Path Integrals and Complex Dynamics

Vladimir G. Ivancevic, Tijana T. Ivancevic

Jeremy Schiff B, Yair Goldfarb, David J. Tannor

Path-integral derivations are presented for two recently developed complex trajectory techniques for the propagation of wave packets: complex WKB and Bohmian mechanics with complex action (BOMCA). The complex WKB technique is derived using a standard saddle-point approximation of the path integral, but taking into account the $\ensuremath{\hbar}$ dependence of both the amplitude and the phase of the initial wave function, thus giving rise to the need for complex classical trajectories. The BOMCA technique is derived using a modification of the saddle-point technique, in which the path integral is approximated by expanding around a near-classical path, chosen so that up to some predetermined order there is no need to add any correction terms to the leading-order approximation. Both complex WKB and BOMCA techniques give the same leading-order approximation; in the complex WKB technique higher accuracy is achieved by adding correction terms, while in the BOMCA technique no additional terms are ever added: higher accuracy is achieved by changing the path along which the original approximation is computed. The path-integral derivation of the methods explains the need to incorporate contributions from more than one trajectory, as observed in previous numerical work. On the other hand, it emerges that the methods provide efficient schemes for computing the higher-order terms in the asymptotic evaluation of path integrals. The understanding we develop of the BOMCA technique suggests that there should exist near-classical trajectories that give exact quantum dynamical results when used in the computation of the path integral keeping just the leading-order term. We also apply our path-integral techniques to give a compact derivation of the semiclassical approximation to the coherent-state propagator.

Evgeny Polyakov, Alexey N. Rubtsov

The derivation of path integrals is reconsidered. It is shown that the expression for the discretized action is not unique, and the path integration domain can be deformed so that at least Gaussian path integrals become probabillistic. This leads to a practical algorithm of sign-problem-free Monte Carlo sampling from the Gaussian path integrals. Moreover, the dynamical influence of Gaussian quantum system (the bath) on any other quantum system can be exactly represented as interaction with classical non-Markovian noise. We discuss the relation of these findings to the Bell's theorem and the Feynman's conjecture on the exponential complexity of the classical simulation of quantum systems. In Feynman's path integral we have quasiprobability distributions for trajectories, and in analitycally continued path integrals we have probability distributions for quasitrajectories.

Osvaldo Civitarese, Manuel Gadella

F. A. Buot

It is suggested that complex problems in ultrasubmicrometer electronics research may greatly benefit from use of the path-integral technique. The use of the Weyl-Wigner formalism of the quantum dynamics of electrons in solids provides a rigorous and straightforward derivation of the path integral in solid-state physics, both from the single-particle and from the many-body field-theoretical description of electron dynamics, without the need to postulate a priori the isomorphism between quantum operators and c-numbers of the base field. A rigorous construction of the path integral in many-body solid-state band theory necessitates a two-stage Weyl correspondence between quantum operators and c-numbers of the base field, namely, the Weyl correspondence of the base field of ``lattice-space'' particle-dynamical variables and that of the continuum many-body field-dynamical variables.

Walter Dittrich, M. Reuter