= Leading high-temperature density kernel
{title2=$\rho(x,x')\simeq Z^{-1}\sqrt{m/(2\pi\beta\hbar^2)}e^{-m(x-x')^2/(2\beta\hbar^2)-\beta V(x)}$}
Neglecting <commutators> between kinetic and <potential energy> permits a leading product approximation to the imaginary-time evolution operator. The kinetic factor is a Gaussian Fourier <integral>; the potential factor multiplies it in position space. The unsymmetrized product places the potential at one endpoint and need not be exactly <Hermitian> at the discarded order. Splitting the potential equally between both endpoints gives a <Hermitian> approximation with exponent $-\beta[V(x)+V(x')]/2$ and the same leading diagonal normalization.
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