= Classical phase-space limit of the thermal kernel
{title2=$Z\simeq\lambda_T^{-d}\int d^dx\,e^{-\beta V(x)}$}
For $H=p^2/(2m)+V(x)$ and a smooth potential on scales larger than the <thermal de Broglie wavelength> $\lambda_T=\sqrt{2\pi\beta\hbar^2/m}$, the diagonal thermal kernel approaches $\langle x|e^{-\beta H}|x\rangle\simeq\lambda_T^{-d}e^{-\beta V(x)}$. Integration over its diagonal gives the classical <phase space> integral with one factor $(2\pi\hbar)^{-1}$ 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.
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