= Normal and Weyl symbols of a harmonic oscillator
For the <quantum harmonic oscillator> $H=\hbar\omega(a^\dagger a+1/2)$, its normal symbol is $H_N=\hbar\omega(\bar\psi\psi+1/2)$. The <Weyl ordering> symbol of $a^\dagger a$ is $\bar\psi\psi-1/2$, so $H_W=\hbar\omega\bar\psi\psi=p^2/(2m)+m\omega^2q^2/2$. Switching from adjacent-label <coherent-state time slicing> to a midpoint <phase-space path integral> must retain this ordering correction. Both correctly normalized constructions give the <thermal partition function of a quantum harmonic oscillator> $Z=[2\sinh(\beta\hbar\omega/2)]^{-1}$; retaining the normal-symbol constant after switching prescriptions would count the zero-point shift twice.
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