t,~x) ψ̂ ∗ (~x)(∂i + ieAi (t, ~x))e−ieχ(t,~x) ψ̂(~x)


+m2 ψ̂ ∗ (~x)ψ̂(~x)
= Ĥ(t, A + ∂χ).

Therefore, by (5.64), we have the following identity, which expresses the gauge
covariance:

Ŵ (χ, t+ )Û (A, t+ , t− )Ŵ ∗ (χ, t− ) = Û (A + ∂χ, t+ , t− ). (5.68)

Using that lim Ŵ (χ, t) = 1l, we obtain

t→±∞

Ŝ(A + ∂χ) = lim eit+ Ĥ0 Û (A + ∂χ, t+ , t− )e−it− Ĥ0

t+ ,−t− →∞

= lim eit+ Ĥ0 Ŵ (χ, t+ )Û (A, t+ , t− )Ŵ (χ, t− )∗ e−it− Ĥ0
t+ ,−t− →∞

= Ŝ(A),

which implies (5.65). (5.66) is a consequence of (5.68).

5.2.17 Energy shift

Suppose that the 4-potential does not depend on time and is given by a Schwartz
function R3 3 ~x 7→ A(~x) = [Aµ (~x)]. We assume that A20 ≤ m2 . The naive (Weyl
ordered) Hamiltonian is
Z 
d~x η̂ ∗ (~x)η̂(~x) + ieA0 (~x): ψ̂ ∗ (~x)η̂(~x) − η̂ ∗ (~x)ψ̂(~x) :


Ĥ =

+(∂i − ieAi (~x))ψ̂ ∗ (~x)(∂i + ieAi (~x))ψ̂(~x)


+m2 ψ̂ ∗ (~x)ψ̂(~x) . (5.69)

It can be compared with the Weyl ordered free Hamiltonian (5.46). We can ap-
ply the formula (A.17) to compute the naive energy shift (the difference between
the ground state energies of Ĥ and Ĥfr ):
q q 
Tr − ∂~ + ieA)
~ 2 + m2 − e2 A2 − −∂~ 2 + m2
0
∞
X
=: e2n En .
n=1

167
In the above sum all the terms with n ≥ 2 are well defined. The term with
n = 1 needs renormalization. The renormalized energy shift is
Z ∞
ren 2 ren 2 µν d~p X
E = −e Π (~
p )Fµν (~
p)F (~
p) + e2n En ,
(2π)3 n=2

where Πren was introduced in (5.52).

6 Dirac fermions
In this section we study the Dirac equation

(−iγ µ ∂µ + m)ψ(x) = 0

and its quantization. Here, m ≥ 0 and γ µ are Dirac matrices.

Note that the Dirac equation is complex, and therefore it describes charged
particles. In particular, one can consider the Dirac equation in the presence of
an external electromagnetic 4-potential A(x) = [Aµ (x)]:

γ µ (−i∂µ + eAµ (x)) + m ψ(x) = 0.

The theory of Dirac fermions is in many ways parallel to the theory of

charged scalar bosons described in Sect. 5.

6.1 Free Dirac fermions

6.1.1 Dirac spinors
We adopt the following conventions for Dirac matrices γ µ , µ = 0, . . . , 3:

[γ µ , γ ν ]+ = −2g µν ,
γ 0∗ = γ 0 , γ i∗ = −γ i , i = 1, 2, 3.

Sometimes we will also need

γ 5 := −iγ 0 γ 1 γ 2 γ 3 .

It satisfies
[γ 5 , γ µ ]+ = 0, (γ 5 )2 = 1l, γ 5∗ = γ 5 .
All irreducible representations of Dirac matrices are equivalent and act on
the space C4 . One of the most common is the so-called Dirac representation
   
0 1 0 0 ~σ
γ = , ~γ = ,
0 −1 −~σ 0
 
0 1
γ5 = .
1 0

168
Here is the Majorana representation:
       
0 0 −1 1 0 σ1 2 −1 0 3 0 σ3
γ =i , γ =i , γ =i , γ =i ,
1 0 σ1 0 0 1 σ3 0
 
0 σ2
γ5 = − ,
σ2 0

and the spinor representation:

   
0 0 1 0 −~σ
γ = , ~γ = ,
1 0 ~σ 0
 
1 0
γ5 = .
0 −1

Above we used the Pauli matrices ~σ = (σ1 , σ2 , σ3 ) defined by

     
0 1 0 −i 1 0
σ1 = , σ2 = , σ3 = .
1 0 i 0 0 −1

They satisfy σi σj = 2iijk σk .

Note useful (representation independent) trace identities:

Tr1l = 4,
Tr(aγ)(bγ) = −4ab,
Tr(aγ)(bγ)(cγ)(dγ) = 4(ab)(cd) − 4(ac)(bd) + 4(ad)(bc).

We also introduce the spin operators

i µ ν
σ µν := [γ , γ ].
2
In the Dirac representation

0 iσ i
 
0i
σ = ,
iσ i 0
 
σk 0
σ ij =  ijk
. (6.1)
0 σk

The operators σ µν form a representation of the Lie algebra so(1, 3) = spin(1, 3).
It is the infinitesimal version of the representation

Spin↑ (1, 3) 3 Λ̃ 7→ τ (Λ̃).

6.1.2 Special solutions and Green’s functions

Note the identity

−(−iγ∂ + m)(−iγ∂ − m) = −2 + m2 .

169
 Therefore, if
(−2 + m2 )ζ(x) = 0,
then (iγ µ ∂µ + m)ζ(x) is a solution of the homogeneous Dirac equation:
(−iγ µ ∂µ + m)(iγ µ ∂µ + m)ζ(x) = 0.
In particular, we have special solutions of the homogeneous Dirac equation
S (±) (x) = (iγ∂ + m)D(±) (x),
S(x) = (iγ∂ + m)D(x),
where D(±) and D are the special solutions of the Klein-Gordon equation intro-
duced in Subsubsect. 2.1.1. We have suppS ⊂ J.
If
(−2 + m2 )ζ(x) = δ(x),
then (iγ µ ∂µ + m)ζ(x) is a Green’s function of the Dirac equation, that is
(−iγ∂ + m)(iγ∂ + m)ζ(x) = δ(x).
In particular, a special role is played by the Green functions
S ± (x) = (iγ∂ + m)D± (x),
c
S (x) = (iγ∂ + m)DF (x),
where D± and DF are the Green’s functions of the Klein-Gordon equation
introduced in Subsubsect. 2.1.1. We have suppS ± ⊂ J ± .
The Dirac propagators satisfy the identities
S(x) = −S(−x) = S (+) (x) + S (−) (x)
= S + (x) − S − (x),
S (+) (x) = S (−) (−x),
+ −
S (x) = S (−x) = θ(x0 )S(x),
S − (x) = θ(−x0 )S(x),
S c (x) = S c (−x) = θ(x0 )S (−) (x) − θ(−x0 )S (+) (x).
Recall that the bosonic causal Green’s function in the momentum represen-
tation can be written as
1
DF (p) = .
p2 + m2 − i0
The Dirac causal Green’s function can be written in a similar way:
−γp + m
S c (p) =
p2+ m2 − i0
1
= , (6.2)
γp + m − i
where i is the shorthand for i0 sgnpγ.

170
6.1.3 Space of solutions
We set αi = γ 0 γ i , i = 1, 2, 3, and β := γ 0 . We obtain matrices satisfying

β 2 = 1l, (αi )2 = 1l, i = 1, . . . , 3;

βαi + αi β = 0, αi αj + αj αi = 0, 1 ≤ i < j ≤ 3;
β ∗ = β, αi∗ = αi , i = 1, . . . , 3.

In the Dirac representation we have

   
1 0 0 ~σ
β= , α
~= .
0 −1 ~σ 0

Using α~ , β we can rewrite the Dirac equation in the form of an evolution

equation:
i∂t ζ(t, ~x) = Dζ, D := α
~ p~ + mβ.
Note that D is essentially self-adjoint on Cc∞ (R3 , C4 ).
The following theorem describes the Cauchy problem for the Dirac equation:
Theorem 6.1 Let ϑ ∈ Cc∞ (R3 , C4 ). Then there exists a unique ζ ∈ Csc ∞
(R1,3 )
that solves the Dirac equation with initial conditions ζ(0, ~x) = ϑ(~x). It satisfies
suppζ ⊂ J(suppϑ) and is given by
Z
ζ(t, ~x) = −i S(t, ~x − ~y )βϑ(~y )d~y . (6.3)
R3

Let WD be the space of space-compact solutions of the Dirac equation, that

∞
is ζ ∈ Csc (R1,3 , C4 ) satisfying (−iγ µ ∂µ + m)ζ = 0.
For ζ1 , ζ2 ∈ C ∞ (R1,3 , C4 ) set

j µ (ζ1 , ζ2 , x) := ζ1 (x)βγ µ ζ2 (x). (6.4)

We easily check that

∂µ j µ (x) = (−iγ∂ + m)ζ1 (x)βζ2 (x) − ζ1 (x)β(−iγ∂ + m)ζ2 (x).

Therefore, if ζ1 , ζ2 ∈ WD , then j µ is a conserved 4-current:

∂µ j µ (x) = 0.

For ζ1 , ζ2 ∈ WD , the flux of j µ does not depend on the choice of a Cauchy

hypersurface S. It defines a scalar product on WD , which will have two optional
symbols: Z
ζ 1 · ζ2 = (ζ1 |ζ2 ) := j µ (ζ1 , ζ2 , x)dsµ (x).
S
In terms of the Cauchy data this scalar product coincides with the natural scalar
product on L2 (R3 , C4 ):
Z
ζ 1 · ζ2 = ζ1 (t, ~x)ζ2 (t, ~x)d~x.

171
 The group R1,3 o Spin↑ (1, 3), acts unitarily on WD by

r(y,Λ) ζ(x) := τ (Λ̃)ζ (y, Λ)−1 x .

We can also parametrize solutions of the Dirac equation by space-time func-

tions. In fact, for any f ∈ Cc∞ (R1,3 , C4 ), let us write
Z
S ∗ f (x) := S(x − y)f (y)dx.

Theorem 6.2 (1) For any f ∈ Cc∞ (R1,3 , C4 ), S ∗ f ∈ WD .

(2) Every element of WD is of this form.
RR
(3) S ∗ f1 · S ∗ f2 = f1 (x)βS(x − y)f2 (y)dxdy.
(4) If suppf2 × suppf2 , then

S ∗ f1 · S ∗ f2 = 0.

6.1.4 Classical fields

#
We will also consider the space dual to WD , denoted WD . In particular, for
∗
x ∈ R , ψ(x), ψ (x) will denote the functionals on WD with values in C4 ,
1,3

called classical Dirac fields, given by

hψ(x)|ζi := ζ(x), hψ ∗ (x)|ζi := ζ(x).

By (6.3), Z
ψ(t, ~x) = −i S(t, ~x − ~y )βψ(0, ~y )d~y .

It is convenient to introduce the Dirac conjugate of the field ψ:

ψ̃(x) := βψ ∗ (x).

(In a large part of the physics literature, ψ̃ is denoted ψ.)

#−1
On WD #
we have the group action R1,3 o Spin↑ (1, 3) 3 (y, Λ̃) 7→ r(y,Λ̃)
:

#−1
r(y,Λ̃)
ψ(x) = τ (Λ̃−1 )ψ(Λx + y).

6.1.5 Smeared fields

We can use the scalar product to pair solutions. For ζ ∈ WD , the corresponding
spatially smeared fields are the functionals on WD given by

hψ((ζ))|ρi := ζ · ρ,
hψ ∗ ((ζ))|ρi := ζ · ρ, ρ ∈ WD .

172
Clearly, for any t
Z
ψ((ζ)) = ζ(t, ~x)ψ(t, ~x)d~x,
Z
ψ ∗ ((ζ)) = ζ(t, ~x)ψ ∗ (t, ~x)d~x.

For f ∈ Cc∞ (R1,3 , C4 ), the corresponding space-time smeared fields are given
by
Z
ψ[f ] := f (x)ψ(x)dx = ψ((S ∗ f )),
Z
ψ ∗ [f ] := f (x)ψ ∗ (x)dx = ψ ∗ ((S ∗ f )).

6.1.6 Diagonalization of the equations of motion

 
ζ↑ 4
Let us use the Dirac representation, denoting elements of C with , where
ζ↓
ζ↑ , ζ↓ ∈ C2 . Ater the space-time Fourier transformation the Dirac equation
becomes

−p0 ζ↑ + ~σ p~ζ↓ + mζ↑ = 0,

0
p ζ↓ − ~σ p~ζ↑ + mζ↓ = 0.

This can be rewritten as

~σ p~
ζ↑ = − ζ↓ ,
−p0 + m
~σ p~
ζ↓ = 0
ζ↑ .
p +m

Using (~σ p~)2 = p~2 we obtain

−(p0 )2 + p~2 + m2 = 0.
p
Set E(~
p) := p~2 + m2 , so that p = (±E(~
p), p~). Define
   
1 0
χ+ := , χ− := .
0 1

Traditionally, one often introduces the following spinors:

√  
E+m χ±
u(p, ±1/2) = √ ~
σp~ , p0 = E(~
p) > 0;
2E E+m χ±
√  ∓~σp~ 
E + m E+m χ±
u(p, ±1/2) = √ , p0 = −E(~p) < 0. (6.5)
2E ±χ±

173
Note that
(u(p, s)|u(p, s0 )) = δs,s0 ,
0
(u(p, s)|u(−p, s )) = 0.
The basic plane waves are defined as
1
|p, s) = p u(p, s)eipx .
(2π)3
By writing (p, s|, as usual, we will imply the complex conjugation. We have
0
(p, s|p0 , s0 ) = p − p~0 )δs,s0 ,
δ(~ sgn(p0 p 0 ) > 0,
0
(p, s|p0 , s0 ) = 0, sgn(p0 p 0 ) < 0.
Note that plane waves diagonalize simultaneously the Dirac Hamiltonian D,
the momentum p~ = −i∂~ and the scalar product:
D|p, s) = p0 |p, s),
~ s) = p~|p, s),
−i∂|p,
XZ

ζ 1 · ζ2 = (ζ1 |p, s)(p, s|ζ2 ) + (ζ1 | − p, −s)(−p, −s|ζ2 ) d~
p.
s
In addition, positive frequency plane waves diagonalize the “upper spin in the
3rd direction” and negative frequency plane waves diagonalize the “lower spin
operator in the 3rd direction”:
 
1 σ3 0
|p, s) = s|p, s), sgnp0 > 0,
2 0 0
 
1 0 0
|p, s) = s|p, s), sgnp0 < 0.
2 0 σ3

6.1.7 Plane wave functionals

Plane wave functionals are the functionals defined by plane waves. One could
doubt whether they deserve a special notation. In the bosonic case the situation
was slightly less trivial, because the pairing was given by the symplectic form.
For fermions the pairing is given by the scalar product, hence it is straightfor-
ward. Anyway, special notation for plane wave functionals is partly motivated
as a preparation for quantization.
Let p ∈ R1,3 be on shell. Anticipating the quantization, we will use different
notation for positive and negative frequencies:
a(p, s) := ψ((|p, s))) (6.6)
Z
d~x
= p u(p, s)e−i~p~x ψ(0, ~x),
(2π)3
b∗ (p, s) := ψ((| − p, −s))) (6.7)
Z
d~x
= p u(−p, −s)ei~p~x ψ(0, ~x).
(2π)3

174
We have
XZ d~
p
u(p, s)eipx a(p, s) + u(−p, −s)e−ipx b∗ (p, s)


ψ(x) = p
s (2π)3
XZ
p |p, s)a(p, s) + | − p, −s)b∗ (p, s) .


= d~
s

6.1.8 Positive and negative frequency subspaces

We define
(+)
WD := {ζ ∈ WD : (p, s|ζ) = 0, p0 < 0},
(−)
WD := {ζ ∈ WD : (p, s|ζ) = 0, p0 > 0}.
(±)
Every ζ ∈ WD can be uniquely decomposed as ζ = ζ (+) +ζ (−) with ζ (±) ∈ WD .
(+)
On WD we keep the old scalar product:

(+) (+)
X Z (+) (+)
(ζ1 |ζ2 ) = (ζ1 |p, s)(p, s|ζ2 )d~ p.
s

(+) (+)
We set ZD to be the completion of WD in this scalar product.
(−)
Instead of WD for quantization we will use the corresponding complex
(−)
conjugate space denoted W D and equipped with the scalar product

(−) (−) (−) (−)

X Z (−) (−)
(ζ 1 |ζ 2 ) := (ζ1 |ζ2 ) = (ζ 1 |−p, −s)(−p, −s|ζ 2 )d~
p.
s

(−) (−)
We set ZD to be the completion of W D in this scalar product.
(+) (−)
The action of R1,3 o P in↑ (1, 3) leaves ZD and ZD invariant.

6.1.9 Spin averaging

1
2m (∓pγ + m) are the projections onto the positive and negative energy states
resp. With E = E(~p) = p0 > 0, we have the identities
 
X 1 E+m −~σ p~
u(p, s)ũ(p, s) =
2E ~σ p~ −E + m
s
−pγ + m m
= = Λ+ ,
2E
 E 
X 1 E−m −~σ p~
u(−p, −s)ũ(−p, −s) =
2E ~σ p~ −E − m
s
−pγ − m m
= = − Λ− .
2E E

175
 In the following spin averaging identities due to H.B.C.Casimir, which are
useful in computations of scattering cross-sections, the trace involves only the
spin degrees of freedom:
X 2 TrB̃(−p+ γ + m)B(−p− γ + m)
ũ(p+ , s+ )Bu(p− , s− ) = ,
4E + E −
s+ ,s−
X 2 TrB̃(−p+ γ − m)B(−p− γ − m)
ũ(−p+ , −s+ )Bu(−p− , −s− ) = ,
4E + E −
s+ ,s−
X 2 TrB̃(−p+ γ − m)B(−p− γ + m)
ũ(−p+ , −s+ )Bu(p− , s− ) = ,
4E + E −
s+ ,s−
X 2 TrB̃(−p+ γ + m)B(−p− γ − m)
ũ(p+ , s+ )Bu(−p− , −s− ) = ,
4E + E −
s+ ,s−

where B is an arbitrary operator on the spinor space and B̃ = βB ∗ β is its

pseudo-Hermitian conjugate.
If we specify B = β, then
X 2 X 2
u(p+ , s+ )u(p− , s− ) = u(−p+ , −s+ )u(−p− , −s− )
s+ ,s− s+ ,s−

E + E − + p~+ p~− + m2 (E + + E − )2 − |p~+ − p~− |2

= = ,
E+E− 2E + E −
X 2 X 2
u(−p+ , −s+ )u(p− , s− ) = u(p+ , s+ )u(−p− , −s− )
s+ ,s− s+ ,s−

E + E − + p~+ p~− − m2 −(E + − E − )2 + |p~+ + p~− |2

= = .
E+E− 2E + E −

6.1.10 Quantization
We would like to describe the quantization of the Dirac equation. As usual, we
will use the “hat” to denote quantized objects.
We will use the formalism of quantization of charged fermionic systems [15].
We want to construct (H, Ĥ, Ω) satisfying the standard requirements of QM
(1)-(3) and a distribution
R1,3 3 x 7→ ψ̂(x) (6.8)
with values in C4 ⊗ B(H) such that the following conditions are true:
(1) (−iγ∂ + m)ψ̂(x) = 0;
(2) [ψ̂a (0, ~x), ψ̂b∗ (0, ~y )]+ = δab δ(~x − ~y ), [ψ̂a (0, ~x), ψ̂b (0, ~y )]+ = 0;
(3) eitĤ ψ̂(x0 , ~x)e−itĤ = ψ̂(x0 + t, ~x);
(4) Ω is cyclic for ψ̂(x), ψ̂ ∗ (x).

176
 The above problem has a solution unique up to a unitary equivalence, which
we describe below.
We set
(+) (−)
H := Γa (ZD ⊕ ZD ).
(+)
Creation/annihilation operators for the particle space ZD ' L2 (R3 , C2 ) are
(−)
denoted with the letter a and for the antiparticle space ZD ' L2 (R3 , C2 )
with the letter b. Thus, for p on the mass shell and s = ± 21 , using physicist’s
notation on the left and mathematician’s on the right, creation operators for
particles/antiparticles are written as

â∗ (p, s) = â∗ |p, s) ,

(6.9)
∗ ∗


b̂ (p, s) = b̂ | − p, −s) . (6.10)

Ω is the Fock vacuum. The quantum field is

XZ d~p  
ψ̂(x) := p u(p, s)eipx â(p, s) + u(−p, −s)e−ipx b̂∗ (p, s) .
s (2π)3

The quantum Hamiltonian and momentum are

Z X 
Ĥ = â∗ (p, s)â(p, s) + b̂∗ (p, s)b̂(p, s) E(~
p)d~
p, (6.11)
s
Z X
~

P̂ = â∗ (p, s)â(p, s) + b̂∗ (p, s)b̂(p, s) p~d~
p. (6.12)
s

We also have the charge operator

XZ  
Q̂ := â∗ (~ p, s) − b̂∗ (~
p, s)â(~ p, s)b̂(~
p, s) d~
p. (6.13)
s

The whole group R1,3 o Spin↑ (1, 3) acts unitarily on H. Moreover, if we set
˜
ψ̂(x) := β ψ̂ ∗ (x), then
˜
[ψ̂a (x), ψ̂b (y)]+ = Sab (x − y), [ψ̂a (x), ψ̂b (y)]+ = 0. (6.14)

We have
˜ (+)
(Ω|ψ̂a (x)ψ̂b (y)Ω) = Sab (x − y),
˜ c
(Ω|T(ψ̂a (x)ψ̂b (y))Ω) = Sab (x − y).

For f ∈ Cc∞ (O, C4 ) we set

Z
ψ̂[f ] := f (x)ψ̂(x)dx,
Z
ψ̂ ∗ [f ] := f (x)ψ̂ ∗ (x)dx.

177
We obtain an operator valued distribution satisfying the Wightman axioms with
(+) (−)
D := Γfin
a (ZD ⊕ ZD ).
For an open set O ⊂ R1,3 the field algebra is defined as
F(O) := {ψ̂ ∗ [f ], ψ̂[f ] : f ∈ Cc∞ (O, C4 )}00 .
The observable algebra A(O) is the subalgebra of F(O) fixed by the automor-
phism
B 7→ eiθQ̂ Be−iθQ̂ ,
where Q̂ will be defined in (6.13). The nets of algebras F(O) and A(O), O ⊂
R1,3 , satisfy the Haag-Kastler axioms.

6.1.11 Quantization in terms of smeared fields

There exists an alternative equivalent formulation of the quantization program,
which uses the smeared fields instead of point fields. Instead of (2.57) we look
for an antilinear function
WD 3 ζ 7→ ψ̂((ζ))
with values in bounded operators such that
(1) [ψ̂((ζ1 )), ψ̂ ∗ ((ζ2 ))]+ = ζ 1 · ζ2 , [ψ̂((ζ1 )), ψ̂((ζ2 ))]+ = 0.
(2) ψ̂((r(t,~0) ζ)) = eitĤ ψ̂((ζ))e−itĤ .
(3) Ω is cyclic for ψ̂((ζ)), ψ̂ ∗ ((ζ)).
One can pass between these two kinds of quantization by
Z
ψ̂((ζ)) = ζ(t, ~x)ψ̂(t, ~x)d~x. (6.15)

6.1.12 Dirac sea quantization

When we quantized a fermionic field we demanded that the quantum Hamil-
tonian Ĥ be positive. In the bosonic case this condition can be dropped if we
start from a positive classical Hamiltonian H. Usually this suffices to guarantee
the positivity of Ĥ. (If we start from a classical Hamiltonian that is not positive
definite, the bosonic quantum counterpart has no chances of being positive).
Suppose now that we drop the positivity requirement of Ĥ in the fermionic
case. Then we have many possible quantizations. Among them one is distin-
guished – it is just the usual second quantization. It means that we consider the
cpl cpl
antisymmetric Fock space Γa (WD ), where WD denotes the completion of WD
in its natural scalar product.
cpl
The Hilbert space WD is equipped with a distinguished family of commuting
self-adjoint operators: the Dirac operator D and the momentum operator −i∂. ~
We can second quantize them using the operation dΓ obtaining the operators
cpl
on Γa (WD ), the Hamiltonian and the momentum
H = dΓ(D), (6.16)
P~ = ~
dΓ(−i∂). (6.17)

178
The number operator will be rebaptized as the charge and denoted

Q = dΓ(1l).

(Let us stress that we do not use “hats” in the above notation).

Let us reinterpret ψ ∗ (x)/ψ(x) (without “hats”) as the creation/annihilation
cpl
operators on the space Γa (WD ). As in (6.14), they satisfy

[ψa (x), ψ̃b (y)]+ = Sab (x − y), [ψa (x), ψb (y)]+ = 0. (6.18)

The plane wave functionals a(p, s), a∗ (p, s), b∗ (p, s), b(p, s) defined as in (6.6)
and (6.7) in terms of ψ(x), ψ ∗ (x), can be used to diagonalize the Hamiltonian,
momentum and charge
Z X
H = (a∗ (p, s)a(p, s) − b(p, s)b∗ (p, s)) E(~
p)d~
p, (6.19)
s
Z X
P~ = (a∗ (p, s)a(p, s) − b(p, s)b∗ (p, s)) p~d~
p, (6.20)
s
Z X
Q = (a∗ (p, s)a(p, s) + b(p, s)b∗ (p, s)) p~d~
p. (6.21)
s

cpl
The vacuum of Γa (WD ) is annihilated by ψ(x), hence also by a(p, s) and
b∗ (p, s). It is the state of the lowest charge possible. Therefore, it will be called
the bottom of the Dirac sea. We will call the above described procedure the
Dirac sea quantization.
The reader should compare the formulas for H (6.19), P~ (6.20) and Q (6.21)
~
with Ĥ (6.11), P̂ (6.12) and Q̂ (6.13). They differ only by the order of a part of
field operators. So formally they coincide modulo an (infinite) additive constant.
The usual quantization, called the positive energy quantization and the Dirac
sea quantization are just two inequivalent representations of canonical anticom-
mutation relations. If WD had a finite dimension (which can be accomplished
by applying both an infrared and ultraviolet cutoff), then the Dirac sea quan-
tization would be unitarily equivalent with the positive energy quantization by
the procedure invented by Dirac and called often filling the Dirac sea. The
Hamiltonians H and Ĥ, and as we see later, the charges Q and Q̂ would differ
~
by a finite constant. The momenta P~ and P̂ would coincide.

6.1.13 Fermionic Hamiltonian formalism

Bosonic quantum fields can be interpreted as a quantization of a classical sys-
tem. In the Hamiltonian (on-shell) formalism this system is described by an
appropriate symplectic space. In the charged case, the symplectic space can be
viewed as a complex space and instead of the symplectic structure it is natural
to consider an appropriate Hermitian form. The spaces YKG and WKG were

179
examples of such spaces. Symmetries are described by symplectic transforma-
tions. The dynamics is generated by a (classical) Hamiltonian – a function on
the symplectic space.
An important element of the Hamiltonian formalism is the “algebra of clas-
sical observables” – the commutative algebra of functions on the symplectic
space equipped with the Poisson bracket. One can ask whether there exists an
analogous structure behind fermionic quantum fields.
Clearly, the space WD , which is equipped with a scalar product, is the ob-
vious fermionic analog of a (complex) symplectic space from the bosonic case.
The fermionic analog of the “algebra of classical observables” considered in the
cpl
literature, eg. [50], is the Z2 -graded algebra of operators on Γa (WD ) equipped
with the graded commutator.
cpl
The space Γa (WD ) is equipped with the fermionic parity operator, which
we denote by I := (−1l)Q . An operator A satisfying IAI = ±A will be called
even/odd. Operators that are either even or odd will be called homogeneous. If
A is homogeneous we will write |A| = 0 if A is even and |A| = 1 if A is odd.
The analog of the Poisson bracket is the graded commutator:
{A, B} := AB − (−1)|A| |B| BA. (6.22)
∗
Note that ψ(x), ψ (x) are odd operators and for such operators {·, ·} coin-
cides with the anticommutator. Thus, to make (6.18) look “classical”, we can
replace [·, ·]+ with {·, ·} in this identity.
Note that the “classical” version of the Dirac theory has a quantum charac-
ter. In particular, the “classical fermionic algebra” is an algebra of operators on
a Hilbert space and symmetries are unitary. Nevertheless, one has a far reaching
analogy with the usual commutative classical mechanics.

6.1.14 Fermionic Lagrangian formalism

The Lagrangian formalism in the bosonic case involves the commutative algebra
of functions on the space-time (the “off-shell formalism”). In the literature one
can also find its fermionic analog. The fermionic Lagrangian formalism involves
the Grassmann algebra generated by anticommuting functions on space-time.
This algebra is generated by anticommuting fields R1,3 3 x 7→ ψ(x), ψ ∗ (x).
(Thus, the anticommutators of the off-shell ψ(x), ψ ∗ (y) are always zero, unlike
in the on-shell formalism).
Note that every Grassmann algebra, besides multiplication, is equipped with
the integral (called sometimes the Berezin integral), the left and the right deriva-
tive. We will use the left derivative as the standard one (see eg. [15]).
The Lagrangian density is an even element of this Grassmann algebra:
1 
L(x) = − ψ̃(x)γ µ (−i∂µ )ψ(x) + i∂µ ψ̃(x) γ µ ψ(x) − mψ̃(x)ψ(x),
2
where as usual ψ̃(x) = βψ ∗ (x). The Euler-Lagrange equations
∂L ∂L
∂ψ̃ L − ∂µ = 0, ∂ψ L − ∂µ =0 (6.23)
∂ ψ̃,µ ∂ψ,µ

180
yield the Dirac equation.
One can define the stress-energy tensor

∂L(x) ν ∂L(x) ν
T µν (x) := − ∂ ψ(x) − ∂ ψ̃(x) + g µν L(x)
∂ψ,µ (x) ∂ ψ̃,µ (x)
1 
= ψ̃(x)γ µ (−i∂ ν )ψ(x) + i∂ ν ψ̃(x)γ µ ψ(x)
2
1 
−g µν


ψ̃(x)γ(−i∂)ψ(x) + i∂ ψ̃(x)γψ(x) + mψ̃(x)ψ(x) .
2
It is conserved on shell
∂ µ Tµν (x) = 0.
The components of the stress-energy tensor with the first temporal coordinate
are called the Hamiltonian density and momentum density.

H(x) := T 00 (x)
1 ∗ ~ ~ ∗ (x)~

= ψ (x)~α(−i∂)ψ(x) + i∂ψ αψ(x) + mψ ∗ (x)βψ(x),
2
P i (x) := T 0i (x)
1
= − ψ ∗ (x)(−i∂ i )ψ(x) + i∂ i ψ ∗ (x)ψ(x) .


2
Note that in (6.24) and (6.24) we put ψ ∗ on the left and ψ on the right.
This is the Wick ordering for the Dirac sea quantization, which can be called
the charge Wick ordering. The Hamiltonian and momentum defined from these
densities coincide with the operators defined by the Dirac sea second quantiza-
tion (6.16), (6.17):
Z
H = H(t, ~x)d~x,
Z
P~ = ~ ~x)d~x.
P(t,

6.1.15 Classical 4-current

The Lagrangian is invariant w.r.t. the U (1) symmetry ψ 7→ e−iθ ψ. The Noether
4-current associated to this symmetry is the 4-current, defined as
 ∂L(x) ∂L(x) 
J µ (x) := i ψ̃(x) − ψ(x)
∂ ψ̃,µ ∂ψ,µ
= ψ̃(x)γ µ ψ(x).

It is conserved on shell and self-adjoint:

∂µ J µ (x) = 0, (6.24)
µ ∗ µ
J (x) = J (x). (6.25)

181
 The sesquilinear form given by J coincides with (6.4):

ζ 1 J µ (x)ζ2 = j µ (ζ 1 , ζ2 , x)

= ζ1 (x)βγ µ ζ2 (x), ζ1 , ζ2 ∈ WD .

The current or the spatial part of 4-current can be expressed in terms of the
α matrices:
J~ (x) = ψ ∗ (x)~
αψ(x).
The 0th component of the 4-current is called the charge density

Q(x) := J 0 (x) = ψ ∗ (x)ψ(x).

The charge is
Z
Q := Q(t, ~x)d~x
XZ
a∗ (~ p, s)b∗ (~


= p, s)a(~
p, s) + b(~ p, s) d~
p.
s

x 7→ Q(t, ~x) is a well defined distribution with values in operators on space

cpl
Γa (WD ). We have the relations

{Q(t, ~x), ψ(t, ~y )} = −ψ(t, ~y )δ(~x − ~y ),

{Q(t, ~x), ψ ∗ (t, ~y )} = ψ ∗ (t, ~y )δ(~x − ~y ),
{Q(t, ~x), Q(t, ~y )} = 0, (6.26)

where the bracket coincides now with the commutator, since Q is even.
For χ ∈ Cc∞ (R3 , R), let αχ denote the ∗-automorphism of the algebra of
operators on WD defined by

αχ (ψ(0, ~x)) := e−ieχ(~x) ψ(0, ~x).

Obviously,

αχ (ψ ∗ (0, ~x)) = eieχ(~x) ψ ∗ (0, ~x).

αχ is called the gauge transformation at time t = 0 corresponding to χ. Set

Z
Q(χ) = χ(~x)Q(0, ~x)d~x. (6.27)

It can be used to implement the corresponding gauge transformation:

αχ (B) = eieQ(χ) Be−ieQ(χ) .

182
6.1.16 Quantum 4-current
Let us try to introduce the quantum 4-current density as an operator valued dis-
(+) (−)
tribution on Γa (ZD ⊕ ZD ) by the antisymmetric quantization of the classical
expression
1 ∗
J µ (x) ψ̂ (x)βγ µ ψ̂(x) − ψ(x)βγ µ ψ ∗ (x) .


:= (6.28)
2
(See Subsubsect. A.1.3 for the definition of antisymmetric quantization. Note
that (βγ µ )∗ = βγ µ , and hence βγ µ is the transpose of βγ µ ). The charge con-
jugation C, which we introduce later on in Subsubsect. 6.2.6, satisfies CΩ = Ω
and C Jˆµ (x)C ∗ = −J µ (x). Therefore, (Ω|J µ (x)Ω) = 0. Hence

˜
Jˆµ (x) = :ψ̂(x)γ µ ψ̂(x):.

Formally, we can check the quantum versions of the relations (6.24) the (6.25).
We have
~
Ĵ (x) = :ψ̂ ∗ (x)~
αψ̂(x):,
and the 0th component of the 4-current is called the charge density

Q̂(x) := Jˆ0 (x) = :ψ̂ ∗ (x)ψ̂(x):.

Formally, the charge density satisfies

[Q̂(t, ~x), ψ̂(t, ~y )] = −ψ̂(t, ~y )δ(~x − ~y ),

∗
[Q̂(t, ~x), ψ̂ (t, ~y )] = ψ̂ ∗ (t, ~y )δ(~x − ~y ),
[Q̂(t, ~x), Q̂(t, ~y )] = 0. (6.29)

For χ ∈ Cc∞ (R3 ) let αχ denote the gauge transformation at time t = 0

defined as a ∗-automorphism of the algebra generated by fields satisfying (5.32),
and hence also (5.33). Assume that χ 6= 0. Let us check whether αχ is unitarily
implementable.
On the level of annihilation operators we have
X Z Z d~xd~ p1
αχ (â(p)) = 3
u(p, s)u(p1 , s1 )ei(~p1 −~p)~x−ieχ(~x) â(p1 )
s1
(2π)
X Z Z d~xd~ p1
+ 3
u(p, s)u(−p1 , −s1 )e−i(~p1 +~p)~x−ieχ(~x) b̂∗ (p1 ).
s
(2π)
1

Let qχ (~
p, s; p~1 , s1 ) denote the integral kernel on the second line above. We need
to check whether it is square integrable. Now

2
X
2 p + p~1 |2 + E(~
|~ p) − E(~
p1 )
|u(p, s)u(−p1 , −s1 )| = . (6.30)
s,s1
2E(~
p)E(~p1 )

183
After integrating in ~x we obtain fast decay in p~ + p~1 , which allows us to control
the numerator of (6.30). We obtain
Z
C
p, p~1 )|2 d~
|qχ (~ p∼ ,
p1 )2
E(~
which is not integrable. Therefore, αχ is not implementable by the Shale-
Stinespring criterion, see Thm A.2.
Formally, with Z
Q̂(χ) := χ(~x)Q̂(0, ~x)d~x, (6.31)

eieQ̂(χ) implements the gauge transformation:

αχ (B) = eieQ̂(χ) Be−ieQ̂(χ) .

But we know that nontrivial gauge transformations are not implementable.

Thus for nonzero χ (6.31) cannot be defined as a closable operator.
However, the (quantum) charge
Z
Q̂ := Q̂(t, ~x)d~x (6.32)

is a well defined self-adjoint operator, which we already discussed before.

For further reference let us express the charge density in terms of creation
and annihilation operators:
Z Z
d~p1 d~
p2
Q̂(x) = u(p1 , s1 )u(p2 , s2 )e−ixp1 +ixp2 â∗ (p1 , s1 )â(p2 , s2 )
(2π)3
Z Z
d~p1 d~
p2
− u(−p1 , −s1 )u(−p2 , −s2 )eixp1 −ixp2 b̂∗ (p2 , s2 )b̂(p1 , s1 )
(2π)3
Z Z
d~p1 d~
p2
+ u(p1 , s1 )u(−p2 , −s2 )e−ixp1 −ixp2 â∗ (p1 , s1 )b̂∗ (p2 , s2 )
(2π)3
Z Z
d~p1 d~
p2
+ u(−p1 , −s1 )u(p2 , s2 )eixp1 +ixp2 b̂(p1 , s1 )â(p2 , s2 ).
(2π)3
~
~ between u(·, ·) and u(·, ·).
To obtain Ĵ (x) one inserts α

6.2 Dirac fermions in an external 4-potential

6.2.1 Dirac equation in an external 4-potential
Let
R1,3 3 x 7→ A(x) = [Aµ (x)] ∈ R1,3 (6.33)
be a given function. In most of this subsection we assume that (6.33) is Schwartz.
The Dirac equation in an external 4-potential A is

γ µ (−i∂µ + eAµ (x)) + m ψ(x) = 0.

(6.34)

184
 If ψ satisfies (6.34) and R1,3 3 x 7→ χ(x) ∈ R is an arbitrary smooth function,
then eieχ ψ satisfies (6.34) with A replaced with A + ∂χ.
Note the identity

− γ µ (−i∂µ + eAµ (x)) + m γ µ (−i∂µ + eAµ (x)) − m

e
= −(∂µ + ieAµ (x))(∂ µ + ieAµ (x)) + m2 + σ µν Fµν (x). (6.35)
2
Let D± (x, y) denote the retarded/advanced Green’s function of (6.35). Then

S ± (x, y) := γ µ (−i∂xµ + eAµ (x)) − m D(x, y)

is the retarded/advanced Green’s function of (6.34), that is, the unique solution
of
γ µ (−i∂µ + eAµ (x)) + m S ± (x, y) = δ(x − y)


(6.36)
satisfying
suppS ± ⊂ {x, y : x ∈ J ± (y)}.
We set
S(x, y) := S + (x, y) − S − (x, y).
Clearly,
suppS ⊂ {x, y : x ∈ J(y)}.
We would like to introduce a field R1,3 3 x 7→ ψ(x) satisfying (6.34). If we
assume that it acts on WD and coincides with the free field ψfr (x) at x0 = 0,
such a field is given by
Z
ψ(t, ~x) = −i S(t, ~x; 0, ~y )βψfr (0, ~y )d~y . (6.37)
R3

6.2.2 Lagrangian and Hamiltonian formalism

(6.34) can be obtained as the Euler-Lagrange of a variational problem. The
Lagrangian density can be taken as
1 
L(x) = − ψ̃(x)γ µ (−i∂µ )ψ(x) + i∂µ ψ̃(x)γ µ ψ(x)
2
−ψ̃(x)eAµ (x)γ µ ψ(x) − mψ̃(x)ψ(x).

The Euler-Lagrange equations (6.23) yield (6.34).

We can introduce the Hamiltonian density

∂L(x) ∂L(x)
H(x) = ψ̇(x) + ψ̇ ∗ (x) − L(x)
∂ ψ̇(x) ∂ ψ̇ ∗ (x)
1 ∗ ~ ~ ∗ (x)~


= ψ (x)~ α(−i∂)ψ(x) + i∂ψ αψ(x)
2
~
+ψ ∗ (x) e~γ A(x)


+ mβ + eA0 (x) ψ(x).

185
The Hamiltonian Z
H(t) = H(t, ~x)d~x

cpl
can be interpreted as a self-adjoint operator on Γa (WD ) that generates the
“classical” dynamics

ψ̇(t, ~x) = i{H(t), ψ(t, ~x)},

where now {·, ·} has the meaning of the commutator.

6.2.3 Classical discrete symmetries

Let κ be a unitary 4 × 4 matrix satisfying

κκ = 1l, κγ µ κ−1 = −γ µ ,

where the bar denotes the complex conjugation. In particular, κβκ−1 = −β.
Note also that
κκu = u, u ∈ C4 .
If ζ solves the Dirac equation with the 4-potential A, then so does κζ with
the 4-potential −A. Thus replacing

ψ(x), ψ ∗ (x), A(x)

with κψ ∗ (x), κψ(x), −A(x)

is a symmetry of the Dirac equation with external 4-potentials (6.34). It is

called charge conjugation and denoted C.
The matrix κ depends on a representation. In the Majorana representation
it is the identity. In the Dirac and spinor representation it can be chosen to
be γ 2 multiplied by an arbitrary phase factor. In fact, in these representations
γ µ = γ µ , except for µ = 2 satisfying γ 2 = −γ 2 . When we consider the Dirac
representation, we will adopt the convention

κ := iγ 2 .

Then κ = κ = κ∗ . The spinor basis that we chose in (6.5) is compatible with κ:

κu(p, s) = u(−p, −s). (6.38)

Choose ξP ∈ {1, −1}. Recall that P denotes the space inversion. Replacing

~
ψ(x), ψ ∗ (x), A0 (x), A(x)

ξP γ 0 ψ(Px), ξP γ 0 ψ ∗ (Px), ~


with A0 (Px), −A(Px)

is a symmetry of (6.34) called parity and denoted P.

186
 Choose ξT ∈ {1, −1}. Recall that T denotes the time reflection. Replacing
(in the Dirac representation)

~
ψ(x), ψ ∗ (x), A0 (x), A(x)

~
with ξT γ 1 γ 3 ψ(Tx), ξT γ 1 γ 3 ψ ∗ (Tx), A0 (Tx), −A(Tx)

is a symmetry of (6.34) called time reversal and denoted T .

The symmetry that is guaranteed by the CPT Theorem consists in replacing

ψ(x), ψ ∗ (x), A(x)

with iγ 5 ψ ∗ (−x), iγ 5 ψ(−x), −A(−x).

It is denoted X . (Note that iγ 5 = γ 0 γ 1 γ 2 γ 3 ).

Assume that ξP ξT = i. (If needed, this can be accomplished by multiplying
ψ by an appropriate phase factor). Then

X = CPT

and we have the relations

C 2 = P 2 = −T 2 = −X 2 = id,
CP + PC = CT + T C = 0,
X P + PX = X T + T X = 0,
CX − X C = PT − T P = 0.

To understand better these relations, let us notice that the automorphisms

P, CT and X anticommute and

P 2 = (CT )2 = −X 2 = id,

where id denotes the identity. Thus together with Spin↑ (1, 3) they represent
the group P in+ (1, 3), see Subsubsect. 1.1.6.
Besides,
(PT )2 = −id
and PT commutes with P, CT , X . Thus it behaves as i · id. Thus the group
generated by Spin↑ (1, 3), C, P and T is P inext (1, 3), see Subsubsect. 1.1.6.

6.2.4 Quantization
We are looking for a quantum field satisfying

γ µ (−i∂µ + eAµ (x)) + m ψ̂(x) = 0

(6.39)

such that
ψ̂(~x) := ψ̂(0, ~x) = ψ̂fr (0, ~x).
Clearly the solution is obtained by decorating (6.37) with hats.

187
 As in the bosonic case, we ask whether the fields are implemented by a
a unitary dynamics. Equivalently, we want to check if the classical dynamics
generated by HInt (t) satisfies the Shale-Stinespring criterion.
Arguments parallel to those of Subsubsect. 2.3.4 show that the classical
scattering operator is unitarily implementable.
An analysis similar to that of Subsect. 5.2.4 shows that the dynamics from
t− to t+ is implementable on the Fock space iff the spatial part of the 4-potential
is the same at the initial and final time:
~ + , ~x) = A(t
A(t ~ − , ~x), ~x ∈ R3 . (6.40)

6.2.5 Quantum Hamiltonian

Formally, we can also obtain the quantum field from a unitary dynamics:
 Z 0   Z t 
ψ̂(t, ~x) := Texp −i Ĥ(s)ds ψ̂(0, ~x)Texp −i Ĥ(s)ds ,
t 0

where the Schrödinger picture Hamiltonian Ĥ(t) and the corresponding inter-
action picture Hamiltonian are
Z
α(−i∂~ + eA(t,
d~x: ψ̂ ∗ (~x)(~ ~ ~x)) + mβ + eA0 (t, ~x))ψ̂(~x) :,


Ĥ(t) =
Z
ĤInt (t) = d~xeAµ (t, ~x)Jˆfrµ (t, ~x).

Note that unlike in the case of charged bosons we use the Wick ordering.
This is because Ĥ(t) differs from Ĥfr by a term involving the 4-current Jˆfrµ (t, ~x),
which is automatically Wick ordered. Therefore, we can assume that both Ĥ(t)
and Ĥfr are Wick ordered, which was impossible for charged bosons.

6.2.6 Quantized discrete symmetries

The discrete symmetries considered in Subsubsect. 6.2.3 remain true when we
decorate the fields with “hats”. Thus on the level of quantum observables the
discrete symmetries are the same as in the classical case.
Let us now discuss the implementation of these symmetries by unitary or
(+) (−)
antiunitary operators on the Hilbert space Γa (ZD ⊕ ZD ). We will discuss
this for free fields, that is, for A = 0. As in the bosonic case, this will imply
some properties of the scattering operator Ŝ(A).
First consider the charge conjugation. We define the following unitary op-
(+) (−)
erator on ZD ⊕ ZD
χ(g1 , g 2 ) := (κg2 , κg 1 ).
We check that
χ|p, s) = | − p, −s), χ| − p, −s) = |p, s).

188
We set C := Γ(χ). We have C 2 = 1l,

C ψ̂fr (x)C −1 = κψ̂fr

∗ ∗
(x), C ψ̂fr (x)C −1 = κψ̂fr (x),

~ ~
C Q̂fr (x)C −1 = −Q̂fr (x), C Ĵfr (x)C −1 = −Ĵfr (x),
C Ŝ(A)C −1 = Ŝ(−A).
(+) (−)
Define the following unitary operator on ZD ⊕ ZD :

π g1 , g 2 := ξP γ 0 g1 ◦ P, ξP γ 0 g 2 ◦ P .

We check that

π|E, p~, s) = ξP |E, −~

p, s), π|−E, −~
p, −s) = ξP |−E, p~, −s).

Set P := Γ(π). We have P 2 = 1l,

P ψ̂fr (x)P −1 = ξP γ 0 ψ̂fr (Px), P ψ̂fr

∗
(x)P −1 = ξP γ 0 ψ̂fr
∗
(Px),

~ ~
P Q̂fr (x)P −1 = Q̂fr (Px), P Ĵfr (x)P −1 = −Ĵfr (Px),
~ −1 = Ŝ(A0 ◦ P, −A
P Ŝ(A0 , A)P ~ ◦ P).

Define (in the Dirac representation) the following antiunitary operator on

(+) (−)
ZD ⊕ ZD :
τ (g1 , g 2 ) := ξT γ 1 γ 3 g 1 ◦ T, ξT γ 1 γ 3 g2 ◦ T .

We check that

τ |E, p~, s) = ξT |E, −~

p, −s), τ |−E, −~
p, −s) = ξT |−E, p~, s).

Set T := Γ(τ ). We have T 2 = −1l,

T ψ̂fr (x)T −1 = ξT γ 1 γ 3 ψ̂fr (Tx), T ψ̂fr

∗
(x)T −1 = ξT γ 1 γ 3 ψ̂fr
∗
(Tx),

~ ~
T Q̂fr (x)T −1 = Q̂fr (Tx), T Ĵfr (x)T −1 = −Ĵfr (Tx),
~ −1 = Ŝ(A0 ◦ T, −A
T Ŝ(A0 , A)T ~ ◦ T).

6.2.7 2N -point Green’s functions

We consider again a Dirac field in an external 4-potential [Aµ (x)]. For yN , . . . y1 ,
xN , . . . , x1 , the 2N point Green’s function are defined as follows:
˜ ˜
ψ̂(y1 ) · · · ψ̂(yN )ψ̂(xN ) · · · ψ̂(x1 )
˜ ˜
   
:= Ω+ |T ψ̂(y1 ) · · · ψ̂(yN )ψ̂(xN ) · · · ψ̂(x1 ) Ω− .

189
 One can organize Green’s functions in terms of the generating function:

Z(g, g̃)
∞ Z
(−1)N ˜
Z
X ˜
:= ··· 2
ψ̂(y1 ) · · · ψ̂(yN )ψ̂(xN ) · · · ψ̂(x1 )
n=0
(N !)
×g(y1 ) · · · g(yN )g̃(xN ) · · · g̃(x1 )dy1 · · · dyN dxN · · · dx1
  Z ∞ Z Z  
˜
= Ω Texp −i ĤInt (t)dt − i g(x)ψ̂fr (x)dx − i g̃(x)ψ̂fr (x)dx Ω ,
−∞

where R1,3 3 x 7→ g(x), g̃(x) ∈ C4 are Grassmann variables anticommuting with

˜
ψ̂(x), ψ̂(x).
One can retrieve Green’s functions from the generating function:
˜ ˜
ψ̂(y1 ) · · · ψ̂(yN )ψ̂(xN ) · · · ψ̂(x1 )
∂ 2N
= (−1)N Z(g, g̃) .
∂g(y1 ) · · · ∂g(yN )∂g̃(xN ) · · · ∂g̃(x1 ) g=g̃=0

We introduce also the amputated Green’s function

˜ ˜
ψ̂(p01 ) . . . ψ̂(p0N )ψ̂(pN ) · · · ψ̂(p1 ) amp
:= γp01 + m · · · γp0N + m γpN + m · · · γp1 + m

˜ ˜
× ψ̂(p01 ) · · · ψ̂(p0N )ψ̂(pN ) · · · ψ̂(p1 ) .

Introduce many particle plane waves

|−p0N 0 , −s0N 0 ; . . . ; −p01 , −s01 ; pN , sN ; . . . ; p1 , s1 )

:= b̂∗ (p0N 0 , s0N 0 ) · · · b̂∗ (p01 , s01 )â∗ (pN , sN ) · · · â∗ (p1 , s1 )Ω,

where all pi , p0i are on shell. Scattering amplitudes are the matrix elements of
the scattering operator Ŝ between plane waves. One can compute scattering
amplitudes from the amputated Green’s functions:
 
−0 −0 − −
−p+0n +0 , −s+0
n +0 ; . . . ; p+
n + , s+
n + ; . . . | Ŝ |−pn −0 , −sn −0 ; . . . ; p n − , sn − ; . . .

−0 −0 − −
· · · ũ(p+ + +0 +0
n+ , sn+ ) · · · u(−pn+0 , −sn+0 )ũ(−pn−0 , −sn−0 ) · · · u(pn− , sn− ) · · ·
= p
(2π)3(n+ +n+0 +n−0 +n− )

˜ +0 −0 ˜ −
× · · · ψ̂(p+
n+ ) · · · ψ̂(pn+0 )ψ(−pn−0 ) · · · ψ̂(−pn− ) · · · amp
.

The scattering operator and Green’s functions satisfy the Ward identities
analogous to those satisfied by charged bosons.

190
6.2.8 Path integral formulation
We have the following formula for the generating function:

Z(g, g̃) (6.41)

−1
= det γ (−i∂µ + eAµ (x)) + m − iγ µ ∂µ + m − i
µ



−1 
× exp ig̃ γ µ (−i∂µ + eAµ (x)) + m − i g
= det (1l + γµ eAµ Sfr
c
)
  −1 
c µ c
× exp ig̃Sfr 1l + γµ eA Sfr g ,

where  has the same meaning as in (6.2).

In terms of path integrals this can be formally written as
R  R

Π dψ̃(x) Π0 dψ(y) exp i L(x) − g(x)ψ̃(x) − g̃(x)ψ(x) dx
y y
R R
.
Π dψ̃(y) Π dψ(y 0 ) exp i Lfr (x)dx
y y0

6.2.9 Feynman rules

The Feynman rules are very similar to those for charged bosons, except that
there are no two-photon vertices. Here are the Feynman rules for Green’s func-
tions.
(1) In the nth order we draw all possible topologically distinct Feynman di-
agrams with n vertices and external lines. All the charged lines have a
natural arrow.
(2) To each vertex we associate the factor −ieγ µ Aµ (p+ − p− ).
c
(3) To each line we associate the propagator −iSfr (p) = −i p2−pγ+m
+m2 −i0
d4 p
(4) For internal lines we integrate over the variables with the measure (2π)4 .

(5) If two diagrams differ only by an exchange of two fermionic lines, there is
an additional factor (−1) for one of them. This implies, in particular, that
loops have an additional factor (−1).
To compute scattering amplitudes with N − incoming and N + outgoing par-
ticles we draw the same diagrams as for N − + N + -point Green’s functions. The
rules are changed only concerning the external lines.
(i) With each incoming external line we associate
1
• fermion: √ u(p, s).
(2π)3
1
• anti-fermion: √ ũ(−p, −s).
(2π)3

(ii) With each outgoing external line we associate

191
 1
• fermion: √ ũ(p, s).
(2π)3

• anti-fermion: √ 1 3 u(−p, −s).

(2π)

Each incoming and outgoing antifermion has an additional factor (−1).

(This follows from the rule (5) above).

6.2.10 Vacuum energy

Formally, the vacuum energy can be computed exactly:
E := i log(Ω|ŜΩ) = i log Z(0, 0)


iTr log γ µ (−i∂µ +eAµ (x)) + m − i − log −iγ µ ∂µ + m − i


=
= iTr log (1l + γ µ eAµ Sfr
c
)
∞
X Dn
= i . (6.42)
n=1
n
Here Dn is the value of the loop with n vertices. Note that n in the denomi-
nator is the order of the group of the authomorphisms of a loop with n vertices,
which is Zn .
Furry’s theorem, proven as in the bosonic case, says that diagrams for
charged fermions of the odd order in e vanish. Hence (6.42) can be written
as
X∞
E= e2n En ,
n=1
2n
where e En = i D2n
2n
.
Just as in he bosonic case, the expressions for En obtained from the Feynman
rules are convergent for n ≥ 3. The gauge invariant part of E2 is convergent. The
computation of E1 will be described below – it needs an infinite renormalization.
There exists a close relationship between the fermionic and bosonic vacuum
energy. To see it, note that using γ 5 γµ (γ 5 )−1 = −γµ , we obtain

E = iTr log − γ µ (−i∂µ + eAµ (x)) + m − i


− log iγ µ ∂µ + m − i . (6.43)

We add up 12 (6.42) and 12 (6.43) and use identity (6.35). We obtain

i  e
Tr − log − (∂µ + ieAµ (x))(∂ µ + ieAµ (x)) + m2 + σ µν Fµν (x) − i0


E =
2 2


+ log − 2 + m − i02

i  
= Tr log 1l + ie∂µ Aµ (x) + ieAµ (x)∂µ + e2 Aµ (x)Aµ (x)
2
e  
+ σ µν Fµν (x) Dfr c
. (6.44)
2
We can compare (6.44) with a similar expression in the bosonic case (5.49).

192
6.2.11 Pauli-Villars renormalization
A single electron loop with two vertices coming from a 4-potential Aµ leads to
a contribution of the form
Z
dp
E1 = Aµ (−p)Aν (p)Πµν (p).
(2π)4
Unfortunately, computed naively, Πµν (p) is divergent.
We will compute it using the Pauli-Villars regularization, similarly as for
charged bosons, see Subsubsect. 5.2.11:

d4 q Trγµ (q + 21 p)γ + mi γν (q − 12 p)γ + mi

Z


X
2
2ΠµνΛ (p) = − Ci ie
(2π)4 (q + 12 p)2 + m2i − i0 (q − 21 p)2 + m2i − i0



i

4d4 q 2qµ qν − 12 pµ pν − gµν (q 2 − 14 p2 + m2i )

X Z
= − Ci ie2
i
(2π)4 ((q + 12 p)2 + m2i − i0)((q − 21 p)2 + m2i − i0)

∞ ∞
e2
Z Z
X 8α1 α2
= Ci dα1 dα2 (pµ pν − gµν p2 )
i
(4π)20 0 (α1 + α2 ) 4

!
m2i

α1 α2 2 i
+4gµν p + +
(α1 + α2 )4 (α1 + α2 )3 (α1 + α2 )2
 
α1 α2 2
× exp −i(α1 + α2 )m2i − i p
α1 + α2
=: (−gµν p2 + pµ pν )2Πgi 2 gd
Λ (p ) + 2ΠµνΛ (p).

We used the identity (A.23).

The gauge dependent part of the vacuum energy tensor up to a coefficient is
the same as for charged bosons and vanishes. We apply the same substitutions
and use the same identities as in the charged boson case:
Z ∞ Z ∞
e2 X 4α1 α2
Πgi
Λ (p 2
) = − C i dα 1 dα2
(4π)2 i 0 0 (α 1 + α2 )
4
 
α1 α2 2
× exp −i(α1 + α2 )m2i − i p
α1 + α2
2 X Z 1 Z ∞
e dρ
= − Ci dv (1 − v 2 )
(4π)2 i 0 0 ρ
(1 − v 2 )p2
  
× exp −iρ m2i +
4
2 X Z 1
(1 − v 2 )p2
 
e 2 2
= Ci dv(1 − v ) log mi + − i0
(4π)2 i 0 4
Z 1
e2 X (1 − v 2 )p2
  
2 1 2
= Ci dv(1 − v ) log 1 + − i0 + log mi .
(4π)2 i 0 4m2i 3

193
Define
Πren (p2 ) Πgi 2 gi

:= lim Λ (p ) − ΠΛ (0) (6.45)
Λ→∞
1
e2 (1 − v 2 )p2
Z  
2
= dv(1 − v ) log 1 + − i0 .
(4π)2 0 4m2
Denote the vacuum energy function for neutral bosons, introduced in (2.137),
by πbren . Let Πren
b denote the vacuum energy function for charged bosons (5.52)
and Πren
f for charged fermions (6.45). Let us note the following identity:
2Πren 2 ren 2 2 ren 2
b (p ) + Πf (p ) = 4e π (p ). (6.46)
This identity can be also derived from (6.44), (5.49) and (2.135).

6.2.12 Method of dispersion relations

The imaginary part of the gauge invariant vacuum energy function can be com-
puted without renormalization:
Z 1
e2 (1 − v 2 )p2
 
ren 2 2 2
ImΠ (p ) = Im dv(1 − v ) log m + − i0
(4π)2 0 4
1
e2 (1 − v 2 )p2
Z  
2 2
= dv(1 − v )(−π)θ − −m
(4π)2 0 4
4e2 π (−p2 + 2m2 ) 1
= − 2 2 3/2
−p2 − 4m2 +2 , p2 ∈ R.
3(4π) (−p )
The full vacuum energy tensor can be obtained by using the once subtracted
dispersion relations, as in (5.58).

6.2.13 Dimensional renormalization

We can also use dimensional regularization to compute Πren µν . We use the Eu-
clidean formalism.
d4 q 2qµ qν − 12 pµ pν − gµν (q 2 − 14 p2 + m2 )
Z
E 2
2Πµν (p) = e Tr1l
(2π)4 ((q + 12 p)2 + m2 )((q − 12 p)2 + m2 )
d4 q 2qµ qν − 12 pµ pν − gµν (q 2 − 14 p2 + m2 )
Z 1 Z
= e2 Tr1l dv 2
−1 (2π)4 (q 2 + p4 + m2 + vqp)2
Z 1 Z
d4 q
= e2 Tr1l dv
0 (2π)4
2
2qµ qν − 12 pµ pν − gµν (q 2 − 41 p2 + m2 ) + v 2 ( 12 pµ pν − gµν p4 )
× p2
. (6.47)
(q 2 + 4 (1 − v 2 ) + m2 )2
Then we use the dimensional regularization. Besides the rules (5.61) and (5.62)
we have a new rule:
Tr1l is replaced by 2d/2 . (6.48)

194
Thus (6.47) is replaced by
1 ∞
2d/2 µ4−d Ωd
Z Z
ΠE,d
µν (p) = e2 dv |q|d−1 d|q|
(2π)d 0 0
2
(2/d − 1)gµν q 2 − 21 pµ pν − gµν (− 14 p2 + m2 ) + v 2 ( 12 pµ pν − gµν p4 )
× 2
2
q 2 + p4 (1 − v 2 ) + m2
Z 1 
4e2 µ2 2π 2−d/2
= dv 2 Γ(2 − d/2)
(4π)2 0 p 2 2
4 (1 − v ) + m
!
 p2  1  1 p2 
2 1
 
2 2 2 2
× gµν (1 − v ) + m − pµ pν − gµν − p + m + v pµ pν − gµν
4 2 4 2 4
1
2e2 µ2 2π
Z   2−d/2
= 2
dv p2 Γ(2 − d/2)(v 2 − 1)(pµ pν − gµν p2 )
(4π) 0 (1 − v 2 ) + m2
4
Z 1 
2e2 2
 p2
2 2

' dv − γ + log(µ 2π) − log (1 − v ) + m (v 2 − 1)(pµ pν − gµν p2 )
(4π)2 0 4
4e2
+ (pµ pν − gµν p2 ). (6.49)
3(4π)2 (2 − d/2)
We can now renormalize (6.49):
ΠE,ren (p2 )(pµ pν − gµν p2 )
 
= lim ΠE,d 2 E,d
µν (p ) − Πµν (0)
d→4
Z 1
1  p2 
= 2
dv(1 − v 2 ) log 1 + 2
(1 − v 2 ) (pµ pν − gµν p2 ).
(4π) 0 4m
Again, this coincides with the Wick rotated result obtained by the Pauli-Villars
method.
Remark 6.3 In the above computations we first try to eliminate gamma matri-
ces. The only remnant of gamma matrices is tr1l, where 1l is the identity on the
space of Dirac spinors, to which we apply the rule (6.48). However, we would
have obtained the same final result if we used eg. the rule Tr1l = 4, since at the
end we apply the normalization condition Πren µν (0) = 0. We use the condition
(6.48), since it is the usual choice in the literature.
Note, however, that in more complicated situations the dimensional renor-
malization can be problematic, especially for fermions in the presence of γ 5 .

6.2.14 Energy shift

Suppose that the 4-potential does not depend on time and is given by a Schwartz
function R3 3 ~x 7→ A(~x) = [Aµ (~x)].
The free Hamiltonian is
Z
d~x:ψ̂ ∗ (~x) α ~ + mβ ψ̂(~x):.


Ĥfr = ~ (−i∂)

195
The naive interacting Hamiltonian is
Z
~ (−i∂~ + eA(~
d~x:ψ̂ ∗ (~x) α ~ x)) + mβ + eA0 (~x) ψ̂(~x):.


Ĥ =

We apply (A.19) to compute the difference between the ground state energies
of Ĥ and Ĥfr , obtaining
 
Tr − α ~ (−i∂~ + eA(~
~ x)) + mβ + eA0 (~x) + α ~ + mβ)
~ (−i∂)
∞
X
= e2n En (A). (6.50)
n=1

Note that we could have assumed that Ĥfr and Ĥ are given by the antisym-
metric quantization, used the formula (A.17), and we would have obtained the
same result for the energy shift. Indeed, formally, the Wick and Weyl quantized
versions of Ĥfr and Ĥ differ by the same (infinite) constant (which was not true
in the bosonic case).
All the terms in (6.50) with n ≥ 2 are well defined. The term with n = 1
needs renormalization. The renormalized energy shift is
Z ∞
ren 2 d~p X
E = e Πren (~
p2 )Fµν (~
p)F µν (~
p) 3
+ e2n En (A),
(2π) n=2

where Πren was introduced in (6.45).

7 Majorana fermions
In this section we consider again the Dirac equation

(−iγ µ ∂µ + m)φ(x) = 0.

We will quantize the space of its solutions satisfying the Majorana condition.
We obtain a formalism that describes neutral fermions.
In the bosonic case we first treated the neutral case and only then the charged
case. In the fermionic case it is convenient to reverse the order.

7.1 Free Majorana fermions

7.1.1 Charge conjugation
Consider a representation of Dirac matrices γ µ . Let κ be a unitary 4 × 4 matrix
described in Subsubsect. 6.2.3. We say that u ∈ C4 is neutral or satisfies the
Majorana condition if u = κu.
Recall that in the Majorana representation κ can be taken to be the identity.
In the Dirac and spinor representation κ := iγ2 .

196
7.1.2 Space of solutions
If a function ζ satisfies the Dirac equation

(−iγ µ ∂µ + m)ζ(x) = 0,

then κζ also satisfies the Dirac equation. Therefore, we can restrict the Dirac
equation to functions ζ satisfying the Majorana condition

κζ = ζ. (7.1)

The space of smooth space compact solutions of the Dirac equation satisfying
(7.1) will be denoted YD . Note that it is a real vector space equipped with a
nondegenerate scalar product
Z
ζ 1 · ζ2 = ζ1 (t, ~x)ζ2 (t, ~x)d~x.

In the Majorana representation the space YD consists simply of real func-

tions. However, we will most often use the Dirac representation, where the
Majorana condition is less trivial.
Let φ(x) be the linear functional on YD defined by

hφ(x)|ζi = ζ(x).

The complexification of YD , that is CYD , can be identified with WD . We can

extend φ(x) to CYD by complex linearity. The subspace YD is then determined
by the condition
κφ∗ (x) = φ(x), (7.2)
where ∗ is the complex conjugation as defined in (5.3).

7.1.3 Smeared fields

Smeared fields are defined very similarly as for Dirac fields. Note that in spite
of the similarity of the formulas, the objects are different: they act on the real
space YD , and not on the complex space WD .
For ζ ∈ WD , the corresponding spatially smeared field is the functional on
YD given by

hφ((ζ))|ρi := ζ · ρ, ρ ∈ YD .

Clearly, for any t

Z
φ((ζ)) = ζ(t, ~x)φ(t, ~x)d~x.

For f ∈ Cc∞ (R1,3 , C4 ) such that κf = f , the corresponding space-time

smeared field is given by
Z
φ[f ] := f (x)φ(x)dx = φ((S ∗ f )).

197
7.1.4 Plane waves
Since we consider neutral fields, the generic name for the momentum variable
is again k, instead of p.
Recall that in the Dirac representation we defined the plane waves u(k, s)
given by (6.5). These plane waves are compatible with the Majorana condition
in the following sense:
κu(k, s) = u(−k, −s). (7.3)
We can introduce the plane wave functionals, where k 0 > 0,

a(k, s) := φ((|k, s)))

Z
d~x ~
= p u(k, s)e−ik~x φ(0, ~x).
(2π) 3

Note that

a∗ (k, s) := φ((| − k, −s)))

Z
d~x ~
= p u(−k, −s)eik~x φ(0, ~x).
(2π) 3

We have
XZ d~k
u(k, s)eikx a(k, s) + u(−k, −s)e−ikx a∗ (k, s)


φ(x) = p
s (2π)3
XZ
d~k |k, s)a(k, s) + | − k, −s)a∗ (k, s) .


=
s

7.1.5 Quantization
To quantize the Dirac equation with the Majorana condition we use the formal-
ism of quantization of neutral fermionic systems [15].
We want to construct (H, Ĥ, Ω) satisfying the standard requirements of QM
(1)-(3) and a distribution
R1,3 3 x 7→ φ̂(x), (7.4)
with values in C4 ⊗ B(H), satisfying the Majorana condition

κφ̂∗ (x) = φ(x), (7.5)

and such that the following conditions are true:

(1) (−iγ∂ + m)φ̂(x) = 0;
(2) [φa (0, ~x), φ∗b (0, ~y )]+ = 2δab δ(~x − ~y );
(3) eitĤ φ̂(x0 , ~x)e−itĤ = φ̂(x0 + t, ~x);
(4) Ω is cyclic for φ̂(x).

198
 The above problem has an essentially unique solution, which we describe
below.
Let ZD ' L2 (R3 , C2 ) denote the fermionic positive frequency Hilbert space
defined in Subsubsect. 6.1.8. We set H := Γa (ZD ). Creation/annihilation oper-
ators on ZD will be denoted â∗ /â. In particular, for k on shell and s = ± 21 , we
have creation operators, written below in both physicist’s and mathematician’s
notation:
â∗ (k, s) = â∗ |k, s) .


(7.6)
The quantum field is
XZ d~k
u(k, s)eikx â(k, s) + u(−k, −s)e−ikx â∗ (k, s) .


φ̂(x) := p
s (2π)3

The quantum Hamiltonian and momentum are

Z X
Ĥ := â∗ (k, s)â(k, s)ε(~k)d~k,
s
Z X
~
P̂ := â∗ (k, s)â(k, s)~kd~k.
s

˜
The whole R1,3 o Spin↑ (1, 3) acts unitarily on H. Moreover, if we set φ̂(x) :=
β φ̂∗ (x), then
˜
[φ̂a (x), φ̂b (y)]+ = 2Sab (x − y). (7.7)
We have
˜ (+)
(Ω|φ̂a (x)φ̂b (y)Ω) = 2Sab (x − y),
˜ c
(Ω|T(φ̂a (x)φ̂b (y))Ω) = 2Sab (x − y).

For f ∈ Cc∞ (R1,3 , C4 ) such that κf = f , we set

Z
φ̂[f ] := f (x)φ̂(x)dx.

If we use the Majorana representation, so that κ = 1l, we obtain an operator

valued distribution satisfying the Wightman axioms with D := Γfin
a (ZD ).
For an open set O ⊂ R1,3 the field algebra is defined as

F(O) := {φ̂[f ] : f ∈ Cc∞ (O, C4 ), κf = f }00 .

The observable algebra A(O) is the even subalgebra of F(O). The nets of alge-
bras F(O) and A(O), O ⊂ R1,3 , satisfy the Haag-Kastler axioms.

199
7.1.6 Quantization in terms of smeared fields
There exists an alternative equivalent formulation of the quantization program,
which uses the smeared fields instead of point fields. We look for a linear function

YD 3 ζ 7→ φ̂((ζ))

with values in bounded self-adjoint operators such that

(1) [φ̂((ζ1 )), φ̂((ζ2 ))]+ = 2ζ 1 · ζ2 ;
(2) φ̂((r(t,~0) ζ)) = eitĤ φ̂((ζ))e−itĤ ;
(3) Ω is cyclic for φ̂((ζ)).
One can pass between these two versions of the quantization by
Z
φ̂((ζ)) = ζ(t, ~x)φ̂(t, ~x)d~x. (7.8)

7.2 Majorana fermions with a mass-like perturbation

7.2.1 Classical fields
The meaning of the expression a mass-like perturbation is slightly different for
fermions, where we perturb m, and for bosons, where we perturb m2 .
“Classical” Majorana fields with a mass-like perturbation satisfy the Majo-
rana condition (7.2) and the equation

(−iγµ ∂ µ + m)φ(x) = −σ(x)φ(x), (7.9)

where we assume that R1,3 3 x 7→ σ(x) is a given real Schwartz function.

Let us define the corresponding retarded and advanced propagators as the
unique distributional solutions of

(−iγµ ∂ µ + σ(x))S ± (x, y) = δ(x − y) (7.10)

satisfying
suppS ± ⊂ {x, y : x ∈ J ± (y)}.
We also set
S(x, y) := S + (x, y) − S − (x, y).
Clearly
suppS ⊂ {x, y : x ∈ J(y)}.
The “classical” Majorana field coinciding with the free field at time t = 0 is
defined as
Z
φ(t, ~x) = S(t, ~x, 0, ~y )βφfr (0, ~y )d~y .

200
7.2.2 Lagrangian and Hamiltonian formalism
The Lagrangian density that yields (7.9) is
1 
φ̃(x)γ µ (−i∂µ )φ(x) + i∂µ φ̃(x)γ µ φ(x) + φ̃(x) m + σ(x) φ(x),


L(x) = −
2
where φ(x) are off-shell fields satisfying the Majorana condition (7.2).
We can introduce the Hamiltonian density

L(x)
H(x) = φ̇(x) − L(x)
∂φ̇(x)
1 ∗ ~


~ ∗ (x)φ(x) + φ∗ (x) m + σ(x) β φ(x),
= φ (x)~
α(−i∂)φ(x) + i∂φ
2
and the Hamiltonian Z
H(t) = H(t, ~x)d~x.

7.2.3 Quantum fields

The quantum fields should satisfy the Majorana condition (7.5), the equation

(−iγµ ∂ µ + m)φ̂(x) = −σ(x)φ̂(x), (7.11)

and they should coincide with the free fields at time t = 0:

φ̂(~x) := φ̂(0, ~x) = φ̂fr (0, ~x).

The quantization amounts to putting “hats” onto (7.11).

We write the Schrödinger picture Hamiltonian as
Z 

Ĥ(t) := :φ̂∗ (~x) αi i∂i + m + σ(t, ~x) β φ̂(~x):d~x.

The interaction picture Hamiltonian is

Z
1
ĤInt (t) = σ(t, ~x):φ̂∗fr (t, ~x)β φ̂fr (t, ~x):d~x.
2
As usual, we define the scattering operator, scattering amplitudes, Green’s
functions, amputated Green’s functions and the generating function.

201
7.2.4 Path integral formulation
The generating function (and hence all the other quantities introduced above)
can be computed exactly. It equals
!1
− iγ∂ + m + σ

 1  2
Z(f ) = det
exp −σ
− iγ∂ + m + σ − i0 −iγ∂ + m − i0
 
i
× exp f (−i∂γ + m + σ − i0)−1 f
2
c c




= det 1l + σSfr exp −σSfr
 
i c c −1
× exp f S (1l + σSfr ) f . (7.12)
2 fr

(7.12) can be expressed in terms of path integrals:

Z  Z 


C Π dφ(x) exp i L(x) − f (x)φ(x) dx .
x

Here, C is a normalization constant, which does not depend on f . As usual, the

formula (7.13) is only symbolic, the full information is contained in (7.12).
One can derive Feynman rules fully analogous to the Feynman rules of
bosonic mass-like perturbations.

7.2.5 Vacuum energy

The logarithm of the vacuum-to-vacuum scattering amplitude can be computed
exactly:

E = i log(Ω|SΩ) = i log Z(0)

i  
= Tr log(1 + σS c ) + σS c
2
∞ ∞
X (−1)n+1 X
= i Tr(σS c )n =: En .
n=2
2n n=2

Note that En = −i D n c n
2n , where Dn = (−1) Tr(σS ) is the value of the loop with
n

n vertices, similarly to the bosonic case (2.135) except for a different sign.

7.2.6 Renormalization of the vacuum energy

The nth order contribution to the vacuum energy has the form
Z
En = π(k1 , . . . , kn )
dk1 dkn−1
×σ(k1 ) · · · σ(kn−1 )σ(−k1 · · · − kn−1 ) ··· . (7.13)
(2π)4 (2π)4

202
For n = 2, 3, 4, En are divergent and need renormalization.
Using the Pauli-Villars method we define for n = 1, 2, 3 the renormalized
vacuum energy functions

π ren (k1 , . . . , kn−1 ) := lim πΛ (k1 , . . . , kn−1 ) − πΛ (0, . . . , 0) .

Λ→∞

Thus

Enren
Z
dk1 dkn−1
= π ren (k1 , . . . , kn−1 )σ(k1 ) · · · σ(kn−1 )σ(−k1 · · · − kn−1 ) ···
(2π)4 (2π)4
Z
dk1 dkn−1
= lim πΛ (k1 , . . . , kn−1 )σ(k1 ) · · · σ(kn−1 )σ(−k1 · · · − kn−1 ) 4
···
Λ→∞ (2π) (2π)4
Z !
−πΛ (0, . . . , 0) σ(x)n dx .

The renormalized scattering operator Ŝren is a well defined unitary operator.

Formally, we have


2 3 4
R R R
Ŝren = e iπ∞ (0) σ(x) dx+iπ∞ (0,0) σ(x) dx+iπ∞ (0,0,0) σ(x) dx Ŝ.
Lren (x) = L(x) + π∞ (0)σ(x)2 + π∞ (0, 0)σ(x)3 + π∞ (0, 0, 0)σ(x)4 .

7.2.7 Pauli-Villars renormalization of the 2nd order term

E3 and E4 are logaritmically divergent. Below we present computations only for
E2 , which is quadratically divergent. As a special case of (7.13) for n = 2 we
write Z
dk
E2 = |σ(k)|2 π(k 2 ) .
(2π)4
Using the Pauli-Villars regularization, as in Subsubsect. 5.2.11, we compute:

203
 4πΛ (k 2 )
− (q + 12 k)γ + mi − (q − 12 k)γ + mi



d4 q X
Z
= −i Ci tr
(2π)4 i ((q + 21 k)2 + m2i − i0)((q − 12 k)2 + m2i − i0)


d4 q X − 4q 2 + k 2 + 4m2i
Z
= −i C i
(2π)4 i ((q + 12 k)2 + m2i − i0)((q − 21 k)2 + m2i − i0)
Z ∞ Z ∞ !
1 X 4α1 α2 k 2 4m2 2i
= − dα1 dα2 Ci + +
(4π)2 0 0 i
(α1 + α2 )2 (α1 + α2 )2 (α1 + α2 )3
 
α1 α2 2
× exp −i(α1 + α2 )m2i − i k
α1 + α2
Z 1 Z ∞ !
1 dρ X 2 2 2 2i
= − dv Ci (1 − v )k + 4m +
(4π)2 0 0 ρ i ρ
1 − v2 2
  
× exp −iρ m2i + k
4
Z 1 X 
(1 − v 2 ) 2
 
1 2 2 2

2
= dv Ci (1 − v )k + 4mi log mi + k − i0 .
(4π)2 2 0 i
4

Note that at the end we use (A.24) besides (A.25), because of the quadratic
divergence.
Finally, the renormalized vacuum energy function is defined as
 
π ren (k 2 ) = lim πΛ (k 2 ) − πΛ (0) (7.14)
Λ→∞
Z 1
m2 1 (1 − v 2 )k 2   (1 − v 2 )k 2 
= − + log 1 + − i0 dv.
(4π)2 0 2 8m2 4m2

A Appendix
A.1 Second quantization
A.1.1 Fock spaces
Let Z be a Hilbert space. Let Sn denote the permutation group of n elements
and σ ∈ Sn . Θ(σ) is defined as the unique operator in B(⊗n Z) such that

Θ(σ)g1 ⊗ · · · ⊗ gn = gσ−1 (1) ⊗ · · · ⊗ gσ−1 (n) , g1 , . . . , gn ∈ Z.

Θ(σ) is unitary. We define the symmetrization/antisymmetrization projections

1 X 1 X
Θns := Θ(σ), Θna := sgnσΘ(σ).
n! n!
σ∈Sn σ∈Sn

204
In what follows we will consider in parallel the symmetric/antisymmetric, or
bosonic/fermionic case. To facilitate notation we will write s/a for either s or
a.
Θns/a are orthogonal projections. The n-particle bosonic/fermionic space is
defined as
⊗ns/a Z := Θns/a ⊗n Z. (A.1)
The bosonic/fermionic Fock space is
∞
Γs/a (Z) := ⊕ ⊗ns/a Z. (A.2)
n=0

The vacuum vector is Ω := 1 ∈ ⊗0s/a Z = C.

We use the convention saying that the tensor products and direct sums used
in (A.1) and (A.2) are completed in their natural topology, so that n-particle
spaces and Fock space are Hilbert spaces. Sometimes we may want a similar
construction without the completion (in particular, if Z is not a Hilbert space).
Then we will speak about algebraic n-particle spaces or algebraic Fock spaces.

A.1.2 Creation/annihilation operators

For g ∈ Z we define the creation operator
√
â∗ (g)Ψ := Θn+1
s/a n + 1g ⊗ Ψ, Ψ ∈ ⊗ns/a Z,
∗
and the annihilation operator â(g) := (â∗ (g)) .
Above we used a compact notation for creation/annihilation operators popu-
lar among mathematicians. Physicists commonly prefer another notation, which
is longer and less canonical, but often more flexible. In order to introduce it, we
need to fix an identification of Z with L2 (Ξ) of some measure space Ξ with its
elements called generically ξ and the measure called dξ. For instance, Ξ can be
Rd with the Lebesgue measure. Every g ∈ Z can be represented as a function
Ξ 3 ξ 7→ g(ξ). Then
Z
â∗ (g) = g(ξ)â∗ (ξ)dξ, (A.3)
Z
â(g) = g(ξ)â(ξ)dξ. (A.4)

We will call the notation on the left of (A.3) and (A.4) “mathematician’s nota-
tion” and on the right “physicist’s notation”.
Sometimes one introduces formal symbols |ξ) treated as vectors, possibly
nonnormalizable, such that for g ∈ L2 (Ξ) we can write
Z
g = |ξ)g(ξ)dξ, g(ξ) = (ξ|g).

205
We have the following dictionary between creation operators written in the
“physicist’s notation” (on the left) and the “mathematician’s notation” (on the
right):

â∗ (ξ) = â∗ |ξ) ,

(A.5)
Z
(ξ|g)â∗ (ξ)dξ = â∗ (g). (A.6)

Let [·, ·]− , resp. [·, ·]+ denote the commutator, resp. anticommutator. Bosonic/fermionic
creation and annihilation operators satisfy the canonical commutation/anticommutation
relations, which in the “mathematician’s notation” read

[â∗ (f ), â∗ (g)]∓ = [â(f ), â(g)]∓ = 0,

Z
∗
[â(f ), â (g)]∓ = (f |g) = f (ξ)g(ξ)dξ,

and in the “physicist’s notation”, at least for Ξ = Rd , have the form

[â∗ (ξ), â∗ (ξ 0 )]∓ = [â(ξ), â(ξ 0 )]∓ = 0,

∗ 0
[â(ξ), â (ξ )]∓ = δ(ξ − ξ 0 ).

A.1.3 Weyl/antisymmetric and Wick quantization

Let
(ξ1 , · · · ξm , ξn0 , · · · , ξ10 ) 7→ b(ξ1 , · · · ξm , ξn0 , · · · , ξ10 ) (A.7)
be a complex function, symmetric/antisymmetric separately wrt the first m and
the last n arguments. Let us introduce the following expression:
Z Z
· · · b(ξ1 , · · · ξm , ξn0 , · · · , ξ10 ) (A.8)

×a∗ (ξ1 ) · · · a∗ (ξm )a(ξn0 ) · · · a(ξ10 )dξ1 · · · dξm dξn0 · · · dξ10 ,

where a(ξ) and a∗ (ξ) are commuting/anticommuting symbols.

In the symmetric case (A.8) can be interpreted as a polynomial on Z ⊕ Z.
Indeed, if we interpret the symbols a(ξ) as the evaluations of g ∈ Z = L2 (Rd ):

ha(ξ)|gi := g(ξ), ha∗ (ξ)|gi := g(ξ),

then (A.8) has the meaning of a polynomial function. It is common to use the
name a polynomial for (A.8) also in the antisymmetric case.
The Wick quantization of (A.8) is the operator on the Fock space given by
the same expression, except that we put the “hats” on a and a∗ . Note that the
creation operators are on the left and annihilation operators are on the right:
Z
b(ξ1 , · · · ξm , ξn0 , · · · , ξ10 )

×â∗ (ξ1 ) · · · â∗ (ξm )â(ξn0 ) · · · â(ξ10 )dξ1 , · · · ξn dξ10 · · · dξm

0
.

206
 In practice we often have some fields, say ϕ1 (ξ), ϕ2 (ξ), that can be written
as linear combinations of a(ξ) and a∗ (ξ), eg.
Z Z
ϕi (ξ) = Ai (ξ)a(ξ) + Bi (ξ)a∗ (ξ).

Their quantizations are denoted by “hats”:

Z Z
ϕ̂i (ξ) = Ai (ξ)â(ξ) + Bi (ξ)â∗ (ξ).

Suppose we have a polynomial

X Z Z
· · · ci1 ,...,im (ξ1 , · · · ξm )ϕi1 (ξ1 ) · · · ϕim (ξm )dξ1 · · · dξm . (A.9)
i1 ,...,im

We assume that the coefficients ci1 ,...,im (ξ1 , · · · ξm ) are symmetric/antisymmetric.

The most natural quantization of (A.9) is the operator on the Fock space given
by the same expression, where we just put “hats” on the fields. It is called the
Weyl quantization in the bosonic case. In the fermionic case this quantization
seems to have no established name, although it would be tempting to call it the
fermionic Weyl quantization. Following [15], we will call it the antisymmetric
quantization.
By inserting (A.9), we obtain a polynomial expressed in terms of a(ξ) and
a∗ (ξ). Its Wick quantization has the traditional notation where the expression
decorated with hats is put between double dots:
X Z Z
: · · · ci1 ,...,im (ξ1 , · · · ξm )ϕ̂(ξ1 ) · · · ϕ̂(ξm )dξ1 · · · dξm :.
i1 ,...,im

For 1st order polynomials their Wick quantization obviously coincides with
their Weyl/antisymmetric quantization:
Z Z
: f (ξ)ϕ̂(ξ)dξ: = f (ξ)ϕ̂(ξ)dξ.

We will often use Wick quantizations of second degree polynomials. For

instance, let c(ξ, ξ 0 ) be a symmetric/antisymmetric function. Then the Wick
and Weyl/antisymmetric quantizations differ by the vacuum expectation value:
Z Z
: c(ξ, ξ 0 )ϕ̂(ξ)ϕ̂(ξ 0 )dξdξ 0 :
Z Z 
= c(ξ, ξ 0 ) A(ξ)A(ξ)â∗ (ξ)â∗ (ξ 0 ) + A(ξ)B(ξ 0 )â∗ (ξ)â(ξ 0 )

±B(ξ)A(ξ 0 )â∗ (ξ 0 )â(ξ) + B(ξ)B(ξ 0 )â(ξ)â(ξ 0 ) dξdξ 0
Z Z Z Z
0 0 0
c(ξ, ξ 0 ) Ω|ϕ̂(ξ)ϕ̂(ξ 0 )Ω dξdξ 0 .


= c(ξ, ξ )ϕ̂(ξ)ϕ̂(ξ )dξdξ −

207
A.1.4 Second quantization of operators
For a contraction q on Z we define the operator Γ(q) on Γs/a (Z) by

Γ(q) = q ⊗ ··· ⊗ q .
⊗n
s/a
Z ⊗n
s/a
Z

Γ(q) is called the second quantization of q.

Similarly, for an operator h we define the operator dΓ(h) by

dΓ(h) = h ⊗ 1(n−1)⊗ + · · · + 1(n−1)⊗ ⊗ h .

⊗n
s/a
Z ⊗n
s/a
Z

dΓ(h) is called the (infinitesimal) second quantization of h.

If h is the multiplication operator by h(ξ), then using physicist’s notation
we have Z
dΓ(h) = h(ξ)â∗ (ξ)â(ξ)dξ.

Note the identity Γ(eith ) = eitdΓ(h) .

A.1.5 Implementability of Bogoliubov translations

Consider bosonic creation/annihilation operators. Let ξ 7→ f (ξ) be a complex
function. Set

â∗1 (ξ) = â∗ (ξ) + f (ξ),

â1 (ξ) = â(ξ) + f (ξ).

A proof of the following well-known fact can be found eg. in [15].

Theorem A.1 There exists a unitary operator U on the Fock space such that

U â∗ (ξ)U ∗ = â∗1 (ξ), U â(ξ)U ∗ = â1 (ξ),

iff Z
|f (ξ)|2 dξ < ∞.

Up to a phase factor
Z

U = exp â(ξ)f (ξ) − â∗ (ξ)f (ξ) dξ .

The following formula is a time-dependent generalization of the well-known

∗ 1 ∗
identity eiâ (f )+iâ(f ) = e− 2 (f |f ) eiâ (f ) eiâ(f ) :
!
Z  Z 
∗
Texp iâ f (t)dt + iâ f (t)dt (A.10)



(f (t1 )|f (t2 ))θ(t1 −t2 )dt1 dt2 iâ∗
R R R R
f (t)dt
= e− e eiâ f (t)dt
.

208
A.1.6 Implementability of Bogoliubov rotations
We will treat simultaneously the bosonic and fermionic case. The upper signs
will always correspond to the bosonic case and lower to the fermionic case.
Let p, q be operators with the integral kernels p(ξ, ξ 0 ), q(ξ, ξ 0 ). We assume
that q(ξ, ξ 0 ) = ±q(ξ 0 , ξ). Set
Z
∗
p(ξ, ξ 0 )â∗ (ξ) + q(ξ, ξ 0 )â(ξ 0 ) dξ 0 ,


â1 (ξ) = (A.11)
Z
q(ξ, ξ 0 )â∗ (ξ 0 ) + p(ξ, ξ 0 )â(ξ) dξ 0 .


â1 (ξ) = (A.12)

Assume that

p∗ p ∓ q # q = 1l, p∗ q ∓ q # p = 0,
pp∗ ∓ qq ∗ = 1l, pq # ∓ qp# = 0,

which guarantees that â∗1 , â1 satisfy the same commutation/anticommutation

relations as â∗ , â.
Here, we use the following notation: For an operator p we will write p∗ for its
Hermitian conjugate, p# for its transpose of p and p for its complex conjugate.
If the integral kernel of p is p(ξ, ξ 0 ), then clearly

p∗ (ξ, ξ 0 ) = p(ξ 0 , ξ), p# (ξ, ξ 0 ) = p(ξ 0 , ξ), p(ξ, ξ 0 ) = p(ξ, ξ 0 ).

Theorem A.2 There exists a unitary U on the Fock space such that

U â∗ (ξ)U ∗ = â∗1 (ξ), U â(ξ)U ∗ = â1 (ξ),

iff q is Hilbert-Schmidt, that means,

Z Z
|q(ξ, ξ 0 )|2 dξdξ 0 < ∞.

The above theorem is called the Shale criterion [48] in the bosonic and the
Shale-Stinespring criterion [49] in the fermionic case. See also eg. [15].

A.1.7 Infimum of a van Hove Hamiltonian

Consider a bosonic Hamiltonian of the form
Z Z Z
H := ε(ξ)â∗ (ξ)â(ξ)dξ + v(ξ)â∗ (ξ)dξ + v(ξ)â(ξ)dξ. (A.13)

Such Hamiltonians are sometimes called van Hove Hamiltonians [13, 15]. As-
sume that ε is positive. We would like to compute the infimum of the spectrum
of H, denoted inf H.

209
 By completing the square we can rewrite (A.13) as

|v(ξ)|2
Z Z

∗ v(ξ)  v(ξ) 
ε(ξ) â (ξ) + â(ξ) + dξ − dξ. (A.14)
ε(ξ) ε(ξ) ε(ξ)

It is easy to see that the infimum of the first term in (A.14) is zero. Hence

|v(ξ)|2
Z
inf H = − dξ. (A.15)
ε(ξ)

A.1.8 Infimum of a Bogoliubov Hamiltonian

Consider a bosonic or fermionic Hamiltonian
Z
h(ξ, ξ 0 ) â∗ (ξ)â(ξ 0 ) ± â(ξ)â∗ (ξ 0 ) dξdξ 0


H :=
Z
g(ξ, ξ 0 )â∗ (ξ)â∗ (ξ 0 ) ± g(ξ, ξ 0 )â(ξ)â(ξ 0 ) dξdξ 0 .


+ (A.16)

We assume that h(ξ, ξ 0 ) = h(ξ 0 , ξ), g(ξ, ξ 0 ) = ±g(ξ 0 , ξ). We will call (A.16)
Bogoliubov Hamiltonians. Note that (A.16) is the Weyl/antisymmetric quanti-
zation of the corresponding classical quadratic Hamiltonian. In the case of an
infinite number of degrees of freedom it is often ill defined, but even then it is
useful to consider such formal expressions.
We have the following formula for the infimum of H [15]:
 12
h2 ∓ gg ∗ ∓hg ± gh#

1
inf H = ± Tr . (A.17)
2 g h − h# g ∗
∗
h#2 ∓ g ∗ g

Here, we write h for the operator with the integral kernel h(ξ, ξ 0 ) and g for the
operator with the integral kernel g(ξ, ξ 0 ).
Consider the Wick ordered version of (A.16):
Z
:H: := 2 h(ξ, ξ 0 )â∗ (ξ)â(ξ 0 )dξdξ 0
Z
g(ξ, ξ 0 )â∗ (ξ)â∗ (ξ 0 ) ± g(ξ, ξ 0 )â(ξ)â(ξ) dξdξ 0 .


+ (A.18)

(In the case an infinite number of degrees of freedom :H: has a better chance to
be well defined compared with H). The formula for the infimum of :H: is more
complicated, but is more likely to lead to a finite expression [15]:
1  !
h2 ∓ gg ∗ ∓hg ± gh# 2

1 h 0
inf :H: = Tr ± ∓ . (A.19)
2 g ∗ h − h# g ∗ h#2 ∓ g ∗ g 0 h#

210
A.2 Miscellanea
A.2.1 Identities for Feynman integrals
Z ∞
1
= i dα exp(−iαA), (A.20)
A − i0 0

pµ = i∂zµ exp(−ipz) , (A.21)

z=0
Z
dp sgn(a)
exp −i(ap2 + bp) exp ib2 /4a .



4
= i 2 2
(A.22)
(2π) (4π) a
Using these identities, a typical evaluation of a loop integral goes as follows:
Z
i P (q)dq
(2π)4 (a1 q 2 + 2b1 q + c1 − i0) · · · (an q 2 + 2bn q + cn − i0)
Z ∞ Z ∞
in+1
Z
= dα 1 · · · dα n dqP (q)
(2π)4 0 0
 
× exp − iα1 (a1 q 2 + 2b1 q + c1 ) · · · − iαn (an q 2 + 2bq q + cq )
Z ∞ Z ∞
in+1
Z 
= dα 1 · · · dα n dqP (i∂ z ) exp − i(α1 a1 · · · + αn an )q 2
(2π)4 0 0

−i(α1 b1 · · · + αn bn + z)q − i(α1 c1 · · · + αn cn )
z=0
Z ∞ Z ∞
in −2
= − dα1 · · · dαn (α1 a1 · · · + αn an ) P (i∂z )
(4π)2 0 0
 (α b · · · + α b + z)2 
1 1 n n
× exp i − i(α1 c1 · · · + αn cn ) . (A.23)
4(α1 a1 · · · + αn an ) z=0
P
If Ci = 0, then
Z ∞X
dρ X
Ci e−iρAi = − Ci log(Ai − i0). (A.24)
0 i
ρ i
P
If in addition Ci Ai = 0, then
Z ∞X
dρ X
Ci 2 e−iρAi = − Ci Ai log(Ai − i0). (A.25)
0 i
ρ i

Z
log(A2 − w2 )dw = w log(A2 − w2 ) − 2w

(A + w)
+A log , 0 < w < A; (A.26)
(A − w)
w3 2w3 2A2 w
Z
w2 log(A2 − w2 )dw = log(A2 − w2 ) − −
3 9 3
3
A (A + w)
+ log , 0 < w < A. (A.27)
3 (A − w)

211
A.2.2 Identities for the dimensional regularization
The Feynman identity:
Z 1
1 1 dv
=
2 . (A.28)
AB 2 1
+ B) + 12 (A − B)v
2 (A
−1

The behavior of Γ around 0:

1
Γ(2 − d/2) ' − γ. (A.29)
2 − d/2
The area of the unit d − 1-dimensional sphere:
2π d/2
Ωd = . (A.30)
Γ(d/2)
Integrals, which can be reduced to special cases of the Euler integral:
Z ∞
td−1 1 2 −2+d/2
2 + A2 ) 2
dt = (A ) Γ(d/2)Γ(2 − d/2), (A.31)
0 (t 2
Z ∞
td+1 1 2 −1+d/2
2 + A2 ) 2
dt = (A ) Γ(1 + d/2)Γ(1 − d/2)
0 (t 2
Z ∞
td−1
= A2 (−1 + 2/d)−1 dt. (A.32)
0 (t2 + A2 )2
Typical integrals:
µ4−d Ωd ∞ |q|d−1
Z

2 d|q|
(2π)d 0 q 2 + A2
1  µ2 4π 2−d/2
= Γ(2 − d/2)
(4π)2 A2
!
1 µ2 4π  1 
≈ 1 + (2 − d/2) log − γ
(4π)2 A2 2 − d/2
2
1  µ 4π 1 
≈ − γ + log + , (A.33)
(4π)2 A2 (2 − d/2)

µ4−d Ωd ∞ |q|d+1
Z
d
(−1 + 2/d)
2 d|q|
(2π) 0 q 2 + A2
A2  µ2 4π 2−d/2
= Γ(2 − d/2)
(4π)2 A2
!
A2 µ2 4π  1 
≈ 1 + (2 − d/2) log − γ
(4π)2 A2 2 − d/2
A2  µ2 4π 1 
≈ − γ + log + . (A.34)
(4π)2 A2 (2 − d/2)

212
A.2.3 Operator identities
If A is a positive self-adjoint operator, then
Z
A dτ
A1/2 = 2
, (A.35)
(A + τ ) 2π
Z
1 dτ
A−1/2 =
(A + τ 2 ) 2π
Z
1 dτ
= −2 τ2 . (A.36)
(A + τ 2 )2 2π
In the following identity κ is a certain operator. It is useful when studying nth
order loop diagrams:
Z n−1 dτ
1  1
Tr κ κ τ2
(A + τ 2 )2 (A + τ 2 ) 2π
Z n dτ
1  1
= − Tr κ . (A.37)
2n (A + τ 2 ) 2π

A.2.4 Coulomb and Yukawa potential

If ρ ∈ Cc (R3 ), then
ρ = −∆f
has a unique solution in functions that decay at infinity given by
Z
−1 1
f (~x) = (−∆) ρ(~x) = ρ(~y )d~y . (A.38)
4π|~x − ~y |
For large |~x|, (A.38) has the asymptotics
Z  
1 1
ρ(~y )d~y + O . (A.39)
4π|~x| |~x|2
More generally

e−m|~x−~y|
Z
2 −1
(m − ∆) ρ(~x) = ρ(~y )d~y . (A.40)
4π|~x − ~y |

A.2.5 Vector fields

~ x) ∈ R3 . We say that it is transversal if
Consider a vector field R3 3 ~x 7→ A(~
~ x) = 0.
divA(~

If it is not necessarily transversal but sufficiently nice, its transversal and lon-
gitudinal part are defined as
~ tr (~x) := A(~
A ~ A(~
~ x) + (−∆)−1 ∂div ~ x), (A.41)
− 21 ~ x),
Alg (~x) := −(−∆) divA(~ (A.42)

213
We have the decomposition
~ x) = A
A(~ ~
~ tr (~x) − ∂(−∆)−1/2
Alg (~x).
We have the identities
Z Z Z
~ 2 ~
A(~x) d~x = Atr (~x) d~x + Alg (~x)2 d~x
2

Z Z Z
~ x) 2 d~x =
∂~ A(~ ~ tr (~x) 2 d~x +
∂~ A ~ lg (~x) 2 d~x
∂~ A




(A.43)
Z Z

2
~ ~ ~ x) 2 d~x,


∂ Alg (~x) d~x = divA(~ (A.44)
Z Z
∂~ A~ tr (~x) 2 d~x = 1 ~ x) 2 d~x.



rotA(~ (A.45)
2

A.2.6 Dispersion relations

The principal value of 1ξ , denoted P 1ξ , is the distribution acting on a test function
f as Z Z − Z ∞ 
f (ξ) f (ξ)
P dξ := lim + dξ.
ξ &0 −∞  ξ
It appears in the Sochocki formula
1 1 1
= lim = ∓iπδ(ξ) + P .
ξ ± i0 &0 ξ ± i ξ
Let f be holomorphic on {Imz > 0} with continuous boundary values at the
real line, denoted f (E + i0), E ∈ R. Let f = fR + ifI be its decomposition
into the real and imaginary part. The following theorem follows easily from the
Cauchy formula and describes what physicists call dispersion relations:
Theorem A.3 Assume that E 7→ f (E + i0) is in C 1 , |f 1+|E|
R (E+i0)|
dE < ∞ and
on the upper half-plane lim f (E) = 0. Then for E ∈ R
|E|→∞
Z
1 fI (ξ + i0)
fR (E + i0) = P dξ,
π ξ−E
Z
1 fR (ξ + i0)
fI (E + i0) = − P dξ.
π ξ−E

Proof. By the Cauchy integral formula

Z R Z π

1 f (ξ + i0) 1 f Reiφ
f (E + i0) = dξ + Reiφ idφ. (A.46)
2πi −R ξ − E − i0 2πi 0 (ReiRφ − E)
Applying the Sochocki formula, we obtain
Z R
1 π f Reiφ
Z

1 f (ξ + i0)
f (E + i0) = P dξ + iRφ − E)
Reiφ idφ. (A.47)
πi −R ξ − E πi 0 (Re

214
 By assumption, the second term on the rhs goes to zero. 2
Sometimes a function f does not have enough decay. We can then use the
so-called once substracted dispersion relations.

Theorem A.4 Assume that R 3 E 7→ f (E + i0) is in C 1 , f1+|E|

R (E+i0)
2 dE < ∞

f (E)
and on the upper half-plane lim E = 0. Then for E ∈ R
|E|→∞

Z  
1 1 1
fR (E + i0) = fR (0 + i0) + P fI (ξ + i0) − dξ,
π ξ−E ξ
Z  
1 1 1
fI (E + i0) = fI (0 + i0) − P fR (ξ + i0) − dξ.
π ξ−E ξ

Proof. We specialize (A.47) to E = 0:

Z R Z π
1 f (ξ + i0) 1
f Reiφ idφ.


f (i0) = P dξ + (A.48)
πi −R ξ πi 0

Then we subtract (A.48) from (A.47) and take the limit R → ∞. 2

References
[1] Asmuß, B. 1990: Zur Quantentheorie geladener Klein-Gordon Teilchen in
äußeren Feldern, PhD Thesis at Fernuniversität Hagen

[2] Araki, H., 2000: Mathematical Theory of Quantum Fields Oxford University
Press
[3] Bär, C., Ginoux, N., Pfäffle, F., 2007: Wave equation on Lorentzian mani-
folds and quantization, ESI Lectures in Mathematics and Physics, EMS.
[4] Baez, J.C., Segal, I.E., Zhou, Z., 1991: Introduction to algebraic and con-
structive quantum field theory, Princeton University Press.
[5] Bialynicki-Birula, I., Bialynicka-Birula, Z. 1975: Quantum Electrodynamics,
Pergamon, Oxford 1975,
[6] Bleuler, K. 1950: Helv.Phys.Acta 23, 567586

[7] Bloch, F., Nordsieck, A., 1937: Note on the radiation field of the electron,
Phys. Rev. 52, 54-59.
[8] Bogoliubov, N.N., Logunov, A.A., Oksak, A.I., Todorov, I.T., 1990: General
Principles of Quantum Field Theory, Kluwer Academic Publishers

215
[9] Brunetti, R., Fredenhagen, K., 2009: Quantum field theory on curved back-
grounds, in Bär, C. and Fredenhagen, K., editors, Quantum Field Theory
on Curved Spacetimes, Lecture Notes in Physics 786, Springer
[10] Cardona, A., Ducourtioux, C., Paycha, S. 2003: From Tracial Anomalies
to Anomalies in Quantum Field Theory, Commun. Math. Phys. 242, 31-65

[11] Carey, A.L., Ruijsenaars, S.N.M. 1987: On fermion gauge groups, current
algebras and Kac-Moody algebras, Acta Applicandae Mathematica 10, 1-86
[12] Deckert, D.-A., Dürr, D., Merkl, F., Schottenloher, M., 2010: Time Evolu-
tion of the External Field Problem in QED J. Math. Phys. 51

[13] Dereziński, J., 2003: Van Hove Hamiltonians—exactly solvable models of

the infrared and ultraviolet problem, Ann. Henri Poincaré 4, 713-738.
[14] Dereziński J., Gérard C. 1997: Scattering Theory of Classical and Quantum
N -Particle Systems, Texts and Monographs in Physics, Springer

[15] Dereziński, J., Gérard, G. 2013: Mathematics of quantization and quantum

fields, Cambridge University Press
[16] Gelfand, I.M. and coathors, 1958: Generalized Functions I-V (Russian),
Fizmatgiz
[17] Greiner, W., Reinhardt, J. 2009: Quantum Electrodynamics, 4th ed.,
Springer
[18] Gravejat, P., Hainzl, C., Lewin, M., Séré 2013: Construction of the Pauli-
Villars-regulated Dirac vacuum in electromagnetic fields, Arch. for Rational
Mach. and An. DOI 10.1007/s00205-012-0609-1

[19] Gupta, S. 1950: Proc. Phys. Soc. A63, 681691

[20] Fujikawa, K., Suzuki, H., 2004: Path Integrals and Quantum Anomalies,
Clarendon Press, Oxford
[21] Gracia-Bondia, J.M. 2000: The phase of the scattering matrix, Phys. Lett.
B 482, 315322

[22] Haag, R., 1992: Local quantum physics, Texts and Monographs in Physics,
Springer.
[23] Haag, R., Kastler, D., 1964: An algebraic approach to quantum field theory,
Journ. Math. Phys. 5, 848-862.

[24] Heisenberg, W., Euler, H., 1936: Folgerungen aus der Diracschen Theorie
des Positrons Z. Phys. 98, 714
[25] Hochstenbach, W. J. M. A. 1976 Field theory with an external potential
Comm. Math. Phys. 51, 211–217

216
[26] Itzykson, C., Zuber, J.-B. 1980: Quantum Field Theory, McGraw Hill
[27] Jauch, J. M., Röhrlich, F., 1976: The theory of photons and electrons, 2nd
edition, Springer.
[28] Kibble, T. W. B., 1968: Coherent soft-photon states and infrared diver-
gences I. Classical currents, J. Math. Phys. 9, 315-324.
[29] Klaus, M. 1980: Nonregularity of the Coulomb potential in quantum elec-
trodynamics Helv. Phys. Acta 53 36–39
[30] Klaus, M. and Scharf, G. 1977: The regular external field problem in quan-
tum electrodynamics, Helv Phys. Acta, 50, 779–802
[31] Labzowski L.N. Klimchitskaya G. Dmitriev Y.: Relativistic Effects in
Spactra of Atomic Systems. IOP Publ. Bristol 1993
[32] Langmann, E. 2001: Generalized YangMills actions from Dirac operator
determinants, J. Math. Phys. 42, 5238
[33] Langmann, E, 1994: Fermion current algebras and Schwinger terms in
(3+1)-dimensions, Commun. Math. Phys. 162, 1-32
[34] Langmann, E., Semenoff, G.W. 1994: QCD1+1 with massless quarks and
gauge covariant Sugawara construction, Phys. Lett. B, 341, 195204
[35] Langmann, E., Mickelsson, J. 1996: Scattering matrix in external field
problems, J. Math. Phys. 37, 3933
[36] Lundberg, L.-E. 1973: Spectral and scattering theory for the Klein-Gordon
equation, Comm. Math. Phys. 31 243–257
[37] Lundberg, L.-E. 1973: Relativistic quantum theory for charged spinless
particles in external vector fields, Comm. Math Phys. 31 295–316
[38] Meissner, K.A. 2002: Classical field theory (in Polish), Wydawnictwo
Naukowe PWN
[39] Molinari, G.L. 2007: Another proof of Gell-Mann and Lows theorem, J.
Math. Phys. 48, 052113
[40] Nenciu, G. and Scharf, G., 1978: On regular external fields in quantum
electrodynamics Helv. Phys. Acta 51, 412–424
[41] Reed, M., Simon, B., 1980: Methods of Modern Mathematical Physics, I.
Functional Analysis, London, Academic Press.
[42] Reed, M., Simon, B., 1975: Methods of Modern Mathematical Physics, II.
Fourier Analysis, Self-Adjointness, London, Academic Press.
[43] Ruijsenaars, S. N. M. 1977: On Bogoliubov transformations for systems of
relativistic charged particles, J. Mathematical Phys. 18, 517–526

217
[44] Scharf, G. 1995: Finite Quantum Electrodynamics. The Causal Approach,
Springer
[45] Schroer, B., Seiler, R., Swieca, J. A. 1970: Problems of stability for quan-
tum fields in external time-dependent potentials Phys. Rev. D 2, 927–2937

[46] Schweber, S. S., 1962: Introduction to non-relativistic quantum field theory,

Harper and Row.
[47] Seiler, R. 1972: Quantum theory of particles with spin zero and one half
in external fields, Comm. Math. Phys. Communications in Mathematical
Physics, 25

[48] Shale, D., 1962: Linear symmetries of free boson fields, Trans. Amer. Math.
Soc. 103, 149-167.
[49] Shale, D. and Stinespring, W.F., 1964: States on the Clifford algebra, Ann.
Math. 80, 365-381.

[50] Srednicki, M., 2007: Quantum Field Theory, Cambridge University Press.
[51] Sternberg, S., 1994: Group Theory and Physics, Cambridge University
Press
[52] Streater, R.F, Wightman, A.S., 1964: PCT, spin and statistics and all that,
W.A.Benjamin, New York-Amsterdam.

[53] Thirring, W., Narnhofer, H., 1992: Covariant QED without indefinite met-
ric, Rev. Math. Phys. SI1 197-211
[54] Wald, R.M., 1994: Quantum field theory on curved space-time and black
hole dynamics, University of Chicago Press, Chicago.

[55] Weinberg, S., 1995: The quantum theory of fields, vol. I, Foundations,
Cambridge.
[56] Wightman, A.: oral communication (2000)
[57] Yennie, D., Frautschi, S. and Suura, H. 1961: The infrared divergence
phenomena and high-energy processes Ann. Phys., 13 379-452.

218