 Coherent-state oscillator vacuum-energy convention

An ordinary normal-ordered coherent-state Gaussian for energy $\epsilon a^{†}a$ gives $(1-e^{-\beta \epsilon }{)}^{-1}$. The physical oscillator with zero-point energy $\epsilon /2$ requires the factor $e^{-\beta \epsilon /2}$. A symmetric determinant prescription may encode that factor implicitly; it must not be counted twice.