Mathematical Methods for Physics and Engineering : A Comprehensive Guide

GROUP THEORY

if matrices are involved. In the notation in whichG={G 1 ,G 2 ,...,Gn}the order

of the group is clearlyn.

As we have noted, for the integers under addition zero is the identity. For

the group of rotations and reflections, the operation of doing nothing, i.e. the

null operation, plays this role. This latter identification may seem artificial, but

it is an operation, albeit trivial, which does leave the system in a physically

indistinguishable state, and needs to be included. One might add that without it

the set of operations would not form a group and none of the powerful results

we will derive later in this and the next chapter could be justifiably applied to

give deductions of physical significance.

In the examples of rotations and reflections mentioned earlier,•has been taken

to mean that the left-hand operation is carried out on the system that results

from application of the right-hand operation. Thus

Z=X•Y (28.4)

means that the effect on the system of carrying outZisthesameaswould

be obtained by first carrying outYand then carrying outX. The order of the

operations should be noted; it is arbitrary in the first instance but, once chosen,

must be adhered to. The choice we have made is dictated by the fact that most

of our applications involve the effect of rotations and reflections on functions of

space coordinates, and it is usual, and our practice in the rest of this book, to

write operators acting on functions to the left of the functions.

It will be apparent that for the above-mentioned group, integers under ordinary

addition, it is true that

Y•X=X•Y (28.5)

for all pairs of integersX,Y. If any two particular elements of a group satisfy

(28.5), they are said tocommuteunder the operation•; if all pairs of elements in

a group satisfy (28.5), then the group is said to beAbelian.Thesetofallintegers

forms an infinite Abelian group under (ordinary) addition.

As we show below, requirements (iii) and (iv) of the definition of a group

are over-demanding (but self-consistent), since in each of equations (28.2) and

(28.3) the second equality can be deduced from the first by using the associativity

required by (28.1). The mathematical steps in the following arguments are all

very simple, but care has to be taken to make sure that nothing that has not

yet been proved is used to justify a step. For this reason, and to act as a model

in logical deduction, a reference in Roman numerals to the previous result,

or to the group definition used, is given over each equality sign. Such explicit

detailed referencing soon becomes tiresome, but it should always be available if

needed.