On the group theoretical approach to relativistic wave equations for arbitrary spin

TL;DR

This paper investigates algebraic methods for formulating relativistic wave equations for particles of arbitrary spin, identifying issues with unitarity and mass spectra, and proposing subsidiary conditions to resolve these problems.

Contribution

It demonstrates that using anti-de Sitter group spin matrices leads to inconsistencies, and shows how subsidiary conditions can restore physical validity.

Findings

Anti-de Sitter group approach causes unitarity violation.

Subsidiary conditions restore unitarity and physical mass spectrum.

Solutions to the equations are analyzed for physical interpretation.

Abstract

Formulating a relativistic equation for particles with arbitrary spin remains an open challenge in theoretical physics. In this study, the main algebraic approaches used to generalize the Dirac and Kemmer Duffin equations for particles of arbitrary spin are investigated. It is proved that an irreducible relativistic equation formulated using spin matrices satisfying the commutation relations of the anti-de Sitter group leads to inconsistent results, mainly as a consequence of violation of unitarity and the appearance of a mass spectrum that does not reflect the physical reality of elementary particles. However, the introduction of subsidiary conditions resolves the problem of unitarity and restores the physical meaning of the mass spectrum. The equations obtained by these approaches are solved and the physical nature of the solutions is discussed.