 Euclidean worldline path integral

A thermal path integral represents the canonical partition function by paths periodic in Euclidean time of circumference $\beta$. For $H=p^2/2+V(x)$, its formal action is $S_E[x]={\int }_0^{\beta }({\dot{x}}^2/2+V(x))d\tau$. The periodic boundary condition results from taking the trace of the evolution kernel. The continuum measure requires a regularization and normalization; finite-dimensional measures make both choices explicit.