Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 304 / 1 / b / ii / Solution

Choose a real lift $\Delta \theta ={\theta }_f-{\theta }_i$. Classical paths fall into winding number sectors $j\in \mathbb{Z}$ and areEach sector has the same fluctuation determinant, so the image-sum form of the propagator isSet $a=\hslash T/(2m{R}^2)$. Applying the Poisson summation formula to the Gaussian $e^{i{x}^2/(4a)}$ giveswhich is exactly the spectral propagator found in part i. Thus the angular-momentum sum is dual to a sum over homotopy classes of classical paths.