452
M. Group Theory
D(a)D(e) by the fundamental property (M.78) of representation matrices. On
the other hand, ae = a by the property of e. So we have D(a) = D(a)D(e),
and hence D(e) = I.
Now let us return to the correspondence between SU(2) and SO(3). V (n ˆ , θ)
corresponds to R(n ˆ , θ), but can an SU(2) matrix be said to provide a valid
representation of SO(3)? Consider the case V (ˆ n = ˆ
z, θ = 2π). From (M.103)
this is equal to
(
)
−1 0
,
(M.107)
0 −1
but the corresponding rotation matrix, from (M.101), is the identity matrix.
Hence our theorem is violated, since (M.107) is plainly not the identity matrix
of SU(2). Thus the SU(2) matrices can not be said to represent rotations, in
the strict sense. Nevertheless, spin-1/2 particles certainly do exist, so Nature
appears to make use of these ‘not quite’ representations! The SU(2) identity
element is V (ˆ n = ˆ
z, θ = 4π), confirming that the rotational properties of a
spinor are quite other than those of a classical object.
In fact, two and only two distinct elements of SU(2), namely
(
)
(
)
1 0
and
−1 0
,
(M.108)
0 1
0 −1
correspond to the identity element of SO(3) in the correspondence (M.106) –
just as, in general, V and −V correspond to the same SO(3) element R(n ˆ , θ),
as we saw. The failure to be a true representation is localized simply to a sign:
we may indeed say that, up to a sign, SU(2) matrices provide a representation
of SO(3). If we ‘factor out’ this sign, the groups are isomorphic. A more
mathematically precise way of saying this is given in Jones (1990, chapter 8).