Causality and dimensionality in geometric scattering

TL;DR

This paper explores how causality and spatial dimensionality influence the geometric scattering matrix for spin-1/2 fermions, revealing that dimensionality affects the phase of the harmonic potential in the geometric action.

Contribution

It introduces a geometric action principle for scattering that incorporates causality constraints and explicitly encodes dimensionality effects.

Findings

Causality constrains scattering trajectories in the geometric framework.

Dimensionality affects the phase of the harmonic potential in the geometric action.

The geometric formulation applies to both zero-range and finite-range potentials.

Abstract

The scattering matrix which describes low-energy, non-relativistic scattering of spin-1/2 fermions interacting via finite-range potentials can be obtained from a geometric action principle in which space and time do not appear explicitly arXiv:2011.01278. In the case of zero-range forces, causality leads to constraints on scattering trajectories in the geometric picture. The effect of spatial dimensionality is also investigated by considering scattering in two and three dimensions. In the geometric formulation it is found that dimensionality is encoded in the phase of the harmonic potential that appears in the geometric action.