Mathematical Methods for Physicists: A concise introduction - Site Map

SOME SPECIAL GROUPS
Table 12.7.
~E ~A ~B ~C ~D ~F
~E ~E ~A ~B ~C ~D ~F
~A ~A ~B ~E ~D ~F ~C
~B ~B ~E ~A ~F ~C ~D
~C ~C ~F ~D ~E ~B ~A
~D ~D ~C ~F ~A ~E ~B
~F ~F ~D ~C ~B ~A ~E
Similarly, re¯ection about axis OP is equivalent to a rotation of 1808 followed by
a re¯ection of the x-axis:
p
1=2 3 =2
~F ˆ R OP …1808† ˆ ~C ~A ˆ p
!:
3 =2 1=2
The group multiplication table is shown in Table 12.7. We have constructed a sixelement
non-Abelian group and a 2 2 irreducible matrix representation of it.
Our group is known as D 3 in crystallography, the dihedral group with a threefold
axis of symmetry.
One-dimensional unitrary group U…1†
We now consider groups with an in®nite number of elements. The group element
will contain one or more parameters that vary continuously over some range so
they are also known as continuous groups. In Example 12.7, we saw that the
complex numbers (1; i; 1; i† form a cyclic group of order 3. These group elements
may be interpreted as successive 908 rotations in the complex plane
…0;=2;; 3=2†, and so they may be written as e i' with ' ˆ 0, =2, , 3=2. If
' is allowed to vary continuously over the range ‰0; 2Š, then we will have, instead
of a four-member cyclic group, a continuous group with multiplication for the
composition rule. It is straightforward to check that the four group axioms are all
Figure 12.5.
449