Relativistic two-body equation based on the extension of the SL(2,C) group

TL;DR

This paper introduces a novel relativistic two-body wave equation derived from extending the $SL(2,C)$ group to $Sp(4,C)$, providing exact solutions for systems with linear confinement potentials.

Contribution

It develops a new group-theoretic framework for two-body relativistic equations and constructs an exactly solvable wave equation with linear confinement.

Findings

Wave equation with linear confinement has an oscillator-like form.

Exact solutions are obtained for the two-body system.

The approach extends the symmetry group to $Sp(4,C)$.

Abstract

A new approach to the two-body problem based on the extension of the $SL(2,C)$ group to the $Sp(4,C)$ one is developed. The wave equation with various forms of including the interaction for the system of the spin-1/2 and spin-0 particles is constructed. For this system, it was found that the wave equation with a linear confinement potential involved in the non-minimal manner has an oscillator-like form and possesses the exact solution.