Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 304 / 1 / ii / Solution

The trace of the Euclidean evolution kernel identifies its initial and final positions. Thus the Euclidean worldline path integral is over periodic paths $x(\tau +\beta )=x(\tau )$, with $\tau \in [0,\beta ]$:The circle has circumference $\beta$, not an arbitrary unit length. The Euclidean action follows by Wick rotation of the unit-mass oscillator action. Equivalently, integration by parts with periodic boundary conditions gives $S_E=\frac{1}{2}⟨x,L_{\omega }x⟩$ for $L_{\omega }=-{\partial }_{\tau }^2+{\omega }^2$. The path integral measure is formal until a regularization and its normalization are specified; the next part supplies a finite-dimensional version.