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Lectures of Quantum Field Theory PDF

790 Pages·2008·3.53 MB·English
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6938tp.indd 1 8/14/08 1:42:10 PM TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Ashok Das University of Rochester, USA World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 6938tp.indd 2 8/14/08 1:42:10 PM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. 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 Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. 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 TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk 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 vii viii Preface 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 ix

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This book consists of the lectures for a two-semester course on quantum field theory, and as such is presented in a quite informal and personal manner. The course starts with relativistic one-particle systems, and develops the basics of quantum field theory with an analysis of the representations of
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