Lectures on Quantum Field Theory

790 Pages·2008·3.63 MB·English

by  Ashok Das

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TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Ashok Das
University of Rochester, USA
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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Published by
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British Library Cataloguing-in-Publication Data
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LECTURES ON QUANTUM FIELD THEORY
Copyright © 2008 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 photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
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ISBN-13 978-981-283-285-6
ISBN-10 981-283-285-8
ISBN-13 978-981-283-286-3 (pbk)
ISBN-10 981-283-286-6 (pbk)
Printed in Singapore.
Lakshmi - Lec on Quan Field Theory.pmd 1 12/15/2008, 3:51 PM
To
My friends and collaborators
Josif and Susumu
and
to
Ever caring and charming
Kiron and Momo
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Preface
Over the past several years I have taught a two-semester graduate
courseonquantumfieldtheoryattheUniversityofRochester. Inthis
coursetheideasofquantumfieldtheoryaredevelopedinatraditional
manner through canonical quantization. This book consists of my
lectures in this course. At Rochester, we also teach a separate course
on quantum field theory based on the path integral approach and
my lectures in that course have already been published by World
Scientific in
A. Das, Field Theory: A Path Integral Approach (Second Edition),
World Scientific, Singapore (2006).
The material in the present book should be thought of as comple-
mentary to this earlier book. In fact, in the present lectures, there is
no attempt to develop the path integral methods, rather we use the
results from path integrals with a brief discussion when needed.
The topics covered in the presentbook contain exactly the mate-
rialdiscussedinthetwo-semestercourseexceptforChapter10(Dirac
quantization)andChapter11(Discretesymmetries)whichhavebeen
addedforcompletenessandarenormallydiscussedinanothercourse.
Quantumfieldtheoryisavastsubjectandonlyselectedtopics,which
I personally feel every graduate student in the subject should know,
have been covered in these lectures. Needless to say, there are many
other important topics which have not been discussed because of
time constraints in the course (and space constraints in the book).
However, all the material covered in this book has been presented in
an informal (classroom like) setting with detailed derivations which
should be helpful to students.
A book of this size is bound to have many possible sources of
error. However, since my lectures have already been used by various
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people in different universities, I have been fortunate to have their
feedback which I have incorporated into the book. In addition, sev-
eral other people have read all the chapters carefully and I thank
them all for their comments. In particular, it is a pleasure for me to
thankMs. JudyMack andProfessorSusumuOkubofortheirtireless
effort in going through the entire material. I am personally grateful
toDr. JohnBoersmaforpainstakingly andmeticulously checking all
the mathematical derivations. Of course, any remaining errors and
typos are my own.
Like the subject itself, the list of references to topics in quantum
field theory is enormous and it is simply impossible to do justice to
everyone who has contributed to the growth of the subject. I have
in no way attempted to give an exhaustive list of references to the
subject. Instead I have listed only a few suggestive references at the
end of each chapter in the hope that the readers can get to the other
references from these sources.
The Feynman graphs in this book were drawn using Jaxodraw
while most other figures were generated using PSTricks. I am grate-
ful to the people who developed these extremely useful softwares.
Finally, I would like to thank Dave Munson for helping out with
various computer related problems.
Ashok Das
Rochester
Contents
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Relativistic equations . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Notations. . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Klein-Gordon equation . . . . . . . . . . . . . . . . . 10
1.3.1 Klein paradox . . . . . . . . . . . . . . . . . . . 14
1.4 Dirac equation. . . . . . . . . . . . . . . . . . . . . . 19
1.5 References . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Solutions of the Dirac equation. . . . . . . . . . . . . . . . 27
2.1 Plane wave solutions . . . . . . . . . . . . . . . . . . 27
2.2 Normalization of the wave function . . . . . . . . . . 34
2.3 Spin of the Dirac particle. . . . . . . . . . . . . . . . 40
2.4 Continuity equation. . . . . . . . . . . . . . . . . . . 44
2.5 Dirac’s hole theory . . . . . . . . . . . . . . . . . . . 47
2.6 Properties of the Dirac matrices. . . . . . . . . . . . 49
2.6.1 Fierz rearrangement . . . . . . . . . . . . . . . 58
2.7 References . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Properties of the Dirac equation . . . . . . . . . . . . . . . 65
3.1 Lorentz transformations . . . . . . . . . . . . . . . . 65
3.2 Covariance of the Dirac equation . . . . . . . . . . . 72
3.3 Transformation of bilinears. . . . . . . . . . . . . . . 82
3.4 Projection operators, completeness relation . . . . . 84
3.5 Helicity. . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.6 Massless Dirac particle . . . . . . . . . . . . . . . . . 94
3.7 Chirality . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.8 Non-relativistic limit of the Dirac equation. . . . . . 105
3.9 Electron in an external magnetic field . . . . . . . . 107
3.10 Foldy-Wouthuysen transformation. . . . . . . . . . . 111
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