Geometry and entanglement in the scattering matrix

• Published in: Annals of physics 2021-10, Vol.433 (C), p.168581, Article 168581
• Main Authors: Beane, Silas R., Farrell, Roland C.
• Format: Article
• Language: English
• Subjects: Effective field theory; Nuclear theory; Quantum entanglement
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/j.aop.2021.168581

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. •Geometrical S-matrix construction of nucleon–nucleon scattering.•Leading order in effective range expansion has UV/IR symmetry not seen in action.•Space of S-matrices maps to the flat torus.•Entanglement needed to probe full manifold of S-matrices.•Inelasticities correspond to the radius of a hyperbolic space.