TRANSLATION OF QUANTUM ELECTRODYNAMICS

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• Publication Date: May 21, 1958
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STAT
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UNCLASSIFIED
UNCLASSIFIED
STAT
AEC-tr-2876 (Pt. I)
UNITED STATES ATOMIC ENERGY COMMISSION
QUANTUM ELECTRODYNAMICS
(Part I)
A. I. Akhiezer and V. B. Berestetsky
Translated by:
CONSULTANTS BUREAU, INC.
Technical Information Service Extension, Oak Ridge, Tenn.
I)
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e7v,
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The set of equations (8.6) can be iolved if
so that the frequency co is given by
Kvantovaya Elekrrodinamika, A. I. Akhiezer and V. B. Berestetsky Moskva,
Gosudarstvennoe Izdatelstvo Tekhniko-Teoreticheskoi Literatury, 1953, 428 pp.
where
Equation (8.9) expresses the well-known relativistic relation between the energy and momentum.
Thus, there exist two kinds of solutions for the Dirac equation corresponding to the two signs in Equation
(8.8). We shall call these positive-and negative-frequency solutions. The general solution to the Dirac equation
can then be written
(8.10)
where
(8.11)
Printed in the U.S.A. Price $ 2.65. Bound in two parts to be sold together.
Available from the Office of Technical Services, Department of Commerce,
Washington 25, D.C.
A solution whose frequency has a definite sign shall also be written 10(i7). where ri = ? = 1.
Of the two spinors that go to make up iP " (k),) namely cp(n)(k) and x (n) (k), one is an arbitrary
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!Nom .1..117.7.4.,Pril?ilert.ontowegali.a...,
4
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1-.
A. I. Akhiezer and V. B. Berestetsky
QUANTUM ELECTRODYNAMICS
State Technico-Theoretical
Literature Press
Moscow, 1953
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ups,ww..16.1161,nanz.
NOTE
This monograph is an orderly development of quantum electro-
dynamics based on the most recent works in this field. It includes
both the elements of the theory and numerous applications to the
calculation of various effects.
The book is intended for readers with sufficient mathematical
background who are familiar with quantum mechanics.
The book will be useful for theoretical physicists and for students
of modern theoretical physics.
1
PREFACE
The concepts of particle and field are fundamental to modern physics. In an earlier phase of physics,
the so-called classical, these two concepts referred to different physical objects; for'instance, the first referred
to the electron, and the second to light.
The further development of physics, however, showed that these concepts reflect different aspects of one
and the same object. The electron has wave properties, and light (the electromagnetic field) manifests those
of a particle, namely the photon. Just as electrons interact by means of the electromagnetic field, so do pho-
tons interact by means of the electron-positron field.
Quantum theory shows the unity of the corpuscular and wave aspects of physical objects. The concepts
of particle and field join in the unified concept of the quantum field which, for example, makes it possible to
describe the processes of production and annihilation of particles within the framework of the existing theory.
At the present time many particles are known, and to these correspond various interacting quantum fields.
Of the many forms of physical interaction existing in nature, however, at the present time only the gravitational
and electromagnetic interactions have been studied in sufficient detail. The theory of the latter interaction is
the subject of quantum electrodynamics, to the systematic exposition of which this book is devoted.
Since electromagnetic interactions are fundamental for the electron and photon, quantum electrodynamics
makes it possible to explain and predict a wide range of phenomena related to the behavior of these particles.
As for the?application of quantum electrodynamics to other particles (nucleons and mesons), it is extremely re-
stricted due to the essential role played by other types of interactions (nuclear or meson interactions) for those
particles. Therefore, meson problems are not treated in this book, and the interaction of nucleons with the
electromagnetic field is treated only in the low-velocity limit.
The formulation of the fundamental equations of quantum electrodynamics, and even the possibility of
separating the interacting fields into the electromagnetic and electron-positron fields, is based on the fact that
the interaction between these fields is a weak one. This situation is expressed by the small magnitude of the
constant a --=-e2 / tic which characterizes the interaction. Thus, the interaction between the fields in quantum
electrodynamics is treated as a small perturbation, and the mathematical method used is perturbation theory
in which all-quantitative results are presented in terms of power series in a.
Since the electromagnetic and electron-positron fields are systems with unlimited degrees of freedom,
the application of perturbation theory gives rise to divergent expressions characteristic of the modern theory,
which are absent only in the first nonvanishing approximation of perturbation theory. The development of
quantum electrodynamics in recent years made it possible to establish principles for regularizing divergent ex-
pressions, so-that it then became possible to calculate higher approximations (the so-called radiative corrections).
This progress is to a great extent due to the new, invariant formulation of perturbation theory. Invariant
perturbation theory made it possible to represent the results in a compact and relativistically invariant form,
which allowed a formulation of the rules for removing singularities. Furthermore, invariant perturbation theory
has significant practical advantages over the earlier methods even for first-order calculations. Therefore, the
whole exposition in this book is constructed on the basis of invariant perturbation theory.
Although it is a completely satisfactory theory in a definite field of physical phenomena, modern quantum
electrodynamics has important drawbacks in that it necessitates the introduction of additional concepts which
are neither contained in the fundamental formulation of the theory nor reflected in its basic equations; these
concepts are necessary in order to remove the divergences which arise in the theory. This state of affairs seems
3
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to arise from profound causes. They lie in the fact that it is sometimes impossible to construct a closed
theory of a limited set of phenomena (in the present case, the pure electromagnetic ones) without accounting
also for a wider class of interactions in nature.
The structure of the present book is the following. The first three chapters are devoted to the theory of
free, i.e., noninteracting particles (electrons and photons). This corresponds to the above-mentioned funda-
mental trait of quantum electrodynamics, which makes it possible to consider the interaction as a perturbation.
In Chapter III the vacuum is defined as the state of the field in which no particles exist. Nevertheless, the
vacuum (as opposed to the metaphysical 'void") has physical properties, and it is necessary to take account
not only of interaction of particles with each other when considering various phenomena, but also their inter-
action with the vacuum (this interaction is considered in Chapter VIII). The interacting field equations are
formulated in Chapter IV, where invariant perturbation theory is developed. Concrete problems reduce to the
calculation of S matrix elements; Chapter V is devoted to a general investigation of this matrix. The, rest of
the exposition is based on the use of the S matrix. Part of Chapter V (Sections 25.-27) is devoted to an analysis
of the divergences in the S matrix and to a description of the methods for removing them. The results of these
sections are dot used in Chapters VI and VII. Therefore, the reader may omit these sections in the first reading,
returning to them before going on to Chapter VIII. Chapters VI and VII consider various concrete phenomena
in the first nonvanishing approximation. The theory of radiative corrections is developed in Chapter VIII and
is based on the methods for removing divergences described in Chapter V. Appendix I describes the general
theory of free fields, important special cases of which are the electron-positron field and the electromagnetic
field. Appendix Ills devoted to the general theory of bound states.
In writing Chapters V and VIII, as well as Sections 32 and-56, we received aid from R. V. Polovin, who
performed many of the calculations. G. Ya. Lyubarsky and L. E. Pargamanik participated in writing Sections
48, 50, and 52 (the latter, Section 52). Appendix II was written by A. D. Galanin. We are sincerely grateful
to all of them.
We express our gratitude to Academician L. D. Landau, Professor I. Ya. Pomeranchuk, and to the members
of the seminars they directed for discussing many problems described in this book.
1\1
PREFACE
TAI3LE OF CONTENTS
CHAPTER I
QUANTUM MECHANICS OF THE PHOTON
1. The Photon Wave Function in Momentum Space
1. Introduction. 2. Wave Function in k-space. 3. Energy.
2. Momentum Eigenstates-
1. Momentum. 2. On the Photon Wave Function in Configuration Space. 3. Plane Waves.
3. Angular Momentum. Spin of the Photon
1. Angular Momentum Operator. 2. Spin Operator. 3. Spin Wave Functions.
4. Angular Momentum and Parity Eigenstates
1. Angular Momentum Eigenfunction. 2. Spherical Vectors. Parity. 3. Expansion in
Spherical Waves. 4. Expressions for the Electric and Magnetic Fields.
5. Potentials
1. Transverse, Longitudinal, and Scalar Potentials. 2. Longitudinal and Scalar Compo-
nents
of the Photon Wave Function. 3. Plane and Spherical Wave. Potentials.
6. The? Two-Photon System... . ? .....
1. Two-Photon Wave Function. 2. Even and Odd States. 3. Classification of States with
a Given Angular Momentum.
CHAPTER II
-RELATIVISTIC QUANTUM MECHANICS OF THE ELECTRON
7. The Dirac Equation
1. Spillers and Pauli Matrices. 2. Dirac Equation. 3. On the Necessity for Four-Compo-
nent Wave Functions. 4. Invariance of the Dirac Equation. 5. The y Matrices. Continuity
Equation. 6. The Transformation Characteristics of Bilinear Combinations of the Wave
Functions.
8 . Electron and Positron States
1. Solutions with Positive and Negative Frequencies. 2. The Wave Function of the Posi-
tron. 3. Positron Parity. 4. Charge-Conjugate Function.
3
11
16
20
28
39
45
53
t
65 ?
9. Momentum and Polarization Eigenstates 74
1. Plane Waves. 2. Polarization States. 3. Sum Over Polarizations. 4. Calculation of
Traces.
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10. Angular Momentum and Parity Eigenstates of the Electron
1. Orbital and Spin Functions. 2. Spherical Spinors. 3. Angular Momentum Elgenfunc-
tion. 4. Parity of a State. 5. Expansion in Spherical Waves.
11. The Electron in an External Field
1. Dirac Equation in an External Field. 2. Separation of Variables in a Central Field.
3. Asymptotic behavior. 4. Level Behavior as a Function of the Potential Well Depth.
12. Electron Motion in the Field of the Nucleus
1. Solution of the Radial Equations for the Coulomb Field. Discrete Spectrum. 2. Wave
Functions of the Continuous Spectrum. 3. Isotope Level Shift. 4. General Investigation
of the Effect of Finite Nuclear Dimensions. 5. On the Existence of Bound States for Large Z.
13. Scattering of Electrons
1. Born Approximation for Scattering in a Coulomb Field. 2. Scattering in a Central Field.
Spinor Scattered Amplitude. 3. Scattering Cross Section In Terms of the Phase Shifts.
4. Azimuthal Asymmetry. 5. Polarization in Scattering. 6. Scattering in a Coulomb
Field.
14. The Nonrelativistic Limit
1. Transition to the Pauli Equation. 2. Second Approximation. 3. On the Applicability
of the Dirac Equation to Nucleons.
CHAPTER III
QUANTIZATION OF THE ELECTROMAGNETIC AND ELECTRON-POSITRON FIELDS
15. Quantization of the Electromagnetic Field
1. Four-Dimensional Form of the Field Equations: Plane Waves. 2. Quantum Conditions.
3. Definition of the Vacuum for the Electromagnetic Field. The Use of the Indefinite
Metric. 4. Wave Function of a System of Photons in Momentum Space and Second Quanti-
zation. 5. Method of Fock Functionals.
16. Commutators of the Electromagnetic Field. The Singular Func
tions D. D(1)
D -
F
1. Quantum conditions for the Potential. 2. Expectation Value of the Operator
f-A (x), A (x')} in the Vacuum State. 3. Chronological Product of the Operators
- v -
A (x) and Av (-x ' ) . The 13- function.
-
17. Quantization of the Electron-Positron Field
1. Variational Principle for the Dirac Equation. Energy-Momentum Tensor. 2. Quantum
Conditions for the Electron-Positron Field.
6
86
96
105
118
130
137
162
173
t.-
18. Ani commutators of the Electron -Positron Field. The Singular Func-
tions (x ), (x ). (x )? (x)
1. Quantum Conditions for the Operators *, *. 2. Definition of the Current. Charge Con-
jugate Operators. 3. Chronological Products of the Field Operators. 4. Oidered Products of
the Field Operators. 5. Representations of the Singular Functions.
181
CHAPTER IV
FUNDAMENTAL EQUATIONS OF QUANTUM ELECTRODYNAMICS
19. Interaction between the Electron-Positron and the Electromagnetic
Fields 200
1. Fundamental Equations of the Interacting Fields. 2. Variational Principle. 3. Charge
Parity. 4. Schroedittger Equation for a System of Fields. "External" Field. "Given" current.
20. Perturbation Theory. Transition from tlie Schroedinger Representa- 210
tion to the Interaction Representation
1. Interaction Representation. 2. Perturbation Theory.
21. Covariant Perturbation Theory. Transition from the Heisenberg
Representation to the Interaction Representation
1. Expansion of the Interacting Field Operators in a Power Series in the Electron Charge.
2. Transition to the Interaction Representation.
CHAPTER V
THE S MATRIX
22. Calculation of the S-Matrix Elements
1. The S Matrix. 2. Matrix Elements of the Field Operators. 3. Representation of the S
Matrix as a Sum of Normal Products.
23. Graphic Representation of the Matrix Elements
1. Graphic Representation of Normal Products. 2. Various Field Interaction Processes,
24. The S Matrix in the lvlomentum Representation
1. General Formula. 2. Example. Furry's Theorem. 3. Summary of the Rules.
25. Analysis of the Singularities of the IS Matrix
1. General Properties of the Diagrams. 2. Ecirthialent Skeleton Diagrams. 3. Possible
Types of Divergences Related to Irreducible Diagrams.
26. Removal of the Divergences from the S Matrix
1. Removal of the Divergence Due to Photon-Photon Scattering. 2. Electron and Photon
Self-Energies. 3. Removal of the Divergence Due to the Electron Self-Energy. 4. Re-
moval of the Divergence Due to the Photon Self-Energy. 5. Removal of the Divergence
Due to the Vertex Part. 6. Removal of Reducible and Overlapping Divergences.
27, Mass and Charge Renormalization.
1. The Renormalization Concept. 2. The. Relation Between the Divergences Due to the
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230
240
245
252
260
273
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Vertex Part and the Electron Self-Energy Part. 3. Removal of Divergences by Means of
Auxiliary Misses. '4". On'the Divergence of the Renormalized Series for the S Matrix.
28. The Probability for Various Processes.
r
1. General probability Formula. 2. Summation and Averaging Over Photon and Electron
Polarizations. 3. Probabilities in the Presence of an External Field. 4. Effective Pertur-
bation Energy.
CHAPTER VI
THE INTERACTION OF ELECTRONS AND PHOTONS
29. Scattering of a Photon by an Electron
1. Impossibility of First-Order Processes for the Free Electron. 2. ,Photon Scattering by a
Free Electron. 3. Angular Distribution and Total Cross Section for Unpolarized Photons.
4. Angular Distribution for Scattering of Polarized Photons.
30. Emission and Absorption of a Photon
1. General Expression for the Matrix Element. 2. Electric Multipole Radiation. 3. Mag-
netic Multipole Radiation. 4. Selection Rules.. Order of Magnitude Evaluations. 5. Ab-
sorption of a Photon.. 6. Photoelectric Effect.
31. Bremsstrahlung
1. General Expression for the Matrix Element. 2. Perturbation Theory for the Wave
Function of an Electron in the Continuous Spectrum. 3. Bremsstrahlung Cross Section.
4. Angular Distribution for Radiation in a Coulomb Field. 5. Bremsstrahlung Spectrum.
6. Screening. 7. Radiation From Electron-Electron and Electron-Positron Collisions.
32. Emission of Long-WavelengSh Photons
1. Infrared "Catastrophe". 2. Investigation of the Divergence in the Low-Frequency
Region. 3. The Bloch-Nordsieck Method. 4. Relation Between the Photon "Mass" and
the Minimum Frequency.
33. Disintegration of an Electron-Positron Pair Into Photons. Pair
Production by Photons
1. Two-Photon Annihilation. 2. Positronium Decay. 3. Three-Photon Orthopositronium
Decay. 4. Single-Photon Pair Annihilation. 5. Pair Production by a Photon in an External
Field. 6. Pair Production in Photon-Electron Collisions.
34. Photon Scattering by a Bound Electron. Emission of Two
Photons
1. Dispersion Equation. 2. Emission of Two Photons. Metastable 2S State in Hydrogen.
-
3. Resonance Scattering. 4. Angular Correlation in Successive Emission of Two Photons.
CHAPTER VII
RETARDED INTERACTION BETWEEN TWO CHARGES
35.. Interaction Function of Two Charges. Retarded Potentials
1. Interaction Matrix of Two.Charges. 2. General Form of the Matrix Element. 3. Re-
tarded Potentials and Transition Currents.
8
285
295
303
315
329
346
356
371
36. Electron and Positron Scattering by an Electron
1. Electron-Electron Scattering. 2. Positron-Electron Scattering.
v
37. Interaction Energy for Two Electrons Up to Terms.in
1. The Breit Formula. 2. The Schroedinger Equation for a Two-Electron System.'
3. Electron-Positron Interaction. 4. Exchange Interaction Between an Electron and
Positron.
38. Positronium
1. The Hamiltonian and Unperturbed Equation. 2. Perturbation Operator. 3. Fine Struc-
ture. 4. Zeeman Effect.
39. Internal Conversion of Gamma Rays
1. Expansion of Retarded Potentials in Spherical Waves. 2. Conversion Coefficient..
3. K-Shell; Reduction to Radial Integrals. 4. K-Shell; Results. 5. Effect of the Finite
Size of the Nucleus.
40. Conversion With Pair Creation. Nuclear Excitation by Elec-
trons
1. Conversion of Magnetic Multipole Radiation. 2. Conversion of Electric Multipole
Radiation. 3. Nuclear Excitation by Electrons. 4, Monochromatic Positrons. 5. Pair Pro-
duction in Particle Collisions.
41. 0-0 Transitions
1. Reduction to the Static Interaction. 2. Conversion and Nuclear.Excitation in 0-0
Transitions.
CHAPTER VIII
RADIATIVE CORRECTIONS. VACUUM ,POLARIZATION.
42. Third-Order S Matrix
1. Third-.'Order Matrix Elements. 2. Calculation of the Matrix Element for Radiative Correc-
tions to Electron Scattering.
43. Vacuum Polarization
1. Calculation of the Vacuum-Polarization Matrix Element. 2. Renormalization of the
Matrix Element.
44. Effective Electron Potential Energy. Magnetic Moment of the
Electron
1. Effective Electron Potential Energy. 2. Radiative Corrections to the Electron Magnetic
Moment.
? 45. Radiative Corrections to Scattering
1. Radiative Corrections to' Scattering of an Electron by an External Field. 2. Radiative
Corrections to Photon-Electron Scattering. 3. Natural Line Width.
46. Radiative Level Shift for Atomic Elections
1. The-Interaction of an Atomic Electron with the Zero-Point Field Oscillations. 2. Radia-
tive Atomic Level Shift. '3. Radiative Level Shift in-Muonium.
380
385
395
407
426
440
446
454
- 460
463
475
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47. Nonlinear Effects in Electrodynamics
1. Scattering of Light by Light. 2. Coherent Nuclear y-ray Scattering. 3. Lagranginn.
Including Nonlinear,Effects. 4. Concluding Remarks.
APPENDIX
I. THE THEORY OF WAVE FIELDS.
4 8 . The Wave Functions of the Field and Their Transformation.......
1. Wave Functions of the Field and the Lorentz Group. 2. Irreducible Finite-Dimensional
Representations of the Lorentz Group. 3. Direct Product of Representations. 4. The
Three-Dimensional Rotation Group. 5. Irreducible Finite-Dimensional Representations of
the Orthochronous Lorentz Group.
49. The Energy-Momentum Tensor and the Current Vector
1. Energy-Momentum Tensor. 2. Current Vector.
50. Relativistically Covariant Field Equations
1. General Form of Relativistically Covariant Field Equations. 2. Invariant Lagrangian.
51. The Mass and Spin of a Particle
1. Mass and Spin Values for a Given Equation. 2. The Spin Values of Particles Described
by a Given Equation.
52. Examples of Wave Equations for Particles of Various Spins
1. Wave Equations with Positive-Definite Charge Density. 2. Wave Equations with Posi-
tive-Definite Energy Density.
53. Field Quantization: Spin and Statistics.
1. The Impossibility of Positive-Definite Charge Density for Particles with Integral Spin.
2. The Impossibility of Positive-Definite Energy Density for Particles with Half-Integral
Spin. 3. Field Quantization for Integral and Half-Integral Spins.
II. BOUND STATE EQUATIONS.
54. The Equation of Motion of an Electron in an External Field with
Radiative Corrections Taken Into Account
1. The Method of Successive Approximations. 2. Electromagnetic Vacuum Expectation
Value. 3. Vacuum Polarization. Electron-Positron Vacuum Expectation Value.
5 5 . The Equation of Motion of Two Interacting Electrons with Radia-
tive Corrections Taken Into Account
56.
5 7 .
1. The Equation of Motion of an Electron in a Real Photon Field. 2. The Equation of
Motion of Two Interacting Electrons.
III. MATHEMATICAL APPENDIX
Calculation of Certain Integrals
1. The Calculation of Integrals over a Finite Invariant Region. 2. Summary of the Integrals.
L-Vec tors and Spherical Functions
1. Irreducible Tensors. 2. L-Vector Algebra. 3. Generalized Spherical Functions.
484
495
503
507
514
515
521
527
536
539
545
2
CHAPTER I
QUANTUM MECHANICS OF THE PHOTON
? 1. The Photon Wave Function In Momentum Space.
1. Introduction
The corpuscular properties of light were historically. the first fundamental fact which established the basis
for the development of quantum theory. The relation between the energy of the light particle, the photon, and
the frequency:_of the electromagnetic field corresponding to
no)
is historically the first relation containing the quantum constant h. However, the systematic quantum mechanics
of the atom was developed before that of the photon. This situation has deep physical meaning. Atomic particles,
the electrons and nuclei, have rest masses different from zero. These particles can possess energies small with
respect to their rest energy, and in this energy region relativistic effects may be neglected. At the present time,
however, only nonrelativistic quantum mechanics may be considered a relatively complete part of quantum
theory. Since the rest mass of the photon vanishes, no nonrelativistic energy region exists for it; the quantum
mechanics of the photon must necessarily be, relativistic from the very start.
Quantum mechanics replaces the particle by the wave function field 41, which determines the probability
distribution and the expectation values of the various physical quantities referring to the particle. The particle
motion is determined by the field equation (Schroedinger equation). The principle of relativity imposes the re-
quirement of-Lorentz invariance on this equation. This requirement is not sufficient for a unique choice of the
equation which would describe the individual properties of a given type of particle (see Section 50). In the case
- _
of the photon, however, the choice is made simpler by the existence of a classical analog (the classical electro-
magnetic field). It is natural to choose Maxwell's equations as the quantum mechanical equations of motion for
the photon; then the wave properties of the photon will be identical with those of the electromagnetic field. We
shall see that together with the quantum postulate (1.1) this is sufficient to construct a theory of photons and their
interaction with charged particles.
Our first problem is the study of photons in the absence of electric charges. Although it is just in interaction*
with other particles that the particle properties appear, such considerations are useful as a preparatory phase for
the study of interactions, particularly, as it shall turn out, since the latter can be treated by perturbation methods.
We shall henceforth use a system of units in which Planck's constant divided by 2ir and the velocity of light
are set equal-to unity;
=-- c 1.
In this system of units; Equation (1.1) can be written
=-- ,
where k is the wave number.
I) [Underlined letters will denote italicized letters in the original Russian- editor's note];
-
(1.2)
11
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2: Wave Function ink-space.
The electromagnetic field in empty space is described by the vectors I (electric field) and II (magnetic
field) which satisfy Maxwell's equations:
nurl E ?
at '
divif 0,
curlly= aE
at
(1.3)
in order to give a 'corpuscular' interpretation to Equations (1.3) let us compare this system to Schroedinger's
equation of ordinary quantum mechanics: this is conveniently done by first subjecting Equation (1.3) to a Fourier
transformation (transition to the space of wave vectors). Let us represent E and H in the form
E = f E (k) eihr dk, I
H = !I(h) eihr dk.
(1.4)
The time-dependent Fourier components E(k) and H (k) satisfy the following set of equations as a result of (1.3)
(the dot indicates differentiation with respect to time)1)
.?
The two vectors I and Lican be replaced by the vectors E and g, eliminating Hwith the aid of (1.5), and
obtaining
[-TE(k)]. (1.7)
Further, we may remove the necessity for a separate reality requirement by performing a substitution which
leads automatically to satisfaction of Equations (1.6):1)
E (k) = N (k) (k) +1* (? k)),
(k) = ? ikN (k)(f (k) ? f* (?k)).
Here N (k) is an arbitrary normalizing factor which, as we shall see below, should be chosen
N (k) Irak-.
4x h
It is not difficult to obtain the equation satisfied by f(). Eliminating H from (1.5), we obtain
k9) E (k) ??=-- 0,
:1
(k) = ? i
E (k) [kH
(k) = 0,
kE (k) = 0,
(kb
(1.5)
which can be rewritten in the form
to which we must add the reality condition of the field
H (? k) = (k),
(1.6)
We shall not consider the question of how the quantities E (y), 11(k), etc., transform on going from one coordi-
nate system to another. We note only that Equations (1.5) are equivalent to the relativistically invariant ones (1.3).
aIItI
acc
- Sanitized CotDv APP
(PE + k)(1- ? lk) E (k) =0.
(1.8)
(1.9)
(1.10)
I) This substitution actually consists of replacing the two real functions! and H by a single complex one
f (r) =
It should be born in mind, however, that f (r) cannot be represented in the form of a linear combination of
E and H . See in this respect, Section 2, paragraph 2.
?
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,:`
Eliminating f ( k) from (1.8), we obtain
thus Equation (1.10) can be written
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? ik) E (k) = ? 21N (k) kf (k);
kf (k).
It is not difficult to see that f. (k) satisfies the equation conjugate to (1.11). Equations (1.5), after expressions
(1.7) and (1.8) have been inserted into them, give
kf (k) = O.
(1.12)
Equations (1.11) and (1.12) are equivalent to Maxwell's equations. We note that (1.11) has the form of a
Schroedinger equation if the energy operator It of the photon in k-space is defined as the operator of multiplication
by the number k, namely
w= k
(1.13)
Equation (1.13) is identical with (1.2). We shall show later that the function f(k) can be interpreted as the
photon wave function in the usual quantum mechanical sense. We shall be able to define photon operators also for
other physical quantities in k -space, for instance the momentum operator, the angular momentum operator, etc.
We note that notall solutions of the Schroedinger equation (1.11) correspond to actual photon states; only
those solutions should be chosen which satisfy the yansversality condition (1.12).
3. Energy.
We shall show that the operator w we have introduced can actually be considered the photon energy operator.
Let us construct an expression for the energy of the electromagnetic field, which we shall designate w:1)
j (E2 - - 112) dr.
(1.14)
This is an integral over space of quantities which are quadratic.in the field vectors. On the other hand, in quantum
mechanics a space integral of expressions quadratic in the wave function is interpreted as the expectation value of
the corresponding physical quantity. Therefore, the *corpuscular" interpretation of Equation (1.14) as the expecta-
tion value of the photon energy is a natural generalization.
Let us show that ;7 can be written in the form
7-zi) f* (k) wf (k) dk, (1.15)
We shall take Heaviside units for E and H.
14'
I
where Ills given by (1.13).
To do this let us insert the expansion (1.4) into (1.14), obtaining
(E (k) E (le) + If(1c) I-I (le)) ei(k+811)1. dk dE dr.
Carrying out the integration over, r , we arrive at
f e'(" dr (2705 8 (k
where 6 (k + le) is the three-dimensional Dirac 6-function; with (1.7), we obtain
1 ? ?
1.1)=47t2 {E(k)E(?k)d- vE(k)E(?k)}dk.
Finally, let us express E and E in terms off according to (1.8). This gives
= 87c8 f N2 ff* (k)f(k)+f* (? f(? k)) dk = 167c2 f N2f* (k)f (k) dk.
If N (k) is chosen according to (1.9), we arrive at (1.15).
Let us consider the monochromatic solution of Equation (1.11):
f (k) =A (k)
where E is the eigenvalue of the photon energy operator
wfo (k) = Pfo (k).
This shows that f (k) fails to vanish only fork =E. The expression for the energy (1.15) then becomes
w p f f* (k) f (k) dk .
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(1.16)
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We require that this be identical with the fundamental quantum condition (1.2). Then the wave function (k)
must satisfy the normalization conditien
f* (k)f (k)dk 7-11.
? 2. Momentum Eigenstates.
1. Momentum
Let us now construct an expression for the momentum of the electromagnetic field, denoted by p ?
p f !EH] dr. (2.1)
We shall interpret p as the expectation value of the photon momentum.
Let us now express p in terms of f (y). Inserting the expansion (1.4) into (2.1), we obtain
=-- [E (k)H (k')Iei(k+k')r dk die' dr = (2708 [E (k) H(? k)1 dk
? (2109 i f [E (k)[4:7E(-- k)]]dk = ? (2708 i -1142- (E (k) (-- k)) dk,
or, writing E and E according to (1.8) in terms off,
(2709 N2 ( (1e) (k) - f* (-k)f(-
(? k)f* (k) - f (k) f (? k)) dk ?
The last two terms vanish on integration, and each of the first two terms gives the same result. This is easy to
show if we replace k by - k in the integrand. Making use of expression (1.9) for N(k), we obtain
p- =---? if :lel, dk
(the index a = 1, 2, 3 denotes the components of the vector f ; the summation convention is used).
It is thus valid to call the operator of multiplication by the wave vector
P
(2.2)
(2.3)
, the photon mmentum operator, and to give k -space the name momentum space. The quantity f* f may be
interpreted as die probability density that the photon possesses a momentum k-, and Equation (2.2) is then the
ustiza quantum meelianical expression for the expectation value.
Thus, the normalization condition (1.17) has a simple and natural physical meaning.
On the Photon Wave Function iu Configuration Space.
_
By performing an inverse Fourier transformation on f (k),
ff (k) Cikr dk -=f (r),
we would be able to determine the photon wave function f(r) in configuration space.1) In view of the normaliza-
tion condition (1.17) for f (k) , f(r) will also be normalized in the usual way, namely,
_ - _ _
f f* (r) f (r) dr =1.
However, the quantity f ?(!) f (I) cannot be interpreted as the probability density, for finding the photon at a given
space point. Indeed, the presence of a photon can be established only by its interaction with charges. This inter-
action is determined by the electromagnetic field vectors E and H at the given point. The latter, however, are
not determined by the wave function f (r) at that point, but by its values in all of space. This is due to the fact
that the Fourier components of the field vectors (see (1.8) ] expressed in terms of f(k) contain the coefficient
jj Thus, the relation between E(r) and f(r) will not be local, but is an integral relation.2) In view of this
situation, the localization of the photon in a region smaller than its wave length has no meaning, and the concept
of-probability density for a localized photon does not exist.
-- This result is strongly related to the behavior of particle densities under Lorentz transformations .(see Section
53). It is impossible to construct bilinear combinations of the electromagnetic field vectors which are four-
vectors satisfying the continuity equation (although the energy-momentum tensor exists).
3. Plane Waves.
- Let us return to a consideration of the photon wave function f(k) in momentum space. Schroedinger's
_ _
equation (1.11) determines its time dependence-
f (k, I)fo(k) e- at.
L. Landau and R. Peierls ,Z.Physik 62,188(1930); see also W. Pauli, Fundamental Principles of Wave Mechanics
(State Tech. Press, 1947).
2). -Formally this can be written
where A is the Laplacian;
sr This is a unique property that follows from the reality of the field (see Section 49).
?
4
E(r)=-- f(r) + complex conjugate,
4
1=7: is actually an integral operator.
4
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The time-independent function ft (k) is restricted only by the tranversality condition (1.12). In particular, we
may consider states for which f is different from zero only in the neighborhood of the point kp in the volume
element dp.. These will be states 'with definite (actually almost definite) momentum, and the wave functions
corresponding to them will be eigenfunctions1) of the momentum operator p. Bearing in mind the normalization
conditiOn (1.17), we can write the momentum eigenfunction in the form*.
Ie,. -4 4,
P
!ER
where e is the photon polarization vector, of modulus unity and perpendicular to p:
-11
letd2=1, etip.O.
(2.4)
This normalization corresponds to one photon in the volume element dp== dpx d2vd2z. In caluclating the
probabilities for various processes, a weighting factor is usually introduced (the number ef states in the interval
de. For the above normalization, this factor is equal to unity:
For a given p, two linearly independent vectorsi are possible. Let the z axis be chosen along p. These
vectors can then be chosen in the following way:
(linear polarization), or
el. =-- 1, els, 0, els = 0,
e2x = 0, e2y = 1, e = 0
2s
1
1
e2,4
I
els= 0,
e2,=.0
(circular polarization). In both cases ei and el are mutually orthogonal, so that
e?ie2= 0.
(2.5)
(2.6)
Thus, the momentum eigensiates are doubly degenerate. A unique characterization of a state requires also
the knowledge of its polarization, and we have, therefore, provided the function (2.4) with the second index 1.
The quantities.px, py, pz and I/ are the complete set of photon quantum numbers (of course the energy is deter-
mined by these:-w fpfj:. The set of functions fpli is sa complete orthonormal set, and an arbitrary function
f ( k) can be expanded in terms of these, writing
I) Actually proper differentials.
2) The factor 1 in (2.4) has no profound meaning and is introduced only for agreement in the future with gener-
ally accepted notation (the wave function always contains an arbitrary phase factor).
Vvirit,
r?
CIL
= ffi; dk.
(2.7)
The expression for the electric and magnetic fields corresponding to a photon state with given momentum
and polarization can, according to (1.4), (1.7) and (1.8) be written
where
Epp.= gpp. (r) (r), I
gerw(r)-i-lepp? (r),
gpiL =4 T v p dp (pr-pt)
n s
= V-1---) di) [1j-- e ei (Pr-Pe).
P
(2.8)
(2.9)
The normalization given in (2.9) corresponds to the existence in all space of a single, photon whose momen-
tum-lies between p and p + dp . If, instead of the normalization (1..17), we were to choose the normalization
f .a(k) flow (k) dk 8 (P P') 8 Ws
then the expressions for the fields would differ from (2.9) by the absence of the factor rq.
- We shall in the future make use also of a normalization for which the photon is found in some large volume
V, so that
Then
1_
;-4.1:(E2+113)dr-,--k.
= V- p i (pr?pt)
8111 I TV. ei?e
(2.10)
(2.11)
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? and
8
p I ?
The decomposition (2.7) corresponds to decomposing an arbitrary electric field satisfying Maxwell's equa-
tions (1.3) into plane polarized waves:
= ciltspit,
where the coefficients c2J are determined from (2.7). Analogous decompositions hold also for the magnetic
field.
3. Angular Momentum. Spin of the Photon.
1. Angular Momentum Operator.
Let us use the photon wave function f (k) to express the angular momentum M of the electromagnetic field,
which we shall identify with the expectation value of the photon angular momentum in the state RV. The
angular momentum of the field, as is well known, is given by the expression
f Er [EHB dr. (3.1)
In terms of the Fourier transforms, we obtain.
f [r LE (k) H (C11 e(h+w) dk die dr.
Let us first integrate over!:
frei(h+r)r dr = ? iV f (1'44') r dr = ? 1(2703 v hi 8 (k
(here Pk. means differentiation with respect to k '). Now let us integrate over k' by parts:
20':
f dk' (Vie 8 (k [E (k)H WA] ? f dk' 8 (k +k') [V le [E(k) H (W)).
-
4
Replacing H by expression (1.7) which relates ti to E, simple operations lead to
Thus,
(1e)\-1 ik'E (kurIki +i [ 7-er E (k)]
(k'
E (k) EE (k) H (ki )11
1e2 I J
? l[kl V h. (
= (27ti8 1 dk dk' 8 (k {ik' (E (k)/i/E: (1e))1
(k')E (k)] + E (k))curik,
Let us now integrate over k'. The last term vanishes, since
ana we obtain
k !l(s) =0,
51= (2708 f dk Ec (k).E (? k))1? 141 E (k)1}
(here V refers to differentiation with respect to lc, and the index c on .Ec indicates that in performing the differ-
entiation this quantity is treated as a constant). Finally, we express E and E in terms off according to (1.8).
This gives
= f dk v (k) f* (k) ? f: (? k)f (? k)-1-
P (k) (k) f (? k))]
Of* (k) f (k)i? If(? k) (? k)1+
+ if* (k)f* (? k)] ? ff(?k)f(k)i) ).
Terms containing both of the arguments k and -k, vanish under integration. For instance,
when-k is replaced by -k becomes
f (? k) f (k)] dk,
21
'ON
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which is the negative of the original expression. The integral
f lk V (f (k)f (? k)))dk
when k is replaced by ?k, becomes
f [kV (f0(? (k)))dk,
and, on the other hand, if we had integrated by parts we would obtain the same expression with the opposite sign.
The integrals of the terms containing the argument-k do not change when 1t is replaced by -k.
Again integrating by parts, we obtain
f [k V (fa (k) (k))1dk.? f lk V (47f)] dk,
and arrive at the following expression for the angular momentum:
f dk _i[kV(ff)]_i[f*ffl dk,
or its components,
dk [? ik jeut3Tfilr (3.2)
where e 7 is the antisymmetric unit tensor of rank three.
r
2. Spin Operator.
Let us introduce the vector operator I, defined by its action on the components of f according to the equa-
__,
tlon
sc fp ?
We notethat the vector product can be written in terms of s in the following way:
Ea% = g
On the basis of (3.3), we can write (3.2) in the form-.
f:(? i [k V] s)f. dk.
(3.3)
(3.4)
(3.5)
We see that expression (3.5)for the expectation value of the photon angular momentum actually has the
structure of a quantum mechanical expectation value with the angular momentum operator of the photon given by
? [kV]-1
(3.6)
We shall show that (3.6) corresponds to the infinitesimal rotation operator of the vector field multiplied by
i. For this purpose let us consider an infinitesimal rotation about the origin in k-space.1) Such a rotation, as is
well known, is defined by the infinitesimal rotation vector
6u=v 6a,
where 6 a is the angle of rotation, and v is the unit vector which defines the rotation axis. For such a rotation, -
the position vector of a point in space changes by an amount
6k= [6a].
An infinitesimal rotation in k -space transforms the vector field f(k) into the field f'(k) related to f (k) by
the expression
f k-1- 819 ? = -1-1(k) [f (k)]. ,
. .
1) V. Sorokin, J. Expt1.-Theoret. Phys. 18, 228 (1948).
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7. : 2.7 171i7:1, ,j3. ....Ili ? 11
'1 ti 1.111, Li 011 la ?Lg..;
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Then at a given point, k the field changes by an amount
Since
(k) = f (k)-f (k).
f 810 f (k) (6k V)f (k)=f (k) (8es f(k)
(in the last term the difference between f (k) and f (k ) may be neglected), we have
_
81(k), [k VD f (k)+ [Sci f (k)).
(3.7)
(We note that for a spherically symmetric field, which can always be written in the form.f(k)= z(k) k , we have
6 f = 0.)
The infinitesimal rotation operator J is given by the expression
6f= (8(a) f= Sa (vi) .f.
Comparing this expression with (3.7) and using the definition of s given in (3.3) we find (note ,that
- svfa , i.e., (v f) = -(s v) 11:
J = - [kV] -Is.
Using expression (3.6) it is easy to show that the components of the angular momentum operator satisfy the
following commutation rules:1)
MA? M", iP42, I
M.M2 ? M2M,, = O. f
It follows from (3.8), as is well known, that the eigenvalues of 112 are given by
(3.8)
(3.9)
where 2J + 1 is a positive integer. (We shall see below that j is also an integer.) The eigenvalues of' the operator
_
14z are
)2.4z =ILI Cm = ? ?1+1, ?
(3.10)
1) These relations are valid for any quantum mechanical system and follow from the general connection between
the angular momentum operators and the infinitesimal rotations.
4
The operator M commutes with the energy. Therefore, photon states with definite values of w, M2 and My
may exist (the quantum numbers p, j, M). We shall now concern ourselves with obtaining the wave functions of -
these states.
_3. Spin Wave Functions.
Equation (3.6) shows that the photon angular momentum operator consists of two terms: The first term is
the same as the usual quantum mechanical orbital angular momentum operator L in the momentum representation:
L. [k VkJ.
(3.11)
The second term s may be called the spin angular momentum operator.
- The separation of the photon angular momentum into its orbital and spin parts has a limited physical mean-
ing. In the first place, the ordinary definition of spin as the angular momentum of a particle at rest is not applic-
able tq the photon, since the photon rest mass is equal to zero. In the second place, states with definite values of
the orbital and spin angular momenta, as we shall see below, do not in general satisfy the transversality condition.
Therefore, only certain superpositions of these states have physical meaning. Nevertheless, the representation of
the angular momentum in the form of two terms is extremely useful from the formal point of view. It allows us to
construct wave functions for photon states with definite values of the angular momentum from the simpler eigen-
functions of the orbital and spin angular momenta.
The vector index a of the photon wave function may be considered an independent variable (it can be
called the spin variable) which takes on three values: a = x, y, z. Correspondingly, we shall introduce the
notation
f (k, a).
(3.12)
The function f (k, a) is a scalar in the generalized spin and momentum space (the space of the variables
kx, ky, kz, a). Various components of the vector fa are now values of the scalar f (k, a) at various points of
sin Space'. The operator L operates only on the variable k, and the operator a [see its definition (3.3) ] operates
only on a. Therefore, the operators L and a commute.
The eigenfunctions of the operators L2 and Lz , corresponding to eigenvalues
.L.9= 1(1+ 1), )
L = in,
shallhe denoted by ci)hn. They are functions only of the variables k and satisfy the equation
L24ti,? =1 (1+ 1) ED/,,,,
The solutions of (3.14), as is very well known, are the spherical. functions
(J)1,?(k). a (k) Y (it),
where n =?? We shall use the spherical functions normalized according to
k
fY ?iY ?,, do =
(3.13)
(3.14)
(3.15)
? (3.16)
25
rJF:
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Thus. the functions a (k) should be normalized so.that
03
f a*(10a(k)Itidk=1.
0
(3.17)
Let us now find the eigenfunctions of the spin operator. Let Xip be an eigenfunction corresponding to the
following eigenvalues of II 2 and!:
all=s(s+ 1), }
4=1s.
(3.18)
The argument of xsp (a) is the spin variable a. We may thus write this function in the form of a vector x
sir
If x is written as a column vector
then Equation (3.3) can be used to obtain the operators 5a and 2 in the form:of the following matrices:
(
0 0 0
sx. 0 ?1);
0 0
0' ?Li o)
0
sa---41 6 ;
o o o
0 0 SI i\. = 0 0 d 7.
P
... 1 0 0
2 0 0)
s2 ?,_ 0 2 0
( ?
0 0 2
(3.19)
From (3.19) it is seen that the quantity s in"(3.18) can take on only the value 1=1. In other words, the photon
spin is equal to one. We shall therefore sometimes suppress the index s in the function xsp, writing
X sp X11 ?
The z component p of the spin can take on three values:
26
= 0, *1.
s (s+ 1) 4,
saR???=--- IVA".
The solution to these equations can be found on the basis of (3.19):
x0=0; xi? (i);
-11- 0 0
(3.20)
The functions (3.20) are mutually orthogonal, since they are eigenfunctions of the Hermitian operator sz belonging
to different eigenvalues. They are normalized so that'
or in vector form
hJ X.*p? (a) ke (a) =
?
?
)(Ale = 811P?'?
The unit vectors xp define a basis in terms of which an arbitrary vector f can be resolved:
ft= ? fiLY
= -
(3.21)
(3.22) ,
We shall call 111 the contzavariant cOmponents of the vector f in the coordinate system defined by this basis. With
the expression x given in (3.20), it is easy to establish the relation between the fP and the cartesian components of
the vector f (Ix, !It 4):
. . to fs,
.? ft' ==r-wi
(3.23)
? In addition to the contravariant components f P, we shall make use of the covariant components 4, which
The-functions xlp satisfy the equations are defined by the requirement that the scalar product of two vectors land a:
27
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fg (f,,-1- ifs) igy)-14(f ? qv) (gx+ igy)
have the following form;
This will be true, as is easy to see, if
. p gp.,
f =( ?1)P f1'.
We note that since the spin operator commutes with the momentum operator, we may speak of eigenstates
of both the momentum and the projection of the spin. The components of the polarization vector in (2.4) may
be chosen so that
(3.24)
sp. ?s_p?
The two possible polarizations correspond exactly to the two values of the spin projection p . The third value is
excluded by the transversality condition. If the z axis is directed along p, the transversality condition excludes
the state x0. The two polarization vectors el and ey in (2.6) are equivalent to xi and x_ L. respectively.
Thus, the value p =1 corresponds to right circular polarization, and pr--- ? 1 to left circular polarization.
? 4. Angular Momentum and Parity Eigenstates.
1. Angular Momentum Eigenfunction.
Let us go on to a considertion of the cigenfunctions fim of the operators Mz and Mz1). These functions
satisfy the equations;
Whitt (k, a) =1 (i -L- 1) fill!. (k, a), (4.1)
Mfr (k, a) =-- Mfjm (k, a).
Instead of solving these equations directly, we may make use of the quantum mechanical rules for composi-
tion of angular momenta in order to determine, the functions fim. Indeed, our problem reduces to the well-known
quantum mechanical problem of constructing wave functions of a system consisting of two noninteracting sub-
systems. In our case the subsystems are the orbital (variables k) and spin (variable a) degrees of freedom of the
photon.
Since the spin angular momentum s =-1, then according to the rules for composition of angular momentum,
the total angular momentum of the photon can take on a value j if the orbital angular momentum 1 is given by
j-1.- c14.0).
1) V. Berestetsky, J. Expt1.-Theoret. Phys. 17, 12 (1947).
(4.2)
thus, in the general case there exist three different wave functions .fim corresponding to three orbital states. We
shall denote then by him. ?. ?
- .
The wave function fiim is, as is well known, a superposition of products of orbital and spin functions
411mx ., in which the projections m and p arc related by the rule
? A
Therefore,
M=--m + p.
fitm (k, a) =:-= X?iL (a).
V4 IJ.-1
(4.3)
lm su
The general expressions f ; or the coefficients C are known from elementary quantum mechanics.1) For
?
our case (s--=-1) their values will be given beloil [see (4.8) 3.
Let us rewrite (4.3) in vector form:
1
film (k). a (k) p..1, dijr"1-11; yl. N-11, (n)
The orbital function thm is expressed in terms of the spherical function Ylm according to (3.15). There-
fore, according to the generai?resolution (3.22), the contravariant components of the vector film are
Correspondingly, the covariant components are
where
(4.4)
(fithr)p. =-- a .m+p. (4.5)
1. J p. p. / 2tf p.? a ?
CiAr '
(4.6)
1 ?? su
For given / and M, the coefficients C jivf? 11' form a matrix of three columns (j--r-- 1, 1 * 1) and three
rows ( p =0, * 1). This matrix is orthogonaLi.e.,
1) See, for instance, L. Landau and E. Lifshiu, Quantum Mechanics (State Tech. Press, 1948); E. Condon and
G. Shortie)+, The Theory of Atomic Spectra (Foreign Lit. Press, 1949).
29
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I
fg = fagz (fx-1- (fx? qv) (gx+ lgy)
have the following form:
This will be true, as is easy to see, if
ftr gly =?_. gp.,
i 11=-1
f ( -1)11 f "11 ?
(3.24)
We note that since the spin operator commutes with the momentum operator, we may speak of eigenstates
of both the momentum and the projection of the spin. The components of the polarization vector in (2.4) may
be chosen so that
The two possible polarizations correspond exactly to the two values of the spin projection pt. The third value is
excluded by the transversality condition. If the z axis is directed along p, the transversality condition excludes
the state x0. The two polarization vectors el and ey in (2.6) are equivalent to xi and x_ I, respectively.
Thus, the value ;1=1 corresponds to right circular polarization, and !iv-- - 1 to left circular polarization.
? 4. Angular Momentum and Parity Eigenstates.
1. Angular Momentum Eigenfunction.
Let us go on to a consideration of the eigenfunctions fim of the operators M2 and Mz1). These functions
-
satisfy the equations:
M2 f jig (k, a) ----- (i -1-1) for (k, (4.1)
(k, a) =-- Mfor (k, a).
Instead of solving these equations directly, we may make use of the quantum mechanical rules for composi-
tion of angular momenta in order to determine the functions fim. Indeed, our problem reduces to the well-known
quantum mechanical problem of constructing wave functions Of a system consisting of two noninteracting sub-
systems. In our case the subsystems are the orbital (variables k) and spin (variable a) degrees of freedom of the
photon.
Since the spin angular momentum s =1, then according to the rules for composition of angular momentum,
the total angular momentum of the photon can take on a value j if the orbital angular momentum i is given by
1.1,1?1 a4:0).
1) V. Berestetsky, J. Expt1.-Theoret. Phys. 17, 12 (1947).
(4.2)
thus, in the general ease there exist
shall denote then by film.
- .
The wave function .film is, as
. ?
OJT m which the projections rn
Therefore, -
three different wave functions 1.)4 corresponding to three orbital states. We
'
is well known, a superposition of products of orbital and spin functions
and are related by the rule
+ p.
filM (k, Phu: sV?
(1)Ini (k) Xst,. (a).
vil.p.= Jr
(4.3)
lm;
The general expressions for the coefficients C - - are known from elementary quantum mechanics.1) For
our case (s=-21) their values will be given below [see (4.8) 1.
Let us--rewrite (4.3) in vector form:
fJzM (k)=--- a (k)1,...
21-111 Yz. N-1, (a) X1,.
The orbital function 4'im is expressed in terms of the spherical function Yim according to (3.15). There-
fore, according to the generai-resolution (3.22), the contravariant components of the vector film are
fiN= adimm-P?IL yi.
Correspondingly, the covariant components are
where
(f114 = .sr+p. ,
J--1
Pe.
= if +IV 8 -IL
!MIL 1) pd. "
?
(4.4)
(4.5)
(4.6)
M-A:AA
For given 1 and M, the coefficients C form a matrix of three columns (J--= 1, 1 * 1) and three
rows ( =iL 0 , * 1). This matrix is orthogonal, i.e.,
1) See, for instance, L. Landau and E. Lifshits, Quantum Mechanics (State Tech. Press, 1948); E. Condon and
G. Shortley, The Theory of Atomic Spectra (Foreign Lit. Press, 1949).
,
29
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. CI - 0 3R0077nni annnR_
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?,1. I': sv? Ci c l? P ; N
" It
(4.m.ht-10;
The explicit expression for the matrix C is given here:
'The Matrix Element' C 4474?P4141* (s=-- 1)
\./
1+1
-
I
1_i.
0
1
,
M) (1-44 + I)11(1
-1-M+1)(/--M)
-1-M)(14-A1+1)
IT(1.?
r (21+ l) (2/ +2)
I 21 (1 + 1)
AI
)1(121(2/1-1)
1)(1?A1+1)
? M) (I ? M)
)1(11-A1+
(2/+1)(1+l)
.0(11- I)
11(1
1(21+1)
, / y - f - mw + m -4- 0
.1 (i+m) (i ? kr+1),
/ (1--h) (1?m -1-1)
V (21+1) (21-F 2)
V 21(/ -1- I) ,
V 2421+ 1)
(4.7)
(4.8)
Since the jliware an orthogonal set of functions (if any of the indices of two functions are different, that
means that thesenctions belong to different eigenvalues of the Hermitian operators 112; Mv,and L 1), the
normalization conditions (4.7) and (3.17) lead to the relation
fdk =8.011,8mm,
(4.9)
1
yo Yz; Y?1= y iY
v 2.
From (4.9) and (3.17) it follows that the, vector spherical functions are an orthonormal set:
Yi*orYipliMi do air8w8Arms .
(4.12)
2. Spherical Vectors. Parity.
We have found a set of eigenfunctions of the operators corresponding to the square of the photon angular
momentum M2 and to its projection Mz. A photon state with definite values of j and M is described by a wave
function which is in general a linear cOinbination of three spherical waves, namely
+1
fJM= PdflAr?
The coefficients of this linear combination are not independent, since the photon wave function must satisfy the
transversality condition (1.12):
fora= O.
Therefore, there are not three, but two different photon states with given quantum numbers j and M. The corres-
ponding wave functions shall be denoted by f( X) where X may take on the values X=1, 0.
J X)
In order to obtain an explicit expression for rim, we note that from the three linearly independent vectors
(X )
in k -space, we can construct three linear com?binations Yjm , with X.= 0, I 1 (we shall call these also
( 1)
spherical vectors) such that they are mutually perpendicular; one of these[ say 2 jm can be made longitudinal,
(1) (0) ?
We shall call the angular part of the function ftim the vector spherical function or spherical vector and lie:, directed along the radiuS vector k, and the two others [Tim and 'Tim ] transverse. Let us find these linear
denote-it by Tjim1) Equations (4.4) or (4.5) will te ITitten in the form combinations.
? ?
where
firm a (A) Yjim (n), (4.10)
(YJIM)p.
Cbilf -11;
(4.11)
Let us bear in mind that according to definitions (3.23) and (3.24), 'the covariant Components of the vector
Y. namely the Y , are given in terms of its cartesian components by the expression
?
1) For a definition of spherical vectors, see G.,Petrashen, Proc. ACad. Sci.,USSR 46, 291 (1945); V. Sorokin,
J. Expt1.-Theoret. Phys. 18, 228 (1948); V. Berestetsky, J. Expt1.-Theoret. Phys. 17(1947); V. Berestetsky, A. Dol-
ginov, and, K. Ter-Martirosyan, J. Expt1.-Theoret. Phys. 20, 527 (1950).
30
Let us make uSe of the well known formula for the expansion of a product nuYimz in spherical functions,
where n (p = 0, I) are the components of a unit vector expressed in terms of the spherical coordinates and
-
rp
by the expression
no = cos 0; n?i= sin tle? i`P ?
1f 2
and yjm is a spherical function.4 In our notation this expansion can be written
1)
See, for instance, H. Bethe, Quantum Mechanics of Simple Systems (United Sci-. Tech. Press, 1935), p. 383.
(This' is a translation of "Quantenmechanik der Ein and Zwef'glectronenprobleme," Handbuch der Physik, 2nd Ed.,,
XXIV, Part 1 (1933) 3.
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31
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y
?= .Y
2J+1 -11 +1 21+1
(4.13)
- 1
which is easily verified from the definitions (4.11) for the spherical vectors and (4.6) for the e gp- as well as
(4.8). Thus, Equation (4.13) is that linear combination of spherical vectors which gives a longitudinal vector.
We therefore define
Yr)
Further, the scalar product of the unit vector A and the spherical vector Tiim
Yfix ? 1 'Pieta _FL m+p.
(4.14)
vanishes, which can be shown by using (4.13) and (418). Thus, /Jim is one of the desired transverse spherical
vectors. Assigning the value X.-- 0 to this one, we have
}Tim yom.
(I)
Finally, the second transverse spherical vector IN will be defined by the equation
=InYSil?
(4.15)
(4.16)
ti)
Using the expansion for no Y , we can express/1m in terms of the spherical vectors,Ijim, namely
jjr ? 2/ 4- r 1+1.M
+ 1 y
V 2/+1 .1.5-1.31*
(4.17)
(X)
From definitions (4.14), (4.15), and (4.16) it can be seen that the vector spherical functions Tim are
normalized in the same way as the ns The functions Tim(X)
, as do the /Am; remain mutually orthogonal for
different j and M. Since, furthermore-, they are mutually perpendicular for different X at every point, we obtain
_
f Yr; Yrk do = 804 xivan, ?
The wave function of a photon in an angular momentum eigenstate can be written
1311 Pi Y311( ? Po }It
(4.18)
(4.19)
where the coefficients pe and pi are arbitrary. This means that the state fim is doubly degenerate. We can
remove this degeneracy by requiring that each state also have a definite Nifty, i.e., be an eigenstate of the in-
'version operator I. For a vector field this is expressed by 1)
1.f (le) = - f ( - 10.
The operator I commutes with the angular momentum and has two eigenvalues, equal to # 1(12.1). Since the
values of a spherical function at the points k and -k are related by
Yhm(? = (? 1)1Yim(n),
the definition (4.11) of the spherical vectors gives
IY (n) (? 1Y+' rpm (n).
For the transverse spherical vectors, as can be seen from (4.15) and (4.17), we have
ivj (?. 0411,1, 1
IYA= (-1)'+1Yrip
(4.20)
Thus, for a given j and M there exist two possible states differing in parity. The wave functions of these states
shall be denoted by limx (X= 0, 1).
The state with X= 1 is called an electric state, and that with X= 0 a magnetic state. These names are
related to the fact that emission of a photon in the corresponding states is determined, as we shall see below(see
Section 30), by the electric or magnetic moment of the system of charges.
If, in addition to the angular momentum and parity, the energy of the photon is definite (or almost definite),
then a (k) differs from zero in a small region kin the neighborhood of that is a (15)= aP ' 6k where, -
ac-
cording to the normalization condition (1.17), 1)
a., ?
pdp?
Thus, the photon state may be uniquely characterized by the four quantum numbers corresponding to the energy_pl
the angular momentum J, the projection of the angular momentum j, and the parity X. Its normalized wave
function caff be written
Yrile-wskp.
1) The minus sign arises because of the change of direction due to inversion.
1) As in Equation (2.4), the factor i has no fundamental significance here..
(4.21)
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We note .that when X=1, the photon cannot be assigned a definite value of!, since according to (4.17) the
? ? (1)? ?
spherical' vector /im is a linear combination of spherical vectors Yjim with different values of 1. This is a
manifestation of the fact that it is in reality impossible to divide the -angular momentum of the photon into an
orbital and spin part, as has been previously mentioned.
For the case j=0 there exists, according to Equation (4.2), only one spherical vector Y020. It is easy to
see that it is a longitudinal one, since according to (4.14) the longitudinal spherical vector
yOicl) Yo3
can always be constructed. It follows then that transverse spherical vectors do not exist for J=O. This result has
a simple meaning. The state with zero angular momentum is a spherically symmetric one, but a spherically
symmetric vector field can only be a longitudinal one.
Thus, the Photon cannot exist in a state with angular momentum zero.
3. Expansion in Spherical Waves.
The wave function of the photon in an arbitrary state j can be expanded in a series of the functions given
by (4.21):
f CNN) f
pork.
In view of the orthonormality of the set (4.21). the expansion coefficients are given by
cp_imx
f ff;jmxdk.
(4.22)
(4.23)
We can make use of expressions (4.22) and (2.7) to expand a state of definite momentum in a series of angular
momentum eigenfunctions,'and vice versa:
cPV.Afx
fp'ildx)
p'imx
fpsilux = Cillorlf)
PP.
where according to (2.4) and (4,21) (cip---=22 gado )
P.
CP.12111 (01--T; .,(x) p
PL
=-- 3MX,* H Vao o
)7 PP ?
(4.24)
The magnitude of I C to,12 determines the probability that the photon is moving in the direction 2 and
has a given polarization 4, if it is known that it has definite angular momentum and parity? Summing over
polarization states, we obtain from (4.24)
a
C"` 1 1'x (P (P )12
4,4 20.1 1111. PIT ly P.
(4.25)
We note that in view of the relation betwe( n two spherical vectors, as given by Equation (4.16), expression
(4.25) does not depend on X: this means that the angular distribution of the photon is determined only by its angulai
momentum, and not by its parity. Here we present explicit expressions for the functions (4.25).
4. Diressions for the Electric and Magnetic Fields.
Let us find the electric and magnetic fields corresponding to the energy, angular momentum, and parity
eigenstates of the photon. According to (1.8) and (4.21), the Fourier components of the electric field are given by
EpjAn, (k) = re V(1) ?ipt? (1)?e ipt
47tall lip dp O
Correspondingly, the Fourier components of the magnetic field are given by
402.17-p Y 31Aix)eiPt).
When (4.26) and (4.27) are inserted in the Fourier series (1.4), there occur integrals of the type
Y1(X) ( IL\) eikr dok.
k
(4.26)
(4.27)
In order to calculate these, We shall use the well-known expansion of a plane wave in spherical functions:
eikr gr (kr) Y 1*?, Yr (L.)
k tn r '
where the radial function gi (kr) is given by
J j(kr)
gi (kr) = (2n)%ii 1+7
(4.28)
(4.29)
rJ 2, (z) is a Bessel function]. The function gi satisfies the following normalization condition, which is not
*1 + /2 _
difficult to obtain from the asymptotic expansion of the Bessel function: -
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,
'I)1
.4-::..`
It 1
E it
1 +
I?
2 it
-.
I +
,
+;
1 'h
i 04..
1 1,
it g
Te ' 1IA
..r 8
-1 E44
+
+
5!
J
-."
,C'
al
+
1
Ttr
+
to .-=.
ii: et .
1 +
-- 1

g It
ift
el . I.
c.,51c1 lg I
c4
+
.1 ---4-
++
.I 1
RItt +
--
-t.
4-
Tt ^
0,
1 1
It It
.... .1734
li +
I
-I-
az
-
1 -
crt 1
F... It
?
C0 IA +
+
Tt :-.-.
I
l a
s
...
I
I
. + ,
It ?74:
7 It
It 1
":'?
2 IA +
+:
n
1
,
7
Li n
I ;
,
C,,
+
It
I
It
I
44-1-1
1
+
':.
".
1 it
?,;:t
I +
t, IA
+
I
t ;..'
I I
. a ,14,1
AO
I
,.,
+
@It
/t
I
? IA
1
t2 It
I
,
I-
ici
It
.i.
u5 IA
?
I
to IA
i 1.
..-.,
1
vt
c"14
I
+
Ti
.1s
?
-
.
.e.
.
114,i .
03
f g1 (kr) 4 (kr) r2 dr = (2los 8 (k ? hi)
Its ?
Using the explicit expressions for the components of the wave vectors (4.11), we obtain
r
f / vp JIM (T iraOk = gi(kr) i vp jut (7).
According to the definition of the transverse wave vectors, it follows from this that
f(1) eundok = g1 (hr)
if ',AI (1) ea' do k.=
Til.?ogj+i(kr)rf, ji(f)-1- ;WI gi_1(kr)Y1,i-i,
Let us represent the electric field J and the magnetic field H, as was done in (2.8), in the form of the
real parts of complex vectors 28 and 21e:
11(r) (r)? Es (r).
The inverse Fourier transformation of expressions (4.26) and (4.17) eyes:
for elzetric states
[
Nix, +1 ? 11 +
4xlis 2/ 1g11 %PI v
r, -1. J+1.
+.1/-1+1
+ 1 (Pr) Y, e-iPt
leadm, +i 1117-5dP
g1 (Pr) Yii.Me-4pt;
for magnetic states
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(4.30)
37
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o 4st81s -g1(12r)V13Ace-iPt
3eivhf .0 ? 117147) r
4tei, L Y 7-2.1+1 gi+iri. i+1.
++ 1g3
yi.
e-ipt.
We note that Equations (4.30) transform to (4.31) when we perform the substitution
--+ ? in; ge i8,
(4.31)
Expressions (4.30) and (4.31) are normalized so that there exists only one photon with an energy between
E and E + dE. If the field is normalized according to (2.10), Les, if we consider the case where one photon is
found in a volume V which is chosen as a sphere of radius. then the expressions for the fields will differ from
(4.30) and (4.31) in the dp is replaced by ir/R.
When normalized for a unit energy interval according to the condition
f f p* fp,ondk =8 (p?p')
the expressions for the fields will differ from (4.30-) and (4.31) by the absence of the factor vai.
? (0)
In view of the transversality of the vector Yim s _Y , the magnetic field in electric states, and the
electric field in magnetic states will be transverse, i.e.,
Hpor, +ir = Epor, or =0.
The electric field in electric states and the magnetic field
the vectors Y1s, it }A, and are therefore, not transverse
71
- 1)
"Li J IA in terms of the longitudinal vector
38
in magnetic states, however, are expressed in terms of
. Expressions (4.13) and (4.17) can be used to express
and the transverse oneY )
We then obtain
M ?
8,411r. +1
g + ) Ykif1)
1
I 1r P3 dP
I ir CI + 1) (bi-'
2 .1+ 1 `
aepjm, o --= i8pim, +1.
J+1 \ v(1) } ??-ipt
(V+ 1g1+1 -T-Tj--+1 gi -1) '
(4.32)
4
The first term in (4.32) gives the radial component of the electric field. At large distances, when pr>> 1
("radiation zone"), the radial component vanishes since the asymptotic behaviors of gi_ and are
identical (this is easily verified by using the asymptotic expressions for the Bessel fun-ctions) and The field becomes
n
ii+t 0 y
0 cos(pr?% ? 7 it) _ .
gPor, +1 =-- ? 13epor, 0 .----zi.--,,, Vp dp jAr e "t
v n r
(pr? 1).
(4.33)
An arbitrary electromagnetic field can be expanded in a series of fields corresponding to angular momentum
and parity eigenstates of the photon:
8 = C8
where the CPPA-X are given by (4.23). In particular, the coefficients (4.24) define the expansion of plane polar-
ized waves in terms of spherical waves:
? 5.. Potentials.
8?1,.(r) =E (e},Y(it* (Ti-)) glum (r)lf do
J1111
Jep0. (r) = (e,,Y,}1S)afpi1Wn(r)1/ dop.
pa
1. Transverse, Longitudinal, and Scaler Potentials.
(4.34)
In the future, in considering the interaction between photons and charges, we shall have need of expressions
for the potentials of the electromagnetic field corresponding to a photon in some definite state. Therefore, we
shall make use of the expressions obtained in the previous paragraph for the electromagnetic field of the photon to
determine the vector A and scalar Ao potentials of the fields E and H from the well-known expressions
E ? ? V A 0,
at
H =curl A.
(5.1)
Let us perform a Fourier transformation, expressing E and H as in Equation (1.4), and the potentials A and
r .A0 in the form
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A = f A (k) efkr dk ,
A0 =f Ao (k) ffir dk.
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F
Then Equation (5.1) is transformed to
E (k) ? (k) ? ikA 0 (k),
If (k) = I EkA (k)].
(5.2)
Since E and H are transverse vectors, it is convenient to separate A Into its transverse and longitudinal parts,
namely
where
A= B?n? (11 =.114i), (5.3)
kB (k) = O.
From (5.2), (5.3). and the transversality condition
nE =0
of the field, it follows that A0 and 9 are related-by
ikA0-1? cp. .0.
(5.4)
Eliminating the magnetic field with the aid Of Equation (1.7), it is simple to obtain an expression relating
the vectors and 2 with w and E ;
B.E, 1
= ? E. f
(5.5)
The coefficients .64 and are related only by Equation (5.4), remaining otherwise arbitrary. This is die
expression of gauge invariance, according to which the fields I and li do not change if an arbitrary function
0 A
, is added to the scalar potential and at the same time VA is added to the vector potential. If the potentials
a t --
ale subjected to the subsidiary condition (Lorentz condition)
then_ Ai-and 9 will satisfy the relation
(INA --12)(49:= 0,
aup = o.
(5a)
-Equations (5.4) and (5.7) for the scalars 9 and Ao are analogous to the system of equations for the trans-
verse vectors E and H. We shall assume that Ao is real,
Then, similarly as was done in (1.8), ive can introduce the complex function f0 (k) defined by
1
A 0(k) = ti.* ..,?(fo(k)+
n k
A o (k) = (fo (k) ? ( ? k))
and satisfying the "Schroedinger equation*
? Oh
1 =
-i- 0.
(5.8)
From these expressions and (1.8), we can find expressions for andj in terms
10) in momentum space;
?
Then according to-(5.3),
of the photon wave function
where_
(5.6)
(5.6) and (5.4), we obtain the following expression for Mak) :
A h* k)),
(5.9)
(6.10)
(k) =(h (k)? (?
4itVi 3/?k
h (k) =1(k) + Info (k).
4,tivirre
? --VT
B - fit (?k)).
(k)403 (.f (k)
41.
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The set of four quantities h, ji, 4, which determines the Fourier components of the potentials can
serve is the photon wave functiOE. Such Tt wave function, however, will satisfy no normalization condition. In
fact, the'expiessio.ti: for the expe.' etatidn value of the photon energy now takes on the form
dk = f (h' h filf-o)dk ,
and the quantity w is independent of (comparesection 15).
Similarly, the expectation values of the momentum and angular momentum of the photon are expressed in
terms of h and 10 by
f k (h*h? fo)dk,
= f (h:Mh.? A;Lf 0) dk,
where M is the angular momentum operator (3.6), and L is the orbital angular momentum operator (3.11).
2. Longitudinal and Scalar Components of the Photon Wave Function.
We obtain the potentials of the electromagnetic field corresponding to momentum and angular momentum
eigenstates of the photon if we require that the wave function (IL .f0) be an eigenfunction of the appropriate
operators. Since we know the wave functions f , we need only determine the function f0.
The eigenfunctions ..fDp of the momentum operator can be written in a form similar to (2.4), namely
for = C , e-iPt
dp
(5.11)
where is an arbitrary constant. Thus, according to (5.10), the form of h for a definite momentum and
polarization is
ie-aPt
(ep.-1-1Cn) 81m.
1r dp
(5.12)
The eigenfunctions of the angular momentum operator, which is identical with the orbital angular momen-
turn operator for the scalar, are, according to (3.15) and (4.21),
f 001 =
riC2 Yjme-iPt
pdp P'
(5.13)
4
For a magnetic state, the scalar part of the wave function vanishes, and we obtain
For an electric state
AVM, o = 0,
fjAr , nt
hpor, ? (VA + lettrim) e-fl't Sky
dp
(5.14)
(5.15)
-Various choices of the arbitrary constant C give various forms for tile function him, + . For C = 0, the
vectors h and f are identical:
hpor. +1 = fpor, +1,
fopim,+1=--- 0.
For C= j the vector 1 is proportional to one of the spherical functions yi _
+ 1
.im, +1? 2J +
hP
1r 1,2 dp.{ 1+1 J. J-1, me-iPt8kp,
fopim, +1 = 1 ?
+1, ore-ipt a
kp.
-3. Plane and Spherical Wave Potentials
(5.16)
(5.17)
Returning from the wave functions (12, J0) to the Fourier components of the potentials (5.8), (5.9), and per-
forming the inverse Fourier transformation, we obtain expressions for the plane wave potentials (momentum and
polarization eigenstates of the photon) and spherical wave potentials (angular momentum and parity eigenstates).
As before, let us express the vectors E and H and the potentials A, A0 in terms of the complex quantities
It and lito: where
A (r) = (r)d- .(r),
A0 (r) Ao(r)-1- it; (r).
We note that the parity of the functions 100,4 is uniquely determined by the angular momentum, being
equal to'( - 1).1.. This is the same as the parity ii-fan electric state. It canThe shown easily that the potentials A and _A0 are or the following foirn:.
42
43'
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for plane waves 1)
= 1 dp
34'. 40i
1
66P1' 4?Virc V p
for spherical waves of the electric type2)
itriar, --=
C .12-)eicrr-Pt),
Cc On*-110;
114Piedif 2i j+ gi+1 (Pr) j+i, (1;7)4-
J-1. Ar
C 4++1 gi+tn,j+1. m 112i+ e' "Pt ,
kph!, +1 ^ C firTICP4 gjYjme?iPt.
When C =0,3) the function APM
?
J-f- 1
1
differs only by a factor Tp-- from him. When
(5.18)
(5.19)
the expression for 4Pildr. +1 contains only one term, which involves the function
and is thus advantageous in many applications (see Sections 30 and 39):
611pjM dp .17-2J+1
, +1 ? gj-irt m e-iPt
4it,1 +1
j 1117(Tp 2' YIN e? t
gitOppf. /if+ 1 4nsis *
For spherical waves of the magnetic type, we have for arbitrary C
4 1 p
41.2
ip
dp yjo)
Apjhf, o ?g4
4n it
dloplAr, 0 0.
(5.20)
(5.21)
1) When normalizing for a volume!, the quantity d.p. should be replaced by (21r )3/ V in (5.18).
2) When normalizing for a sphere of radius Ft, the quantity dp in (5.19), (5.20), and (5.21) should be replaced by
(W2r)
3) This special case has been considered by Heider [W. Heider, Proc. Cambridge Phil. Soc. 32, 112 (1936) ].
44
? 6. The Two-photon System.
1. Two-photon wave function.
Up to this point we have been considering only a single photon and the electromagnetic field corresponding
to it. The method of treating the wave function in momentum space makes it possible to consider also an arbi-
trary number of photons similarly as is done for a system of particles in quantum mechanics. In this section we
shall undertake an investigation of the two-photon system.
The arguments of the two-photon wave function arc the momenta ici, k2 and spin variables al, al of both
particles. We shall write the wave function
f(ki, al; k2, a2).
The square of its absolute value determines the probability of finding one photon with a momentum ki and
polarization in the direction given by al (al =x, y, z), and the other with a momentum k2 and polarization
given by al (al= y, z). Instead of representing the wave function as a scalar in the space of all the variables
k 2, al, a2, we can consider it a tensor function in the space of momenta k1, k 2 , SO that:
f (kr, al; k2, a2) k2),
(6.1)
Just as we treated the one-photon wave function (see Section 3) either as a scalar .L (k. a) or as a vector ict(k).
In the first approximation the photons can be considered noninteracting particles.') Then the energy of a
two-photon system is the sum of the energies of each photon, and their wave function satisfies the Schroedinger
equation
. of
Tt= (ki k2V.
(6.2)
In addition to this equation, the wave function must satisfy two other conditions. The first of these is the trans-
versality condition for each photon; it can be formulated by the relations
1 (k).,f.,?, ----- o, }
.,
(14 2) Cl2 fat Ct = ?
(6.3)
If conditions (6.3) are satisfied, the probability .vanishes .for polarization of the first photon along k1 or of
the second along k2.
The second condition is a symmetry condition which follows from the identity of the photons. Photons satisfy
Bose-Einstein statistics and their wave function should therefore be symmetric with respect to exchange of the
particles. Thus,
We shall see (see Section 47) that in view of the interaction of photons with electrons, there exists also a
weak interaction of photons with each other.
45
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'Or in tensor form
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f(k1, as; k2, a2)=f (k2, a2; lel, al),
f?,?,(ki, kit). f,,.,,, (k2, k1).
(6.4)
Instead of the variables 1(1, k2 we can introduce the total momentum of the system (the momentum of the
center of mass)
and the relative momentum 2k, where
Kt="-k1?i?te2
k? kt? k2
2 ?
In these variables the wave function of the system can be written
f(ki, al; k2, a2)-=cp(K)f(k; al, a2).
The function f (k; cxi, al) f iaz (k), in view of the transversality condition (6.3) and symmetry condition
(6.4), satisfies the relations
f = 0, ,I
a,
f(k; ai, (? k; a2, al).
(6.5)
(6.6)
When considering a state with a definite total momentum K, it is always possible (except when1_3.1 and Is"
are parallel) to transform to a coordinate system in which
K=0; ki=k; k2=?k.
The function f (lc; a1, cx2) is a two-photon wave function in this coordinate system. Two photons with zero total
momentum are experimentally observed in the decay of electrically neutral systems at rest (neutral meson,
positronium).
46
ii
2. Even and Odd States.
Let us attempt to find the wive functions of states with definite angular momentum and parity for the two-
photon system") whose total mOmentum K = 0. The angular momentum operator M of the system is the sum of
the orbital and spin momenta:
where
L---=71[1eVO
is the orbital angular momentum of the relative motion, and
+ Si
Is the sum of the spin operators of both photons.
The eigenfunctions of the operators 1.2 a n d Lz are, as in the case of a single photon, the spherical functions
orPs?,= a (k) Ys?, (n).
The eigenvalues of the square of the spin angular momentum are
s 2 = s ( s ? 1),
where, according to the rules for composition of angular momenta, the number s may take on the values
s = 0, 1, 2.
, Since for a given value of s, the eigenvalue of the z component of the spin
-1) L. Landau, Proc. Acad. Sci. USSR 60, 207 (1948).
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may take on the values
and there exist only nine different spins states of a two-photon system.
The spin wave functions
Xvik(att as)
of the system are bilinear combinations of the spin wave functions of each of the photons:
XiLi (at) (pi (as) (P,=O,?1).
Let us construct the following normalized combinations which have definite symmetries with respect to exchange
of the spin variables al, a2:
(at) kJ, (as) (111 = Ps; 3 &Defiant),
ZL( (al) Xis.. (c110 Xp, (at) 42 (as)) Oh *14; 3 fgetctii3ns
y 2
1 ?
r -2 fXiLi (at) Xpi (as)? XiL; (at) XPI (as)) (111* IS; 3 fefient).
1
(6.7)
The first six ?functions in (6.'7) are symmetric, and the last three are antisymmetric. Since the states with different
quantum numbers 11 and the same quantum number s should have the same symmetry, we can identify the three
antisymmeuic spin functions with the functions xs p Ci 4. ps) for mi.
The six symmetric functions (6.7) refer to _s?-0, 21).
1) Of these, the four functions for which Pi + p96 0 can be identified with the wave function x for _ame2,
+ Pa. The other two functions
and
X0 (al) XO (al)
1
(Xt (al) X-I (as) + X-i (al) Xt (as))
are linear combinations of the wave functions Xso and Ms.
We note that the separation of the spin states into symmetric and antirymmetric has actual significance.
As-for the classification of states in terms of spin values, it has no profound physical meaning since, as will be
seen later, the transversality condition requires a superposition of states with j..= 0 and .1.= 2.
The wave function of the two-photon system which corresponds to definite values of the angular momentum
L and its projection hi, namely
fjv (k; ali al)
is, according to the rules for composition of angular momenta, a linear combination of products of orbital and
spin functions
44(n) X?y? (03! x
Eigenstates of the angular momentum can be further classified in terms of their parity. The effect of the
inversion operator I on. the tensor function is determined in the following way:1)
that is,
lf(k; a1, c(s)==f( ?k; a1, as).
Thus, the inversion operator acts only on the orbital part of the wave function. The parity of the state is deter-
mined by ( -1)1, so that
Y tin ( it) (? 1)1Y tin (n).
On the other hand, replacing Is. by is equivalent to interchanging the momenta of the two photons. According
to_the symmetry condition (6.6) of the wave function, we have
Yto (n) (av as) = YINS Xst(as,
We see, therefore, that there is a relation between the parity of a state and the symmetry properties of the
spin function: "
X.( a1, 42
?.1)/ Xsp. (air al),
(6.8)
1)Sinco under inversion all the directions are reflected, the components of a tensor of odd tank change sign (see
Section 4), and those of an even rank do not.
?4
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that is, even state (1 = 2n, where n.is any jump!) are symmetric with respect to the spin variables, and odd
states (1 n' 2n-' 1) are antisymmetrie. In other words, the tensor f ala2(k) is a symmetric tensor in even states,
and is ,antisymmetric in odd states. In the former ease it has six components, and in the second case, three, and
these correspond to the number of symmetric and antisymmetric functions (6.7).
3. Classification of States With a Given Angular Momentum.
Let us start with an analysis of the odd states. An antisymmetric tensor of second rank, as is well known,
can he written in the form
An. = esig.T.f7)
7
(6.9)
where e a.tasY is the unit antisymmetric tensor, and f is a vector. Therefore, odd states can be described by
? ?7
a vector wave function f (k). According to the results of Section 4, there exist three linearly independent vector
(X)
functions f jm, corresponding to given values of the quantum numbers J and M, such that two of these are trans-
verse (X = 01), and one is longitudinal (X = -1). The transversality condition (6.5) when applied to the anti-
symmetric tensor (6.9) becomes
Ikfl =0.
Thus, the vector f should be longitudinal, and for given J and M there therefore exists only one odd state with
-.9v1
_41(.= aYSit).
According to (4.14) and (4.13), Y(i .1j th
? contains the values 1 = J ? 1, Since 1 is odd in e case we are
con-
sidering, J must-be even:
2n; 1=2n-1-1. (6.10)
We note that since the tensor wave function f %al of the two-photon system is bilinear in the components
of the photon vector functions f1 and f5, the antisymmetry of falal means that the vectors fl and_f_2 do not have
components along the same axis. Since the component f a determines the probability for polarization along the
axis a, this means that two photons in an odd state are polarized perpendicular to each other. We note that a
system decaying into two photons, for instance a neutral meSon or positronium (see Section 33) in the singlet
ground state, is odd, and the above results are applicable, to these decays.
Let us go On to a consideration of even states. Since they correspond to the values s = 0, 2, then according
?
to the rules for the composition of angular momenta with given J and M, there are six different wave functions
-
corresponding to values of 1 given by
,
. 1=1?1; 1?2; (s=2),
1=] (.?=0).
a/
Since in this case 1 tan take on only even values, the number of wave functions is given by
4 for j =- 2n,
2 for =2n+1.
In addition, we must take account of the transversality condition (6.3). To do this we shall make use of
the possibility of constructing functions (fim) ala ("spherical tensors") in the form of bilinear combinations of
spherical vectors: 2
(6.11)
Here MI + M2 = M and the values J1 and J2 should be such that the value I can be obtained from them according
to the rules for composition of angular momentum. Corresponding to the three values of X1 and X1 one can ob-
viously construct six different symmetric bilinear combinations, which correspond to the six tensor wave functions
ajm) %eel. Of these six combinations, three do not contain a longitudinal vector and satisfy the transversality
Condition. The remaining three functions ("longitudinal") cannot correspond to actual photons.
In order to find the parity of the functions which satisfy the transversality condition, let us construct an
explicit expression for the longitudinal spherical tensors. We shall choose ji = 0 in (6.11), without loss of gener-
ality. Then
and since (see (4.14)):
12=1; M2= M; M1= O,
the longitudinal spherical tensors will be of the following form:
(fikeic,..= na,(410,2+ne..(4111)ast
A=O, -?1.
(6.12)
( X)
When X- = 1, the function 'I'm contains [see (4.13), (4.17) spherical functions with 1 values given by
1 =L 1; when X = 0, these are given by / =J. Since (6.12) contains the vector n, and In is? n, we obtain
(- 1)1f(ii
inA)f
This means that the number of even longitudinal states is
? for
for X = O.
51
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2 for j 2n,
1 for f=2n -F 1.
Subtracting this number from the total number of even states, we obtain the number of even states which satisfy
the transversality condition. There are, then,
2 for.
1 for . J =2.n+1.
The cases1=0 and1=1 require separate consideration. When j the rule for composition of angular
momenta allows only two states (both even): ?
7=0; ?and= 2; s = 2.
On the other hand, there exists only one spherical vector with J=.0 (longitudinal). Therefore, only the value
X =- 1 is possible in Equation (6.12). This means that of the two tensors (4.) (sot, one is longitudinal and the
other is therefore transverse. Thus, there exists only one even state of the system with zero angular momentum.
In this cue, as is not difficult to prove, the wave function which satisfies the symmetry and transversality condi-
tions is
(lode's. = a (k) (fte.ng.? 8,007
(6.13)
One may conclude from (6.13) that in this case the photons are polarized parallel to each other. Indeed, let the
z axis be directed along k; then Ac
._Y
?
When J = 1, only one even state is possible; 1 = 2, s= 2. But an even state is already contained among the
(a)
functions of (6.12) (namely ([.)a as' which do not satisfy the transversally condition. This means that
--
there exist no even states with j=1. According to (6.10), there also exist no odd states with j=1. Thus, a two-
photon system can never be in a state with unit angular momentum, and a,system with unit angular momentum.
cannot decay into two photons. -
In conclusion we present a summary of the classification of two-photon states.
52 ?
Angular
Momentum
No. of even
states
-}
No, of odd
states
0
1
2n
2n+1
1 1
_ _
2 1
1 ?
CHAPTER II
RELATIVISTIC QUANTUM MECHANICS OF THE ELECTRON
f 7. The Dirac Equation.
1. Spinors and Pauli Matrices.
In nonrelativistic quantum mechanics the motion of the electron is described by a two-component wave
function,-.riamely a spinor (51). The components of the spinor are usually denoted by 9X, where the index X
takes on the values X = 1/1. The spinor is written in the form of a column vector
Under a rotation of the coordinate axes, the spinor components transform according tot)
p=(1+--)'
(7.1)
(7.2)
where ?co' denotes the spinor (p (at the same point r in space), except that its components are given in terms of a
coordinate system rotated relative to the original one by an infinitesimal angle 6 about the axis directed along
the vector 6, and a is the set of Pauli matrices2)
- -
== 0
C1 1)
clx
1 0) (7.3)
a --
_1)
According to the general relation between the infinitesimal rotation operator and the angular momentum,
the matrix 1/2 a is the intrinsic angular momentum operator of the electron, or the spin. The transformation law
(7.2) is a result of the fundamental physical fact that the electron spin is 1/2.
See, for instance, L. Landau and E. Lifshits, Quantum Mechanics, Section 54.
s) The expressions for the ai (i = x, y, z) are independent of the coordinate system. Therefore, a is not a
vector. Nevertheless, vectornotatiort, which is extremely convenient, leads to no confusion.
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The matrices (7.3) satisfy the commutation rules of the components of the angular momentum, namely
aceaf/?tiox Vas
(7.4)
(and similar expressions obtained by cyclic permutation of the indices). In addition, they satisfy the relation
It follows from (7.4) and (7.5) that
clerk akai =28i?.
? 8
-4-k ? ik+
(7.5)
(7.8)
where elm is the unit antisymmetric tensor.
As do other types of quantities (for instance tensors), spinors differ in the way they transform uncle reflec-
tions. A spinor co which transforms according to
=
(7.7)
where 91 is the same spinor, but with its components given in the reflected coordinate system, shall be called a
polar spinor (or simply a spinor). A spinor x, on the other hand, which under reflection transforms according to
(7.8)
X =
? *all be called a pseudospinor. In either case, double reflection leads to the identity transformation.
We note that as opposed to tensors, spinors can have transformation rules under reflection4 other than (7.7)
or (7.8). This is because according to (7.2), under a rotation of 21r, the spinor components transform according
X 2,tiX Xt
eP =e cp
-Thus, if we define the double reflection as a rotation through 2rr, rather than the identity transformation, under
reflection the spinor transforms accOrding to
See S Pauli, Relativistic Theory of Elementary Particles (Foreign Lit. Press, 1947); G. Zharkov, J. Expt1.-
Theoret. Phys. 20, 492 (1950); Yang.and.Tiomno?Phys. Rev. 79, 495 (1951).
Indeed, it can-be shown from (7.2) :and (7:3)-lhat under rptation through a finite angle 6 about thez axis, the
transformation is given by
a
-4
Tif e 2 eh.; e
64
(7.8')
We shall-henceforth use the first definition of the double reflection, and thus, use the transformations given by
(7.7) and (7.8')..
T.-Dirac Equation.
The relativistic equations which perform the same function for the electron as Maxwell's equations do for
the photon were found by Dirac.1) These are the following system of first order homogeneous differential equations
for the two spinor functions cp and xl)
?at = nvP aPX,
ax
it=? ntx + app.
Here p=-: -i V is the momentum operator, and m is the electron mass.
The set of two spinors (p and x can be written in the form of a single four-component quantity
/P , if th
5r- = 4,2
XI','
7,-1/21
(7.9)
which we shall call a bispinor.
In order to write (7.9) in the form of a single equation for 0, we introduce the four-dimensional matrices
1) P. Drrac, Quantum Mechanics (United Sci. Tech. Press, 1937).
2) We note that Maxwell's equations can also be written in a form similar to (7.9), namely
aE.311
= (sP) 11; = ? (sp) E,
where s is the photon spin operator (3.3) (the photon mass is zero).
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(7.10)
(The symbols 1 in the expression for 8 represent two-dimensional unit matrices.) Using this notation, we can
write (7.9) in the form')
a+
i -67 = (19 + Pm) its.
Equation (7.11) is in the form of a Schroedinger equation,
in which the Hamiltonian is the expression
(7.11)
(7.12)
H op ?Pnt.
Let us point out the properties of the matrices a- and 8. From the definitions (7.10) and from (7.5), we obtain
aiak+ 424= 2800
132=1,
(7.13)
in the future it will be convenient aleo to introduce the following additional four-dimension matrices
1) In this equation we use ordinary matrix multiplication, so that
Similarly,
56
4
(a+)i = I ctiktPlet
k=1
4
((ra)j = irk aki ,
k=1
4
(174' = 'rePt ?
4=1
(7.14)
Here, as in (7.10), the number 1 denotes the two-dimensional unit matrix.4 The matrices Z satisfy the same
equations (7.4), (7.5), and (7.6) as do the or In addition, as can be seen from the appropriate definitions, the
following relations hold:
PE ?zp.o,
p2=1.
The relations between the matrices ai and Ek are easy to obtain if we bear in mind that
p Z
Thus, replacing a by E in (7.4), (7.5), and (7.6), we arrive at
aaar ? aro.. 21E3,
atEk Eic(k = 84kP leiketz,
[AM =
3. On the Necessity for Four-Component Wave Functions.
(7.15)
(7.16)
(7.17)
Let us make some remarks that will clarify the structure of the Dirac equation. The transition from the
spinor co to the bispinor 11) is analogous to transition from the three-dimensional vector a to the four-dimensional
vector a = (a; 5) (N is the time component of the four-dimensional vector). Just as a and as transform independ-
ently of each other-under space .rotation
a --,-- a' ? [aal,
a a
0
(7.18)
We note that when the number 1 is replaced by these two-dimensional unit matrices, ox is transformed into
p and ol into 8.
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(61s the infinitesimal rotation vector, as in (7.2) ), so the two spinors and x which combine to make up the
bispinor.* transform independently according to Equation (7.2) under rotation. Under a general Lorentz transfor-
mation, however, a and an no longer transform inde'pendently. If de is an infinitesimal vector giving the velocity
of the new coordinate system mlative to the original one, then
a= a' --I- 80a; ,
ao= do -1- 80d.
(7.19)
It is easy to see that the spinors and (01 and x cannot transform independently under general Lorentz trans-
formations. In fact, the four matrices ax,ay., az and 1 are a. Complete linearly independent set of two-dimen- ?
sional matrices. Therefore, a transformation lin?ear in 60 which reduces to the identity when 50 = 0, can only
be of the form
cp.--=(1?F-Tv 30cr) (7.20)
where v is a constant. But such a transformation equation cannot be valid, since it is not invariant with respect
to reflection. To show this, let us compare it with (7.2). In that equation the infinitesimal rotation vector
is a pseudovector, whereas the infinitesimal velocity vector 60 in (7.20) is a polar vector. Therefore, if (7.2) is
to be valid in two coordinate systems which are reflections of each other, then the second term in (7.20) has
different signs in these two systems. At the same time the set of two spinors and x, one of which is a pseudo-
spinor, can transform under Lorentz transformation according to
}
We thus see that the electron wave function must consist of two spinors (one bispinor).
4. Invariance of the Dirac Equation..
(7.21)
Let us now go on to a proof that the Dirac equation is invariant with respect to Lorentz transformations. Let
us consider inversions, rotations, and actual Lorentz transformations in turn.
1) Inversions. If the bispinor consists of a spinor 9 and a pseudospinor x, then (7.7) and (7.8) can be
written { I); ? a (pi ? PO) ? Pitt)( I +--a)'= 0.
we write the Dirac equation (7.11) in the form
(Po? ccP ?1314)11' = 0.
(7.23)
Let us now perform an inversion of the space axes. Then pn and m, since they are scalars, do not change, so that
pn=opt0 ; the numerical matrices 8 and a are independent of the coordinate system and also remain invariant.
The components of the vector p, however, change sign under inversion; therefore,
Thus, Equation (7.23) is transformed to
ap ? ccp/,
(V0+11? ? Pm) =
Let us multiply this equation,on the left by B. With the aid of (7.13) we obtain
(1;0? Pm) =
which is the same as Equation (7.23), but in the transformed coordinate system.
2) Rotations. Both spinors which go to make up V) transform according to (7.2). From the definition (7.14)
we can write the transformation law in the form
(7.24)
It follows from (7.24) that the matrix 1/2 E is the intrinsic angular momentum (spin) operator of the electron.
Let us insert (7.24) into the Dirac equation (7.23). Then since p is a vector, it transforms according to (7.18),
and we obtain
where 8 is the matrix given by (7.10). Introducing the notation
58
.0
1.?a?.1= Po,
(7.22)
Let us multiply this equation on the left by (1 2 6 E) and neglect all terms quadratic in 6. We then obtain
(Pio? ccp' ? Pm) =
= ? ma (1P ? PE) ? ((a E) (IP') (Gclif) (a V} (le/ ? rap' ccY?
59
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? The first term on the right vanishes according to (7.15). The second term can be rewritten with the aid of
? (7.17), so that
?72 R8z )(9') ?(sp') (BE)) = a?
This cancels with the third term, and we arrive again at Equation (7.23).
3) Lorentz transformation. The transformation (7.21) with v = 1 can be written
Inserting (7.25) into (7.23), and using Equation (7.19) to transform p and ps, we arrive at
(PO+ aoP' ? 801)10)? Pm} (1 +4 Bocg)4/ = 0.
Multiplying this equation on the left by (1 + 1/2 61a), we obtain, to terms of first order in 6,,
(PO up'? Pnt) or' =
= exP + 4- f(szp') (80e) -f- (boa) (ale)) 4"? 4/.
(7.25)
According to (7.13) the first term on the right vanishes, and the second term (that in the braces) cancels With the
last one, so that we again arrive at Equation (7.23).
5. The y Matrices. Continuity Equation.
The Dirac equation can be written in a more elegant form if we introduce the four matrices yi (1= 1, 2,
3, 4), which are defined by
Ti = 012i (I = 11 21 3)1 }
14 = P.
(7.26)
We shall denote the set of three matrices ys, ys, Vs by the symbol 7. The matrices yi satisfy the following
relations:
Tax ? "(kit = 28ik(i, k == 1, 2, 3, 4).
Just AkaitsthettiltilFeriiiiiiiiritiocrudecl, the yi are Hermitianr. that
(7.27)
iti 74== Is
(7.28)
(y i is the Hermitian conjugate of the matrix, and yi is the transposed matrix).
Let us multiply the Dirac equation (7.23) on the left by US. Using the definitions (7.26), we obtain
If we introduce the abbreviated notation
(iT4P0 ins) =0.
1414 Ell 'IP +14P4 TP1
for the "scalar product", where
P4-= iPot
then the Dirac equation takes on the form
YIP + n1) +
(7.29)
(7.30)
The operator iyp in (7.30) is not self-conjugate. Therefore, the complex.conjugate wave function le
satisfies an equation which is not the same as (7.30). If we introduce the function ! , defined by
47=94% = tTP,
then 0 will satisfy an equation that can be written in the form
R?i1p+m)=0.
_
In (7.32) the differential operators pi are assumed to act on the function -1i to their left.
_17
Let us prove the validity of Equation (7.32). Since
p
(7.30
(7.32)
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-AMA
41=-014,
(..%111.141..q. CiPS tiliP (9,PATil.t.PP ttP t sitgtto tate 110tt ind malti w.f.: fuT((71.:23.
ti1.19 Aigtht ly.,4ituI 1nkitts. co.f0.1-2r0),, we zatti.v.c at Equation (( 7:321).
11,Rttilfiffill1W,PY LE 'ti.) (PP tt 141tt And ftuatipp(CP.:a0)) (on itheileft i,
iby
41%W W,9(9,11.41AP
41(tcori)-+-1qp3,)16,-,g.
(I11.1.9 611).P.E.g9.r1p ACSfi&iii cktimPA.0.9.P ikt.s trlAiht..) LSItir&
Plc g:?9sneow?4,95scsok, yritsAki
.
0,5sit.)0
09 h
01610.---
9I? OM) 0111%g' ilggiNgSa A? A &PA.0114.4Xy e..4.u.4191), lt in the form
obtgin g.>;pig..??ion? for th.4 s.i?igityjg imJ thg gurignt 1, r.ormy
/j =. 6,0 ___-_-,--... flegli, .
(7.33)
(7.34)
6. The Transformation Characteristics of Bilinear Combinations of the Wave Functions.
The yi matrices can be used to give a unified form to the transformation equations (7.24) and (7.25),
namely
4,.=. (1 ? (4.35)
where Pik is the four-dimensional antisymmetric infinitesimal rotation tensor
The equation which is the complex conjugate of (7.35) is
oti=qs
Multiplying this on the right by 13, we obtain
A4X (80)x?
=Tr' (1 ? Aik1k7i).
(7.36)
These transformation equations can be used to prove that the product is a four-dimensional scalar.
Indeed, according to (7.35) and (7.36)
orlp to a term linear in Aik,
= v(1 ? (1 ? Aikiak) ,
--= Te'oe' ? (*fax?FWD 4/-
Bift since Akis an antisymmetric tensor and yiyk + ykyi= 26.1k, we have
?
(14.=
. )!
?
(7.37)
In a similar way we can prove that 7i yi is a four-dimensional vector. In fact, according to (7.35) and
(7.36)
-a
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TI;Th =Tile Ajkci (Wil t+ MOO +1
.or, by making use of the properties of the 7 matrices given in Equation (7.2'7) and the antisymmetry of pik, we
obtain
iTh= 7/7411? 41071/ efkil?
(7.38)
But (7.38) is exactly the expression of the transformation properties of a four-dimensional vector under infinitesi-
mal rotation.
Similarly, it is easy to show that
(1 *
is a tensor of second, rank, that
(1*.1
is a tensor of third rank, and that
41i7k7j1m41 (1k#.10m)
is a tensor of fourth rank.
Due to the commutation rules of the yi , it is impossible to construct higher rank tensors out of products
of * and *. We note that the above four-drmensional tensor of fourth rank is equivalent to the pseudoscalar
and that the third rank tensor is equivalent to the pseudovector
where
16
= TiY27574.
^ V71.09,1
(7.39)
-A 8. Electron and Positron States.
1. Solutions with Positive and Negative Frequencies.
The general solution of the Dirac equation (7.11) can be written in the form of a Fourier integral
f + (k) elk* dk
(8.1)
The Fourier transform * (k) is the wave function of the electron in momentum space. Inserting (8.1) into (7.11),
we obtain
. a+ (k)
= (ak m) 4,
or, writing (k) in terms of its component spinors,
This can be written in the form
=--- (12 (k)
X (k)
.a(tk) me? (k)-1- k/. (k),
i O(k)=
We shall attempt to find solutions to this system in the form
tle k) %/do (1e)e-iwts
cp (k) =Po (k) e-
X (k) Xo (k)
so that after this is substituted into (8.3) we obtain
(ak + pm? 0,
(8.2)
(8.3)
(8.4)
(8.5)
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function of k, and the other is expressed in terms of the first by means of Equation (8.6).1) For instance, if
CO and x (?) are given, we obtain cp (?) and x CO as follows:
m
cp(?)
(8.12)
Solutions with positive and negative frequencies belong to different eigenvalues of the self-conjugate oper-
ator H (7.12). Thus, they are mutually orthogonal,2)
? Or
+M. (k) (k) =?Mt (k) cP(-) (k) X(+). (h) (k) = 0,
j
(8.13)
4)(+)* (r) 4,(-) (r) dr = 0. (8.14)
Equation (8.13) is easy to prove with the aid of (8.12).
We note that the resolution (8.10) is relativistically invariant. Under Lorentz transformations the sign of
the frequency will not change. This can be seen from the fact that the lowest positive frequency is m, and the
_ highest negative frequency is ? m, so that these are separated by an interval of 2m, whereas the Lorentz transfor-
mations contain a continuous parameter.
2. The Wave Function of the Position.
The occurrence of two kinds of solutions has a fundamental meaning for the theory of the electron. Let us
consider the particular solution Of the Dirac equation in the form of a monochromatic wave
= ! 0 (r) e?ful?
Inserting (8.15) into (7.11), we find that 0 o(r) is an eigenfunction of the Hamiltonian, namely
(8.15)
1) The fact that not one but two components are arbitrary, is due to the form of the determinant of (8.6). Instead
of considering a fourth order determinant, we are dealing with the second order determinant (8.7), which contains
the matrices a . The fact that this determinant is independent of these matrices, shows that when it vanishes, so
do its third order minors.
2) The symbol ? cp represents the scalar product of spinors given by
c
ecp = "Ay
67,
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I-1,6= 040 =-- '710?o?
(8.16)
? In quantum mechanics the eigenvalnes of the Hamiltonian are the particle energies. In the relativistic
quantum mechanics of the electron it Is impossible to maintain this interpretation literally. Indeed, the occur-
rence of negative frequencies would mean both that there exist electron states with negative energies, and that
there exists no lowest energy state. This would mean that in interacting with other particles the electron could
transfer unlimited energy, going over into lower and lower energy states. The physical absurdity of this conclu-
sion makes it necessary to change the fundamental quantum mechanical rules for calculating a physical quantity
from the known solutions of the corresponding wave equation.
The, new rules should, first of all, assign positive energies to negative frequency states. We shall assume
that the electron energy for both signs of the frequency is given by
a
In this way, however, there correspond to a given energy value (as well as to the values of other physical quantities
which characterize completely the dynamic state of the electron, for instance, the momentum and spin orientation
or angular momentum and parity) two different states, namely the states with different signs of their frequencies
(quantum number 71).
Therefore, the new rules should, in addition, give some physical meaning to the quantum number n . Such
meaning can be obtained from the assumption that the Dirac equation refers to electrons with both positive and
negative charge, or to the electron and the positron. It then becomes possible to interpret the quantum number' 71
as characterizing the charge state of the electron. When 71-= 1 the electron has a charge e, and when 71 ?1 it
has a charge -e. We shall see later that this hypothesis makes it possible to construct a theory of electrons and
positrons and their interaction with the electromagnetic field.
, In order to formulate the new quantum mechanical rules in correspondence with the above demands, let us
establish the relation between solutions to the Dirac equation which have different signs for their frequencies. The
wave function with negative frequency, iccording to (8.5) satisfies.the equation
(4Ck + PM + (k)
Let us consider the complex conjugate equation
(a*k? p*m e),F(-)* (k),= 0.
Let us multiply Equation (8.17) on the left by the matrix
0 o 0 1\
0 " 0 ?1 0
C = 113 y ( 0 ? 1 0 0
o o of
(8.17)
(8.18)
This matrix, as can be seen from Its definition, has the following properties:
CP := ?PC,
(We note that 8* = 8; a* x = as az; a* y = ay.)
Equations (8.19) can be used to write (8.17) in the form
(? ak -I- Pm ? a) CIA-r(k). 0.
(k) = 01,(-)* (? k).
11?('-') k) = CM* (k).
Let us now define that -function
Then according to (8.19)
From (8.20) and (8.22) it follows that 0 f P) (k) satisfies the equation;
(ak + Pm ? e)tlicP) (k) 0.
(8.19)
(8;20)
-(8.21)
(8.22)
(8.23)
Going over from the momentum-space wave function 0 (P) (k). to the configuration-space wave function
(P)_(1) by means of the Fourier transformation
we find that the function
4013) (r) (k) eikr d k,
4.0) (r) = co-). (r)
satisfies the Dirac equation for a positive-frequency wave function.
(8.24)
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Thus, if wc consider the electron wave function with n. ?3, not the function H, but. the function
(P), then the Hamiltonian H may still be considered the energy operator. Its eigenvalues are now positive.
We may call 0 (P) the positron wave function!) The assignment of a charge whose sign is opposite to the elec-
tron charge-to the state with 1 is merely a formal procedure so long as we restrict our considerations to the
free electron and do not consider its interaction with the electromagnetic field. Later, however, we shall see
isee Sections 11 and 19), that when the transformation (8.24) is applied to the equations of motion of the elec-
tron in the electromagnetic field it is found that the state whose wave function is 0 (P) actually corresponds to
the charge2) e.
3. Positron Parity.
Let us now go on to a consideration of what happens to the operators for the other (in addition to the energy)
physical quantities, for the electron in a charge state with, T1 - 1. In doing this we can make use of the fact
that the fundamental quantities, the momentum, the angular momentum, and the parity, are related to definite
transformations of the wave function, in particular, to translations, rotations, and reflections. Let us first establish
the ,form of the transformation operator for the function (8.24), and then let us go in the usual way to the operator
corresponclin,g to the physical quantity. Let the transformation for the function * be of the form.
KV.
Then the function* (P) will transform according to
VP) ?,---- OV?r = CK*4e1(?)* ---= CK*CVPY .
Thus, the transformation operator for the positron function is
K(P=CK*C.
1
(8.25)
It follows from (8.25) that the momentum and angular momentum operators are of the same form for the
positron and the electron. In particular, the infinitesimal translation operator is
or
or = (1+ipP)
K 1 + ipp,
hall sometimes denote the positive-frequency ware function CO by the symbol *(e).
2)
The transformation (8.21) is not the only one for which the equation ct.:* is transformed into HIP =co.*.
The transformation * p .will also perform this function. But if we demand, in addition, that the sign of the
charge changes in the equation of motion of the electron in the electromagnetic field [transition from Equation
(11.1) to (11.4)3, then (8.21) is the only transforrhation possible: ..-?
70
where. p Is the momentum operator and p is the translation vector. Since
and
it fcillows from (8.25) that
and, therefore,
=
CC* C2 I,
KO)
0131 p.
Similarly, the infinitesimal rotation operator is given by
a) 4/,
where 6 is the infinitesimal rotation vector, and I is the rotation operator related to the angular momentum
operator.M by
J =--- iM.
The angular momentum operator of the electron consists of two terms,
1
L-FT z,
where L is the orbital angular momentum
and Vi is the spin operator. Since
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= frpl,
(8.26)
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?
we have in this case
and, therefore,
We obtain a different result for the inversion operator.1) According to (7.22) the inversion operator is
(8.27)
where Ir is the operator which changes the sign of the coordinates according to
Since according to (8.19)
cpc ? f3,
it follows from (8.25) that the inversion operator 1(p) differs from the electron inversion operator I in the follow -
ing way:
(8.28)
In order to investigate the meaning of (8.28) we need only compare the electron and positron states whose wave
functions coincide. Then the energy, momentum, and angular momentum of both particles are the same, whereas
the parities are opposite.2)
V. Berestetsky, J. Expt1.-Theoret. Phys. 21, 43 (1951).
2) We note that if the inversion properties of spinors are defined according to (7.8'), that is if I = ilrI3, then
I()) = I. This has no effect on physical properties related to the parity conservation law. The pariEjt operator
s of the. electron-positron system 110) is the same for both definitions.
,
ID
4.---Charge-Conjugate Function.
It should be emphasized that in defining the positron wave function according to Equation (8.24) we are
not actually remaining within the realm of permissible quantum mechanical transformations. Part of the solution
of the Dirac equation is subjected to a nonlinear transformation (the transition to the complex conjugate is a non-
linear operation). As a result the general solution to the Dirac equation, which can now be written
(8.29)
= ve)-F
cannot be, thought of as a wave function. Furthermore, a superposition of states with opposite signs of their
charges, such as
'1,(e) ?ti)o)
does not form a general solution of the Dirac equation.1) This situation, of course, is of no significance so long
as we are consideririg free electron states with a definite charge. In the following we shall see also that the
necessity for using the transformation of (8.24) does not interfere with the development of a general theory of
electron systems and their interaction with the electromagnetic field.
We note also that (8.24) introduces only an apparent asymmetry in our method for considering electron and
positron states. Let us consider the following transformation of a general solution tope Dirac equation:
or in other words
where
= Cr
7-=
. pc.
(8.30)
(8.31)
It is easy lo see that *' also satisfies the Dirac equation (7.30). The solution (8.29) transforms according io
Thus, s ? sliffers from rI) by an interchange of the electron and positron wave functions. The function *' is called
the solution which is charge-conjugate to *. In Section 19 we shall show the invariance of the theory with re-
spect to the transformation (8.30) under more general conditions.
This is-true because a solution with positive frequencies is not a complete set of functions.
?
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9. Momentum and Polarization Eifonstates.
IL. Blanc Waves.
?
In this section. and the one folioyring we shall consider electron states with a given sign of the charge and
with a given energy. Then. the wav.e function.of the electron is a solution of the equation
et,
(9.1)
where H is the Hamiltonian given by (7.12). Here is an electron wave function r (01 or a positron wave
function PP (0) . We shall drop the time-dependent factor, since it is always of the form
e?iit.
States with definite values of n and c are degenerate. We can distinguish these states by demanding that
their wave functions be eigenfunctions also of other operators which commute with each other and with the
Ha miltonian.
Let us start with a consideration of momentum eigenstates. The corresponding wave functions shall be
denoted by OR. They satisfy the equation
whence
P4)13 Flipp
gip ?-??=?- AeiPr,
(9.2)
where A is a constant bispinor.
The wave function (9.2) should be normalized in some definite way. The normalization conditions can
be chosen in various ways. If we consider the electron to be localized in some large volume V, then the normali-
zation condition is
f 1,612(11:=
In this case
where u is a unit bispinor satisfying the condition
'74
U ipr
ITV e
(9.3)
(9.4)
Normalizing for an electron in an unbounded volume, according to the condition
f14,1,12dr =-- 1
?????..
tli-e-momentum must be considered not exactly, but "almost" definite. With the aid of the Fourier transformation
(8.1), this normalization condition can be written in the form
.P1013 f (k) {2 dk 1.
If (k) differs from zero only in a small (three-dimensional) region dp in the neighborhood of
P
then
k =PI
(k) = ap, k (2r9ft jrd?p,
where u is the same unit bispinor as in (9.3) and (9.4). Inserting this expression in the Fourier integral (8.1), we
?
obtain
tit Irdp fp/.
TP ue .
(2,0% ?
(9.5)
The wave function (9.5) normalized "in an interval dp " differs by a factor Ntirfi from that normalized
"in_a unit interval of momentum", i.e., according to
f(6%6/ dr ._,--. (p ? p').
--- 2. Polarization States.
----- The bispinor u, in addition to the normalization condition (9.4), must satisfy Equation (8.3), or
(cep ? pm ? u==0.
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Let us write u in the form of a column vector consisting of two spinors v and w:
(9.6)
where N is the normalizing constant. The spinors v and ware related by Equation (8.6). The spinor v may be
considered arbitrary, and then w is given, according to (8.12), by
We shall choose the spinor v as a unit spinor, that is as one satisfying the condition
v*v -= 1.
From (9.8), (9.7), and (9.4) it follows that the constant N in (9.6) is givan by
N= 1

11 1 +p2
(9.7)
(9.8)
(9.9)
For a given momentum, there are two possible different electron states corresponding to two linearly inde-
pendent two-component spinors v. We shall speak of these two states as different polarization states.
Two linearly independent spinors v can be chosen, for instance, as eigenfunctions of the operator 112 oz.
We shall consider the spinor index A an independent variable which can take on the two values A = 1/2. The
eigenfunctions which belong to the eigenvalue ? of the operator //2 az, shall be denoted v (A). From the
?11
eigenvaiue equation
vi, (A) =--- p.vp, (A),
and expression (7.3) for the matrix az; we find that p can take on the values
and that
'16
=714
(9.10)
(9.11)
?
Oft
i;
Let us note that the second spinor w Is not necessarily in this case an elgenfunetion of the operator a?
Indeed, according to (9.7)
I 1
'w (A) =-'1, vv. (A') =t m (crPh.w.
t -t- m
If, however, we choose the z axis so that it is directed along the vector p. then
Paz,
(crP)4== 0,
and w is also an eigenfunction of az, namely
1
? a 'W (A) 11'wv (A).
2 z m
These results follow from the fact that the electron spin operator
does not commute with the Hamiltonian
ap p m ,
whereas the operator pE does commute with the Hamiltonian. Therefore, the eigenfunctions of H
ir?N(vP-)
are not not eigenfunctions of Ez, though they may be eigenfunctions of pE, so that
(9.12)
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States for which the bispinor u is defined by (9.10) and (9.11), or by (9,10) and (9.12) will be called spin
or polarization eigenstates. In the latter ease they are in addition eigcnstates of the projection of the spin along
the direction of motion. In both cases, as can be easily shown, the bispinors u belonging to different values of
I/ are orthogonal;
le 1111(X) it, (X) -
Pi
3. Sum Over Polarizations.
(9.13)
In the future we shall often come across the following problem. Given an expression Mpu', of the form
A41,10 up./ Tap. = up/(v') Tvr,up. (v),
(9.14)
where T is some four-dimensional matrix (whose elements are Tv ); the index v takes on the values v = 1, 2,
3, 4, and is related to the spin index X in the following way (see page 55).
the bispinors u and u' refer to different values of the momentum, namely p and respectively; and ii is re-
lated to u? by Equation (7.31). We wish to obtain the sum
S= IM'
p.=_S',=
2
ahem.
Tr.flT4f3
(9.16)
and T is the Hermitian conjugate of T. Expression (0.15) contains products of the matrix elements of the oper-
ators 13T and T +8 connecting the states it and le, namely
(te I PT I ix) (it I T hi!).
HIV
The summation method consists of applying matrix multiplication rules to (9.15). However, direct use of these
multiplication rules is impossible, since the bispinors u entering the expression do not form a complete set. In
-
fact, as can be seen from the Dirac equation in the form (7.30), u satisfies the equation
_p
where
(iyp --I- in) u ,---- 0,
.17) == +14P4, P4 -----' /11.2
that is, till is an eigenfunction of the operator iyp belonging to the eigenvalue ? m. We arc considering tvio
eigenfunctions it of this operator (II = ? 14). However, since the operator iyp is a four-dimensional matrix, it
has not two, but four eigenfunctions. The second pair belongs to the eigenvalue + m. Therefore, by considering
the functions nit A()), which satisfy')
? AuiLe, (A
of the squares of the absolute values of (9.14) over the different electron spin states.
We shall" here present a convenient method for calculating such sums. we obtain a system of functions u.p A which is a complete system.
Since The functions-up A which enter into (9.15) have A = m. However, the sum can be extended also to
A = m if we note?that the operator
u = u*p, ui T u ul*p T u,
the desired sum can be written in the form
7P.
lo Vl ?
S = (1111, p Tu )(uT+ pd,?) = (u ,T u )(u Tui
it IL /L 11 111'9
1,411 Plil
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1) The eigenfunctions tip A with A = ?ni arc simply related to the bispinors u (?) corresponding to the negative-
frequency solutions of the Dirac equation;
(9.15)
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)
UP, ?in ?
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11P
2m
is a projection operator: that is,it multiplies the function um m by unity, and the function so _ m by zero:
Thus,
m?/Tp
2A.
S)(?
PIA/T m 2A UttA. UpA m 2A U14/A')?
RV AM
(9.17)
Before making Use of the completeness of the system uu A, let us exhibit the orthogonality and normaliza-
tion properties of these functions. We shall prove the relation
A 2 2
Up.- Allie As iliLA 0.) tip./At (v) ==.7 uAlts ?
(9.18)
For thit purpose we note that the functions up A (_h e eigenfunctions of the non-Hermitian operator iyp) are not
orthtigonal in the ustial sense, i.e.,
it is true, however, that
0
p.A p! Af (A # A').
Up.A111.1 =0' #
Which follows from the fact that according to (7.32) A satisfies the equation
--u
TIILA yip ? A) = 0.
To establish the orthogonality properties of the functions with different p, we shall make use of the fact that the
A are eigenfunctions of the operator .1/2 E p , namely
es
.4
The complex conjugate of this equation is
E
*
E P =-?
Multiplying this on the right by 8, and noting that B commutes with Z, we obtain
? 1
Up .A ? I
2
Thus, in addition to the usual orthogonality relations for functions with different values of p, these functions also
satisfy
isia up., A, =-? 0 * IL')?
Filially, in order to find the normalization of the functions uo A, we shall make use of expressions (9.6), (9.7), and
(9.8) which are valid for all the u if m is replaced by A. We then obtain
p A
u - u u* pu .N2(e ?w* w ).N2(i
?A pat ?A ?A ?A ?A. pat pit
whereas according to (9.4)
Therefore,
P2
(c
p2
11*A 11 A = N2 (V*A V A --1"'" W?.A ? ? - wiut) Ar (1 (i l?
iL ?
1-1p.AU?A
1 A2
(g -I- Ay A
_22
1? (a + A.)2
?
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Thus, we have proved Equation (9.18).
In view of the completeness of the set of um A, it follows from (9.18) that
E
?1 ?u (V) (,)=
which
A IA
which shall be useful for our purposes.
Let us rewrite (9.17) in the form
v
S E (vi) IT (m _(yo-u AN) ><
, .
NV AA' ressvo,
IT (m - iip')Lav.4?A, ( )
At- ,AA/
Equation (9.19) can be used to sum over ?', and A, A'. We obtain as a result
or
S=
1 VI
ilag/ 2.d IT (In (nt ?
1
S = Sp T (m? T
(in ?
4ae
where Sp denotes the trace (sum along the main diagonal) of the matrix.
4. Calculation of Traces.
(9.19)
(9.20)
Usually the matrix whose trace we want to calculate, as in (9.20), is a sum of terms each of which is a
product of a number of y (j= 1, 2, 3, 4). To calculate the trace we shall make use of the fact that
SP Tit'f32- ? ?
Is a tensor of rank n, and therefore,
u?Aterj, . . ..fintsep. = Sp if, ? . ?
is also a tensor. Since the y; have the same form in any coordinate system, the tensor Sp 2. An yf,4...yi should
i
also be independent of the coordinate system. The Only tensor that has this property is 6th' Therefore, the
desired tensor is composed of terms 6ik.
Then it follows immediately that if n is an odd number
If n is an even number, then
(n = 2k --I- 1).
SP i ? ? ? aPatkatfo ? ? ? I
(9.21)
where j,k, m, . . . is some definite permutation of the indices .11, . . j_ and thea are to be found. The
sum is taken over all possible pairs of combinations ik,I m, ., and the number of tenns in the sum, which is
the number of such combinations, is clearly
711
= 1 ? 3 ? 5 ... (n ? 1).
212(.-)
In order to determine the _ap, all that is necessary is to give each pair of indices ik etc., the same values. Since
and
-= lTj?lj (i nt
\ is a four-dimensional tensor of rank n (where 11 isthenumberof factors). This follows from the fact that according
the trace will be multiplied by i 1, depending on whether it takes an even or an odd number of transpositions of
to Section 7 the indices of the yi to bring the product
'Wigan& ? ? ?
83
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Into the form
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IA ? ? ? Ti,,'
Further, since the trace of the unit matrix is equal to 4,
it follows that
Sp 1= 4,
SP ? ? lj n= I-4- 8 ik8lin ? ? ? (it -= 2k). (9.22)
The rules for finding the coefficients _ap can be formulated in the following way.
Each matrix vi is made to correspond to a point j on a circle. The points on the circle are ordered in the
same way as the matrices in the trace. Pairs of points are joined by straight lines. Then to each line joining
points i and k, corresponds a factor 6 and to each way of joining the points (or in other words to each way of
breaking up the indices J, k, I, m, n, p, into pairs j ?j. / ?jr. n ? p, . ..) there corresponds a term
( ?1) 7 6 6 6 ..., in the series for thc trace, where P is the number of intersections of these lines. Thus,
113. JUI np. ?
The values of Spy; . y with n = 2, 4, I. are
I c.
olliak =80,
1 c
T 8fk8tm+ 8i7,&41-8it8km,
Sp lakTamTris = 8i1c8Irts8rR+ 8ft:611181ms 840/a6tnr 8im8klar8
- F- 'dia8kr8lm ?6il8kr8ins 8i?i8kR8/r 8081117;81R
80,81r8ms 8{18ktn8rs? ail8AR8inr
6fr6k16m8-8ir0ka812n afs87clic81r?
(9.23)
We note that the matrix, yin can be simplified if among the indices ja there exist two which are
Y.11.
equal, h., -= J8 =_J and the surri over thEm is taken. Equation (7.27) can be used to transform the product
84
111 ? ? ? Tin
41
so that the two matrices y occur next to each other. 'Chen, since
Tiyi 4,
the trace of this matrix is 11(11 a tensor of rank n, but one of rank n ? 2. For instance, if the two matrices yj are
separated by one, two, or three factors, the following relations hold (summation over j from j = 1 to J = Cis
Implied):
Ifferkii= 48a,
TiTiTkTai= ?2Takii. 1
(9.24)
? 10. Angular Momentum and Parity Eigenstates of the Electron.
1. Orbital and Spin Functions.
Let us go on to a consideration of eigenstates not only of the energy, but also of the angular momentum,
The eigenfunctions of the square of the angular momentum M2 and of its projection Mz, belonging to the eigenz
valnes
*2=i(i+
M, ==M1
shall be denoted by 0 ;,,?. They satisfy the equations
IPA
ht906m = 1)oom, I
mhim /14,11/kr,
(10.1)
where M is the angular momentum operator given by (8.26).
As in Chapter I we shall not solve Equation (10.1) directly, but shall make use of the quantum Mechanical
rules for composition of angular momentum to find the desired wave functions.
We shall consider the two spinors making up the wave function *im separately. Let
(Tim )
YIP!'
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? Both 9* and xiiA satisfy Equations (10.1) with the four-dimensional matrix E replaced by the two-dimensional
- one a; it.,wit1iiiic angular momentum operator ti written in the form
The eigenfunctions of the orbital and spin angular momentum are known. We shall denote the eigenvalues
of L2 and L as previously, by
L2=1(1+1),
nt
and the orbital wave function by This latter can be represented in the form
(Pim = a (r)r
"' (??)'
where a (r) is a radial function which shall be determined later. The eigenfunctions V,L (A) of the spin angular
momentum belonging to the eigenvalues of 1/2 az and( 1/2 a) 2 equal to
az
( )2 (
7+.1),
were found in Section 9. According to (9.10) and (9.11)
We note that an arbitrary spinor 9 can be expanded in a series of orthonormal spinors v 1) according to
?11
1) It may be said that the vti form a "spinor basis" analogous to the in Section 3.
86
(10.2)
We shall call the 911 the contsavariant components of the spinor. Inserting the explicit expression for .y (X), we
find that
(1) =,P1.
which means that the conaavariant components of the spinor defined by (10.2) are the same components as we
have been using on the basis of the 'definition (7.1)..
In addition to the contravariant spinor components we can make use also of the covariant components,
which are defined by the condition that the scalar product of two spinors 9 and is be of the form
'7=111197w
Since the scalar product of two spinors is given by the relations)
cpq =_? cp1A71-1/. T-1/11-4-1/2,
we have
PIJ = 1)P.-thCP-11.
2. Spherical Spinors.
(10.3)
Let us now return to the problem of determining vim and xjm. We shall do this by the same method as
we used for the photon in Section 4.
The function 9 may be considered a scalar function in the generalized space of the Coordinates icy/. and
spin variable A of the electron. [Different spinor components 9X W ata given point in position space correspond
to values of the scalar function 9 (r X) at different points of the generalized spin space. j Our'problem reduces to
finding the-wave function of a system consisting of two noninteracting subsystems (the orbital and the spin degrees
of freedom). The angular momentum of the spin subsystem is always 1/2. Therefore, according to ihe rules for
composition of angular momentum, a given total angular momentum J can be obtained only for two values of the
orbital angular momentum
The wave?functions of two different states corresponding.to the same values of j and M shall be denoted by
1) See, for instance, L. Landau and E. Lifshits, Quantum Mechanics (State Tech. Press, 1948) Section 54.
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?fix
These iiinctions can be written in the form of a superposition of products of orbital and spin functions / in and
U:
:4
. ?
tpiim? Cr4h ill (Di (r) u (A) =-- a (r) CP2' '4` 1' 1, M-pfir
alsip.=M IL=
(10.4)
The coefficients entering into (10.4) differ from those in (4.3) by the fact that now $ =1/2. Equation (10.4)
can be written in the form
cfrim= a (r) Rom (Lr ),
where 0,1/ NI is a spit= whose contravariant components, according to (10.2), are given by
?
_ Cr. M-14 aylL
AV .M?p..
(10.5)
(10.6)
The quantity QJ ti shall be called a spinor spherical function or spherical spinor,i) The covariant compouents
of a spherical spinet are, according to (10.3),
Qfiri Ft= ( +t: ?LY1. lif+p
We give here the Valt14-4 of the coefficients C-I131;111 (we shall at times also use the notation
_a&
110 1st ):
? LI.
1
I t_4
I
7..T
1
t?m?!
iii-A1+1
2/?1
21+1
i'mlf+4.
V
21+1
2/-1-1
(10.7)
rhe ,.,..crutitke vibw. cLet.m.t.see also V lbci, Introduction to Quantum Mechanics (Commis-
tztc,-,vritg 19M); V. Berestetsky, A. Dolginov, and K. Ter-Martirosyan,
L Ther Et
M
?
SS
-
?-?
They are normalized so that
(4101 1
(10.8)
The spherical spinors are an orthonormal set of functions, In fact, for different values of J. M. or!, the
functions nil/Aare mutually orthogonai since they are eigenfunc. tioris of Hernitian operators and belong to differ-
ent eigenvirua In view of (10.8) they are also normalized:
id? 7.=:811,83.1,8111fro
(10.9)
3. Angular Momentum Eigenfunction.
In order to determine the radial dependence of the coilm, i.e., the function a (r) in (10.5), we shall make
use of the Fourier transformation for
(r) =f cp (k) eikrdk.
To the angular momentum eigenfunctions there corresponds a momentum-space function cp (k) which is an eigen-
function both of the square of the angular momentum and of its projection. The angular momentum operator in
momentum space has the same structure as in configuration space, namely
where a has the previous meaning, and
We can therefore write immediately
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1
M=L+Tcr,
L ?
'
(ppm (k) = a (0) k\ j?
,
? (10.10)
89
1
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-
The function a (D, except for its normalization which we shall consider later, is determined by the fact that
since the electron has a definite energy ? , it must also have a definite momentum 2 given by
p
We shall therefore assume that ..a .(k) differs from zero only in a small region dn in the neighborhood of k 2.
Inserting (10.10) into the Fourier integral, we have
vim = a (p) p9 dp 5OP" Q jut (n) do.
Expanding ciPI:Ir in a series of spherical functions [see (4.28) ), we obtain
(f) = a (p) p2 dp E (pr) Y1',' (-r-r)Y. (n) Q11(n) do.
I' tit'
If we insert the expression for the spherical spinors in terms of spherical functions [see (10.6) into this expression,
then it is easy to see that
and, therefore,
(Lr)S gip, do _1,91'm ()
cp jar .---- a (p) p2 dp gt (pr) JiM
Comparing (10.11) with (10.5), we see that
a (r) = a (p) p2 dp g1 (pr).
Since viim (k) is given by (10.10), we obtain s
Xim VI) 'fil!?1)nkt !;I) gilm
(10.12)
On the other hand according to (10.1) )(Dm, as well as pm, is an eigenfunction of the operators 142 and M.
,
and belongs to the same eigcnvalues 1 and M.. This means iliat its angular dependence is given by a spinor
spherical function. From this we can conclude that the product
n) Qji (n)
which enters into (10.12) is a spherical spinor. Since according to (10.6) 1l4114 contains the spherical function
Yjta. it follows [see (4.13)1 that
contains the spherical function Yrra., where I' =-- 1* 1. Of these two values of 1'. only one; namely
=2J-1=
{
1+ 1 for 1=] 1
1- 1 for
(10.11) is compatible with the rules for composition of angular momenta with a given value of J. Thus, it follows that
To determine the second part xim of the electron wave function, we make use of the fact that in momen-
tum space x (k) and cp (k) are related simply by Equation (8.12), so that
90
oh
X(k)=-E4-7i-np(k).
crn@ Jim = qQjvx (1' = 2J? 1, q =1).
(10.13)
The validity of (10.13) is also easy to see with the aid of direct application of (10.6) and (4.13). Then the co-
efficient q is seen to be unity. In configuration space we obtain from (10.12) and (10.13) an expression for the
x Jim, which is analogous to the transition from (10.10) to (10.11):
)(Jim = a (p)p2dp +P In gr (pr) 541, M (et= 2J-1).
(10.14)
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4. Parity of a State.
? Let us note that cp,m4 and xiim belong to. different eigenvalues of LI (the former to 1 (1 + 1), and the
. ?
latter to 1 '(1 t + 1) ]. This means that the electron wave funttion
4, (4' j1M)
Xjiltf
is not an eigenfunction of the orbital angular momentum. It is similarly easy to see that L2 does not commute
with the Hamiltonian H. We thus arrive at the conclusion that the separation of the angular momentum into its
orbital and spin parts has only limited physical meaning for the electron, as for the photon. This separation,
however, becomes valuable in the nonrelativistic limit. As is seen, for instance, from (10.12), the ratio of
x to co goes to zero as k?.- 0. This means that for low energies we may make use of a wave function consisting
only of two compopents (since the other two are small). Then 1 takes on the meaning of the orbital angular
momentum. In the general case, however, the index! on the wave function serves only to denote the two differ-
ent electron states with the same values of j and M.
The quantum number 1 can be uniquely related to the parity of the state. Since the inversion operator,
according to (8.27) and (8.28), is different for the electron and positron, we shall first consider electron states
+ 1) .
Let us apply the operator of (8.27) to the wave function Since
where 1 denotes the two-dimensional unit matrix, we have
frYitilr
jib =_- I I-)
JIM rr X gm r?XjIM ?
Further, since .1 contains the spherical function Y1 , , and since
X and the function
?
Ir Yim =L- (--1 )1 Ylns,
Yrtn=- (? l)1' (? 1)1+1 yym,
- ' .
we arrive atl)
1) We note that because the factor 13 appears in the inversion operator, the wave functions Ojim would not be
eigenfunctions of the operator I if (ppm and xiim belonged to the same value 1.
?-
92
?
'2_ ;,Ta ,-' : 111
4.?
I 03, 1)1tgtit?
(10.15)
Equation (10.15) determines the parity of the state in terms of the quantum number 1. Since the two values I
for a given j differ by unity, of the two electron states belonging to the same values of j and M one is even and
the other is odd.
For positron states, it follows from (8.28) that
(10.16)
In the expression we obtained for the wave function [see (10.11), (10.14) ] the factor a (p) has so far re-
mained unknown. It can be determined from the normalization conditions, which can be written
or, using (10.10) and (10.12),
From this we obtain
f 1040E12 dr -,---(2n)2 f (k)12 ilk.-- 1,
(2108 P2 dP I a (17)12[1 nt)2]= 1-
1 1 1
a (p)---- y ? (1 + nt)
Pio-. 2 e Irp2 dp
(10.17)
Inserting (10.17) into (10.11) and (10.14), we finally obtain an expression for the wave function of the
electron which is an eigenfunction of the angular momentum, parity, and energy (in order to emphasize the
latter, we shall provide the ftinction with a fourth quantum number c), namely
1
s" (20% if (a m) p d I
2
g1(pr)Q31, m
T/
I (s? m) pdc
g' (pr) Q1(1' = 2]?
X?jim? (2)'1,
2
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(10.18)
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7
We here note the expressions for the wave functions for other normalization requirements. If the normalization
conditions are such that
J. 41: Pm +iv jim dr =8 (e_ e),
then the wave function li?sc PM differs from (10.18) only by the absence of the factor . If the wave function
is normalized for a sphere of radius R, dp should be replaced by j-, or de by 5, in (10.18).
? ?
. 5. Expansion in Spherical Waves.
We have now constructed two different complete sets of functions *pti and ;pc pm. Any solution of the
Dirac equation (with a given sign of its frequency) can be expanded in eiTher one of these sets. Let *(k) be' the
wave function of an electron in momentum space. Then
(k) = c7.1,? (k),
where n represents the set of quantum numbers pi/ or E J1M.
If the (k) are normalized so that
we obtain
(2708 f4 (k) (k) dk =
Cn=-- (2708 (k) tr.(k) dk .
(10.19)
(10.20)
? In particular, we can expand a momentum and polarization eigenstate in angular momentum and parity
eigenfunctions, obtaining
or the inverse expansion
94
I &NI, ?
'rp? IVAN
? gAr
? PI' di
?sjiht ? Cishif T
2111.
According to (10.20), (10.18), and (9.5)
or since._
and
we have
(crtm)* (1 [ (Pi) +w--12_71 (-0 lifh"
ap
w ?
Is' + M 14
Ril' M = ( cr 12.) fajil
C1Ar (CA)* = dop.
(10.21)
The quantities IC=1-1-MI 2 determine the angular distribution of electzons in angular momentum eigenstates. If we
?EP
sum this quantity over spins, we obtain
liCgri2=.1 Qjim12 do. (10.22)
IL
For a given j, equation (10.22) does not depend on 1, since
Q*. Q j ? *
Jim tm ? (Capmern)(angivm)=1Q3111?12.
An expansion of the form (10.19) is valid also for configuration-space wave functions, so that
Oaen??
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(1)(r) C"41? (r),
(10.23)
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s
'-
where the coefficients C11 are given by (10.20) as previously. In particular, using (10.21) and the explicit ex-
pressions for 0 and 0 we obtain the following expansion for an electron plane wave in terms of spherical
Pit
waves:
(
ilepr ._-=1(1712;tm (r--;-3 vi T if In
jar 1 ? 7 g p (Pr)Qar m kr)
( r \
)11 1 - I --ni-i g i (Pr) Rolf ("ri) , (1'=2J ?1).
11. The Electron in an External Field.
1. Dirac Equation in an External Field.
(10.24)
The theory of the interaction between the electron and the electromagnetic field will be developed in later
chapters. This interaction leads in general to the creation and annihilation of photons and electron-positron
pairs. Within the framework of the single-electron theory, however, which we are now considering, we can treat
a more limited type of problem. In this type of problem the number of particles does not change and the inter-
action can be introduced on the basis of the external field concept.
The equation of motion of an electron in a given external field is easily obtained in the same way as it is
done in nonrelativistic quantum mechanics. Let A be the four-dimensional potential of the external electromag-
netic field (A is the vector potential, and Ao = ? iA4 is the scalar potential). We obtain the desired equation if
we replace the four-dimensional momentum operator p by p ? eA in the Dirac equation:
p p ? eA .
Obviously, the equation remains relativistically invariant, since the transformation properties of p and A are the
same.
Thus, the Dirac equation in an external field becomes
p ? fit.)11). O.
Let us restrict our considerations to the case in which the external field is time independent. Then there
exist stationary solutions of the form
tp (r, t) = 4'o(r) e7?wt,
where 00 Cr) is an eigenfunction of the Hamiltonian, .i.e.,
96
Fftlko==401
H p Pm eA0? eaA. 1 (11.2)
A general solution of (11.1) may be represented as a superposition of wave functions with various frequencies to.
There is a significant difference between the values of the frequencies in this case and in the free-electron
case. In the absence of an external field, as we have seen in Section 8, we obtain a continuous frequency spectrum
with a discontinuity between ?m and m, namely
Or
m < oo,
eo ni the
coefficient in the exponent of (11.13) is seen by (11.14) be imaginary. In this ease the second of the boundary
conditions (11.12) is satisfied independent of thc value of c The asymptotic expression for the radial functions
(11.13) can then be written, as is easy to sec, in the form '
where
rg=-- c11/ 1 +?iTta cos (pr +8'),
rf sin (pr -F 8%,
p A =17'2? m2, ,
and ci is a constant determined by the normalization. The phase angle 6' depends on the external field as well
as on the energy and angular momentum of the electron.
When c < m, the quantity A is real, and therefore, the second terms in (11.13) increase exponentially..
Then the solutions given by (11.13) satisfy the boundary condition at infinity only if
II(e) = (11.17)
The roots of (11.17) determine the possible energy values. Thus, as has been previously, asserted,. we obtain a
continuous spectrum when > m, and a discrete spectrum when c < m., The latter may be absent if (11.17)
has no roots.
4. Level Behavior as a Function of the Potential-Well Depth.
_ Let us now, return to a consideration of the possibility of separating the solutions of the Dirac equation in
an external field into electron and positron states. For this purpose let us consider the frequency spectrum in the
special case of a spherically symmetric "potential well"
V = ? Vo for
V = 0 for
where Vo is a constant. This simple example can be used to clarify several general aspects of the discrete spec-
trum.Equations (11.1) become
'L. Shiff, H. Snyder, and J. Weinberg, Phys. Rev. 57, 315(1940).
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,.?
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.;
-` d " le ?
;Tr- (rg) + 7 (rg)? (co 4- m vo) (rf) = 0,
-0(rf) ? 2T`...(rf)-1- (o) ? m+ Vo) (r g) = 0;
i+ii(rf) ? 2-:.(rf) -I- (co ? m)(rg).?0;
r < ro.
The first of the boundary conditions (11.12) now refers to the first two of these equations (the region r < ro), and
the second one refers to the second two equations. In addition, f and g should satisfy the continuity condition at
r ro.. Instead of the second boundary condition Of (11.12);
we shall use the condition
r f,rg* co
fr. g =-- 0 for r R (R >>r0.).
This corresponds to placing the system into a large spherical "box" with impenetrable walls. In principle, chang-
ing the boundary condition in this way has no particular meaning. It is, however, convenient in practice because
it makes the whole spectrum discrete (with very small separation between adjoining frequencies in the region
I co I > mc2). In this way we can follow the variation of each level as we vary the depth of the potential well
Yo'
We shall restrict ourselves to the case x rf from Equations.(11.18), we obtain a second
order equation for rg, namely
d2 (rg) ...L.
dr2 [(0) 4_ vor_ #12] (rg) 0, r
- ;
The function rf.is determined by sz :from (11.18),in both regions* It then also satisfies the boundary conditions
at _r =s0 and at r = R. Thus. the ?condition at the boundary r =Ai, when written in the form
(L) ? f
0%
g)r=r0+0
contains no arbitrary constants. It is the characteristic equation which determines the possible values of a) and
replaces (11.17) in our case.
Figure 1 is a schematic diagram of the dependence of the frequency spectrum on the potential well depth
_Nfo for a fixed potential well radius j, as obtained from the solutions of Equation (11.19). We see from the graph
that when Vo < Vo (1), no bound states (I a) I < m) exist. The frequency spectrum consists, as in the absence of
an external field, of two regions co > m and co < m (we shall call them the upper and lower continua). When
Vo > Vo (1), the lowest level of the upper continuum lies below in,. which means that there exists a single bound
state; when .y = yo (2), there appears a second bound state, etc. The,frequency of each bound state decreases
continuously as V.0 increases, becoming negative at Vo = V'o . Nevertheless, these states can be considered
electrou states, since by adiabatically varying the external field we can "return" this state to the upper continuum.
.44 :difficulty arises at Vo = Vo (k), when the level crosses the boundary co =? m and joins the lower continuum,
which is the set of positron states. The problem of the behavior of an electron in a potential well whose depth is
greater than Vo 04) cannot be solved wi,thin the framework of single-particle quantum mechanics. Later (see
Section 17) we shall show that in this case the external field absorbs the electron.
?_ 12. Electron Motion in the Field of the Nucleus.
- 1. Solution of the Radial Equation for the Coulomb Field. Discrete Spectrum.
A most important application of the Dirac equation is the study of the motion of the electron in the field
of die nucleus. The latter is not strictly a central field. The deviation from spherical symmetry is due to the
fact that nuclei generally possess electric quadrupole and magnetic dipole (as well as higher multipole) moments.
If we neglect these effects (which produce a hyperfine structure in the electron levels), then we can consider
electron states which are simultaneously eigenstates of the energy, angular momentum, and parity. The wave
function then is of the form (11.6), and the problem reduces to, the solution of ,the ,equations for the radial- tune,?
tons (11.11).
At large distances the nuclear field is.Coulombicl)
:Ze2
r '
(r > ro), (12.1) '
1). In this section and the two following, we are using Gaussian units, rather than Heaviside.
105
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where Zc is the nuclear charge, and 12 is its radius. We shall first consider the problem in the approximation
in which the finite nuclear dimensions may be neglected, and shall consider expression (12.1) valid in all
space. =? V xi? Zie4. (12.6)
In agreement with the asymptotic behavior (11.13) of the radial functions, we shall attempt to find
solutions of Equations (11.11) in the form
+ F2),
F2).
m Ze2\ F
-p)
Ze3
T F2.
t
(12.2)
I
variable
(12.3)
29
(12.4)
(12.5)
Only the + sign In (12.6) is compatible
forth use the positive value of 7..
Thus, p 7F1 and p?Y_Fx are finite
we obtain a second order equation for
where
The solution of (12.7) which is finite for
which can be represented by the series
F
Thus,
F2 =
with the first boundary condition
at p = 0. If we eliminate
p Y,, whose form is
Id' d 1
dp a j
b=214-1; a=i?Zs'1..
p = 0 is, as is well known,
00
P(b) v r(dn)P's
b;
of (11.12). We shall, therefore, hence-
one of the functions, say Ft. , from (12.4),
(P-7F2) = Of (12.1)
the confluent hypergeometric function,
ru=--11 1 - F iie-lr(Fi
rf =-- 311 ? Tali e-lr (Fi?
Inserting (12.2) and (12.1) into (11.11), and introducing the independent
p = 21r
we arrive at the following equations for F1 and F2:
Ze2\
d23 ( a
1 ? p )Ft?AT,
dp
dF3 "L? Z?e2) F1-1?
dp P AP
The solution of (12.4) for small p is of the form
Ft= aiPT,
a2pT. j
(a, ircij Z.d r + n) nle
n=0
c (21r)T F ? Z el ,21+1; 2kr) , (12.8)
Inserting (12.5) into (12.4) we obtain algebraic equations for .11 and a2, namely
? Zei a2 ? O.
Setting the determinant of this Set of equations equal to zero, we arrive at
106
wr*r..............,....ron?ClbraiMaltery.iyarrYWIrataTegr.r.an204.*rarrra r?
(IV
where c is a constant. Further, according to (12.4) F1 can be expressed in terms off2 andthing a recursion
P '
formilla for the hypergeometric function
we obtain
P4F(a, b; P)-'4V(4-1-11 p)--F
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?
y? Ze2
171? c m(2
?T (7- + 1? Ze2, 2y-1-1; 2Xr).
?11.1-Ze27
(12.9)
_ ' .itt,
?
.
We have seen that when c < in, that is for real values of A. these solutions have meaning only for those
discrete values of E which are given by (11.17). In order to find those values, we shall make use of the asymp-
totic ,cxpression fpr Ihelygergcometric funstion
F (a, b; e-ixa P(b)
r (b ? ra
?I.
r (b) ?b
ep
P (a) r
It follows from this that in order for the exponentially increasing terms in Fi and F2 to vanish,
/? Ze2-L
1
1
?0,
P(i ? Ze2 T1)
1
Ze27 r(T ? ze2_,_ .4_1) (Ze2Ln ? IL) P ? ze21_.)
. X
,
=0.
(12.10)
If Ze2?m ?X # 0. then for K < 0 the setorid of donditions(4 with th
.11) is identical wie first. Since the poles of
- X ? ? ?
the r-function are the negative integers and zeio, (12.11) becomes
y ? Ze2?
A =
$
? ? m
where n' is a nonnegative integer. When ' x Ze2 r (12:6) gives
Ze2
which
-
which means that n' = 0. Then the second of conditions (12.11) is not satisfied: Thus,
n' =0, 1, 2, ... when x < 0,
1, 2,... when x >0. f
(12.12)
(12.13)
1
- Solving (12.12) for c, we obtain
rn
, Z2e4
(11'
?
(12.14)
Equation (12.14) gives the fine structure of the hydrogen atom. When Ze2 o,
aX
= ( c7', Li) is the expectation value of the operator L in the state (1).
a Ap
We see that in quantum electrodynamics it is not the operator a which vanishes, but its expectation
x p
value in the states (P.
Or
From (15.27) and (15.27') it follows that
(4), (ct_ich(c8+1c4)(p)-FR, (ct-f-ict).(cs?ic4)41)
((i), (c:c3 ct c4) 10) = < c: c e > < c t c4 > --= O.
We note that it does not follow from this that 123 = t14 = 0. On
exists no state of the field for which .113 = N4 = 0.
Let us now determine the eigenvalues of the energy operator (15
making use of (15.29), we find that the eigenvalues of the energy are
146
(-15.29)
the contrary, as we shall see later, there
.25). Noting that cxcx+ = cxtex + 1 and
VI 1
= IitoNk -I- 2i ?2 fito.
=1. 2 if7X.
(15.80)
We see that if we ignore the divergent sum E h td the electromagnetic field energy is actually given
k X
as the sum of the energies of separate photons. The integer ix Nkx gives the number of photons with wave
vector k, polarization X (X 1, 2), and energy lita.
The energy of the field is given only by the transverse oscillations (A = 1, 2). Oscillations with X = 3, 4
and the numbers of longitudinal and scalar "photons" N3 and N4 corresponding to them do not enter into the ex-
pression for the energy.
Similarly we find the eigenvalues of the momentum operator, which is given by the expression
The eigenvalues of the momentum are
p fik (cxcif
k,
tik + -21- fik.
k, X
(15.31)
(15.32)
We see that if we ignore the sum EkX 1/2 Ilk, the momentum of the field is the sum of the momenta of the
, ?
separate photons, and that as in classical 'electrodynamics, the momentum depends only on the transverse oscilla-
tions of the field (X = 1, 2).
Equation (15.30) shows that the energy of the field does not vanish when all photon occupation numbers
Nx (X = 1, 2) are zero. The state with Nx = 0 (X = 1, 2) is the vacuum state of the electromagnetic field.
We may, therefore, say that the quantization rules (15.23) lead to an infinite vacuum energy (this energy is
called the zero-point energy).
One might think that the zero-point energy plays no role and can simply be dropped, since it does not enter
into the differences of the energy eigenvalues, which are all that are of importance in energy transfer., This con-
clusion, however, would be incorrect, since as we shall see later (see Chapter VIII), the vacuum oscillations are
of importance in many effects having to do with the interaction of Charged particles with the electromagnetic
field.
In Section 17 we shall see that the vacuum must be defined in the same way also for the electron-positron
field, and that the quantum rules for the electron-positron field lead to infinite energy and infinite charge 'of the
electron-positron vacuum. Just as the zero-point oscillations of the electromagnetic field cannot be ignored,
neither can those of the electron-positron field, since their effect is felt in many phenomena related to interac-
tions between fields.
The interaction of fields with zero-point vacuum oscillations leads to?fundamental difficulties involving
divergent expressions for the energies and probabilities of various interaction processes between electrons and the
electromagnetic field. Nevertheless, as we shall see later (Sections 26, 27), general rules can be formulated for
uniquely separating finite and physically meaningful quantities out of the divergent expressions.
The fact that the vacuum possesses physical properties and cannot simply be considered "empty" space is
an extremely important result of quantum electrodynamics. Effects related to the interaction of electrons with
the electromagnetic vacuum and of the electromagnetic field with the electron-positron vacuum will be given
special consideration in Chapter VIII.
Let us now sum up. It can be said that quantization of the electromagnetic field reduces to considering the
potential as an operator of the form.')
i) We assume here and in the future that h = c = 1.
_
147 ?
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1 1
A =--,-_. E -0 (Ck) eikx C+?
1.17 7; .
1..).
where the operators ex and ex+ satisfy the quantum conditions (15.23).
a a
The operator Ap, just as the tensor field operator Fp I, ? As,, ? ax u Ap., acts on the wave func-
tion p
which describes the state of the field and depends on the number of photons N. Only those wave functions
cp are admissible which satisfy condition (15.27).
As a function of space and time, the operator Au satisfies D'Alembert's equation
AIL.? (p.=1, 2, 3, 4).
(15.34)
This equation, together with the quantum conditions (15.23) and the subsidiary condition (15.27), is the
complete set of equations of quantum electrodynamics for the free electromagnetic field.
From these equations we can now obtain Maxwell's equations, which can be written
We see that the operator
__Lay n
az, I ?XI, " I ax., ==
a
axp.
? a- -I--
OX, r vit oxv
(15,35)
as the ccarresponding quantity in
a
classical electrodynamics, vanishes. As for ? Fp , the situation is not exactly the same as in classical
a?xv
electrodynamics, where ? Fu = 0; in quantum electrodynamics only the expectation value of the operator
ax u ?11
F in the state vanishes.
u 111/
This is related to the fact that in quantum electro.slynamies the subsidiary con-
dition is not ?a A = 0, but is (c1), ? A cl,) = 0; it is this last equation together with (15.34) which leads
a
ax 11 .a ?p x p
? p
'to the second of Equations (15.35).
By taking the expectation value of the first of Equations (15.35) for a state (I), we obtain the classical
Maxwell's equations for the expectation value of the operator F , namely
148
a a a
ax. =-- LI)) f (13* L grD dq, (15.39)
Here 4'4. is the complex conjugate of p, and dq is the product of the differentials of the variables on which 4'
depends; it is assumed here that 4' is normalized according to
(ED, 4:13) 43* dq 1. (15.39')
< L > Le0 = (D+L1 dq, (15.40)
where
and 4 is in general an arbitrary Hermitian operator. The normalization condition can now be written
(41,(13).= f (1704(1)dq==t_-1. (15.401
We see that the norm of the wave function corresponding to this definition of the expectation value may
be negative (an indefinite metric in Hilbert space).
The expectation value of an operator corresponding to any physical quantity should obviously be real. We
shall prove that the expectation value of L is real if it satisfies the equation
where
L*=-- (15.42)
and L+ is the Hermitian conjugate of L. For simplicity let us consider the case of a discrete variable. In this case
< L > nmi
41:1?11??, L,,,i
Since 4 is Hermitian by definition, the complex conjugate of < L> can be written
(13n= < L* > ?
non/ ona
1)--W. Pauli, Revs. Mod. Phys. 15, 175 (1943).
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Thus, 11(15.41) is satisfied, then < L = < L>, and < L> is real.
The operator I,* = 1-1L+ 71 shall be called the conjugate of L, and an operator satisfying Equation
(15.41) shall be called self-conjugate.
When using the indefinite metric in Hilbert space, that is when the norm of a vector is defined not as
f
(I)* (1.) dq,
but as
f(1)1:11(1)(141 operators which are self-conjugate in the sense of (15.41) play the
same role as Hermitian operators in the usual scheme in which (15.39') is used as the definition of the norm.
The general definition of the expectation value in the form of (15.40), based on the use of the indefinite
metric, can be employed in quantizing the electromagnetic field. In this way we avoid the difficulties related
to the longitudinal and scalar 'photons', in particular we eliminate the above-mentioned problem of the im-
possibility of the usual definition i of the expectation values in the state (I) (No, N4) describing the longitudinal
and scalar oscillations.
Quantization on the basis of the indefinite metric is performed as follows.')
We start by expanding the potential A in plane waves in the form
NI 1
Ai(x). I
V .44 if-270-(ckxeikx+ck*xe-ikw)eki, .1=1, 2, 3,
k,
1 VI I
A o(.r)24? (C kxelk? Cekxe?ikal) e
y10
(3.548)
and consider the operators ckx cx and c?kx orx to be conjugate in the sense of (15.42), and to satisfy the
quantum conditions
Ecjc;1= 1, [coeol
.TheHermitian operator n entering into (15.42) is defined by the conditions
which can be written
ej= ct, 1=1, 2, 3,
c*0= ?c+?
o
rict,
S. Gupta, Proc. Phys. Soc. (London) 63A, 681 (1950).
(1-5.44)
415.45)
(15.45')
4.7
We note that conditions (15.44) differ from (15.23) and (15.23') in that they conyin the symbol* instead +.
- From (15.44) and (15.45') it follows that the eigenvalues of the operators c? sIci = c+ici = 1, 2, 3) are
Nj = 0, 1. and the eigenvalues of co. ce=-- -co+ce are - = 0, -1, - 2,...; N1 and NI are the numbers of
"Transverse" photons, and N3 and No are the numbers of longitudinal and scalar *photons".
- We shall assume that the operator n is diagonal in the representation in which c? ici and c' are diagonal.
It follnws from (15.45') that its diagonal elements can be written 2. 1'.
(N1, No, N0, N I Yil N1, No, No, N0)------ileIN,11N.11.1.4,
where the ?I Nj satisfy the conditions
112.734.1 11Ni J=1, 2, 3,
114+1 TIN.?
These conditions show that the operator 11 can be considered the unit operator with respect to the variables
Nj = 1, 2, 3), and to act as( - 1)E0 with respect to the variable No; in other words, the matrix elements of
n-are
(Ni, N2, No , No J i N11, Na',Ne', = )11e8 24,148 N24811.8 48 No ?roe
(15.46)
Let us use (N.0 = 1, 2, 3, 0) to denote the eigenfunction of a state,containing NI photons of the j-th
kind:: Then from (15.40') and (15.46) we find that these functions should be normalized according to
(Arj), (N;)) ---- a . 1 = 1 , 21 3, ?
(4:130 (No), (1?0(ND)=(-1)1411N
0 0.
(15.47)
-Using (15.44) and (15.45) it can be shown that the application of the operators .cx and cx? to 4,4J and 4,0
gives- the followifig results:
cilki (NJ).- YlVi.cIzoi (N.1-1),
c;(13.1(IVi)=Y j =1, 2, 3,
co'to (No) ? irgoito (No -- 1),
4:1)0 (No) VAio Pko (No + 1).
0 = DF(x ? x') 8, (16.25)
1 iota-ado d,:k
Dr (x) (2708 e
(16.26)
In addition to the chronological product, we shall also define the so-called ordered (or normal) product of
A (x). and Ay (x'). In this product, denoted by N (Au (x) Ay(x')), the photon emission operators cx+are on
II
the left and the absorption operators cX are on the right;
N(Al, (x) A, (x')) =? 1
2V IfcicAiei (km+ ki?3')
kl,
CCke(k--ww7)? cte,, etre -k34 e-i (kaH-le 601)
1) R. Feynman, Phys. Rev. 76, 769 (1949).
(16.27)
169
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3
We-note that the ?vactium expectation Value of N (Ail (x) Av (x')) vanishes.
Comparing (16.27) and (16.23) for the ordered and chronological products of Ail (x) and Av (x'), and
using the quantum conditions (15.23), we arrive at
P (A1,(x) A, (x')) ? N(At, (x) A, (x')) DF (x ? x') 8.
(16.28)
This relation will be of importance in investigating the scattering matrix (see Section 22).
Let us examine the function DE (x) in more detail. Integrating (16.26) over the angle between k and x .
we can write DE in* the form
where
DF (x)=-? ulir (8+ (r?it1)? 8+ (?r?It1)1,
1 c"3
8+ (a) ei*EdE=8 17-c
The last equation means that integrals containing 6+ in the form
cc
ff(a)8+ (a)da,
(16.26')
(16.29)
(where
1(a) is a function with no poles on the real axis and which behaves at infinity so that the integral exists) are to
be calculated according to
03 0.
(a) 6+ (a) cl:t =ft(a) 14.t da =1 (0)-F. 11(a) ,
?
-co
(16.30)
where the contour of integration .0 passes above the pole at a = 0, and P is the Cauchy principal part of the
integral. This follows immediately from the definition of the function 6+. In fact,
170
Co co A
ft(a) 8+ (a) da = lim Ida fr (a) f ei?E d4
14 A4.03
-CO ?-.03 0
1
lim 11(a) eied da.
la
A.4.co t
If-we displace the contour of integration into the upper half-plane and go to the limit A --1? co, the integial
with
the term elat? vanishes, and we obtain the first part of (16.30). Further displacing the contour of integration
from the real axis, and passing above the pole a = 0 in the remaining integral, we obtain the second part 01.
(16.30).
Using (16.29), DE may be written
1
. ? DF (x). (27o2Ir [Ira (r r_it 1c8(r+Ii1)+
1 1 -1- 8 (x2).
2 [Ica (1'2-42) +;4--7--la ? ? 2/c1
= (21921
(16.31)
Froin-this it can be seen that DE is invariant under Lorentz transformation, and satisfies the wave equation when
x2 # 0.
We shall later need to Fourier analyze DE, and therefore, let us consider
I
DF (x) ?-.-- f DF (p) e4 d4p,
DF (p)------: kof f D7 (X) C.iPX d4X?
Inserting (16.26) into this and noting that
we-obtain
01 (?-p) tdBX =6 (k
(2
CO
(21104 f (2?k. {8 (k f e-i (wit I-Put)dt).
_
The integral "in the braces can, according to (16.29), be written
+03 .00
f ei (Pa-w it dt ei(P0-`0't di + e-i(Po+w)t dt
'o
= 16+ (Po? w)-1-6+ (?Po?(01.
(16.31')
A ?
171
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A
Therefore,
(P) j[ I6+ (PO? IPI)+8+ (?PO? I Pp] =
1 I 11_1_ 1 Po+Ipl1
(21-75--31 1.7L?Pc)-1P11 ' Pn-11P1+1t8(P?+1111)
2 r
n8 (P3?Pg) ? 1 p
(27c)4i p2 ,pgi1
---(2%) (-2)'
(16.32)
where p2 = p2?
p._
In integrating over Po in (16.31') we must make use of (16.30). This can be done either by integrating
along the real po axis and using the principal value of the integral, or by dropping the 6 -function in (16.29) and
Integrating along a contour passing above the pole pc = I p I and below the pole po = I p I . This procedure
can be formulated differently: instead of passing aFove the' pole A) = p , we lay move?this pole into the
lower half-plane, and similarly, instead of passing below the pole- po =7--- I p I, we may move this pole into the
upper half-plane. This procedure can be formalized in the following way:?in integrating (16.31') over po, the
space part p2 should be replaced by p2 ?it, where c is an infinitesimal positive number, and the contour of inte-
gration shOtild be closed in the lo4ierhalf-p1ane ift> 0, and in the upperhalf-plane if.t 4 0 (see Fig. 4).
With this rule we can write DE in the form
Fig. 4
2 1 1
/0. (16.33)
= (21-7 pi P2 P2 ?
We note that g (x) is continuous for != 0 and
r t 0, whereas the derivative ?a D_F(x) at t = 0 has
a discontinuity of? 21 V ems_ 2/8(x). Therefore,
V hd
at != 0
?
62 ?7
?dt2 u- (x)=-1? 218 (x) 8(1) ? 218 (x).
For t 9L 0,we have according to (16.26)
ODE? (x) 0,
and, therefore, for all values of t ,217 satisfies the equation
172
Dar (x) = 2/8(x).
(16.34)
It is easy to go to (16.33) from this equation, bearing in mind that
8(x) (2%1)4S elP7' (141)*
We note that from (16.4), (16.22), and (16.26) it follows that
DF(x)=-D(1) (x)? is D (x), (16.35)
where c = 1 for t > 0.
? 17. Quantization of the Electron-Positron Field.
1. Variational Principle for the Dirac Equation: Energy-Momentum Tensor.
In Chapter II we studied the properties of the individual electron. To study an arbitrary system of non-
interacting electrons and positrons we may, Just as in the case of a system of photons, go over to an occupation-
number representation. For a system of photons, as was shown in Section 15, the transition to this representation
is equivalent to quantizing the electromagnetic field. For a system of electrons, the transition to the occupation-
number representation is equivalent to quantizing the electron-positron field, that is,,the field given by the Dirac
wave functions 7,0 and V. In doing this, the Dirac equation is considered a field equation like Maxwell's equa-
tions, rather than the equation of a single particle; quantization of the electron-positron field means that the
wave functions and V are considered operators which operate in occupation-number space and satisfy definite
commutation rules. The dependence of these operators on space and time is given by the Dirac equation.
We note that the Dirac equation can be obtained from a variational principle
if the Lagrangian is taken as
8 f Ld4x--,-- 0,
a4i 0,17 ?) 1
L TT? OA", ?xv
(17.1)
In the variation of L, the functions zy and are to be considered independent.
?Having an expression for the Lagrangian, we can use the general equations of field theory (see Section 49)
to define the energy-momentum tensor and the current vector of the electron-positron field. The energy-momen-
turn tensor T is given by
173
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Ts,
aL L6 \
rdi- 04' ? a 1,41
r., kT1'.. ()xi,
- ()xi,
r Ox,,
and satisfies the continuity equation
a
?T
The energy density and momentum density are given, respectively, by
I (i34 O3.
a4, _Lae ,0
g= 21 Vt.
(17.2)
(17.3)
The total energy and total momentum of the electron-positron field are given by the integrals of wand g
over all three-space, namely
Iv c/Bx ? :30 ex,
1 - a
g dsx f tiereuvii (18x.
The tensor T is not symmetric. It is, however, possible to form a symmetric tensor
which satisfies the same continuity equation
1
Op,, T,,,,),
+T )
IL.-- 2 iv. .11
and leads to the same total energy and momentum as does Tpv.
174
(17.3')
Let us now find the current density of the electron-positron field. It is defined by the general elation
(sec Section 49)
jp.= ie qr-q
.\.
? C? 31-q-c dq,.
Using the Lagrangian of (17.1), we must consider the variables 0 and 0,- not 0 and 0*, as independent; therefore,
the current four-vector is obtained in the form')
ie aL sTsle aL)..= kg-71,1,44
d4,
\a 4
a
d?xp, dxp.
(17.4)
(The last term is an abbreviated form of writing ei-ia (yi) as .) The charge density and the current three-
vector are given by
The total charge of the field is
p = =-- ale+, = = era+ ,
We see that the charge density is positive definite (compare Section 52).
2. Quantum Conditions for the Electron-Positron Field.
(17.5)
(17.5')
Let us no go on to a study to the quantum conditions of the electron-positron field. As has been mentioned
above, we must consider the field components 4 p operators acting in occupation-number space and satisfying
certain definite quantum conditions. These quantum conditions, however, differ from those for the electro-
magnetic field as given by (15.23). Indeed, it follows from (15.23) that the number of particles in any given
state may be arbitrary, whereas the Dirac equation describes spin 1/2 particles, which behave according to Fermi-
Dirac statistics, so that the number of particles in a given state may be either one or zero (the Pauli exclusion
principle).
In order to quantize the electron-positron field we shall expand the general solution of the Dirac equation
)(7.30) in the orthonormal set of functions 0j+) and 0 n(-Y, where 0 p(+) and 0 ?(- are the solutions with positive
and negative frequencies, respectively:
This definition of the current differs from that in Section 7 by the fadtor e.
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11
= ca,l-V,s+)+
(17.6)
where the an and an+, as well as the bn and be are considered Hermitian conjugate operators satisfying the
following quintum -conditions:
(ara'o) = 8?,
{141):
farga) = 0,
(4'4') = 0,
fbrb?) ?0,
{14134) =0,
(17.7)
farbs} ?0,
(arb:) = 0,
(arl.b,)= 0,
fa:b: =0,
where = AB + BA (the bracket {AB) is called the anticommutator bracket of the operators A and B).
When no external fields are present, we can expand ;it and ti in plane waves which are eigenstates of the
momentum and polarization: We shall write this expansion in the form
p
p
2
r=21
-I +
far (p) u (p)
r=1
b,. (p) Vra (p)
(17.8)
(x`
1/17
1
1171
where V is the normalizing volume and summation over r (for the values L= 1, 2) indicates summation over the
two spin states; u- and v- ( u3, u4) are constant spinors satisfying the following orthogonality and normaliza-
tion condition.
4
UrP (P)* tee (A = Ors,
P=1
4
10; (P)*V8p (P) = Bra,
p.1
(17.9)
In the scallr product px = px pot, the fourth component of E is defined as p4 =1Po ie = I42 + M2: The
- -
spinors tr. and v-r
are related to .u-and v- by
176
Ur- = ?vr or*p.
0,711
r Jpx,
Without loss of generality we may consider the solution vte - JPX to be the charge conjugate of
so that [see (8.30))
v = = C'- u
(this is true because C is a unitary matrix arid, therefore, the orthonormality conditions for v follow from those
for u).
We note also the easily obtained relation
4
ur*u; =BP, (p,=---- 1, 2, 3, ? 4
r=1 P
where u3 = !1 ti= v2
From the quantum conditions (17.7) it follows that the eigenvalues of
= aar, I?1; = bAr
(17.12)
are either zero or one, and that the nonvanishing matrix elements of the operators a, a +, b , b in the r -r -r
?
repre-
sentation in which N+r and N are diagonal are given byl)
(a0N+_1, N+ = N;t1
r
(a:)NA. 20.-1 ,
r r
(br)N,-: _1. NI
(b;.4-)N_ N_ 1/p--7H.T
r ? ?1 r ri
(17.13).
and we may, therefore, say that the operaiors ar and br decrease, whereas the ar+ and br+ increase, the numbers
N and N - by unity.
-r -r
We shall now show that the quantum conditions (17.7) lead to the correct corpuscular picture. For this
purpose let us determine the energy, momentum, and charge of the field. 'Like the field components, these quan-
tities are operators. We shall determine them with the aid of (17.3'), (17.5'), and the expansion (17.8), where
the a a+, b' b r" satisfy the quantum conditions (17.7).
rr?
Using the normalization conditions (17.9), which are valid for all it is easy to show that
?
1) I This is easily shown by considering a +a , a 4. and using (17.7) together with the diagonality of a +arj . I r
_ -
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2
=-. 11 (ar+a, ? brlht),
r=1
2
(a,t a,. ? brbn,
p r= t
2
Q e (a;Far brb:).
r r=1
Inserting (17.12) and making use of (17.7)1 we obtain
1
p r=1
P ip(N+N? 1),
P r=12
Q=e/l(A/7?N,7 +1). j
p r=i
(17.14')
Expressions (17.14) and (17.14') for the energy are not positive definite. In other words, the energy of the
electron-positron field, both in the classical theory and when the ar and br satisfy the quantum conditions.(17.7),
can have both positive and negative values.
We shall now show that the vacuum state of the electron-positron field can be defined so that the energy
of the field is positive definite.
Let us define the vacuum state as that in which the energy has the :lowest possible value, that is as the
state in which all the N and N are zero, or in other words' when
aar >0
< b7br >0 0,
< ara,t >0 1, I
< brb: >0=1.
According to (17.14') the energy and the charge in the vacuum state are given by
The symbol < L >0 denotes the expectation value of L in the vacuum state.
178
Clo
p r:1
a
I c?
p r=1
?
I
(1'7.16)
Therefore, the energy, momentum, and charge of the quantized electron field differ from their values in the
vacuum by
2
t=7=11E(N;?1.-1-Ntr),
? r==1
P=IiP(A4- -1-Nno
p
Q=1Ie (N7 ?N,7).
? r=1
?
(17.17)
We see that the numbers ?Nr+ and Nr - which can take on the values 0 and 1 enter as sums in the expressions
for the energy and momentum and as differences in the expression for the total charge. These numbers are,
therefore, interpreted as the numbers of electrons and positrons having energy c, momentum p, and a definite
spin orientation.
Thus, the quantum conditions (17.7) lead indeed to the correct corpuscular picture, since the energy,
momentum, and the charge of the field is given by the sum of the energies, momenta, and charges of the separate
electrons and positrons.
If the Dirac equation (7.30) is considered a wave equation determining the various states of an individual
electron, Equation (17.16) can be interpreted in the following way: the vacuum is the state in which all the
negative energy levels are filled. This infinite "negative background" of electrons is not in itself observed, but
under the influence of various external fields the electron can undergo a transition from a negative energy state
(17.15) to a positive energy state (this necessitates, obviously, an energy no less than 2mc2), leaving a "hole" in the
infinite negative sea, and this hole behaves like a particle with a positive energy and a charge with the opposite
sign but the same magnitude as the electron charge.
Such a "hole" in the infinite sea of negative energy electrons can be interpreted as a positron, and the
creation of a "hole' represented by an electron going from a negative to a positive energy state can be interpreted
in terms of electron-positron pair creation.1)
In quantizing the electron-positron field we started with the expansion (17.6) of a general solution of the
? Dirac equation in a series of the orthonormal set of functionsn(4-) and On (?) corresponding to positive and
negative frequencies. This expansion is possible in the absence of external fields (we have considered this in
detail already) and when the external fields are sufficiently weak.
1) It is, of course, possible to treat electrons as "holes' in the infinite positron sea. Later we shall go through a
more detailed investigation of the symmetry of the theory with respect to replacing e by ? e.
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In a very strong external field we nave at the difficulty of separating the frequency spectrum into "posi-
tive" and "negative" frequencies, as was pointed out at the end of Section 11. If the external field is varied
adiabatically, for instance, by varying the potential well depth Vo in the example of Section 11, the frequency
4 which for Vo = 0 belongs to the "upper continuum' > m), moves into the "lower continuum" (co < - m)
(k)
for some value yo > yo ? . When the frequency ab crosses the boundary of the "lower continuum" we must start
considering the state corresponding to this frequency a positron state. Therefore, in the series expansion for the
field operator * given by (17.6), the operator an corresponding to this frequency (we shall denote it by ao) be-
comes an emission operator instead of an absorpon operator. We shall denote this operator by b'o+, so that
ao loot+, v0> VP.
When Vo < Voi, while each electron and positron state can be associated adiabatically with some free
electron or free positron state, the wave function (I)0 in the vacuum state satisfies
an4)0= 01 b,,(1)0=-- 0. (17.15')
This means according to (17.15) that in the vacuum state all the occupation numbers vanish:
cDo o, o, ...; o, o, o, ...), vo< yr.
(k)
When yo > Ito , (17.15') are replaced by
awl:10=0, n *0,
b0=0, Vo>14,k)
boLi-(1'0=--- 0
(11.15?)
(n = 0 corresponds to the frequency 4). These conditions mean that the vacuum now represents that state in
which one of the positron levels is filled, namely
(N.(0, o, o, ...; 0, 0, 1-, 0, ...), 110> V?).
(k)
If there is an electron in the field, then the wave function for Vo < Vo - is of the form
_ _
180
(Di o, 1+, ...; 0, (.1, ...), Vo < VPk)
(k)
and for yo > Vo it is =0, 0, ? * ? If. ...; O. 0, . ? ? 1 ? )1 VO > V.
(k)
Thus, the behavior of an electron in a strong field VD > Vo - is equivalent to the behavior of an electron
in the presence of a positron.
As will be shown later, electrons and positrons can decay into photons or be absorbed by. an external field.
We, therefore, see that an electron in a strong field (V0 > Vo(k)) can be absorbed by the field.
This result is simple to interpret in the language of hole theory. We have said above that the vacuum state
is defined as that state in which all the levels with L < - in arc filled. The occurrence of a new unfilled level
with coo < - in in the field Vo > Vo(-I')means that an electron can go over to this lower level in this field, so that
it is absorbed.
? 18. Anticommutators of the Electron-Positron Field. The Singular Functions
(x(1)(x ), A-F (x)(2c).
1. Quantum Conditions for the Operators *,
Let us find the quantum conditions satisfied by the field operators Op. For this purpose let us use the
quantum conditions (17.7) satisfied by the a, ra r +, b, b+. rr
It follows from (17.7) that
(4la (X) (x1)} =-? 0; F1'et (x) (x')} (X')} = 0.
In addition,
2
u
'4 ' v ,44
I N1 ctr? ip (cc ?co') r?r ?ip (x
(4(x)fp (x')) ?t? 77 hJ
ocivoe
r r=1 ? p. r=1.2
where px = px ? pot. Let us determine the functions
_ -
We shall first find
2
1-71 llar7/0.eiP
1 xi
V .4.4
1) See Sections 33 and 40.
?
r=1
2
114 r?r? ?
VaVi3e,
p r=1
and
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2
Vra?Vte.?
r=1
? (18.1)
We introduce the operator
(18.2)
(18.3)
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1
t-1
II
Since the spinors*
11+
us andys
1 , i)-r ,
lccP m eh e
114 belong to states with negative
ur, r=1, 2,-
r=3, 4,
(18.4)
(18.5).
in2.
frequencies, then obviously
t
I
and, therefore,
But according to (17.11),
4
r y 1 r?
110Up = 2; (ccp + nip -I- u?up .
r=1 r=1
r=1
and, therefore,
2
r
U Up (ccp ?I? n113 e) op?
r=1
Multiplying this equation on the right by ( y4) px and bearing in mind the relation between the a and y matrices,
we finally obtain
The sum
2
I177,?"Orx.
r=1
2
E 1 .
r=1
can be found in a similar way. We introduce the operator
1 ,
e /1/77-1-- In2
1) Actually these are bispinors; see Section 7.
182
?
(18.6)
(18.7)
""rt-le
and in the same way as previously we obtain*
2
Evar-Thr = (-441)v? m)fx?
r=1
Let us now find *S+ andS- Using (18.6) and (18.8) we obtain
??13.
? (x)
=
IV
Introducing the two functions*
p
eiP23= 4-m) 3E efPx1
e? Px = (1"
Ox V 2c
(18.8)
(18.9)
1) Equations (18.6) and (18.8) can be obtained also from the results of Section 9, paragraph 3. We write
2
..r7r m?ur, A=n6
? ua
E ?11aPa an ?tir, A
u, up
2A ---
r=1 r r, a
== E Cm 2P?P.) A74' A'
aT
r,
- and since according to (19.19) (with the substitutions
we obtain
2
v T, vf p, ulth (v).
e r
A A?r, B
? u ? up A ?
4.4 T
E 474 = ? (41Pv ? m)o
Equation (18.8) is obtained similarly.
P. Dirac, Proc. Cambridge Phil. Soc. 30, 150 (1934).
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A (x) =L17 NI
P
P
we write ? iS+ and .--1S in the form
?aB ?aB
If
? as (x) =1.,
eip. sin el 1 f sin Ica
(18.10)
(18.11)
where
(x) = ? !leo) (x).
We note that A (x) and A( 1)(x) may be considered generalizations of D (x) and D(1)(x) which were intro-
duced in Section 16; if the mass m is set equal to zero in A (x) and A( 1)(x), we obtain D (x) and
It is easy to see that A (x) and A(1)(x) satisfy the equations
? m2) A (x) = 0, (p ? m2) A(') (x) = 0. (18.16)
is A A(i) Lorentz The function
invariant
___. eipa
1 (27c)3
1 I COS ti drip,
?
(i,
a
crr,
eip,
c (2isr . a
? (x)? iA(x)),.
m)ap [A(1) (x)? (x)].
Since an invariant, (x) and (x) are under transformations.
where
Using these formulas, we obtain the following expression for the anticommutator?
(4). (x) )j:
(4)? (x)-tfip (4) = ? iSgp (x ? xf), (18.12)
a
(x) .3;1-p (x) s;,-0 (x) ? (7, ? A (x).
il
(18.13)
+ _
The functions ?iS as (x ? x') and ?j.8 (x ? x') have simple physical meaning: they are the vacuum state
?
expectation values of tio? (x) th (x. ) and 48 (xr) 4,5(x). Indeed, using the expansion (17.8) and
relation (17.15) it is easy to show that
<
4). (x) (x) >0
=
is:13 ?
<
(x) 4). (x) >0
=
isc; (x? x'). I
It follows from, this and from (18.11) that
184
< (x), fp (x/)] >0 = ? cx ?
(18.14)
(18.15)
- A (x) vanishes at t= 0, and, therefore, also everywhere outside the light cone when x2 = x2 ? t2 >0. The invari-
ance
of these functions is also explicitly exhibited when they are written
A (x) __,.27ti
(rfeiPxs(p)84-
(p2m2)d4p,
A(1) (x) = TIT---ou if eiPx8(p2+ m2) d4p, 1
Eo
where 6 (p) = .
Ro Ti
(18.17)
2. Definition of the Current. Charge Conjugate Operators.
We have seen above that expression (17.4) for the current leads to infinite values of the vacuum charge and
current. It is possible to define the current somewhat differently in the quantum theory of the electron-positron
field, in particular so that the vacuum charge vanishes. Indeed, according to (18.12) the operators (x) 4, (x')
_?_
and ? (1)0 f, (x) differ only by a c-number, and therefore, the current may be defined as1)
(x) = 1444 (x)] E2-?le (Ydep (TE (x) (x) th (x) TE (x)) =
= (-4 (x) IA) (x) ? (x) 7,..4,(x)),
(18.18)
? where is the transpose of y , that is (y ) as = (yii)s a. This expression, as well as (17.4), satisfies the con-
tinuity equation, but leads to the following expression for the total charge:
1) W. Heisenberg, Z.Physik 90, 209, 692 (1934).
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2 2
Q ? f I 4 (x) d's =-- 4- V 141 04. , ? (14., brp e (N: ? Arr).
p r p r=1
(18.19)
This expression for Q does not contain an infinite vacuum charge, and is identical with the previously obtained
expression (17.17).
Let us now introduce the operators
cf -1 4),
(18.20)
where the?matrix C' is defined by (8.31) and (8.19). We shall call and 0' the charge conjugates of * and *.
They satisfy the equations
(i, k + m) 4/ = 0, 1
Using (8.19) and (8.31) it can be-shown that Cr is a unitary antisymmetric matrix, that is.
and that it satisfies
It follows from these expressions that
ci+c' 1,
(E1-1) 1L c'tI-T/71,4r,
? (C-') C't-P = tYtY) I
and therefore, (18.18) can be written in a more symmetric form, namely
186
(18.21)
(18.20')
(18.22)
11, (x) =Ic (4; (x) Iv (x) ? (x) 41/ (x)).
(16.23)
This expression is invariant under the operation ol replacing * and by their charge conjugates *' and and
of changing the sign of the charge c.
We shall show that the whole theory of the quantized electron-positron field is invariant with respect to
the transformation
, , e ?e.
(18.24)
For this purpose we note that it follows from (17.8) and (17.10) that transformation from *, to the charge
conjugates *', *'
(x) = c' (x) =
is equivalent to the transformation
It is easy to see that the operators
a , b , a +, b +, namely
tIr1.
tor
2
v (P) (P) CiPx br (P) ur (P) eipx }
P r=1
2
710- I (ar(p);r(p)eiPx?Eb (P)Tir (A) e-41"1
p r=1
a a'br a+ ??).a+ b+,
t* l r
tor, br' = ar, br+ = a+
r ?
(18.25)
a', a+, b', b' satisfy the same quantum conditions as do the
r I I I
= (brV) =
{aa} = tar'+a's+) {he.'+We+}
Therefore the anticommutator
th (xi)) , namely
{tg(x) Cx')) is given by the same equation as the anticommuta-
(44,(x)if;(4) IS (x?x').
187.
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Furthermore, under (18.25, the vacuum expectation values of 4). (x) fp (Xi) do not change, since the
primed quantities a', b', a+, b'L.+ satisfy (17.15):
rrr
- - _
< 41-811. >07.-=o Ov
< nriari+ >0 < blrbir4 >0 1.
Finally, since (18.25) transforms N+ into N and -kir into N r+ according to
- - _
1\4.1" -+ Air N r+
it is necessary, in order to leave the current invariant, also to change the sign ole. Thus, the invariance of the
theory under (18.24) has been proved.
3. Chronological Products of the Field Operators.
It will be necessary in the future to have expressions for the various types of products of the electron-
positron field components.
First we shall calculate the chronological products (denoted P) introduced in Section 16, in which by defi-
nition, the operators are chronologically ordered from left to right. Let us consider the chronological product of
*a (I) and * (x')
13 -
(x) p(x'), t > t' ,
P (4 ?1 (x) (x'))
(x') (x), t <
and let us find its vacuum expectation value.
Using (17.8), (17.15), and (18.3), we obtain
Noting that
188
1 1 XI k V---
i ?r r ip (x-x') -
17 Li 41 VeVoe .= ?LS?, )C), tf t.
v r=1
?i4, (x) = ? (y, ? m)0. (AO) (x) ?.i (x)),
(x)),
? iS-0? (x) = ? m) 0. (A(') (x) i A (x)),
(18.26)
(18.27)
and tnaking use of (18.10), let us write
_ where')
o in the form
S( x)=.---- nt) AF (x),
ltl
(x) =kw el" 'Op, C =?? Tip2 ni2
and 6 (t, = 1 if t> t', and .6 (t, t') = - 1 if t.<
In a similar way we find that
(18.28)
(18.29)
(18.28')
- -
The vacuum expectation values of P ( 4 4) and P (4' 4) vanish. We note thatF(x) may be con-
-
sidered a generalization of the function D- (x) introduced in Section 16; if we set m = 0 in (18.29), we obtain
D( x).
In caluclating the chronological products, we expanded ft and 4 in plane waves. We may, however, start
with the general expression (17.16) for * in terms of an arbitrary complete orthonormal set of functions. Such
an expansion is necessary in investigating the interaction between the electron and the electromagnetic field if
the electron is in a stationary external field which cannot be considered a perturbation and must be included in
an exact way in the electron wave functions. In this case* and * must be expanded in the eigenfunctions of the
electron in the external field under consideration. Going through the same considerations as we used to obtain
(18.28), it is easy to show that the vacuum expectation values of 4' and IP are given in general by
42 (Jc) 'ti? (f), 0,
and along the contour C if j< 0 (see Fig. 5).
We note that we proceeded similarly in considering the Fourier transform of 17t (x).
The integration procedure can also be formulated in the following way: the mass m Should be considered
complex with an infinitesimal negative imaginary part,
m m is.
190 \
? ?????????????????.??????????????50.11
(18.32)
This prcicedure can be used both for 46.-F (x)
+.0 and for D-(x), except that in the second case it is
Vp2+ -to necessary to set m ='0.
o
We shall henceforth use this procedure, writing
F .
in the form
I
(p)27 (27,4 p2+ , m-o? m-10.
(18.33)
From this and from (18.29) it follows that
2 1 (4. P. ? In). p _+ ft
F
Sao kp)=-- 7 (2T94 p2 + n1 2
(18.31) and (18.33) can be used to show that tr- (x) satisfies the equation
(18.34)
(0 ? m2) (x) 2/ 8 (x). (18.35)
4. Ordered Products of the Field Operators.
It us now consider the ordered products of the electron-positron field components.
Let us write * and * in the form
tis(x),- u (x)-1-7%; (x), (x) u (x)-1- v (x),
(18.36)
where u (25.) and v(x) are those parts of expression (17.8) for # which contain the positive and negative frequen-
cies, respectively (that is, the terms with e and those with e - ipx ), v (x) and u (x) play the same roles for
the operator Using (17.13) we may say that u(x) and v (/) are electron and positron annihilation operators,
and that u and v (I) are electron and positron emission operators.
InZthe ordered products all the annihilation operators should be on the right and all the creation operators
should be on the left.
Ordering the operators in this way, as we shall see later (see Section 22), is very convenient in determining
matrix elements of products of operators, since in this case the annihilation operators annihilate only those particles
which are,:in the original state, and the emission operators create only such particles which are found In the final
state.
The ordered product of operators *2, *2, which we 011 denote by N (CA ...), is given by the distribu-
tive law -
N(4142(u 7/) 18 ? ? ? ) N0E4)2134)8 ? ? ? ) +N(4)11);\-48 ? ? ? )
and the rule according to which
N(UV 81,XY W,
(18.37)
(18.38)
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where each of the operators U, V1 Z is either an emission operator (u, v) or an absorption operator (u, v); on
the'right side of (18.38) these operators are ordered so that the emission operators are' on the left and the absorp-
?ti9n operators on the right, and 6 is + 1 or - 1 depending on whether the permutation UV...Z 'XY... W is even
or odd. P.
Some examples of N-products are
We note that
IV (4 (x) 47(y)) = N Ku (x) +Tr (x)) (V (.10+ t7 =
=
(t) (Y) (x)) =1-1 (3') u (x) v ( .Y) u (x) +71 (AT' (x) (x) v
IV (41(x) 4/ (.0) = (x)+ (.1/),
IV (4-1(x) -4-1 (A) = (x) t-f) (11).
N(4; (x) 400)-1-NO4(Y) ('x))= O.
It is easy to see that expression (18.18) for the current can be written in the form of the N-product
In fact,
But
.Therefore,
192
j = leN (Ty A)) =--
it:
ie
J1=-- e a; p cppl ?(i) to i+ Vs) (Up Vp)?(Up + Vp) (UK+V?)]
ie
- ,
(Iwo ftlatip U?Vp V?Vp ? Upti, ? I./Avg Vpd, iffiVil
U, Vp + Vp ?=?-?? 0, V,Up + UpV, =?:?- 0,
Uoti p Uptle ??=??? V,Vp VpVE.
(18.39)
(18.39')
(18.40)
=-- le (id ,p(ri ?up-I-T.-107p+ ;up ?
Equation(18.40) gives the same expression:
ie (T1,),(3N ((ph p) ie (yd?0 ( uo-r; vop
The ordered product of operators can be simply expressed in terms of their chronological product. We
shall show, first, that
(it P (x) thi (4) ? N($ (x) 41P (XI)) ----- (x'-x).
Using (17.8) and the quantum condition (17.7), we obtain
8 (i) (P (-4; (x) 4/P (4) N(41a (x) 4'P (4) =
7-1
2
P r=1
From the definition of S- "[see (18.27), (18.28)1 we obtain (18.41).
_
(18.41)
(18.41')
Later, in addition to the chronological product1 we shall need the so-called I-product, which is defined in
the following way :1)
T(+1 412* ? ? ) = 4/ is ? ? ? ?
(18.42)
where the operators *. *. ... are chronologically ordered, so that t, > t, ..., and 6 is + 1 or - 1 depending
1 1 12 -11 -12 P.
on whether the permutation *I*2 ...*, ... is even or odd (among the *1*2 ... there may be both operators
-1-2
of the type *, and those of the type * ).
1) G. Wick, Phys. Rev.-80, 269 (1950).
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Obviously, the chronological product differs from the T-product only by the factor 6 :
Equation (18.41) can be rewritten
T (4,?(x) 4/13 (4)? N( (x) th (4) = Sr, (x' ? x).
Similarly, it can be shown that
- N(4,0 (x') if;? (x)) -_, (x' - x),
T (xf) 4 (x)) ? N(?Pfl (xl) 4 (r)) = 0,
T (4;? (x)io (4) ? (x)trip (x)') = 0.
(18.43)
(18.44)
(18.45)
(18.46)
(18.47)
where X = _
These relations will be proved by introducing the function
0 (x) iA(x).
'
eli"??11-g?
(x),
p. and
r =14
cf3p.
(18.10')
x, We obtain
x(x,
Since
x (x, ? 0 =AM (x)-111
we have
A(t) (x) f x (x, 0 x (x, ?
1
A (x)=-y fx (x, 1)- x (x,
If, in the integral defining x (x, t), we integrate over the angle between
1 0
x(x, .--U (r,
(2102r Or
5. Representations of the Singular Functions.
In conclusion to this section we shall consider various representations of the singular functions A (x) and
A
A (i) (x), as well as the related functions (x) and A? (x) (see below).
The functions A (x) and A (i) (x) can be simply expressed in terms of the cylindrical functions J1 (z),
_ _
(z) and Ki (z):
194
A (x) m2 1 j (m) I + 4?1X
2
=.____ * 8(A) ? IT-It n-l/ -57 1
m2 Nt (n(K) , A>0,
1 4n m "VT,
110)(x) = ni2 Kt On i(jr1) , )1/4 < 0,
27c2 mIrIAI) ?
(18.48)
where
CO
U (r, t) =
dp es (pri.v p, +yet)
1
?
es (tiiinht tcoshi) dt,
?co
(We have performed the substitution 2, = m slth . "The last integral can-be evaluated in terms of one of the
following Hankel functionsI):
CO
go1) (x) --1717 f eat'alt di, x> 0,
1) See Watson, Theory of Bessel Functions (Foreign Lit. Press, 1949).
1915,?
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co
(x)? 4-5 f e-fait??ht dt, x> 0,
-?07
CO
H(1) (Ix) f e? t dt, O.
Weed, replacing E by t + a, U (r, t) can be written
where
OD
U (r, t)=em (Aim i+B...bo de,
_co
A = r cosh a ?1? /shill cc,
B = llinh a + icor, a, Aa ? B2 ri
Let us now choose a so that one of the coefficients Aor B vanishes. There are then three possibilities:
_ _
1) t > r, in which case we set tanh a = A =0, B = + 17-7
- - t -- -
2) - r < t< r, in which case we set tanh a = - =0 .I 2 ? 2
, ;
r
3) t < - r, in which case we set tanh = - ? , A = 0,B = ,
- 1t2 .Z.
t
We then arrive at the following result:
11rf-41)(m17/2?r2), t > r,
co
U (r, f eke (tkInh E kcal E) cit 7r1-ir ?12), I 0,
0, x > 72 ? ? ?
This expression shows that (t) is first acted upon by V (t ) when t = ? co, and then by values of this operator
at later times. Therefore, expression (2,1..34) can be used if we are careful about the correct order of V (t) for
different times: earlier values of this operator should act first. To express this fact, we shall apply the chrono-
logical operator P, already discussed in Sections 16, 18, and 20, to the right side of (21.34), obtaining
V (t)dt
(I) P (e 2?3 ) (I) .(? co).
(21.36)
In this form the expression is always valid. The exponential appearing here can be expanded in a power
series, bearing in mind the correct order of the operators. This series can be written
-i V(t)dt
- ) (? oo).=
co
.E(- )Ju . . . rit?P V (II) V (t2) . . . V (in)} .1) (? 00)
ni
n=0 -co -co
and is identical with the previously obtained Equation (20.15).
(21.37)
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CHAPTER V
THE S MATRIX
? 22. Calculation of the -S Matrix Elements.
1. The S Matrix.
Let us now consider the following general question. Assume that at the "initial" time t = ? co we are
given a state of the electron-positron and electromagnetic fields; in other words, at t = ? co we are given the
number of electrons, positrons, and photons in various individual states. We wish to find the state of the fields at
t = co. This problem is most easily handled with the aid of the interaction representation, in which the state of
the fields is described by the wave function 4) (t) which satisfies (20.6) and (21.32). We take the general solution
(21.36) or (20.7), (20.15), and set t = co, obtaining
We see that the operator
co
?i V (t) \
(? 0'4
co
f v at)
S ? - Pe
f
The S matrix can be expanded in a power series in the electron charge c. Clearly this expansion can be
writtcni)
S =
co
(? Icy& j-
ni ,
co
I dx ? ? ?
where
-co
dx?P (V(n) (xi) V(o) (x2) . . . V(?) (x?)), (22.4)
V" (x) ? (x) 40) (x).
If we write the interaction energy e V (o) ? ej(o) (x) A(o) (x) in the form of a sum
?Il? ?
e V(0) V(e) +V(0, (22.5)
(21.1) where V (?e) and V (i) are the interaction energies of the electron-positron field with the external fields (as well
as photons) and the zero-point oscillations of the electromagnetic field, respectively, then it is. easy to obtain a
power series expansion in the external fields. This expansion can be written
(22.2)
transforms the initial state of our system, which is given by 43( ? co), into the final state, described by (I) ( + co).
Thus, the problem reduces to determining S, called the scattering matrix.
The final state 41( + co) can be considered a superposition of asset of mutually orthogonal states x. If we
are interested in the transition from the state 40 to the state x, we obtain the probability of such transition from
(x, ( + co)), which can be written,according to (22.1),
(x, ( co)) = (x, s (? co)).
(22.3)
Thus, the probabilities for various separate processes are determined by those elements of the S matrix which
connect the appropriate initial and final states. -
230
CO CO co
73.
S = So+ S1-1- . . . =114 ir+n i.dx,n+n X
,,,.on.0 -co -co
X P (V(e) (x1) . . . (x?,) V&) (xtd+i)
t.) We note that the various terms of (22.4) can be written
.cr?,+,)).
fKr, ... (x1, x2, ...) P (A(xi) A, (x2) ...) dx1 dx2
(22.6)
where K ; contains the chronological product of current density operators. Since j satisfies the continuity
Ilv??
equation, we have
aKi4v ? ? ?
axiv ax2,
? =0.
We used this expression in Section 16 for deriving Equation (16.21).
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Here the zeroth order term Se, which does not contain V (?e) (x), determines the scattering of electrons by the
zero-point oscillations of the electromagnetic field. The first order term S1
CO
Sj = ? i f Vie (..x) dx ,
co
VF (x) ( dxi dx,P (x) V(0 (x1) . . . V(0 (x)),
nl
n=o 00
CO
(22.7)
determines the scattering due to the external field in the first Born approximation (for the external field) with
the interaction ofthe dleetrons and positrons with the zero-point field oscillations taken into account. The suc-
ceeding terms of the series correspond to higher Born approximations with this same zero-point interaction taken
into account.
We must bear in mind that unlike the interaction energy between the electron-positron field and the zero-
point oscillations of the electromagnetic field, which can be treated as small perturbations, the interaction energy
with the external field V (?e) (x) can not always be considered a small perturbation. In particular, this cannot be
_
done in determining the energy eigenvalues of the electron in an external field. In these cases one should use the
wave functions of the electron in the external field to determine the current operator which enters into the ex-
pression for the S matrix.
2. Matrix Elements of the Field Operators.
Let us now establish the rules for calculating those S matrix elements which connect any two given states.
We shall start with the general power series expansion in! as given by (22.4), in which the interaction energy
(i)
e V (o) is not separated into V (?e) and V ? as_ described in the previous paragraph. The various terms of (22.4)
are integrals whose integrands contain sums of products of the operators * (x), * (x), A (x)
It is easy to see that *(x) is an electron annihilation and positron creation operator, *(x) is an electron
creation and positron annihilation operator, and A (x) is a photon emission and absorption operator, so that
(x) = u (x) :17(x),
A (x) = a (x) + a+ (x),
(22.8)
where u (or u) is an electron annihilation (or creation) operator, v (or v) is a positron annihilation (or creation)
operator, and a (or a`f') is a photon absorption (or emission) operator.
To prove this assertion, let us consider those matrix elements of * (x), *(x), and A (x) which correspond to
_
transitions from an initial state ( ? co),7..=_ g:11{ N+, N_, N} to a final state ,t.( + co) cl) N'_, ,
where N+, N, N are the sets of electron, positain, and photon occupation numbers, respectively, in
the initial state, arid N'+?N'_, N', are. the sets for the final state. Equation (17.8) can be used to
write the matrix elements of * (x) and ?0(x) in the form
I) Since henceforth only the free-field operators enter the expressions, we shall not use the index zero as we did
in Section 21.
232
(1(+ 00), 4) (x) (E) (? ("3) )=5-k7 E u' (p) eiPai (4) ( 00), ar(p)(1) (?c?))+
r=1
2
14- (1) (? co) ) ;
r r=1
2
(4) (+ co), (x) (1)(? co))= 14-v- Tir(P)e-iPz(4)(+ co), art (P)(1)(? cc))
r r=1
2
-I- I rrr(p) ei Px (10 (-1- co),
r=1
The matrix elements of
(22.9)
(22.10)
ar (p), a + (2), br (2), br+ (E) are given by (17.13). If in the initial state
r
there is only one electron with momentum a and polarization r, the only term different from zero in (22.9) will
be
((+ co), a r(P) ??) )= (ar (P) )01 = 1,
(22.11)
which means that in the final state the number of electrons with momentum E and polarization r is zero. If in
the initial state there was no positron, but in the final state there is one positron with momentum E and polariza-
tion r, then the only term in (22.9) which does not vanish is
(c1) (+ oo), 13;4- (At' (? co)) = (1),t (p))10= 1.
(22.12)
In other words *(x) is an electron annihilation and positron creation operator, as stated above. Similarly, it can
_
be shown that *(x) is a positron annihilation and electron creation operator.
It follows from (22.9), (22.11), and (22.12) that the matrix elements for annihilation of an electron with
momentum p and creation of a positron with momentum p are, respectively,
1
(+ 00), ti) (x) 434. (n)(-00) --= ur (p) cipx,
(22.13)
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(4)1,... (+o?), 4'(x)4)0,7 (r) co) ) vr (22.14)
These matrix elements are .normalized Dirac plane waves with moment p and ?2 corresponding to positive and
negative energies.
Similarly, it can be shown that the matrix elements for creation of an electron with momentum p and
annihilation of a positron with momentum p are, respectively
I ?
(4) + (-1? co), (x) 4' + r ?ipx
(r) 00) ) 57-71/ (P) e
Ro; (,)(+ co), (X) 4) c (p)(? C?)) (P) x ?
(22.15)
(22.16)
Equations (22.15) and (22.16) are based on an expansion of 0 and in plane waves. Such expansions, as
has been noted in Section 17, can be used only when there is no external field or when the *external field can be'
treated as a small perturbation. In general 0 should be expanded, as in (17.6) in electron eigenfunctions in the
external field. In this case the annihilation and creation operators for the electron are given by
_
1
(43 0:-I- tl) ( co), (X) ( I 3 1: (? ??)) = ?i1;E) (X),
R ig- (+ C(1), -tti (X) (I) 0,-11- (--- CC))) =7:141+1 (X),
(22.17)
where on(+) is the nbrmalized wave function of the electron in the n-th state. Similar formulas are obtained for
annihilation and creation of positrons.
Let us now consider the matrix elements of A (x). Using the plane wave expansion (15.33),( EI) ( + co),
11
A (x) ( - w)) can be written*
-
(-1-43( co), At, (x) (I) (? oo)) { exveikx (+ 00), ckx
V
J. X
ex e-ikx (4) (d- 00)) 0;;x4a (? CO) )1 e
112w II
k, X
The nonzero matrix elements of ckX and c+kX are, according to (15.24),
(22.18)
[In equations (22.18)-(22.22) only the k's appearing beneath the two summation signs should be boldface -
_
editor's note.]
234
a
(CIA) N N. =-- Nhx,
kX
(CZ) N +1. == 17. + 1
kX le).
(22.19)
Therefore, the matrix elements of A (x), corresponding to absorption or emission, respectively, of a photon
P -
whose momentum is k and whose polarization is e, are given by
((Do(+ oo), Ay. (x) (big(? 09)) --- eu.elkX
-07-0V
(4)ik (+ oo), (x) 1)0A, (? 00)) ?
1r2il e e-ikx.
w
(22.20)
(22.21)
If, instead of a plane wave expansion for the electromagnetiC field, we use a spherical wave expansion,
then (22.20) and (22.21) can be replaced by
(%(-F 00), Ap. (r) 4)17; C?)) = (A.IMX)F0
(41)170(d- 00), A (x) if)ok (-00)) = (Apir),V,
(22.22)
where-A .MX are the normalized momentum and parity eigenstates of the photon [see (5.19) and (5.21)1
--J
_
Let us now return to the general expression (22.4) for the S matrix. Since the operators 0, 0, A are sums of
single-particle creation and annihilation operators, each term of (22.4) can be written as a sum of operators for
the creation or annihilation of single electrons, positrons, and photons in various states. We must clarify the con-
ditions under which this type of product has nonzero matrix elements corresponding to some definite process
i f. If, for instance, there is one electron and no photon in state i, and there is a photon and an electron in
state 7, then obviously one of the annihilation operators annihilates the electron, instatei , and two creation
operators create the electron and photon in state f; all the other operators can be separated into pairs, with the operators
of each pair creating and annihilating the same particle.
The virtual processes of successive creation and annihilation of an individual particle make the calculation
of the.S matrix elements extremely complex. We shall, therefore, try to transform the S matrix to a form in
which-the virtual processes need not be considered. Clearly, the problem reduces to representing the S matrix as
a sum of products of creation and annihilation operators, such that in each term the creation operators are on the
left of the annihilation operators. In calculating the matrix elements of such products, the annihilation operators
will annihilate only those particles which exist in the initial state, and the creation operators will create those
which are in the final state. As for the virtual creation and annihilation processes, they will not enter explicitly
into fhe_ considerations.
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7
Products in which the creation operators are on the left of die annihilation operators shall be called ordered
or normal products (compare Sections 16 and 18). It is clear that if we know the commutators and anticommu-
tators of the field operators, we can always write the S matrix in the form of a sum of ordered products multiplied
by some numerical coefficients. Ordering die field operators in the necessary way gives rise to additional terms
containing commutators and anticommutators; these terms contain, though not in an explicit way, the virtual
creation and annihilation processes.
3. Representation of the S Matrix as a Sum of Normal Products.
We shall now show that it is possible to represent the S matrix elements as sums of normal products of
creation and annihilation operators by Using algebraic techniques.1)
First we shall give a general definition of the normal product of operators, which is a generalization of the
definition of the ordered product given in Sections 16 and 18.
We shall denote the various terms of (22.8) which are either annihilation or creation operators by U, V,
W. ..., Z. If we are given a product of operators UVW Z, we shall call 62XY W the normal product corres-
ponding to' the given one, where X, Y, W is the same set of operators as in the original product but ordered
so that the creation operators are on the left of the annihilation operators, and the coefficient 62 is + 1 if the per-
mutation of the electron-positron operators necessary to achieve this ordering is even, and is ? I. if this permuta-
tion is odd. As for the various separate creation (or annihilation) operators, they may be permuted in any con-
venient way among themselves.2) We shall denote the normal product by the symbol N, so that
N(UV 81, XY . . . W. (22.23)
The normal product of operators U VOW Z, where u + v is a sum of a creation and an annihi-
lation operator, is defined according to the distributive law
N (U . . . V cpW . . . Z) N (U . . . V uW . . . Z) N (U . . . VW Z).
? ?
The same law is valid if * is replaced by *= u + v or A = a + a+.
Let us now consider the general expression (22.4) for the S matrix. Recalling [see (18.40)] that the current
operator can be written in the form of an N-product
(x) = ieN (4; (x) i4 (x) )
and that, therefore,
eV" (x) =? (x) A (x) = ? ieN (.4 (x) A (x) (x)), (22.24).
1) G. Wick, Phys. Rev. 80, 268 (1950).
2) This is due to the fact that the electron-positron operators anticommute, and the photon operators commute.
236
where
we shall write the S matrix in the form
s(ti)? (?e)n
nl
?co
A (x) =.4,Ap.(x),
x dx,1 P I N (-v A x (4) x
N(tT) (X2) A (X2) 4$ (x2)) ? ? ? Na (Xn) A (x.) (x.)).
(22.25)
The normal products N(0 (x.) #As(x.) (x.) ) entering into this expression must be chronologically ordered.
? ?
? It is convenient to use, instead of P (UV ... Z), the so-called T-product, which differs from the P-product
by the factor 6 = ? 1, namely
T(UV Z) =SpXY ... NV, (22.26)
where 6 ,as in (22.23), is determined by the permutation only of the electron-positron operators in (22.26); the
operators X, Y, W on the right side of (22.26) are chronologically ordered. I)
Since the electron-positron operators enter in pairs into (22.25), the operator P can be simply replaced by
T, so_that
soo= $dx, . ? ? fdx?TIN(47(xi) A (xi) 4)(xi)) ? ? ? N(tIT (x.) A (x.) (x.)} ?
nl
_co
(22.27)
A T-product such as that in the integrand of (22.27), whose ihdividual factors are N-products, will be called
a mixed T-product. We shall now show that a mixed T-product can be written in the form of a sum of simple N-
products of creation and annihilation operators: Let us consider the difference between the T- and N-product of
two operators U and V (U and V can be *, *, A operators or their component parts u, u, v, v, a, a+). Let us de-
note This difference by U--c\r?c , writing 2)
1) It can be shown that the T-produCt, unlike the P-product, is relativistically invariant.
2) Instead of c, we shall often use any other lower case Latin letter.
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U?V? = T (UV) ? N(UV),
(22.28)
which we call ilie contraction of the operators U and V.
In Sections 16 and 18 we determined one contraction of photon and electron-positron operators and saw
that they are c-numbers.
If we know the contractions of the operators, it is simple to transform a T-product into a sum of N-products.
This is done with the following two theorems.
I. If U, V. X, y,?Z are 0. 7/5. A operators or their component parts u, u, v, v, a, a+, then the T-product
of these operators is the Sum Of their N-products with all possible contracted pairs; in other words,
T (UV . . . XYZ) N (UV . . . XYZ)+ N(UaVaW . . . XYZ) -F
N (UaVIVa XYZ) . . . N(UaVbWb . . . XaYar).
(22.29)
Here the different superscripts denote different contractions. The contraction of nonadjacent operators is
defined in accordance with (22.23) as follows: if the contracted operators are photon operators, they are merely
placed adjacent to each other; if they are electron-positron operators, they are placed adjacent, and the N-product
must be multiplied by 62 to indicate whether an even or odd permutation of the electron-positron operators is
necessary.
For instance, if all the operators U, V, ... are electron-positron operators, then
N(UaVWbXbYaZ) == ? (Uan (WbXb) N (VZ).
Here the contractions UaYa and W,-bX-b are o-numbers which we have taken out of the .Nrproduct.
N (UaVWbXrZb) (UaYa) (We) N (VX).
II. A mixed T-product, for instance T (UV N (WXY) ... Z), can be resolved into a sum of N-products
_ _ _ _
similar to (22.29), except that the terms involving the contraction of operators within a given N-product should
be drooped ( in the example T (UV N (WXY) ... Z).we need not take into account the contractions Wa Xa,
Theorem I is proved as follows: note first that permutation simultaneously of the factors in the T-product
on the left and the N-products on the right of (22.29) does not change this relation; we may, therefore, assume,
without loss of generality, that the operators in (22.29) are chronologically ordered from right to left. If this is
true, we shall say that the operators are T-ordered. Then the symbol T on the left side of (22.29) can be elimi-
nated. Let us now order the operators on the left side of (22.29) so that all the creation operators are on the left
of the annihilation operators. The .operators are then called N-ordered. To do this let us take the furthest left
N-unordered creation operator and interchange it successively with all annihilation operators on its left. We
shall then obtain additional terms containing contractions between operators that have been interchanged,
according to
238
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UV = T (UV) = N (UV) --I- U?Va = thVU --FU?V?
Let us now perform this operation with the other unordered creation operators. We shall then have expressed
the original T-product on the left side of (22.29) as a sum of N-products (we can clearly set the symbol N in front
of each of the products). Although these N-products may enter both with positive and negative' signs, if we re-
order the factors within the N-products so that they are again T-ordered, then obviously all the N-products will
enter with a positive sign. We then obtain an expression for the T-product in the form of a sum of N-products,
which differs from (22.29) in that the right side will not contain all possible contractions between the factors, but
only the contractions between pairs of N-unordered operators. Since the contraction between operators which are
both N-ordered and T-ordered vanishes, we may add to the right side terms containing all possible contractions
-
between pairs. This proves Theorem I.
Theorem II is proved similarly. In performing the proof, we need only bear in mind that it is unnecessary
to interchange operators within a given N-product, since these operators are already N-ordered; therefore, these
contractions do not enter into the expression.
In conclusion to this paragraph we present a summary of the formulas for the contractions of various operators:
(Y) (x) = - (y? x);
A: (x) A (y) ? x);
44 (x) = 0, 1
(x)3 (Y) =0,
Ai! (x)oe: (y) = O.
(22.30)
(22.31)
(22.32)
(22.33)
Equations (22.30) -(22.32) were obtained in Sections 16 and 18; the first two equations of (22.33) follow from
(18.39); the third of equations (22.33) is self-evident.
Theorems I and II and these formulas simplify the problem of representing the integrand of expression
(22.2-7) for the S matrix as a sum of normal products of creation and annihilation operators. ?
4-TrAll the ilos and Tp's in equations (22.30)-(22.33) should be boldfaced - editor's note.]
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We nom in conclusion that in the expression for the vacuum expectation value of products of the potential
(16.2) or (22.32), both the longinulinai and scalar components enter similarly to the transverse ones. It is the
former, as we shall see lat.-r. which cause the static (Coulomb) interaction of charges. Previously, before the
development of covariant perreoatien theory, the longitudinal and transverse degrees of freedom of the electro-
magnetic field were mated in csnatially different ways. With the aid of a certain canonical transformation')
the iongimdinal degrees of freedom were eliminated. Then terms corresponding to the Coulomb interaction ap-
peared in the liarliiTnnian? and the free electromagnetic field contained only transverse degrees of freedom. We
shall not make use of this now obsolete method, since it lacks the advantage of relativistic Invariance (which Is
extremely il--cortant for the elimination of divergences in the S matrix) and requires separate consideration of
the =sic and retarded (doe to emulsion and absorption of virtual photons) interactions. In the method we shall
durrie below for calculating the S matrix elements, the interaction of charges is characterized uniformly by
toe D-
23. Graphic Representation of the Matrix Elements.
1. Grap'lic ".-Iresm.tion of Nor=1 Products.
In the preceding paragaph -we have shown that the separate terms S (-n) of the expansion of the S matrix in
powers of the elem-on charge are integrals of mixed T-products of 0, 0, A operators, and that these T-products
can he resolved trri narr.-al piljel:CZ of the same operators. Each of the normal products into which the integrand
of the S (-n) matrix is resolved r-lbe represented in the form of a diagram which is constructed in the following
'way.4
(n)
The forn.-din-i--"Onal vecm-s x over which the integration in the expression for S - is per-
ford are represeat..,,--d by poinrs on the diagram (these points shall be called vertices or corners of the diagram).
A contraction cf pLccon operators A (x) and A (y) will be represented by a dotted line connecting the
vertices x and y.
A connactba of oper----rz--0 (x) and 0 (y) will be represented by a solid line connecting the vertices x
and y and directed from x to y.
- -
The operator A (x) whic-nis not contracted will be represented by a dotted line starting at x and leaving
the diagram (going to
Operators (x) and * (x) -"-h-lc:u are not contracted will be represented by solid lines from x out of the
diacrz-=-?-?'%.71,(x).-this Elle is directed from x to "infinity", whereas for 0(x), it is directed from
-infinity" to .x. lincontracted ooeraton shall be called free.
Since 0(x) is the electron amihiladon and positron creation operator, a solid line directed from "infinity"
to the vertex x is a graphic representation either of an electron which exists before the scattering process, or a
positron created as a result of scattering.")
Similarly. since -0- (x) is a positron annihilation operator and an electron emission operator, a solid line
tram a vertex x to 'infinity' can represent both an electron created as a result of scattering, or a positron which
existed before the scattering.
In addition, since A (x) is a photon emission and absorption operator, a dotted line connecting a vertex x
with "q_nfinity'? can reprnt a photon either emitted or absorbed as a result of scattering.
E. Fermi, Revs. Mod. Phys. 4, 87 (1932).
2) For more details see Chapter VII.
This method was developed by Feynman [Phys. Rev. 76, 749, 769 (1949); a Russian translation can be found in
thesympwinns Problems of Modern Physics Ser. 3, No. 11, pp. 25, 371.
4) 'Th.. poihility of such a representation of the motion of a positron (as an electron moving "backwards in time')
was-first firs' caned by G. Zisman (J. Expt1.-Theoret. Phys. (USSR) 10, 1063 (1040) ).
? 240
A dotted line connecting a vertex x with "infinity" will also be used to represent the "qcternal" electro- ?
magnetic field acting at the point x.
Lines connecting vertices of a diagram and representing contractions of operators can be interpreted in the
following way. Since a contraction of operators contains a product of emission and absorption terms, lines con-
necting vertices can be associated with virtual particles) created or annihilated in the scattering process. An
electron line is directed from its point of creation to its point of annihilation (and vice versa for a positron).
It is clear that at each vertex of a graph there appear two electron lines and one photon line.
Corresponding to the expansion of the S matrix in a series of powers of e, we shall say that a scattering pro-
cess or an interaction process is an n-th order effect if the matrix element corresponding to this process is pro-
portional to c-n. Obviously, all n-th order processes are described by the matrix S ( - , the n-th term in the power
series expansion of S. A diagram representing one of the normal products into which the integrand of the ex-
pression for S - is expanded contains n vertices. We shall call this an n-th order diagram.
PS
Fig. 8
YP2
A single diagram which represents some normal
product of field operators can in general describe several
different scattering processes (see, for instance, Diagram
3 of Fig. 10).
If normal products of operators can be described by
diagrams which differ from each other-only in the indices
associated with the vertices, they are called equivalent.
All these products clearly describe the same set of scat-
tering processes (actually the same scattering process).
(3).
Let us consider, as an example, the matrix S
which describes third order scattering processes. Expand-
ing the integrand into normal products according to
Theorems I and II of Section 22, we obtain, among other terms,
?0 1
dx f dx2 dx2 N 1) A b
(r1) tli (X1)4 (X2) X
3!
X :4b (X2) VI (X2) -471c (x3) A (x8) cr (4),
(23.1)
where the indices a, b, c, indicate the various contracted pairs.
The diagram corresponding to this term is shown int Fig. 8. Assuming that the external lines of the
diagram represent an,electron and an external electromagnetic field, we may say that this diagram represents the
following process: an electron emits a virtual photon (at x1), is then scattered by the external field A (x3) (at x3),
_
and finally absorbs (at x2) the virtual photon it emitted previo-usly. To this process there correspond six equiva-
lent terms in the decomposition of the integrand of 0 )into normal products. Dropping the symbol N and sup-
pressing the arguments of the operators, these equivalent terms can be written
1) The momentum of a virtual electron, as opposed to a real one, does not satisfy the relation p2 + m2 = 0; the
momentum of a virtual photon, as opposed to a real one, does not satisfy the relation Et= 0.
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1
4,(4:i.c-fitto(44144;
.....
4) (I AcpX4)1141)(4) ;
where we have represented the contractions be-
tween electron operators by solid lines, and those
between photon operators by dotted lines, as in
the diagram. Diagrams corresponding to these
terms will obviously differ only in the indices
associated with the points x1.
/ y?Ili( (0)(0(0
r=1,
;1 A _ A _'A
2) 5) (tit. 4)2? 9) cP) ;
Noting that we can move the electron oper-
ators about in the N-product (multiplying, at the
sanie time, by (5 = ? 1) and making use of
opAcpqAcp)
(4.4149g ll(P)(4) ;
E .
,go
rat, E.1
(22.30) and (22.31), it is easily shown that all
six terms are equal. Thus, the matrix element
corresponding to the process represented by the
(cpAcp)(cpA)
diagram of Fig. -8 is six times the matrix element in (23.1).
?
r2, 4=4.
Henceforth, we shall use r to designate the number of equivalent N-products in the integrand of the ex-
_
(n)
pression for S ? . Since the diagrams corresponding to these differ only in that the indices of the vertices are
permuted, r = ntig, where g is the number of permutations of the indices which will not change the form of the
N-product. For instance, in diagram 4 of Fig. 10; g = 2, and in diagram 6 of Fig. 23, g = 4.
2. Various Field Interaction Processes.
Let us go on to a construction of the diagrams and a calculation of the various interaction processes between
the electron-positron and electromagnetic fields.
We shall start with first order effects. In this case, there is obviously only the one diagram shown in Fig.
9. It represents the scattering of an electron or positron in an external field, the emission or absorption of a pho-
ton by an electron (or positron), electron-positron pair creation or annihilation.
On the right of the diagram we indicate symbolic-
ally the integrand of the expression for S (1) (without the
symbol N and without the factor e/1 t), as well as the
numerical coefficient E = .=-, where r is the number of
n
normal products of the given type in the N-product de-
composition of S
Let us now consider second order effects. In this
Fig. 9 case only six topologically inequivalent diagrams are pos-
sible and these are presented in Fig. 10. Next to the
diagrams we describe the scattering processes correspond-
ing to them.
On the right of each diagram
(without the symbol E or the factor
tractions between the operators; the
is a symbolic representation of the integrand of the matrix element in S(2)
e2/2:). The lines connecting various of the factors ?0, A designate the con-
number L gives the number of N-products in the decomposition of S (2), and
Figure 11 gives fifteen diagrams showing all possible third-order effects. Since the number of these effects
is so great, we do not describe them;,merely giving the expressions for the integrands of S (3)' indicating the con-
;
tracted pairs, and the values of r and I =
242
3'
4
el)A0(< p
(p4(p)(Acp)
t--2, 4=1
r I,
Two simultaneously occurring first order effects.
?
Scattering of an electron by an electron by an electron (or position by a
,positron), or an electron by a positron.
Scattering of a photon by an electron; emission of two photons; two-fold
electron (or positron) scattering in an external field; electron (or positron)
bremsstrahlung; pair creation; two-photon pair annihilation.
Interaction between a photon and the electron-positron vacuum (photon
"self-energy").
Interaction between an electron and the zero-point oscillations of the
electromagnetic field (electron "self-energy").
Creation of a virtual pair and a virtual photon followed by annihilation
of the pair and the photon (vacuum fluctuation).
Fig. 10
The diagrams under the number, 10 differ in the direction of the arrows around the electron loop; according
to Furry's theorem (see Section 24, below) these diagrams need not be considered, since the matrix element cor-
respondfng to them vanishes; for the same reason diagrams 15 can also be ignored.
The diagrams presented in Figs. 10 and 11 represent normal products of the field operators A, 0, 0 in
general form, and illustrate many processes. These operators represent the sums of creation and annihilation
operators for particles in various states. Therefore, in considering any concrete physical process, the normal
.produc.t which corresponds to it in the integrand of the expression for S (1-1) (that is, the normal product for the
process being investigated) can in general be broken up into several terms each of which contains products of
creation and annihilation operators for the particles taking part in the process being considered. These terms,
which differ in the order in which the creation and annihilation operators of various particles appear, can also
be represented by diagrams which are topologically equivalent and differ from each other only in the order in
which the electron and photon lines occur in the diagram.
Let us consider, for instance, the emission of a k photon in an external field A (?e) (x). This is a second
order process, and the normal product corresponding to it in the integrand of the expression for S (2) can be written
NE-pi:VA (.rdcP(adj;(x2) 1(a.2)(p(a.3)]
In this expression A should be written
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I;
A"(x)+A("(x)+;1'(x),
(lc)
where A (?e) is the external field, A ? is the emission and absorption operator of the k photon, and A' is the sum
of similar operators referring to other photons. The normal product then breaks up into several terms, but the
nonzero matrix elements for the process will clearly have only two terms, namely ?
4-1-1(4PtrdcPC;)FfiGrillekxdcp(xzli-tiRi(zdii"kx,)(pCx,*sz)/20)(rz)cp(x2g.,
Each of these terms can be represented graphically, using diagram 3 of Fig*. 10; which represents the general
normal product before the concrete form for A (x) has been inserted into it. These diagrams are shown in
244
6
8
Vo020(00
r=1,
(Alie0s5b)Vo
y
r3. 4- 4-
10
004;A19)00
r-6,
(df4)
A
r=6, t= 1
(29)
r = 6. E- 1
(q41(-0 41)4
r=6,
(010(0-0)Wcio
r = 6, F,.I
?A...** ?A
,citmo(pl)w,p)
r6, 4=I
Fig. 11
(040)0pA05
r-6, 4-1
- A
(-)tit;dip)
(.-,AAF(ploq,Aip)
r - 3, &=.
p)01)110(rp Atli)
=.1-
(00 0' soqiv
r - 6, 4=1
qii'Vq7;444
=I
(ibaiNfpler
(k)
Fig. 1.2, and differ from each other only in the order in which the photon lines corresponding to A(& and A ?
appear.
If we were interested in a process in which three
photons take part, then after the normal product was broken
up into terms containing emission and absorption operators
of the separate photons, we would obtain six terms for the
I I nonzero matrix elements. These terms can be graphically
/1\k represented, and the diagrams corresponding to them will
differ only in the order of their photon lines.
Similarly, if several electrons or positrons take part
In the process, the normal product can be represented as a
sum of separate terms, each of which contains creation and
annihilation operators for the electrons and positrons participating in the process, and which differ in the order of
these operators. The various terms of the normal product can be represented by diagrams which are topologically
equivalent and which differ only in the order of the lines corresponding to the various electrons and positrons.
In contradistinction to the case of several photons, in which the various graphs correspond to matrix ele-
ments all of which have the same sign, in the case of several electrons the various graphs correspond to matrix
elements which may have different signs. This is related to the fact that the electron and positron annihilation
and creation operators anticommute with each other, whereas those for the photons commute (we arc assuming
that the electrons are in different states).
This way of breaking up the normal product into separate terms containing annihilation and creation oper-
ators for particles in different states can be used in the general case when several photons and electrons participate
in the process. Then the matrix element corresponding to the process under investigation is written in the form
Fig. 12
sMf =1114Mb
(23.2)
where the:individual terms of the sum differ from each other in the order of the creation and annihilation oper-
ators for the particles of interest, and may also differ in their sign. As was explained above, the diagrams corres-
ponding to the individual terms of (23.2) are topologically equivalent, and differ only in the order of their elec-
tron and photon lines.
? 24.---The S Matrix in the Momentum Representation.
-^
1. General Formula.
To determine the probabilities for various processes with the aid of the S matrix, as well as for a general
examination of its properties, it is convenient to go to momentum space.
(
We shall start from Equation (23.2), which defines the matrix element,Sn).? for an arbitrary process
f
(n)
i f by-means of a sum of terms M.? each of which contains matrix elements of annihilation and creation
?1 ?4- f
(n)
operators for the particles participating in the process under investigation. The M.? differ among themselves
elec-
tron the and
rdpehirtionnwlhiniee.
which theseoperators appear and are represented by diagrams which differ in the order of the
The matrix elements of the annihilation and creation operators are given by Equations (22.13)-(22.16),
(22.20), and (22.21). We shall write these equations here again, assuming that the normalizing volume is V = 1.
The matrix_elements shall be denoted by the symbol of the corresponding quantity, with a superscript (+) or(-)
depending on whether the particle is an electron or a positron.
- 245
-
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The matrix elements of *(x) and *(x) corresponding to annihilation and efeation of an electron with mo-
mentum:2 are given by
151(4.)(x) =U' (p) e'',() (p)
(24.1)
Similarly, the matrix elements of *(x) and ?0(x) corresponding to creation and annihilation of a positron
with momentum 2 are given by
= 1(p) e7inx 71;(-) (x) =-1-7 (P) eii'm ?
Here u?r (2) and v? (2) are constant spinors satisfying the normalization conditions (17.9).
-a - a
(24.2) ,
The matrix elements of A (x) corresponding to absorption and emission of a photon with a fou, momentum
k and a polarization vector e are given, respectively, by
1
4. (x) = ? ep.eikx,
1/2a)
1
312-0; ee-4kx
(24.3)
(
Let us establish the form of the Mn).? in momentum space. For this purpose we shall represent the con-
-1 f
?
tractions of operators, i.e., the functions '12 S?F
(x) and 112 D?(x), as well as the external potential A (?e) (x) in
in the form of Fourier Integrals, and we shall insert these together with expressions (24.1)-(24.3) into Equation
(22.25) for the matrix S (?n);;more exactly speaking, we shall insert these expressions into that one of the normal
(n) (n)
products, in the decomposition of the integrand of the expression for S , which corresponds to M
?I f.
The contractions of the operators are given by the Fourier integrals
1- Sup (x) Srp (p) efl"' dsp ,
1 r
DF (x) = -21- DF (p) ,
I
where, ; ccording to (18.34) and (16.33),
246
(24.4)
F
S (11)
I F
D ( P)
1 (ITI&Pp.-171).tp,
(24.5)
=
(2704 1).4 + M2
1 1
)
(2704 F
p2 PO).
The Fourier expression for the external potential A(2) (x), which we shall treat as ac-number, shall be
written ?11
AL") (x) =?(2.1 )4 al, (q) eiqx d4 ,
a p.(q) = I AIV (x) 41' x j
(24.6)
(n)
Inserting (24.4), (24.6), and (24.1)-(24.3) into the normal product corresponding to M.? , we shall
f.
? ?
first integrate over x'1, x2, ..., eipx.
Collecting all factors .with a given x (here E are the four-vector mo-
_ _
(e) F
menta of the free particles, as well as the variables of integration in the Fourier representations of A ?
-
11,
and D-F ), we obtain e?i(E2)?xj , where the number of vectors in the sum EE is obviously three, i.e., the number
of lines which meet at the vertex x of the diagram.
The integral of e?i (E2)x?J is (2.ir )46 (E2), where 6(E2) is the four-dimensional 6 -function. Thus, in inte-
grating over xi., x2, ...?, x, we obtain a product of ri four-dimensional 6 -functions.
We note that each line of the diagram corresponds to some four-dimensional vector 2. Since and
which are related to internal lines of the diagram, depend on the differences between the coordinates at the ends
of these lines, two of the 6 -functiOns corresponding to some internal line contain the vector E associated with this
line, but with opposite sign. This makes it possible to interpret the vector E corresponding to an internal line as
the four-dimensional "momentum" of a virtuall)particle "emitted" at one end and "absorbed" at the other end of
the internal line. The external lines of the diagram, that is those lines which leave the diagram, correspond to
the four-dimensional momenta of real particles taking part in the process.
In order to obtain a final expression for M(. n)? , we must now integrate over the mcimenta E. Denoting
the momenta related to internal electron and photon lines of the diagram by p and kJ:. respectively, we obtain
? n ?
the following general expression in the form of a momentum-space integral for a.)
f
1) We recall that for virtual particles there exists no definite relation between the time and space components of
(see the note on p.241 ).
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Hit:Illy, the symbol t denotes the definite order of the operators in (24,7); the operators (acting on the
spinor indices) arc ordered front right to left in the sante way as they are encountered when moving along the
AIM/.
ten
X
(? i)F (2T)4 (n- F)
it
d'qi...d4qa111-1P,(Ep)0{
d4pid4p2...d4pF
( a (pi)
d4k1d4k2. ? .(1414F X
'
X
direction of an electron line of the diagram.
2. Example: Furry's Theorem.
In order to prove the validity of this last rule, let us consider the concrete example of an electron scattered
in an external field with emission and absorption of a virtual photon. The diagram representing this process is
v (pi)
v 20)4
X II (it (PO v (pi) .7,---/:2wf) .( 4) 11F6 (75.2-2 +11,2)11(1% /741 Iv))
^
(24.7)
I:
where the integration is taken over the 4F variables 2.1, 2.2, ..., EF due to factors of the type .S, the 4F vari-
ables k2, 1> rri is similar to 1E1 and 1221 respectively. Since S- and D- behave for 1 E I like
I E. I ? I and I 21 - I, it follows from (25.2) and (25.6) that for 1E1 >> mthe regularized values 65! and
_
6D- behave,up to a factor In-2- , like 1E1 - I and 1E21 - I, respectively; this behavior is the same as that of
m
F
S-F and D-. -
- -
_
Similarly, it is easy to show that the operator A given by (25.9) behaves, after regularization, in the
limit ILI ?m like In .
In the general case of an arbitrarily complex self-energy or vertex part, the following asymptotic expres-
sions for the regularized values of S- , D- , A are valid:
_ _
IL
n2 \
(p) SP (p)f8 G72-132), DRFt (p),DF (p)fn / n2 \ A--Tij /r 1-1;), I P2
Heref f , f are dimensionless functions of-2- which can be written aspolynomials in ln (in the expres-
sion for A it is assumed that I pi I - Ipil E 121 >> m2)?
Thus, having removed the divergence in any internal part of a diagram and having gone on to integration
over the variables external to it, we arrive at no divergences other than those considered in Section 25.
We note, however, that the above considerations become inapplicable as the order of the diagram approaches
22
Infinity, since in this case the polynomials itiln ?r can become infinite series, which can change the asymptotic
behavior of the functionsf, f f .
--S' -D, -y
- -
If, for instance, it is found that these functions are given by the series
f
coon(In?
e- /12 n
ft
m2 p2
?\i),
n=0
the character of the divergence changes greatly on going from the internal parts of the diagram to the diagram
as a whole.
27. Mass and Charge Renormalization.
1.- The Renormalization Concept.
We shall now go on to a discussion of the physical concepts at the basis of the above regularization method.
In studying the divergence due to the photon self-energy part, we have shown that it leads to infinite pho-
ton self-energy, and on this basis we have replaced the divergent operator 6/1 (k) corresponding to the dotted line
oft Fig. 25, diagram 4 .by zero. If, however, the dotted line of Fig. 25, diagram 4 does not represent a photon,
but some given external electromagnetic field, then the divergent operator 6 (k) cannot be eplaced by zero.
273,
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.?,????????????????????
In this case tire can obtain a finite expression having physical meaning from 6 (k) according to the above rules;
this will then describe vaCuum polarization.
(e)
It is known that the D'Alcmbertfan, when applied to a potential A - (x) describing a given external field,
-11 -
leads to the current I(& (x) which gives rise to the field. Thus, - 06 A (x) is a correction to the external cur-
-II - II-
rent I (-e)(x), which is due to the interaction of this current with the zero-point oscillations of the electron-
-II -
positron field. In Section 43 it will be shown that
? OSA.p(x)--= a,2 11:)(x)-1-1fP(x),
(P)
where a2 is a quadratically infinite constant, and X (x) is finite. If we remove the divergence from SA ,
II-
then the regularized operator 6ApR satisfies the relation
? 0(3417(x) 'fr.) (9.
???
infinite constants related to the irreducible diagrams for the electron and photon self-energy, the vertex parts,
and the photon-photon scattering parts. These constants EL, E0, no, Lo, Mo enter the operators E (W,
fl -e
(W k), A and the photon-photon scattering matrix clement in the following way:
-
E (Wet p)= + ?110 + ER (Wet p),
II( W,,, IT 2 + II0k2 + R (Wp, k),
ip.(V, Pp p2, k)r= Loyp.-F Ala? (VI P11 P29 k),
(k1, h2,k8, k4)-= Alo-I-MR (kj, k2, ka, k4).
(27.2)
We have seen above that ;, Ho, Lo, Mo are logarithmically divergent, El is linearly divergent, and II2 Is
.quadratically divergent. As for the infinite constants Me and n2, they are independent of the photon momenta
and Can thus be simply dropped from considerations of gauge invariance. We may thus say that there are only
four types of infinite constants, namely El, E0, Ilo, Le.
We shall now show that two of these constants, namely the linearly divergent one ; and the logarithmi-
(27.1) cally divergent one Ilo, can be eliminated if the form of the interaction between the electron-positron and the
electromagnetic fields is altered slightly.
Thus far, we have used the following expression for the interaction energy density;
We may, therefore, say that the regularization procedure removes an infinite term a2I -e (x) from this correction
P
to the 'external current; this infinite term is proportional to the original external current I- (x).
-
In particular, if we have a charge e (for instance an electron), then its interaction with the zero-point
oscillations of the electron-positron field causes this charge to change by an infinite amount proportional to e.
The regulaiization procedure consists of not taking account of this addition, assuming that it has no physical
meaning. It may be said that the addition to .the charge of the electron cannot be separated from the charge it-
self, and that the regularization process reduces essentially to renormalization of the charge: the sum of the
hypothetical electron charge which does not interact with the zero-point oscillations of the electron-positron
field, and the infinite correction, which modem theory gives to this charge in view of this interaction, is actually
finite and is the total experimentally observed electron charge. ?
A similar situation arises in removing divergences due to the electron self-energy part. .We have seen above
that these divergences lead to an infinite electromagnetic mass of the electron, that is, to an infinite correction
to the mass of the electron due to its interaction with the zero-point oscillations of the electromagnetic field.
The regularization method consists of ignoring this correction, assuming that it cannot be separated from the total
electron mass. Thus, we may say that the regularization process reduces to a renormalization of the electron
mass: the sum of the mass of the "bare" hypothetical electron, which does not interact with the zero-point
oscillations of the electromagnetic field, and its infinite electromagnetic mass, which the theory predicts as a
result of this interaction, is actually finite and is the total experimentally observable electron mass.
Thus, the physical concepts underlying the above regularization method are essentially contained in the
renormalization of the constants m and e.
We shall now attempt a more rigorous formulation of the renormalization procedures for the mass and
charge of the electron. Let us first recall that in quantum electrodynamics we have to deal with six types of
S. Gupta, Proc. Phys. Soc. (London) A 64, 426 (1951); F. Dyson, Phys. Rev. 83, 608 (1951).
274
?
V (x) ? (x) (x).
(27.3)
Let us now add to this expression the two terms -6 m * * and - 16 f F2 , where F is the electromagnetic
4 - p v v
field tensor, and 6m and Of are certain constants (infinite, but independent of * and F ); this means that we
In/
shall-cOnsider the interaction energy density given by
V* (x) = ?h(x) (x) ? (x) (x) ? 14-8fF;,2.? (x). (27.4)
If this expression is inserted for V (x) into the general formula (22.2) for the S matrix, the form of the S matrix
is changed, as is that of the operators E (W Z E) and II (W k). It is clear that the second term in (27.4) will
-e -
affect electron transitions, and since it does not contain the electron momentum, appropriate choice of 6m will
make it possible to remove the divergent term El. in the expression for! ( e, p). Similarly, the third term in
(27.4) will affect photon transitions, and since it is proportional to the square ofthe field tensor, or the square of
the photon momentum, appropriate choice of Of will make it possible to remove the divergent term 110k2 in the
expression for 11 (W, k).
?E
1
Thus, the addition of the two terms - 6 rn-*- * and - Of F2 to the expression for the interaction
4 - pv
energy density, makes it possible to eliminate the two divergent constants Ei and Ho from the theory. ,But the
physical meaning of such an addition to the interaction term is that the electron mass and the electromagnetic
field are being renormalized. Indeed, the Lagrangian of the free electron-positron field contains the term
275
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- m *, and, therefore, the additional term -6m II* can be thought of as the renormalIzation of the electron
1
mass. The Ligrangian of the free electromagnetic field is equal to -F F , and therefore, the correction
1 4 pv pv
-- 6 fF F causes the renormalization of the electromagnetic field, that Is, replacement of F by the
4 -Pu Pv p V
expression
(1 +
This field renormalization can also be thought of as renormalizing the electron charge, that is to say ,replacing
the charge e by the expression
e* =-- e (1 71-8f)-71.
We thus see that the divergent terms El and 11 are effectively removed by renormalizing the electron's
mass and charge. (We note that the removal of 112 and Mo is equivalent to renormalizing the photon mass, since
gauge invariance follows from the fact that the photon mass vanishes).
2.. The Relation Between the Divergences Due to the Vertex Part and the Electron Self-Energy Part.
After renormalizing the charge and mass of the electron, we are left with the two infinite constants ; and
Lo. We shall now show that ?in any given matrix element they lead to expressions which cancel each other. We
first make note of the identity
--1 1 OSIP(P)
Wic) 2 ()pi,. 2
On the basis of this identity, it can be shown that
(27.5)
O2 (2) (2704 A(p!)(V8,P)) (27.6)
aPp.
(2)
where E (2) (W1, 2) and A (V3, 2, E) are operators referring to the electron self-energy part VII and the vertex
"
part V3 shown in Fig. 27 (see Equations (25.3) and (25.9) 1.
Equation (27.5) has a simple graphical interpretation. Since S-(2) is represented by an electron line, we
1) J. Wird, Phys. Rev. 73; 182 (1950).
276
_________
acci
- Sanitized Coiv APP
may say that -
a2
-17,7
P ?4-1:124-.8gfp
l )
sF(p)
Fig. 27 Fig. 28
should be represented by a vertex with one photon and two electron lines (the electron
momenta are equal; see Fig. 28). It is then simple to establish the following general rule for the graphic repre-
sentation of the derivative of a matrix element with respect to an electron momentum., In differentiating any
matrix element by an electron momentum, we obtain a series of expressions which can be represented by the set
of diagrams in each of which an electron line of the original diagram (corresponding to the original matrix ele-
ment) is replaced by a vertex with one photon and two electron lines. For instance, the matrix element E (W, 2)
z(w,
represented by the diagram W of Fig. 29a has a derivative
grams are shown in Fig. 29b.
02P
Fig. 29
rp
consisting of the three terms whose dia-
If we use these diagrams to represent the derivatives of matrix elements, it is easy to show that
e a E*
? ( 2 tt) op
(27.7)
where E is the sum of the expressions E (W, 2) for all the electron self-energy proper parts W, and A* is the
11
sum of the expressions A (V, 2, 2) for all the proper vertex parts V.
11
F'
We note that the operator V' is related to the modified function S- = S- + 6S-F which accounts for all
the radiative corrections by the expression') ?
1) A part of a diagram is called proper if it cannot be separated into two disconnected parts by the omission of
a single line.
2) See Section 45, Equation (45.20), and Dyson's article cited on p. 258.
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?
?217
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's"= 1SF -1- I SPE* 1 7 SF'.
(27.8)
Let us now illustrate the cancellation of divergences due to vertex parts and electron self-energy parts by
two examples (we shall assume that the charge and mass renormalization have been previously performed).
Let us first consider the radiative corrections to
electron scattering in an external field up to third-prder
terms. The diagrams representing these processes are
2 shown in Fig. 30. Here diagrams 2 and 3 show third-
order effects, and diagram 1 Illustrates the nonzero
matrix element in the first approximation. We should,
however, bear in mind that this diagram also describes
a third-order effect, since according to (25.10), ac-
counting for the radiative corrections reduces not only
to replacing S- and D- by the modified functions
F'F'
S- = S-F + 6S-F and D- = D- + 613-, but also assumes
that the electron wave function operator ? is replaced
by some modified operator. 0' = 0 + 60 . (This replacement is equivalent to considering, in addition to diagram
1 with the unmodified 0, also diagrams in which the electron lines Ei and E2 have self-energy parts W1. See Fig. 27).
1 F
The replacement of 0 by 0' can be performed in the following way. Since is the vacuum expec-
tation value of P(0 ),
I A
Fig. 30
0 (t', (x?x'),
1 F1
we may say that j? eS- is the vacuum expectation value of P(*'*'), given by
-
0 = g (I' , 54' (x ?
On the other hand, in the approximation we are now considering [see (25.2)1
,
TS' (P) SF (P) -F (P) s=a Sr (P) SIP (P) E P9 -2- SP (P),
where the electron self-energy part W1 is that indicated in Fig. 27. Since
278
vomeariersaa.?
'412?42=74-477,4=4M1.?-
(27.9)
(27.9')
(27.10)
r.
C
and
we have-
E (W1, p)= Xi+ (Pt,? Pi)t)(*),..p.+ ER (Wit 12)
( )\ OP.
(2704 AJA ? (2704 043).(p.,
e IL Y I Po' rai
E = El ?PITY Oa) (Pit' 13?) 11L+ ER (W P) =
?.=.? ? (2104 e2a) (11? im)-F ER (W" p).
? (27.11)
itenormalizing the electron mass we can drop the linearly divergent constant Ei from (27.11), and use the follow-
ing expression for E 2):
Noting that
P) = ? (2704e248) ER (W11 p).
1 1 -
?SF (p)1,10-1
2 ? (2704
and inserting (27.11) into (27.10), we obtain
TI Si" (P) = SF (P) e21-(08) SF (P) 71 SF (P) ER (Wit P) (p).
?
For the free electron the last term in (27.12), as we know, vanishes so that
SF' ( p) = (1? e2 Lr) SF (p), p = Po.
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From this and from (27.9') it follows that
c-
(27.11')
(27.12)
(27.13)
* ' (1 ? e21,r)16 *-/--z (1 ? e21,7)) ? ; (27.14)
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(The last equation is valid if 1 Is only formally infinite; actually it gives a small correction proportional to
e 2).
Let us now determine the sum of the matrix elements corresponding to diagrams 1, 2, and 3 of Fig. 30.
This sum can be written in the form
S72). r=e-u2' a's (q) .1; + I Pt, P2) at, (q) g 8, q P2 ?P12
(27.15)
where 9R8 ? is the matrix element corresponding to diagram 3.* In renormalizing the electron charge
we will remove' the divergence in W13. We shall therefore consider already finite.
We must now show that the divergences in the first two terms of (27.15) cancel each other, This, however,
follows directly from (27.14) and the form of Ala), which can be written
11
(V,p13 ps)=-- L(08)Tp.-F A% (112 P13 PO' (27.15')
Indeed, by inserting this expression into (27.14) and (27.15) we obtain a finite expression, namely
-- ? (9)
s(3)
0)
f eu2a(q)u1(1?e2 L. o esu-2L0(8)a" (q)u1+ e8 uokotap.(9) Ili -Li gm=
-
- (
eu2a (q) u A? a (
2 p. s)R Ul 03R.
(27.16)
(3)
Thus, we see that the divergence in ?Si I- due to the vertex part cancels that which remains in the electron
self-energy part after mass renormalization (thislatter divergence is contained in the modified free-electron
operator). This can also be put differently; the divergence due to the vertex part is removed after the renormal-
ization of the free-electron wave function. As a second example of the way the vertex parr divergence cancels
with that remaining in the electron self-energy part after mass renormalization, let us consider the polarization
of the vacuum up to terms in e4. The diagrams describing this process are shown in Fig. 31.
The functions II (W, k) corresponding to these
diagrams [see Equation (25.6)]shall be denoted by
11 (k),D(k), and II (k). The total function
a ? C
a
Fig. 31
u' is the spinor amplitude of
280
(k) corresponding to vacuum polarization is clearly
given by
11(k) = Ea (k)+ 21Ib (k) If (k). (27.17)
?
?
:In order to remove the divergences in II (k), let us first renormalize the electron charge. Then II (k) be-
- a ?
Comes finite. As for lib and 11c, they contain no principal divergences [that is, divergences related to the diagram
as a whole), although they do contain divergences related to the internal parts of the diagram (II (R) contains
?
the divergence related to the internal electron self-energy part W1 and lie (k) contains a divergeTice related to
the two- vertex parts at 1 and 2]. We shall now show that these divergences due to the vertex and electron self-
energy parts cancel in the function 11 (k) after mass renormalization. We shall do this by using Equation (25.2)
1 F
for ?26S? (2.). According to (25.10) and (25.2), Ilb (k) can be written
lTEb(k)=? e2 ip.4. SF (p)E(Wi, (p)Tv.- SF (p k) a4p.
(27.18)
Let us now perform the mass renormalization. Then E (W1, 2) is written in the form given by (27.11'). Inserting
(27.11') into (27.18) and bearing in mind the definition of 11a (k) [see (25.7)], we obtain
Ilb (k) , e242).110 (h) lIbR (k),
where IIbR (k) is finite [we have made use of the fact that
?
using Equations (25.10) and (21.15'), II (I-1) can be given by
.
?2S (2n)4 (p ? tm)- J.
11 (k) = 2e2 L1110 (k)llcR
Similarly,
where IIcR (k) remains finite. Inserting these expressions for IIb (k) and II (k) into (27.17), 11 (1> m, this equation can be written
2,5 4 in
4itroz. a ? m.
In the nonrelativistic limit, when to '? m, (30.34) leads to
a ?321/1?ic 2 roc, ,5 a 4)
M 7/2
(30.35)
(30.36)
1
We note that (30.36), as (30.34), is valid only when w ? I = a2m, where I is the ionization potential of the
2 ?
K- electron.
The cross section for the photoelectric effect with the nuclear Coulomb field taken into arc-mint can be
obtained only in two limiting cases.In the nonrelativistic case, use of the exact nonrelativistic wave functions
in (30.33) leads to (30.36) multiplied by
1) SeeW. Heitler, The Quantum Theorty of Radiation (State Tech. Press, 1940).
314
where
ft4R=: 2irjr7 *e?it arctg.E
1-e2'
/ Za
=V .7=7
(!is the velocity of the K-electron).
In the ultrarelativistic case, the result obtained differs from (30.35) by the factor,
(30.37)
IUR 1-----.e-"'zcL+2(z2)1(1-In 2x). (30.38)
In the following table we present the exact values for the cross section for the photoelectric effect, as
obtained by numerical integrationl) for several cases in which the approximate formulas are not applicable.
Values of
z5-3-8ag rg (137)4
26
50
I 82
0,69
18
12,2
2,2
1. 1,05
0,80
0,60
We also present the expression for the angular
distribution of the photoelectrons, in the same approx-
imation as is used for (30.36) (co > I)
da 3 sln20 cos2cp
do 475 (1? v cos 4)4$
(30.39)
where 9-isthe angle between the photon propagation vector k and the momentum of thephotoelectron, and
r-
co is the angle between the plane defined by p and k and that defined by k and e.
? 31. Bremsstrahlung
1. General Expression for the Matrix Element.
In collisions involving an electron and another charge (or system of charges), in addition to electron
scattering, photon emission may also take place. Such a process is called bremsstrahlung or deceleration radiation
If the electron collides with a heavy particle (a nucleus or an atom), the effect of the latter can be treated as an
external field. Then the bremsstrahlung is described by the matrix element (30.1) in which the initial and final
states belong to the continuous spectrum*.
In view of the complex character of the electron wave functions in the field of the nucleus (see Sections 12,
13), the integral in (30.1) can be calculated analytically only for low energies.? when the nonrelativistic approx-
imation may be used.
I)-- -
H. R. Hulme, I. Mc. Dongall, R. Buckingham,arui it Fowler, Proc. Roy. Soc. 149,131 (1935).
2)The final state may belong to the discrete spectrum. We shall not consider this case.
?Recently results for the high energy limit (e? in) and small electron-scattering angles have also been obtained.
See fs,4aximon, and Bethe, Phys. Rev. 87,156 (1952); Davies and Bethe,loc. cit.; Bethe, Maximon, and Low, Phys.
Rev. 91,417 (1953).
315
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We shall present these results somewhat later, first considering another somewhat simpler case which has a wide
range of applications.
Let us assume that the external field is such that it can be treated by perturbation methods (the criterion
for the applicability of perturbation theory is clearly the same as that for the Born approximation for a Coulomb
field, namely Ze2/v > m and c >> in (31.14) becomes
de, e r -1-
e2e2
2 " 2111[ 2ee
? in --
03 Et ei?2 3 nue 2
==4,1; du f u hu 2et l? U'\
)
u 3 k In
whereu- =?
el ?
(31.17)
We note that the probability that the electron radiates a given fraction of its energy (that is, the radiation
probability for a given value of (0/6.1) increases approximately as the logarithm of ci/m. For low frequencies
(31.17) is essentially inversely proportional to w, and rod ouj diverges logarithmically as cu-.-0, just as in the
nonrelativistic case.
Let us also give the expression for the cross section for electron energy loss: due to radiation, that is for the
quantity
it-vs
1 .
tu?
tl .
Inserting claw ?from (31.14), we obtain
=.(1). t 12q+4m2 el + I Pi I (81. +61PIDIn2 (In ei 4" I \2 4 -1-
2
38iPi
m 3
where F is the function defined by
.1inp2t \wipti(.1.+IptD)},
21 m2
a;
F = In (1 +y)
al y 14Y
(31.18)
n?
For x>1 we can use the same series for F (1/x), and the relation
F (x) FG) + (In x)2.
For low energies, when 1p11 ? m , it follows from (31.18) that
_
and for very high energies, when c i >> m,
16 -
(I) (I)
3 '
(I) =-- 4 (In 21-L. ? -)f.
m 3
(31.19)
(31.20)
Thus for low energies,the ratio of the mean energy radiated to the initial energy is constant. For high
energies, this ratio increases in proportion to the logarithm of the energy.
6. Screening
All the above results from (31.12) on, are obtained for the Coulomb field, whose Fourier components are
given by (31.9). Since nuclei are ordinarily surrounded by eiectrons, the nuclear field is Coulombic only for
distances-smaller than the radius of the K-shell; at large distances the field is partially or entirely screened. In
order to clarify the role of distance from the nucleus in radiation processes, let us consider the general expression
for the Fourier component a? as given by (31.8):
ia? fA(e)e- iqr dr.
4
The region which essentially determines the value of this integral is Oven by the inequality
1
l,
since for larger valuesof r the oscillations of e-iqr-- become important. The minimum value of q which corresponds
to the maxim urn effective distance, namely q min, can be found from the conservation laws. For low energies
and small distances (p > m)
m2co
qmin 2ate2 ?
(31.21)
1
It is seen from this that for sufficiently low frequencies q min can be greater than the dimensions of the atom,
and the Probability for emission of such photons will be much less than that calculated according to (31.14).
Therefore when w the product axlcrui vanishes, which differs from its behavior in a pure Coulomb field.
For sufficiently high energies, minq becomes small even for frequencies of the order of the original
electron energy. Indeed if wand c 2 are of order 6J>> m , then according to (31.21)
When cl ki 137 ----
must be taken into
radius of the atom,
hydrogen atom.
e I
2
qmIn nt
1
,the quantity becomes equal to the radius of the IS-shell, and therefore screening
q min
account. When el > 13 ,--cr the quantity-1?.- becomes greater than the effective
3 CI min
which, according to the Thomas-Fermi model is _40/ei3 , where go is the radius of the
We present here the expression for co
to the Thomas-Fermi atomic potentiall)):
where
with screening taken into account (a? is calculated according
1 aw _2 ) (t. 4
?=--,Z2ar? ? 2 [(a- + c2)(4) (C)? ?4 in Z)? 3 (4 2 _ y in Z)j,
o e2 to 1 2
=100
In (0
(31.22)
where (pi and 432 are the functions shown in Fig. 36 in the interval 0 < C < 20. When C ? 1 (low frequency
- or high cnergy) we have so-called complete screening. In this case
and
2
(1)1 (0) 41n 183, 4)2 (0) =-- (Di (0) ?
1 dw 2
1
de Z2arl-- 4 [(?2-1- 1 e:.,) In 183 Z-V3+ `15.2].
to 0 2 3 9
.1
1)
SCC the reference on p 319.
,324
(31.23)
When C > 1, we return to the case of no screening as given by (31.17).
-
_ 20
-19
18
17
16
-15
cl
-
-
IP
N4%Ns??,-
....,...
0 2 4
6 8 10 12 14 16 18 2
Fig. 36
When -2. < C < 16 Equation(31.17)can
1
still be used if 2.
1
is replaced by c (c)
In the last factor, where c (c) is given by
the table below:
2
15
10
5
1 \
\
\
\\20
?N11.30
\
i'
5
20
I
- -
I-,
----
J 0.2.3 04 0.5 0.6 0.7 0.8 0.9 1717fi
Fig. 37.
C
1 2
2.5
1 3
4 1 5
1 6
1 8 1 10 1 15
c(C) I
0.21
0.16 I
0.13
0.09
I 0.065
0.05
I 0.03 I 0.02
1 0.01
The cross section for energy losses to radiaiion in the case of complete screening is
[41n (183 Z-Vs)21
(31.24)
We see that the screening eliminates the logarithmic increase of the cross section with energy which was given
by (31.20):-
data ei - Fri re
Figure 37 shows the dependence of w on for various values of -11::-L-n
dw 6 I
? 1-M Ell
?
(the latter are given by the numbers on the curves). These curves include screening as well as the deviation from
the Born approximation (for low energies) in the region of the low-wavelength limit. Figure 38 shows the
c
dependence of ? on
of screening.1). (1)
7: Radiation from Electron-Electron and Electron-Positron Collisions.
for various substances with screening included. The highest curve represents absence
Let is now consider photon emission arising from electron-electron collisions. In this problem we may
no longer replace the effect of one of the particles by an external field. It is therefore necessary to consider
those S matrix elements involving one photon and four electron (two initial and two final) states. These are
contained in the third-order matrix S. . We shall calculate the matrix element directly from the general
rules formulated in Sections 23 and 24.
Figure 39 shows diagrams corresponding to individual terms of this matrix element. Altogether there are
eight diagrams. Next to each line segment we have indicated the appropriate four-momenta: pi and pl are
the initial-electron momenta, p2 and .$ arc the final electron momenta, and k IS the momentum of the emitted
See the reference on p. 369.
325
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/5
10
5
H20
Cu
-411,
0.1 02 0.5 2 5 10 20 50100200 5(101000 6,-m
in
P
q2
Fig. 38
/ 0
/ ? 1
19;
a)
//'q3
pg,/f3
???? K
f2
b P;
fs
K
/ 3
P,
P2
q4
14 pit
,.????? K
pg
K
2 / f6
,'q6
P2
f P"
f2
13;
Ps
p;--4---7--.4-- g ,
/
/ n
/ i 8
// fa
P2
-h ...1? ......* '...."..%
P1'
Fig. 39
PS
photon; the f5 are the four-momenta of the virtual electrons, and:Ithe qs are those of the virtual photons
(s = 1,2...,8). The last four diagrams (e,t,g,h) differ from the first four (a,b,c,d) by the interchange of
ps and p '2 ("exchange" diagrams). The terns corresponding to these enter the matrixelementi with opposite
signs. In diagrams a and c the scattering takes place "first" and the radiation "second;' and. in diagrams b and
d, vice versa. In diagrams a and b "the first electron radiates" and in diagrams c and, d, the second one does.
The values of f and a1 are easily obtained from the conservation laws at each of the vertices. Thus,
-s
?
q = q 2 = Pi,
f P2+ k,
q11.= q4= P2?
f 2 ?k,
f 4 = P; k.
(31.25)
Moving along the electron lines, and replacing the line segments and vertices by the appropriate amplitudes
and operators-, we obtain the following expression for the matrix element (in each term the first factor refers.
to the upper electron line, and the second one, to the low,er one):
snj { mN-N
2-- e Y) --F(u27011)
1/. 2w #22
91
(7-121i fij;2217:21 eAui) (7141tu;)-?(11-21tu )-1
q2 1 u2e
(172101) 112ii 2
1/4 ? M ^
q4m 2 e ? U2 e
+ m2
2 TSUI +
?? 171
171
(6'6 ?m ^ 1
u2ii .
2 \ q 7
eut 2 ( U2n141) kli2itUl) 2 U2 e ^
#1 m
/1+1112
q6
1 (umui)--(- /8 ? m ^
7,- twit + m2 etti.)}(2T048 (pi+ pi? P2?p2 ? k).
The expression for the cross section corresponding to the matrix element (31.26) is extremely complicated
(after summing over electron spills and photon polarizations, there arise many terms containing the trace of
a product of six y-inatrices). We shall restrict ourselves to presenting the results for two limiting cases:
the nonreiativistic
!Pt
and the. ultra-relativistic
In the nonrelativistic case the cross section can also be obtained by using Equation (30.14) for electric
quadrupole radiation (the dipole moment of a two-election system is zero) i)? This can be done by using the fact
that in the nonrelativistic approximation a system of two particles call be replaced by a single particle in an
1)111 Section .30 we made use of the approximation in which ca ? 1, where r represents significant distances for a
y,iven plobleni-7- For bound states, r represents atomic (nuclear) dimensions. In the present case, r- vr, where v
Is the velocity-of the electron, and r is the "collision time" given by r ???? . Thus, the approximation of
mnitipole iitcliation and the nonrelativistic limit agree in that they require v ?1.
-
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external field (the Cdttlomb interaction field for electrons). Then the wave functions of the initial and final
states can be chosen according to (31.3), in which
?
(ow ? e-fri e-f,,
y2
where r is the relative position vector, and p is the relative momentum (the + sign corresponds to total
spin 0, and the?sign corresponds to total spin 1).
The expression for the differential cross section can be written1)
16
da== ar2 14 (p2 ? p2)2 -I- 3 [pip 121 --I- f]44 fpip218
o q1 1 2 2 q s8
-F 264 VI ?14)2 1P1p2 14 -1- 1?5017 ?fi22)4 1PIP212 + 12(p ?/4)61-
1
q1/44 [36 1p1p214 4- 39 VI ip1p212 + 12 (p7._p94j dP
2 1/11)(P21.
The energy of the emitted photon is
q ?p2; s ?pc-FP2.
?2
PI P2
el 22 m_ m
(31.27)
where et and ez are the initial and final electron energies in the center-of-mass system.
The angular distribution of the radiation in the center-of-mass system of two electrons is determined by its
quadrupole character ( a ks iy2(8)I 2).
Integration of (31.27) over the angles, which should be performed only over a hemisphere in view of the
indistinguishability of the particles, gives
where
dew ---=-- -T-6- ar o (2? -r)2
4 2 3x2 12 (2 ? 7 (2 x)2x2? 3x4
(1 ? x)"' (2 ? x)8
1
x in 1/1?xxdx
El
(31.28)
1) E. Lifshits, J. Expth-Theoret. Phys. 18 ,562 (1948); B.Fedyushin, J. Expt1.-Theoret. Phys. 22, 140 (1952).
328
4?????:,
?
The cross section for energy loss due to radiation by an electron colliding with an electron at rest is given
by the equation
2
? CO da,,, 8a ro,
ct ,
which of the sameorder as that obtained in the field of the nucleus for,Z = 1 (sec 31.19).
(31.29)
Since the atom of the nucleus whose charge is Z e has Z electrons, the energy loss due to radiation b an
electron colliding with electrons in the shells is 1 /Z of the losses due to collisions with the nucleus ).
In the ultrarclativistic case the cross section for radiation due to collision with a stationary electron is given
by')
E2.) (2 ln 2ele2
dw 1( 2 +
da =-- 2arg --Et
?2 t -3- etez k 1) (1'70 iPti1P21
, 2
cl) 4ar e In ? 1,05).
in
(31.30)
Up to a factor in the logarithm, the cross section do to is the same as that for radiation by an electron in the
field of a nucleus with z = 1. This result is easy to understand on the basis of the following considerations. As we
have seen [sec (31.21)], for high energies the efficiency of momentum transfer to the nucleus becomes extremely
small, and therefore the mass of the nucleus becomes less and less important.
For collision of an electron with a positron, the ultrarclativistic case gives exactly the same result as for
electron-electron collision. In the nonrclativistic case, hoWever, dipole radiation can take place. In this case
the cross section for radiation is
32 m rho
daor- - -
0 3 610,
+lii
20)
CI
?
2(0
Cl
(31.3])
This last equation Is written on the assumption that before the collision one of the particles was at rest. In the
center-of-mass system ,(31.31) is the same as the cross section for radiation in the field of the nucleus (with Z
1) in the nonrclativistic approximation [see (31.15)].
? 32. Emission of Long-Wavelength Photons
1. Infrared "Catastrophe"
In the previous section we saw that the matrix element which gives the radiation by an electron in an
external field approaches infinity as w-3/2 .vhen Me photon energy approaches zero. Therefore the probability
than an electron will emit a photon whose energy lies between w and w dw is proportional, for low photon
energies ,to
Garibyan , Bull. Acad. Sci. Armenian SSR 5,3 (1952).
329
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-
dco
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(32.1)
and the total emission probability diverges logarithmically as fa This divergence in the low-energy region
is called the infrared catastrophe. It should he noted, however, that this situation has nothing in common with the
fundamental divergences of quantum electrodynamics in the high-momenturn regions of the virtual particles,
and is related to the fact that ordinary perturbation theory based on the series expansion of the S matrix in
powers of e is not valid for processes involving long-wavelength photons. Indeed by repeating the considerations
which led to (32.1), it is easy to show that if the probability wt for the emission of a single long-wavelength
photon is proportional to e21n4-0) , where c is an energy of the same order of magnitude as that of the electron,
then the probability w2 for the emission of two photons is proportional to (02 ln 2. Therefore,the order of
magnitude of the ratio of the probabilities is given by
e ? ?2-tvt e2 In
(01
(32.2)
It is this ratio, and not the quantity e2,as we have thus far assumed, which is the power series parameter in the
application of perturbation theory to processes involving the interaction of the electron with long-wavelength
photons. Strictly speaking, since g is not small compared to unity when co?s-0, perturbation theory is not
applicable to these cases.
The inapplicability of ordinary perturbation theory is related to the fact that the number of photons
emitted by an electron per unit energy interval approaches infinity as co-0, whereas perturbation theory assumes
that the radiation of a single photon is always more probable than that of two or more.
In order to show that as the number of photons emitted actually approaches infinity, we note that If the
the energy and momentum of the photon are much smaller than the kinetic energy and momentum change of
the electron, and if the photon wavelength is much larger than the classical electron radius, then we may consider
the electron motion. to be given and may use classical electrodynamics. Assuming for simplicity that the
velocity of the electron is small with respect to that of light, we can make use of the following formula for
the intensity d gt.e of the dipole radiation in a freqtiency interval dial)
?
(11 2 dU),
where .d.tuis the Fourier component of the second time derivative of the dipole moment,
CO
faeiwt dt.
27: ,
-CO
If co-4- 0,then
1) See, for instance, L. Landau and E. Lifshits, Field Theory (State Tech. Press, 1948).
030
(32.3)
cerharl in Part - Sanitized CoDy Approved for Relea
(II and 42 are the values of the dipole moment before and after radiation). In the case in which we are
Interested, d ev and
r (t2 TO where y, and v2 are the electron velocities before and after
- 2
radiation. Therefore
2 e2
dtf, 0 -Jr (v2 --- v1)24541).
We see that the intensity bf the radiation pet unit frequency interval
co--?0. It follows from this that the mean number of photons emittea, namely
(32.4)
de
has a finite nonzero limit as
1 dgu,
h co d w
, approaches
infinity as as was asserted above.
Since the probability for an electron transition from a state with momentuin pi to one with momentum
P2 is always finite, the probability of simultaneous emission of an infinite number a photons with infinitesimally
? ?
small frequencies (co--?0) is also finite and nonzero. Therefore the probability for radiating one or a finite nunilh
of photons as co-6-0 actually vanishes, and does not become infinite as is assumed by perturbation theory.
1 dgw
The quantity is the average number of photons with frequency w emitted by an electron into
d co
the frequency interval dw. We shall now find the probability that an electron emits some arbitrary number
n of long-wavelength photons whose frequencies lie in the interval w.-s w co2. Assuming, as before, that
fico tho e20.) ro
1
mc3 1 I
(32.5)
where C is the kinetic energy of the electron, ,pis its momentum change ?X is the photon wavelength, and.ro is
the classical electron radius, we may assume that the emission of photons does not effect the motion of the
electron, that is, we may consider its motion given. Under these conditions successive photon ,emission events
are statistically independent, and therefore the probability for photon emission is given by Poisson's formula
T
Wn --i?n ,
(32.0)
where W mii is the average number of emitted photons whose frequencies lie in the given interval co, (or:
w2.
When conditions (32.5) are satisfied W can be found from classical electrodynam?ics. In particular, if dff =
ti-)31
= Itundtado is_the classical radiation intensity in the frequency interval (co,to-i- dco) and the solid angle. do'), men
cu tu,
W dw do.
CO. 001
Here n is tlre_unit vector in the ditection of radiation.
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(32.7)
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r.4
?
We shall now show how to find I It is known that
d am =cH,2R2dwdo,
where H is the Fourier component corresponding to the frequency w of the magnetic field H at the point R; the
quantity Ho) is related to the Fourier component of the potential Ace by the expression H cd= tAc,.). Using
the Lienard-Wiechert potentiall), it is easy-to show that
Co
A. -= 2rccR
.e I?V (1) ei (cut-kr ")I di, ?w=---. 21t1? Je
e
-Co
i(co t?kr(oi
(32.8)
If viand v2 are much smaller than the velocity of light,then (32.11) reduces to (32.4)1).
Let us now determine the probability for electron scattering in an external field accompanied by radiation
of n long-wavelength photons. Assuming that conditions (32.5) are satisfied, which means that the recoil
effect of the radiation on the electron motion is extremely small, we can write the probability for electron
scattering with radiation of n long-wavelength photons in the form
svndw,
(32.12)
where chits = f dos is the probability for elastic scattering of the electron into a solid angle dos, as defined in
Section 13. Clearly the frequency interval (tai, cl.,t2) we must use should contain the frequency zero. The average
number of photons emitted in such an,interval is infinite according to (32.7), and therefore ml = 0. In other
words, the probability for electron scattering with the emission of a finite number of lopg-waVelength photons
is zero. In particular, the probability for pure elastic scattering vanishes. Since, on the other hand,
where r = r (t) is the equation of the electron. trajectory, and v (1) is the electron velocity (9 ca is the Fourier IX)
211 11
U .:0
component of the scalar potential). Therefore
d wn c11-1 .12 R 2 dco do.-= cR2k2 (IA.(2 ? I cpw12) dw do
CO Co
ec f (I'c2
?co ?co
(32.9)
v (1) (if) ? 1) ei (w (1?t.)-4 tr("r(P)1) dt k2 do) do
1
(we have made use of the relation kA w +?c tarp w = 0).
If the frequency co satisfies the condition
(O'C 11
where r is of the same order of magnitude as the time during which the electron scattering takes place, then
on integrating (32.9) we may consider that
r(t)c---Jvii-Fa, ?co > m) Equation (33.18) can be written
?32 e+E- (In 2E,,,4-riez-
do, -= 4(I) de+ E2+ + +co3
=--- Z2a/02.
These equations, as those in the bremsstrahlung problem, are restricted by the condition
? Za .> Xi.
(1) f? 2
I CO I" " )
I (1) I (g,
CI tri - E(y))2 ?
(38.26)
Measurements of the effect of the magnetic field on positronium decay made possible the experimental
determination of the energy difference= Eio Eoo of the ground states of ortho- and parapositronium, using
Equation (38.26)1). The measured value of agrees with the theoretical one given by (38.21).
,? 39. Internal Conversion of Gamma Rays.
1. Expansion of Retarded Potentials in Spherical Waves.
Due to its electromagnetic interaction with electrons, a nucleus in an excited state can undergo transition to
a lower energy state by transferring its excitation energy to the electrons in the atomic shells or by creating an
electron-positron pair. This process is called internal conversion of gamma rays (this terminology indicates that
the excited nucleus may also lose energy by emission of photons). We can treat internal conversion as a special
case of the interaction of a nuclear proton with an electron, as described in Section 35, using the general expre\isson
(35.15).
Let us-denote the initial state of the electron by ?Pi, and its final state by 02; similarly for the proton we shall
use 4t1 and 11/2. According to (35.8) and (35.15) the matrix element of the effective perturbation energy is
ft,-rt I _
U= -IX TIT2 (r1) 1.r (r) ? r2 t1)2 (r2)1,1), (r0) d r, dro
I rt
cc pr; (r)'F(r 1) (r 2) c (r 2)
ri-r, I
- OF; (r1) hr1 (r3)) (''4(1-2.) cc2th (r.))) dr dr
rol 1 21
I) M. Deptsh and E. Dutit, Phys. Rev. 84, 601 (1951).
(39.1)
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== H, ? F.,2 =-- et). ? a1,
Ei, 2 are the proton energies and ? b 2 the electron energies.
The calculation of these integrals is simplified due to the difference in the sizes of the regions in Which the
proton and electron motions take place. For the proton this region is of the order of the nuclear radius. For the
electron, on the other hand, the region is much larger. Therefore, in integrating (39.1) we may assume that the
contribution comes primarily from the region in which 11 R
(39.33)
and the Coulomb wave functions may be used in (39.33). Thus, the integrals for the 12, should have a lower
limit r2 = R. The finite dimensions of the nucleus lead to corrections which are insignificant for small values of
Z, but can be as great as 30-40 per cent when Z - 80-90.
? 40. Conversion With Pair Creation. Nuclear Excitation by Electrons.
1. Conversion of Magnetic Multipole Radiation.
If the excitation energy (1.) of the nucleus is greater than 2m, then in addition to conversion on an atomic
shell, electron-positron pair creation may also take place. The conversion coefficient with pair creation can be
found from the general formula (39.12), with 71,2 an electron Wave function, and 02 a negative-frequency wave
function corresponding to a positron. The matrix element entering (39.12) can be reduced to integrals of the
radial wave functions, as was done in the previous paragraph for K-shell conversion.
We shall restrict our considerations to cases in which the electron and positron may be considered free
particles') (applicable to low-Z nuclei).
. .
We hail choose the wave functions in the form of plane waves
T.2 e
t!1 I = 1Wir
(40.1)
Here e, is the electron momentum, R is the positron MOIlldllt11111, and u and v are unit bispinor amplitudes.
When we insert (40.1) into (39.12) we obtain the following expression for the differential conversion co-
efficient of an electric or magnetic multipole with creation of an electron in the momentum interval 5_12 p and a
positron in the momentum interval dp
where
dp _ s+)z e iqs?tt r-A) l?
iv ar ,
{40.2)
1) V. Beresteisl:y and I. !?411111011.L.vivh, I. Ixp1.1.-.Theoret. Phys. 12, 591 (1949); I. Shapiro, J. Expt1.-Theoret.
Phys. 1.0, 597 (1.949), M. 12():?e, Phy:.;:ev. 76, 6723(1949).
426
and E denotes summation over the electron and positron spin states.
Let us first perform the calculation for the case of a magnetic multipole. In this case, according to (39.10),
BT,11 = ? Yilr GL (tor).
Expanding eic-Lr in spherical functions, we arrive at
where -
Thus,
NrY(n) G d
c rLm L,
L(cor) gL(qr) r2 dr -=
(47c)2/ \L
(w2 ? q2) co '
dfl) =a 1 ( ? 2L 2
n2((.02_?,2)2(0u)
dp + dp_ u*ccYni(-1-) 8 (co ?e_ ?E+).
The summation over spin states may be performed in the usual way. We have
/I u*ccYVIdvI9=--
Re"111.--),-/7+)1 YDA(12+
-
e_e+ +
(P- Y2f)(P+ YVir) (P _YE)(p +YVAI)) .
(o)
This expression can be simplified. Since E = q - E and q y = 0, we may write
Further,
P_ Yi = ?p+ Yr.
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2
_P ---= Pq.q ? P + ?
(40.3)
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On the other hand, conservation of energy
leads to
Thus,
.,-- + nig --I- If(q ?p_r itt2
ho2 q2).
p +(I = ?
2
EjeczY21012 =
1 r q2 y(par 12 2 ip+ yzif
2
Let us insert this expression into (40.3) and transform from the variables 2. 4., 2 to the variables q, +. We
shall write the products of differentials in the form
dp + dp = dp + dq = pa+ dp + do q2 dq do,
and the argument of the 6-function.in (40.3) in the form
o) ? e_ ? 4 = ? e + ? qa + + ma ? 2qp + cos 5,
where .9- is the angle between q and . If we choose the z axis along q, then
do+= d cos U dp
and we can eliminate the 6 -function according to
8 (co ? e+ ? e_) d cos() 6-
P +4'
The angle .9- is now determined by the conservation of energy:
428
?
(co ? e )2= qa+ ea+ ? 2qp
Equation (40.3) then becomes
dpv = a
7c2 (c02 q2)2 (
)2L+1
q
X I (1)2 112 I YVM 12 2p24. YV2ir sin2 ide dq dcp 4. do q.
(40.4)
(40.5)
Let us integrate (40.5) over dv+ anddo q . Taking the value of sin2 .9- from Equation (40.4), we obtain the
-
following expression for the conversion coefficient for pair creation with a positron energy e+ and a total momen-
tum q:
2a (72L -1 I 1,2 + p2 2
dp)(e +, q) =-- n (j) de dq
(021; +1 4 z to)2? q2) (0,2_ q2)2 ?
Integrating this equation over q between the limits -qmi'n and -qmax, where
qmax=IP++P_I,
we'obtain the energy distribution of the conversion positrons:4
a
t'' .1"I ?
dPi,)(e+)? 7.:(,)2L+1 ) [P-+ --I- P2 ?2 (L 1) mai In M2 + P -4- 6 1-6
mu)
12r, 1
? [(P P -)2 (I) 4 --p_) + -2- f(a+c- p+p_ _m2) ( p++p_)2(L-1)._
(e+e_ ?pp- _ m2)P \2(L-1), 1 2
P-i ? I p p2_ ? 2 (L ? 1)1;121
L-1
X - i(n+ p (1) p_r21} de+.
n
n=i
(40.6)
(40.7)
4 When L_=-- 0 the last term in (40.7) which contains the sum over n from 1 to L - 1 is set equal to zero.
429
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We present explicit expressions for the cases of a magnetic dipole (L = 1) , quadrupole (I, = 2), and
octupole (L = 3):
mu)
04?) (e +){w2 (11-1+ ? 2M2) In m2 + P - ? -- -1-
1w5mm
P +P _ (w2_ 3P2 )) +;
a
(s+) =;?-77- (034(p2++ p2 ? 4m2) In ni2 P + +e-
mm
p +p [032 (a)2+ 8m2) ? 2w2 (p24. +pl) ?5 (4. p2)2]) dc.
(40.8)
The total conversion coefficient $ (0) is obtained by integrating (40.7) from m to co- m. The results of
L - -
_
this integration can not in general be expressed in terms of elementary functions. Later we shall present the re-
sults of a numerical integration. First let us investigate the ultrarelativistic case, w>> m , when we may make
the approximations k c E_ c ?
Assuming, in addition, that Hw ) 2 ? L, we obtain
As an example of an application of the formulas obtained for the conversion coefficient, let us determine
the cross section for formation of a deuteron and creation of an electron-positron pair in slow neutron capture by
a proton. Noting that in this process there is a transition from the IS to the 3S state, we conclude that the transi-
tion is magnetic dipole. Therefore,
a =php1
where a is the cross section for the above process, and a is the photocapture cross section, which for slow
neutrons is a pi 0.3 ? 10 -24 cm2. The deuteron binding energy is I c J = 2.15 Mev, so that w = 4.2 m.
(
Numerical integration for this value of :a gives the value 3 ? 10 - 4 for $ 0). Therefore,
a = 0.9 ? 10'28=2.
2. Conversion. of Electric-Multipole Radiation.
Let? us now consider an electric multipole. Inserting the expression
p(10,) 2-237t In.
? BPAr = GL (cur) Yrag 71-
- 2L -F
1 GL-1 Ow) YL, L-1, MX
3 4n + 5
L-1 (40.9) 'IL +.1 +
[
2 4 (21+ I) 411 2 (2n + I) (2n + 3)1 ?
U . t
In the other limiting case / ? HI) )2 (also with w>> m ), we find the following asymptotic expression for
( 0 )
0 :
0) aL.
n -U72- +2 +11,
(40.10)
into (39.12), and again using the wave functions (40.1), we obtain
B ) ?
1 ( q z\ 1
illEY Lidev ?112L +1 (-L YL L-1 Iftl*ad.
(q2 ? w2) ) '
Using this expression in (9.12), and integrating over the angle between E + and q, we arrive at
where y is Euler's constant. dc3S12=
The distribution in the angle x between the particles of the pair can be obtained by going from the variabll
q to x in Equation (40.6), and integrating the resulting expression over ?+. The relation between x and q is
given by
11?4%
(12 = p2+ p2 ? 2p .1.p _ cos x.
aq2L +1
20)2. b? (0)2_q2 (L+ Elv--.EYL3[11*V-Y2L + 17 L-1 M(11*(r012 X
q ?
X as_ dedqdoqdp,
where E denotes summation over electron spins. The summation is
(40.11)
431
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tog
I, .12= L YL;iiI2(e+e_ ? in2) (2L 1) I YL, L-1, at 12 X
ge
? w
X (6.1.e_ ? p p 1112)-1f (2 L 1) ? ( YLM Y.1, L-1, M-1-- Y'Lm Y -1. it) X
.(02
X (e-P + +P -)?(2L (K, L-1, ittP4
L-t. ArP-)( Yr,, sip +)1.
The scalar products (Y p+ ) which enter into this exvression are easily calculated by resolving
- L - 1, M
into longitudinal and transverse spherical vectors. For the longitudinal vector we have
- L, L 24,1
and for the transverse one,
Y(-1) p
? + COS ? - 11
LAI p + ? ,
Y( I)
1.31 P - = (q ? p + cos 0) Yhm,
Y9111+ =
I YZTP + 12 = 1 Y (II 12 sin2 a.
Inserting these expressions into (40.11) and integrating over dg and do q, we obtain
1
2a q
2E-1 L 4 ((3L+ 1)0)2 2Ls+e-) (02 ____ q3
49) (a+1 (L + 1) 1w2L+1 2 2
1
2 tw2E+E- 92L-3 [3L + 1
? ((LL e+e _ Lw2 + (14- 1) m2) -I-
q2
(032? q2)2 0)2L-1 3
-I- (02 (2Le+e _ -1- CL -1- 1) tit2) (.2 _ .7.2).11 de dq.
'Integration over q leads to the energy distribution of the conversion positrons:
432"
(40.12)
a
(e 4.) ? 7.0,2L+1 (0,2(1-.1) [e2+ e2 2 (L -- 1) m2] In E?? 1417- -I-
2(L1-1 [(p+ p (p+I (a:.(1.?+e-1))2 nt2] [(7+4.p...)2(L-1)___
?????
- (p+ -p _)2(r--0] +4 Ke+e- +P+P-- m2)(P++P-)2(L-2)
(ea_ p p in2) p + p (L-2) [e2+ e2
2 (L ? 1),2] X
L-1
2(7,-n-1)
X E ? [(p+ P (P+ P-)21) de.
n=1
Equation (40.13) is valid for all L, except L == 1. In the latter case (electric dipole), we have
no + p+P
t+E_ - + 2p+ p_} de.
diaill)(8+)=L {W.I.-Fel) In mco
We present here the expressions for the electric quadrupole and octupole, as given by (40.13):
QM a ? 2in ) In '112 p?p-
2
MO)
dV2 (E+)=-- ( 0)2 (a2+ e2
4-4P+P- (e+e-
dp(31)
? 4m2) in M2 C?C- P
mto
vo)7
? p+ p_ [20)4 ? w2m2 ? 8,714 ?(9w2 -I- 4m2) e _ 462+ 62 ] )de
- + ?
? m2)) de +;
(40.13)
(40.14)
(40.15)
As for magnetic multipoles, we shall present the expression for the total internal conversion coefficient
only in the limit of ti) ? m. In this case, if ?w 2 >> L, then
. -
L-1
Pm_ 2a 2w 23 _j_ 5L +1 4n+5
L in 12 ' 4 (L +1) (2L +1) E 2(2n + I)(2n+3)}?
n=1
This formula is valid for L a- 2. When L = 1,
po) 2a (in 20) ___5 \
? ? 3Tc m 3).
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(40.16)
(40.17)
433
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In the other limiting case when L2 > > ) 2 ? 1, we obtain the same expression as (40.10) for the
rn
magnetic Inultipole.
We present below a table of values for the internal conversion coefficient with pair creation for electric and
magnetic multipoles, as obtained by numerical integration of expressions (40.7) and (40.13).
3. Nuclear Excitation by Electrons.
The process which is the direct opposite of internal conversion of gamma rays is nuclear excitation by an
electron. In this case the energy of the final electron state is lower than that of the initial one, E 2 < E 1, and the
final energy of the nucleus is greater than the initial, I"::2 > El. Using the results of Section 39 we may write down
the expression for the probability of exciting a nucleus, changing its angular momentum by L, and its parity by
+? L X +
I ) together with the transition of an electron from an initial state (1) to a final state (2);
0) tx) 00)
wL= Wad,tg,
(40.18)
where wL rad (x)
is the probability that the nucleus undergoes radiative transition from the excited (final) state to
-
the initial one, and Bis the "conversion coefficient" given by
2
,) 4 w. ,f 1'h dr'.
,
(40.19)
As in treating conversion with pair creation, we shall restrict our considerations to the free-electron approxi-
mation.1) Let pi and p2 be the electron momenta in the initial and final states, and let ui and n2 be their corres-
ponding plane-wave amplitudes. According to (40.18) and (40.19), the differential cross section for the process
under consideration is
(X) ? _ I N1
27_4 ? ? (I))
.19 MC1.15' o f eig,. 11* _0.)
4 v 2
id
, mruid r ,
where vi is the initial electron velocity,
(40.20)
1 .
and E denotes summation over initial and final electron spins (the factor m front of the summation comes
2
from averaging over initial spins). A comparison of (40.20) and (40.2) shows that the last factors in these equa-
tions, namely
1) G. Wick, Ricerca sci. (Italy) 11, 49 (1940); K. Ter-Martirosian, J. Expt1.-Theoret. Phys. 20, 925 (1950).
434
in (40.20), and
in (40.2), differ only by the substitution
Therefore, (40.20) can be written
feioru;Bilits dr 12
Ifei(P +P-)r BPmv dr 12
da2) I w(k) 2 I dp2
-2- mad? ke2 ? el ?
(40.21)
where b(X ) can be found from expressions (40.3), (40.11) for the differential conversion coefficients by writing
-L
them in the form
(X) (X)
_da 2_,24.) 6 (?4. + E_ - co)
Transforming from E. 2 to q in (40.21), and integrating over angles, we obtain
do) = wDadb24 (pi, dq
where
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(40.22)
Declassified in Part - Sanitized Copy Approved for Release 2013/08/13: CIA-RDP81-01043R002200190006-7
436 '
6 7 10 15 20
0.675.10-3 0.864. 10-3 0.134 ? 10-2 0.193 .10-2 0.236. 10-2
0.471 ? 10-3 0.637. 10-3 0.108 ? 10-2 0.164 ? 10-2 0.206 ? 10-2
0.346.10-3 0.491 ? 10-3 0.902. 10-3 0.141.10-2 0.186 ? 10-'2
0.262 .10-3 0.390 ? 10-'3 0.770 .10-3 0.129 10-2 0.1G9.10
0.203 .20-3 0.297. 10-3 0.668. 10-3 0.117.10-2 0.157 .10-2
0.680 ? 10-4 0.130.10-3 0.372 ? 10-3 0.792.10-2 ? 0.115 ? 10-2
?
U)
?)
1
0
?
C)
to
..3.
6
co
1
0
?
_.
?::=>
CO
6
:,
1
0
?
c?
CNI
6
.1
1
01
?
--4.
v.
?-?
6
co
1
CD
?
co
0
?-?
6
,
1
0
?
00
"rt?
C.)
c;
whir
3 4
1 0 710. 10-4 0.257.10-3
2 0.273 .10-4 0.141 ? 10-3
3 0.114 ? 10-1 0.833 .10-4
4 0..503 ? 10-5 0.511 ? 10-4
5 0.229. 10-3 0.323 .10-4
10 0.58 ? 10-7 0.409. 10-5
Total Conversion Coefficients
CD
C..I
10
..
I/
1
0.
?
D)
?-.
CO
6
:1
1
0
'
I.0
10
Cs3
c::,
11
.
0
?
C.)
(N
01
6
II
I
0
?
CO
0)
?-?
6
71
I
0
?
CD
00
?-?
6
II
I
0
?
t?-?
*7 I
?-?
6
IIIIII
CD
?
I."--
CO
C.I
6
11
i
.0
'0
CD
CV
6
0
?
IN
..
0)
6
:1
I
0
17,1
10
?-?
6
:.
CD
?
0)
t-..
?-?
6
II
I
0
v..
DI
?-?
6
CI
0
?
ID
to
?--,
6
11
I
0
C.,
0
?-?
6
CI
0
?
CI
CO
?-?
6
It
I
0
CD
10
CO
6
0
0
?
0
0
CI?
6
M
i
0
?-.?
10
co
0
????
t--.
I
c:D
?
CD
. ?-
,_.
6
..:,
I
6
?
--?
0
,_.
6
., I.,
: I
6 o
? ?
CC co
.1.t--
I,..? io
6 6
,s,
1
6
?
to
co
-r.
o
CO
I
6
?
-r
N.
?....
CD
CD
CI m
1 1
0 0
N. ?-?
CV ?-?
?-. CO
6 6
?,, ,,,
1 1
0 0
.-.
? ?
CO --.
C) CO
0 10
6 6
c., :,
t III!!
= 6
"
? co
N. CO
CD CO
6 6
co ',
1 1
0 0
1.0 C0
CD ?-?
t 0 -}:
6 6
10 t?
1 1
0 0
? ?
CO CO
I- It)
CO Cs1
6 6
I-, :.,
6 co
"
It) Cr?
CO 0
?-? ?--
6 6
el
1
0
C I
?-?
CP,1
6
..?
1
0
CO
N.
0:
6
HO
10
1
0
?
N.
1--?
.--?
6
o
?
00
CO
tr...:
6
-r?
1
G
?
CO
0
co
6
6
?
,-I.
SO
I .
6
...,
I
6
?-?
c;
0
.._,
-4,
co
CC IC
1 III
6 6
?i
co
cl.--
6 6
? cs,
-:.
6
co
co
6
-,.
00
r.,c.)
.?
6
It
I
g.?
6
..r"3
co
-4,
?
IN
PP(p4, q)- 4d-(1)(e+' q)
?-e+, dq
(X)
and de is given by (40.6) and (40.12).
? L
In order to obtain the total cross section o(X ) for excitation of the nucleus, (40.22) must be integrated
over q from q . to q , these limits being determined by the conservation laws. Since the total cross section
?nun ?max
cannot depend on the direction of the initial electron momentum, (40.22) can also be averaged over directions
of the vector.2.1.:
qma.
(X) 0,) _1 rd o1
0L ? wLrad.p1,
- 4t 41s
This expression can be written
where
qmln
w rad. ri)
ay b/
8Tc/4 (61))
dfiD
b(1)(e)- (e)
..= de
and d 8(X) is given by (40.7) and (40.13) with 2+ and R_replaced by pi and E2.
? L
4. Monochromatic Positrons.
(40.23)
If an atomic shell has an unfilled state, then it is possible to produce a pair in which the electron will
occupy this bound state, and the positron ...Jill have a well-defined and definite energy. This is the phenomenon
of internal conversion with emission of monochromatic positrons. In calculating the internal conversion coef-
ficient in this case, Equation (39.13) can be used directly with 4/2,a wave function of the discrete electron
speqtrum, and 01. a negative-frequency wave function of the continuous spectrum, normalized for a unit energy
interval. Equations (39.25) and (39.28) are then valid for the K-shell, and the radial functions of the continuous
spectrum sic and f belong to the negative frequencies. In the approximation of low Z and energies which are
high compared with the K-shell binding energy, we obtain
437
?
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436
(Total Conversion Coefficients
6 7 10 15 2u
0.675 ? 10-3 0.864 ? 10-3 0.134 ? 10-2 0.193 .10-2 0.236. 10-2
0.471 ? 10-3 0.637 ? 10-3 0.108 .10-2 0.164 10-2 0.206 ? 10-2
0.346 ? 10-3 0.491 ? 10-3 0.902.10-3 0.141? 10-2 0.186 ? 10-2
0.262 ? 10-3 0.30 ? 10-3 0.770 ? 10-3 0.129 ? 10-2 0.1G9 ? 10-2
0.203. 20-3 0.297. 10-3 0.6G8.10-3 0.l17.10-2 0.157.
0.680 .10-4 0.130 .10-3 0.372 10-3 0.92.10-2 ? 0.115 ? 10-2
?
Ira
0,
I
0
Ch
CO
-1.,
c 1,
V, fn
I I
0 0
-- -4.
00
0, Cy
d c D
M
I
0
-14
'.4.
?.
ci
M
I
0
C.?3
0
?
d
,
I
0
cr.,
.cr
CNI
c i
3 4
1 0710. 10-4 0.257. 10-3
2 0.273 ? 10-4 0.141 ? 10-3
3 0.114 ? 10-1 0.833 ? 10-1
4 0.503.10-3 0.511 ? 10-4
5 0.229. 10-5 0.323. 10-4
10 0.58 ? 10-7 0.409. 10-5
Total Conversion Coefficients
;)(p4, 4,12)(c+' q)
dq
and di:1is given by (40.6) and (40.12).
?L
In order to obtain the total cross section ct(X ) for excitation of the nucleus, (40.22) must be integrated
over q from ?mm ?max these limits being determined by the conservation laws. Since the total cross section
cannot depend on the direction of the initial electron momentum, (40.22) can also be averaged over directions
of the vectorzi
expression can be written
where
(X) I (X) 1 1 d
?
p1 4j?b (Xk),piq) dq.
--= ?2 IDLrad.--F ?rc L
qmin
CX) wL
? 2 I (e1),
8npi
b2) (a)
de
and d 8(X) is given by (40.7) and (40.13) with E+ and E_replaced by EA and E2.
? L
4. Monochromatic Positrons.
(40.23)
If an atomic shell has an unfilled state, then it is possible to produce a pair in which the electron will
occupy this bound state, and the positron will have a well-defined and definite energy. This is the phenomenon
of internal conversion with emission of monochromatic positrons. In calculating the internal conversion coef-
ficient in this case, Equation (39.13) can be used directly with Ova wave function of the discrete electron
spectrum, and ?Pi a negative-frequency wave function of the continuous spectrum, normalized for a unit energy
interval. Equations (39.25) and (39.28) are then valid for the K-shell, and the radial functions of the continuous
spectrum zx and fx belong to the negative frequencies. In the approximation of low Z and energies which are
high compared with the K-shell binding energy, we obtain
437
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pp,: 2a (Zar (L+ 1) (D2 + 4Lm2 ( 2m )-6-111
+ 1) to2
2a (Za)3 (1 21,64-'12
?
co
As a numerical example, we present the ratio between the probability for conversion with emission of
monochromatic positions(with creation of a K-electron) and the total probability for conversion with pair crea-
tion.1) According to calculations performed with the exact radial wave functions for w --z--- 1.4 Mev. this ratio
for an electric dipole is I/3. As co decreases, this ratio increases.
The process which is the inverse of this One is the following. A positron collides with an electron in the
K-shell of an atom, and this pair is absorbed by the nucleus, so that no photons are observed and the nucleus is
excited. The probability for this process can be written
w
(40.24)
where ft is the coefficient for internal conversion with creation of monochromatic positrons, and w is the proba-
bility for nuclear excitation by a photon of the appropriate energy. -Y
Since the energy the nucleus acquires in this process is quite large, fission may result. For instance, the
cross section for uranium fission by this pair-absorption process is of the order of 10 31, cm . -
5. Pair Production in Particle Collisions.
A process similar to internal conversion with pair creation is pair creation on collision of two charged
particles. This process can be represented by the diagrams shown in Fig. 50. The upper two solid lines of each
diagram represent the colliding particles, and the
lower ones represent the pair produced. The diagrams
can be used to construct the matrix element for the
process of pair creation on collision of two electrons.
The cross section for this type of process has not been
calculated. Its order of magnitude can be evaluated
from the fact that we are dealing with a fourth-order
process, and, therefore, the cross section is about c(
times that for radiation in the same two-electron col-
lision. If one of the electrons is at rest before the col-
lision, conservation of momentum and energy neces-
sitates that the second electron have an energy greater
than 7m.
a)
Fig. 50
b)
If one of the particles can be replaced by an equivalent external field, then the upper lines of Fig. 50 may
be removed. The diagrams so obtained are shown in Fig. 51, where the external dotted line represents the ex-
ternal field. We note that if the external field is accounted for in the electron wave functions, the diagrams of
Fig. 51 are transformed to the internal conversion diagrams of Fig. 47.
L. Si iv, Proc. Acad. Sci. USSR 64, 321 (1949).
2) R. Present and S. Chew, Phys. Rev. 85, 447 (1952).
438
*IP
?
Using Fig. 51 it is easy to construct the matrix
element for pair creation by an electron in the field of
the nucleus. We present here only an approximate ex-
pression for the cross section of this process in the ultra-
relativistic case (6 ? m), where 6 is the initial elec-
tron energy:
ro2z2a2 )8
u In
(40.25)
>a)
Fig. 51
We note that when Z = 1, pair production in collision of two electrons will be the same order of magnitude in
a
this limiting case. The cross section of (40.25) has a coefficient which is ? times less than that of the
4 r
bremsstrahlung cross section (31.20); the power of the logarithm, however, is in this case higher [Equation (40.25)
does not take account of screening].
Figure 51 can also be used when considering the process of pair creation in collision of two heavy particles
moving at a. nonrelativistic velocity, since a system of two nonrelativistic particles is equivalent to a single
particle in an external field. This process is equivalent to conversion with pair creation for an electric dipole
(in this case the particle states belong to the continuous spectrum). An approximate expression for this cross
section isl)
= /VC
2 2 m24z22 ZiA42 Z2Mi )2
M2 T2
Mt
(40.26)
(MI and M2 are the masses of the particles; the particle with mass M1 is at rest, and that with mass M2 has a
kinetic energy T2, which is assumed much greater than 2m).
Fig. 52
It is also possible that both colliding particles can
be replaced by external fields, so that the diagram
illustrating the process degenerates to that shown in Fig.
52. In this diagram both dotted lines correspond to an
external field. Its structure is the same as that for pair
creation by two photons. This case occurs in collision
of very high-energy particles, so that after pair creation
their states of motion may be considered unchanged 2.1
(i.e., with no change of direction or velocity). The
matrix element for this process is
^ TS
y2 - u 122>r e u a q, 4-71727 q2
See W. Heitler, The Quantum Theory of Radiation (State Tech. Press, 1940) p. 220.
2) L. Landau and E. Lifshits, Physik.. L Sowjetunion 6, 244 (1934).
439
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where a and a are the Fourier components of the external field which is a superposition of the fields of two
511 q 2
charges moving at equal velocities (we note that it is just the superposition of the field that leads to pair produc-
tion; a single free particle, naturally, cannot create pairs). The expression for the cross section for the case in
which one of the particles is at rest is the same as (40.25), except that the nuclear charge should be replaced by
the product of the charges Z = ZiZ2.
Any radioactive decay with sufficient energy can be accompanied by pair production. This process can
also be treated as pair production by an external field h.i.ving frequencies greater than 2m. Thus, when a nucleus
disintegrates into two parts, the probability is given by3)
??
2.2ftt 2 2E( 2E 1092\
? v 1 n ?1
M n '
Z 2A 2
(40.2'7)
where Z' = Z1( 1 ), A1 and A2 are the atomic masses of the fragments, E and v are the energy and
velocity of the smaller fragment, and M is its mass. The probability for pair creation in 8 -decay is about*
10 - 6 - 10 7.
? 41. 0 -0 Transitions. A21 (r2) ao (r2)r2 ? vX (r2),
A20
1(r2) (r2)-
where
e
A21 (r f ,1? (r , dr,
1 ..t 4n I rt?r2i
(41.2)
_
The case L = 0 corresponds to spherical symmetry of a nuclear transition current distribution. Therefore,
its space (Jzi) aiidtithe(Lh ) components should be of the form
J21 (r2) ?=1(r2)1'2,
Ai(r2)---= g (r2).
The potentials related to these currents have a similar structure:
1. Reduction to the Static Interaction.
The general expressions derived in Section 39 for the probabilities of conversion or nuclear excitation be-
come invalid if the angular momenta of the initial and final nuclear states are zero. When L = 0 Equation
(39.11) vanishes, in agreement with the fact that there exists no photon state with angular momentum zero.
Nevertheless the element of S(2) for transitions between two such states is nonzero. Indeed, as has already
been shown in Section 39, the integrals in (39.11) are taken over the region outside the nucleus, whereas the
general expression for the matrix element, according to (39.32), contains an integral also over the region occu-
pied by the nucleus, and this integral is in general 'different from zero when L = 0. Thus, the matrix element
for the effective interaction energy is of the form
eiw I
ri?r2
Ut-->1= .121.(r1)J21 (r2),lic ? r21dridr2,
6
where j21 is the four-dimensional electron transition current, and j2i. is the four-dimensional nuclear transition
current the integral is taken over the region C2 occupied by the nucleus. This matrix element can also be
written in the form of (35.16), namely,
ui -+ r ----= fi 21 (r1) A21 (r1)dr1,
2
I) A. Migdal, Bull. Acad. Sci. USSR 4, 2, 287 (1940).
2) L. Tissa, I. Expt1.-Theoret. Phys. 7, 690 (1937); E. Feinberg, Proc. Acad. Sci, USSR 23, 778 (1939).
440
(41.1)
[This is easily seen by using (39.2) in (41.2).]
Inserting (41.3) into (41.1) we obtain
.
U4.4. f = r (JICP0 -1-j21 VA) dr.
Introducing the new function 9 by the relation
?=--- ju)A9
integrating the term containing 7/X by parts, and using the continuity equation
we obtain
di,vj21 == ? it!) r
211
Lii f= f j1pdr= e f tr2 (141 dr.
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(41.3)
(41.4)
(41.5)
_
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Wc note that transformation of (41.4) to (41.5) is possible only because the general expression (41.1) for the
matrix element is gauge invariant.
Let us determine the potential co by making use of the fact that (p0 satisfies the equation
and, therefore,
Ach (02170
= A (To + iwA) ?g? ?To
Since A 21 = , it follows from the Lorentz condition for the potentials that
Therefore,
AA =? kuTo
AT= g eV. 2.4r
1.
Equations (41.5) and (41.6) can also be written in the form
drtdr2
Ui_*f?--= a PF*2(rt) 4(r2)11rt (ri) (r2)
r2I
(41.6)
(41.7)
We'note that when L = 0 the transition is caused by the electrostatic (Coulomb) interaction of the charges.
Equation (41.5) is conveniently written in the form
Ui4f= f .114:b dr,
where cl is the electrostatic potential of the electron distribution, and satisfies the equation
A(I)= ?
(41.8)
(41.9)
rt
- Sanitized Copy AIDID
Va.
In this equation the right side may not have spherical symmetry. In view, however, of the spherical
symmetry of the charge density in (41.8), a nonzero contribution is obtained only from the symmetric part of
the potential. Therefore, the right side of (41.9) may be averaged over angles.
2. Conversion and Nuclear Excitation in 0-0 Transitions.
Let us find the probability for conversion and for nuclear excitation in 0-0 transitions.1) The electron
wave function hardly changes in the region occupied by the nucleus. Therefore, the right side of (41.9) may be
considered constant, and we may use the equation
where
The general solution of this equation is
1 d2 (r)
r dr2
epo,
Po = +; (0) +1 (0).
c2 r2
epo --6?.
The constant c1 is of no significance, since (41.8) vanishes for constant cl) in view of the orthogonality of the
nuclear wave functions. The constant c2 is zero, which follows from the fact that ck is finite when r =-- 0. Thus,
where
2n
U1 9.r
e2 * 2
Qo = f 721F1r dr.
(41.10)
The quantity Q0 is the "zero-pole moment" of the nucleus, in analogy with the quadrupole moment. Since
when L = 0 no photon is radiated, we cannot introduce a conversion coefficient. To order of magnitude,
Q0 ? R2e2,
1.) Yukawa and Sakata, Proc. Phys.-Math. Soc. Japan 17, 10 (1935); K. Ter-Martirosian, J. Expt1.-Theoret.
Phys. 20, 925 (19-50).
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whew R is the radius of the nucleus.
The probability for conversion on the K-shell is thus given in the following way:
/2% \2
w= 27r --)1Q012
3 I i& (0)121q't (?) 12,
(41.12)
where s (0) is the value of the K-electron wave function in the nucleus, and Vi (0) is the value, also in the
nucleus, of the continuous-spectrum electron wave function normalized for a unit energy interval, and correspond-
ing to the energy c=co-t-Ic
k and the angular momentum j
The probability for conversion with pair production is
dw (2Tr) I Q01 2 E I Po I 2 dPd1)1,3-
(2
In the free-electron approximation,
11'0,
where u and v are the positive- and negative-frequency plane-wave amplitudes, and Z denotes summation over
_ _
electron and positron spins. We may perform the summation, obtaining
so that
:444
EIP012=1 ."
4e_c Sp p (tp_ p(-ip -in) ? e_ e+ 1712
? E_E
rho Q01-?
IP? 11P+ Ida+ (10+ (c_c+ ni2?P-P+)?
Similarly, the differential cross section for unclear excitation is given by
do = -11-8 I Q01:2 pP24111 (s let: I? in2
(41.13)
(41.14)
The total cross section for nuclear excitation is
2n n I
a in 12 2%
? _ 9 I s? I I Pt ele2-1-
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(41.15)
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F.'
CHAPTER VIII
RADIATIVE CORRECTIONS. VACUUM POLARIZATION.
42. Third-Order S Matrix
1. Third-Order Matrix Elements
Let us now go on to the consideration of concrete physical effects due to the interaction of an electron
with the zero-point oscillations of the electromagnetic field, as well as those due to the interaction of an
external electromagnetic field with the zero-point oscillations of the electron-positron field.
We shall start by considering radiative transitions to electron scattering in an external field. Since the
scattering is a first- order process, the effects in which we are interested arise in third-order perturbation theory;
we shall restrict- nur considerations to the third order. Diagrams illustrating such third order effects are shown
(e)
in Fig. 53(A - denotes the external electromagnetic field).
vertex
(24.7)%
cr*
Our previous results can be used immediately to exclude
diagrams 3,4, and 5, since 3 and 4 contain electron self-
energy parts along the free-electron path, and 5 represents
two independent processes one of which (corresponding to the
closed electron loop) is a correction to the vacuum-vacuum
transition probability which must be considered,zero for
P2
3 1 4 5 I
,XCCN /1\ask ^1/4
physical reasons.
Thus the third-order processes which determine the
interaction of the electron with the zero-point oscillations
of the electromagnetic field as well as the interaction of
the external electromagnetic fieI4 with the zero-point
Fig. 53 oscillations of the electron-positron field, correspond only to
diagrams 1 and 2. The first of these will be called a
Ailgram (radiative correction proper),and the second, a vacuum polarization diagE n. According to
the matrix elements corresponding to these diagrams are
_ fi (112 kn) r7 (I) 61-14)?m 1114k
11 (P2? k) 2 + m2 (2v)4 (Pt ? k)2 ni21P' k2)
osesti-2 P
M3) ? ( ? e8?u2Sp { 1 m?;Is (g) m -1A
q*4 (p 02+ m2 (2704 /5i_T_Trii- iv.) -pi
where a (q) is the Fourier transform of the external potential A(c) (x):
- - I-1 -
446
a11. (q) f 4e) (31) d4x,
(42.1)
(42.2)
s
and th and th are the spinor amplitudes for the electron states with momenta pi and p2; in addition, q =
The extra factor of -1 in (42,2), as compared with (42.1), is due to the fact that diagram 2 contains a closed
electron loop. This is also the reason that (42.2) Involves the trace (spur) of the matrices belonging to this
closed-loop (see Section 24).
where
and
2. Calculation of the Matrix Element for Radiative Corrections to Electron Scattering').
Let us now calculate M(i3). We shall write
m(2).__ les ?1/41u,
I ? (2104 ?
i,le22:::ki))2?+ :2 #a'
= (ii2? In) (is (9) UP.;
0,) (1)t ? ) ? m (12
(p ? nt2 k2
? in) y ? i [ip. (432 m) tis (q) y 4-
+ (q) ? m) 1LJ ./a ? italz (q) 1r1 1j.'
d4k
1 (k2-2p1k) (k2-2p2k)k2 '
(k2_2pik) (k2 ? 2p2k) k2 I
kc,d4k
Jaz= f kakid4k
2pik) (k2__. 21I2k) te2 ?
(42.3)
(42.4)
(42.5)
Only the third of these integrals is divergent for large litl; according to the classification of divergences
in Section 25. this integral is logarithmically divergent (diagram 1 is an irreducible vertex part). Therefore in
the future wesha il have to renormalize Jar , by subtracting its value for q = 0, writing
j VTR J OT 414.0
(42.6)
(seo(26.11); in this case pi and 1)2 are free-electron momenta].
We note that I diverges for small (Lc' ?This represents the infrared catastrophe due to the inapplicability of
perturbation theory in the lowlkl region (see Section 32).
Let us now use Equations (28.12) to simplify the operators multiplying the integrals J, Jo ,j0. .in (42.4).
The operator multiplying J can be written
ii1.(tp2 ? m) a (q) (11,1 ? In) Tp.=
2pi a (q)p2? 2int fa (q) p2 p2a ,(q)J-2im fa (q) p, + pia (q)J-2m2a(q).
/Tice R. P-Cynnian, Phys. Rev. 76, 769 (1949).
(42.7)
14'7
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Since this expression is multiplied on the left by 172 and on the right by ut in the matrix element, It is
clear thit the operator on the right can be replaced by -m. In the same way, -In can be substituted for
11;2 on the left. (If ifit, rather than 432 is on the left, or if i2, rather than i1:01, is on the right, then before
performing this substitution we must write = ?4,132 =151 + CI.). After performing these substitutions we
_ _ _ _ _ _
may rewrite (42.'7) in the form
Tp, (43a ill) (q)*(41? nt) ip. 4 m2It (q)? 2 '4/1 (q)q.
Let us now make use of the relation')
A A A A
q a (q) q =-- q2a (q). (42.8)
Then the final expression for the operator multiplying ifs
(42.9)
in) 71 (q) (ii) 1? m) ?cp, 4m2S (q) -I- 2 q2 (q).
We may similarly transform the operators multiplying J a and J a T in (42.4), obtaining
111(1,2? in) (4) TaTp. Tp.'faii (9) (1/11 ? in) i,. =
(42.9')
= ?4 ma, (q) -F 21 fq'it (q) T.? Tort (q)
Tgair (q) ?2(q)y0.
(42.9")
Inser?ting (42.8), (42.9'), and (42.9") into (42.4), we arrive at
A
91
A A A A = [4m2a (0+ 2q2a (q)JJ -I- 1 (4 nza. (q) H- 21 (i ?a (q) q ? q a (q) i?)] J (42.10)
(q) i0J0.
Let us now calculate the integrals:J.
1
ab
it can he shown that
Ja, Jar. By using
dx
the
1
n?b?
1
dy
Identities')
1
2x dx (42.11)
[ax b (1 ? x)I2 '
1
[nx
b (1 ? x)18
(42.12) (42.12)
(42.12')
(42.13)
(see Section 56), where it is shown
(k2 ?2ptk)
where-
Therefore, we can write the integrals
-
The integrations over k in these. expressions
that -
(k2? 2p2k) =
(k2-2pyk)2 '
2x dx
(k2-2pyk)2
Py=01-1--(1?Y)
J, J0,
0
0
dd:
dy
are
= _.-
le .01
in in the form
22: ddxx
ft 2x dx
performed
(k2 ?2p,k)8
Px=xPy?
fir
:22 :22k:
88
vesk_akt2dp4:k)3 ?
in the Appendix
rrThis formula is derived by multiplying the equation
on the left by ii, and using the relation
448
?
;(q) t? (q) 2n (q) q
,
112q111 == 112 (P2 -fit) 112 = 0.
d4k 7t21
4, I
(k2 ?-? 2p,x/e)3 2K;
n2i
(k2 _kg -
1) R. Feynman, Phys. Rev. 76, 769 (1949). In some other cases the identity
is also Iisful.
a1a2 ?-? ? an
? (n ?1)1 1 dxt I dx2. ? ? ? O,&_1 (x,_2 ? 4-
? .
O
? ? ? ? at (I ?xt))-".
?
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f ick,d4k (k2-- 2pa,k) 7c21,nr,n (._._ d) -4.. Ao (N) + I] ? 2/4 n21 PmerPxt?
8 _ 4 l 2 .
I
(42.14)
The first two of these integrals are absolutely convergent, whereas the third diverges logarithmically for
large Ott . This last integral is therefore taken over some finite region (N) and contains a constant Ao(N), which
behaves as in In N as N9.co (the number N increases as the region of integration becomes larger).
Using Equation (42.14) and noting that p x (1 -y) p2J, we rewrite (42.13) in the form
x
where
=--
-
1
J =f I dy,
2
1
= Tc2/
f dy,
2
n2/
I 1?dy,
IgT 2
1
2x dx
p2 x
1.
? 2xpx, dx
2 24,
2 I
Pa, Py
1
1 1 1
Pa'aPMT dx
'0 Px
1
Plicrillit
P y
2 ?
(42.13')
(42.15)
We note that I diverges logarithmically for small x. This divergence, however, is not at all connected
with the divergences relatad to high momenta of the virtual particles, but, as will be seen later, is connected
with the infrared catastrophe. This, as we know (see Section 32), is due to the inapplicability of perturbation
theory to emiscion and absorption of low-momentum photons.
We shall therefore restrict our considerations in this Section to the interaction of the electron with photons
whose frequency is greater than some minimum value cumin. In practice it is more convenient, however, to
perform the integration over photon momenta using another method which is equivalent to the condition
ca>wmin' ? this method does not involve restricting the photon frequencies, but assumes that the photon has some
extremely small nonzero mass X. In Section 32 we established the relation between x and wmin, and saw that
if the electron momentum is small compared to m, then
450
ln 4=ln 2%0+-56..
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?
In assigning a nonzero mass x to the photon, we must replace the function D-(p) as defined by (24.5) by
thelunction
Then instead of expression (42.5) forj we obtain
Noting that
2/ I
(2704 p2 + x2 I 12 > O.
J
d4k
(k2-2ptk)(k2? 2p2k) (k2+ 42)
1
2x dx
(k2 ?2pyky (k2 4-x9? f 02_ 2pxk + Lox I Pal = XP yl L a, = (1 ? X) X2
and
we write! in the form
d4k n2i 1
(k2- 2pxx Lx)8 2 pc.2
1
dy,
where -I is no longer given by (42.15) but by
f (A) 2x dx
? (1 ? x) _ x2 2 I
P y
which does not diverge as x -*O. Performing this integration, we obtain (neglecting terms of order X)
and therefore
/ (X) ? -1-p (In V? ? In A)
J = ? Tc2i -f!-c- 1/1 ? ? In A).
(42.16)
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We thus see that the divergence as 3-40 in expression (42.15) for I is removed oy assigning a nonzero
rest mass to the photon. It then follows that this divergence is indeed related to the infrared catastrophe, as was
stated above.
We shalt later eliminate the photon mass X in (42.16), and consider the interaction of the electron with
long-wavelength photons.
2 2
Let us transform (42.16) to another form. Noting that pi = p2 = - m2 and 2pict = - $12 (q = P2-1:1). we
can write p in the form 2 - - - - -
- Y
where 0 is related to q by
P2y= ? f pi+ q (1 ? y)12 nt2 q2 (1 ? y) y
[1 ?4y (1 ? y) sin20],
? q2= 4 m2 sin20.
2y ?1. ta?lortaA
p2 = m2 COS? 0
COS2
Eliminating/ by means of a new variable g
we obtain
and
Since
we finally arrive at
452
e e
270/ I
J .1. =---- ln ?tic? 1 In ? dc }.
m2sin 20 1 co s 0 . m
o o
cos e
In cos 0 clt== fE tan E
'0
J = (
2Tc21 fg taut at + 0 In ?mA } ?
m2 stn 20 .
(42.17)
(42.17')
it)
(42.16')
Let us now calculate I(/ and J-0 as given by (42.13') and (42.15). Noting that
? r
we obtain
where
= (l P2a = (P la + P2a)+ Zrint0
7c21 0
=--- ni2 sIn 20 (Pl? 4- P2a
.1t2i1 [
jay 80t _71 A0 (N) -1- .71_In m1 0 (Pt.+ P21) (Ptt Por)
2m2 sin 20
(8a, -I- _Mgt, )(I? Ocot (1)).
Inserting (42.16') and (42.18) into (92.10), we obtain the following expression for 9t:
(42.16)
= (4m2+ 2q2)ci (q) J -1-[141n a (q) (j, p2) ? 2(131+ ..P2) (q) ;*7 (42.19)
2i 'a (9) CPst -1- /70 61-1- PO lz (q)(i)1 -140] Kt
+ 2/7 (q) ;K2? 4c2 (q) K3,
7;2"
K1 =nt2 sin 20'
K2 = ? 12L (1 ? 0 cot 0),
2q2
Ka = [1 ? 0 COE 0 ? A0 (N) ? In ntl.
Replacin.,, the operator ip, on the right and the operator i62 on the left by -m as we did in deriving
(92.9), we call write the expression in square brackets in (42.19) in the form
4 itna (q) (p p2) ? 2 (p^i + ^PO 2q^ (q) (I) -F 1)2)-F
1
(p1 1,2) a(q) (13 1+ ? j2- (20m2+ 7q2) (q) int (q) (q)'41.
Inserting this expression into (42.19) and noting that according to (42.17')
We final
4m2 /72 = 4,112 COS2 0,
/ U
7l = 47z2ia (q) f? --(in ? ? 1)? ? fE tan th. ? ?tarp, -r- ? /la vv.) ?
tan20 , ',. tan20 2 4 -
;.? A % 20 I nr , 2 t 0 A . I A /NT%
0
3 1 20
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; 453
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We thus see that the divergence as x in expression (42.15) foil is removed oy assigning a nonzero
rest mass to the photon. It then follows that this divergence Is indeed related to the infrared catastrophe, as was
stated above.
We shall later eliminate the photon mass
long-wavelength photons.
Let us transform (42.16) to another form. Noting that pf =p = - r12 and 2piq = - q2(q = p2-pi), we
can write p,2 in the form
X in (42.16), and consider the interaction of the electron with
? ? [ pi+ q(1 ?y)12=-? m2-1- q2 (1 ? y)y
m2 [1 ? 4y (1 ? sin2
? q2 4m2 sin2 0.
2y ? 1 =tan' 0 tane,
p2 m2 cos 0
v cog e
where 0 is related to q by
Eliminating/ by means of a new variable g
we obtain
and
J=
Since
we finally arrive at
452
8
In cos e I In ?Xdi }.
m22sni2n120 { f cos 0
8
fg tan E dt,
f
in _cos E
COS 0
m22s12n120 { .10f tan t dt
0 In ?A }.
(42.17)
(42.17')
(42.16')
Jet us now calculate! and0 as given by (42.13') and (42.15). Noting that
Pl/t/ =YPlo ? (I P2a = (Pia P2a) 2tange
tao (Pi. ?
we obtain
nzi 0
(42.18)
(42.19)
nt2 511120
Te2i
J., =-- So,
(Pia + P2e),
1 ? 1 0 (PtoP27) (//tt /12t)
-i-
A0 in
{
Inserting (42.16') and (42.18)
=-- (4 nt2 -1- 2q2)
? (N)+ ? ml 2nt2 sin 20
[
?cot (1)).
9i:
(80, -I14;4 )(I ?
into (42.10), we obtain the following expression for
(q) J -1-[i4 nta (q) (P1 + P2)? 2 (1;1-Fil2) (09 +
where
+ 21; (q) (/;t+ /30 + (/3I -I- Pt!) 71 (q)(P1+/;2)]
+ 2 '4.csi (q) 4:1(2? (q) K3,
7'2"
K1 =m2 sin 20 '
K2 7-2j
. (i 0 COL 0),
2q2
K8 = [1 0 COI 0 --- Ao (N) ? In
. A
Itcplacini., the operator i pi on the right and the operator tp2 on the left by -m as we did in deriving
(92.9), we can write the expression in square brackets in (92.19) in the form
4 ima (9)(P + ? 2 (Pi + P2) a (q) q 2q a (q) (p -F PO -1-
+ (13 1+ I; 2) lz (9) (131+ 132) = ? (20m2+ 7q2 )/z (q) ['q a (q) ? a (q) qj.
Inserting this expression into (42.19) and noting that according to (42.17')
we finahy-obriiiir
= 4itta (q) ftz72121 (In!
4m2 q2 4m2 cos2 0,
1) tan20fg tat t dt ? -2-tan0 + j- A? (N)
4
2
sin 20.
2nt
3 1 20
(42.2
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This expression must now be renormalized . Since 11, diverges logarithmically, jtt is
regularized by subtracting its value at q = 0, that is at 0 = 0, from the factor multiplyins a (q), as was shown in
Section 26 [see (26.11); in our case p1 and pz are free-electron momenta].
The regularized expression for Sa, is
ri 20
9(fi=4n2ja (q)Rtan20
0
1)(in 1)--iat4Er fg tancito tad)]
2
^ A A A 20
+sin 20'
n [a (q) '1? a (a)]
Inserting (42.21) into (42.3), we obtain the matrix element corresponding to Fig. 53, diagram 1, namely
mo) . en ?of
'1 ? I u2"Rul'
which does not take into account the interaction between electrons and long-wavelength photons.
(3)
Before analyzing this expression, we shall calculate M.
.
? 43. Vacuum Polarization
1. Calculation of the Vacuum-Polarization Matrix Element
(42.21)
(42.21')
The matrix element M(2s) which describes the polarization of the electron-positron vacuum (diagram 2 on
Fig. 53) is given by Equation (42.2). Introducing the notation
_sp I i (4)?
j (p q)2 4. m2 Iv p2 + m2 lir. }d4p,
(3).
we shall rewrite M2- in the form
The tensor T
diagram which does
by a new potential
, .154
ay (q)
Ul.
(43.1)
(43.2)
has a simple physical meaning. Indeed, replacing Fig. 53, diagram 2, by a skeleton
not contain the photon self-energy part, we must replace the external potential WI? (1)
6 A (x) which will give the matrix element NI(s) to first order:
? ? ?
? ul, Say. (q). f (x)
.
R. Feymnan. See the Reference on p. 447.
?
?
?
?
?
Comparison of this equation with (43.2) shows that
. e2 1
8a,. (q) = ? Ti,?ct,(q).
(43.4)
The potential 6 A (x) should be considered a correotion,due to the polarization of the electron-positron
vacuum, to the "given" glxternal potential 1-e) 0i). If we apply the operator ?0 to ,6 A (x), we obtain the
correction to the external current J (x) which is dUe to its interaction with the zero-point oscillations of the
electron-positron field. Denoting the Fourier transform of this correction by (5 J (q), Equation (43.4) leads to
P
. e2
Ta (q).
(43.5)
-Let us now calculate T the tensor which determines vacuum polarization. Replacing 2 by p? q in
Us ? ?
(43.1), we write T in the form 2
where
1(13 F m 1(13 ? 4)? In
Sp Tis. I d4p
q)2 + no (p q)2 + m2
=KSp ? nt) y (? i4- ? nt) i ?
+ W. SP {(i in) Wail,. + SP f (43.0
KaT =
1.
dap
Rp qy 11121Rp .jT q)2 M211
pa d4p
RP + + m21[(P + m2]
PoPT d4p
[(p + 12_ q)2 + no] [(p q)2 + no]
(43.3)1
To calculate these integrals we set?x (1 +n) in the first of Equations (42.11), obtaining
?
+1
1 1 r dq
('t+
_-b)2
a ? 2
(43.6)
(43.7)
455
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It then follows that
where
Therefore
that
+I
Rp 4- -12-17)2 + ? ;?? q)2 ni2 1-7 I r
-1
KJ= - 42-
-1
qa (1 ?12) - tn2.
dip
Rp--2L qny 2
+ 1
padip
L r
PaP d4 P
[
1 ) -F L I
(p_ ql 2 3
L12
(43.7')
The integrals over 2 in these expressions are calculated in the Appendix (see Section 56) where it is shown
d4p
(j)1.(1 2 2 ? Tc2i [In L-F- A0 (N)J,
p +Li
pc, dip . 1
f RP ?+ L 0 j2 ? 7-war In L ---F-A0(N) 4_
-2-91
Pa P'T d4P
01 -2-8.41n L?L (L 4- q2112){110(N) -F ?
7
q1)2+Lr
q2712 [Ao(N) A2 (N ))? qpq,712[ In L A0 (N) -?5].?
(43.8)
(43.8')
(43.8")
All these integrals diverge for large ipl , and therefore the integral is taken over some large but finite region
(N)[ the constants Ao and A2 in (43.8) behave as in N and N2 for large NJ.
_
_
Equations (43.8), (43.81), and (43.8") must now be averaged overq (with -1 .5 n 5_ 1) and inserted into
(43.6), after which the tensor T must be renormalized. This renormalization is performed by subtracting the
-1.1v
first three terms in the expansion of Tii ,in a.power series in q. We note that this subtraction is in agreement
_
456
with the fact that according to (43.2) M2(0 contains e in the denominator: since M2(3) must be finite for q =' O.
the tensor Tgvshould have at least a third-order zero at q = 0.
This subtraction is also in agreement with the concept of charge renormalization. Indeed, subtraction
of a term proportional to 9.2 from T v according to (43.5), is the same as subtracting from 6 J p a term proportion
to the original
K. K0 Ku.,.
currentAsee Section 27).
Since
Kai. are multiplied in (43.6) by expressions which are, respectively,, quadratic in q, linear in
{
q, and independent of q, the term-ir2lAo (N) can be dropped in (43.8),the term-7r21 1'2 ncio Ao (N) + 01
' 4 .
can be dropped in (43.8'), and the term
1 ,, 5
iS? {? L -F (L-1- ?4 q21-)[440(N) -1 ? ?1 q212[Ao(N) -F (7)
2 4
-4- -2-1 A2 (N)}-1LC:Fig,q,12 [Ao(N )+-6-51?
can be dropped in (43.8").
After these subtractions we obtain the following expressions for K, K0, K07.:
1
K ? 1 In {L (1 ?12) q2 mai diri,
2 , 4
70i
in [71 (1 _12) q2 ma]di,
4 -1
1
? ,A21
K? =-- 4-gag, j 12 In 14 (1 ? 19)q2-km21thri+
1
-1---_j 8at 11.14- (1 ?12) q2 m21 In 14.(1 _12) q2 noldm
(43.7")
Since the integrand-of Ka is an odd function of n we obtain, as may have been expected [compare (26.101
Noting further that
"--*1
_.1
K,
1
? 71) q2+ m2] chi = 4 In m ? 4 (1 ? OCOt (i)
2In 1 4 4- 4 -y ? _
[_4. (1 _ .12) q2 .4..m2]d1 _3_ ? cot 2 0 (1 ?0cot
'457
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where. 0 is related to q2 by ? q2= 4tn2s1n20, we can write K and K
?ar
In the form
K ? ?2? ic2i121,
Ka, 7c2ig.qJ?2?i? 711* It2i(30t R m2 ?1? 711" (12) Ri? 92R2].
According to the above concepts we can drop terms independent of 0 (q= 0) in R1 and R2,since they vanish when
p V is renormalized. Making use of Equation (9.23) we can write T in the form
v
-= 4/121 (qpg, ap.,q2)( 411_,2-3;q2)
(1? Ocot
2. Renormalization of the Matrix Element.
We must now renormalize T :that is, we must subtract from it the first three terms of its power series
? 11 11
expansion in qa .
Since T
? pv contains the factor gp q,v ? 6uv qz ,which is quadratic in qa , renormalization reduces to
subtracting a constant from the factor 411-12-29..2... (1 ? 0 cot 0). Noting that 0 = arcian -- q2 1/2 ?
..
?3q2
it is easy to see that this constant is
-b--. Therefore, the final renormalized expression for T p v is 1)
1
4m2-2q2
T R 47C2i (q q ? 8 q2) [, (1 ?0 cot 0)?
11 1.100 3q2
(43.9)
e2
(p)
If we multiply T1.tv R by (2704 v (q), then according to (43.5) We obtain the correction ? (q) to the
11?
Fourier transform of the original "external" current JO (q) which is due to its interaction with lie zero-point
?11 ?
oscillations of the electron -positron field:
1 2 2 1
JO') (q) = e2 (q q ? q2) [4m 2 ? 3 2q (1 ? 0 cot 0)? a (q)
(2Tc) '
*
It is easily shown that this correction satisfies conservation of charge
R. Ferman. See the reference on p. 447.
458
q 1,4P) (q) = 0.
(43.10)
Noting that qv (q) = 0 and qza (q) = J (q), we 'rewrite J(0 (q) in the form
? P ?
j(p)(q)._ 02 r 4,0 2q2
(2g).2 L _42 (1 ?11 cot 0)
Expandtng T.tivRin a power series in q, and considering only the fourth-order terms, we obtain
= 421
? T57?.712 pg v?ap.1q2) q2.
In this case the correction to the Fourier transform of the current is
ea
j ( 0,1(4 (q)
s 60n2m2 7
Fre,m this we obtain the following expression for the current as a function of the space-time variables:.
e2
fie) (x).
607r.vi2
( 171. I I)
( 1'1.
(4:1. I :)
If we keep sixth-order terms in the expansion of R T in a power series in q then instead of (43.12) wc
?
obtain ?
e2 e2 / 1 \ 2
60ht2M2 j11?) (V) 680v2 6:2 ) 4?) (x)'
lnserpeg.E9uation (43.9) for Tv R into (43.2), we obtain the following expression for the matrix
elemeni
?
a" (q)
2 (2704 2 Al ? 44 1.cati
(43.13)
To determine the probabilit/ for the qrsresswepresented by the diagrams of Fig. 53, we clearly need to find
the total matrix 'Clement ..8.4 =-- M1 + M 2 . Using Equations (42.21'), (42.21), (43.13), and (43.9), we can
write the total matrix elenftrik in the form
.03.6?2.
SO) ?
f 1 M(8) ?F M ? (8) tt2 -z SO) (a) u19
2 ?
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(43.14)
459
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where
S(U) (q) = I ?Ti,,., R]
= 1)? pa
f rt 20 ? ?tan20 r1E tan E dEi
2
[77,(0? (q) Ti'av`q) q 42) X
0 q'4imp
X [4"1-2-3q22q2 (I ? 0 eot
9 )
(43.15).
13)
Later we shall need an expansion of s (q) in powers of qa up to'quadratic terms. Such an expansion for
Is given by
tirct/m 3\
[q a n2 ^ A A
In ? -8-- (q)? a (q) q 1.
Inserting (43.16) and (43.9') into (43.15), we obtain up to third-order terms
4 ^ q. t.n m / _^ -
SW (q) ?(41 (02 37.0q2 a, ) T 8 5 )Th 2m I q a (q) ( I }
(we have made use of the fact that q a (q) = 0).
(43.16)
(43.17)
44. Elective Electron Potential Energy. Magnetic Moment of the Electron.
1. Effective Electron Potential Eriagy:
In the preceding Section we have calcu:ated the matrix element Si(3)f ( , which gives the radiative
s)
onrrections to electron scattering in an external field Ae) (x) If we add to S. ?he matrix element
P
?. f= Ti_ Le a_ (9) 21 (1-.= p2?p1), which gives eiceiron scatterint, ih the first approximation, the matrix
element Su) aso obtained gives the interaction of the electron with the zero-point oscillations of the field
? f
UI) to tertns of order e4:
where
scv = 172e (q) + it a (q) u
"2 Tea (q)+ i [ey (q) 8 LI (q)]) up
(44.1)
(44.2)
Equation (44.1) shows that 6 U(q) maybe treated as the Fourier transform of the correction to the electron potential
energy due to interaction between the electron and the zero-point, field oscillations. We shall call this correction
?
the effective potential energy_ which gives the 'interaction of the electron with,the zero-point field oscillations.
Since S(a)(q), according to (43.15), can be written
where
we may write
s (q) F .,(q)
F v (11) = (2e:P Iry f(tan220 I) (In if_ 1)_Itano_
8
2 t t AO j__ i t's ^ 20.
tan 20 it " "?.1 -1-- 87n k9T, ? IS) -- y -L- (q q., ?
sin 20 P? (72 II v
0
[4m2 ? 2q2 1
(1 ?0 cot
?3q2
SU (q) 4.13 F ?(q) a ,,(q).
(44.3)
(44.4)
If we know its.Fouriertransform 6U (q) ,we can find the effective electron potential energy 6 U (x) as a
function of the space-time coordinates, namely
SU (x) ? (2nI)4 1
a., (q) f ,4(,;* (y) e?igY dty,
we can write (44.5) in the form
SU (x) = (3F, (q) eig 11) (y) d4 q d4 y
1
fi3F,,(x ? y).4(:) (y)'d4y,
Noting that
where
?
F Y(x)=-- I F v(q) eicix d49.
(44.5)
(44.6)
461
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We see that the effective electron potential energy can be written in the form of an integral operator
acting on the given external field A(e) (x).
-v -
For actual calculations, another form of 6 U (x) is moreconvedient. Noting that
we can rewrite (44.5) in the form
icpc 1
q
-v
?i a ' it a\. (e)
8U (X) 13 F., () a.,(q)eivel4q=713FAT Tr. )A (x)*
where E and H are the electric and magnetic fields, and yo and A are the scalar and vector potentials.
2. Radiative Corrections to the ElectroaLv_laEnetic Moment.
It follows from (44.9) that an electron in a constant magnetic field gains an additional energy
2 e
(44.7) Rea2 2nz Pall .27t 137 2nt
1 1 e Ff.
This equation can be used if Fv(q) can be written as a convergent power series in q; then F -:- ? Is a
(
differential operator acting on.Av(e) (x).
It is possible to represent Fv(q) as a power series in q if the fiele A(x) does not vary too rapidly in space
and time. We shall assume that the external field varies so slowly that we may neglect all terms higher than
the quadratic in the expansion of Fv(q) in a power series in q. According to (43.17) this expression for
F (q) can be written
-v -
? e3 { 4 ,2 (in t!../._ j_ i)}.11n
tq (4r.)2 8 5 )11-1- 2m `III ?
Thus the effective potential energy .5 U (x) is
where F (x)
v
en 4 I t 3 I
(. n (e(x) ?) 1
a V (X) = ? ln ? PT (x)) ,
(41.02 dm2 X 8 5 ' 0 un (44.8)
a (0
(x)
ax ?v
a
A(4 (x) is the field tensor.
axv -p -
Let us now introduce the matrices
It can easily be shown that
7 1112 at), "(2.'fa =01, 7 "fa oa?
0102 = jaw 020e kir GS al
or in other words, that the ai matrices satisfy the same commutation relations as the electron spin operators. With
the aid of these matrices we-can write 6 U(x) in the form
462
e3 e3 43
81./ (x) ? (47)2m (palf?ipecE) (47c)2 3m2 (I n 8 5 ) (0?-4CA), (44..9)
1 ?
?
due to Its interaction with the zero-point oscillations of the electro-
magnetic field. But the energy of an electron in a magnetic field isl)
Um golf,
where ? is the magnetic moment of the electron. We may therefore say that due to the electron's interactiun
with the zero-point oscillations of the electromagnetic field, it gains an additional magnetic moment equal
ez e
to Or 2)
bir 21:_n
1 e2 ett a
rt 41ctic? 2/c110'
(44.10)
eri
where go = - is the Bohr magneton, and et = ---1-- is the fine structure constant. Thus up to terms
2mc ", 401c 137
of order e4, the magnetic moment of the electron is
11=tio (1 +),
?a)
The "anomalous" magnetic moment A p of the electron has been experimentally observed.
We note that the interaction of the electron with the zero-point field oscillations taking account of fourth7
-N.
order effects, gives the following value for the magnetic moment of the electron:4)
a
=0(i +-2,97 CI)
zli it ?
Equation (4'4:9) for the effective potential energy contains the nonexistent phot9n "mass" X, which we
introduced in connection with the low-frequency divergence of the matrix element Si3). We shall now show
how to remove this "mass" when actually calculating radiative corrections to electron scattering in an external
field or radiative s-hifts of atomic levels.
? 45. Radiative Corrections to Scattering
1. Radiative Correcttis to Scattering of an Electron by an External Field
In Section 13 we found the cross section for electron scattering by an external field. In this section we
shall show how to obtain the radiative corrections to the scattering, which are due to the interaction of the
electron with the zero-point oscillations of the electromagnetic field and to the polarization of the electron-
Tthe nonrclativistic approximation 8 becomes the unit matrix.
2) This result is due to Schwinger[Phys . Rev. 75, 1912 (1949)].
34)) See the article-by Kusch and Foley in Atomic Electron Level Shifts, p. 52.
R. Karplus and N. Kroll, Phys. Rev. 77, 536(1950).
463
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positron vacuum1). We shall assume that the external field can be treated as a perturbation.
Since radiative corrections are a third-order effect, and since scattering takes place even in the first
order, in addition to radiative corrections we must take into account second-order scattering processes (scattering
in the second Born approximation).
Diagrams representing the processes of Interest are shown in Fig. 54 (diagram i 1 and 2 illustrate the first
and second Born approximation, and diagrams 3 and 4 illustrate radiative corrections and vacuum polarization).
The matrix elements corresponding to these diagrams, according to the rules of Section 24, are
S(I) ^ ?
r ? eu2a (q) up
47in( f (p.

Aio) ? r 1 k
1 (27Ti ip. 12 M (il) 161i m 11411\
. (P .? ley
2
Olt? 102 + Mg IP' 1.7) Ulf
1) K ^ /. A
(-27?t)4 "2 VIII (q)2? 7,, f sp ( P ? m m
M9 IC
, t (p in2 p2 d4p) Ui,
where u1 and Liz are the spinor amplitudes of the electron states with momenta pi and pz and q =
lq-Pt-P,
3
pg
14.1)
(45.1)
Fig. 54
We shall also be'interested in elastic2) scattering of electrons; In the Coulomb field of the nucleus. In this
case, as we know,
Ze
a (q) a q 27c 6 (s1? e2)--= 27tiP (ei? e2),
q 11
(45.2)
where Zeis the charge of the nucleus, 5 =p2-pi and ci and c 2 are the electron energies before and after scatterirq
Since ci = c 2 and q4 = 0, we may write
q2---= 4p1 sln2 ,
where 9. is the scattering angle, and I prIpil p21. As in Section 32, we shall make use of the quantity 4,,
which is related to q by Equation (32:51), namely
1) J. Schwinger, Phys. Rev. 75, 1912 (1949).
2) Actually, we shall be interested in scattering which is almost elastic; see below.
464
(it' ? ?4:112 II)
TI
If I ? v2
sin ?
where v is the ratio between the electron velocity and the velocity of light. We note that the quantity 0
introduced in (42.17') is given by 0 i 4, (in the present case it is 9., rather than 0 , which is real).
(1)
Inserting (45.2) into the first of Equations (45.1), we obtain the following expression for S,
Ze2 ?
(notio (e2 61)?
(1"
(45.3)
(3)
We have calErlated M(is) i
and M2 n general form in Sections 42, 43 [see (42.21), 43.2), (43.9)). In ordet
to find 1?113)and 14 for the case of -Coulomb scattering -A in Equations (42.21) and (43.2) should be replaced by
expression (45,2) and 6 should be replaced by i t. Noting also that -.a`(q) is proportional to 8 ,and that therefore
A A A A A A
a (q) q ? q a (q) = 2 = 4ti Ze q 8 (e2?
(it(
we finally arrive at the following expressions for MINand MP:
11 41( 24, coth ,E)) (1 -.1 ? In -I. ? -; oinhq)
A,.(8) . en .7:e ?-?
+2, c011121)
fu built u (Jul
q 0? - - u., )(a, --
(95.4)
Ze
.A1(:'11)
?1 2-n- u 011.11(1
? (i
coth(j))
71)
8 (ei ? G2). I
z)
The matrix element MI( can be determined by the second of equations (45.1). We shall not, however, go
into a detailed calculation of the integral in the expression for M12),but shall merely present the final result'.
IfZ-9 (x) is a screened Coulomb potential given by
_ _
(A.) Ze- e-Tir,
?ID:r '
where n is the screening constant, then for sufficiently small ii
M(1") d ( (I? J) -I- Pei (i+J) itif; (et ? e2),
1) R. Dalitz, Proc. Roy. Soc. A 206, 509 (1951).
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(45.5)
Where
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0
ic2i ,
1= 0 In (21 p 1 sln T\
P71
21p 13 sin2-2-
n3n2/
I ? J = 0 ( 1 0 1) + 41p13 cos 0 In (sIn2 ?6)' 11---). O.
2
2 ?
41p12cos2 ?T sin -2- 2
(45.5')
where A, B-, and C do not contain Dirac matrices and are given by1)
1
A c, -17 A = 2Tc1 Z e2 (Pcoth 49(1 ? cot
ti2 (I)) ?
94 27s q2
?) 1
? -I-(l ? 24)coth 2)(l-F in +-Titan?' (I) ?I? (1 ? X
1/4 coth 21' (
A
1 mil 24) In
C hi (I -- v2C2)dC (I) --IT
sin -I co: _0 (1 ? v2v) v._ coo ? t7rf
2 2
?--47.2a2ie (/+ J),
' ,Inon lima!, '
We see that I diverges logarithmically for a pure Coulomb field. As will be shown later, however, it is
possible to go to the limit n..?, 0 in the expression for the scattering cross section. ,
8 ?
- (.1.: 41.
The differential cross section for elastic scattering of an electron' can be written, according to the general ? 4:2:2irit (II) ? J)
formulas of Section 28, in the form
d3s
1 1 -1 -1? u2Si.4./ui 12 prdo,
21tJ 7
(45.6)
(i) (2) (3) (4
where is the sum of the matrix elements S L44., Mi, M M2 with the factor 6 (el? c2) omitted,
J = v is the electron current density, (the normalizing volume is assumed equal to unity( do is the element
P.2 cif) P s
of solid angle about the electron momentum after scattering and I
Pt ___
? (27)3 d = (2n)?
Is the density of the final states per unit energy interval and unit solid angle. Equation (45.6) includes summation
over electron spins in the initial and final states.
Before calculating dol. let us recall that MP contains the photon "mass" X which we introduced in order
to avoid the infrared catastrophe. In eliminating this quantity, we must bear in 'mind that the probability for
purely elastic electron scattering is strictly speaking zero, and that only scattering with some energy loss has
physical meaning. We shall therefore add to dos the differential cross section for inelastic electron scattering
with a very small energy loss A c (the magnitude of A e is assumed small compared with ithe electron energy:
A c 8 3-1 Z4Ry, 1 0,
0,
Z4 Ry
?
mc < iPctE > =
h
{-4a2
e
+40?Z4 ,Ry
it (i+1)(21-1-1) '
n3 1(21 +1) ' =1?'121
1 ? 1/9
(46.2V,
2
where n and 1 are the principal 'And orbital quantum numbers, a and ame2 . Finally, for
hydrogenwe obtain In == 7.6876. We can use these levels to determine the shifts of the hydrogen levels
c 0
whose frequencies are
(2sv.) = 1034 Mc
17 ,
Ev (2p./.) = + 8 .
(46.28,
The shift of the s-level is of order of magnitude a2E1, where E1 is the energy of the ground state.
It is well known that in the ordinary Dirac theory, the 2s and and 2p1 states have the same energy. We
/2 -
see that due to the interaction of the electron with the zero-point field oscillations the 2.s.,, state actually
/ 2
lies above the 2pj. state, and that the energy difference between these levels is about 1051 Mc.
-
Equations (46.25)- (46.27) give the radiative level shifts for the hydrogen atom up to terms of ordera2Ei.
It can be shown that the corrections to the hydrogen level shifts, up to terms of ordera3ED are given for ns state.
by')
.Z3a4 ( 11 1 5
(6E?) n3 1 + 1-fg In 2+ To) Ry.
(46.29
When n = ?this gives -7 Mc. The difference between the 2.si, / and 2pi , levels up to terms of orders is
/2 - 2
1057.19 Mc. This result is in good agreement with the experimental value of 1057.77.t 0.1 Mc2).
3. Radiative Level Shift in Muonium.
The role of vacuum polarization is small in the radiative shift of atomic electron levels. It is therefore
of interest to consider a case in which vacuum polarization is fundamental. Such a case is that of the level shil
in muonium3).
TFICKarplus, A. Klein, J. and J. Schwinger Phys. Rev. 86,288 (1952).
Salpeter, Phys. Rev. 89, 92 (1953).
3) A. Galanin and I. Pomeranchuk, Proc. Acad. Sci. USSR 86 , 251 (1952).
482
Let us consider a negative p-meson captured into an orbit about k. proton. The lifetime of this system
is sufficient for the observation of its spectrum, since it is determined by.the lifetime of a free p-meson
(2 ? 10-6 see).
The radius of the normal orbit of such an "atom" will be smaller than the Compton wavelength of the
electron, Since at distances of the order of the Compton wavelength of the electron the Coulomb field of the
proton is altefed by the polarization of the electron- positron vacuum, we may expect that the muonium spectrum
will exhibit a "fine structure" whose level splitting will be of the order of a= 1/137 of the energy of the ground
state, whereas the "ordinary" fine structure is an effect of order a2 (because of the, large orbit of the electron, the
part of the ordinary level shift which is due to vacuum polarization is an effect of order a3 and is about 3 per cent
of the total radiative shift).
Let 9(q) be the Fourier transform of the external potential. According to (43.11), this potential Induces
a polarization potential in the vacuum given by
a [-3q 4m2-2q2 1
cp tq) = ? ..(1 ? 0 ?t 0) ? ? (q),
rc 2
?
where m is the reciprocal of the Compton wavelength of the electron, q2 =q2 - co2, w is the frequency,q is
the wave vector of the external potential, and - qz= 4m2sin2 0. In the case of the Coulomb field
and
cp eqZ,
(v) (q) a eZ (4m2-2q3 [1 11.41,12+ q4
It 3q2 q
arcsinli
Transforming to configuration space and integrating over angles, we obtain
to
op kr/.
0)) f 7. C2 21:X { 4i/12 ? 2q2
4rer it2 3q2 1
ji4M2 + 612 arrilith
g 1
19114_11
2ni J 9 J ?
1711
r2rn J '
1 )5l"lqIrtj
1-?
ll
Accordingly, the shift of the n-th level is given by
03
8 fine2 72c: 1 4/03; 2q2 1/411/24. (72
arcsinli Igli } X
k I 91 2m . 9
x s'n Igir 106,(r)I2 dV dig'
171
(46.30)
(46.30')
(46.31)
483
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e2z2
Dividing6 En by the energy of the ground state Et (where Am. is the radius'of the p-meson orbit in
Oral'
muonium) and using ap Z as the unit of length, we obtain
where
co
8E0i4a f
Et f (r) ?t (2ex) dx
f (x) 417[(1 ? 2x2)(1 Yl ?X2
arcilohX)--1?
X
J (k) = f y sin ky lea(y)dy
0
and Ry) is the normalized radial function. The coefficient C IS
m aZ
where mil is the reduced mass of the p-meson, and m is the electron mass. For the first levels we have
16k k (1 ? 3k2 -1- 2k4) m k (1 ? k2)
I (k) = A 0(k) =-
(-.1- k2p P 2 (I +k2)4 9 421 V?1 (1 h2)4 ?
(46.32)
Inserting the expressions for J11/(2 ex) into (46.32) and performing an approximate integration, we obtain the level
shifts in electron volts for various Z (these are always negative):
1
6
20
Is
1.8
320
20 000
2s
0.2
47
3 250
2p
0.014
27
2 550
The splitting of then. = 2 level for Z = 1 is 0.19 ev, which is approximately 25 times greater than the
"ordinary" fine structure.
? 47. Nonlinear Effects in Electrodynamics
I. Scattering of Light by Light
The interaction of the electromagnetic field with the zero-point oscillations of the electron-positron
I field leads not only to vacuum polarization, but also to several nonlinear effects in electrodynamics. Primary
I
1
484
among such effects is scattering of light by light.
Photon-photon scattering can be illustrated by six diagrams, one of which is shown in Fig. 58. (The other
diagrams are obtained from Fig. 58 by permuting 1(2, 1(3,1(4) The scattering process represented by these diagrams
can be interpreted in the following way. A photon with momentum lit creates a virtual pair, the electron
(or positron) absorbs the photon with momentum k2 and emits one with momentum 1(3, and finally the virtual pair
annihilates into a photon with momentum 144. Another interpretation is also possible'): the pliotonslq and 1s2
create two virtual pairs ,and the electron of the first pair and positron of the second one annihildte,giving off
the photon h3, and the positron of the first pair and electron of the second one annihilate,giving off the photon
k4.
It is clear that from the classical point of view photon-photon scattering is
a typical nonlinear effect, since according to Maxwell's equation for the electro-
magnetic field in vacuum, waves with different frequencies propagate independen-
tly; in other words ,according to the classical electrodynamic description of the
vacuum, spontaneous frequency changes are impossible.
(4)
Let us write the matrix element S 1,4f corresponding to the diagram of
Fig. 58 for the general case in which electfornagnetic potentials A p (x1), A vfx0,
Ax fx3). An (x4) act at points xt, x2, x3, x4. According to the rules of Section
(a)
24, Sf can be written
CO Co uo CO
Fig. 58.
442).-f ? te4 //Xi f dX2 dX2 dX4 (Xi) Av(x2)24),(x8)A?(x4)X
-Co -CO Co
X SP Irp.42- SF(x2? xi) T., sF(x8 ? x2) Ix -SF(x.i 'fa X
X 22- SF(xt _x4)},
(47.1)
1
where = Tit = ?6- =- ?4 (here r is the number of normal products of the appropriate form in the general
4!
expansion of the fr-matrix into different N- products); the minus sign in (47.1) arises from the fact that the
diagram contains a closed electron loop.
Transforming to the momentum representation (see (24.7)), we obtain
42,
(fir f d'/e1 d4k d4k2 dik 4eiur,+k,+k?-Fko x
2
X a1 (k1) a.,(k2)ax (/es) a0 (k) (k1, k2, k, k4),
where
ks, k4).
d4p Sp {?m (I) ? IZ2) ? 1(11 ? Z2 ? Z4) ? \/
ie. + m21 (p k2)2 nt2 IX(p ? k j)2 In2
X I a (p k2 --1 ki 4)2 -I- '
IT-This interpretation corresponds to the diagram obtained by interchanging 1> m, the differential scattering cross section for small angles is 2)
ITT Euler, Ann. Physik 26,308 (1936).
2) A. A. Akhiezer, Physik Z. Sowjetunion 11,263 (1937); Karplus and Neuman, Phys. Rev. 80,380 (1950).
486
(47.5)
The integral cross section is
ct4 1 ,
do ? ill- II tio.
wz
, awl
08 (c))
'
where a is a numerical constant
-
We see that when to >> m, the scattering cross section does not depend on the electron mass and is
proportional to the square of the frequencyl).
Since the scattering cross section is proportional to co6 for co > En, it is a maximum roughly when co - m. The order of magnitude of the maximum cross section
4 ? 10-31 cm2.
The above results refer to the reference system in which the total momentum of the photon is zero. We
shall now show that the scattering cross section d a is invariant with respect to Lorentz transformations. We note
that in a time dt the photon with momentum k1 scatters do [1-cos 03 _nz sy photons with momentum_112,where
112 is the.density of the latter and 1-cos 0 is the relative velocity of the photons (0 is,the angle between
and k2). The total number of photons with momentum It2 scattered in a volume y during a time dl is
(4.7.6)
(47.7)
inversely
to2 for
is
do [1 ?cos 0 I dt nin2V,
where n1 is the density of photons with momentumk1. This number is clearly invariant, and since V dt is
invariant, so is do [1? cos 01 ni n2. In other words,
and
?-
doll ?cos 0[ not, ------- 2 &Tonle",
do ? don nmil2^
1Z1112 1 ? cos 0
2
(the index zero denotes the reference system in which the total photon momentum is zero).
But it is known that the particle density transforms according to
no= rI ? v2
n ? vj
...can be shown by dimensional analysis that the second of these statements follow from the first,
(47.8)
(47.8')
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yherel
ikl
is the photon current density ind v ? -1.'k2 is the relative velocity of the two reference
I
systems In which the total photon momentum is zero and klk+ik+2.k2i
Inserting (47.8') into (47.8), we obtainl
da dc70,
so that do is in fact an invariant.
In order to express a in terms of kp k2, k3, 1C4, we can use the relation
whence it follows that
Therefore for low frequencies
and for high freque-nqes
(0)1 00.02 (k 1 k 2)2 40)2,
2'0 wial2 11 ? cost)).
ca.-- (AO 11 ?cos .013
t. 2
c8
w1to211 ? cos OJ ?
(47.9)
(47.10)
(47.11)
2. Coherent Nuclear y - Ray Scattering
If in Fig. 58 not all four, but only two,of the dotted lines are photons and the other two represent the external
electrostatic field due to a nucleus, then the diagram will represent coherent photon scattering by nuclei with no
change in frequency.
It can be shown2) that the integral cross section for coherent photon scattering by nuclei is given by the
following formula. If to > m,
a 0.4 (aZ)4r26,
(47.13)
e2
where eZ is the nuclear change, and r0 =
-27 1
When Z = 92, Equation (47.13) gives a cross section of the order of 6 '10 cm2 .This is about 8000
of the cross section for pair production by photons in the field of nucleus. Coherent photon scattering by nuclei
becomes comparable to scattering by electrons at an energy of the order of 10" ev. It can, however, be observed
also for lower energies due to its characteristic angular distribution, which has a sharp maximum in the region of
small scattering angles.
For small angles the differential scattering cross section is of the form
where
do 7 \ 2
d9 lLaY /41 (2-)2 F2 CI) 0
)'
m
x2
F(x)--=-- 0.116 ?Inx+-2---1-Ri? ? ?
For Z = 92, co = 300 Mev, and 0 =OM?, this gives 3-10-21 cm2/ steradian, whereas the Compton scattering is
7.10-24 cm2/ steradian. When 0 = 0'
C12:
? ( 1 \ 2 Co )4
do 3-7-2-) (Za)41.
3. Lagrangian Including Nonlinear Effects.
The existence of nonlinear effects in electrodynamics clearly indicates that the linear Maxwell equation
for the field in vacuum should be replaced by a nonlinear equation. In particular, the Lagrangian density of the
classical electromagnetic field
Lo ? H2)
should be replaced by some more complicated function.
We shall assume that the field varies sufficiently slowly in space and time. This means that the change
of the field F (E, H) over a length of order ? and during a time interval of order ?1-2 should be small
mc mc
compared with the field itself, or
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?
fi A E.' ..e.1%." I ri
--.1grau a- it ithic21(1dFt I ?C IFI.
me
(47.14)
Fields satisfying these conditions will not in general create real pairs, since the energy of a photon associated
with such fields is much less than m1). We can therefore speak of a pure electromagnetic field without charged
particles.
If the fields satisty (47.14), the Lagrangian L will depend only on the field components, 0114 not on their
time or space derivatives. Since the Lagrangian should be invariant with respect to Lorentz transformation, the
fields should enter only in the form of the two combinations E2 ? H2 and (EH)2, which are, as is well known,
the only invariants on the field.
In the case of weak fields the Lagrangian can be expanded in powers of E2 ? H2, (EH)2 and we may
write
1
L=-- (E2 ? H2) + L' , , I el I/ & 0
?
=-- -72- (E2 + 112) M-572 Mi lk112) (3E2-1-H2)+7 (EH)2).
a (E2?H2)2 f3 (EH ) 2 --I- I (E2 ? + (E 2 ?112)(Ell)2 ...} (47.15) w
in the fields1)._
1 e4
? 21_ f(p rj?,
36070 7171 I --r- 7 (EH )2
If we know the Lagrangian density, it is simple CO find the energy density w of the field:
01
E ?L
OE ?
Inserting (47.15) and (47.18) into this expression we obtain
We shall show that if we know S(4iLf, we can find the coefficients c,8, y,6,.... of this expansion.
_ _
In order to establish the relation between SqLf and L, let us bear in mind that if the S matrix is expanded
in a power series of the interaction energy betweel: th-e- electron-positron and electromagnetic fields, then in the
first approximation we obtain
where
SO)r = V (t) di,
-03
V (I) ? (x) x
(47.16)
(47.16')
and L' =j A is the difference between the Lagrangians of the coupled and free fields [see (19.13)1 Now in
- P
considering the nonlinear interactions between the electromagnetic fields, we may assume that they are described
by an additional term L' to the Lagrangian Lo of the noninteracting electromagnetic fields. According to (47.16)
and (47.16') this additional term should be related to STf by the expression
???
ji 411 (x)d4x.
(47.17)
(a)
If we make use of this formula, then a knowledge of S __),f is sufficient to find the coefficients a,8 ,... We
shall not go through the calculations, but shall merely -pres-ent the expression for up to fourth order terms in
We are not considering the case of pair production by a large number of low-energy photons.
490
a
(47.18)
(47.19)
The first term in this expression is the classical energy density of the field, and the second term is a correction
due to the nonlinear effects.
The nonlinear effects can be accounted for with the aid of a field-dependent dielectric constant and a
field-dependent magnetic permeability of the vacuum. We shall determine these quantities by introducing the
vectors D and B according to:
Dr=aL B.= ?az'
aE, a H
From (47.15) and (47.18) we obtain
1 e4/4:c
(2 (E2? H2) Ei + 7 Hi (EH)) 7
I 457F m4
7c
Bi= Hi -I- I Tic,-c 84/4 {2 (E2? H 2) Hi? 7 Ei(EH)).
In microscopic electrodynamics the vectors D, B, E, H are related by expressions of the form
D= akEk, 8
(47.20)
(47.21)
where eik and p., are the dielectric constant and magnetic permeability tensors. Comparison with (47.21) shows
that for Tteak an slowly varying fields the dielectric Constant and magnetic permeability are given, respectively,
by
1)-7-See V. Weisskcpf, Kgl. Danske Videnskab. Seiskap 14,1 (1936).
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i
ork6U..47-57c
, 1
k ,77t
e41,1it
? (2 (E2 ? 1-P) 80.1- 71-41-4),
In I
elblre
Ini (2 (El ?PP) 8/k ? 7E/Ek).
A dielectric constant and a magnetic permeability which are field-dependent and different from unity lead
to nonlinear effects such as refraction of light in an electric field and scattering of light by light.
Equation (47.1E) gives the additional term of the Lagrangian for weak and sufficiently slowly varying fields
It can be shown that for a field which is not weak, but remains slowly varying, the additional term in the
Lagrangian is given by')
1- d -1-
(Elf) E*2 ?IP' 21r;411* c. C. = ? Ili c`ls
cos'Of E*3 ? 2i5I14' ? c. C.
m4 712 2
St:3 e
E4, __eE 1. ?
eH
(47.23)
And C..C. denotes the complex conjugate.
Analysis of this expression shows that for extremely strong fields the most important term in L' is of the
form
El el El eFf
247ez :72.2 , or I. in e I
In
247:2 m2 '
(47.24)
It folloWs from this that the ratio oil' to the classical Lagrangian 1.1 increases only logarithmically with the field
stt'ength for \*exy strong has., and the coefficient of the logarithm is proportional to the fine structure constant:
e1H1
CT
alt2
111171 OTC 41CilliPC2aity !the %N./mations of fc3orczodynantics is only a small correction even if the field is much
grO;LICT 4TS "tcritir;a111' 31 E is dhccn as he ficld ,at the "edge of the electron", namely
E
v.. agot..fibtrs 4170 rta. filIty.t1c ((a36.);; .a. Sdhwinger,, Phys. Rev. 82;664 (1951).
-- 2
(Ep ? is the classical electron radi L'
us), then ?-becomes
4irM C
LI 1 1 ,
-- in 137,
Go 3r: 137
L'
so that even in this case ==101 (Ih, ko? 1), hi >0,
= 1
y(jk, I?k0? 1), k1 t' -t" 2 . This postulate is in agreement with the principle
of relativity, since in a time I t' ? t" I a perturbation which propagates from the first point with the velocity
of light will travel a distance I t'? t" which is less than the distance I x' ? x" I between the points, and
_ _ _ _
Particles satisfying Bose-Einstein statistics are sometimes called bosons, and particles satisfying Fermi-Dirac
statistics are sometimes called fermions.
2) The commutator of the operators *% and *. is the expression
and the anticommutator is the expression
Nip iv] =
g'j} = +.11' '11 ebt
s I j I -?
We shall denote the commutators and anticommutators by the brackets ( ] and ( ]+, respectively.
3) This theorem is due to Pauli (see Footnote 2 on p. 521).
therefore, the second point is not effected by the perturbation from the first point. Therefore, measurements of
two events which are separated by a space-like interval cannot be influenced by each other. It follows from this
postulate that the c-numbers given by the quantum brackets must vanish if I x' ?x" I2- I t' -t" I 2> 0.1)
_
We-shall now show how to find the desired c-numbers. Let us start by constructing a c.'-numberscalar which
vanishes for I x' ? x" 2- I t' -t" I 2 > 0. Since the components of the quantized field *k satisfy the second-
order equation? _
(.327.,,,5?.__.n12,4 = 0,
ax;
(53.12)
as functions of space and time, the desired scalar must satisfy the same equation.
Since the scalar depends only on the combination x2? t2, it clearly satisfies an ordinary second-order dif-
ferential equation. It then follows that there exist only two linearly independent scalars depending on x2? t2
_
and satisfying (53.12). The plane wave expansion of these scalars is of the form
_ina, sin pnt
`) V
Pa
6,(1)e'-'' Cos pnt
V 4.1
Po ?
(53.13)
where V is the normalizing volume, and pn = Ip + m2. Going to the limit V -4- co and replacing summation
over 2 by integration over Vd3p / (27r )2-, we obtain
1I ? cos pot
Po
A (x)=
Po ?
, (x) (27,3 d8pesPa'
(2708 topeip, sin pnt
The scalar character of these functions is seen from the fact that d3E/E0 is an invariant: in fact,
?
d3p
?
? ? --
Po 2 .11 ? ? ? a (p2 d4p.
(53.13')
Since p (x) = 0 when t = 0, the invariant function p (x) vanishes also when x2? t2> 0. In other words,
A (x) vanishes_ for space-like intervals x2? t2 >0. On the other hand, the function ,6 (i) (x), does not satisfy
this condition. Therefore, the desired scalar function can only be A (x).
The construction of c-number vectors and tensors which vanish for x2? t2> 0 presents no difficulty, since
they can all be obtained by single or multiple differentiation of p (x) with respect to the coordinates )1a.
1) We note that for I x' ? x' ' I 2? I
, ' ? t''' I 2> 0, not only must the commutators vanish for fields correspond-
ing t_ _
to bosons, iiit so must the anticommutators for fields corresponding to fermions. If this were not true such
quantities as the charge and current densities of fermions would not commute at space-like separated points.
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Having gone through these preliminary remarks, let us-consider the quantum bracket (# (x'), (x")) ,
? ?
whcrej contains quadtities from only one of the previously described classes + 1, ? 1, + E. ? 6. According to
the multiplication table (53.1) and relations (53.2), the product, # belongs to class ?1 if j + k is a .' _
half-
integer, or for half-integral spin, and it belongs to class + 1 if j +1c is an integer, or for integral spin. Therefore,
the quantum bracket must transform under Lorentz transformations like an odd-rank tensor in the case of half-
integral spin, and like an even-rank tensor in the case of integral spin. In other words,
for half-integral spin, and
(41 114 (x")] ? Q2n +1 x")
[4,3 (x'), 4,; (x01 ? ......
(53.14)
(53.15)
for integral spin, where Q2n +1 and Q2n are differential operators containing, respectively, odd-and even-order
derivatives with respect to x , multiplied by certain constant coefficients. The explicit forms of 02n and
Q2n may be established if we know the wave equation of the field, the form of the energy-momentum tensor,
and that of the current vector. The operators Q,n and Q2n are then found from the requirement that the
energy and momentum of the field, as well as the charge, can be written in the form of sums over the energies,
momenta, and charges of the individual particles (see Sections 15 and 17, in which the forms of Q2n 4. I and
Q2n are established for the electromagnetic and electron-positron fields) .1)
Let us now consider the expression
= (x'), 4,.*; (x")1? Prj (x"), 4 (x')I ?,
which is symmetric with respect tothe events x' and x". Since A (x) is an even function of the space coordinates
_ _ _
and an odd function of time[see (53.13') ) it follows from the symmetry of X that X is a product of an even num-
ber
of space derivatives of A (x' ?x"), and an odd number of time derivatives. This is in agreement with Equa-
tion (53.14) for half-integral spin, but is in contradiction with (53.15) for integral spin, unless X vanishes.
_
We thus see that for integral spin
For particles with spin 0 and 1 the commutation rules (53.15) are of the form
Nej (x1), 7.7? (x")] ?= [(rv ac4r}.,) (rv
where x = #* A and A = II a ,,_ ll is the matrix which defines the invariant bilinear form(, #) =
?_k_cLI I. L'.
# = a E., cp = (the g. are basis vectors). [V. Karpman, j. Expt1.-Theoret. Phys. 21, 1337 (1951).]
ii ii 1
-- ?- ?
528'
[?.5 (x1), 4, (x")1? [4)i (x"), +; (x')1? = 0.
(53.16)
We have so far not considered the difference between Bose-Einstein and Fermi-Dirac statistics. If the
particles are described by Bose-Einstein statistics, the commutators of the fields are c-numbers and the minus
sign should be taken on the brackets in Equation (53.16); if, on the other hand, the particles are described by
Fermi-Dirac statistics, then the anticommutators are c-numbers and the positive sign should be taken in Equation
(53.16). If-we take the positive sign on the brackets in (53.16), we arrive at ,an algebraic contradiction, since
the left side of (53.16) is essentially positive for x' = x" . We have thus concluded that for integral spin, field
quantization according to Fermi-Dirac statistics is impossible. In other words, particles with integral spin must
always satisfy Bose-Einstein statistics.
As for-the quantization of half-integral spin fields, such fields may be formally quantized according to
Bose-Einstein statistics, but as we have seen above, the energy of the system will not be positive definite. It is
clear from physical considerations that the energy of noninteracting particles must be considered positive. There-
fore, half-integral spin fields must be quantized according to Fe,mi-Dirac statistics and we must require all (or
almost all) states with negative energy to be filled. Then particles in positive energy states will not undergo
transitions to occupied negative energy states.
If the state with all negative energies occupied is interpreted as the vacuum state of the half-integral spin
field, then an unoccupied negative energy state can be considered as a state of a particle whose charge has the
same absolute value as, but opposite sign from, a particle in a positive energy state (compare Section 17). We
thus see that half-integral spin particles must satisfy Fermi-Dirac statistics.
II. BOUND STATE EQUATIONS.
? 54. The Equation of Motion of an Electron in an External Field, With Radiative
Corrections Taken Into Account.
1. The:Method of Successive Approximations.
In?Section 21, perturbation theory was applied to the equations of quantum electrodynamics in the Heisen-
berg representation, and equations for the S matrix were derived. We shall now show that by applying perturba-
tion theory to the same equations we can obtain not only the elements of the S matrix, but also bound-state equa-
tions (the motion of a particle in a given field or the equation for interacting fields). The formal method used
here is characterized by its great simplicity; it can also be used for constructing the S matrix.1)
We shall start by considering the simple problem of electron motion in a given electromagnetic field.
Let us write the Dirac equation in the form
where
A. Galanin, I. Expd.-Theoret. Phys. 22, 448 (1952).
(54.1)
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L Ip ? tea + in
(54:2)
a
is the differential operator Ta = - y OX , a = y a (where ? is the given external electroinagnetid
11 11
-11 ?
potential), = y A (where A is the electromagnetic radiation potential operator), and * is the electron-
-
positron field operator.
We shall use the method of successive approximations, assuming e * to be a small term in Equation
(54.1). As for the external. potential I the assumption that it is small is more conveniently used later.
The electromagnetic field operators A satisfy the equation
11
0 AIL leiTh?
Equation (54.1) can be written in the form of an equivalent integral equation
111=1110-1- L-1 (lA) 4',
where *0 satisfies the homogeneous equation
(54.3)
(54.4)
L4)0= O. ?(54.5)
The inverse operator L -1 is given by an infinite series in powers of ft` (it is assumed that this series has
meaning), namely
The inverse operator
is defined as follows. Let
? '528
L-1 =(ir; in)-1 OP in)-1 (lea) (43 4- in)-1 ? (54.6)
in)- I = ? in) (FP -I- ,n)-1
f (x) =_- (210- 4 f elk Xf A44 k,
(54.7)
and then
(ip + n)-1f (x),...(270 -4 I eikx f
_F J k
(54.8)
where, In accordance with the explanation in Section 18, the integration over k0 avoids the pole at -14 + k2 +
_ _
. + m2= 0 by replacing m by m -16 (where 6 > 0), and 6 is then allowed to approach zero.
?
We shall solve the integral equation (54.4) by the method of successive approximations. In the first approxi-
mation, let us replace *0 on the right side by #, so that
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tP = 440+ L-1 (jek) 41o. (54.9)
Inserting (54.9) into the right side of (54.4), we obtain
11= 1110 -1-- L-1 (LA) 4)0+ L-1 (iek) L-1 (icA )16.
Continuing this iteration procedure without limit, we obtain the infinite series
= 11'0 L` 1 (icA)1140 L-1(icA) L-1 (icA)1110
L-1(iek) L-1 (kik) L-1 (ie A) t6+ ? ? ?
We may sum this series symbolically, writing
41= L1 (icA) ( L ? (jek)1Po'
Applying the operator L to both sides, and bearing in mind (54.5), we obtain
where
tIA = w ti)0
W = iek +(lA) (L ? 1 (ieA).
(54.10)
(54.12)
(54.13)
2. Electromagnetic Vacuum Expectation Value.
In order to apply the equation obtained to concrete physical problems, we must find the expectation value
of (54.12) in the electromagnetic vacuum state.
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Before doing this, we must make several remarks as to the meaning of the operator * in Equation (54.1).
The operator * is defined by the series
ai'',(x),
where an Is the annihilation operator for the electron in the state whose wave function is * (x) and a+ is the
n - n
electron-creation operator for the same state n. The functions 1,6 _ are a complete orthonormal set. If the calcu-
lation is performed in the interaction representation, it is convenient to choose the On as plane waves. Here,
however, in using the Heisenberg representation, it is most convenient to choose the il)n as the set of functions
s that are the exact solutions of the problem under consideration. For instance, if we are dealing with the problem
of the stationary states of the electron in the hydrogen atom, the On are the eigenfunctions for this problem, but
with all radiative corrections taken into account (we may assume from physical considerations that such a set
?n exists). If, however, we wish to consider other problems such as the problem of electron motion in a given
field accompanied by photon emission, then the On should be chosen as the exact solutions of this particular
problem. In this case the index n would depend also on the state of the photon emitted by the electron. The
action of the operator a+ on the occupation-number dependent wave function of the system would mean creation
of an electron in state n in this case; but due to the equations of motion, the creation of such an electron is auto-
_
matically associated with the creation of a photon. Thus, although the action of * on the wave function of the
system involves only changing the occupation numbers for the electron, due to the equations of motion this ac-
tion can be accompanied also by a change in the photon occupation numbers.
Let us now return to Equation (54.12), first treating the operator W only up to terms quadratic in%
ieAtP0 Ye A) L-1 (MA) 4.
(54.14)
We shall consider the problem of stationary electron states. in an external field (for instance, the hydrogen
atom). We must, therefore, take the expectation value of (54.14) in the electromagnetic vacuum state. The
left-side will ther not be altered. The expectation value of the first term on the right side vanishes, and the ex-
pectation value of the operator multiplying *0 in the second term shall be denoted by
We thus obtain the relation
U = < (icA)L-1(leA)>. (54.15)
1,4) = U/4), (54.16)
where we have replaced *0 by *, on the right side, which is valid up to terms of order eI. We can now already
tieat at a c-number. In calculating U1, we must bear in mind that the differential operator l contained in
L I acts on the function A on the right of L I. If we move the coordinate-dependent part A through L
from right to left, we must make use of the relations
and
p e-tkx=_-
(54.17)
L' eik?? e-iktc (L ? irt) -1. (54.18)
After the coordinate-dependent part of A, which is represented by a Fourier integral, is moved through
11
L I, we make use of the Fourier representation of the function D- (see Section 16). We shall here assume that
A satisfies Equation (54.3) without the right side, i.e., that it satisfies the free-field equation, since the form
11
of the function D- was established just for a free field. We shall show below how the result changes when the
right side of (54.3) is taken into account.
We will thus obtain the first part of the operator U1, which we shall denote by U'i
? e2i ? A
16,4 p. (IP tea+ nty d4k,
(54.19)
where the integration over k0 avoids the poles as has been indicated above.
Before calculating the second part of U1, which is due to the presence of the right side of Equation (54.3),
let us examine Equation (54.19) in more detail. We note that in (54.19) the operator p is a differential opera-
tor acting on 1, whereas in Section 42 Eli was assumed to be a number. 11
Let us show that (54.19) gives the same result as the previous one in the first approximation in it. To do
this let us expand U'l in a series, and keep only the first power of 1:
I
10 -I- U11
Ul U1 11
where
"/ e21
u 10 ----
1670 f I p.(iP ik m)_1 14-2 d4k
and
, e21 U11 r (IP ?, t/1)_ . A A
I (tea) (ip ? ik m)-1 k-2 d4k.
?16 ?xt j
A. Abrikosov and I. Khalatnikov, J. Expt1.-Theoret. Phys., 21, 429 (1951).
(54.20)
(54.21)
(54.22)
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Inserting Wu into (54.16), we may consider * a plane wave, since accounting for the difference between
and a plane wave would, mean considering the next power of the external field '2. Replacing Sin (54.22) by
a plane wave with wave vector q , we obtain C*2= -u2e1P2I; 01 =122e"-IP-1)--c ; /*ace )
Old Iii.106)= Cu-
1
e21 r
16n4 ./ ,(tp2? tk
nt)l (iecr0)(6;1? m) 1I.,k-2 u1y,
(54.23)
where put and 22v = + ctu, are now numbers, and are equal to the electron momenta in states 02 and 2.
In this form, (54.23) agrees with the previously obtained result [see (42.1) ].
If we insert U'in into (54.16), we may not assume that ip on the right side is a plane wave, but that
LO? = 0 for /= 0. If we write 00 in a Fourier integral, we can consider the action of U'20 on one of the
Fourier components 00. Then the differential operator p can be replaced by a number (but? + m2 # 0,
since *a is not a plane wave). 11
According to the method described in Sections 26 and 27, we must remove the infinities from U'10. The
renormalized operator U'lk (let us denote it by U'20R) will have a second-order zero at 117=-. - m. It follows
from this that U' 0ioR o is of the form
A A
U000 = (11;? in) f (it; #1)N= UP ni)f (P)(leal.'?0,
where f(') does not in general vanish for iP = - m. When we calculate the matrix element of U12000, we
multiply on the left by rp; in this case this latter function must be considered a plane wave, since we are con-
sidering only linear terms We then obtain a vanishing result. Thus, the operator U'20 gives no contribution
to (54,19);,up to terms linear in S.
If we want to calculate (54.19) with an accuracy up to T, then we must add to (54.20) the following
term of the power series expansion of (54.19) in S:
01 .^ . .^ ?-
ui2= 167t4
m)-1 (ie) ? m)-11pir2 d4k.
To this expression corresponds the diagram of Fig. 60.
In addition, when calculating U'u00, we must write *0 in the form
cleo .1/00+ (ip's mri (ie) 0.6,
(54.24)
(o)
On),
where 00 is a plane wave. When calculating the matrix element (V/02 I trii in addition to (54.23) we
obtain the terms
.1.
(+T I U1I1(iPs nt)-1(ka)1410(2) (Teg I(iea-)(64, m)-1 1)1111V:I ,
(Fig. 61).
----Finally, there remains a finite term also from Ulio R, We obtain
_
(4'o I Ui-ohliP0) = (ticl OP -I- #02 f (f.,)1+0)? 61-(?)1(jea)f (1;)(ieS)1 05.
(54.25)
(54.26)
Equation (54.26) corresponds to the diagram of Fig. 62. No simple renormalization diagrams such as that shown
in Fig. 63 occur in this method of calculation.
- Thus, by expanding (54.19) in a power series in 71, we obtain the same results as we obtained previously,
from the S-matrix theory. However, Equation (54.16) contains more, at least in principle. This lies in the fact
that this equation gives the motion of an electron in the external field with the first-order radiative corrections
taken into account, whereas from S-matrix theory we were able to obtain only the matrix elements of the effect-
ive potential U between states described by plane waves'.
Fig. 60
Fig. 61.
Fig. 62 Fig. 63
In the present problem there is, to a great extent, little difference between the two methods, but if we
consider bound states of two interacting particles (rather than the motion of a single particle in a given field),
then itis easier to use the present method to obtain the appropriate equations. Let us now return to the general
equation (54.12). Let us take the expectation values of both sides of (54.12) in the electromagnetic vacuum
state.--
I`P=+0.
(5i.27)
-In order to obtain an equation for * from (54.27), we must express 00 in terms of 0. This can be done by
. -
writing
W>%
and applying the same iteration process as that which led to (54.12), considering 4 a known function, and *0
unknown.
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As a result we obtain
where
= U+,
u w - < > < w < w >
series in T, the terms of odd power pc vanish due to the symmetry in k-space and the terms of even power in k
_ _ _
contain the trace of an odd number of y matrices, so that they also vanish. The linear term in 'ci gives the
(54.28)
polarization current considered in Section 43. Indeed, if a is treated as a plane wave with propagation vector
--
ti (that is a = ao ?), then up to terms linear in at) we obtain
(54.29)
If we write W in the form of a power series in eA, and use only up to terms of some power of e2;
then we see chat the transition from < W > to U corresponds to eliminating from our considerations those many
diagrams which can be divided into parts each of which is contained in a U operator of lower order in e2 and
which are connected by only one electron line. This electron line is given by the operator (ip - ie a + us)
which means that it may contain several vertices with the external field. We shall call such diagrams degener-
ate. They enter into the S tnatrix just as do other diagrams.
It is simple to see how degenerate diagrams appear in the S matrix.
Let us represent U in the form of a series U = U1 + U2 where the index denotes the power of e2 in
the operator. The function 0 can then be written in the series
4,0 -F- L-lUit1)0 4- L-Il12.16+
When we calculate the matrix element CpIU I'P) with the aid of (54.30), we obtain
u +) (T0 u +0) + (tT0 u2 +0) + cji0i u L? u +0) +
+(''0I u ' U1; .6) 4- ? ? ?
The operators 1.111, -1111, U2L IU1, etc., are those that give the degenerate diagrams.
3. Vacuum Polarization. Electron-Positron Vacuum Expectation Value.
(54.30)
We shall now take account of the right side of Equation (54.3). In considering the single-electron problem
(the two-electron problem will be considered in Section 55), we must find the 'expectation value of (54.3) in the
electron-positron vacuum state. In the first approximation we may replace 0 in (54.3) by 0o, so that L00 = 0,
for 1 0. Then the right side of (54.3) can be written in terms of the function S. This function, differs,
however, from that introduced in Section 18 in that it refers to an electron moving in the given external field
A A
By making use of the relation between S-F and the inverse operator (ip - ie + m) I, we can obtain the follow-
ing
expression for (54.3) in the first approximation in e2;
0 Av.= 1:To Sp Ii( i? ii;? nt)-11c14k,
(54.31)
where the differential operator acts only on
Let us expand - iet+ m) -1 in a power series inci. The term independent of Ce vanishes. In
fact, if we write (m - -1 [we note that p must be considered zero if 'c't is omitted from (54.31)] as a power
SP [1' - ik ntri (tea%) (?ik nt)ildske-fqx
0A# = -161-Tc,
Let us denote the polarization current by j . The solution of the equation
-1)
0
can be written in a-space in the form
A0. =A?? (I-2 .1 pp. (0 A=O).
(54.32)
(54.33)
We have inserted the first part of (54.33), that is A? into (54.15) [making use of the fact that A? satisfies
F -11II
the free-field equation and writing k 2 instead of D- in (54.19)). We must insert the second part of (54.33)
into the first term on the right side of (54.14). As a result we obtain the second part of the operator U of (54.29),
which is of first order in e2 and it;
e21 A
.k. _1 . A
161c4 q2 T,,. Sp (Iq ? m) (teao) C? 1 + m)-1
(54.34)
which is in agreement with the result of Section 42 [see (42.2)].
In the further expansion of (54.31) in powers of it*, all terms of even order in 'it vanish by Furry's theorem
1 -
[ if the current were written in the symmetric form j = ? (1,by - y 0'), where 0' is the charge con-
-11 2 11
jugate function, even powers of a would cancel automatically, which would prove Furry's theorem; see also Sec-
tion 24].
Successive approximations in powers of e2 are obtained by considering that 0 in (54.3) satisfies the exact
equation (54.1). By generalizing (54.31) we obtain (for simplicity we set =c 0)
o Ap. 16en4 f Sp (Tp.(j? ? ie A --I- m)-1] d4k.
,
(54.35)
The operator whose trace is being taken is defined as a power series in A. If we take only the linear term in 2,
then Equation (64.35) gives a linear equation for l (it is necessary, of course, to renormalize according to Sec-
tions 26 and 27).
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In the momentum representation this equation is
where according to (43.11)
P2 [1 ?
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e2
F (132)] AP.= Cl/
F op) 4m2 ? 2p2( 0 )
3p2 y tan 0
(54.36)
Since the equation for A is now different (instead of 22A = 0, we have obtained (54.36)1 the function
F II 11
D- is also different. This means that in obtaining the expectation value, as for instance, in (54.15), we must '
_
replace k -2 by
Hg. 64
ea
Expanding [1 ? ?470 F (k2)j-1 into a series in powers of e2, we obtain a sequence of terms corres-
ponding to diagrams whose photon lines contain one, two, etc., successive simple closed loops.
The remaining terms of the power series in 2 give nonlinear expressions in (54.35). From these nonlinear
terms one may again obtain linear equations for A of the form (54.36), but then the function F (E2) will be
11
more complicated. This is done by combining all but one of the factors A (according to Furry's theorem there
is an odd number of them) into pairs, and replacing each pair by its electromagnetic vacuum expectation value.
This gives rise to diagrams with mom complicated closed loops (Fig. 64).
If this is not dOne, then (54.35) gives a nonlinear equation for A, which describes processes such as
11
scattering of light by light. When A in (54.35) may be considered a c-number and a slowly varying function
11
of the space-time coordinates, the operator on the right side of (54.35) can be calculated without an expansion
in powers of Al? The result can be formulated as a change in the Lagrangian for the free electromagnetic field
which is due to the interaction with the electron-positron vacuum, and is the same as that given in Section 47.
? 55. The Equation of Motion of Two Interacting Electrons, with Radiative Cor-
rections Taken Into Account.
1. The Equation of Motion of an Electron in a Real Photon Field.
We can now go on to the two-electron problem (we shall speak of two electrons, but the interaction may
also be between two positrons or an electron and a positron). The interaction between electrons is accomplished
by photon exchange. These photons are virtual, but they can be considered real photons from the point of view
of a single electron. Therefore, we must first formulate an equation of motion, analogous to (54.28), in the
field of an arbitrary number of real photons.
1) J. Schwinger, Phys. Rev. 82, 664 (1951).
538
4
?
where
Equation (54.1) for a = 0 and in the presence of real photons, can be written
L,4= (55.1)
1p m ? leA,.
(55.2)
i??
and Ar is a real photon field. Therefore, the desired equation is the same as (54.28) with the external field re-
placed by the real photon field. When taking the vacuum expectation value of some operator, for instance, one
of the form of (54.15), we must interchange the photon emission operators in 'it and L. Clearly Lr does not
commute with an emission (or absorption) operator for a photon whose momentum is Ile same as that of one of
the real photons in 21... The contribution of these momenta in (54.15) approaches zero as the normalizing volume
A
increases. We may, th- erefore, assume that Lr commutes with A.
Let us now write out some of the terms of the expansion of U in powers of Ar . For simplicity, we shall not
write the brackets < >, but shall use 2' to denote the 2 operators for which the exp- ectation value is taken. We
shall also drop the index r on, so that 2 without a dot will denote a real photon field. Further, we shall
-
A
in-
dudel the term ieA of L in U. As a result we obtain
where
?r
(IP ? in) (Yr =
U = (MA) (4; -F ni) (ie A') (ie A') (fp -1- In)-2 (MA.') 01 in)-1 (ie A') X
X (1^P-1-171)-' (iek.)-1- ? ? ? -1- ie A (ieA) ql.)-1 X
X (ie (i I; --I- m)-1 (iA') . . . ieA nt)-1 (iA) -1- . . .
(55.3)
(55.4)
Here the first bracket contains no real photons, the second one contains one photon, the third one contains
two, etc. The degenerate diagrams, for instance, the diagram corresponding to the term (leA ) (ip + m) -1
(ie2') (1+ (ie2), are dropped in agreement with what has been said above.
2. The Equation of Motion of Two Interacting Electrons.
The wave function of a system of two electrons depends on the coordinates of two events and on two spinor
indices which we shall suppress, this wave function can be written in the form
(1) (x1, x2) == ciAi (xl) (x2),
(55.5)
where each function t/). (x1) or 11) (x2) satisfies Equation (55.3) with an operator U of the form of (55.4). Let us
- -
apply the operator (i'1 + in) X (42 + in) to both sides of (55.5), where 91 acts on )..ti and the first spinor index
of zb (xi, x2), and% acts on x2 and the second spinor index.
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537
7W.1
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Since 0. (x1) and 0 (x2) satisfy Equation (55.3), we obtain*
? ?
@Pi in) (IP2 in) 4, (x1, x2) -='? U1U241 (x1, x2).
(55.6)
where U1 is obtained from U of (55.4) by replacing A by Al (x1), and U2 is obtained by replacing A by A2 (x2).
Equation (55.6) can be solved by the method of successive approximations in the same way as we solved
the equation for a single particle. We first obtain
where
(IP m) (02 -I-- in) 'le (x1, x2) = W12+0 (XI/ x2),
W12= U1U2 U1U2 [(iji1 -1- in) (tio^ 2-1- in) ? U1U21-1U1U2,
and then the final equation
where
-I- in) (62 -I- in) L'e (xp x2) =-- U12+ (x1, x2),
U12= ? [(1p, ? in) (4;2+ in) + ]-1 .
(55.7)
(55.8)
(55.9)
(55.10)
Here is the electromagnetic vacuum expectation value if there are no real photons in the problem, for in-
stance, when considering the problem of stationary states of the system of two particles.
The transition from (55.8) to (55.10) involves the elimination of degenerate diagrams. These are now de-
fined as those diagrams which can be divided into parts connected only by two electron lines belonging to differ-
ent electrons. Examples of degenerate diagrams are shown in Fig. 65.
The final result for the operator U.12 given by
Equations (55.10), (55.8); and (55.4) can be written in
a much more simple way, namely
J12 < W1W2>
(55.11)
where W1 and W2 are obtained from (54.13) by replacing
A by Al (x1) and dA2 (x2), respectively, and assuming that
in taking the expectation value all degenerate diagrams
are dropped.
Let us investigate Equation (55.9) in more detail for the simple case when
U2 = (ie; (xi)) (1eA.2(x2)).
Fig. 65
(55.12)
i) Strictly speaking, Equation (55.6) describes the motion of particles iihich cannot annihilate. If the possibility
of annihilation were taken into account, we would obtain a nonhomogeneous equation for * (x1, x2).
538
a
We shall write the equation
(ipAi m) + m) 4, (xi, x2) = ? e2A;(zi) A; (X2) 4,(xi, x2)
(55.13)
in the momenturn-iepresentation. To do this, let us multiply both sides of (55.13) by ...e1(2-11-cl+ E221-2), and
integrate over x1 and x2. The left side becomes
(1; 1+ in) (iiia+ in) (ps. Pt,),
A
where 0 (p1, 22) is the Fourier transform of 0 (x1, x2), and Ei and 22 are now matrices, rather than differential
operators. The right side becomes
? e2 f A (x) cr2) ei(Pixi+P'x')kle (xi, x2) d4x1 d4x2.
i
Since
A; (.V1) g; (X2) = ? dro k-27(vlyv2)e-{?Xt-Xild4k,
we finally obtain 1)
OP 1+ in) (4;2+ in) 4, (pi, p2) = k-21N2?+(p1? k, p2+ k)d4k,
(55.14)
where the indices (1) and (2) on y denote action on the first and second spinnr indices Of b, respectively.
In principle these equations can be used to determine the energy levels of positronium. We shall not,
however, go into this problem here.
The method 'of this section is easily generalized to more than two electrons.
III. MATHEMATICAL APPENDIX
? 56. Calculation of Certain Integrals.
1. The Calculation of Integrals over a Finite Invariant Region.
? 6.1:1
We shall show how to calculate the integrals occurring in Sections 42 and 43.. 14;ie -shali?l" perform the integra-
tion over a finite invariant region (N) which is characterized by the number N, and shall consider the limit
E. Salpeter and H. Bethe, Phys. Rev. 84, 1232 (1951); A. Galanin, J. Expt1.-Theoret. Phys. 23, 448 (1952).
539
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.?111
N ce to correspond to unbounded four-dimensional k-space. Such a finite invariant region may be defined,
for instance, by the inequalities
1k21