Perturbatively Defined Effective Classical Potential in Curved Space

Abstract

The partition function of a quantum statistical system in flat space can always be written as an integral over a classical Boltzmann factor $\exp[ -βV^{\rm eff cl({\bf x}_0)]$, where $V^{\rm eff cl({\bf x}_0)$ is the so-called effective classical potential containing the effects of all quantum fluctuations. The variable of integration is the temporal path average ${\bf x_0\equiv β^{-1}\int_0^ βdτ{\bf x}(τ)$. We show how to generalize this concept to paths $q^{\mu }(\tau )$ in curved space with metric $g_{μν(q)$, and calculate perturbatively the high-temperature expansion of $V^{\rm eff cl(q_0)$. The requirement of independence under coordinate transformations $q^{\mu }(\tau )\to q^{\prime \mu }(\tau )$ introduces subtleties in the definition and treatment of the path average $q_0^{\mu }$, and covariance is achieved only with the help of a suitable Faddeev-Popov procedure.

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