Quantum Mechanics for Mathematicians

1

1

− 1

Φ

~v7→u~vu−^1

u

−u

inSO(3)

inSp(1) =Spin(3)

Figure 6.1: Double coverSp(1)→SO(3).

Bothuand−uact in the same way on~v, so we have two elements inSp(1)
corresponding to the same element inSO(3). One can show that Φ is a surjective
map (any element ofSO(3) is Φ of something), so it is what is called a “covering”
map, specifically a two-fold cover. It makesSp(1) a double cover ofSO(3), and
we give this group the name “Spin(3)”. This also allows us to characterize
more simplySO(3) as a geometrical space. It isS^3 =Sp(1) =Spin(3) with
opposite points on the three-sphere identified. This space is known asRP^3 , real
projective 3-space, which can also be thought of as the space of lines through
the origin inR^4 (each such line intersectsS^3 in two opposite points).

Digression.The covering mapΦis an example of a topologically non-trivial
cover. Topologically, it is not true thatS^3 'RP^3 ×(+1,−1).S^3 is a connected
space, not two disconnected pieces. This topological non-triviality implies that
globally there is no possible homomorphism going in the opposite direction from
Φ(i.e.,SO(3)→Spin(3)). This can be done locally, picking a local patch in
SO(3)and taking the inverse ofΦto a local patch inSpin(3), but this won’t
work if we try and extend it globally to all ofSO(3).

The identificationR^2 =Callowed us to represent elements of the unit circle
groupU(1) as exponentialseiθ, whereiθwas in the Lie algebrau(1) =iRof
U(1).Sp(1) behaves in much the same way, with the Lie algebrasp(1) now the