Use of SU(2) & SO(3)

Every Lie group homomorphism gives rise to a Lie algebra homomorphism. In the case of a simply-connected matrix Lie group G, a Lie algebra homomorphism also gives rise to a Lie group homomorphism. In fact, for a simply-connected matrix Lie group, there is a natural one-to-one correspondence between the representations of G and the representations of the Lie algebra g. Each of the representations π_m of su(2) was constructed from the corresponding representation Π_m of the group SU(2).

SU(2) is simply connected but SO(3) is not (SU(2) can be thought of (topologically) as the three-dimensional sphere S³ sitting inside R⁴. It is well known that S³ is simply connected). There exists a Lie group homomorphism Φ which maps SU(2) onto SO(3) and which is two-to-one. Therefore, SU(2) and SO(3) are almost isomorphic.

Consider the space V of all 2x2 complex matrices which are self-adjoint (i.e., A^* = A) and have trace zero. This is a three-dimensional real vector space with the following basis: A₁ = (0,1: 1,0) ; A₂ = (0,-i : i,0) ; A₃ = (1,0 : 0,-1) . Define the inner product on V by <A, B> = 1/2 trace(AB). {A₁, A₂, A₃} is an orthonormal basis for V. Next we are going to identify V with R³. Suppose U is an element of SU(2) and A is an element of V. Consider UAU^-1, trace(UAU^-1) = trace(A) = 0 and (UAU^-1)^* = UAU^-1. So UAU^-1is again in V. Because the map A --> UAU^-1 is linear. Therefore, we can define a linear map Φ_U of V to itself by Φ_U = UAU^-1. Given A, B ∈ V, <Φ_U(A), Φ_U(B)> = <A, B>. Thus, Φ_U is an orthogonal transformation of V.

Once we identify V with R³ using the above orthonormal basis, we may think of Φ_U as an element of O(3). Since Φ_U1U2 = Φ_U1Φ_U2, we see that Φ (the map U --> Φ_U) is a homomorphism of SU(2) into O(3). SU(2) is connected, Φ is continuous, and Φ_I is equal to I, which has determinant one. It follows that Φ must map SU(2) into the identity component of O(3), namely SO(3). However, the map Φ_U is not one-to-one, since for any U ∈ SU(2), Φ_U = Φ_-U. Actually Φ_U is a two-to-one map of SU(2) onto SO(3) (recall that every element of O(3) has determinant ± 1).

If we have the basis E₁ = 1/2(i,0 : 0,-i) ; E₂ = 1/2(0,-1: 1,0) ; E₃ = (0,i : i,0) for su(2) and the basis F₁ = (0,0,0 : 0,0,1 : 0,-1,0) ; F₂ = (0,0,-1 : 0,0,0 : 1,0,0) ; F₃ = (0,1,0 : -1,0,0 : 0,0,0) for so(3). Then, we have [E₁ , E₂] = E₃ , [E₂ , E₃] = E₁ and [E₃ , E₁] = E₂ , and similarly with the E's replaced by the F's. Thus the linear map Φ : su(2) --> so(3) which takes E_i to F_i will be a linear algebra isomorphism.

Let σ_m = π_m º Φ^-1 be the irreducible complex representations of the Lie algebra so(3) (m ≥ 0). If m is even, then there is a representation ∑_m of the group SO(3) such that ∑_m(exp X) = exp(σ_m(X)) for all X in so(3). If m is odd, then there is no such representation of SO(3).

Representation of su(2) ≅ so(3) in Physics

Representation of su(2) ≅ so(3) in Physics are labeled by the parameter l = m/2. In terms of this notation, a representation of so(3) comes from a representation of SO(3) if and only if l is an integer. The representations with l an integer are called "integer spin"; the others are called the "half-integer spin."Consider the path in SO(3) consisting of rotations by angle 2πt in the (x, y)-plane, which comes back to the identity when t = 1. However, this path is not homotopic to the constant path.

If one defines ∑_m along the constant path, then one gets the value ∑_m(I) = I, as expected. If m is odd and one defines ∑_m along the path of rotations in the (x, y)-plane, then one gets the value ∑_m(I) = -I. There is no way to define ∑_m(m odd) as a "single-valued" representations of SO(3).

An electron is a "spin-1/2" particle, which means that it is described in quantum machines in a way that involves the representation σ₁ of so(3). In the quantum machines, one finds statements to the effect that performing a 360º rotation on the wave function of the electron gives back the negative of the original wave function.This reflects that if one attempts to construct the nonexistent representation ∑₁ of SO(3), then when defining ∑₁ along a path of rotations in some plane, one gets that ∑₁(I) = -I.

A Unitary Representations of SO(3)

Consider the unit sphere S² ⊂ R³, with the usual surface measure Ω. Any R ∈ SO(3) maps S² into S² . For each R, we can define Π₂(R) acting on L²(S², dΩ) by [ Π₂(R) f](x) = f(R^-1x). Then, Π₂ is a unitary representation of SO(3). Here, L²(S², dΩ) has a very nice decomposition as the orthogonal direct sum of finite-dimensional invariant subspaces. This decomposition is the theory of "spherical harmonics" in physics.