 Solution

Apply the Poisson summation formula in the convention supplied, takingCompleting the square in its Fourier transform and evaluating the analytically continued Gaussian integral givesThus the spectral sum becomesThe square-root branch is fixed by continuation from positive imaginary-time damping. Each term is the classical contribution of part (b), multiplied by the same fluctuation prefactor. Since the action is quadratic, writing $\theta ={\theta }_m+\eta$ with $\eta (0)=\eta (T)=0$ removes the cross term and leaves a free Gaussian fluctuation integral independent of $m$. This real-time winding representation of a quantum rotor kernel is exact, rather than only a leading semiclassical approximation.