Coherent-state thermal boundary conditions ($\psi (\beta )=\zeta \psi (0)$)

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.