The first term is a gauge choice for the <Spin coherent-state Berry phase>. For a closed spin trajectory, changing the surface used to evaluate the solid angle changes the action by
$$
\Delta S_B=4\pi S.
$$
The path-integral phase $e^{iS_B}$ must be independent of that choice, so
$$
e^{i4\pi S}=1.
$$
Hence
$$
\boxed{2S\in\mathbb Z},
$$
and the spin is integer or half-integer.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 337 / 2 / b / i / Solution

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The spin coherent-state path integral contains a Berry phase equal, modulo gauge choices, to the spin $S$ times the solid angle swept on the Bloch sphere. Single-valued path-integral amplitudes require $2S\in\mathbb Z$.


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