Quantum Mechanics for Mathematicians

space of all pure imaginary quaternions, which can be identified withR^3 by

w=





w 1

w 2

w 3



∈R^3 ↔~w=w 1 i+w 2 j+w 3 k∈H

Unlike theU(1) case, there’s a non-trivial Lie bracket, the commutator of quater-
nions.
Elements of the groupSp(1) are given by exponentiating such Lie algebra
elements, which we will write in the form

u(θ,w) =eθ~w

whereθ∈Rand~wis a purely imaginary quaternion of unit length. Since

~w^2 = (w 1 i+w 2 j+w 3 k)^2 =−(w^21 +w^22 +w^23 ) =− 1

the exponential can be expanded to show that

eθ~w= cosθ+~wsinθ

Takingθas a parameter, theu(θ,w) give paths inSp(1) going through the
identity atθ= 0, with velocity vector~wsince

d

dθ

u(θ,w)|θ=0= (−sinθ+~wcosθ)|θ=0=~w

We can explicitly evaluate the homomorphism Φ on such elementsu(θ,w)∈
Sp(1), with the result that Φ takesu(θ,w) to a rotation by an angle 2θaround
the axisw:

Theorem 6.1.
Φ(u(θ,w)) =R(2θ,w)

Proof.First consider the special casew=e 3 of rotations about the 3-axis.

u(θ,e 3 ) =eθk= cosθ+ksinθ

and
u(θ,e 3 )−^1 =e−θk= cosθ−ksinθ

so Φ(u(θ,e 3 )) is the rotation that takesv(identified with the quaternion~v=
v 1 i+v 2 j+v 3 k) to

u(θ,e 3 )~vu(θ,e 3 )−^1 =(cosθ+ksinθ)(v 1 i+v 2 j+v 3 k)(cosθ−ksinθ)

=(v 1 (cos^2 θ−sin^2 θ)−v 2 (2 sinθcosθ))i

+ (2v 1 sinθcosθ+v 2 (cos^2 θ−sin^2 θ))j+v 3 k

=(v 1 cos 2θ−v 2 sin 2θ)i+ (v 1 sin 2θ+v 2 cos 2θ)j+v 3 k