Click here to download this book from filefactory.com its free 4 all
lectures on quantum electrodynamics given by Freeman Dyson at Cornell
University in 1951. Two generations ago, graduate students (and their professors)
wishing to learn the new techniques of QED passed around copies of
Dyson's Cornell lecture notes, then the best and fullest treatment available.
Textbooks appeared a few years later, e.g. by Jauch & Rohrlich [25] and
Schweber [6], but interest in Dyson's notes has never fallen to zero. Here is
what the noted theorist E. T. Jaynes wrote in an unpublished article [26] on
Dyson's autobiographical Disturbing the Universe, 1984:
But Dyson's 1951 Cornell course notes on Quantum Electrodynamics
were the original basis of the teaching I have done since.
For a generation of physicists they were the happy medium:
clearer and better motivated than Feynman, and getting to the
point faster than Schwinger. All the textbooks that have appeared
since have not made them obsolete. Of course, this is
to be expected since Dyson is probably, to this day, best known
among the physicists as the man who rst explained the unity of
the Schwinger and Feynman approaches.
As a graduate student in Nicholas Kemmer's department of theoretical
physics (Edinburgh, Scotland) I had heard vaguely about Dyson's lectures
(either from Kemmer or from my advisor, Peter Higgs) and had read his
classic papers [27], [28] in Schwinger's collection [4]. It never occurred to
me to ask Kemmer for a copy of Dyson's lectures which he almost certainly
had.My interest in the legendary notes was revived thirty years later by the
Kaiser and Schweber books. Within a few minutes Google led to scans of
the notes [29] at the Dibner Archive (History of Recent Science & Technology)
at MIT, maintained by Karl Hall, a historian at the Central European
University in Budapest, Hungary. He had gotten permission from Dyson
to post scanned images of the Cornell notes. Through the e orts of Hall,
Schweber and Babak Ashra these were uploaded to the Dibner Archive. To
obtain a paper copy would require downloading almost two hundred images,
expensive in time and storage. Was there a text version? Had anyone retyped
the notes? Hall did not know, nor did further searching turn anything
up. I volunteered to do the job. Hall thought this a worthwhile project, as
did Dyson, who sent me a copy of the second edition, edited by Michael J.
Moravcsik. (This copy had originally belonged to Sam Schweber.) Dyson
suggested that the second edition be retyped, not the rst. Nearly all of
the di erences between the two editions are Moravcsik's glosses on many
calculations; there is essentially no di erence in text, and (modulo typos) all
the labeled equations are identical.
Between this typed version and Moravcsik's second edition there are few
di erences; all are described in the added notes. (I have also added references
and an index.) About half are corrections of typographical errors. Missing
words or sentences have been restored by comparison with the rst edition;
very infrequently a word or phrase has been deleted. A few changes have
been made in notation. Intermediate steps in two calculations have been
corrected but change nothing. Some notes point to articles or books. No
doubt new errors have been introduced. Corrections will be welcomed! The
young physicists will want familiar terms and notation, occasionally changed
from 1951; the historians want no alterations. It was not easy to nd the
middle ground.
I scarcely knew LATEX before beginning this project. My friend (and
Princeton '74 classmate) Robert Jantzen was enormously helpful, very generous
with his time and his extensive knowledge of LATEX. Thanks, Bob.
Thanks, too, to Richard Koch, Gerben Wierda and their colleagues, who
have made LATEX so easy on a Macintosh. George Gratzer's textbook Math
into LATEX was never far from the keyboard. No one who types technical
material should be ignorant of LATEX.
This project would never have been undertaken without the approval
of Prof. Dyson and the e orts of Profs. Hall, Schweber and Ashra, who
made the notes accessible. I thank Prof. Hall for his steady encouragement through the many hours of typing. I thank Prof. Dyson both for friendly
assistance and for allowing his wonderful lectures to become easier to obtain,
to be read with pleasure and with prot for many years to come.
Originally, the typed version was meant to serve as an adjunct to Karl
Hall's scanned images at the Dibner site. Bob Jantzen, a relativist active
in research, insisted that it also go up at the electronic physics preprint site
arXiv.org, and after a substantial amount of work by him, this was arranged.
A few weeks later the alert and hardworking team at World Scientic1 got
in touch with Prof. Dyson, to ask if he would allow them to publish his
notes. He was agreeable, but told them to talk to me. I was delighted,
but did not see how I could in good conscience prot from Prof. Dyson's
work, and suggested that my share be donated to the New Orleans Public
Library, now struggling to reopen after the disaster of Hurricane Katrina.
Prof. Dyson agreed at once to this proposal. I am very grateful to him for
his contribution to the restoration of my home town.
David Derbes
Laboratory Schools
University of Chicago
loki@uchicago.edu
11 July 2006
Contents
1 Introduction 1
1.1 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Subject Matter . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Detailed Program . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 One-Particle Theories . . . . . . . . . . . . . . . . . . . . . . 3
2 The Dirac Theory 5
2.1 The Form of the Dirac Equation . . . . . . . . . . . . . . . . 5
2.2 Lorentz Invariance of the Dirac Equation . . . . . . . . . . . 7
2.3 To Find the S . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 The Covariant Notation . . . . . . . . . . . . . . . . . . . . . 11
2.5 Conservation Laws. Existence of Spin . . . . . . . . . . . . . 12
2.6 Elementary Solutions . . . . . . . . . . . . . . . . . . . . . . . 13
2.7 The Hole Theory . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.8 Positron States . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.9 Electromagnetic Properties of the Electron . . . . . . . . . . 16
2.10 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . 18
2.11 Solution of Radial Equation . . . . . . . . . . . . . . . . . . . 20
2.12 Behaviour of an Electron in a Non-Relativistic
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.13 Summary of Matrices in the Dirac Theory in
Our Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.14 Summary of Matrices in the Dirac Theory in the
Feynman Notation . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Scattering Problems and Born Approximation 31
3.1 General Discussion . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Projection Operators . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Calculation of Traces . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Scattering of Two Electrons in Born Approximation.
The M ller Formula . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 Relation of Cross-sections to Transition Amplitudes . . . . . 41
3.6 Results for M ller Scattering . . . . . . . . . . . . . . . . . . 43
3.7 Note on the Treatment of Exchange E ects . . . . . . . . . . 44
3.8 Relativistic Treatment of Several Particles . . . . . . . . . . . 45
4 Field Theory 47
4.1 Classical Relativistic Field Theory . . . . . . . . . . . . . . . 47
4.2 Quantum Relativistic Field Theory . . . . . . . . . . . . . . . 51
4.3 The Feynman Method of Quantization . . . . . . . . . . . . . 52
4.4 The Schwinger Action Principle . . . . . . . . . . . . . . . . . 53
4.4.1 The Field Equations . . . . . . . . . . . . . . . . . . . 55
4.4.2 The Schrodinger Equation for the State-function . . . 55
4.4.3 Operator Form of the Schwinger Principle . . . . . . . 56
4.4.4 The Canonical Commutation Laws . . . . . . . . . . . 57
4.4.5 The Heisenberg Equation of Motion
for the Operators . . . . . . . . . . . . . . . . . . . . . 58
4.4.6 General Covariant Commutation Laws . . . . . . . . . 58
4.4.7 Anticommuting Fields . . . . . . . . . . . . . . . . . . 59
5 Examples of Quantized Field Theories 61
5.1 The Maxwell Field . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1.1 Momentum Representations . . . . . . . . . . . . . . . 63
5.1.2 Fourier Analysis of Operators . . . . . . . . . . . . . . 65
5.1.3 Emission and Absorption Operators . . . . . . . . . . 65
5.1.4 Gauge-Invariance of the Theory . . . . . . . . . . . . . 67
5.1.5 The Vacuum State . . . . . . . . . . . . . . . . . . . . 68
5.1.6 The Gupta-Bleuler Method . . . . . . . . . . . . . . . 70
5.1.7 Example: Spontaneous Emission of Radiation . . . . . 71
5.1.8 The Hamiltonian Operator . . . . . . . . . . . . . . . 74
5.1.9 Fluctuations of the Fields . . . . . . . . . . . . . . . . 75
5.1.10 Fluctuation of Position of an Electron in a Quantized
Electromagnetic Field. The Lamb Shift . . . . . . . . 77
5.2 Theory of Line Shift and Line Width . . . . . . . . . . . . . . 79
5.2.1 The Interaction Representation . . . . . . . . . . . . . 80
5.2.2 The Application of the Interaction Representation to
the Theory of Line-Shift and Line-Width . . . . . . . 82
5.2.3 Calculation of Line-Shift, Non-Relativistic Theory . . 87
5.2.4 The Idea of Mass Renormalization . . . . . . . . . . . 88
5.3 Field Theory of the Dirac Electron, Without Interaction . . . 91
5.3.1 Covariant Commutation Rules . . . . . . . . . . . . . 92
5.3.2 Momentum Representations . . . . . . . . . . . . . . . 94
5.3.3 Fourier Analysis of Operators . . . . . . . . . . . . . . 94
5.3.4 Emission and Absorption Operators . . . . . . . . . . 95
5.3.5 Charge-Symmetrical Representation . . . . . . . . . . 96
5.3.6 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . 97
5.3.7 Failure of Theory with Commuting Fields . . . . . . . 98
5.3.8 The Exclusion Principle . . . . . . . . . . . . . . . . . 98
5.3.9 The Vacuum State . . . . . . . . . . . . . . . . . . . . 99
5.4 Field Theory of Dirac Electron in External Field . . . . . . . 100
5.4.1 Covariant Commutation Rules . . . . . . . . . . . . . 101
5.4.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . 104
5.4.3 Antisymmetry of the States . . . . . . . . . . . . . . . 105
5.4.4 Polarization of the Vacuum . . . . . . . . . . . . . . . 106
5.4.5 Calculation of Momentum Integrals . . . . . . . . . . 111
5.4.6 Physical Meaning of the Vacuum Polarization . . . . . 115
5.4.7 Vacuum Polarization for Slowly Varying
Weak Fields. The Uehling E ect . . . . . . . . . . . . 119
5.5 Field Theory of Dirac and Maxwell Fields
in Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.5.1 The Complete Relativistic Quantum
Electrodynamics . . . . . . . . . . . . . . . . . . . . . 120
5.5.2 Free Interaction Representation . . . . . . . . . . . . . 122
6 Free Particle Scattering Problems 125
6.1 M ller Scattering of Two Electrons . . . . . . . . . . . . . . . 126
6.1.1 Properties of the DF Function . . . . . . . . . . . . . 128
6.1.2 The M ller Formula, Conclusion . . . . . . . . . . . . 129
6.1.3 Electron-Positron Scattering . . . . . . . . . . . . . . 130
6.2 Scattering of a Photon by an Electron. The Compton E ect.
Klein-Nishina Formula . . . . . . . . . . . . . . . . . . . . . . 130
6.2.1 Calculation of the Cross-Section . . . . . . . . . . . . 133
6.2.2 Sum Over Spins . . . . . . . . . . . . . . . . . . . . . 134
6.3 Two Quantum Pair Annihilation . . . . . . . . . . . . . . . . 139
6.4 Bremsstrahlung and Pair Creation in the Coulomb Field of
an Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7 General Theory of Free Particle Scattering 145
7.1 The Reduction of an Operator to Normal Form . . . . . . . . 148
7.2 Feynman Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.3 Feynman Rules of Calculation . . . . . . . . . . . . . . . . . . 155
7.4 The Self-Energy of the Electron . . . . . . . . . . . . . . . . . 158
7.5 Second-Order Radiative Corrections to Scattering . . . . . . . 162
7.6 The Treatment of Low-Frequency Photons. The Infra-Red
Catastrophe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8 Scattering by a Static Potential. Comparison with
Experimental Results 183
8.1 The Magnetic Moment of the Electron . . . . . . . . . . . . . 189
8.2 Relativistic Calculation of the Lamb Shift . . . . . . . . . . . 191
8.2.1 Covariant Part of the Calculation . . . . . . . . . . . . 193
Covariant Part of the Calculation . . . . . . . . . . . . . . . . 193
8.2.2 Discussion and the Nature of the -Representation . . 196
8.2.3 Concluding Non-Covariant Part of the Calculation . . 198
8.2.4 Accuracy of the Lamb Shift Calculation . . . . . . . . 202
Notes 205
References 210
Index 215
ISBN-13 978-981-270-622-5
ISBN-10 981-270-622-4
ISBN-13 978-981-270-661-4 (pbk)
ISBN-10 981-270-661-5 (pbk)
Preface
Both Kaiser's admirable Drawing Theories Apart [8] and Schweber's masterfulQED and the Men Who Made It [7] refer frequently to the famouslectures on quantum electrodynamics given by Freeman Dyson at Cornell
University in 1951. Two generations ago, graduate students (and their professors)
wishing to learn the new techniques of QED passed around copies of
Dyson's Cornell lecture notes, then the best and fullest treatment available.
Textbooks appeared a few years later, e.g. by Jauch & Rohrlich [25] and
Schweber [6], but interest in Dyson's notes has never fallen to zero. Here is
what the noted theorist E. T. Jaynes wrote in an unpublished article [26] on
Dyson's autobiographical Disturbing the Universe, 1984:
But Dyson's 1951 Cornell course notes on Quantum Electrodynamics
were the original basis of the teaching I have done since.
For a generation of physicists they were the happy medium:
clearer and better motivated than Feynman, and getting to the
point faster than Schwinger. All the textbooks that have appeared
since have not made them obsolete. Of course, this is
to be expected since Dyson is probably, to this day, best known
among the physicists as the man who rst explained the unity of
the Schwinger and Feynman approaches.
As a graduate student in Nicholas Kemmer's department of theoretical
physics (Edinburgh, Scotland) I had heard vaguely about Dyson's lectures
(either from Kemmer or from my advisor, Peter Higgs) and had read his
classic papers [27], [28] in Schwinger's collection [4]. It never occurred to
me to ask Kemmer for a copy of Dyson's lectures which he almost certainly
had.My interest in the legendary notes was revived thirty years later by the
Kaiser and Schweber books. Within a few minutes Google led to scans of
the notes [29] at the Dibner Archive (History of Recent Science & Technology)
at MIT, maintained by Karl Hall, a historian at the Central European
University in Budapest, Hungary. He had gotten permission from Dyson
to post scanned images of the Cornell notes. Through the e orts of Hall,
Schweber and Babak Ashra these were uploaded to the Dibner Archive. To
obtain a paper copy would require downloading almost two hundred images,
expensive in time and storage. Was there a text version? Had anyone retyped
the notes? Hall did not know, nor did further searching turn anything
up. I volunteered to do the job. Hall thought this a worthwhile project, as
did Dyson, who sent me a copy of the second edition, edited by Michael J.
Moravcsik. (This copy had originally belonged to Sam Schweber.) Dyson
suggested that the second edition be retyped, not the rst. Nearly all of
the di erences between the two editions are Moravcsik's glosses on many
calculations; there is essentially no di erence in text, and (modulo typos) all
the labeled equations are identical.
Between this typed version and Moravcsik's second edition there are few
di erences; all are described in the added notes. (I have also added references
and an index.) About half are corrections of typographical errors. Missing
words or sentences have been restored by comparison with the rst edition;
very infrequently a word or phrase has been deleted. A few changes have
been made in notation. Intermediate steps in two calculations have been
corrected but change nothing. Some notes point to articles or books. No
doubt new errors have been introduced. Corrections will be welcomed! The
young physicists will want familiar terms and notation, occasionally changed
from 1951; the historians want no alterations. It was not easy to nd the
middle ground.
I scarcely knew LATEX before beginning this project. My friend (and
Princeton '74 classmate) Robert Jantzen was enormously helpful, very generous
with his time and his extensive knowledge of LATEX. Thanks, Bob.
Thanks, too, to Richard Koch, Gerben Wierda and their colleagues, who
have made LATEX so easy on a Macintosh. George Gratzer's textbook Math
into LATEX was never far from the keyboard. No one who types technical
material should be ignorant of LATEX.
This project would never have been undertaken without the approval
of Prof. Dyson and the e orts of Profs. Hall, Schweber and Ashra, who
made the notes accessible. I thank Prof. Hall for his steady encouragement through the many hours of typing. I thank Prof. Dyson both for friendly
assistance and for allowing his wonderful lectures to become easier to obtain,
to be read with pleasure and with prot for many years to come.
Originally, the typed version was meant to serve as an adjunct to Karl
Hall's scanned images at the Dibner site. Bob Jantzen, a relativist active
in research, insisted that it also go up at the electronic physics preprint site
arXiv.org, and after a substantial amount of work by him, this was arranged.
A few weeks later the alert and hardworking team at World Scientic1 got
in touch with Prof. Dyson, to ask if he would allow them to publish his
notes. He was agreeable, but told them to talk to me. I was delighted,
but did not see how I could in good conscience prot from Prof. Dyson's
work, and suggested that my share be donated to the New Orleans Public
Library, now struggling to reopen after the disaster of Hurricane Katrina.
Prof. Dyson agreed at once to this proposal. I am very grateful to him for
his contribution to the restoration of my home town.
David Derbes
Laboratory Schools
University of Chicago
loki@uchicago.edu
11 July 2006
Contents
1 Introduction 1
1.1 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Subject Matter . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Detailed Program . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 One-Particle Theories . . . . . . . . . . . . . . . . . . . . . . 3
2 The Dirac Theory 5
2.1 The Form of the Dirac Equation . . . . . . . . . . . . . . . . 5
2.2 Lorentz Invariance of the Dirac Equation . . . . . . . . . . . 7
2.3 To Find the S . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 The Covariant Notation . . . . . . . . . . . . . . . . . . . . . 11
2.5 Conservation Laws. Existence of Spin . . . . . . . . . . . . . 12
2.6 Elementary Solutions . . . . . . . . . . . . . . . . . . . . . . . 13
2.7 The Hole Theory . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.8 Positron States . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.9 Electromagnetic Properties of the Electron . . . . . . . . . . 16
2.10 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . 18
2.11 Solution of Radial Equation . . . . . . . . . . . . . . . . . . . 20
2.12 Behaviour of an Electron in a Non-Relativistic
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.13 Summary of Matrices in the Dirac Theory in
Our Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.14 Summary of Matrices in the Dirac Theory in the
Feynman Notation . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Scattering Problems and Born Approximation 31
3.1 General Discussion . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Projection Operators . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Calculation of Traces . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Scattering of Two Electrons in Born Approximation.
The M ller Formula . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 Relation of Cross-sections to Transition Amplitudes . . . . . 41
3.6 Results for M ller Scattering . . . . . . . . . . . . . . . . . . 43
3.7 Note on the Treatment of Exchange E ects . . . . . . . . . . 44
3.8 Relativistic Treatment of Several Particles . . . . . . . . . . . 45
4 Field Theory 47
4.1 Classical Relativistic Field Theory . . . . . . . . . . . . . . . 47
4.2 Quantum Relativistic Field Theory . . . . . . . . . . . . . . . 51
4.3 The Feynman Method of Quantization . . . . . . . . . . . . . 52
4.4 The Schwinger Action Principle . . . . . . . . . . . . . . . . . 53
4.4.1 The Field Equations . . . . . . . . . . . . . . . . . . . 55
4.4.2 The Schrodinger Equation for the State-function . . . 55
4.4.3 Operator Form of the Schwinger Principle . . . . . . . 56
4.4.4 The Canonical Commutation Laws . . . . . . . . . . . 57
4.4.5 The Heisenberg Equation of Motion
for the Operators . . . . . . . . . . . . . . . . . . . . . 58
4.4.6 General Covariant Commutation Laws . . . . . . . . . 58
4.4.7 Anticommuting Fields . . . . . . . . . . . . . . . . . . 59
5 Examples of Quantized Field Theories 61
5.1 The Maxwell Field . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1.1 Momentum Representations . . . . . . . . . . . . . . . 63
5.1.2 Fourier Analysis of Operators . . . . . . . . . . . . . . 65
5.1.3 Emission and Absorption Operators . . . . . . . . . . 65
5.1.4 Gauge-Invariance of the Theory . . . . . . . . . . . . . 67
5.1.5 The Vacuum State . . . . . . . . . . . . . . . . . . . . 68
5.1.6 The Gupta-Bleuler Method . . . . . . . . . . . . . . . 70
5.1.7 Example: Spontaneous Emission of Radiation . . . . . 71
5.1.8 The Hamiltonian Operator . . . . . . . . . . . . . . . 74
5.1.9 Fluctuations of the Fields . . . . . . . . . . . . . . . . 75
5.1.10 Fluctuation of Position of an Electron in a Quantized
Electromagnetic Field. The Lamb Shift . . . . . . . . 77
5.2 Theory of Line Shift and Line Width . . . . . . . . . . . . . . 79
5.2.1 The Interaction Representation . . . . . . . . . . . . . 80
5.2.2 The Application of the Interaction Representation to
the Theory of Line-Shift and Line-Width . . . . . . . 82
5.2.3 Calculation of Line-Shift, Non-Relativistic Theory . . 87
5.2.4 The Idea of Mass Renormalization . . . . . . . . . . . 88
5.3 Field Theory of the Dirac Electron, Without Interaction . . . 91
5.3.1 Covariant Commutation Rules . . . . . . . . . . . . . 92
5.3.2 Momentum Representations . . . . . . . . . . . . . . . 94
5.3.3 Fourier Analysis of Operators . . . . . . . . . . . . . . 94
5.3.4 Emission and Absorption Operators . . . . . . . . . . 95
5.3.5 Charge-Symmetrical Representation . . . . . . . . . . 96
5.3.6 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . 97
5.3.7 Failure of Theory with Commuting Fields . . . . . . . 98
5.3.8 The Exclusion Principle . . . . . . . . . . . . . . . . . 98
5.3.9 The Vacuum State . . . . . . . . . . . . . . . . . . . . 99
5.4 Field Theory of Dirac Electron in External Field . . . . . . . 100
5.4.1 Covariant Commutation Rules . . . . . . . . . . . . . 101
5.4.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . 104
5.4.3 Antisymmetry of the States . . . . . . . . . . . . . . . 105
5.4.4 Polarization of the Vacuum . . . . . . . . . . . . . . . 106
5.4.5 Calculation of Momentum Integrals . . . . . . . . . . 111
5.4.6 Physical Meaning of the Vacuum Polarization . . . . . 115
5.4.7 Vacuum Polarization for Slowly Varying
Weak Fields. The Uehling E ect . . . . . . . . . . . . 119
5.5 Field Theory of Dirac and Maxwell Fields
in Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.5.1 The Complete Relativistic Quantum
Electrodynamics . . . . . . . . . . . . . . . . . . . . . 120
5.5.2 Free Interaction Representation . . . . . . . . . . . . . 122
6 Free Particle Scattering Problems 125
6.1 M ller Scattering of Two Electrons . . . . . . . . . . . . . . . 126
6.1.1 Properties of the DF Function . . . . . . . . . . . . . 128
6.1.2 The M ller Formula, Conclusion . . . . . . . . . . . . 129
6.1.3 Electron-Positron Scattering . . . . . . . . . . . . . . 130
6.2 Scattering of a Photon by an Electron. The Compton E ect.
Klein-Nishina Formula . . . . . . . . . . . . . . . . . . . . . . 130
6.2.1 Calculation of the Cross-Section . . . . . . . . . . . . 133
6.2.2 Sum Over Spins . . . . . . . . . . . . . . . . . . . . . 134
6.3 Two Quantum Pair Annihilation . . . . . . . . . . . . . . . . 139
6.4 Bremsstrahlung and Pair Creation in the Coulomb Field of
an Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7 General Theory of Free Particle Scattering 145
7.1 The Reduction of an Operator to Normal Form . . . . . . . . 148
7.2 Feynman Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.3 Feynman Rules of Calculation . . . . . . . . . . . . . . . . . . 155
7.4 The Self-Energy of the Electron . . . . . . . . . . . . . . . . . 158
7.5 Second-Order Radiative Corrections to Scattering . . . . . . . 162
7.6 The Treatment of Low-Frequency Photons. The Infra-Red
Catastrophe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8 Scattering by a Static Potential. Comparison with
Experimental Results 183
8.1 The Magnetic Moment of the Electron . . . . . . . . . . . . . 189
8.2 Relativistic Calculation of the Lamb Shift . . . . . . . . . . . 191
8.2.1 Covariant Part of the Calculation . . . . . . . . . . . . 193
Covariant Part of the Calculation . . . . . . . . . . . . . . . . 193
8.2.2 Discussion and the Nature of the -Representation . . 196
8.2.3 Concluding Non-Covariant Part of the Calculation . . 198
8.2.4 Accuracy of the Lamb Shift Calculation . . . . . . . . 202
Notes 205
References 210
Index 215
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