Quantum Mechanics (1) · Problem 5.8

Rotations and angular momentum

Physics — Quantum Mechanics · irreps-o3-from-su2

Problem

Prove the following theorem: If 2j2j is an odd integer, then Dj(u)D^j(u) is a faithful representation of SU(2)SU(2), and hence is not a representation of O+(3)O^+(3). If 2j>02j > 0 is an even integer (and hence j>0j > 0 is an integer), then Re(θ)→Dj[ue(θ)]R_{\mathbf{e}}(\theta) \to D^j[u_{\mathbf{e}}(\theta)] is a faithful representation of O+(3)O^+(3). Except for the trivial identity representation, all irreps of O+(3)O^+(3) are of this form.

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