Non-Grassmann mechanical model of the Dirac equation

TL;DR

This paper introduces a novel spinning-particle model for the Dirac equation that avoids Grassmann variables by using an inner anti-de Sitter space to describe spin, leading to new insights into space-time interpretation and classical solutions.

Contribution

It presents a new non-Grassmann mechanical model for the Dirac equation with detailed classical and quantum analysis, connecting spin, inner space, and classical orbit analogies.

Findings

Model reproduces Dirac equation via canonical quantization.

Classical solutions reveal spin and space-time interpretation.

Analogies with two-body confining systems and elliptic orbits.

Abstract

We construct a new example of the spinning-particle model without Grassmann variables. The spin degrees of freedom are described on the base of an inner anti-de Sitter space. This produces both $\Gamma^\mu$ and $\Gamma^{\mu\nu}$\,-matrices in the course of quantization. Canonical quantization of the model implies the Dirac equation. We present the detailed analysis of both the Lagrangian and the Hamiltonian formulations of the model and obtain the general solution to the classical equations of motion. Comparing {\it Zitterbewegung} of the spatial coordinate with the evolution of spin, we ask on the possibility of space-time interpretation for the inner space of spin. We enumerate similarities between our analogous model of the Dirac equation and the two-body system subject to confining potential which admits only the elliptic orbits of the order of de Broglie wave-length. The Dirac equation dictates the perpendicularity of the elliptic orbits to the direction of center-of-mass motion.