Noncommutativity due to spin

TL;DR

This paper introduces a classical model of spin-induced noncommutativity that maintains Lorentz invariance and leads to modified quantum equations, revealing spin-dependent nonlocality effects.

Contribution

It presents a novel classical framework for spin noncommutativity that extends to relativistic quantum equations while preserving Lorentz invariance.

Findings

Noncommutativity depends on particle spin, with no nonlocality for spin zero.

Modified Pauli and Dirac equations incorporate spin noncommutativity.

Nonlocality varies with spin, affecting spatial uncertainty relations.

Abstract

Using the Berezin-Marinov pseudoclassical formulation of spin particle we propose a classical model of spin noncommutativity. In the nonrelativistic case, the Poisson brackets between the coordinates are proportional to the spin angular momentum. The quantization of the model leads to the noncommutativity with mixed spacial and spin degrees of freedom. A modified Pauli equation, describing a spin half particle in an external e.m. field is obtained. We show that nonlocality caused by the spin noncommutativity depends on the spin of the particle; for spin zero, nonlocality does not appear, for spin half, $\Delta x\Delta y\geq\theta^{2}/2$, etc. In the relativistic case the noncommutative Dirac equation was derived. For that we introduce a new star product. The advantage of our model is that in spite of the presence of noncommutativity and nonlocality, it is Lorentz invariant. Also, in the quasiclassical approximation it gives noncommutativity with a nilpotent parameter.