Coherent-state time slicing

Insert the coherent-state resolution of identity between short-time evolution operators. For a normally ordered Hamiltonian, the kernel uses the next bra label and current ket label: $⟨{\psi }_{j+1}|e^{-ϵH}|{\psi }_j⟩=\exp ({\bar{\psi }}_{j+1}{\psi }_j-ϵH({\bar{\psi }}_{j+1},{\psi }_j)+O({ϵ}^2))$. Together with the Gaussian measure this yields the first-order derivative term $\bar{\psi }{\partial }_{\tau }\psi$. The adjacent-label prescription retains operator ordering information that a formal continuum expression alone can hide.

Insert coherent-state completeness between short factors of $e^{-\beta (H-\mu N)}$. The adjacent bra and ket labels determine the normal-ordered symbol $H_N({\bar{\psi }}_{j+1},{\psi }_j)$, while overlaps generate the derivative term. Bosonic labels are complex and periodic in imaginary time; fermionic labels are independent Grassmann variables and antiperiodic. The continuum shorthand is defined by that temporal ordering and measure, not an arbitrary substitution of commuting fields for operators.^[ref]

The thermal trace closes bosonic coherent-state paths periodically and fermionic coherent-state paths antiperiodically: $\psi (\beta )=\zeta \psi (0)$ and $\bar{\psi }(\beta )=\zeta \bar{\psi }(0)$, with $\zeta =1$ for bosons and $-1$ for fermions. The minus sign follows from the fermionic coherent-state trace, not from choosing a different Hamiltonian. These boundary conditions determine the allowed Matsubara frequencies.

An antiperiodic free fermion field with inverse propagator ${\partial }_{\tau }+\xi$ has a thermally normalized determinant $1+e^{-\beta \xi }$. The factor counts empty and occupied states of one level. Its logarithmic Matsubara sum is meaningful only with a regularization fixing the determinant normalization; an unqualified infinite product is not convergent.

The formal product over fermionic Matsubara frequencies is divergent before its normalization and time-slice prescription are fixed. The coherent-state thermal trace fixes the single-level factor to $1+e^{-\beta \xi }$, counting its empty and occupied states. A symmetrically regularized product can instead expose $2\cosh (\beta \xi /2)$; the additional factor $e^{-\beta \xi /2}$ restores the physical empty-state energy convention. Dropping that factor can incorrectly shift the grand potential and particle occupation.