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For the quantum harmonic oscillator $H=\hslash \omega (a^{†}a+1/2)$, its normal symbol is $H_N=\hslash \omega (\bar{\psi }\psi +1/2)$. The Weyl ordering symbol of $a^{†}a$ is $\bar{\psi }\psi -1/2$, so $H_W=\hslash \omega \bar{\psi }\psi =p^2/(2m)+m{\omega }^2{q}^2/2$. 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 $Z=[2\sinh (\beta \hslash \omega /2){]}^{-1}$; retaining the normal-symbol constant after switching prescriptions would count the zero-point shift twice.

Let $K=H-\mu N$, take $M$ slices of width $ϵ=\beta /M$, and insert the coherent-state resolution of identity between factors of $e^{-ϵK}$. If $H$ is in normal ordering, each short-time matrix element isCombining the overlap with the Gaussian weight leaves ${\sum }_{j,n}{\bar{\psi }}_{n,j}({\psi }_{n,j}-{\psi }_{n,j-1})$ in the action. The coherent-state time slicing prescription fixes which adjacent labels appear in $H$, rather than allowing an arbitrary ordering change after taking the continuum limit.

The twist $\zeta$ in the thermal trace closes the path with the coherent-state thermal boundary conditions: periodic for bosons, antiperiodic for fermions. Taking the regulated continuum limit givesThe action is dimensionless because $\tau$ has inverse-energy units. For fermions, $\bar{\psi }$ and $\psi$ remain independent Grassmann fields. If the original Hamiltonian is not normally ordered, first express it in normal order, retaining all constants.

For the quantum harmonic oscillator, set $\mu =0$ and $\Omega =\hslash \omega$. The normal symbol is $H_N(\bar{\psi },\psi )=\Omega (\bar{\psi }\psi +1/2)$, so the thermal coherent-state time slicing givesThe connection with a phase-space path integral uses the canonical real coordinatesTheir derivative term satisfies $\bar{\psi }\dot{\psi }=-ip\dot{q}/\hslash$ up to total derivatives that vanish for periodic paths. The normal and Weyl symbols of a harmonic oscillator must be distinguished: in midpoint phase-space time slicing, the Weyl ordering symbol of $a^{†}a$ is $\bar{\psi }\psi -1/2$. Thus the appropriate midpoint Hamiltonian is $H_W=p^2/(2m)+m{\omega }^2{q}^2/2$, with no additional constant. The change from the adjacent-label normal prescription to the midpoint prescription includes this ordering correction.

In physical imaginary time $u=\hslash \tau$, the resulting phase-space path integral isGaussian momentum integration in a phase-space path integral produces the usual oscillator configuration-space path integral. An exact check follows directly from the coherent kernel $⟨{\psi }^{\prime }|e^{-\beta \Omega a^{†}a}|\psi ⟩=\exp (e^{-\beta \Omega }{\bar{\psi }}^{\prime }\psi )$:This is the thermal partition function of a quantum harmonic oscillator. Keeping the $\Omega /2$ normal-ordering constant a second time after switching to the Weyl symbol would double count the zero-point energy.