Group theory in physics

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 precis

Group Theory in Physics

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97 98 99 00 01 02 EB 9 8 7 6 5 4 3 2 1

C o n t e n t s

T h e

1

2

B a s i c F r a m e w o r k 1

T h e c o n c e p t of a group 1

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 4

(a) R o t a t i o n s 5

(b) T r a n s l a t i o n s 9

T h e g r o u p of t h e Schr5dinger e q u a t i o n 10

(a) T h e H a m i l t o n i a n o p e r a t o r 10

(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 11

(c) T h e scalar t r a n s f o r m a t i o n o p e r a t o r s P ( T ) 12

T h e role of m a t r i x r e p r e s e n t a t i o n s 15

T h e 1 2 3 4 5 6 7 S t r u c t u r e of G r o u p s 19 Some e l e m e n t a r y considerations 19

Classes 21

I n v a r i a n t s u b g r o u p s 23

Cosets 24

F a c t o r g r o u p s 26

H o m o m o r p h i c a n d isomorphic m a p p i n g s 28

Direct p r o d u c t s a n d semi-direct p r o d u c t s of groups 31

Lie G r o u p s 35 1 Definition of a linear Lie g r o u p 35

2 T h e c o n n e c t e d c o m p o n e n t s of a linear Lie group 40

3 T h e c o n c e p t of c o m p a c t n e s s for linear Lie groups 42

4 I n v a r i a n t i n t e g r a t i o n 44

R e p r e s e n t a t i o n s o f G r o u p s - P r i n c i p a l Ideas 1 2 3 4 5 47 Definitions 47

E q u i v a l e n t r e p r e s e n t a t i o n s 49

U n i t a r y r e p r e s e n t a t i o n s 52

R e d u c i b l e a n d irreducible r e p r e s e n t a t i o n s 54

Schur's L e m m a s a n d 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 rep- r e s e n t a t i o n s 57

iii

iv GROUP T H E O R Y IN PHYSICS

6 C h a r a c t e r s 59

R e p r e s e n t a t i o n s o f G r o u p s - D e v e l o p m e n t s 1 2 3 65 P r o j e c t i o n operators 65

Direct p r o d u c t representations 70

T h e Wigner-EcLurt T h e o r e m for groups of coordinate transfor- m a t i o n s in ] R 3 73

T h e W i g n e r - E c k a r t T h e o r e m generalized 79

Representations of direct p r o d u c t groups 83

Irreducible representations of finite Abelian groups 85

Induced representations 86

G r o u p T h e o r y in Quantum Mechanical Calculations 93 1 T h e solution of the SchrSdinger equation 93

2 Transition probabilities and selection rules 97

3 T i m e - i n d e p e n d e n t p e r t u r b a t i o n t h e o r y 100

Crystallographic Space Groups 1 2 3 103 T h e Bravais lattices 103

T h e cyclic b o u n d a r y conditions 107

Irreducible representations of t h e 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 s y m m o r p h i c space groups 121

(a) F u n d a m e n t a l t h e o r e m on irreducible representations of s y m m o r p h i c space groups 121

(b) Irreducible representations of the cubic space groups O~, O~ and O 9 126

Consequences of the f u n d a m e n t a l theorems 129

(a) Degeneracies of eigenvalues and the s y m m e t r y of e(k) 129 (b) Continuity and compatibility of the irreducible repre- sentations of G0(k) 131

(c) Origin and orientation dependence of the s y m m e t r y la- belling of electronic s t a t e s 134

The R o l e o f L i e A l g e b r a s 1 2 3 4 5 135 "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 t h a t correspond to general linear Lie groups 145 (a) T h e existence of a real Lie a l g e b r a / : for every linear Lie group G 145

(b) T h e relationship of t h e real Lie a l g e b r a / : to the one- p a r a m e t e r subgroups of G 148

C O N T E N T S v

The Relationships b e t w e e n Lie Groups and Lie Algebras E x -

I n t r o d u c t i o n 153

S u b a l g e b r a s of Lie algebras 153

H o m o m o r p h i c a n d isomorphic m a p p i n g s of Lie algebras 154

R e p r e s e n t a t i o n s of Lie algebras 160

T h e adjoint r e p r e s e n t a t i o n s of Lie algebras a n d linear Lie groups168 Direct s u m of Lie algebras 171

10 The T h r e e - d i m e n s i o n a l R o t a t i o n Groups 1 2 3 4 5 175 Some p r o p e r t i e s reviewed 175

T h e class s t r u c t u r e s of SU(2) a n d SO(3) 176

Irreducible r e p r e s e n t a t i o n s of t h e Lie algebras su(2) a n d so(3) 177 R e p r e s e n t a t i o n s of t h e Lie groups SU(2), SO(3) a n d 0 ( 3 ) 183

Direct p r o d u c t s of irreducible r e p r e s e n t a t i o n s a n d t h e Clebsch- G o r d a n coefficients 186

Applications to a t o m i c physics 189

11 The Structure of S e m i - s i m p l e Lie Algebras 1 2 3 4 5 6 7 8 9 10 193 An outline of t h e p r e s e n t a t i o n 193

T h e Killing form a n d C a r t a n ' s criterion 193

Complexification 198

T h e C a r t a n s u b a l g e b r a s and roots of semi-simple complex Lie algebras 200

P r o p e r t i e s of r o o t s of semi-simple complex Lie algebras 207

T h e r e m a i n i n g c o m m u t a t i o n relations 213

T h e simple roots 218

T h e Weyl canonical form of L 223

T h e Weyl group o f / : 224

Semi-simple real Lie algebras 228

12 R e p r e s e n t a t i o n s of S e m i - s i m p l e Lie Algebras 1 2 3 4 235 Some basic ideas 235

T h e weights of a r e p r e s e n t a t i o n 236

T h e highest weight of a r e p r e s e n t a t i o n 241

T h e irreducible r e p r e s e n t a t i o n s o f / : - A2, t h e complexification of s = su(3) 245

Casimir o p e r a t o r s 251

13 S y m m e t r y s c h e m e s for the elementary particles 255 1 L e p t o n s a n d h a d r o n s 255

2 T h e global internal s y m m e t r y group SU(2) a n d isotopic s p i n 256

3 T h e global internal s y m m e t r y group SU(3) a n d strangeness 259

vi GROUP T H E O R Y IN P H Y S I C S

1 Definitions 271

2 Eigenvalues and eigenvectors 275

B V e c t o r S p a c e s 1 2 3 4 5 6 7 279 The concept of a vector space 279

Inner product spaces 282

Hilbert spaces 286

Linear operators 288

Bilinear forms 292

Linear functionals 294

Direct product spaces 295

C C h a r a c t e r T a b l e s f o r 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 2 9 9 D P r o p e r t i e s o f 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 3 1 9 1 The simple complex Lie algebra Al, l >_ 1 319

2 The simple complex Lie algebra Bz, l > 1 320

3 The simple complex Lie algebra Cl, 1 > 1 322

4 The simple complex Lie algebra D1, 1 > 3 (and the semi-simple complex Lie algebra D2) 324

P r e f a c e

ace to my three-volume work Group

garded by physicists as merely providing a very valuable tool for the eluci- dation of the symmetry aspects of physical problems However, recent de- velopments, 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 ever- growing 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 commu- nication 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 intro- duction 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 daunt- ing 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

(1984)"), although some new material has been included The intention has been to concentrate on introducing and describing in detail the most impor- tant 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 cover- age 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

viii GROUP T H E O R Y IN P H Y S I C S

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 Exam- ples 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 appli- cations, 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

To my wife Elizabeth and my daughters

Rebecca and Jane

This Page Intentionally Left Blank

W i t h 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 t h a t 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

are satisfied:

"group" if the following four "group axioms"

(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 multipli- cation 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

( T T ' ) 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 t h a t the product

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 ex- amples 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 t h a t the group multiplication operation need not have any connection with ordinary multiplication, take G to be the set of all real num- bers 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)

[~

M6 = - 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

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, M21 = M2, M31 = M3, M~ -1 = M4,

M 5 1 = M6, M61 = M5, M~-i = MT, 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 m a t r i x 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 t h a t these sets do form groups Consider first the set U ( N ) As (ulu2) t = u2u 1 and (ulu2) t t -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 S U ( N ) the same considerations apply, but in addition if ul and u2 both have determinant 1, Equation (A.4) shows t h a t 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 t h a t is the case for u

The set of groups SU(N) is particularly important in theoretical physics SU(2) is intimately related to angular m o m e n t u m 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 S O ( 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 t h a t 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 S U ( N ) respectively t h a t consist only of real matrices (All that has to

be observed to supplement the arguments given in Example IV is t h a t the product of any two real matrices is real, t h a t 1N is real, and t h a t the inverse

of a real m a t r i x 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

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 non- commuting pairs of products of Example III which makes that group non- Abelian 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 multiplica- tion 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 pre- cedes 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 o f 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

IR 3, as most applications in atomic, molecular and solid state physics involve only transformations of this type

THE BASIC FRAMEWORK 5

origin O t h a t 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 T h e n 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

(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 ~

y t _ y c o s O + z s i n O ,

zt = - y s i n O + z c o s O ,

Figure 1.2: The plane containing the axes Oy, Oz, Oy ~ and Oz ~ corresponding

to the rotation of Figure 1.1

so t h a t

[1 o o 1

R ( T ) = 0 c o s 0 s i n 0 ( 1 3 )

0 - s i n 0 cos0 The m a t r i x R ( T ) obeys the orthogonality condition R ( T ) = R ( T ) -~ be- cause rotations leave invariant the length of every position vector and the angle between every pair of position vectors, t h a t is, they leave invariant the scalar product r l.r2 of any two position vectors (Indeed the name "or- thogonal" stems from the involvement of such matrices in the transforma- tions being considered here between sets of orthogonal axes.) The proof t h a t

R ( T ) is orthogonal depends on the fact that rl.r2 can be expressed in ma- trix form as r l r 2 Then, if r~ = R ( T ) r l and r~ = R ( T ) r 2 , it follows t h a t

The simplest example of an improper rotation is the spatial inversion op- eration 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,

(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 con- sidered to be the product of the spatial inversion operator I with a proper rotation For example, for the reflection in the Oyz plane

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 "crys- tallographic 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

8 GROUP THEORY IN PHYSICS

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;

C-1

left-hand screw senses 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

C4~ C2d C4y C2c E C2y C2x C2z

C2c C4~ C2d C~ 1 C2~ C2z E C2y C2d C4y 1 C2c C4y C2z C2x 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

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 IR 3 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 non- symmorphic 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 ( T 1 ) ] t ( T 1 ) } f f - R(T1)r' + t(T1), which give

r" R(T1)R(T2)r + [R(T1)t (T2) + t(T1)] (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 ) - l r ' - R ( T ) - l t ( 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 T h e g r o u p o f t h e S c h r S d i n g e r e q u a t i o n (a) The H a m i l t o n i a n 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

H e = er are the only allowed values of the energy of the system Secondly, the time de- velopment 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 "single- particle" SchrSdinger equation That is, it will be supposed that either the

system contains only one particle, or, if there is more than one particle in- volved, then they do not interact or their inter-particle interactions have been

T H E B A S I C F R A M E W O R K 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 sig- nificant 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 func- tions, 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 0 2 c92

H ( r ) = - ~ - - ( _~-~o + ~ + ~ ) + V(r), (1.10)

Oz ,~# ax" uy-

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,

Let H ( { R ( T ) I t ( T ) } r ) be the operator t h a t is obtained from U ( r ) by substi-

tuting 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

h 2 02 02 02

/ - / ( { 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~ =

12 GROUP THEORY IN PHYSICS

and hence in this case

T h e o r e m I The set of coordinate transformations that leave the Hamilto- nian 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 trans- formations T1 and T2, then it is invariant under their product T1T2 (Invari- ance 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 coor- dinate 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 IR 3

( c ) T h e s c a l a r t r a n s f o r m a t i o n o p e r a t o r s 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 IR 3 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

THE BASIC FRAMEWORK 13

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

= (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

of the sets of brackets on the left-hand side, giving

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

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 I I Each operator P(T) is a unitary operator in the Hilbert space

L 2 with inner product (r r defined by

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

/?/??

( P ( T ) r P ( T ) r = r ( r ' l ) r '') dx dy dz

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

J = det Oy/Ox" Oy/Oy" Oy/Oz"

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

(1.21)

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 Equa- tion (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 up- per 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 dy" dz",

o o o o o o

from which Equation (1.20) follows immediately

T h e o r e m I I I 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

THE BASIC F R A M E W O R K 15

the last equality being a consequence of the fact that r is

by definition the function obtained from r by simply replacing the compo- nents of r by the components of {R(T1)[t(T1)}-lr Thus, on using Equation

(1.9),

T h e o r e m I V The set of operators P(T) that correspond to the coordinate transformations T of the group of the Schr5dinger equation forms a group that is isomorphic to the group of the Schr5dinger equation

orem, may be taken to specify the group multiplication operation, so that the associative law of axiom (b) is satisfied The previous theorem then implies that group axiom (a) is fulfilled, and with P(E) being the identity operator it also implies that the inverse operator P(T) -1 may be defined by

16 GRO UP T H E O R Y IN PHYSICS

D e f i n i t i o n Representation of a group

If each element T of a group G can be assigned a non-singular d x d matrix F(T) contained in a group of matrices having matrix multiplication as its group multiplication operation in such a way that

for every pair of elements T1 and T2 of G, then this set of matrices is said to provide a d-dimensional "representation" r of G

E x a m p l e I A representation of the crystallographic point group Da

The group D4 introduced in Example III of Section2 has the following two- dimensional representation:

F(C2d) - Ms,

where M1, M 2 , a r e the 2 • 2 matrices defined in Example III of Section 1 That Equation (1.25) is satisfied can be verified simply by comparing Tables 1.1 and 1.2

It will be shown in Chapter 4 that every group has an infinite number

of different representations, but they are derivable from a smaller number

of basic representations, the so-called "irreducible representations" A finite group has only a finite number of such irreducible representations that are essentially different

The representations of the group of the Schrbdinger equation are of partic- ular interest The intimate connection between them and the eigenfunctions

of the time-independent Schrbdinger equation is provided by the notion of

"basis functions" of the representations

D e f i n i t i o n Basis functions of a group of coordinate transformations G

A set of d linearly independent functions r (r), r Cd(r) forms a basis for a d-dimensional representation I' of ~ if, for every coordinate transforma- tion T of G,

d P(T)~bn(r) E r ( T ) r ~ m ( r ) , n - 1, 2 , , d (1.26)

m - - 1 The function Cn(r) is then said to "transform as the n t h row" of the repre- sentation r

The definition implies that not only is each function P(T)~bn(r) required

to be a linear combination of r r Cd(r), but the coefficients are required to be equal to specified matrix elements of F(T) The rather unusual ordering of row and column indices on the right-hand side of Equation (1.26)

THE BASIC FRAMEWORK 17

ensures the consistency of the definition for every product TIT2, for, according

E x a m p l e I I Some basis functions of the crystallographic point group D4

The functions r (r) = x, r = z provide a basis for the representation F

of D4 that has been constructed in Example I above, as can be verified by inspection (This set has been deduced by a method that will be described in detail in Chapter 5, Section 1.)

T h e o r e m I The eigenfunctions of a d-fold degenerate eigenvalue c of the time-independent SchrSdinger equation

H ( r ) r = er form a basis for a d-dimensional representation of the group of the Schr5dinger equation ~

tions of H ( r ) with eigenvalue e, so that

H ( r ) r = er n - 1, 2 , , d, and any other eigenfunction of H(r) with eigenvalue e is a linear combina- tion of r (r), r Cd(r) For any transformation T of the group of the Schr5dinger equation, Equation (1.23) implies that

18 GRO UP THEORY IN PHYSICS

At this stage the F(T)mn are merely a set of coefficients with the m, n and T dependence explicitly displayed For each T the set F(T)mn can be arranged

to form a d • d matrix F(T) It will now be shown that

From Equation (1.27), with T replaced by T1, T2 and TIT2 in turn,

be shown that the familiar categorization of electronic states of an atom into

"s-states", "p-states", "d-states" etc is actually just a special case of this type

of description More precisely, every s-state eigenfunction is a basis function

of particular representation of the group of rotations in three dimensions, the p-state eigenfunctions are basis functions of another representation of that group, and so on

Having established a prima facie case that groups and their representations play a significant role in the quantum mechanical study of physical systems, the next chapters will be devoted to a detailed examination of the structure of groups and the theory of their representations So far only a brief indication has been given of what can be achieved, but the ensuing chapters will show that the group theoretical approach is capable of dealing with a very wide range of profound and detailed questions

Chapter 2

T h e S t r u c t u r e of G r o u p s

1 S o m e e l e m e n t a r y c o n s i d e r a t i o n s

This section will be devoted to some immediate consequences of the definition

of a group t h a t was given in Chapter 1, Section 1 As many statements will

be made about the contents of various sets, it is convenient to introduce an abbreviated notation in which "T E S" means "the element T is a member of the set S" and "T ~ S " m e a n s "the element T is not a member of the set S" The associative law of Equation (1.1) implies that in any product of three

or more elements no ambiguity arises if the brackets are removed completely Moreover, they can be inserted freely around any chosen subset or subsets

of elements in the product, provided of course that the order of elements is unchanged The proof t h a t

E It will be shown in Section 4 t h a t if g and s are the orders of ~ and $ respectively, then g/s must be an integer

19

E x a m p l e I Subgroups of the crystallographic point group D4

The group D4 defined in Chapter 1, Section 2 has the following subgroups: (a) s = 1 (i.e g/s = 8): {E};

(b) s = 2 (i.e g/s = 4): {E, C2x}, {E, C2y}, {E, C2z}, {E, C2c}, {E, C2d};

(c) s = 4 (i.e g/s = 2)" {E, C2x, C2y, C2z}, {E, C2y, C4y, c4yl}, {E, C2y,

c2~, c2d};

(d) s = 8 (i.e g / s = 1)" {E, C2x, C2y, C2z, C4y, c ~ l , c2c, C2d}

The following theorem displays an interesting property of multiplication

in a group

T h e o r e m II For any fixed element T' of a group G, the sets {T'T; T E G}

and {TT'; T E ~} both contain every element of G once and only once (Here

{T'T; T E G} denotes the set of elements T ' T where T varies over the whole of

G For example, in the special case in which G is a finite group of order g with elements T1, T 2 , , Tg and T' = Tn, this set consists of TnT1, TnT2, , TnTg

The interpretation of { T T ' ; T E G} is similar The theorem is often called the

"Rearrangement Theorem", as it asserts that each of the two sets {T'T; T E

G} and {TT'; T E G} merely consists of the elements of G rearranged in order.)

Proof An explicit proof will be given for the set {T'T; T E G}, the proof for the other set being similar

If T" is any element of G, then with T defined by T = ( T ' ) - I T '' it follows that T t T = T" Thus {T'T; T E G} certainly contains every element of G at least once Now suppose that {T'T; T C G} contains some element of G twice (or more), i.e for some T1, T2 C G, T'T1 = T'T2, but T1 =fi T2 However, these statements are inconsistent, for premultiplying the first by (T') -1 gives T1 = T2, so no element of 6 appears more than once in {T'T; T E G}

The Rearrangement Theorem implies that in the multiplication table of a finite group every element of the group appears once and only once in every

T H E S T R U C T U R E OF G R O U P S 21

row, and once and only once in every column This provides a useful check

on the computation of the multiplication table Tables 1.1 and 1.2 exemplify these properties

Whereas in ordinary everyday language the word "class" is often synonymous with the word "set", in the context of group theory a class is defined to be a special type of set In fact it is a subset of a group having a certain property which causes it to play an important role in representation theory, as will be shown in Chapter 4

As a preliminary it is necessary to introduce the idea of "conjugate ele- ments" of a group

D e f i n i t i o n Conjugate elements

An element T ~ of a group G is said to be "conjugate" to another element T

of ~ if there exists an element X of G such that

If T ~ is conjugate to T, then T is conjugate to T t, as Equation (2.2) can be rewritten as T = X-1TI(X-1) -1 Moreover, if T, T p and T" are three elements of G such that T ~ and T" are both conjugate to T, then T t

is conjugate to T" This follows because there exist elements X and Y of G such that T ~ = X T X -1 and T " = Y T Y -1, so that T ~ = X ( Y - 1 T " Y ) X -1 = ( X Y - 1 ) T " ( X Y - 1 ) -1 (by Equation (2.1)), which has the form of Equation

(2.2) as X Y -1 E G It is therefore permissible to talk of a set of mutually

conjugate elements

D e f i n i t i o n Class

A class of a group ~ is a set of mutually conjugate elements of G (For extra precision this is sometimes called a "conjugacy class" )

A class can be constructed from any T E G by forming the set of products

X T X -1 for every X E G, retaining only the distinct elements This class

contains T itself as T = E T E -1

E x a m p l e I Classes of the crystallographic point group D4

For the group D4 this procedure when applied to C2~ gives (on using Table 1.2)"

(a) Every element of a group 6 is a member of some class of 6

(b) No element of G can be a member of two different classes of ~

(c) The identity E of G always forms a class on its own

T h e o r e m I I

its own

If (~ is an Abelian group, every element of G forms a class on

P r o o f For any T and X of an Abelian group G

X T X -1 = X X - 1 T = E T = T,

so T forms a class on its own

T h e o r e m I I I If G is a group consisting entirely of pure rotations, no class

of {~ contains both proper and improper rotations Moreover, in each class

of proper rotations all the rotations are through the same angle Similarly,

in each class of improper rotations the proper parts are all through the same angle

P r o o f If T and T' are two pure rotations in the same class, Equations (1.4) and (2.2) imply that R ( T ' ) = R(X)R(T)R(X) -~, s o that

det R ( T ' ) = det I t ( T ) (2.3) and

tr R ( T ' ) = tr R ( T ) (2.4)

THE S T R U C T U R E OF GROUPS 23

(see Appendix A) Equation (2.3) shows that T and T' are either both proper

or are both improper Moreover, for any proper rotation T through an angle

0 (in the right- or left-hand screw sense) tr R ( T ) = 1 + 2 cos 0 (cf Equation (10.4)) Equation (2.4) then implies t h a t all proper rotations in a class are through the same angle 0 Finally, by expressing any improper rotation T as the product of the spatial inversion operator I with a proper rotation through

an angle 0, it follows t h a t t r R ( T ) = - { 1+ 2 cos 0}, so all proper parts involved

in a class are through the same angle 0

It should be noted t h a t the converse of the last theorem is not necessarily true, in t h a t there is no requirement for all rotations of the same type to be

in the same class Indeed, in the above example of the point group D4, the proper rotations C2x and C2y are in different classes, even though they are rotations through the same angle 7r

Invariant subgroups are sometimes called "normal subgroups" or "normal divisors" Because of the occurrence of the same forms in Equation (2.2) and Condition (2.5), there is a close connection between invariant subgroups and classes

T h e o r e m I A subgroup $ of a group G is an invariant subgroup if and only

if $ consists entirely of complete classes of G

Proof Suppose first t h a t $ is an invariant subgroup of G Then if S is any member of ,S and T is any member of the same class of ~ as S, by Equation

(2.2) there exists an element X of G such t h a t T - X S X -1 Condition (2.5)

then implies t h a t T E S, so the whole class of G containing S is contained in

S

Now suppose t h a t $ consists entirely of complete classes of G, and let S be

any member of S Then the set of products X S X -1 for all X E G forms the class containing S, which by assumption is contained in S Thus X S X -1 E S

for all S E $ and X E ~, so $ is an invariant subgroup of G

This theorem provides a very easy method of determining which of the

24 GROUP THEORY IN PHYSICS

subgroups of a group are invariant when the classes have been previously calculated

E x a m p l e I Invariant subgroups of the crystallographic point group D4

For the crystallographic point group D4 it follows immediately from the lists

of subgroups and classes given in Sections i and 2 that the invariant subgroups are {E}, {E, C2y}, {E, C2x, C2y, C2z}, {E, C2y, C4~, C ~ 1 }, {E, C2y, C2c, C2d},

and D4 itself (The subgroup {E, C2x} is not an invariant subgroup as C2x is part of a class {C2~, C2z } that is not wholly contained in the subgroup The same is true of {E, C2c} and {E, C2d }.)

For every G the trivial subgroups {E} and ~ are both invariant subgroups

D e f i n i t i o n Coset

Let S be a subgroup of a group G Then, for any fixed T E G (which may or may not be a member of S), the set of elements ST, where S varies over the whole of $, is called the "right coset" of S with respect to T, and is denoted

by ST Similarly, the set of elements TS is called the "left coset" of 8 with respect to T and is denoted by T8

In particular, if ,S is a finite subgroup of order s with elements $1, $2, .,

Ss, then S T is the set of s elements S1T, S2T, , SsT, and T,S is the set of s elements TS1, T S 2 , , TSs In the following discussions two sets will be said

to be identical if they merely contain the same elements, the ordering of the elements within the sets being immaterial

E x a m p l e I Some cosets of the crystallographic point group D4

Let G be D4 and let S = {E, C2x} Then from Table 1.2 the right cosets are

and the left cosets are

T H E S T R U C T U R E O F G R O U P S 25

This example shows that the right and left cosets S T and T S formed from the same element T C G are not necessarily identical The properties of cosets are summarized in the following two theorems The first theorem is stated for right cosets, but every statement applies equally to left cosets It is worth while checking t h a t the above example of the point group D4 does satisfy all the assertions of this theorem

T h e o r e m I

(a) If T E S, then S T = S

(b) If T ~ S, then S T is not a subgroup of G

(c) Every element of G is a member of some right coset

(d) Any two elements S T and S ' T of S T are different, provided that S # S'

In particular, if S is a finite subgroup of order s, S T contains s different elements

(e) Two right cosets of S are either identical or have no elements in common (f) If T ' E S T , then S T ' = S T

(g) If G is a finite group of order g and S has order s, then the number of distinct right cosets is g / s

T C S Thus if T 9~ $, S T cannot be a subgroup of ~

(c) For any T E S, as T = E T and E E 8, it follows that T E S T

(d) Suppose that S T SPT and S =/= S' Post-multiplying by T -1 gives

S - S', a contradiction

(e) Suppose t h a t S T and S T p are two right cosets with a common element

It will be shown that S T = S T p Let S T - S~T ' be the common element

of S T and S T ' Here S, S' E S Then T ' T -1 = ( S ' ) - I S , so T ' T -1 c S ,

and hence by (a) S ( T ' T -1) = S As $ ( T ' T -1) is the set of elements of the form S T t T -1, the set obtained from this by post-multiplying each member by T consists of the elements S T ~, that is, it is the coset S T t

Thus S T = S T '

(f) As in (c), T' E S T ' I f T C S T ' then S T ' and S T have a c o m m o n element and must therefore be identical by (e)

26 G R O U P T H E O R Y I N P H Y S I C S

(g) Suppose t h a t there are M distinct right cosets of S By (d) each contains

s different elements, so the collection of distinct cosets contains M s

different elements of G But by (c) every element of G is in this collection

of distinct cosets, so M s - g

The property (f) is particularly important It shows t h a t the same coset

is formed starting from a n y member of the coset All members of a coset therefore appear on an equal footing, so t h a t a n y member of the coset can be taken as the "coset representative" t h a t labels the coset and from which the coset can be constructed For example, for the right coset {C4y, C2d} of the point group D4, the coset representatives could equally well be chosen to be

Cay or C2d As the number of distinct right cosets is necessarily a positive integer, property (g) demonstrates t h a t s must divide g, as was mentioned in Section 1

T h e o r e m II

identical (i.e

subgroup of G

The right and left cosets of a subgroup $ of a group G are

S T = T S for all T E G) if and only if S is an i n v a r i a n t

P r o o f Suppose t h a t S is an invariant subgroup It will be shown t h a t if

T t E S T then T ~ E T S (A similar argument proves t h a t if T ~ E T S then

T ~ E S T , so, on combining the two, it follows t h a t S T = T S ) If T ~ E S T

there exists an element S of S such t h a t T ~ = S T Then T - 1 T ~ - T - 1 S T ,

which is a member of S as S is an invariant subgroup Thus T - 1 T ~ E S, so

T ' = T ( T - 1 T ~) must be a member of T S

Now suppose t h a t S T - T ? for every T E G This implies t h a t for any

S E S and any T E G there exists an S ~ E S such t h a t T S = S~T, so

T S T -1 = S ' and hence T S T -1 E S Thus S is an invariant subgroup of G

Of course, in the above example concerning the point group D4, the sub- group S - { E, C2x } was carefully chosen so as n o t to be an invariant subgroup,

in order to demonstrate t h a t right and left cosets are not always identical

Let S be an i n v a r i a n t subgroup of a group G Each right coset of S can be considered to be an "element" of the set of distinct right cosets of S, the internal structure of each coset now being disregarded W i t h the following definition of the product of two right cosets, the set of cosets then forms a group called a "factor group"

D e f i n i t i o n Product o f right cosets

The product of two right cosets ST1 and S T 2 of an i n v a r i a n t subgroup S is defined by

S TI S T2 - S(T1T2) (2.6)

THE S T R U C T U R E OF GROUPS 27

Proof of consistency It will be shown t h a t Equation (2.6) provides a mean- ingful definition, in that, if alternative coset representatives are chosen for the cosets on the left-hand side of the equation, then the coset on the right-hand side remains unchanged Suppose that T~ and T~ are alternative coset repre- sentatives for ST1 and 8T2 respectively, so that T~ E ST1 and T~ c ST2 It has to be proved that S(T~T~) = S(T1T2) As T~ E ST1 and T~ C ST2, there exist S , S ' E S such that T~ = ST1 and T~ = S'T~ Then T{T~ = STIS'T2

But T1S r C TIS, so, as S is an invariant subgroup, T1S r E ST1 Consequently there exists an S" e S such t h a t T I S ' = S"T1 Then T~T~ = (SS")(TIT2),

so t h a t T~T~ E S(TIT2) and hence, by property (f) of the first theorem of Section 4, $(T~T~)= S(TIT2)

T h e o r e m I The set of right cosets of an invariant subgroup 8 of a group forms a group, with Equation (2.6) defining the group multiplication opera- tion This group is called a "factor group" and is denoted by ~ / S

Proof It has only to be verified that the four group axioms are satisfied (a) By Equation (2.6), the product of any two right cosets of S is itself a right coset of 8 and is therefore a member of ~ / 8

(b) The associative law is valid for coset multiplication because, if ST, S T '

and S T " are any three right cosets,

The coset S ( T -1) is a member of Q / $ as T -1 C G

If {~ is a finite group of order g and S has order s, part (g) of the first theorem of Section 4 shows t h a t there are g/s distinct right cosets Thus G/S

is a group of order g/s with elements S T I , $ T 2 , ,STs, (T1, T 2 , ,T8 being

a set of coset representatives) As S itself is one of the cosets, one can take

TI = E

28 GROUP THEORY IN PHYSICS

S E 8C2~

E x a m p l e I A factor group formed from the crystallographic point group D4

Let G be Da and let S = {E, C2y}, which is an invariant subgroup of G (see Example I of Section 3) Then G/S is a group of order 4 with elements

$ E = 8C2y = {E, C2y},

Let G and G ~ be two groups A "mapping" r of G onto ~ is simply a rule by which each element T of G is assigned to some element T ~ = r of G', with every element of G t being the "image" of at least one element of G If r is a

one-to-one mapping, that is, if each element T t of ~, is the image of only one

element T of G, then the inverse mapping r of G ~ onto G may be defined

by r = T if and only if T' = r

D e f i n i t i o n Homomorphic mapping of a group ~ onto a group ~'

If r is a mapping of a group G onto a group G t such that

r162 = r T~) (2.7) for all T1, T2 E ~, then r is said to be a "homomorphic" mapping

On the right-hand side of Equation (2.7) the product of T1 with T2 is evaluated using the group multiplication operation for G, whereas on the left-hand side the product of r with r is obtained from the group multiplication operation for G p Although these operations may be different, there is no need to introduce any special notations to distinguish between them, because the relevant operation can always be deduced from the context and there is really no possibility of confusion

E x a m p l e I A homomorphic mapping of the point group D4

Let ~ be D4 and let G p be the group of order 2 with elements +1 and - 1 ,

T H E S T R U C T U R E OF GROUPS 29 with ordinary multiplication as the group multiplication operation Then

is a homomorphic mapping of G onto G ~, as may be confirmed by examination

of Table 1.2 For example, r162 ( + 1 ) ( - 1 ) = - 1 , while Table 1.2 gives r - r 1) - - 1

Clearly, if g and g~ are the orders of ~ and G ~ respectively, then g g~ Actually, the First Homomorphism Theorem, which will be proved shortly,

implies that if g and gr are both finite, then g/g~ must be an integer One

major example of a homomorphic mapping has already been encountered in the concept of a representation of a group Indeed, the definition in Chapter

1, Section 4 can now be rephrased as follows:

D e f i n i t i o n Representation of a group G

If there exists a homomorphic mapping of a group G onto a group of non-

singular d • d matrices F(T) with matrix multiplication as the group multi- plication operation, then the group of matrices F(T) forms a d-dimensional representation F of ~

There is no requirement in the definition of a homomorphic mapping that the mapping should be one-to-one However, as such mappings are particu- larly important, they are given a special name:

D e f i n i t i o n Isomorphic mapping of a group G onto a group G ~

If r is a one-to-one mapping of a group ~ onto a group G ~ of the same order

such that

r )r r T1, T2 E {~, then r is said to be an "isomorphic" mapping

In the case of representations, if the homomorphic mapping is actually isomorphic, then the representation is said to be "faithful"

Clearly, if r is an isomorphic mapping of G onto G r, then the inverse mapping r is an isomorphic mapping of G r onto ~ (There is no analogous result for general homomorphic mappings, as r is only well defined when r

is a one-to-one mapping.)

Although two isomorphic groups may differ in the nature of their elements, they have the same structure of subgroups, cosets, classes, and so on Most important of all, isomorphic groups necessarily have identical representations The following theorem clarifies various aspects of homomorphic mappings

As it is the first of a series of such theorems, it is often called the "First Homomorphism Theorem", but the others in the series will not be needed in this book