Superintegrable relativistic systems in scalar background fields

TL;DR

This paper explores superintegrable relativistic systems influenced by scalar background fields, identifying conserved quantities from symmetries and providing exact solutions for quantum wavefunctions via the Klein-Gordon equation.

Contribution

It introduces new superintegrable relativistic models with scalar backgrounds and derives exact quantum solutions, expanding understanding of symmetry and integrability in relativistic quantum mechanics.

Findings

Conserved quantities arise from Poincaré, dilation, and conformal symmetries.

Explicit solutions to the Klein-Gordon equation are obtained.

Examples of superintegrable systems in relativistic settings are demonstrated.

Abstract

We consider a relativistic charged particle in a background scalar field depending on both space and time. Poincar\'e, dilation and special conformal symmetries of the field generate conserved quantities in the charge motion, and we exploit this to generate examples of superintegrable relativistic systems. We also show that the corresponding single-particle wavefunctions needed for the quantum scattering problem can be found exactly, by solving the Klein-Gordon equation.