Elementary Particle Theory Volume 2 Quantum Electrodynamics 1st Edition Eugene Stefanovich Elementary Particle Theory Volume 2 Quantum Electrodynamics 1st Edition Eugene Stefanovich Elementary Particle Theory Volume 2 Quantum Electrodynamics 1st Edition Eugene Stefanovich

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Electrodynamics 1st Edition Eugene Stefanovich Digital
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7. Eugene Stefanovich
Elementary Particle Theory

8. De Gruyter Studies in
Mathematical Physics
|
Edited by
Michael Efroimsky, Bethesda, Maryland, USA
Leonard Gamberg, Reading, Pennsylvania, USA
Dmitry Gitman, São Paulo, Brazil
Alexander Lazarian, Madison, Wisconsin, USA
Boris Smirnov, Moscow, Russia
Volume 46

9. Eugene Stefanovich
Elementary
Particle Theory
|
Volume 2: Quantum Electrodynamics

10. Mathematics Subject Classification 2010
Primary: 81-02, 81V10, 81T15; Secondary: 47A40, 81T18
Author
Dr Eugene Stefanovich
San Jose, California
USA
eugene_stefanovich@usa.net
ISBN 978-3-11-049089-3
e-ISBN (PDF) 978-3-11-049320-7
e-ISBN (EPUB) 978-3-11-049143-2
ISSN 2194-3532
Library of Congress Control Number: 2018016481
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available on the Internet at http://dnb.dnb.de.
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Typesetting: VTeX UAB, Lithuania
Printing and binding: CPI books GmbH, Leck
www.degruyter.com

11. Contents
List of figures | IX
List of tables | XI
Postulates, statements, theorems | XIII
Conventional notation | XV
Preface | XVII
1 Fock space | 1
1.1 Creation and annihilation operators | 1
1.1.1 Sectors with fixed numbers of particles | 1
1.1.2 Particle observables in Fock space | 3
1.1.3 Noninteracting representation of Poincaré group | 3
1.1.4 Creation and annihilation operators for fermions | 4
1.1.5 Anticommutators of particle operators | 6
1.1.6 Creation and annihilation operators for photons | 7
1.1.7 Particle number operators | 7
1.1.8 Continuous spectrum of momentum | 8
1.1.9 Normal ordering | 9
1.1.10 Noninteracting energy and momentum | 11
1.1.11 Noninteracting angular momentum and boost | 12
1.1.12 Poincaré transformations of particle operators | 13
1.2 Interaction potentials | 15
1.2.1 Conservation laws | 15
1.2.2 General form of interaction operators | 17
1.2.3 Five types of regular potentials | 20
1.2.4 Products and commutators of regular potentials | 23
1.2.5 More about t-integrals | 25
1.2.6 Solution of one commutator equation | 27
1.2.7 Two-particle potentials | 28
1.2.8 Momentum-dependent potentials | 31
2 Scattering in Fock space | 33
2.1 Toy model theory | 33
2.1.1 Fock space and Hamiltonian | 33
2.1.2 S-operator in second order | 35
2.1.3 Drawing diagrams in toy model | 36

12. VI | Contents
2.1.4 Reading diagrams in toy model | 39
2.1.5 Scattering in second order | 40
2.2 Renormalization in toy model | 41
2.2.1 Renormalization of electron self-scattering in second order | 41
2.2.2 Renormalization of electron self-scattering in fourth order | 43
2.3 Diagrams in general theory | 46
2.3.1 Products of diagrams | 46
2.3.2 Connected and disconnected diagrams | 47
2.3.3 Divergence of loop integrals | 50
2.4 Cluster separability | 52
2.4.1 Cluster separability of interaction | 52
2.4.2 Cluster separability of S-operator | 54
3 Quantum electrodynamics | 57
3.1 Interaction in QED | 57
3.1.1 Why do we need quantum fields? | 58
3.1.2 Simple quantum field theories | 58
3.1.3 Interaction operators in QED | 60
3.2 S-operator in QED | 62
3.2.1 S-operator in second order | 62
3.2.2 Covariant form of S-operator | 66
3.2.3 Feynman gauge | 68
3.2.4 Feynman diagrams | 70
3.2.5 Compton scattering | 72
3.2.6 Virtual particles? | 73
4 Renormalization | 75
4.1 Two renormalization conditions | 75
4.1.1 No self-scattering condition | 75
4.1.2 Charge renormalization | 78
4.1.3 Renormalization by counterterms | 78
4.1.4 Diagrams of electron–proton scattering | 79
4.1.5 Regularization | 80
4.2 Counterterms | 81
4.2.1 Electron’s self-scattering | 81
4.2.2 Electron self-scattering counterterm | 83
4.2.3 Fitting coefficient (δm)2 | 84
4.2.4 Fitting coefficient (Z2 − 1)2 | 85
4.2.5 Photon’s self-scattering | 86
4.2.6 Photon self-energy counterterm | 87
4.2.7 Applying charge renormalization condition | 89
4.2.8 Vertex renormalization | 90

13. Contents | VII
4.3 Renormalized S-matrix | 93
4.3.1 “Vacuum polarization” diagrams | 93
4.3.2 Vertex diagram | 93
4.3.3 Ladder diagram | 95
4.3.4 Cross ladder diagram | 98
4.3.5 Renormalizability | 101
A Useful integrals | 103
B Quantum fields of fermions | 107
B.1 Pauli matrices | 107
B.2 Dirac gamma matrices | 108
B.3 Dirac representation of Lorentz group | 109
B.4 Construction of Dirac field | 112
B.5 Properties of functions u and v | 114
B.6 Explicit formulas for u and v | 115
B.7 Useful notation | 118
B.8 Poincaré transformations of fields | 119
B.9 Approximation (v/c)2
| 120
B.10 Anticommutation relations | 122
B.11 Dirac equation | 123
B.12 Fermion propagator | 125
C Quantum field of photons | 129
C.1 Construction of photon quantum field | 129
C.2 Properties of function eμ(p, τ) | 130
C.3 Useful commutator | 131
C.4 Commutator of photon fields | 133
C.5 Photon propagator | 133
C.6 Poincaré transformations of photon field | 135
D QED interaction in terms of particle operators | 139
D.1 Current density | 139
D.2 First-order interaction in QED | 142
D.3 Second-order interaction in QED | 142
E Relativistic invariance of QFT | 155
E.1 Relativistic invariance of simple QFT | 155
E.2 Relativistic invariance of QED | 156
F Loop integrals in QED | 163
F.1 Schwinger–Feynman integration trick | 163

14. VIII | Contents
F.2 Some basic four-dimensional integrals | 164
F.3 Electron self-energy integral | 167
F.4 Vertex integral | 170
F.4.1 Calculation of M | 172
F.4.2 Calculation of Mσ | 173
F.4.3 Calculation of Mστ | 174
F.4.4 Complete integral | 175
F.5 Integral for ladder diagram | 178
F.5.1 Calculation of LI | 179
F.5.2 Calculation of LII | 181
F.5.3 Calculation of LIII | 182
F.5.4 Complete integral | 184
G Scattering matrix in (v/c)2
approximation | 185
G.1 Second perturbation order | 185
G.2 Vertex contribution in fourth order | 187
H Checks of physical dimensions | 191
Bibliography | 193
Index | 195

15. List of figures
Figure 1.1 Operators in “index space” (page 20)
Figure 2.1 Diagrams for operators V1 and V1(t) (page 37)
Figure 2.2 Normal ordering of the product of two diagrams (page 38)
Figure 2.3 Renorm diagrams in Vc
Vc
Vc
Vc
(page 44)
Figure 2.4 Diagram of the counterterm Q2 (page 44)
Figure 2.5 Renorm diagrams in Vc
Vc
+ Vc
Vc
Vc
(page 45)
Figure 2.6 To the proof of Lemma 2.3 (page 49)
Figure 2.7 Generic diagram in a hypothetical theory (page 49)
Figure 3.1 Second-order diagram for e−
+ p+
scattering (page 72)
Figure 3.2 e−
+ γ scattering diagrams (page 73)
Figure 4.1 e−
+ p+
scattering diagrams up to the fourth order (page 80)
Figure 4.2 Electron self-scattering diagrams (page 81)
Figure 4.3 Photon self-scattering diagrams (page 86)
Figure 4.4 “Vacuum polarization” diagrams (page 89)
Figure 4.5 Vertex diagrams (page 91)
Figure 4.6 Ladder diagram (page 95)
Figure 4.7 Cross-ladder diagram (page 98)
Figure A.1 To the calculation of integral (A.10) (page 104)
Figure F.1 Wick rotation in the integral (F.6) (page 164)
Figure F.2 Integration area in (F.45) (page 179)
https://doi.org/10.1515/9783110493207-201

17. List of tables
Table 1.1 Types of potentials in the Fock space of QED (page 22)
Table 1.2 Commutators, t-derivatives and t-integrals (page 26)
Table 3.1 Components of Feynman diagrams (page 71)
https://doi.org/10.1515/9783110493207-202

19. Postulates, statements, theorems
Theorem 1.1 Wick theorem (page 10)
Lemma 1.2 Commutator of renorm operators (page 20)
Lemma 1.3 Product of regular operators (page 23)
Theorem 1.4 Properties of phys operators (page 23)
Lemma 1.5 Product of phys operators (page 24)
Lemma 1.6 Commutator [phys,renorm] (page 24)
Lemma 1.7 Commutator [unphys,renorm] (page 24)
Lemma 1.8 Commutator [phys,unphys] (page 25)
Lemma 1.9 t-derivative of operator (page 25)
Lemma 2.1 Disconnected part of product (page 47)
Theorem 2.2 Multiple commutator (page 48)
Lemma 2.3 Number of independent loops (page 49)
Theorem 2.4 Convergence of loop integrals (page 51)
Theorem 2.5 Cluster separability of smooth potentials (page 54)
Theorem 2.6 Smoothness of products of potentials (page 54)
Theorem 2.7 Smoothness of commutators of potentials (page 55)
Statement 4.1 Absence of self-scattering (page 77)
Postulate 4.2 Charge renormalization condition (page 78)
https://doi.org/10.1515/9783110493207-203

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