 Solution

Let $\epsilon =\hslash \omega$, use $M$ time slices and $q=e^{-\beta \epsilon /M}$. The exact normal-ordered coherent Gaussian has cyclic matrix $A=1-qP$, where $P$ shifts one slice. Its eigenvalues are $1-q{e}^{2\pi in/M}$ andRestoring the constant energy givesconsistent with the coordinate path integral. Alternatively, pairing positive and negative Matsubara frequencies symmetrically yields a determinant proportional to $x{\prod }_{n\ge 1}[1+(x/2\pi n{)}^2]=2\sinh (x/2)$. This symmetric prescription already incorporates the zero-point convention and must not receive the vacuum factor twice. A naive unregulated frequency product can hide this difference.