Group theory in physics

Group Theory in Physics

An Introduction

J.F. Cornwell

School of Physics and Astronomy
University of St. Andrews, Scotland

ACADEMIC PRESS
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Contents
Preface

vii

T h e Basic F r a m e w o r k
1
T h e concept of a group . . . . . . . . . . . . . . . . . . . . . . .
2
G r o u p s of coordinate t r a n s f o r m a t i o n s . . . . . . . . . . . . . .
(a)
Rotations ..........................
(b)
Translations . . . . . . . . . . . . . . . . . . . . . . . . .
T h e g r o u p of the Schr5dinger equation . . . . . . . . . . . . . .
(a)
The Hamiltonian operator . . . . . . . . . . . . . . . . .
(b)
T h e invariance of the H a m i l t o n i a n o p e r a t o r . . . . . . .

(c)
T h e scalar t r a n s f o r m a t i o n operators P ( T ) . . . . . . . .
T h e role of m a t r i x representations . . . . . . . . . . . . . . . .
The
1
2
3
4
5
6
7

Structure of Groups
Some e l e m e n t a r y considerations . . . . . . . . . . . . . . . . . .
Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Invariant subgroups . . . . . . . . . . . . . . . . . . . . . . . .
Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F a c t o r groups . . . . . . . . . . . . . . . . . . . . . . . . . . . .
H o m o m o r p h i c and isomorphic m a p p i n g s . . . . . . . . . . . . .
Direct p r o d u c t s and semi-direct p r o d u c t s of groups . . . . . . .

1
1
4
5
9
10
10
11
12

15
19
19
21
23
24
26
28
31

Lie G r o u p s

35

1
2
3
4

35
40
42
44

Definition of a linear Lie group . . . . . . . . . . . . . . . . . .
T h e connected c o m p o n e n t s of a linear Lie group . . . . . . . .
T h e concept of compactness for linear Lie groups . . . . . . . .
Invariant integration . . . . . . . . . . . . . . . . . . . . . . . .

R e p r e s e n t a t i o n s of Groups - Principal Ideas

47

1
2
3
4
5

47
49
52
54

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Equivalent representations . . . . . . . . . . . . . . . . . . . . .
U n i t a r y representations . . . . . . . . . . . . . . . . . . . . . .
Reducible and irreducible representations . . . . . . . . . . . .
Schur's L e m m a s and the o r t h o g o n a l i t y t h e o r e m for m a t r i x representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii

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GROUP THEORY IN PHYSICS

iv
6

Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Representations of Groups - Developments
1
Projection operators . . . . . . . . . . . . . . . . . . . . . . . .
2
Direct product representations . . . . . . . . . . . . . . . . . .
T h e Wigner-EcLurt Theorem for groups of coordinate transfor3
mations in ] R 3 . . . . . . . . . . . . . . . . . . . . . . . . . . .
T h e Wigner-Eckart Theorem generalized . . . . . . . . . . . . .
Representations of direct p r o d u c t groups . . . . . . . . . . . . .
Irreducible representations of finite Abelian groups . . . . . . .
Induced representations . . . . . . . . . . . . . . . . . . . . . .

65
65
70

G r o u p T h e o r y in Quantum Mechanical Calculations
1
T h e solution of the SchrSdinger equation . . . . . . . . . . . . .
2
Transition probabilities and selection rules . . . . . . . . . . . .
3
Time-independent p e r t u r b a t i o n theory . . . . . . . . . . . . . .

93

93
97
100

Crystallographic Space Groups

103

1
2
3

T h e Bravais lattices . . . . . . . . . . . . . . . . . . . . . . . .
103
The cyclic boundary conditions . . . . . . . . . . . . . . . . . .
107
Irreducible representations of the group T of pure primitive
translations and Bloch's T h e o r e m . . . . . . . . . . . . . . . . .
109
Brillouin zones . . . . . . . . . . . . . . . . . . . . . . . . . . .
111
Electronic energy bands . . . . . . . . . . . . . . . . . . . . . .
115
Survey of the crystallographic space groups . . . . . . . . . . .
118
Irreducible representations of symmorphic space groups . . . .
121
(a)
Fundamental theorem on irreducible representations of
symmorphic space groups . . . . . . . . . . . . . . . . .

121
(b)
Irreducible representations of the cubic space groups
O~, O~ and O 9 . . . . . . . . . . . . . . . . . . . . . . .
126
Consequences of the fundamental theorems . . . . . . . . . . .
129
(a)
Degeneracies of eigenvalues and the symmetry of e(k) . 129
(b)
Continuity and compatibility of the irreducible representations of G0(k) . . . . . . . . . . . . . . . . . . . . .
131
(c)
Origin and orientation dependence of the s y m m e t r y labelling of electronic states . . . . . . . . . . . . . . . . .
134

The R o l e o f Lie A l g e b r a s
1
2
3
4
5

73
79
83
85
86

135

"Local" and "global" aspects of Lie groups . . . . . . . . . . .
135
T h e m a t r i x exponential function . . . . . . . . . . . . . . . . .
136
O n e - p a r a m e t e r subgroups . . . . . . . . . . . . . . . . . . . . .
139
Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
140
T h e real Lie algebras that correspond to general linear Lie groups 145
(a)
The existence of a real Lie a l g e b r a / : for every linear
Lie group G . . . . . . . . . . . . . . . . . . . . . . . . .
145
(b)
The relationship of the real Lie a l g e b r a / : to the oneparameter subgroups of G . . . . . . . . . . . . . . . . .
148

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CONTENTS

v

The Relationships between Lie Groups and Lie Algebras E x 153
plored
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
153
Subalgebras of Lie algebras . . . . . . . . . . . . . . . . . . . .
153

H o m o m o r p h i c a n d isomorphic mappings of Lie algebras . . . .
154
Representations of Lie algebras . . . . . . . . . . . . . . . . . .
160
T h e adjoint representations of Lie algebras and linear Lie groups168
Direct sum of Lie algebras . . . . . . . . . . . . . . . . . . . . .
171
10 The Three-dimensional Rotation Groups
1
2
3
4
5

11 The
1
2
3
4
5
6
7
8
9
10

Structure of Semi-simple Lie Algebras

193

An outline of the presentation . . . . . . . . . . . . . . . . . . .
T h e Killing form and C a r t a n ' s criterion . . . . . . . . . . . . .
Complexification . . . . . . . . . . . . . . . . . . . . . . . . . .
T h e C a r t a n subalgebras and roots of semi-simple complex Lie
algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Properties of roots of semi-simple complex Lie algebras . . . .
T h e remaining c o m m u t a t i o n relations . . . . . . . . . . . . . .
T h e simple roots . . . . . . . . . . . . . . . . . . . . . . . . . .
T h e Weyl canonical form of L . . . . . . . . . . . . . . . . . . .
T h e Weyl group o f / : . . . . . . . . . . . . . . . . . . . . . . . .
Semi-simple real Lie algebras . . . . . . . . . . . . . . . . . . .

193
193
198

12 Representations of Semi-simple Lie Algebras
1
2
3
4

Some basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . .
T h e weights of a representation . . . . . . . . . . . . . . . . . .
T h e highest weight of a representation . . . . . . . . . . . . . .
T h e irreducible representations o f / : - A2, the complexification
of s = su(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Casimir operators . . . . . . . . . . . . . . . . . . . . . . . . . .

13 S y m m e t r y schemes for the elementary particles

1
2
3

175

Some properties reviewed . . . . . . . . . . . . . . . . . . . . .
175
T h e class structures of SU(2) and SO(3) . . . . . . . . . . . . .
176
Irreducible representations of the Lie algebras su(2) and so(3) . 177
Representations of the Lie groups SU(2), SO(3) and 0 ( 3 ) . . . 183
Direct products of irreducible representations and the ClebschG o r d a n coefficients . . . . . . . . . . . . . . . . . . . . . . . . .
186
Applications to atomic physics . . . . . . . . . . . . . . . . . .
189

200
207
213
218
223
224
228

235
235
236
241
245

251

255

Leptons and h a d r o n s . . . . . . . . . . . . . . . . . . . . . . . .
255
T h e global internal s y m m e t r y group SU(2) and isotopic s p i n . . 256
T h e global internal s y m m e t r y group SU(3) and strangeness . . 259

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GROUP T H E O R Y IN PHYSICS

APPENDICES

269

A Matrices
1
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . .

271
271
275

B V e c t o r Spaces
1
The concept of a vector space . . . . . . . . . . . . . . . . . . .
2
Inner product spaces . . . . . . . . . . . . . . . . . . . . . . . .
3
Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Linear functionals . . . . . . . . . . . . . . . . . . . . . . . . .
6
Direct product spaces . . . . . . . . . . . . . . . . . . . . . . .
7

279
279
282
286
288
292
294
295

C C h a r a c t e r T a b l e s for t h e C r y s t a l l o g r a p h i c P o i n t G r o u p s

299

D P r o p e r t i e s of t h e C l a s s i c a l S i m p l e C o m p l e x Lie A l g e b r a s

1
The simple complex Lie algebra Al, l >_ 1 . . . . . . . . . . . .
2
The simple complex Lie algebra Bz, l > 1 . . . . . . . . . . . .
3
The simple complex Lie algebra Cl, 1 > 1 . . . . . . . . . . . .
4
The simple complex Lie algebra D1, 1 >__3 (and the semi-simple
complex Lie algebra D2) . . . . . . . . . . . . . . . . . . . . . .

319
319
320
322

References

327

Index

335

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324


Preface
ace to my three-volume work Group

Theory in Physics, thirty years or so ago group theory could have been regarded by physicists as merely providing a very valuable tool for the elucidation of the symmetry aspects of physical problems. However, recent developments, particularly in high-energy physics, have transformed its role,
so that it now occupies a crucial and indispensable position at the centre of
the stage. These developments have taken physicists increasingly deeper into
the fascinating world of the pure mathematicians, and have led to an evergrowing appreciation of their achievements, the full recognition of which has
been hampered to some extent by the style in which much of modern pure
mathematics is presented. As with my previous three-volume treatise, one of
the main objectives of the present work is to try to overcome this communication barrier, and to present to theoretical physicists and others some of
the important mathematical developments in a form that should be easier to
comprehend and appreciate.
Although my Group Theory in Physics was intended to provide a introduction to the subject, it also aimed to provide a thorough and self-contained
account, and so its overall length may well have made it appear rather daunting. The present book has accordingly been designed to provide a much more
succinct introduction to the subject, suitable for advanced undergraduate and
postgraduate students, and for others approaching the subject for the first
time. The treatment starts with the basic concepts and is carried through to
some of the most significant developments in atomic physics, electronic energy
bands in solids, and the theory of elementary particles. No prior knowledge
of group theory is assumed, and, for convenience, various relevant algebraic
concepts are summarized in Appendices A and B.
The present work is essentially an abridgement of Volumes I and II of
Group Theory in Physics (which hereafter will be referred to as "Cornwell
(1984)"), although some new material has been included. The intention has
been to concentrate on introducing and describing in detail the most important basic ideas and the role that they play in physical problems. Inevitably
restrictions on length have meant that some other important concepts and
developments have had to be omitted. Nevertheless the mathematical coverage goes outside the strict confines of group theory itself, for one soon is led
to the study of Lie algebras, which, although related to Lie groups, are often
vii

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viii

GROUP T H E O R Y IN PHYSICS

developed by mathematicians as a separate subject.
Mathematical proofs have been included only when the direct nature of
their arguments assist in the appreciation of theorems to which they refer.
In other cases references have been given to works in which they may be
found. In many instances these references are quoted as "Cornwell (1984)",
as interested readers may find it useful to see these proofs with the same
notations, conventions, and nomenclature as in the present work. Of course,
this is not intended to imply that this reference is either the original source or
the only place in which a proof may be found. The same reservation naturally
applies to the references to suggested further reading on topics that have been
explicitly omitted here.
In the text the treatments of specific cases are frequently given under the
heading of "Examples". The format is such that these are clearly distinguished
from the main part of the text, the intention being that to indicate that the
detailed analysis in the Example is not essential for the general understanding
of the rest of that section or the succeeding sections. Nevertheless, the Examples are important for two reasons. Firstly, they give concrete realizations of
the concepts that have just been introduced. Secondly, they indicate how the
concepts apply to certain physically important groups or algebras, thereby
allowing a "parallel" treatment of a number of specific cases. For instance,
many of the properties of the groups SU(2) and SU(3) are developed in a
series of such Examples.
For the benefit of readers who may wish to concentrate on specific applications, the following list gives the relevant chapters:
(i) electronic energy bands in solids: Chapters 1, 2, and 4 to 7;
(ii) atomic physics: Chapters 1 to 6, and 8 to 10;
(iii) elementary particles: Chapters 1 to 6, and 8 to 13.

J.F. Cornwell
St.Andrews
January, 1997

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To my wife Elizabeth and my daughters
Rebecca and Jane

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This Page Intentionally Left Blank

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Chapter 1

The Basic Framework
1

The concept of a group

The aim of this chapter is to introduce the idea of a group, to give some
physically important examples, and then to indicate immediately how this
notion arises naturally in physical problems, and how the related concept of a
group representation lies at the heart of the quantum mechanical formulation. 9
With the basic framework established, the next four chapters will explore in

more detail the relevant properties of groups and their representations before
the application to physical problems is taken up in earnest in Chapter 6.
To mathematicians a group is an object with a very precise meaning. It
is a set of elements that must obey four group axioms. On these is based
a most elaborate and fascinating theory, not all of which is covered in this
book. The development of the theory does not depend on the nature of the
elements themselves, but in most physical applications these elements are
transformations of one kind or another, which is why T will be used to denote
a typical group member.
D e f i n i t i o n Group g
A set g of elements is called a "group" if the following four "group axioms"
are satisfied:

(a) There exists an operation which associates with every pair of elements T
and T ~ of g another element T" of g. This operation is called multiplication and is written as T " = T T ~, T" being described as the "product
of T with T t'' .
(b) For any three elements T, T ~ and T" of g

(TT')T" = T(T'T").

(1.1)

This is known as the "associative law" for group multiplication. (The
interpretation of the left-hand side of Equation (1.1) is that the product

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2

GROUP THEORY IN PHYSICS
T T ~ is to be evaluated first, and then multiplied by T" whereas on the
right-hand side T is multiplied by the product T ' T ' . )

(c) There exists an identity element E which is contained in ~ such that
TE = ET = T

for every element T of G.
(d) For each element T of G there exists an inverse element T -1 which is
also contained in G such that
T T -1 = T - 1 T = E .

This definition covers a diverse range of possibilities, as the following examples indicate.
E x a m p l e I The multiplicative group of real numbers
The simplest example (from which the concept of a group was generalized)
is the set of all real numbers (excluding zero) with ordinary multiplication
as the group multiplication operation. The axioms (a) and (b) are obviously
satisfied, the identity is the number 1, and each real number t (~ 0) has its
reciprocal 1/t as its inverse.
E x a m p l e I I The additive group of real numbers
To demonstrate that the group multiplication operation need not have any
connection with ordinary multiplication, take G to be the set of all real numbers with ordinary addition as the group multiplication operation. Again
axioms (a) and (b) are obviously satisfied, but in this case the identity is 0
(as a + 0 - 0 + a = a) and the inverse of a real number a is its negative - a
(as a + ( - a ) = ( - a ) + a -- 0).
E x a m p l e I I I A finite m a t r i x group
Many of the groups appearing in physical problems consist of matrices with
matrix multiplication as the group multiplication operation. (A brief account
of the terminology and properties of matrices is given in Appendix A.) As an
example of such a group let G be the set of eight matrices

M1 =
M4 =

M7 =

[ I
[ ]
[ ]
1
0

-1

0
1

0
1

0

0

1

1
0

M2=

'

'

[1 0]
[o_1]
[ 1
,

0

'

M5 =

Ms =

-1

1

0

0
-1

-1
0

,

M3 -

- 10

M6 =

[~
-1

- 10 1 '
0

'

"

By explicit calculation it can be verified that the product of any two members
of G is also contained in G, so that axiom (a) is satisfied. Axiom (b) is

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THE B A S I C F R A M E W O R K

3

automatically true for matrix multiplication, M1 is the identity of axiom (c)
as it is a unit matrix, and finally axiom (d) is satisfied as

M~ -1 = M1,
M51 = M6,

M21 = M2,
M61 = M5,

M31 = M3,
M~-i = MT,

M~ -1 = M4,
M s 1 = Ms.

E x a m p l e I V The groups U(N) and SU(N)
U(N) for N > 1 is defined to be the set of all N • N unitary matrices u
with matrix multiplication as the group multiplication operation. SU(N) for
N >_ 2 is defined to be the subset of such matrices u for which det u = 1,
with the same group multiplication operation. (As noted in Appendix A, if
u is unitary then det u = exp(ia), where c~ is some real number. The "S" of
SU(N) indicates that SU(N) is the "special" subset of U(N) for which this a
is zero.)
It is easily established that these sets do form groups. Consider first the
set U(N). As (ulu2) t = u2u
t t1 and (ulu2) -1 = u 2 1 u l 1 , if Ul and u2 are both
unitary then so is UlU2. Again axiom (b) is automatically valid for matrix
multiplication and, as the unit matrix 1N is a member of U(N), it provides
the identity E of axiom (c). Finally, axiom (d) is satisfied, as if u is a member
of U(N) then so is u - 1.
For SU(N) the same considerations apply, but in addition if ul and u2
both have determinant 1, Equation (A.4) shows that the same is true of ulu2.
Moreover, 1N is a member of SU(N), so it is its identity, and u -1 is a member

of SU(N) if that is the case for u.
The set of groups SU(N) is particularly important in theoretical physics.
SU(2) is intimately related to angular momentum and isotopic spin, as will
be shown in Chapters 10 and 13, while SU(3) is now famous for its role in the
classification of elementary particles, which will also be studied in Chapter
13.
E x a m p l e V The groups O(N) and SO(N)
The set of all N • N real orthogonal matrices R (for N >_ 2) is denoted
almost universally by O(N), although O(N, IR) would have been preferable as
it indicates that only real matrices are included. The subset of such matrices
R with det R - 1 is denoted by SO(N). As will be described in Section 2,
O(3) and SO(3) are intimately related to rotations in a real three-dimensional
Euclidean space, and so occur time and time again in physical applications.
O ( N ) and SO(N) are both groups with matrix multiplication as the group
multiplication operation, as they can be regarded as being the subsets of U(N)
and SU(N) respectively that consist only of real matrices. (All that has to
be observed to supplement the arguments given in Example IV is that the
product of any two real matrices is real, that 1N is real, and that the inverse
of a real matrix is also real.)
If T1 T2 = T2T1 for every pair of elements T1 and T2 of a group G (that is, if
all T1 and T2 of ~ commute), then G is said to be "Abelian". It will transpire

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GROUP T H E O R Y IN PHYSICS

M1

M2
M3
Ma
M5
M6
M7
Ms

M1
M1
M2
M3
M4
M5
M6
M7
Ms

M2
M2
M1
Ma
M3
M7
Ms
M5
M6

M3
M3

Ma
M1
M2
M6
M5
Ms
M7

Ma
Ma
M3
M2
M~
Ms
M7
M6
M5

M5
M5
Ms
M6
M7
M3
M1
M2
Ma

M6
M6

M7
M5
M8
M~
M3
Ma
M2

M7
M7
M6
Ms
M5
Ma
M2
M~
M3

Ms
Ms
M5
M7
M6
M2
Ma
M3
Ma

Table 1.1: Multiplication table for the group of Example III.

that such groups have relatively straightforward properties. However, many
of the groups having physical applications are non-Abelian. Of the cases
considered above the only Abelian groups are those of Examples I and II
and the groups V(1) and SO(2) of Examples IV and V. (One of the noncommuting pairs of products of Example III which makes that group nonAbelian is MsM7 = M4, MTM5 = M2.)
The "order" of G is defined to be the number of elements in G, which may
be finite, countably infinite, or even non-countably infinite. A group with
finite order is called a "finite group". The vast majority of groups that arise
in physical situations are either finite groups or are "Lie groups", which are a
special type of group of non-countably infinite order whose precise definition
will be given in Chapter 3, Section 1. Example III is a finite group of order
8, whereas Examples I, II, IV and V are all Lie groups.
For a finite group the product of every element with every other element
is conveniently displayed in a multiplication table, from which all information
on the structure of the group can subsequently be deduced. The multiplication table of Example III is given in Table 1.1. (By convention the order of
elements in a product is such that the element in the left-hand column precedes the element in the top row, so for example M5Ms = M2.) For groups
of infinite order the construction of a multiplication table is clearly completely
impractical, but fortunately for a Lie group the structure of the group is very
largely determined by another finite set of relations, namely the commutation
relations between the basis elements of the corresponding real Lie algebra, as
will be explained in detail in Chapter 8.

2

G r o u p s of c o o r d i n a t e t r a n s f o r m a t i o n s

To proceed beyond an intuitive picture of the effect of symmetry operations,
it is necessary to specify the operations in a precise algebraic form so that
the results of successive operations can be easily deduced. Attention will be
confined here to transformations in a real three-dimensional Euclidean space
IR3, as most applications in atomic, molecular and solid state physics involve

only transformations of this type.

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THE BASIC FRAMEWORK

5

Z

J2"
f

J

y

Figure 1.1: Effect of a rotation through an angle 0 in the right-hand screw
sense about Ox.

(a)

Rotations

Let Ox, Oy, Oz be three mutually orthogonal Cartesian axes and let Ox ~,
Oy ~, Oz' be another set of mutually orthogonal Cartesian axes with the same
origin O that is obtained from the first set by a rotation T about a specified
axis through O. Let (x, y, z) and (x', y', z') be the coordinates of a fixed point
P in the space with respect to these two sets of axes. Then there exists a real

orthogonal 3 x 3 matrix R ( T ) which depends on the rotation T, but which is
independent of the position of P, such t h a t

r'= R(T)r,

(1.2)

ix] ix]

where
r/=

y1

and r -

Z!

y

.

Z

(Hereafter position vectors will always be considered as 3 • 1 column matrices
in matrix expressions unless otherwise indicated, although for typographical
reasons they will often be displayed in the text as 1 x 3 row matrices.) For
example, if T is a rotation through an angle 0 in the right-hand screw sense
about the axis Ox, then, as indicated in Figures 1.1 and 1.2,
X !

"-

X~

yt

_

ycosO+zsinO,

zt

=

-ysinO+zcosO,

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6

GROUP T H E O R Y IN PHYSICS

9

Y
s"

I

I X

%J

y

,,

r

Y

O

Figure 1.2: The plane containing the axes Oy, Oz, Oy ~ and Oz ~ corresponding
to the rotation of Figure 1.1.
so that

[1 o o 1

R(T) =

0

cos0

sin0

0

-sin0

cos0

.

(1.3)

The matrix R ( T ) obeys the orthogonality condition R ( T ) = R ( T ) -~ because rotations leave invariant the length of every position vector and the
angle between every pair of position vectors, that is, they leave invariant
the scalar product r l.r2 of any two position vectors. (Indeed the name "orthogonal" stems from the involvement of such matrices in the transformations being considered here between sets of orthogonal axes.) The proof that
R ( T ) is orthogonal depends on the fact that rl.r2 can be expressed in matrix form as rlr2. Then, if r~ = R ( T ) r l and r~ = R(T)r2, it follows that
r~.r2 = r-~r~ = Y~R(T)R(T)r2, which is equal to Ylr2 for all rl and r2 if and
only if R ( T ) R ( T ) = 1.
As noted in Appendix A, the orthogonality condition implies that det R ( T )
can take only the values +1 or - 1 . If det R ( T ) = +1 the rotation is said to
be "proper"; otherwise it is said to be "improper". The only rotations which
can be applied to a rigid body are proper rotations. The transformation of
Equation (1.3) gives an example.
The simplest example of an improper rotation is the spatial inversion operation I for which r' = - r , so that

[ 10 0]

n(I)

=

0

0

-~

0

0

.

-1

Another important example is the operation of reflection in a plane. For
instance, for reflection in the plane Oyz, for which x' = - x , y' = y, z' = z,

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THE BASIC F R A M E W O R K

7

the transformation matrix is
0
0

00]

1
0

0
1

The "product" T1T2 of two rotations T1 and T2 may be defined to be the
rotation whose transformation matrix is given by
(1.4)

R(T~ T2) = R(T1 )R(T2).

(The validity of this definition is assured by the fact that the product of
any two real orthogonal matrices is itself real and orthogonal.) In general
R(T~)R(T2) r R(T2)R(T~), in which case T~T2 =/= T2T1. If r ' = R(T2)r
and r " = R(T~)r', then Equation (1.4) implies that r " = R(T~T2)r, so the
interpretation of Equation (1.4) is that operation T2 takes place before 7"1.
This is an example of the general convention (which will be applied throughout
this book) that in any product of operators the operator on the right acts first.
With this definition (Equation (1.4)) every improper rotation can be considered to be the product of the spatial inversion operator I with a proper
rotation. For example, for the reflection in the Oyz plane

[100] [1 0 0][1 0 0]
0
0

1
0

0
1

--

0
0

-1
0

0
-1

0
0

-1
0

0
-1

,

and, as the second matrix on the right-hand side is the transformation of
Equation (1.3) with 0 = ~, it corresponds to a rotation through ~ about Ox.
If a set of matrices R ( T ) forms a group, then the corresponding set of
rotations T also forms a group in which Equation (1.4) defines the group multiplication operator and for which the inverse T -1 of T is given by R ( T -1) =
R ( T ) -1. As these two groups have the same structure, they are said to be
"isomorphic" (a concept which will be examined in more detail in Chapter 2,
Section 6).
E x a m p l e I The group of all rotations

The set of all rotations, both proper and improper, forms a Lie group that is
isomorphic to the group 0(3) that was introduced in Example V of Section
1.
E x a m p l e I I The group of all proper rotations
The set of all proper rotations forms a Lie group that is isomorphic to the
group SO(3).
E x a m p l e I I I The crystallographic point group D4
A group of rotations that leave invariant a crystal lattice is called a "crystallographic point group", the "point" indicating that one point, the origin
O, is left unmoved by the operations of the group. There are only 32 such

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8

GROUP THEORY IN PHYSICS

%
"%
%

/

%
%
\

/
I

i r

\

'

~T

/

J

/

/

f

/

J

/

Jo

P

,,/

,,r

l/" x

p
e,

x

0

Figure 1.3: The rotation axes Ox, Oz, Oc and Od of the crystallographic point
group D4.
groups, all of which are finite. A complete description is given in Appendix C.
The only possible angles of rotation are 27r/n, where n - 2, 3, 4, or 6. (This
restriction on the value of n is a consequence of the translational symmetry
of a perfect crystal (cf. Chapter 7, Section 6). For a "quasicrystal", which
has no such translational symmetry, this restriction no longer applies, and so
it is possible to have other values of n as well, including, in particular, the
value n - 5.) It is convenient to denote a proper rotation through 27r/n about
an axis Oj by Cnj. The identity transformation may be denoted by E, so
that R ( E ) - 1, and improper rotations can be written in the form ICnj. As
an example, consider the crystallographic point group D4, the notation being
that of SchSnfliess (1923). D4 consists of the eight rotations:
E: the identity;
C2x, C2y, C2~" proper rotations through 7r about Ox, Oy, Oz respectively;

C4y, C-1.
ay proper rotations through 7r/2 about Oy in the right-hand and
left-hand screw senses respectively;

C2c, C2d" proper rotations through lr about Oc and Od respectively.
Here Ox, Oy, Oz are mutually orthogonal Cartesian axes, and Oc, Od are
mutually orthogonal axes in the plane Oxz with Oc making an angle of 7r/4
with both Ox and Oz, as indicated in Figure 1.3. The transformation matrices
are
R(E)

R(C2y)

=

=

[100]
0
0

1
0

0
1

,

[_1o o]
0
0

1
0

0
-1

[1 0 0]

R(C2~)=

,

R(C2z)

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0-1
0
0

=

0
-1

,

[_1 o 0]
0
0

-I
0

0
1

,


THE B A S I C F R A M E W O R K

a(c~)

C2~
C~y
C2z
C4y

%1
C2~
C2d

0
0

0
1

-1

0

1

0

0

-1
0

0
0

[[0 01]1

R(C4y)

E

9

0
1

E
E
C2~
C2y
C2z

C4y
C4~
C2c
C2d

C2~
C2~
E
C2z
C2y

C2d
C4~

C2y
C2y
C2z
E
C2x
C41
C4y
C2d

C4y1

C2c

C2c

,

R(C4~ )

=

0
0

C2z
C2z
C2y
C2x
g
C2d
C2c
C~ 1
C4y

=

C4~
C4y
C2d
C4y1
C2c

C2y
E
C2~
C2z

1
0

0

0

[[0 0 1]]
-1

R(C2d)

0
1

0-1
1
0

C~ ~
C{~
C2~
C4y
C2d
g
C2y
C2z
C2x

C2c
C2c
C{1
C2d
C4y
C2z
C2x
E
C2y

0
0

C2d
C2d
C4y
C2c
C4y1
C2x
C2z
C2y
E

Table 1.2: Multiplication table for the crystallographic point group D4.
The multiplication table is given in Table 1.2. This example will be used
to illustrate a number of concepts in Chapters 2, 4, 5 and 6.

(b)

Translations

Suppose now that Ox, Oy, Oz is a set of mutually orthogonal Cartesian axes
and O~x I, 01y I, 0 lz t is another set, obtained by first rotating the original set
about some axis through 0 by a rotation whose transformation matrix is
R(T), and then translating 0 to O / along a vector - t ( T ) without further
rotation. (In IR3 any two sets of Cartesian axes can be related in this way.)
Then Equation (1.2) generalizes to
r ' = R ( T ) r + t(T).

(1.5)

It is useful to regard the rotation and translation as being two parts of a single
coordinate transformation T, and so it is convenient to rewrite Equation (1.5)

as
r ' = {R(T) It(T) }r,
thereby defining the composite operator {R(T)It(T)}. Indeed, in the nonsymmorphic space groups (see Chapter 7, Section 6), there exist symmetry
operations in which the combined rotation and translation leave the crystal
lattice invariant without this being true for the rotational and translational
parts separately.
The generalization of Equation (1.4) can be deduced by considering the
two successive transformations r' = {R(T2)[t(T2)}r =- R(T2)r + t(T2) and
r ' = {R(T1)]t(T1)}ff- R(T1)r' + t(T1), which give
r" -- R(T1)R(T2)r + [R(T1)t (T2) + t(T1)].

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(1.6)

10

GRO UP THEORY IN PHYSICS

Thus the natural choice of the definition of the "product" T1T2 of two general
symmetry operations T1 and T2 is
{R(T~)It(T~)} = {R(T~)R(T2)I R(T1)t(T2) + t(T1)}.

(1.7)

This product always satisfies the group associative law of Equation (1.1).
As Equation (1.5) can be inverted to give
r = R(T)-lr '- R(T)-lt(T),
the inverse of {R(T)It(T)} may be defined by
{R(T)It(T)} -1 = { R ( T ) - I I - R ( T ) - I t ( T ) } .

(1.s)

It is easily verified that

{R(T1T2)It(T1T2)} - 1 = {R(T2)It(T2)}-I {R(T1)It(T~ )} -1,
the order of factors being reversed on the right-hand side.
It is sometimes convenient to refer to transformations for which t(T) = 0
as "pure rotations" and those for which R(T) = 1 as "pure translations".

3
(a)

T h e g r o u p of t h e S c h r S d i n g e r e q u a t i o n
The Hamiltonian operator

The Hamiltonian operator H of a physical system plays two major roles in
quantum mechanics (Schiff 1968). Firstly, its eigenvalues c, as given by the
time-independent SchrSdinger equation
He=

er

are the only allowed values of the energy of the system. Secondly, the time development of the system is determined by a wave function r
which satisfies
the time-dependent SchrSdinger equation

H e = ihor
Not surprisingly, a considerable amount can be learnt about the system by
simply examining the set of transformations which leave the Hamiltonian
invariant. Indeed the main function of group theory, as it is applied in physical
problems, is to systematically extract as much information as possible from
this set of transformations.
In order to present the essential features as clearly as possible, it will
be assumed in the first instance that the problem involves solving a "singleparticle" SchrSdinger equation. That is, it will be supposed that either the
system contains only one particle, or, if there is more than one particle involved, then they do not interact or their inter-particle interactions have been

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THE BASIC FRAMEWORK

11

treated in a Hartree-Fock or similar approximation in such a way that each

particle experiences only the average field of all of the others. Moreover, it
will be assumed that H contains no spin-dependent terms, so that the significant part of every wave function is a scalar function. For example, for an
electron in this situation, each wave function can be taken to be the product
of an "orbital" function, which is a scalar, with one of two possible spin functions, so that the only effect of the electron's spin is to double the "orbital"
degeneracy of each energy eigenvalue. (A development of a theory of spinors
along similar lines that enables spin-dependent Hamiltonians to be studied is
given, for example, in Chapter 6, Section 4, of Cornwell (1984).)
With these assumptions a typical Hamiltonian operator for a particle of
mass p has the form

h 2 02

02

c92

H(r) = - ~ - - ( _~-~o + ~
+ ~)+
uyOz
,~# ax"

V(r),

(1.10)

where V(r) is the potential field experienced by the particle. For example,
for the electron of a hydrogen atom whose nucleus is located at O,

h2 02

02

H(r) = - ~ p (0-~x2 + ~

02

+ ~z2) - e2/{x 2 + y2 +

z2}1/2.

(1.11)

In Equations (1.10) and (1.11) the Hamiltonian is written as H(r) to emphasize its dependence on the particular coordinate system O x y z .

(b)

T h e i n v a r i a n c e of t h e H a m i l t o n i a n

operator

Let H ( { R ( T ) I t ( T ) } r ) be the operator that is obtained from U(r) by substituting the components of r' - {R(T)[t(T)}r in place of the corresponding
components of r. For example, if H(r) is given by Equation (1.11), then
h2

02

02

02

H({R(T)]t(T)}r) = - ~ ( ~ z , 2 +~y,2 +~z,2 )-s

(1.12)

/ - / ( { R ( r ) l t ( r ) } r ) can then be rewritten so that it depends explicitly on r.
For example, in Equation (1.12), if T is a pure translation x p = x + tl, y~ =
y + t2, z' - z + ta, then

g2
H({R(T)It(T)}r)

so that

-

02

02

-~02)

-2-~(~x2__ + ~5y2 + Oz
--e2/{(x q- tl) 2 -t- (y + t2) 2 + (z q- t3)2} 1/2.

H ( { R ( T ) I t ( T ) } r ) :/: H(r),

whereas if T is a pure rotation about O, then a short algebraic calculation
gives
h2

02

02

S({R(T)lt(T)}r) = -~(-5~x2 + ~

02

+ --~) - e:/{x: + y: + z2} 1/:
Oz

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GROUP THEORY IN PHYSICS

12
and hence in this case

H({R(T)It(T)}r ) = H(r).
A coordinate transformation T for which

H({R(T)It(T)}r ) = H(r)

(1.13)

is said to leave the Hamiltonian "invariant". For the hydrogen atom the above
analysis merely explicitly demonstrates the intuitively obvious fact that the
system is invariant under pure rotations but not under pure translations.
The following key theorem shows how and why group theory plays such a

significant part in quantum mechanics.
T h e o r e m I The set of coordinate transformations that leave the Hamiltonian invariant form a group. This group is usually called "the group of the
Schr5dinger equation", but is sometimes referred to as "the invariance group
of the Hamiltonian operator".

Proof It has only to be verified that the four group axioms are satisfied.
Firstly, if the Hamiltonian is invariant under two separate coordinate transformations T1 and T2, then it is invariant under their product T1T2. (Invariance under T1 implies that H(r") = H(r'), where r " = {R(T1)It(T1)}r', and
invariance under T2 implies that H(r') = H(r), where r ' = {R(T2)It(T2)}r,
so that H ( r " ) = H(r), where, by Equation (1.7), r " = {R(T~T2)It(T1T2)}r).
Secondly, as noted in Section 2(b), the associative law is valid for all coordinate transformations. Thirdly, the identity transformation obviously leaves
the Hamiltonian invariant, and finally, as Equation (1.13) can be rewritten
as H(r') = H({R(T)It(T)}-lr'), where r' = {R(T)It(T)}r , if T leaves the
Hamiltonian invariant then so does T -1.
For the case of the hydrogen atom, or any other spherically symmetric
system in which V(r) is a function of Irl alone, the group of the SchrSdinger
equation is the group of all pure rotations in IR3.
(c)

The

scalar transformation

operators

P(T)

A "scalar field" is defined to be a quantity that takes a value at each point
in the space ] a 3 (in general taking different values at different points), the
value at a point being independent of the choice of coordinate system that
is used to designate the point. One of the simplest examples to visualize is

the density of particles. The concept is relevant to the present consideration
because the "orbital" part of an electron's wave function is a scalar field.
Suppose that the scalar field is specified by a function ~p(r) when the
coordinates of points of IR3 are defined by a coordinate system Ox, Oy, Oz,
and that the same scalar field is specified by a function r
p) when another
coordinate system Otx ~,O~y~,O~z~is used instead. If r and r ~ are the position

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THE BASIC FRAMEWORK

13

vectors of the same point referred to the two coordinate systems, then the
definition of the scalar field implies that

r

:

%b(r).

(1.14)

Now suppose that O'x', O'y', O'z' are obtained from Ox, Oy, Oz by a coordinate transformation T, so that r ' - {R(T)[t(T)}r.
Then Equation (1.14) can be written as
r
which provides

the function r
component of r
For example, if

= r

(1.15)

a concrete prescription for determining the function %9' from
namely that r
is the function obtained by replacing each
in ~(r) by the corresponding component of { R ( T ) I t ( T ) } - l r '.
r
= x2y 3 and T is the pure rotation of Equation (1.3), as

{R(T)]t(T)}-~r '

=

R ( T ) - l r ' = R:(T)r'

=

(x',y'cosO - z' sin0, y' sin0 + z'cos0),

then

r (r') = x'2(y ' cos0- z' sin 0)3.
It is very convenient in the following analysis to replace the argument r'
of ~' by r (without changing the functional form of ~'). Thus in the above

example
r (r) = x(y cos 0 - z sin 0) 3,
and Equation (1.15) can be rewritten as
r

= ~b({R(T)lt(T)}-~r).

(1.1.6)

As r
is uniquely determined from r
for the coordinate transformation T, ~' can be regarded as being obtained from ~ by the action of an
operator P(T), which is therefore defined by ~b'= P(T)~b, or, equivalently,
from Equation (1.16) by
(P(T)r

= r

The typography can be simplified without causing confusion by removing one
of the sets of brackets on the left-hand side, giving

P(T)~;(r) = r

(1.17)

These scalar transformation operators perform a particularly important role
in the application of group theory to quantum mechanics. Their properties
will now be established.
Clearly P(T1) = P(T2) only if T1 = T2. (Here P(TI) = P(T2) means that
P(T1)%b(r) = P(T2)r

for every function r
Moreover, each operator
P(T) is linear, that is

P(T){ar

+ be(r)} = aP(T)r

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+ bP(T)r

(1.18)


14

GRO UP THEORY IN PHYSICS

for any two functions r
and r
and any two complex numbers a and b,
as can be verified directly from Equation (1.17); (see Appendix B, Section 4).
The other major properties of the operators P(T) are most succinctly stated
in the following four theorems.
T h e o r e m II Each operator P(T) is a unitary operator in the Hilbert space
L 2 with inner product (r r defined by

/?/?/

(r ~b) =

o~ r (r)~b(r)

(1. 19)

dx dy dz,

where the integral is over the whole of the space IR 3, that is,
(P(T)r

P(T)r = (r r

(1.20)

for any two functions r and r of L2; (see Appendix B, Sections 3 and 4).

Proof With r '1 defined by r " - { R ( T ) ] t ( T ) } - l r , from Equations (1.17) and
(1.19)
(P(T)r P ( T ) r

=

/?/??
oo

However,

oo

r (r'l)r

'')

dx dy dz.

(1.21)

(x)

dx dy dz - J dx" dy" dz", where the Jacobian J is defined by
J = det

[OlOx" OzlOu"Ox/Oz"]
Oy/Ox" Oy/Oy" Oy/Oz"

Oz/Oz" Oz/Oy" Oz/Oz"

As r = R ( T ) r " + t(T), it follows that Ox/Ox"= R(T)11, Ox/Oy"= R(T)12
etc., so that J - det R(T) - =kl. In converting the right-hand side of Equation (1.21) to a triple integral with respect to x", y", z", there appears an odd
number of interchanges of upper and lower limits for an improper rotation,
whereas for a proper rotation there is an even number of such interchanges.
(For example, for spatial inversion I, x" - - x , y" -- - y , z" - - z , so the upper and lower limits are interchanged three times, while for a rotation through
7r about Oz the limits are interchanged twice.) Thus in all cases Equation
(1.21) can be written as

(P(T)r162

= / ? f ? / ? r162
oo

oo

dy" dz",

oo

from which Equation (1.20) follows immediately.
Theorem III

For any two coordinate transformations T1 and T2,

P(TIT2) = P(T1)P(T2).

(1.22)

Proof It is required to show that for any function r
P(TIT2)r
=
P(T1)P(T2)r
where in the right-hand side P(T2) acts first on r
and

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