Introduction To Symmetry And Supersymmetry In Quantum Field Theory

388 Pages·1990·125.044 MB·English

by  J. Lopuszanski

An Introduction to
SYMMETRY and SUPERSYMMETRY
/-it in
QUANTUM FIELD THEORY
Jan topuszanski
World Scientific
An Introduction to
SYMMETRY and SUPERSYMMETRY
in
QUANTUM FIELD THEORY
This page is intentionally left blank
An Introduction to
SYMMETRY and SUPERSYMMETRY
in
QUANTUM FIELD THEORY
Jan topuszanski
Institute of Theoretical Physics
University of Wroclaw
World Scientific
Singapore • New Jersey • LLoonn don • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Fairer Road, Singapore 9128
USA office: 687 Hartwell Street, Teaneck, NJ 07666
UK office: 73 Lynton Mead, Totteridge, London N20 8DH
AN INTRODUCTION TO SYMMETRY AND SUPERSYMMETRY
IN QUANTUM FIELD THEORY
Copyright © 1991 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced
in any form or by any means, electronic or mechanical, including photo­
copying, recording or any information storage and retrieval system now
known or to be invented, without written permission from the Publisher.
ISBN 9971-50-160-0
9971-50-161-9 (pbk)
Printed in Singapore by JBW Printers and Binders Pte. Ltd.
ACKNOWLEDGEMENTS
These notes originated in a course of lectures given by me in the Spring
Semester of 1984 at the University of Goettingen on symmetry and supersym-
metry in quantum field theory. The very warm hospitality extended to me
there by Dr. Helmut Reeh and his constant friendly interest in my work are
cordially acknowledged. I am grateful to Dr. Jerzy Lukierski for critical read­
ing of my manuscript and to Dr. Detlev Buchholz for assisting me in putting
Section 6.2.9 into a proper shape, and to both of them for valuable advice
as well as many remarks and comments. I wish to thank Drs. Wlodzimierz
Garczynski, Andrzej Hulanicki, Witold Karwowski, Jan Mozrzymas, Zbigniew
Oziewwicz, Maciej Przanowski, Helmut Reeh and Erhard Seiler as well as my
students in Wroclaw, Wojciech Hann and Robert Olkiewicz for discussions and
advice I made use of while writing this book. And last but not least I wish
to record my gratitude to Dr. K. K. Phua, Editor-in-Chief of World Scientific,
who encouraged me to write this book.
The research reported here was supported in part through funds provided
by the Polish Department Program C.P.B.P. 01.03.
J. Lopuszanski
V
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TABLE OF CONTENTS
Acknowledgements v
1. Introduction 1
1.1. Introductory remarks about symmetries in physics 1
1.2. Introductory remarks about quantum free theory,
in particular axiomatic quantum field theory 8
References and Comments to Chapter 1 13
2. Example of a Classical and Quantum Scalar Free Field Theory 15
2.1. Classical scalar free field 15
2.2. Quantum scalar free field 18
References and Comments to Chapter 2 27
3. Scene and Subject of the Drama. Axioms 1 and 2 28
3.1. Scene of the drama 28
3.1.1. Axiom 1 28
3.1.2. Complete normed linear space and its application
in quantum field theory 28
3.1.3. Operators in the Hilbert space and their
application in quantum field theory 33
3.2. Subject of the drama 38
3.2.1. Axiom 2 38
3.2.2. The space of test functions S 39
n
3.2.3. The space of tempered distributions S„ 40
vii
Vlll Table of Contents
3.2.4. Some explanatory comments 43
References and Comments to Chapter 3 44
4. Principle of Relativity. Causality. Axioms 3, 4 and 5 45
4.1. Basic geometrical transformations of the field 45
4.1.1. Axiom 3 45
4.1.2. The group SL(2,C) 46
4.1.3. The symplectic tensor and the contravariant
components of the spinor 54
4.1.4. The Lorentz group 56
4.1.5. The representation spaces of the SL(2,C) group 57
4.1.6. The proper spinorial group ^o 64
4.1.7. Infinite dimensional representations of the^o group.
Stone's Theorem. Common domain of definition
of the generators of ^o 66
4.1.8. Projective representations of P\ 71
4.1.9. The transformation properties of the fields 73
4.2. Spectral condition in Minkowski momentum space 75
4.2.1. Axiom 4 75
4.2.2. Some explanatory remarks 78
4.2.3. Properties of the vacuum. The set Dn 79
4.3. Causality 81
4.3.1. Axiom 5 81
4.3.2. Some explanatory comments 82
4.3.3. The Theorem on Spin and Statistics 83
4.3.4. Example of a free neutral spinor field 85
References and Comments to Chapter 4 86
5. Irreducibility of the Field Algebra and the Scattering Theory.
Axiom 6. Axiom 0 89
5.1. Irreducibility of the field algebra 89
5.1.1. Weaker version of Axiom 6 89
5.1.2. Associative, involutory, normed algebra 89
5.1.3. Irreducibility and cyclicity of a vector
with respect to a field algebra 90
5.2. Scattering theory 91
5.2.1. General remarks concerning the scattering theory 91
5.2.2. Outline of the mathematical formalism of
the scattering theory 93
Table of Content* IX
5.2.3. Stronger version of Axiom 6 101
5.2.4. Scattering states 101
5.2.5. The weaker version as a consequence of the
stronger version of Axiom 6 102
5.3. The 5-matrix 102
5.4. Superselection rule. Axiom 0 104
References and Comments to Chapter 5 106
6. Preliminaries about Physical Symmetries 108
6.1. General theory of physical symmetries 108
6.1.1. Wightman functional 108
6.1.2. Definition of a symmetry group 110
6.1.3. Antilinear operations and PCT symmetry 113
6.1.4. Relativistic geometrical symmetries 115
6.1.5. Borchers' classes. The global symmetries of the
5-matrix 115
6.1.6. Poincare symmetry is a global symmetry of the
5-matrix. Conserved quantities 124
6.1.7. Internal symmetry is a global symmetry of the
5-matrix 125
6.2. Currents and charges 126
6.2.1. Classical theory of charges and currents.
Noether's Theorem. Some remarks concerning the
quantal case 126
6.2.2. Quasilocal and (strictly) local states 129
6.2.3. Some theorems concerning the sesquilinear forms of
the field operators 131
6.2.4. Definition of the translationally covariant,
locally conserved quantum current 133
6.2.5. Inverse of the quantal Noether Theorem 135
6.2.6. Outline of the proofs of Theorems 6.2.3-6.2.13 142
6.2.7. Some characteristic features of the charges 154
6.2.8. Spontaneously broken symmetry.
Goldstone's particles 157
6.2.9. Identically conserved currents, Gauss' Law
and gauge charges 161
6.2.10. Translationally noncovariant currents.
Currents associated with the Lorentz group 167