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.

In a temporal winding sector of a quantum rotor, decompose the lifted angle into a straight winding path and a fluctuation vanishing at both endpoints. The cross term in the kinetic action vanishes, so all sectors have the same Gaussian path integral. With inverse-energy imaginary time its value is the displayed diagonal free-particle kernel on the line. Integrating the initial angle separately supplies $2\pi$. Thus this fixed-endpoint factor is an amplitude density, not an additional rotor partition function or an uncounted angular zero mode.

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.

For the quantum rotor, low temperature makes the angular-momentum sum converge fastest: n=0 gives $Z\to 1$. High temperature makes the temporal winding sum converge fastest: m=0 gives $Z\sim \sqrt{2\pi I/(\beta {\hslash }^2)}$. At low temperature many winding sectors contribute and cancel the vanishing fixed-endpoint prefactor. A zero-winding approximation valid at high temperature cannot be extended to low temperature merely because winding paths look geometrically longer.

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.