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 affects spin interactions and quantum transitions, providing bounds on noncommutativity.

Contribution

It presents a novel nonrelativistic wave equation and pseudoclassical model for spinning particles in noncommutative space, extending the understanding of spin interactions under space noncommutativity.

Findings

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

Showed that noncommutativity allows forbidden EPR state transitions in a magnetic field.

Estimated an upper bound on the noncommutativity parameter based on transition probabilities.

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. We extract $\theta$-modified interaction between a nonrelativistic spin and a magnetic field from such a 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. This allows us to estimate an upper bound on the noncommutativity parameter.