Geometry and entanglement in the scattering matrix

TL;DR

This paper introduces a geometric, holographic framework for nucleon-nucleon scattering using the S-matrix as the fundamental object, revealing connections between entanglement, unitarity, and RG fixed points in a novel operator space.

Contribution

It develops a geometric embedding of unitarity constraints in a flat torus, linking entanglement to geodesics and providing an integrable system at low energies with holographic error correction insights.

Findings

Low-energy S-matrix exhibits conformal invariance and integrability.

Entanglement corresponds to geodesics on the flat torus.

Inelasticity relates to hyperbolic space radius and unitarity violation.

Abstract

A formulation of nucleon-nucleon scattering is developed in which the S-matrix, rather than an effective-field theory (EFT) action, is the fundamental object. Spacetime plays no role in this description: the S-matrix is a trajectory that moves between RG fixed points in a compact theory space defined by unitarity. This theory space has a natural operator definition, and a geometric embedding of the unitarity constraints in four-dimensional Euclidean space yields a flat torus, which serves as the stage on which the S-matrix propagates. Trajectories with vanishing entanglement are special geodesics between RG fixed points on the flat torus, while entanglement is driven by an external potential. The system of equations describing S-matrix trajectories is in general complicated, however the very-low-energy S-matrix -- that appears at leading-order in the EFT description -- possesses a UV/IR conformal invariance which renders the system of equations integrable, and completely determines the potential. In this geometric viewpoint, inelasticity is in correspondence with the radius of a three-dimensional hyperbolic space whose two-dimensional boundary is the flat torus. This space has a singularity at vanishing radius, corresponding to maximal violation of unitarity. The trajectory on the flat torus boundary can be explicitly constructed from a bulk trajectory with a quantifiable error, providing a simple example of a holographic quantum error correcting code.