Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 81 / 3 / c / Solution

Analytically continuing the free-particle propagator to imaginary time gives the Gaussian heat kernelThe two image endpoints have displacements $2rL$ and $2rL-2q$ from the starting point. Inserting their kernels into the thermal trace of an interval image kernel givesThe prefactor comes from the normalization of the free-particle propagator; it cannot be dropped from a thermal trace.