This is the orthogonal transformation ofR^3 given by
v=
v 1
v 2
v 3
→
cos 2θ −sin 2θ 0
sin 2θ cos 2θ 0
0 0 1
v 1
v 2
v 3
(6.2)
The same calculation can readily be done for the case ofe 1 , then use the
Euler angle parametrization of equation 6.1 to show that a generalu(θ,w) can
be written as a product of the cases already worked out.
Notice that asθgoes from 0 to 2π,u(θ,w) traces out a circle inSp(1). The
homomorphism Φ takes this to a circle inSO(3), one that gets traced out twice
asθgoes from 0 to 2π, explicitly showing the nature of the double covering
above that particular circle inSO(3).
The derivative of the map Φ will be a Lie algebra homomorphism, a linear
map
Φ′:sp(1)→so(3)
It takes the Lie algebrasp(1) of pure imaginary quaternions to the Lie algebra
so(3) of 3 by 3 antisymmetric real matrices. One can compute it easily on basis
vectors, using for instance equation 6.2 above to find for the case~w=k
Φ′(k) =
d
dθ
Φ(cosθ+ksinθ)|θ=0
=
−2 sin 2θ −2 cos 2θ 0
2 cos 2θ −2 sin 2θ 0
0 0 0
|θ=0
=
0 −2 0
2 0 0
0 0 0
= 2l 3
Repeating this on other basis vectors one finds that
Φ′(i) = 2l 1 ,Φ′(j) = 2l 2 ,Φ′(k) = 2l 3
Thus Φ′is an isomorphism ofsp(1) andso(3) identifying the bases
i
2
,
j
2
,
k
2
and l 1 ,l 2 ,l 3
Note that it is thei 2 ,j 2 ,k 2 that satisfy simple commutation relations
[
i
2
,
j
2
]
=
k
2
,
[
j
2
,
k
2
]
=
i
2
,
[
k
2
,
i
2
]
=
j
2
6.2.4 The spin group andSU(2)
Instead of doing calculations using quaternions with their non-commutativity
and special multiplication laws, it is more conventional to choose an isomorphism