Quantum electrodynamics - V.N.Gribov, J.Nyiri.pdf

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QUANTUM ELECTRODYNAMICS
Gribov Lectures on Theoretical Physics
V. N. Gribov
J. Nyiri
Contents
Particles and their interactions in relativistic quantum
mechanics
1.1 The propagator
1.2 The Green function
1.2.1 The Green function for a system of particles
1.2.2 The momentum representation
1.2.3 Virtual particles
1.3 The scattering amplitude
1.3.1 How to calculate physical observables
1.3.2 Poles in the scattering amplitude and the bound states
1.4 The electromagnetic field
1.5 Photons in an ‘external field’
1.5.1 Relativistic propagator
1.5.2 Relativistic interaction
1.5.3 Relativistic Green function
1.5.4 Propagation of vector photons
1.6 Free massive relativistic particles
1.7 Interactions of spinless particles
1.8 Interaction of spinless particles with the electromagnetic field
1.9 Examples of the simplest electromagnetic processes
1.9.1 Scattering of charged particles
1.9.2 The Compton effect (photon–π-meson scattering)
1.10 Diagrams and amplitudes in momentum representation
1.10.1 Photon emission amplitude in momentum space
1.10.2 Meson–meson scattering via photon exchange
1.10.3 Feynman rules
1.11 Amplitudes of physical processes
1.11.1 The unitarity condition
1.11.2
S-matrix
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Contents
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1.11.3 Invariant scattering amplitude
1.11.4 Cross section
1.11.5 2
→
2 scattering
1.11.6
π
−
π
−
scattering
1.11.7
π
+
π
−
scattering
1.12 The Mandelstam plane
1.13 The Compton effect (for
π-mesons)
2
1
Particles with spin
2
. Basic quantum electrodynamic
processes
85
1
2.1 Free particles with spin
2
85
2.2 The Green function of the electron
98
2.3 Matrix elements of electron scattering amplitudes
100
2.4 Electron–photon interaction
102
2.5 Electron–electron scattering
105
2.5.1 Connection between spin and statistics
106
2.5.2 Electron charge
111
2.6 The Compton effect
112
2.6.1 Compton scattering at small energies
121
2.6.2 Compton scattering at high energies
123
2.7 Electron–positron annihilation into two photons
125
2.7.1 Annihilation near threshold
128
+
e
−
annihilation at very high energies
2.7.2 e
128
2.8 Electron scattering in an external field
130
2.9 Electron bremsstrahlung in an external field
132
2.9.1 Emission of a soft photon by a low energy electron
133
2.9.2 Soft radiation off a high energy electron
135
2.10 The Weizs¨cker–Williams formula
a
137
3
3.1
3.2
3.3
General properties of the scattering amplitude
Symmetries in quantum electrodynamics
3.1.1
P
-conservation
3.1.2
T
-invariance
3.1.3
C-invariance
The
CP T
theorem
3.2.1
P T
-invariant amplitudes
Causality and unitarity
3.3.1 Causality
3.3.2 Analytic properties of the Born amplitudes
3.3.3 Scattering amplitude as an analytic function
3.3.4 Unitarity
3.3.5 Born amplitudes and unitarity
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Contents
3.3.6 How to restore perturbation theory on the basis of uni-
tarity and analyticity, or perturbation theory without
Feynman graphs
4
4.1
Radiative corrections. Renormalization
Higher order corrections to the electron and photon Green
functions
4.1.1 Multiloop contributions to the electron Green function
4.1.2 Multiloop contributions to the photon Green function
Renormalization of the electron mass and wave function
Renormalization of the photon Green function
Feynman rules for multiloop scattering amplitudes
Renormalization of the vertex part
The generalized Ward identity
Radiative corrections to electron scattering
4.7.1 One-loop polarization operator
4.7.2 One-loop vertex part
The Dirac equation in an external field
4.8.1 Electron in the field of a supercharged nucleus
Radiative corrections to the energy levels of hydrogen-like
atoms. The Lamb shift
Difficulties of quantum electrodynamics
Renormalization and divergences
5.1.1 Divergences of Feynman diagrams
5.1.2 Renormalization
The zero charge problem in quantum electrodynamics
References
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4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5
5.1
5.2
Foreword
The idea of this book is to present the theory of quantum electrodynamics
in the shortest and clearest way for applied use. At the same time it
may serve as a general introduction to relativistic quantum field theory
within the approach based on Green functions and the Feynman diagram
technique.
The book is largely based on V. N. Gribov’s lectures given in Leningrad
(St. Petersburg) in the early 1970s. The original lecture notes were col-
lected and prepared by V. Fyodorov in 1974.
We were planning several modifications to the work. In particular,
Gribov intended to include discussion of his new ideas about the structure
of the theory at short distances, the problem he had been working on
during his last few years. His death on 13 August 1997 prevented this,
and I decided to stay as close as possible to the version completed by
early 1997 and already checked by him.
In preparing the book, I got invaluable help from many of our friends
and colleagues. I would like to express my gratitude to those who read,
commented on, and provided suggestions for improving the manuscript,
especially to A. Frenkel. I would also like to thank C. Ewerz and especially
Gy. Kluge for their help in preparing the figures.
I am deeply indebted to I. Khriplovich and, most of all, to Gribov’s
former students, Yu. Dokshitzer, M. Eides and M. Strikman. They per-
formed the enormous work of checking the manuscript by going metic-
ulously through the whole book several times. They compared the text
to their own notes taken at Gribov’s university courses and restored the
Gribov lectures as fully as possible. They found and corrected inconsis-
tencies and errors. It was more than mere scientific editing. Among their
objectives was to preserve in the English text the unique style of Gribov
the lecturer, a style that is remembered by his disciples and colleagues
with admiration.
J. Nyiri
Budapest
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