Path Integrals for Potential Scattering

TL;DR

This paper derives two path integral representations for the T-matrix in nonrelativistic potential scattering, ensuring they reproduce the complete Born series and incorporate energy conservation, with potential applications in real-time evaluations and high-energy approximations.

Contribution

The paper introduces novel path integral formulations for the T-matrix that include phantom degrees of freedom and Faddeev-Popov constraints, advancing the theoretical framework for potential scattering analysis.

Findings

Derivation of two path integral representations for the T-matrix.

These representations reproduce the complete Born series.

Development of high-energy approximations and systematic expansions.

Abstract

Two path integral representations for the $T$-matrix in nonrelativistic potential scattering are derived and proved to produce the complete Born series when expanded to all orders. They are obtained with the help of "phantom" degrees of freedom which take away explicit phases that diverge for asymptotic times. In addition, energy conservation is enforced by imposing a Faddeev-Popov-like constraint in the velocity path integral. These expressions may be useful for attempts to evaluate the path integral in real time and for alternative multiple scattering expansions. Standard and novel eikonal-type high-energy approximations and systematic expansions immediately follow.