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.

In a bosonic coherent-state path integral, normal ordering determines which fields occur on the two neighboring time slices. For unnormalized coherent states, the short-step kernel obeyswhere $K$ is the normal-ordered grand Hamiltonian and $ϵ$ has inverse-energy units. Creation operators supply the later conjugate field and annihilation operators the earlier field.

The overlaps yield the temporal derivative term in the continuum action, but replacing the two times by one too early can lose equal-time ordering information. Retaining the discrete prescription fixes contact terms and the meaning of occupation insertions. A continuum expression is therefore used together with its time-slice convention, especially when evaluating products at coincident times.

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.