Functional Integration and Semiclassical Expansions

J. Florencio

We present a functional-integral method that is suitable for obtaining nonperturbative solutions to the partition function of fermion systems. The density matrix is cast as an expansion about the classical path contributions, and the quantum corrections can be calculated systematically by using retarded propagators in the space holomorphic functions. It is shown that the semiclassical expansion amounts to a high-temperature expansion, where the first nonvanishing quantum correction is of O(${\mathrm{\ensuremath{\beta}}}^{2}$). Each term of the semiclassical expansion may contain the coupling parameter in all orders of perturbation theory. The method can also be extended to boson systems.

Cécile DeWitt-Morette

Joachim Ankerhold

B. Mehlig

We derive a semiclassical expansion of quantum spectral functions for integrable systems. This complements a previous analysis of chaotic billiards [B. Mehlig et al., Phys. Rev. Lett. 75, 57 (1995)]. We discuss similarities and differences between the integrable and the chaotic case. As a numerical example, we consider a two-dimensional circular billiard.

Zhuangqi Cao, Cheng Yin

Maurice M. Mizrahi

This paper shows how to calculate the terms of a semiclassical (WKB) expansion of the quantum-mechanical propagator corresponding to the quartic anharmonic-oscillator potential, V=mω2q2/2 +λq4/4. This nonperturbative treatment expresses each term in the series as a path integral, which is then evaluated in the framework of a formalism, introduced by C. DeWitt-Morette, which does not entail the usual time-slicing operation followed by a limiting procedure. The Gaussian measure used absorbs all the quadratic terms in the expansion of the action functional about a classical path. The covariance of this Gaussian measure is Feynman’s Green function for the small-disturbance operator of the system. This function can be obtained by varying the constants of integration in the classical solution, and therefore the coefficients of the expansion depend only on this classical solution. If the latter is chosen to be the one which tends to its harmonic counterpart when λ→0, then it is seen that the propagator also tends to its harmonic counterpart when λ→0.