Rotor winding-sector path integral ($\varphi (\beta )-\varphi (0)=2\pi m$)

The thermal rotor path integral traces paths closed modulo $2\pi$. Lift them to the line and sum endpoints differing by $2\pi m$. In inverse-energy time $0<\tau <\beta$, the kinetic exponent is $I\int {\dot{\varphi }}^{2}d\tau /(2{\hslash }^2)$. Integer $m$ counts trajectory windings, not angular-momentum eigenstates.

Poisson resummation relates the momentum sum ${\sum }_n{e}^{-\beta {\hslash }^2{n}^2/(2I)}$ to the winding sum $\sqrt{2\pi I/(\beta {\hslash }^2)}{\sum }_m{e}^{-2{\pi }^{2}I{m}^2/(\beta {\hslash }^2)}$. Gaussian Fourier transformation proves equality, including the prefactor. The momentum form converges rapidly at low temperature and the winding form at high temperature.

Writing $\varphi ={\varphi }_0+2\pi m\tau /\beta +\eta$ with endpoint-fixed $\eta$ separates winding action and Gaussian fluctuations. The latter give the free return kernel on the lifted line, $Z_0=\sqrt{I/(2\pi {\hslash }^2\beta )}$. The starting-angle integral supplies $2\pi$. Zero winding is not the same as the zero-momentum ground-state sector.