Exact Path-Integral Representations for the $T$-Matrix in Nonrelativistic Potential Scattering

TL;DR

This paper introduces exact path integral representations for the nonrelativistic T-matrix, capturing both the full Born series and eikonal approximation, using phantom degrees of freedom and a Faddeev-Popov constraint.

Contribution

It presents novel path integral formulations for the T-matrix in potential scattering that unify different approximation regimes and incorporate a new constraint for energy conservation.

Findings

Path integral representations reproduce the complete Born series.

The formulations include the eikonal approximation as a special case.

An approach for stochastic evaluation of the real-time path integral is discussed.

Abstract

Several path integral representations for the $T$-matrix in nonrelativistic potential scattering are given which produce the complete Born series when expanded to all orders and the eikonal approximation if the quantum fluctuations are suppressed. They are obtained with the help of "phantom" degrees of freedom which take away explicit phases that diverge for asymptotic times. Energy conservation is enforced by imposing a Faddeev-Popov-like constraint in the velocity path integral. An attempt is made to evaluate stochastically the real-time path integral for potential scattering and generalizations to relativistic scattering are discussed.