Quantum Electrodynamics

557 Pages·1975·13.181 MB·English

by  Iwo Bia၂ynicki-Birula, Zofia Biłynicka-Birula and D. ter Haar (Auth.)

QUANTUM
ELECTRODYNAMICS
BY
IWO BIALYNICKI-BIRULA
University of Warsaw and University of Pittsburgh
and
ZOFIA BIALYNICKA-BIRULA
Institute of Physics, Polish Academy of Sciences
Translated from the Polish by
EUGENE LEPA
PERGAMON PRESS
OXFORD · N EW YORK · TORONTO
SYDNEY
Ρ WN — P O L I SH S C I E N T I F IC P U B L I S H E RS
W A R S Z A WA
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Pergamon Press Ltd. and PWN —Polish Scientific Publishers
Copyright© by PWN —Polish Scientific Publishers
Warszawa 1975
All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted,
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First English edition 1975
This is a translation from the Polish edition
Elektrodynamika kwantowa, published by Panstwowe Wydawnictwo Naukowe
Library of Congress Cataloging in Publication Data
Bialynicki-Birula, Iwo.
Quantum electrodynamics.
(International series of monographs in natural philosophy, v. 70).
Translation of Elektrodynamika kwantowa. 1974
1. Quantum electrodynamics. I. Bialynicka-Birula, Zofia, joint author. II. Title.
QC680.B513. 1974 537,6 74-4473
ISBN 0-08-017188-5
Printed in Poland
PREFACE
SHOULD we today, in the fashion of the Greek sages, wish to construct
the world out of elements, we would name two of them without hesita­
tion—electricity and magnetism. For electromagnetic interactions are
a source of forces in a vast number of physical systems, and accordingly
no doubt deserve to be so singled out. The quantum theory of these
interactions, called quantum electrodynamics, underlies the foundations
of most modern areas of physics. The whole of optics and electrody­
namics, atomic and molecular physics, solid-state physics, the physics
of fluids, gases, and plasmas are all, actually speaking, special applica­
tions of quantum electrodynamics to selected physical systems. In the
physics of the atomic nucleus and elementary particles, strong and weak
interactions do play a primary role, it is true, but electromagnetic
interactions here, too, occupy a distinguished place as the sole and
universal "measuring instrument". Man and all instruments "see and
feel" by means of electricity and magnetism. Thus far, there has been
no other method of accelerating, sorting, and recording particles, apart
from methods based on electromagnetic interactions.
Quantum electrodynamics concerns itself first and foremost with
the mutual interaction of electrons and photons and their interaction
with given electromagnetic fields and given currents. Since nuclear
(strong) interactions do not come within its compass, atomic nuclei
in quantum electrodynamics are treated as indivisible carriers of electric
charge and magnetic moment. Such replacement of atomic nuclei by
given sources of electromagnetic fields is a good approximation in
atomic physics.
The description of the electromagnetic interactions of muons in
an approximation in which we disregard weak interactions does not
differ from the quantum electrodynamics of electrons.
Xii PREFACE
Quantum electrodynamics also concerns itself with the description
of the electromagnetic properties of other elementary particles. This
is hindered, however, by the lack of a complete theory of the other
interactions.
The mathematical structure of quantum electrodynamics, as of
every fundamental theory, is highly complex. It would be unreasonable,
therefore, to apply the vast apparatus of complete quantum electro­
dynamics to every physical system. In most cases, this would make it
altogether impossible to obtain results. For this reason, most areas
of theoretical physics which arise out of quantum electrodynamics
in practice develop on the basis of their own, usually phenomenolo­
gical, models of the physical systems considered. It is our belief that
each such model can be justified on the basis of quantum electrody­
namics, but showing these connections would go beyond the bounds
of our book. In it we shall confine ourselves merely to giving the formu­
lation of quantum electrodynamics in its most general and its most
abstract form: relativistic quantum field theory. For pedagogical rea­
sons, we precede this general formulation with several introductory
chapters, in which step by step we shall introduce the concepts and
methods of this difficult theory. Such a gradual, inductive introduction
of quantum electrodynamics is also advisable for another reason.
Tn contrast to classical mechanics, quantum mechanics, thermodynam­
ics, or classical electrodynamics, in quantum electrodynamics we have
a theory that is not complete. Owing to the enormous mathematical
difficulties, it is not yet known whether the set of postulates adopted
is not inconsistent and whether it determines the theory uniquely. All
we know is an approximate scheme for carrying out calculations based
on perturbation theory. The results of these calculations are in amazing
agreement with experimental results but the problem of the convergence
of the series obtained has not been solved as yet. Moreover, the results
of investigations into the mathematical structure of the theory indicate
that perturbation series are divergent asymptotic series.
Quantum electrodynamics thus constitutes a programme rather
than a closed theory. As we try to show in the early chapters of this
book, this programme rests on two pillars: on the theory of the quantum
Maxwellian field interacting with given (external) classical sources
PREFACE Xlll
of radiation and on the relativistic quantum mechanics of electrons
interacting with a given (external) classical electromagnetic field. Both
of these theories are closed and relatively straightforward, mathemat­
ically speaking. It is not until the complete relativistic theory of the
mutual interactions of electrons and photons, which is a synthesis of
these two theories, is being formulated that insurmountable difficulties
arise. As a consequence of these difficulties, numerous monographs
and papers have presented many diverse formulations of quantum
electrodynamics. These formulations in the final account lead to the
same results, but they are by no means equivalent in the mathematical
respect, lii our book we give only one formulation of quantum electro­
dynamics; others may be found in the books of Akhiezer and Bereste-
tskii, Bjorken and Drell, Bogolyubov and Shirkov, Jauch and Rohrlich,
Heitler, and others. None of the existing formulations of quantum
electrodynamics is satisfactory from the mathematical point of view.
In this theory we must on many occasions renounce mathematical
rigour and resort to heuristic considerations, which are frequently justi­
fied solely by the agreement of their results with experiment.
The very formulation of the theory thus is a controversial matter,
one to which we devote much more space in our book than we do to
concrete applications of the theory. For it is our conviction that the
computational scheme of renormalized perturbation theory is quite
simple and upon mastering this scheme, the reader will be able to follow
the calculations given in the monographic and original literature.
For this reason, we have confined applications to a minimum and little
space is devoted to them.
Our book is addressed to readers familiar with quantum mechanics
and classical electrodynamics at the level of university courses. The
necessary material from the domain of quantum mechanics is given,
for instance, in Chapters 1-20 of the textbook by E. Merzbacher,
and that from classical electrodynamics in Chapters 1-12 of the textbook
by J. D. Jackson.
The scheme on p. xiv illustrates the logical interconnections between
the various chapters of the book.
Chapter 1 has been given over to the fundamental principles of
quantum theory formulated in a general, abstract fashion. Chapters 2-5
XIV PREFACE
are easier to grasp than the others and may be read independently of
the rest of the book since they constitute a self-contained vahσle made
up of two independent parts. The part comprising Chapters 3 and 4
contains primarily the theory of the electromagnetic field interacting
with given sources of radiation. This theory is applicable to the study
of systems in which the influence of the radiation on the sources may
be neglected. An example of such a system is black-body radiation.
If the reader wishes to confine himself to only this part of the book,
he may omit the final three sections (Sections 13-15).
The part composed of Chapters 2 and 5 concerns the quantum me­
chanics of particles. Chapter 5 expounds the relativistic theory of mutu­
ally non-interacting electrons, moving in a given electromagnetic field.
It may be applied, for instance, when one considers the motion of
electrons in the field of a nucleus.
Small print has been used in the text for those parts of the material
which give the justification for assertions made or contain ancillary
material going beyond the logical mainstream of the exposition. The
references are given at the end of the book, the listing being made with
the authors in alphabetical order. In view of the large number of no­
tions and symbols used, we felt it necessary to give a detailed subject
index and an index of the principal symbols. The subject index indi­
cates the page on which the given concept is defined or explained, but
the pages where further reference to it is made are not listed. It frequently
PREFACE XV
happens that in various parts of the text, the same term denotes dif­
ferent, though related, concepts. In such cases, several page numbers
are given. Most terms Hsted in the index are italicized in the text. In the
index of symbols bold-face print is used for the number of the formula
which defines the given symbol explicitly or implicitly. Furthermore,
we sometimes give the numbers of other formulae in which the given
quantity plays a cardinal role.
Footnotes referring to the history of quantum electrodynamics
are marked by sharps (#) whereas footnotes explaining the text are
indicated by t, *, §, etc.
In the course of our work on the first and second editions of this
book, our colleagues and co-workers made many suggestions and
criticisms. We are indebted to them for pointing out errors and mistakes
and for proposing several improvements.
This text has come into being in the creative scientific atmosphere
in the Warsaw School of Theoretical Physics, created by the late Pro­
fessor Leopold Infeld, to whose memory we dedicate this book.
INTRODUCTION
ÍN THE introduction we shall discuss the system of units and the
notation used throughout the text.
Units
We shall use a system of units based on Planck's constant h, the
velocity of light c, and an arbitrarily chosen unit of length / (most
frequently it will be 1 cm, though sometimes it will be 1 fermi, i.e.
10"^^ cm). Table 1 lists the dimensions of the physical quantities which
occur most frequently, in hcl units.
The strengths of electromagnetic fields and charges are defined
so that the factor 4π appears in the formula expressing Coulomb's
law, but does not figure in Maxwell's equations, just as in the choice
of the Heaviside-Lorentz electromagnetic units. To simplify the notation,
TABLE 1
Physical quantity ñcl
Length I
Time
Velocity c
Mass ñc-Ί-'
Momentum
Angular momentum h
Energy
Action h
Electric charge
Strength of electromagnetic field
Wave function 3
/-2
2 QUANTUM ELECTRODYNAMICS
we shall omit all h's and c's in the formulae. In order to read the value
of a physical quantity given in this abbreviated notation, first comple­
ment its dimensions by writing in the missing powers of h and c in
TABLE 2
Velocity of light c = 2.99792 χ 10^*^ cm-sec"^
Planck's constant ñ = 6.5822χ 10-^^ MeV-sec = L0546x lO-^^ erg.sec
Fine-structure constant α = β^Ι4πη€, 1/α = 137.036
Electron mass m = ñjÁ^ c = 0.51100 MeV/c^
Compton wavelength of electron Ac — filme = 3.8616 χ 10~^^ cm
Classical radius of electron ro = e^jAnmc^ = 2.818 χ 10"'^ cm
accordance with Table 1. For instance, an energy of (10~^^ cm)~^
should be read as hc(10-^^ cm)"^ = 200 MeV.
Table 2 lists the relations between ñcl units and other units used in
quantum electrodynamics.
Vectors, Tensors, and Spinors
The components of various geometrical objects will be labelled with
various types of indices. Lower-case Roman letters from the middle
of the alphabet—i,J,k,l,m,n—label the components of vectors and
tensors in three-dimensional space.
Lower-case Greek letters label the components of vectors and tensors
in four-dimensional space-time.
Lower-case Roman letters from the beginning of the alphabet—
a, b, c, d, e,/—label the components of bispinors.
Upper-case Roman letters label the components of spinors.
Vectors are denoted by bold-face Roman letters in three-dimensional
space, and by ordinary Roman letters in four-dimensional space-time.
In Hilbert space, upper-case Greek letters are used for vectors.
The scalar product of three- and four-dimensional vectors is indicated
by a dot. For example:
k- X = /:ΙΧΙ+/Γ2^2+^3·^3,
k ' X = RQ XQ — k^Xi — ^2 — ^3 ^3 ·
The vector product is denoted by the symbol x. The scalar product
of the vectors Ψ and Φ in Hilbert space is written as the symbol {Ψ\Φ).
INTRODUCTION 3
The summation convention holds for all components of geometrical
objects. This means that the summation is to be performed over repeat­
ed indices. For instance:
k^x'' = koX^-hki^x^ +k2X^ + k2X^,
Completely antisymmetric tensors, with coordinates equal to 0 and
±1, in three-dimensional space and four-dimensional space-time are
denoted by, respectively, eij^, ε^'"^, and e^,^i>- We adopt the following
convention:
^123 = 1,
ε _ J — — £oi23 ·
V V
The tensors ε^'"^^ and ε^^χ^ can be used to define β'" and the duals
of arbitrary antisymmetric tensors and h''\
Symmetrization and antisymmetrization are denoted (together with
a factor of 2) by brackets:
^(μν) = ^μν-\-^νμ^
^[μν\ = ^μν~ ^νμ'
Fourier Transforms
Fourier transforms of a function will be denoted by the same
symbol as the function but with a tilde over it. We shall adopt the
following convention concerning the factors 2π and the signs in the
exponent:
d'p
/(P) = Srf'xe-'P-/(x),