Scattering using real-time path integrals

TL;DR

This paper introduces a novel real-time path integral method for directly calculating scattering observables, avoiding imaginary time and applicable to arbitrary short-range potentials, with results matching traditional solutions.

Contribution

The paper presents a new interpretation of path integrals as potential functionals on continuous paths, enabling direct real-time scattering calculations for short-range potentials.

Findings

Method accurately computes half-shell transition matrix elements.

Results agree with Lippmann-Schwinger equation solutions.

Method can be extended to complex systems.

Abstract

Background: Path integrals are a powerful tool for solving problems in quantum theory that are not amenable to a treatment by perturbation theory. Most path integral computations require an analytic continuation to imaginary time. While imaginary time treatments of scattering are possible, imaginary time is not a natural framework for treating scattering problems. Purpose: To test a recently introduced method for performing direct calculations of scattering observables using real-time path integrals. Methods: The computations are based on a new interpretation of the path integral as the expectation value of a potential functional on a space of continuous paths with respect to a complex probability distribution. The method has the advantage that it can be applied to arbitrary short-range potentials. Results: The new method is tested by applying it to calculate half-shell sharp-momentum transition matrix elements for one-dimensional potential scattering. The calculations for half shell transition operator matrix elements are in agreement with a numerical solution of the Lippmann-Schwinger equation. The computational method has a straightforward generalization to more complicated systems.