Dirac equation on coordinate dependent noncommutative space-time

TL;DR

This paper develops a Lorentz-covariant noncommutative Dirac equation for spinning particles on coordinate-dependent noncommutative space-time, ensuring key physical properties like probability conservation and standard dispersion relations are maintained.

Contribution

It introduces a new noncommutative Dirac equation that preserves Lorentz symmetry and fundamental physical properties in a coordinate-dependent noncommutative space-time.

Findings

The noncommutative Dirac equation maintains Lorentz covariance.

Probability current density is conserved in the noncommutative framework.

Standard energy-momentum dispersion relations hold in the noncommutative setting.

Abstract

We consider the consistent deformation of the relativistic quantum mechanics introducing the noncommutativity of the space-time and preserving the Lorentz symmetry. The relativistic wave equation describing the spinning particle on coordinate dependent noncommutative space-time (noncommutative Dirac equation) is proposed. The fundamental properties of this equation, like the Lorentz covariance and the continuity equation for the probability density are verified. To this end using the properties of the star product we derive the corresponding probability current density and prove its conservation. The energy-momentum tensor for the free noncommutative spinor field is calculated. We solve the free noncommutative Dirac equation and show that the standard energy-momentum dispersion relation remains valid in the noncommutative case.