Relativistic Spinning Particle without Grassmann Variables and the Dirac Equation

A A Deriglazov

doi:10.1155/2011/680367

Abstract

We present the relativistic particle model without Grassmann variables which, being canonically quantized, leads to the Dirac equation. Classical dynamics of the model is in correspondence with the dynamics of mean values of the corresponding operators in the Dirac theory. Classical equations for the spin tensor are the same as those of the Barut-Zanghi model of spinning particle.

Highlights

We present the relativistic particle model without Grassmann variables which, being canonically quantized, leads to the Dirac equation
Starting from the classical works 1–6, a lot of efforts have been spent in attempts to understand behaviour of a particle with spin on the base of semiclassical mechanical models 7–24
Antisymmetric symmetric classical bracket arises in the classical mechanics of even odd Grassmann variables

Summary

Discussion

Starting from the classical works 1–6 , a lot of efforts have been spent in attempts to understand behaviour of a particle with spin on the base of semiclassical mechanical models 7–24. Since the quantum theory of spin is known it is given by the Pauli Dirac equation for nonrelativistic relativistic case , search for the corresponding semiclassical model represents the inverse task to those of canonical quantization: we look for the classicalmechanics system whose classical bracket obeys 1.1 for the known left-hand side. Canonical quantization of the model immediately produces the Pauli equation 1.6 We generalize this scheme to the relativistic case, taking angular-momentum variables as the basic coordinates of the spin space. On this base, we construct the relativistic-invariant classical mechanics that produces the Dirac equation after the canonical quantization, and briefly discuss its classical dynamics

Algebraic Construction of the Relativistic Spin Space

Dynamical Realization

They imply xμ μν pν