Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2014 / Paper 1 / 2 / Solution

A Wick rotation $t=-i{\tau }_E$ changes real-time evolution into $e^{-H{\tau }_E/\hslash }$. At ${\tau }_E=\beta \hslash$, taking the trace integrates equal endpoints:Thus temperature sets the imaginary-time circumference, and bosonic coordinate paths are periodic. The measure is fixed by the short-time propagator. The trace sums paths, not just a single stationary trajectory; a quadratic action is special because the saddle plus Gaussian fluctuations is exact.