PURE AND APPLIED PHYSICS A SERIES OF MONOGRAPHS AND TEXTBOOKS CONSULTING EDITOR H. MASSEY University College, London, England Volume 1. L FRANKLIN, Electron Impact Phenomena and the Properties of Gaseous Ions. KOPFERMANN, Nuclear Moments.
English Version Pre- pared from the Second German Edition by E. THIRRING, Principles of Quantum Electrodynamics. Translated from the German by J. W i t h Additions and Corrections by WALTER E.
RACAH, Irreducible Tensorial Sets J. MULLINEUX, Mathematics in Science and Technology E. WIGNER, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. With Additions and Corrections by E.
Translated from the German by 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 www.com Originally Published in 1955 under the title EINFÜHRUNG in die QUANTENELEKTRODYNAMIK by Franz Deuticke, Vienna. Copyright©, 1958, by ACADEMIC PRESS INC. Ill FIFTH AVENUE N E W YORK 3, N.
ACADEMIC PRESS INC., PUBLISHERS 40 PALL MALL, LONDON, S. 1 ALL RIGHTS RESERVED NO PART OF THIS B O O K M A Y B E REPRODUCED IN A N Y FORM, B Y PHOTOSTAT, MICROFILM, OR A N Y OTHER MEANS, W I T H O U T WRITTEN PERMISSION FROM T H E PUBLISHERS. Library of Congress Catalog Card Number: 58-10414 P R I N T E D I N THE U N I T E D STATES OF AMERICA www.com 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 www.com 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.com 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. 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.
We are indebted to Professors H., and to Professor F. Scarf for reading part of the manuscript and for valuable criticism.com NOTATION Hubert Space Operators in Hubert space will be denoted by capital Roman letters, e.; 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) = Y2 (1 + e(a)) δ function: δ(α) = 0 for a ^ 0, j dab{a) = 1 θ{α) = f dßö(ß), δ ( - α ) = δ(α) J— 00 j 5 ( « ) = -δ(«)/α da S(/(«)) =Σ|/'(αοΓΊ«(«-«ο) «0 with f(ao) = 0.com Xll NOTATION Operators Transposed operator: 0 (c' | 0T \ c") = (c" \0 \ c') T 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 = ekyk (N.2) 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, yaßta · However, spin indices will usually be suppressed, so that we will write 7*7V fo r T«/3 Ύβδφδ, or ΨΨ — Ψαψι t (N.3) Tr M = Maa 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 - 1 0 0 x1 = x g = 0 0 - 1 0 x = y 3 0 0 0 - 1 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 www.com NOTATION xiii letters, contravariant indices are raised, and covariant indices are lowered ik i ik ~. 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) =(k>idke~ikX- &Λ) Differential and integral symbols Partial differentiation d/dxlf(x) is sometimes denoted by dj and sometimes by an index following a comma fti. An x will stand for differentiation with respect to proper time. The symbol/ θμ g will be useful and stands for/#,M — f^g.
If not otherwise 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 www.com 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 2. Conversely, if / da if is 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 dkf - f dak dif = 0.6) The proof proceeds by showing that didkf — dkdif = 0 implies that the left side of Eq.6) is independent of the surface σ. Hence one may evaluate the integrals on a spacelike surface defined by a con- stant time t.