Quantum Mechanics for Mathematicians

6.2.2 Rotations and spin groups in four dimensions

Pairs (u,v) of unit quaternions give the product groupSp(1)×Sp(1). An
element (u,v) of this group acts onq∈H=R^4 by left and right quaternionic
multiplication
q→uqv−^1

This action preserves lengths of vectors and is linear inq, so it must correspond
to an element of the groupSO(4). One can easily see that pairs (u,v) and
(−u,−v) give the same linear transformation ofR^4 , so the same element of
SO(4) and show thatSO(4) is the groupSp(1)×Sp(1), with the two elements
(u,v) and (−u,−v) identified. The nameSpin(4) is given to the Lie group
Sp(1)×Sp(1) that “double covers”SO(4) in this manner, with the covering
map

Φ : (u,v)∈Sp(1)×Sp(1) =Spin(4)→{q→uqv−^1 }∈SO(4)

6.2.3 Rotations and spin groups in three dimensions

Later on we’ll encounterSpin(4) andSO(4) again, but for now we’re interested
in the subgroupSpin(3) that only acts non-trivially on 3 of the dimensions,
and double covers notSO(4) butSO(3). To find this, consider the subgroup
ofSpin(4) consisting of pairs (u,v) of the form (u,u) (a subgroup isomorphic
toSp(1), since elements correspond to a single unit length quaternionu). This
subgroup acts on quaternions by conjugation

q→uqu−^1

an action which is trivial on the real quaternions (sinceu(q 01 )u−^1 =q 01 ). It
preserves and acts nontrivially on the space of “pure imaginary” quaternions of
the form
q=~v=v 1 i+v 2 j+v 3 k

which can be identified with the vector spaceR^3. An elementu∈Sp(1) acts
on~v∈R^3 ⊂Has
~v→u~vu−^1

This is a linear action, preserving the length|~v|, so it corresponds to an el-
ement ofSO(3). We thus have a map (which can easily be checked to be a
homomorphism)

Φ :u∈Sp(1)→{~v→u~vu−^1 }∈SO(3)