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[-βV[sup eff cl](x[sub 0])], where V[sup eff cl](x[sub 0]) is the so-called effective classical potential containing the effects of all quantum fluctuations. The variable of integration is the temporal path average [formula]. We show how to generalize this concept to paths q[sup μ](τ) in curved space with metric g[sub μν](q), and calculate perturbatively the high-temperature expansion of V[sup eff cl](q[sub 0]). The requirement of independence under coordinate transformations q[sup μ](τ)→ q′[sup μ](τ) introduces subtleties in the definition and treatment of the path average [formula], and covariance is achieved only with the help of a suitable Faddeev–Popov procedure. [ABSTRACT FROM AUTHOR]