Path-integral formulation of scattering theory

Abstract

A new formulation of nonrelativistic scattering theory is developed which expresses the S matrix as a path integral. This formulation appears to have at least two advantages: (1) A closed formula is obtained for the 8 matrix in terms of the potential, not involving a series expansion; (2) the energy-conserving 8 function can be explicitly extracted using a technique analogous to that of Faddeev and Popov, thereby yielding a closed pathintegral expression for the T matrix. The introduction of the concept of the classical interaction picture provides considerable physical insight into this formulation. This formulation also suggests a successionof improvements to the eikonal approximation, the first of which is discussed explicitly.

Path integrals for potential scattering

“…In the present work we will derive a path-integral representation of the T matrix in potential scattering which is similar to the one given by Campbell et al [18] many years ago (see also Refs. [19,20]).…”

Scattering theory with path integrals

“…The path-integral expressions (12) and (8) are both exact but emphasize the initial and final momentum, respectively, as do the starting points (9) and (1). A more symmetrical form would be useful, in particular when approximations are made.…”

“…Here we start from the standard expression for the T -matrix 1 for scattering from a local potential V (x) (see e. g. Ref. [12], eq.…”

“…Following the pioneering work of Ref. [1] there has been renewed interest in the last few years to use it also for scattering problems [2,3,4] in nonrelativistic physics. This offers the chance of finding new approximation methods [5,6] or trying to evaluate the involved path integrals by stochastic methods although the oscillating nature of the real-time path integral presents a great challenge (see e.g.…”

Variational approximations in a path integral description of potential scattering

“…Oddly enough, scattering has mostly remained outside the path-integral approach and it is only at zero energy that bounds for the scattering length have been obtained from the path integral in the imaginary-time formulation [14,15]. Expanding on previous attempts [16] a realtime path-integral representation for the nonrelativistic T -matrix in potential scattering has recently been derived [17]. In this formulation the particle travels mainly along a simple reference path while quantum fluctuations around this path are taking into account by functional integration over velocities.…”