Lie Groups and Lie Algebras for Physicists

358 Pages·2015·1.77 MB·English

by  Ashok Das

Lie Groups and
Lie Algebras
for Physicists
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Lie Groups and
Lie Algebras
for Physicists
Ashok Das
Susumu Okubo
University of Rochester, USA
World Scientific
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LIE GROUPS AND LIE ALGEBRAS FOR PHYSICISTS
Copyright © 2014 Hindustan Book Agency (HBA)
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To
Mary Okubo
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Preface
Group theory was discovered by E´variste Galois in the 19th century
for the case of finite dimensional symmetric group. It has been suc-
cessfully generalized subsequently for the infinite dimensional case
(i.e. Lie group and Lie algebra) by Sophus Lie. Before the advent
of quantum mechanics in the early 20th century, group theory was
thoughtbymanyphysiciststobeunimportantinthestudyofphysics.
It is indeed of some interest to note the following anecdote narrated
by Freeman Dyson (as quoted in Mathematical Apocrypha, p 21, by
S. A. Kranz), “In 1910, the mathematician Oswald Veblen (1880-
1960) - a founding member of the Institute for Advanced Study - and
the physicist James Jean (1877-1946) were discussing the reform of
the mathematics curriculum at Princeton University. Jeans argued
that they ‘may as well cut out group theory,’ for it ‘would never be of
any use to physics’.” The real fundamental change in thinking truly
occurred with the development of quantum mechanics. It was soon
realizedthatadeepknowledgeofgrouptheoryandLiealgebrainthe
study of of angular momentum algebra is crucial for real understand-
ingofquantummechanicalatomicandnuclearspectralproblems. At
present,grouptheorypermeatesproblemsinpracticallyeverybranch
ofmodernphysics. EspeciallyinthestudyofYang-Millsgaugetheory
andstringtheorytheuseofgrouptheoryisessential. Forexample,we
note that the largest exceptional Lie algebra E appears in heterotic
8
string theory and also in some one-dimensional Ising model in statis-
tical mechanics, some of whose predictions have been experimentally
verified recently.
Our goal in this monograph is to acquaint (mostly) graduate
students of physics with various aspects of modern Lie group and
Lie algebra. With this in view, we have kept the presentation of
the material in this book at a pedagogical level avoiding unnecessary
mathematical rigor. Furthermore, the groups which we will discuss
in this book will be mostly Lie groups which are infinite dimensional.
We will discuss finite groups only to the extent that they will be nec-
vii
viii Preface
essary for the development of our discussions. The interested readers
on this topic are advised to consult many excellent text books on
the subject. We have not tried to be exhaustive in the references.
Rather, we have given only a few references to books (at the end of
each chapter) that will be easier to read for a student with a physics
background. We assume that the readers are familiar with the mate-
rialofatleastatwosemestergraduatecourseonquantummechanics
as well as with the basics of linear algebra theory.
We would like to thank Dr. Brenno Ferraz de Oliveira (Braz-
ing´a) for drawing all the Young tableau diagrams in chapter 5.
A. Das and S. Okubo
Rochester
Contents
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Introduction to groups. . . . . . . . . . . . . . . . . . . . . 1
1.1 Definition of a group . . . . . . . . . . . . . . . . . . 1
1.2 Examples of commonly used groups in physics. . . . 3
1.2.1 Symmetric group S . . . . . . . . . . . . . . . 3
N
1.2.2 One dimensional translation group T . . . . . 6
1
1.2.3 One dimensional unitary group U(1) . . . . . . 9
1.2.4 U(N) and SL(N) groups . . . . . . . . . . . . 10
1.2.5 O(N) and SO(N) groups . . . . . . . . . . . . 13
1.2.6 Symplectic group Sp(2N) . . . . . . . . . . . . 20
1.3 Group manifold . . . . . . . . . . . . . . . . . . . . . 24
1.4 References . . . . . . . . . . . . . . . . . . . . . . . . 28
2 Representation of groups . . . . . . . . . . . . . . . . . . . 29
2.1 Matrix representation of a group . . . . . . . . . . . 29
2.2 Unitary and irreducible representations. . . . . . . . 41
2.3 Group integration. . . . . . . . . . . . . . . . . . . . 51
2.4 Peter-Weyl theorem. . . . . . . . . . . . . . . . . . . 55
2.4.1 Fully reducible representation . . . . . . . . . . 55
2.4.2 Unitary representation . . . . . . . . . . . . . . 57
2.5 Orthogonality relations. . . . . . . . . . . . . . . . . 58
2.6 Character of a representation . . . . . . . . . . . . . 61
2.7 References . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Lie algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1 Definition of a Lie algebra . . . . . . . . . . . . . . . 63
3.2 Examples of commonly used Lie algebras in physics. 66
3.2.1 Lie algebra of gl(N) and sl(N) . . . . . . . . . 66
3.2.2 Lie algebra of so(N) . . . . . . . . . . . . . . . 67
3.2.3 Lie algebras of u(N) and su(N). . . . . . . . . 71
3.3 Structure constants and the Killing form. . . . . . . 78
3.4 Simple and semi-simple Lie algebras . . . . . . . . . 87
3.5 Universal enveloping Lie algebra. . . . . . . . . . . . 97
ix