On the $\star$-product quantization and the Duflo map in three dimensions

Abstract

We analyze the star product induced on the algebra of functions on R^3 by a suitable reduction of the Moyal product defined on F(R^4). This is obtained through the identification of R^3 with the dual of a three dimensional Lie algebra. We consider the su(2) case, exhibit a matrix basis and realize the algebra of functions on its dual in such a basis. The relation to the Duflo map is discussed. As an application to quantum mechanics we compute the spectrum of the hydrogen atom.

Misprints corrected. Journal ref. added

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• Quantum gauge theories on noncommutative 3-d space
• Noncommutative $R^d$ via closed star product
• Matrix Bases for Star Products: a Review
• Noncommutative gauge theories on ${\mathbb{R}}_{\lambda }^3$: Perturbatively finite models
• Closed star product on noncommutative ${\mathbb{R}}^3$ and scalar field dynamics
• Doubling, T-Duality and Generalized Geometry: a Simple Model
• Noncommutative Gauge Theory and Gravity in Three Dimensions
• Involutive representations of coordinate algebras and quantum spaces
• Dirac equation on coordinate dependent noncommutative space-time
• Non-commutative Fourier transform for the Lorentz group via the Duflo map
• $\mathcal{P}\mathcal{T}$-symmetric quantum field theory on the noncommutative spacetime
• Derivation based differential calculi for noncommutative algebras deforming a class of three dimensional spaces
• T-Dualities and Doubled Geometry of the Principal Chiral Model
• Influences of the coordinate dependent noncommutative space on charged and spin currents
• On the physics of the $s{o}_q(4)$ hydrogen atom
• Extensions of the Duflo map and Chern-Simons expectation values
• Field Theory with Coordinate Dependent Noncommutativity