(8171) Propagator in a circle

In this exercise you are requested to find the propagator \( U(\theta|\theta_0) \) for a particle of mass \( M \) that is free to move on a circle, given the time \( t \). The exercise demonstrates the equivalence of two different strategies.

(1) Derive the Poisson resummation formula \( \sum_n f(x_n) = \sum_m F(k_m) \). Tip: the LHS is the integral over a product of \( f(x) \) with a comb function. Use the convolution theorem to find the zero Fourier component of this product.

(2) Obtain a result for the propagator via spectral decomposition, where the summation is over the angular momentum \( m \).

(3) Use the Poisson resummation formula to write the result as a summation over the winding number \( n \).

(4) Show that the latter result is directly obtained from the Feynman path integral.