Does statistical mechanics equal one-loop quantum field theory?

The ``thermodynamic'' partition function ZT({β})=Jₙexp(-{β}Eₙ) is compared to the Euclidean ``quantum'' path integral ZQ({β})=Fd[{φ}]exp(-S) over (anti)periodic fields {φ}({τ}+{β})=±{}{φ}({τ}). We assume (1) free spin-0 or spin-(1/2) fields and (2) an ultrastatic spacetime. Our main result is that ZT({β}) does not equal ZQ({β}). Nevertheless, they are simply related: we prove that lnZQ({β})=lnZT({β})+(A+B lnμ²){β}. Thus, the logarithms of the two partition functions differ only by a term proportional to {β}. The constant A arises from vacuum energy and the constant B from the renormalization-scale ({μ}) dependence of ZQ. We derive a simple formula for A and B in terms of the ``energy'' {ζ} function ζE(z)=Jₖ{E}ₖ^-z. In particular we show that A and B are determined by the behavior of the energy {ζ} function near z=-1: for small {ε}, ±{}ζE(-1+{ε})=(1/4)Bε^-1+( 1/2)A+O({ε}) (where the upper sign applies to bosons and the lower sign applies to fermions). We also give a high-temperature expansion of Z({β}) in terms of ζE(z). Finally we argue that ZT and ZQ are interchangeable in any situation where gravitational effects are unimportant. This is because adding a term linear in {β} to lnZ is equivalent to shifting all energies by a constant; but if gravity is neglected, then the physics only depends upon the difference between energies, which is unchanged.