Principles of Quantum Electrodynamics

228 Pages·1958·5.456 MB·English

by  WALTER E. THIRRING (Eds.)

PURE AND APPLIED PHYSICS
A SERIES OF MONOGRAPHS AND TEXTBOOKS
CONSULTING EDITOR
H. S. W. MASSEY
University College, London, England
Volume 1. F. H. FIELD and J. L FRANKLIN, Electron Impact Phenomena
and the Properties of Gaseous Ions. 1957
Volume 2. H. KOPFERMANN, Nuclear Moments. English Version Pre-
pared from the Second German Edition by E. E.
SCHNEIDER. 1958
Volume 3. WALTER E. THIRRING, Principles of Quantum Electrodynamics.
Translated from the German by J. BERNSTEIN.
With Additions and Corrections by WALTER E.
THIRRING. 1958
IN PREPARATION
U. FANO and G. RACAH, Irreducible Tensorial Sets
J. IRVING and N. MULLINEUX, Mathematics in Science and Technology
E. P. WIGNER, Group Theory and its Application to the Quantum
Mechanics of Atomic Spectra. With Additions and Corrections
by E. P. WIGNER. Translated from the German by J. J. GRIFFIN
FAY AJZENBERG-SELOVE (ed.). Nuclear Spectroscopy
ACADEMIC PRESS INC., NEW YORK AND LONDON
PRINCIPLES OF QUANTUM
ELECTRODYNAMICS
WALTER E. THIRRING
Universität Bern, Switzerland
TRANSLATED FROM THE GERMAN BY
J. BERNSTEIN
The Institute for Advanced Study
Princeton, New Jersey
WITH CORRECTIONS AND ADDITIONS BY
WALTERE. THIRRING
1958
ACADEMIC PRESS INC., PUBLISHERS
NEW YORK · LONDON
Originally Published in 1955 under the title
EINFÜHRUNG in die QUANTENELEKTRODYNAMIK
by Franz Deuticke, Vienna.
Copyright©, 1958,
by
ACADEMIC PRESS INC.
Ill FIFTH AVENUE
NEW YORK 3, N. Y.
ACADEMIC PRESS INC.
(London) LTD., PUBLISHERS
40 PALL MALL, LONDON, S. W. 1
ALL RIGHTS RESERVED
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS,
WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.
Library of Congress Catalog Card Number: 58-10414
PRINTED IN THE UNITED STATES OF AMERICA
FOREWORD TO THE GERMAN EDITION
Elementary particles, their properties and interrelationships, have
in recent years come to the forefront of fundamental research in
physics. The only theory which one has at one's disposal, as yet, to
describe the behavior of such systems is the quantum theory of
fields. Although this theory represents one of the most fundamental
that we possess—it not only unifies elementary quantum mechanics,
but it is also the first theory that brings together quantum theory
and relativity—it is still not an area in which most physicists feel at
home. The reason may be that in field theory one needs to draw on a
considerable amount of higher mathematics, hiding much of the de-
velopment behind a dense smoke screen of formalism. Hence, one
may get the impression that field theory is a dry mathematical scheme
in which work may be done when one has mastered the necessary
rules, but which does not require any special physical insights. In
this book we shall concentrate on one of the best understood parts of
quantum field theory, quantum electrodynamics. We shall endeavor to
emphasize the physical basis of the theory and to avoid purely mathe-
matical details. For this reason, the book should not be taken as a
handbook of field theory, but rather as a compendium of the most
characteristic and interesting results which have been obtained up
to now.
The advances which have been made most recently in quantum
electrodynamics depend essentially on the new formal structure which
the theory has been given. One may now condense its starting points
into a few fundamental postulates from which everything else may be
deduced. As we shall learn, correspondingly significant simplifications
have been made in the computation of specific processes. On these
aesthetic developments we shall put special emphasis in the body of
the book. Not only will the mathematics be made as simple as pos-
sible, but it is hoped that the connections with the current literature
will be made easier.
As for mathematical background, some analysis and linear algebra
are necessary for the text. Less familiar tools, such as Dirac y matrices
and invariant Green's functions, are discussed in the two appendices.
The notation is explained in a separate section. As far as physics is
v
VI FOREWORD TO THE GERMAN EDITOR
concerned, the reader will need a knowledge of special relativity and
quantum mechanics. The notation and concepts of Dirac's book,
"Principles of Quantum Mechanics,,, (Oxford, 1947) will be used.
Further applications of the theory can be found in W. Heitler,
"Quantum Theory of Radiation" (Oxford, 1954). The mathematical
aspects of the theory are treated in a more elementary fashion by G.
Wentzel, "Quantum Theory of Fields" (Interscience, 1949). Other
references are given only for details which are not treated in the text.
Therefore neither are the references complete nor is attention paid to
priority.
To illuminate the physical background the book starts with a
chapter in which the orders of magnitude of the various effects, to be
calculated later in detail, are discussed. They will be estimated by
heuristic arguments which may seem somewhat arbitrary at the be-
ginning. Such arguments become more convincing when they are
backed by calculations, and the reader should return to this section
after having worked through the rest of the book. For practicing the
calculational techniques, problems with solutions are given for each
part of the book.
FOREWORD TO THE ENGLISH EDITION
In the six years that have passed since the German edition of this
book was written, no essential new discoveries have been made on the
subject of the book. However, considerable progress has been made
in the understanding of the physics underlying the post-war develop-
ments. Since the main purpose of the original book was the discussion
of the physical principles involved, it was necessary to do considerable
rewriting and expansion of the text to justify the publication of an
English edition. In particular, the section on renormalization theory
had to be brought up to date and discussed more elaborately.
In the meantime, two other books in this field have appeared,
namely, J. M. Jauch and F. Rohrlich, "The Theory of Photons and
Electrons/' (Addison-Wesley, 1955), and H. Umezawa, "Quantum
Field Theory," (Interscience Publishers, 1956). In these books many
mathematical and formal details are elaborated. We did not, there-
fore, endeavor to achieve more completeness in these respects since
the references above can be consulted for this purpose. However, we
have tried to give a reasonably detailed discussion of physical con-
cepts which are not treated adequately in the literature.
The English edition has been prepared in collaboration with Dr.
J. Bernstein. We are indebted to Professors H. Feshbach and F. Low,
to W. H. Nichols, S.J., and to Professor F. Scarf for reading part of
the manuscript and for valuable criticism.
vii
NOTATION
Hubert Space
Operators in Hubert space will be denoted by capital Roman
letters, e.g., 0, A, Tik , Q, etc.; ordinary numbers, such as eigenvalues,
coordinates, and indices, will be denoted by small letters like o', x,
k, and a. Vectors in Hilbert space will be written in Dirac fashion
as | ) ; conjugate vectors, as ( | . Generally, the eigenvector associated
with an eigenvalue o' will be denoted by | o'). The symbol for the
product of two vectors will be ( | ) ; and for an operator and a vector,
0 | ). Both operations may be combined into ( | 0 | ). We shall also
use the following notations and definitions.
Ordinary numbers
Complex conjugate: a*
Real part: Re a
Imaginary part: Im a
Signum function: e(a) = 1 for a > 0
= -lfora < 0
Step function: 0(a) = 1 for a > 0
= 0 for a < 0
θ(α) = Y (1 + e(a))
2
δ function: δ(α) = 0 for a ^ 0, j dab{a) = 1
θ{α) = f dßö(ß), δ(-α) = δ(α)
J— 00
j5(«) = -δ(«)/α
da
S(/(«)) =Σ|/'(αοΓΊ«(«-«ο)
«0
with
f(ao) = 0. (N.l)
xi
Xll NOTATION
Operators
Transposed operator: 0T (c' | 0T \ c") = (c" \0 \ c')
Hermitian conjugate operators: 0*; (c' | 0* | c") — (c" \ O \ c')*
Inverse operators: 0_1, 0~λ0 = 1
Commutators: [A, £]_ or [A, B] = AB - BA
Anticommutators: [A, B] or {A, B} = AB + BA
+
Defining equations
Hermitian operators: 0^ = 0
Unitary operators: Of = 0~l
Symmetric operators: 0T = 0
Operators representing interacting fields will be distinguished by
bold face type: A, ψ
Spin space
The operators acting in spin space are the Dirac Y'S and expressions
containing them. For y invariants, that is, scalar products constructed
from a four vector and a y vector, we introduce the notation
p = P*7fc e = eyk (N.2)
k
and so forth. Vectors comprising the spin space (spinors) are denoted
by ψ or u. Matrix indices are always lower case Greek letters, for
example, y ßta · However, spin indices will usually be suppressed,
a
so that we will write 7*7V for T«/3 Ύβδφδ, or
ΨΨ — Ψαψι
t
(N.3)
Tr M = M
aa
The rest of the notation is the same as in Hubert space.
Ordinary space
We use real world-coordinates with the metric
Ί 0 0 0 ' x° = t
0 - 10 0 x1 = x
g =
0 0 - 10 x = y
0 0 0 -1 3
X = Z.
The space part of a four vector is designated by setting a bar
under the letter, as x. Tensor indices are usually lower case Roman
NOTATION xiii
letters, contravariant indices are raised, and covariant indices are
lowered
ik i ik ~.i
9 Pk = V Ç 9ki = δι .
The scalar product atf = aob° — ab is sometimes denoted by round
brackets (ab) and sometimes by writing ab.
If a vector is a sum of vectors b = c + d then we write (a, c + d)
for (ab).
Momentum space
For the Fourier transform we write
/(*) = j£yf*kë~axfW> /(fc) = jdxe<kxf(x)
where dk is the four-dimensional volume element and k any four
vector; sometimes we will use p instead of k.
The Fourier representation of the four-dimensional δ-function δ(χ)
f dx'f(x')6(x - x) = f(x)
is given by
b{x) dke ikX &Λ)
=(k>i ~ -
Differential and integral symbols
Partial differentiation d/dxlf(x) is sometimes denoted by dj and
sometimes by an index following a comma f .
ti
i2 ik^ o &
\J = g*% àk = ^ - Δ.
2
An x will stand for differentiation with respect to proper time. The
symbol/ θ g will be useful and stands for/#, — f^g. If not otherwise
μ M
indicated all integrations will run from — oo to °o.
We write the four-dimensional volume element as dx = dxdxdxdx.
The surface element of a three-dimensional surface, a covariant vector
directed normal to the surface, we denote by
άσ{ = (dxldx2dxz, dxdxdx, dx°dx1dxz, dx°dx1dx2).
Any surface for which dai is timelike for all points will be called
spacelike. The four-dimensional generalization of Gauss' theorem
XIV NOTATION
can be given by / dxdif = I da if, where σ is the surface of the four-
dimensional volume under consideration. For a divergence-free vector
/'(/*,* = 0) which vanishes sufficiently strongly at infinity (spatially)
the value of the integral / daif taken over a spacelike surface does
not depend upon the particular choice of surface. This follows di-
rectly from the relation
[ darf - [ daif = \ dV dif (N.5)
where V is the volume between σι and σ . Conversely, if / da if is
2
independent of σ, then dif = 0. In this case we are entitled to call
daif a scalar, since observers in different Lorentz frames would
/
obtain the same values by integrating over a surface defined by
t = constant in their frames. In the same way, tensors of higher rank
can only be obtained by integrating divergence-free expressions.
If a function / vanishes sufficiently strongly on infinitely remote
parts of spacelike surfaces then one has the lemma
[ dai d f - f da dif = 0. (N.6)
k k
The proof proceeds by showing that didf — ddif = 0 implies that
k k
the left side of Eq. (N.6) is independent of the surface σ. Hence one
may evaluate the integrals on a spacelike surface defined by a con-
stant time t. In this case only dao differs from zero and the terms with
di,2,3 reduce by the generalized Gauss theorem to vanishing surface
integrals. To complete the proof we note the obvious fact that Eq.
(N.6) holds for i = * = 0.
As a final matter of notation we remark that if a surface σ or, more
generally, any region AV, contains a point x, then this will be denoted
by x C σ or x C V.
Frequently recurring symbols
A (x) electric vector potential
D(x) invariant function
e elementary charge
el small displacement