Does statistical mechanics equal one-loop quantum field theory?

Abstract

The ``thermodynamic'' partition function ${Z}^{T}$(\ensuremath{\beta})=${\mathcal{J}}_{n}$exp(-\ensuremath{\beta}${E}_{n}$) is compared to the Euclidean ``quantum'' path integral ${Z}^{Q}$(\ensuremath{\beta})=Fd[\ensuremath{\varphi}]exp(-S) over (anti)periodic fields \ensuremath{\varphi}(\ensuremath{\tau}+\ensuremath{\beta})=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}(\ensuremath{\tau}). We assume (1) free spin-0 or spin-(1/2) fields and (2) an ultrastatic spacetime. Our main result is that ${Z}^{T}$(\ensuremath{\beta}) does not equal ${Z}^{Q}$(\ensuremath{\beta}). Nevertheless, they are simply related: we prove that ln${Z}^{Q}$(\ensuremath{\beta})=ln${Z}^{T}$(\ensuremath{\beta})+(A+B ln${\ensuremath{\mu}}^{2}$)\ensuremath{\beta}. Thus, the logarithms of the two partition functions differ only by a term proportional to \ensuremath{\beta}. The constant A arises from vacuum energy and the constant B from the renormalization-scale (\ensuremath{\mu}) dependence of ${Z}^{Q}$. We derive a simple formula for A and B in terms of the ``energy'' \ensuremath{\zeta} function ${\ensuremath{\zeta}}^{E}$(z)=${\mathcal{J}}_{k}$${E}_{k}$${\mathrm{}}^{\mathrm{\ensuremath{-}}z}$. In particular we show that A and B are determined by the behavior of the energy \ensuremath{\zeta} function near z=-1: for small \ensuremath{\epsilon}, \ifmmode\pm\else\textpm\fi{}${\ensuremath{\zeta}}^{E}$(-1+\ensuremath{\epsilon})=(1/4)B${\ensuremath{\epsilon}}^{\mathrm{\ensuremath{-}}1}$+( 1/2)A+O(\ensuremath{\epsilon}) (where the upper sign applies to bosons and the lower sign applies to fermions). We also give a high-temperature expansion of Z(\ensuremath{\beta}) in terms of ${\ensuremath{\zeta}}^{E}$(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 \ensuremath{\beta} 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.