 Solution

First lift the angle to the real line. The time-sliced configuration-space path integral for a free coordinate with inertia I has kinetic action $\int dtI{\dot{\varphi }}^2/2$. The rotor kernel sums lifted endpoints separated by ${\varphi }_F-{\varphi }_I+2\pi m$, because all such endpoints represent the same final angle on the circle.

Continue real time as $t=-i\hslash \tau$. Then $e^{-iHt/\hslash }=e^{-H\tau }$ and the end of the thermal interval is $\tau =\beta$, in inverse-energy units. ThusThe trace integrates the initial angle over one copy of the circle, not the whole lifted line. It yields the rotor winding-sector path integralThe integer m is temporal winding number, not the angular-momentum label n. Locally the measure has the free-kernel normalization $[I/(2\pi {\hslash }^2\Delta \tau ){]}^{1/2}$ per time slice. Summing topological sectors and retaining this normalization are necessary for the correct thermal trace.