Formulation, Interpretation and Application of non-Commutative Quantum Mechanics

Abstract

In analogy with conventional quantum mechanics, non-commutative quantum mechanics is formulated as a quantum system on the Hilbert space of Hilbert-Schmidt operators acting on non-commutative configuration space. It is argued that the standard quantum mechanical interpretation based on Positive Operator Valued Measures, provides a sufficient framework for the consistent interpretation of this quantum system. The implications of this formalism for rotational and time reversal symmetry are discussed. The formalism is applied to the free particle and harmonic oscillator in two dimensions and the physical signatures of non commutativity are identified.

11 pages

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Cited by in corpus (76)

• Voros product, Noncommutative Schwarzschild Black Hole and Corrected Area Law
• Topics in Noncommutative Geometry Inspired Physics
• A comparative review of four formulations of noncommutative quantum mechanics
• Path integral action of a particle in the non commutative plane
• Exotic galilean symmetry and non-commutative mechanics
• Fundamental length in quantum theories with PT-symmetric Hamiltonians
• A new approach to non-commutative inflation
• A unifying perspective on the Moyal and Voros products and their physical meanings
• Noncommutative quantum mechanics -- a perspective on structure and spatial extent
• Tunneling of massive particles from noncommutative inspired Schwarzschild black hole
• The problem of coexistence of several non-Hermitian observables in PT-symmetric quantum mechanics
• Minimal areas from q-deformed oscillator algebras
• Class of invariants for the 2D time-dependent Landau problem and harmonic oscillator in a magnetic field
• A squeezed review on coherent states and nonclassicality for non-Hermitian systems with minimal length
• Spectrum of the three dimensional fuzzy well
• Semiclassical and quantum description of motion on noncommutative plane
• Connecting dissipation and noncommutativity: A Bateman system case study
• Self-quartic interaction for a scalar field in an extended DFR noncommutative spacetime
• Particle production and transplanckian problem on the non-commutative plane
• Non-commutative Quantum Mechanics in Three Dimensions and Rotational Symmetry
• Spectral triplets, statistical mechanics and emergent geometry in non-commutative quantum mechanics
• Entropy and Information of a harmonic oscillator in a time-varying electric field in 2D and 3D noncommutative spaces
• Some aspects of quantum mechanics and field theory in a Lorentz invariant noncommutative space
• Noncommutative oscillators from a Hopf algebra twist deformation. A first principles derivation
• Wigner Functions for Noncommutative Quantum Mechanics: a group representation based construction
• Scattering in three-dimensional fuzzy space
• On the role of Schwinger's SU(2) generators for Simple Harmonic Oscillator in 2D Moyal plane
• Triply Extended Group of Translations of ${\mathbb{R}}^4$ as Defining Group of NCQM: relation to various gauges
• On the Plethora of Representations Arising in Noncommutative Quantum Mechanics and An Explicit Construction of Noncommutative 4-tori
• Voros product, noncommutative inspired Reissner-Nordstr{ö}m black hole and corrected area law
• Commutation relations of operator monomials
• Fingerprints of the quantum space-time in time-dependent quantum mechanics: An emergent geometric phase
• On the algebraic structure of rotationally invariant two-dimensional Hamiltonians on the noncommutative phase space
• Effects of Twisted Noncommutativity in Multi-particle Hamiltonians
• Bound state energies and phase shifts of a non-commutative well
• Classical dynamics on three dimensional fuzzy space: Connecting the short and long length scales
• The entropy of dense non-commutative fermion gases
• Emergent entropy of exotic oscillators and squeezing in three-wave mixing process
• Time-dependent q-deformed bi-coherent states for generalized uncertainty relations
• A note on broken dilatation symmetry in planar noncommutative theory
• Squeezed coherent states for gravitational well in noncommutative space
• Path integral action of a particle in a magnetic field in the noncommutative plane and the Aharonov-Bohm effect
• Landau levels in a 2D noncommutative space: matrix and quaternionic vector coherent states
• Variations on the Planar Landau Problem: Canonical Transformations, A Purely Linear Potential and the Half-Plane
• Classical limits of quantum mechanics on a non-commutative configuration space
• Path integral action and Chern-Simons quantum mechanics in noncommutative plane
• Spectral distances on doubled Moyal plane using Dirac eigen-spinors
• Time reversal and rotational symmetries in noncommutative phase space
• A General formulation of the Moyal and Voros products and its physical interpretation
• The N=1 Supersymmetric Landau Problem and its Supersymmetric Landau Level Projections: the N=1 Supersymmetric Moyal-Voros Superplane
• Revisiting Connes' Finite Spectral Distance on Non-commutative Spaces : Moyal Plane and Fuzzy Sphere
• Coherent States on Hilbert Modules
• Spectral Distance on Lorentzian Moyal Plane
• On the uniqueness of the unitary representations of the non commutative Heisenberg-Weyl algebra
• Pinhole interference in three-dimensional fuzzy space
• On the noncommutative eikonal
• Quantum dot with spin-orbit interaction in noncommutative phase space and analog Landau levels
• Minimal momentum estimation in noncommutative phase space of canonical type with preserved rotational and time reversal symmetries
• Voros product and noncommutative inspired black holes
• The Klauder-Daubechies Construction of the Phase Space Path Integral and the Harmonic Oscillator
• An Unfolded Quantization for Twisted Hopf Algebras
• Path-integral action of a particle in the noncommutative phase-space
• Dirac equation with a magnetic field in 3D non-commutative phase space
• Noncommutative inspired Schwarzschild black hole, Voros product and Komar energy
• Connes spectral distance and nonlocality of generalized noncommutative phase spaces
• Time dependent transitions with time-space noncommutativity & its implications in Quantum Optics
• New parameters of Non-commutativity in Quantum Mechanics
• Noncommutative coherent states and related aspects of Berezin-Toeplitz quantization
• Semi-Classical Localisation Properties of Quantum Oscillators on a Noncommutative Configuration Space
• Thermal effective potential in two- and three- dimensional non-commutative spaces
• Statistical interparticle potential on noncommutative space
• Connes distance of $2D$ harmonic oscillators in quantum phase space
• Connes spectral distances, quantum discord and coherence of qubits
• Noncommutativity in Configuration Space Induced by A Conjugate Magnetic Field in Phase Space
• Cryptohermitian Hamiltonians on graphs. II. Hermitizations
• Dynamic and static properties of Quantum Hall and Harmonic Oscillator systems on the non-commutative plane