Twist Deformation of Rotationally Invariant Quantum Mechanics

Abstract

Non-commutative Quantum Mechanics in 3D is investigated in the framework of the abelian Drinfeld twist which deforms a given Hopf algebra while preserving its Hopf algebra structure. Composite operators (of coordinates and momenta) entering the Hamiltonian have to be reinterpreted as primitive elements of a dynamical Lie algebra which could be either finite (for the harmonic oscillator) or infinite (in the general case). The deformed brackets of the deformed angular momenta close the so(3) algebra. On the other hand, undeformed rotationally invariant operators can become, under deformation, anomalous (the anomaly vanishes when the deformation parameter goes to zero). The deformed operators, Taylor-expanded in the deformation parameter, can be selected to minimize the anomaly. We present the deformations (and their anomalies) of undeformed rotationally-invariant operators corresponding to the harmonic oscillator (quadratic potential), the anharmonic oscillator (quartic potential) and the Coulomb potential.

arXiv comment20 pages

References in corpus (37)

• String Theory and Noncommutative Geometry
• The quantum structure of spacetime at the Planck scale and quantum fields
• Berry phase correction to electron density of states in solids
• Non-Commutative Quantum Mechanics
• Magnetic fields, branes and noncommutative geometry
• The exotic Galilei group and the "Peierls substitution"
• New concept of relativistic invariance in NC space-time: twisted Poincaré symmetry and its implications
• The Quantum Hall Fluid and Non-Commutative Chern Simons Theory
• Exotic galilean symmetry in the non-commutative plane, and the Hall effect
• The non-commutative Landau problem
• Hall Effect in Noncommutative Coordinates
• Comments on the Hydrogen Atom Spectrum in the Noncommutative Space
• Topics in Noncommutative Geometry Inspired Physics
• Representation theory of finite W algebras
• Color Magnetic Flux Tubes in Dense QCD. II: Effective World-Sheet Theory
• Interactions and non-commutativity in quantum Hall systems
• Spectrum of the non-commutative spherical well
• Twisted Noncommutative Field Theory with the Wick-Voros and Moyal Products
• Noncommutative field theory with the Wick-Voros product
• Quantum Fields on the Groenewold-Moyal Plane
• Noncommutativity due to spin
• Non-Pauli Transitions From Spacetime Noncommutativity
• Inequivalence of QFT's on Noncommutative Spacetimes: Moyal versus Wick-Voros
• Wigner Oscillators, Twisted Hopf Algebras and Second Quantization
• Noncommutative relativistic particles
• Thermodynamics of a non-commutative fermion gas
• Projection on higher Landau levels and non-commutative geometry
• Dynamical noncommutativity
• Lectures on Hopf Algebras, Quantum Groups and Twists
• Noncommutative quantum mechanics as a gauge theory
• Noncommutative version of an arbitrary nondegenerated mechanics
• On the quantum dynamics of non-commutative systems
• Bound state energies and phase shifts of a non-commutative well
• Noncommutative spacetime and the fractional quantum Hall effect
• Noncommutative Geometry and Anyonic Field Theory in the Magnetic Field
• Noncommutative quantum mechanics: uniqueness of the functional description
• The Groenewold - Moyal Plane and its Quantum Physics

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• Noncommutative oscillators from a Hopf algebra twist deformation. A first principles derivation
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• The entropy of dense non-commutative fermion gases
• Energy corrections due to the noncommutative phase-space of the charged isotropic harmonic oscillator in a uniform magnetic field in 3D
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