Thermal trace as a closed Euclidean path integral ($Z={\int }_{x(\beta \hslash )=x(0)}\mathcal{D}x{e}^{-S_E[x]/\hslash }$)

Taking the trace of the position-space imaginary-time kernel identifies its endpoints and integrates over the common position. The resulting closed paths have imaginary-time length $\beta \hslash$. Bosonic coherent fields are periodic, while fermionic coherent fields are antiperiodic. This temporal condition follows from the trace, independently of the spatial boundary conditions.

For $H=p^2/(2m)+V(x)$ and a smooth potential on scales larger than the thermal de Broglie wavelength ${\lambda }_T=\sqrt{2\pi \beta {\hslash }^2/m}$, the diagonal thermal kernel approaches $⟨x|e^{-\beta H}|x⟩\simeq {\lambda }_T^{-d}{e}^{-\beta V(x)}$. Integration over its diagonal gives the classical phase space integral with one factor $(2\pi \hslash {)}^{-1}$ per coordinate-momentum pair. Exchange and boundary corrections must be small; a positive imaginary-time weight by itself does not make an arbitrary quantum system classical.