General Solutions of Relativistic Wave Equations II: Arbitrary Spin Chains

TL;DR

This paper develops a framework for relativistic wave equations for arbitrary spin chains using harmonic analysis on complex spheres, providing explicit solutions and a Lagrangian formalism for these systems.

Contribution

It introduces a new construction of relativistic wave equations on complex homogeneous spaces for arbitrary spins, including explicit solutions and a Lagrangian approach.

Findings

Constructed relativistic wave equations for arbitrary spin chains.

Identified the Minkowski space and complex sphere as the optimal homogeneous space.

Derived general solutions via hyperspherical function expansions.

Abstract

A construction of relativistic wave equations on the homogeneous spaces of the Poincar\'{e} group is given for arbitrary spin chains. Parametrizations of the field functions and harmonic analysis on the homogeneous spaces are studied. It is shown that a direct product of Minkowski spacetime and two-dimensional complex sphere is the most suitable homogeneous space for the physical applications. The Lagrangian formalism and field equations on the Poincar\'{e} and Lorentz groups are considered. A boundary value problem for the relativistically invariant system is defined. General solutions of this problem are expressed via an expansion in hyperspherical functions defined on the complex two-sphere.