Quantum Mechanics for Mathematicians

Given such a construction ofSpin(n), we also need to explicitly construct
the homomorphism Φ, and show that its derivative Φ′is an isomorphism of
Lie algebras. We will see that the simplest construction of the spin groups
here uses the groupSp(1) of unit-length quaternions, withSpin(3) =Sp(1)
andSpin(4) =Sp(1)×Sp(1). By identifying quaternions and pairs of complex
numbers, we can show thatSp(1) =SU(2) and thus work with these spin groups
as either 2 by 2 complex matrices (forSpin(3)), or pairs of such matrices (for
Spin(4)).

6.2.1 Quaternions

The quaternions are a number system (denoted byH) generalizing the complex
number system, with elementsq∈Hthat can be written as

q=q 0 +q 1 i+q 2 j+q 3 k, qj∈R

withi,j,k∈Hsatisfying

i^2 =j^2 =k^2 =− 1 ,ij=−ji=k,ki=−ik=j,jk=−kj=i

and a conjugation operation that takes

q→q ̄=q 0 −q 1 i−q 2 j−q 3 k

This operation satisfies (foru,v∈H)

uv= ̄vu ̄
As a vector space overR,His isomorphic withR^4. The length-squared
function on thisR^4 can be written in terms of quaternions as

|q|^2 =qq ̄=q^20 +q 12 +q^22 +q^23

and is multiplicative since

|uv|^2 =uvuv=uvv ̄u ̄=|u|^2 |v|^2

Using
qq ̄
|q|^2

= 1

one has a formula for the inverse of a quaternion

q−^1 =

̄q
|q|^2
The length one quaternions thus form a group under multiplication, called
Sp(1). There are also Lie groups calledSp(n) for larger values ofn, consisting
of invertible matrices with quaternionic entries that act on quaternionic vectors
preserving the quaternionic length-squared, but these play no significant role in
quantum mechanics so we won’t study them further. Sp(1) can be identified
with the three dimensional sphere since the length one condition onqis

q^20 +q^21 +q^22 +q 32 = 1

the equation of the unit sphereS^3 ⊂R^4.