Perturbatively Defined Effective Classical Potential in Curved Space

The partition function of a quantum statistical system in flat space can always be written as an integral over a classical Boltzmann factor \exp[ -\beta V^{\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 \beta ^{-1}\int_0^ \beta d\tau {\bf x}(\tau). We show how to generalize this concept to paths $q^\mu(\tau)$ in curved space with metric g_{\mu \nu (q), and calculate perturbatively the high-temperature expansion of V^{\rm eff cl(q_0). The requirement of independence under coordinate transformations $q^\mu(\tau)\to q'^\mu(\tau)$ 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.