Fermionic one-level thermal determinant
= Fermionic one-level thermal determinant
{title2=$\det_{\rm AP}(\partial_\tau+\xi)=1+e^{-\beta\xi}$}
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.