A Novel "Magnetic" Field And Its Dual Non-Commutative Phase Space

TL;DR

This paper explores a new non-commutative phase space with an operator-based magnetic field, revealing unique particle interactions, modified properties, and potential links to real-world Berry curvature phenomena.

Contribution

It introduces a novel operatorial non-commutative phase space with an internal magnetic field affecting particle dynamics and properties, expanding understanding of non-commutative geometries.

Findings

Particle experiences an internal magnetic field affecting its symplectic structure.

Interaction with external magnetic fields modifies charge and mass properties.

Dynamics can reduce to Hall-type motion under certain conditions.

Abstract

In this paper we have studied a new form of Non-Commutative (NC) phase space with an operatorial form of noncommutativity. A point particle in this space feels the effect of an interaction with an "{\it{internal}}" magnetic field, that is singular at a specific position $\theta^{-1}$. By "internal" we mean that the effective magnetic fields depends essentially on the particle properties and modifies the symplectic structure. Here $\theta $ is the NC parameter and induces the coupling between the particle and the "internal" magnetic field. The magnetic moment of the particle is computed. Interaction with an {\it{external}} physical magnetic field reveals interesting features induced by the inherent fuzziness of the NC phase space: introduction of non-trivial structures into the charge and mass of the particle and possibility of the particle dynamics collapsing to a Hall type of motion. The dynamics is studied both from Lagrangian and symplectic (Hamiltonian) points of view. The canonical (Darboux) variables are also identified. We briefly comment, that the model presented here, can play interesting role in the context of (recently observed) {\it{real}} space Berry curvature in material systems.