= Thermal normalization of a fermion determinant
{title2=$\det_{\rm AP}(\partial_\tau+\xi)=1+e^{-\beta\xi}$}
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.
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