A novel “magnetic” field and its dual non-commutative phase space

In this Letter 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 “internal  ” magnetic field, that is singular at a specific position θ⁻¹${\theta }^{\mathrm{-1}}$. By “internal” we mean that the effective magnetic fields depends essentially on the particle properties and modifies the symplectic structure. Here θ 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 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) real space Berry curvature in material systems.