The ‘‘thermodynamic’’ partition function $Z^T$(β)=${\mathcal{J}}_n$exp(-β$E_n$) is compared to the Euclidean ‘‘quantum’’ path integral $Z^Q$(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 $Z^T$(β) does not equal $Z^Q$(β). Nevertheless, they are simply related: we prove that ln$Z^Q$(β)=ln$Z^T$(β)+(A+B ln${\mu }^2$)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 $Z^Q$. We derive a simple formula for A and B in terms of the ‘‘energy’’ ζ function ${\zeta }^E$(z)=${\mathcal{J}}_k$$E_k$${\mathrm{}}^{\mathrm{-}z}$. In particular we show that A and B are determined by the behavior of the energy ζ function near z=-1: for small ε, ±${\zeta }^E$(-1+ε)=(1/4)B${\epsilon }^{\mathrm{-}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 ${\zeta }^E$(z). Finally we argue that $Z^T$ and $Z^Q$ 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.

• Received 27 January 1986

DOI:https://doi.org/10.1103/PhysRevD.33.3640