Semiclassical series at finite temperature

We derive the semiclassical series for the partition function of a one-dime nsional quantum-mechanical system consisting of a particle in a single-well potential. We do this by applying rile method of steepest descent to the p ath-integral representation of the partition function, and we present a sys tematic procedure to generate the terms of the series using the minima or t he Euclidean action as the only input. For the particular case of a quartic anharmonic oscillator. we compute the first two terms of the series, and i nvestigate their high and low temperature limits. We also exhibit the nonpe rturbative character of the terms, as each corresponds to sums over infinit e subsets or perturbative graphs. We illustrate the power of such resummati ons by extracting from the first term an accurate nonperturbative estimate of the ground-state energy of the system and a curve for the specific heat. We conclude by pointing our possible extensions of our results which inclu de field theories with spherically symmetric classical solutions. (C) 1999 Academic Press.