Quantum mechanics with coordinate dependent noncommutativity

Abstract

Noncommutative quantum mechanics can be considered as a first step in the construction of quantum field theory on noncommutative spaces of generic form, when the commutator between coordinates is a function of these coordinates. In this paper we discuss the mathematical framework of such a theory. The noncommutativity is treated as an external antisymmetric field satisfying the Jacoby identity. First, we propose a symplectic realization of a given Poisson manifold and construct the Darboux coordinates on the obtained symplectic manifold. Then we define the star product on a Poisson manifold and obtain the expression for the trace functional. The above ingredients are used to formulate a nonrelativistic quantum mechanics on noncommutative spaces of general form. All considered constructions are obtained as a formal series in the parameter of noncommutativity. In particular, the complete algebra of commutation relations between coordinates and conjugated momenta is a deformation of the standard Heisenberg algebra. As examples we consider a free particle and an isotropic harmonic oscillator on the rotational invariant noncommutative space.

35 pages, new material concerning the trace functional, new physical example and new references added

References in corpus (6)

• Position-dependent noncommutativity in quantum mechanics
• Hydrogen atom on curved noncommutative space
• Geodesic equation in $k$-Minkowski spacetime
• Noncommutativity due to spin
• Star products made (somewhat) easier
• Path integral representations in noncommutative quantum mechanics and noncommutative version of Berezin-Marinov action

Cited by in corpus (18)

• Hydrogen atom on curved noncommutative space
• Energy-dependent noncommutative quantum mechanics
• Noncommutative $R^d$ via closed star product
• Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry
• Twists, realizations and Hopf algebroid structure of kappa-deformed phase space
• Natural Cutoffs via Compact Symplectic Manifolds
• $\kappa$-Deformed Phase Space, Hopf Algebroid and Twisting
• Symplectic realisation of electric charge in fields of monopole distributions
• $k$-Minkowski-deformation of $U(1)$ gauge theory
• Universal $\kappa$-Poincaré covariant differential calculus over $\kappa$-Minkowski space
• Dirac equation on coordinate dependent noncommutative space-time
• Recurrence relations for symplectic realization of (quasi)-Poisson structures
• Canonical Structure of Noncommutative Quantum Mechanics as Constraint System
• Conserved symmetries in noncommutative quantum mechanics
• Field Theory with Coordinate Dependent Noncommutativity
• Higher Order Theories and its Relationship with Noncommutativity
• Noncommutative Geometry and dynamical models on U(u(2)) background
• Noncommutative Classical Dynamics on Velocity Phase Space and Souriau Formalism