Quantum Mechanics for Mathematicians

Recall that anySU(2) matrix can be written in the form

(

α β

−β α

)

α=q 0 −iq 3 , β=−q 2 −iq 1

withα,β∈Carbitrary complex numbers satisfying|α|^2 +|β|^2 = 1. A somewhat
unenlightening formula for the map Φ :SU(2)→SO(3) in terms of such explicit
SU(2) matrices is given by

Φ

(

α β

−β α

)

=





Re(α^2 −β^2 ) Im(α^2 +β^2 ) −2 Re(αβ)

−Im(α^2 −β^2 ) Re(α^2 +β^2 ) 2 Im(αβ)

2 Re(αβ) 2 Im(αβ) |α|^2 −|β|^2





See [82], page 123-4, for a derivation.

6.3 A summary

To summarize, we have shown that in the three dimensional case we have two
distinct Lie groups:

• Spin(3), which geometrically is the spaceS^3. Its Lie algebra isR^3 with
  Lie bracket the cross-product. We have seen two different explicit con-
  structions ofSpin(3), in terms of unit quaternions (Sp(1)), and in terms
  of 2 by 2 unitary matrices of determinant 1 (SU(2)).


• SO(3), which has a Lie algebra isomorphic to that ofSpin(3).

There is a group homomorphism Φ that takes the first group to the second,
which is a two-fold covering map. Its derivative Φ′is an isomorphism of the Lie
algebras of the two groups.
We can see from these constructions two interesting irreducible representa-
tions of these groups:

• A representation onR^3 which can be constructed in two different ways: as
  the adjoint representation of either of the two groups, or as the defining
  representation ofSO(3). This is known to physicists as the “spin 1”
  representation.


• A representation of the first group onC^2 , which is most easily seen as
  the defining representation ofSU(2). It is not a representation ofSO(3),
  since going once around a non-contractible loop starting at the identity
  takes one to minus the identity, not back to the identity as required. This
  is called the “spin^12 ” or “spinor” representation and will be studied in
  more detail in chapter 7.