 Solution

The Euclidean saddle in winding sector $m$ is a constant-speed lifted path, ${\varphi }_{\mathrm{cl}}(\tau )={\varphi }_0+2\pi m\tau /\mathcal{T}$. Write $\varphi ={\varphi }_{\mathrm{cl}}+\eta$ with Dirichlet fluctuation endpoints $\eta (0)=\eta (\mathcal{T})=0$. The cross term in the kinetic action is proportional to $\int {\partial }_{\tau }\eta d\tau =0$, and the flux term of $\eta$ vanishes for the same reason. ThereforeDefine the fluctuation prefactor byIt is the free-line diagonal Euclidean propagator for the angular coordinate with inertia $I$, or equivalently the Gaussian bridge amplitude about a given classical lifted path. It is independent of $m,{\varphi }_0$ and flux. It is not the complete zero-flux ring partition function, which still includes every winding and the initial-angle integral. Translational independence supplies the factor ${\int }_0^{2\pi }d{\varphi }_0=2\pi$, leavingThe momentum-winding duality of a quantum rotor uses a Gaussian expansion that is exact here because the kinetic action is quadratic and the flux coupling is a total derivative. It is not merely a leading semiclassical approximation.