Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 337 / 1 / b / Solution

Letwhere $dg$ is Haar measure. Left invariance gives $hK{h}^{-1}=K$ for every $h\in SU(2)$. Since the spin-$S$ representation is irreducible, Schur lemma impliesInserting this coherent-state resolution between $N$ short time steps and taking the continuum limit gives, for a noninteracting spin with zero Hamiltonian,up to the normalization generated by $c$. A Hamiltonian would add $-i\int ⟨g|H|g⟩dt$ in the exponent.

Fordirect differentiation and $S_3|\uparrow ⟩=S|\uparrow ⟩$ giveThe $\dot{\psi }$ term is a total derivative and records only the arbitrary phase used to represent a ray. It cancels against the endpoint-state phases, so the physical Spin coherent-state path integral depends only on the path on the two-sphere. For example, choosing the section $\psi =-\varphi$ gives the Spin coherent-state Berry phase proportional toChanging the surface used to fill a closed path changes its solid angle by $4\pi$. Single-valuedness of the path-integral phase requiresThus the Wess-Zumino coefficient is quantized in integer or half-integer units.