On Hierarchy and Equivalence of Relativistic Equations for Massive Fields

TL;DR

This paper establishes a novel correspondence between solutions of Dirac and Klein-Gordon equations, offering new insights into their relationship, spinor transformations, and potential extensions to curved space-time and interactions.

Contribution

It introduces a non-canonical method to relate solutions of relativistic equations for different spins, revealing a hierarchy and equivalence that deepen understanding of their structure.

Findings

A chain of solutions linking Dirac and Klein-Gordon equations is constructed.

Comparison with the massless case highlights differences in solution structures.

Examples of non-trivial singular solutions are provided.

Abstract

A non-canonical correspondence of the complete sets of solutions to the Dirac and Klein-Gordon free equations in Minkowski space-time is established. This allows for a novel viewpoint on the relationship of relativistic equations for different spins and on the origin of spinor transformations. In particular, starting from a solution to the Dirac equation, one obtains a chain of other solutions to both Dirac and Klein-Gordon equations. A comparison with the massless case is performed, and examples of non-trivial singular solutions are presented. A generalization to Riemannian space-time and inclusion of interactions are briefly discussed.