Classical description of spinning degrees of freedom of relativistic particles by means of commuting spinors

TL;DR

This paper explores classical models using commuting spinors to describe relativistic particles with spin, demonstrating how these models can reproduce the Dirac equation and discussing their potential for higher spin interactions.

Contribution

It introduces two classical gauge models with commuting Majorana spinors for describing spin-half and higher spins, analyzing their quantization and interaction capabilities.

Findings

Both models reproduce the Dirac equation upon quantization.

The first model allows minimal electromagnetic interaction but yields reducible representations.

The second model is suitable for higher spins but faces challenges with electromagnetic interactions.

Abstract

We consider a possibility to describe spin one-half and higher spins of massive relativistic particles by means of commuting spinors. We present two classical gauge models with the variables $x^\mu,\xi_\alpha,\chi_\alpha$, where $\xi,\chi$ are commuting Majorana spinors. In course of quantization both models reproduce Dirac equation. We analyze the possibility to introduce an interaction with an external electromagnetic background into the models and to generalize them to higher spin description. The first model admits a minimal interaction with the external electromagnetic field, but leads to reducible representations of the Poincare group being generalized for higher spins. The second model turns out to be appropriate for description of the massive higher spins. However, it seams to be difficult to introduce a minimal interaction with an external electromagnetic field into this model. We compare our approach with one, which uses Grassman variables, and establish a relation between them.