Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 75 / 3 / a / Solution

A Euclidean configuration-space transition kernel is obtained by slicing $\tau$ into $M$ steps $\delta =\tau /M$, inserting position resolutions and using the free Gaussian kernel. With initial $q_i=\pm a$ and final $q_f=a$,The normalization means the limit of $(m/(2\pi \hslash \delta ){)}^{M/2}\int {\prod }_{j=1}^{M-1}d{q}_j$ with exponent $-{\sum }_j[m(q_{j+1}-q_j{)}^2/(2\delta )+\delta V(q_j)]/\hslash$, in a consistent time-slice prescription. This fixes the endpoint normalization that an informal continuum symbol alone leaves unspecified. The Wick rotation converts oscillatory real-time weight into the positive-potential Euclidean weight; classical extrema of this action organize the semiclassical approximation.