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Lecture Notes in Physics Edited by H. Araki, Kyoto, J. Ehlers, M~Jnchen,K. Hepp, Z~Jrich R. Kippenhahn, MLinchen, H. A. Weidenm~ller, Heidelberg and J. Zittartz, K61n
220 Walter Dittrich Martin Reuter
Effective Lagrangians in Quantum Electrodynamics
Springer-Verlag Berlin Heidelberg New York Tokyo
Authors Walter Dittrich Martin Reuter Institut fQr Theoretische Physik der Universit~t Tebingen D-7400 Tf3bingen, ER.G.
ISBN 3-540-15182-6 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-38?-15182-6 Springer-Verlag New York Heidelberg Berlin Tokyo Library of Congress Cataloging in Publication Data. Dittrich, Walter. Effective Lagrangians in quantum electrodynamics. (Lecture notes in physics; 220) Bibliography: p. 1. Quantum electrodynamics. 2. Lagrangian functions. I. Reuter, Martin, 1958-. II. Title. II1. Series. QC68O.D53 1985 537.6 85-2527 ISBN 0-387-15182-6 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Veflag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2153/3140-543210
PREFACE
With
these notes we would
subject
of effective
Although
this topic
vides
Lagrangians
of QED which
Moreover,
studying vacuum problems
considerations
To make
In contrast
our computations
general
concepts
these sections
in typing
T~bingen,
for similar
and other more
compli-
electrodynamics
as possible,
In particular,
has the
many of them
this is true
The reader who is mainly
could omit the rather
technical
for the
interested
derivations
in a first reading.
We wish to thank Christel skill
are
can still be done analytically.
as transparent
3rd and 4th sections.
tech-
texts.
in QED, where matters
to the latter,
in great detail.
calculational
preparation
(QED).
it also pro-
found in standard
can be a helpful
that many calculations
are presented
important
in quantum chromodynamics
cated theories. advantage
of several
to the
electrodynamics
in its own right,
are usually not
fairly well understood,
an introduction
for quantum
is interesting
us with an example
niques
2nd,
like to provide
Kienle
the various
September
1984
for her endless
versions
patience
and
of the manuscript.
W. Dittrich M. Reuter
of
in
TABLE OF CONTENTS
(I)
Introduction
I
(2)
The Electron Propagator in a Constant External Magnetic Field
28
(3)
The Mass Operator in a Constant External Magnetic Field
37
(4)
The Polarization Tensor in a Constant External Magnetic Field
56
(5)
One-Loop Effective Lagrangian
73
(6)
The Zeta Function
98
(7)
Two-Loop Effective Lagrangian
121
(8)
Ren