Bosonic coherent-state path integral ($S={\int }_0^{\beta }d\tau [\bar{\psi }{\partial }_{\tau }\psi +K(\bar{\psi },\psi )]$)

In a bosonic coherent-state path integral, coherent-state resolutions between imaginary-time evolution factors turn a normal-ordered grand Hamiltonian $K=H-\mu N$ into its complex symbol. Coherent-state overlaps supply the first-order temporal term. A thermal trace makes the fields periodic in imaginary time. The chemical potential couples to number, and interactions retain their normal-ordering convention.

Bosonic traces impose periodic complex fields and inverse determinants. Fermionic traces impose antiperiodic Grassmann fields and determinants. The distinction encodes quantum statistics, not a freely chosen boundary condition.

Physical imaginary time runs over $[0,\beta \hslash ]$ with an action divided by $\hslash$. Rescaling it by $\hslash$ gives a variable in $[0,\beta ]$ and the operator ${\partial }_{\tau }+\epsilon$. Matsubara eigenvalues in the latter variable have energy units.

With exact one-slice weight $q=e^{-\beta \epsilon /M}$ and periodic shift matrix $P$, $\det (I-qP)=1-q^M$. This finite-dimensional identity fixes the normal-ordered thermal determinant without an ambiguous unregulated frequency product.

An ordinary normal-ordered coherent-state Gaussian for energy $\epsilon a^{†}a$ gives $(1-e^{-\beta \epsilon }{)}^{-1}$. The physical oscillator with zero-point energy $\epsilon /2$ requires the factor $e^{-\beta \epsilon /2}$. A symmetric determinant prescription may encode that factor implicitly; it must not be counted twice.