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The propagator is the quantum amplitude for evolving from one angular position to another. It evolves a wavefunction by $\Psi ({\varphi }_F,t)={\int }_0^{2\pi }d{\varphi }_{I}K({\varphi }_F,{\varphi }_I;t)\Psi ({\varphi }_I,0)$. The path integral weights each trajectory by $e^{iS/\hslash }$, with $S=\int dtI({\partial }_t\varphi {)}^2/2$. Since the configuration space is a circle, paths must include all possible lifts of the final angle to the real line.

Take $t=-i\hslash \beta$ by Wick rotation. Let $\tau =t_E/\hslash$, which runs from zero to $\beta$, and let the dot denote $d/d\tau$. The dimensionless Euclidean exponent is then $I{\int }_0^{\beta }{\dot{\varphi }}^{2}d\tau /(2{\hslash }^2)$. Tracing the kernel identifies final and initial angles modulo $2\pi$, while the lift can change by $2\pi m$. Thus the rotor winding-sector path integral isThe integration measure is fixed by the short-time free-rotor kernel. The dot on $\varphi$ is essential; it is present in the PDF but lost in the TeX aid.

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.