Mathematical Methods for Physicists: A concise introduction - Site Map

ISOMORPHIC GROUPS
Table 12.4.
E 0 A 0 B 0 C 0
E 0 E 0 A 0 B 0 C 0
A 0 A 0 B 0 C 0 E 0
B 0 B 0 C 0 E 0 A 0
C 0 C 0 E 0 E 0 B 0
is, we make no a priori assumption about the signi®cance of the two elements of
our group. One of them must be the identity E, and we call the other X. Thus we
have
E 2 ˆ E; EX ˆ XE ˆ E:
Since each element appears once and only once in each row and column, the
group multiplication table takes the form:
E
X
E E X
X X E
We next consider some groups of operators that are isomorphic to G 2 . First,
consider the following two transformations of three-dimensional space into itself:
(1) the transformation E 0 , which leaves each point in its place, and
(2) the transformation R, which maps the point …x; y; z† into the point
…x; y; z†. Evidently, R 2 ˆ RR (the transformation R followed by R)
will bring each point back to its original position. Thus we have
…E 0 † 2 ˆ E 0 , RE 0 ˆ E 0 R ˆ RE 0 ˆ R; R 2 ˆ E 0 ; and the group multiplication
table has the same form as G 2 : that is, the group formed by the set of the two
operations E 0 and R is isomorphic to G 2 .
We now associate with the two operations E 0 and R two operators ^O E 0 and ^O R ,
which act on real- or complex-valued functions of the spatial coordinates …x; y; z†,
…x; y; z†, with the following e€ects:
^O E 0 …x; y; z† ˆ …x; y; z†;
^OR …x; y; z† ˆ …x; y; z†:
From these we see that
… ^O E 0† 2 ˆ ^O E 0; ^O E 0 ^O R ˆ ^O R ^O E 0 ˆ ^O R ; … ^O R † 2 ˆ ^O R :
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