Quantum and pseudoclassical descriptions of nonrelativistic spinning particles in noncommutative space

TL;DR

This paper develops a noncommutative space version of the Pauli equation for spinning particles, introduces a pseudoclassical model, and explores how noncommutativity enables forbidden spin transitions in magnetic fields.

Contribution

It constructs a $ heta$-modified nonrelativistic wave equation and pseudoclassical model for spinning particles in noncommutative space, linking them through quantization and analyzing novel spin interactions.

Findings

Derived a $ heta$-modified Pauli equation from the nonrelativistic limit of the $ heta$-modified Dirac equation.

Showed that noncommutativity allows forbidden EPR spin transitions in magnetic fields.

Established a $ heta$-modified Heisenberg model for coupled spins in external magnetic fields.

Abstract

We construct a nonrelativistic wave equation for spinning particles in the noncommutative space (in a sense, a $\theta$-modification of the Pauli equation). To this end, we consider the nonrelativistic limit of the $\theta$-modified Dirac equation. To complete the consideration, we present a pseudoclassical model (\`a la Berezin-Marinov) for the corresponding nonrelativistic particle in the noncommutative space. To justify the latter model, we demonstrate that its quantization leads to the $\theta$-modified Pauli equation. Then, we extract $\theta$-modified interaction between a nonrelativistic spin and a magnetic field from the $\theta$-modified Pauli equation and construct a $\theta$-modification of the Heisenberg model for two coupled spins placed in an external magnetic field. In the framework of such a model, we calculate the probability transition between two orthogonal EPR (Einstein-Podolsky-Rosen) states for a pair of spins in an oscillatory magnetic field and show that some of such transitions, which are forbidden in the commutative space, are possible due to the space noncommutativity.