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Abstract:
The ‘‘thermodynamic’’ partition function ZT(β)=Jnexp(βEn) 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 spin0 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 renormalizationscale (μ) dependence of ZQ. We derive a simple formula for A and B in terms of the ‘‘energy’’ ζ function ζE(z)=JkEk 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)Bε1+( 1/2)A+O(ε) (where the upper sign applies to bosons and the lower sign applies to fermions). We also give a hightemperature 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.