The Groups SO(3) and SU(2)

For physical and mathematical motivation for this section (which involves nonrelativistic quantum mechanics) we refer the reader to some remarks of Feynman and of Weyl. Specifically, Feynman, in his section entitled “Degeneracy,” shows that a process involving a specific choice of direction in space requires that the process be described not by a single wave function ψ but rather by a multicomponent column vector of wave functions Ψ = (ψ1, …, ψN)T. He then indicates, roughly speaking, that since the physics cannot depend on the choice of cartesian coordinates (x1, x2, x3) of space, the N-tuples must transform under some representation ρ: SO(3) → U(N) of the rotation group SO(3) of space. This is not quite accurate; since eiγΨ represents the same wave function (when γ is a constant), ρ is only a “ray” representation, ρ(g)ρ(h) = eiγ(g,h) ρ(gh) for a function γ(g, h). Weyl shows that this can be made into a genuine representation, except that it is (perhaps) double-valued. We shall show in this section that there is a natural 2 : 1 homomorphism π of the special unitary group SU(2) onto SO(3), thus yielding a (perhaps double valued) representation of SU(2) into U(N).