Semiclassical Series at Finite Temperature

Abstract

We derive the semiclassical series for the partition function of a one-dimensional quantum-mechanical system consisting of a particle in a single-well potential. We do this by applying the method of steepest descent to the path-integral representation of the partition function, and we present a systematic procedure to generate the terms of the series using the minima of the 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 investigate their high and low temperature limits. We also exhibit the nonperturbative character of the terms, as each corresponds to sums over infinite subsets of perturbative graphs. We illustrate the power of such resummations 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 out possible extensions of our results which include field theories with spherically symmetric classical solutions.Comment: 25 pages + 5 figure

Erratum: Thermal and quantum fluctuations around domain walls [Phys. Rev. D65, 065021 (2002)]

“…Taking the field theory lane of that two-way road, we have recently studied the problem of quantizing the simplest of field theories, one-dimensional quantum mechanics, whose (scalar) "field" lives in zero spatial dimension [5][6][7][8]. Working in imaginary time, i.e., at finite temperature, and with smooth potentials, we were able to show that a full semiclassical series could be constructed from the knowledge of semiclassical propagators in the classical backgrounds of the solutions of the equations of motion, and gave a general prescription for deriving them from those backgrounds.…”

“…The latter would require finding all classical solutions that satisfy the finite temperature boundary conditions, and a general procedure for constructing semiclassical propagators in their backgrounds. As we have already stated, in quantum mechanics the general procedure does exist, and it is often possible to find all classical solutions [5][6][7][8]. In field theories, however, finding all classical solutions is already a difficult problem, and even if we find them all, no general procedure to obtain semiclassical propagators in their backgrounds is known.…”

“…We note that even in one spatial dimension, where the domain wall has finite Euclidean action, and therefore contributes to the semiclassical approximation, we abdicate the idea of "solving" the theory semiclassically, because there are many other solutions whose semiclassical propagators cannot be obtained from a general recipe. In particular, solutions which only depend on Euclidean time do contribute: they are given by elliptic functions, similar to the ones in quantum mechanics [5][6][7][8], but whose fluctuation problems are much harder to solve.…”

Semiclassical calculation of decay rates

“…Surprisingly, the first term of the semiclassical series can already produce accurate results. As an example, let us consider the single-well quartic potential: [8]. We see that the semiclassical quadratic approximation is in good agreement with numerical techniques that used optimized perturbation theory, even for large values of the coupling.…”

Semiclassical approximation for the partition function in QFT

“…In one-dimensional quantum mechanics, strong results on ordinary differential equations lead to a simple expression for the semiclassical propagator in terms of the classical solution. Using the semiclassical propagator, one can construct the whole semiclassical series, either in quantum mechanics [1] or in quantum statistical mechanics [2].…”