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Published by
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Library of Congress Cataloging-in-Publication Data
Names: Vergados, J. D.
Title: Group and representation theory / by J.D. Vergados (University of Ioannina, Ioannina, Greece).
Description: New Jersey : World Scientific, 2017. | Includes bibliographical references and index.
Identifiers: LCCN 2016040287 | ISBN 9789813202443 (hardcover : alk. paper)
Subjects: LCSH: Symmetry (Mathematics) | Group theory. |
Transformations (Mathematics) | Lie algebras.
Classification: LCC QA174.7.S96 V47 2017 | DDC 512/.2--dc23
LC record available at https://lccn.loc.gov/2016040287
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
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Printed in Singapore
To the Students of all Greek Universities who greatly
appreciated the Greek version of this book and thus
encouraged the present edition.
To the Graduate Students of Fall 2012 and Spring 2016 of
Nanjing University who greatly helped in preparing the initial
Lecture Notes.
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Preface
This book was initiated in a set of lectures I have given at Nanjing Uni-
versity, since 2012. Most of the first chapter of the current volume is taken
from a set of lectures on The Standard Model and Beyond delivered to the
first-yeargraduatestudentsofDepartmentofPhysicsofNanjingUniversity
during the Fall of 2012 and later appeared as “Lecture Notes in Particle
Physics: TheStandardModelandBeyond,NanjingUniversityPress,Nan-
jing, China, 2013”.
The other chapters of this volume resulted from a set of lectures on
Elementary Group Theory and Applications which have been delivered in
the Spring Semester of 2016 to advanced undergraduate and graduate stu-
dents of Nanjing University. I am indebted to the students who attended
the class for comments, suggestions and criticism. I am particularly in-
debted to Ruo Fei Xie, who acted voluntarily as my Teaching Assistant
(TA), read the notes carefully, made useful suggestions and transformed
mostofthetablesoftheappendixinLaTeXform. Inaddition,Iamhappy
to acknowledge his excellent drawing of the Dynkin and the weight chain
diagrams.
IacknowledgesupportformyvisittoNanjingUniversitybythePhysics
Department and Professor Yeukkwan Edna Cheung. I am thankful to her
and Dr Konstantin G. Savvidy for their hospitality.
A major part of this volume, mainly chapters 4–8, was initiated during
avisitoftheauthorattheUniversityofSouthCarolinaduringthespringof
2015andwasalmostcompletedwhileonavisittoARCCentreofExcellence
in Particle Physics (CoEPP), University of Adelaide, Australia during the
Fall–Winter of the same year.
vii
viii Group and Representation Theory
The final touch was put at the Center for Axion and Precision Physics
(CAPP), IBS, KAIST, S. Korea, during a visit in the Fall of 2016.
The author is indebted to distinguished S.C. Professor Frank
Avignone and the Director of the Adelaide node of CoEPP, Professor
Anthony Thomas and the director of CAPP, Professor Y. Semertzidis, for
their support and hospitality.
A guide for teachers and students
There exist many similar titles in the market. This book differs from other
existingbooksinthesensethatittriestobridgemoreefficientlythegapbe-
tween the mathematical structure, Lie groups and Lie algebra, with actual
applications. It has tried not to fall short of adequately describing realistic
applications or to introduce more or less ad hoc the basic conclusions of
the relevant mathematics into the applications.
Furthermore, this book is organized so that it can be used by an ad-
vancedundergraduateorbeginnergraduatestudentwithoutgoingthrough
the full mathematical machinery. This can be achieved by going through
the rather elementary chapters 1, 2 and 10. Only familiarity with quan-
tum mechanics and an introduction to mathematical methods of physics
is a prerequisite. By studying only this part the reader will be rewarded
by obtaining the solution of the Kepler’s problem without solving a dif-
ferential equation via the Ru¨nge-Lenz vector as well as by doing the same
thing for all differential equations encountered in quantum mechanics. An
instructormayalsoselecttostudythesechaptersinthefirsttermandcon-
tinue with the more elaborate mathematical structure in the second term.
Alternatively, he may skip chapters 1, 2 and 10 altogether and start with
chapter 3.
There exist many useful tables in the appendix, which can be used by
an experienced reader. They were constructed, however, mainly for the
reader who has understood the main text.
J.D. Vergados
KAIST, Daejeon, S. Korea, October 2016
Contents
Preface vii
1. Elements of Group Theory 1
1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Matrix groups . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 The exponential of a matrix . . . . . . . . . . . . 5
1.2.2 Determination of the independent parameters . . 9
1.3 The structure constants . . . . . . . . . . . . . . . . . . . 12
1.4 The metric tensor and Casimir operators . . . . . . . . . . 13
1.4.1 The metric tensor . . . . . . . . . . . . . . . . . . 13
1.4.2 The Casimir operator . . . . . . . . . . . . . . . . 14
1.5 Representations . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.1 The fundamental representation . . . . . . . . . . 17
1.5.2 The adjoined representation . . . . . . . . . . . . 23
1.6 More representations . . . . . . . . . . . . . . . . . . . . . 25
1.7 Irreducible representations . . . . . . . . . . . . . . . . . . 26
1.8 Some further examples . . . . . . . . . . . . . . . . . . . . 31
1.9 The proper orthogonal groups O(3) and O(4) . . . . . . . 33
1.10 RealizationofSO(4)=SO(3)×SO(3)andsomeapplications 36
1.10.1 Application to classical mechanics . . . . . . . . . 37
1.10.2 Application to quantum mechanics. . . . . . . . . 39
1.11 The proper Lorentz transformations — The group SO(3,1) 41
1.12 Symmetries and conservation laws — Noether’s theorem . 42
1.12.1 Global transformations in classical physics . . . . 42
1.12.2 Global transformations in quantum mechanics (in
the Heisenberg picture) . . . . . . . . . . . . . . . 43
ix
