Position-dependent noncommutativity in quantum mechanics

Abstract

The model of the position-dependent noncommutativety in quantum mechanics is proposed. We start with a given commutation relations between the operators of coordinates [x^{i},x^{j}]=ω^{ij}(x), and construct the complete algebra of commutation relations, including the operators of momenta. The constructed algebra is a deformation of a standard Heisenberg algebra and obey the Jacobi identity. The key point of our construction is a proposed first-order Lagrangian, which after quantization reproduces the desired commutation relations. Also we study the possibility to localize the noncommutativety.

published version, references added

References in corpus (23)

• String Theory and Noncommutative Geometry
• Noncommutative Field Theory
• Quantum Field Theory on Noncommutative Spaces
• The quantum structure of spacetime at the Planck scale and quantum fields
• Hydrogen Atom Spectrum and the Lamb Shift in Noncommutative QED
• Non-Commutative Quantum Mechanics
• The exotic Galilei group and the "Peierls substitution"
• Exotic galilean symmetry in the non-commutative plane, and the Hall effect
• Galilean-Invariant (2+1)-Dimensional Models with a Chern-Simons-Like Term and D=2 N oncommutative Geometry
• Symmetry, Gravity and Noncommutativity
• Deformed Special Relativity and Deformed Symmetries in a Canonical Framework
• Anyon wave equations and the noncommutative plane
• A Consistent Noncommutative Field Theory: the Wess-Zumino Model
• Non-relativistic anyons, exotic Galilean symmetry and noncommutative plane
• Nonrelativistic anyons in external electromagnetic field
• Noncommutative relativistic particle on the electromagnetic background
• Star products made (somewhat) easier
• Noncommutative Planar Particle Dynamics with Gauge Interactions
• Canonical quantization of so-called non-Lagrangian systems
• Path integral representations in noncommutative quantum mechanics and noncommutative version of Berezin-Marinov action
• Scattering of spin 1/2 particles by the 2+1 dimensional noncommutative Aharonov-Bohm potential
• Dirac Quantization Condition for Monopole in Noncommutative Space-Time
• Born series and unitarity in noncommutative quantum mechanics

Cited by in corpus (45)

• Hydrogen atom on curved noncommutative space
• Composite system in rotationally invariant noncommutative phase space
• Squeezed coherent states for noncommutative spaces with minimal length uncertainty relations
• Thermodynamical properties of graphene in noncommutative phase-space
• PT-symmetric noncommutative spaces with minimal volume uncertainty relations
• Noncommutativity due to spin
• A new bound on polymer quantization via an opto-mechanical setup
• Energy-dependent noncommutative quantum mechanics
• Strings from position-dependent noncommutativity
• Aharonov-Bohm effect in a Class of Noncommutative Theories
• Exotic galilean symmetry and non-commutative mechanics
• Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry
• Hermitian versus non-Hermitian representations for minimal length uncertainty relations
• Quantum mechanics with coordinate dependent noncommutativity
• Dynamics of a Dirac Fermion in the presence of spin noncommutativity
• Modification of Schrödinger-Newton equation due to braneworld models with minimal length
• Effect of noncommutativity on the spectrum of free particle and harmonic oscillator in rotationally invariant noncommutative phase space
• Dynamical Noncommutative Graphene
• A squeezed review on coherent states and nonclassicality for non-Hermitian systems with minimal length
• Dynamical noncommutativity
• Dynamical noncommutative quantum mechanics
• $L_{\infty }$-Bootstrap Approach to Non-Commutative Gauge Theories
• Noncommutative phase space with rotational symmetry and hydrogen atom
• Rotationally invariant noncommutative phase space of canonical type with recovered weak equivalence principle
• Noncommutativity in (2+1)-dimensions and the Lorentz group
• Klein-Gordon theory in noncommutative phase space
• Spin Non-commutativity and the Three-Dimensional Harmonic Oscillator
• Holographic duals to Poisson sigma models and noncommutative quantum mechanics
• Recurrence relations for symplectic realization of (quasi)-Poisson structures
• Path Integral for Non-Paraxial Optics
• Dirac equation, hydrogen atom spectrum and the Lamb shift in dynamical noncommutative spaces
• On the algebraic structure of rotationally invariant two-dimensional Hamiltonians on the noncommutative phase space
• Squeezed atom laser for Bose-Einstein condensate with minimal length
• System of interacting harmonic oscillators in rotationally invariant noncommutative phase space
• Deformed classical-quantum mechanics transition
• Time reversal and rotational symmetries in noncommutative phase space
• From Noncommutative Sphere to Nonrelativistic Spin
• Gauge invariance and classical dynamics of noncommutative particle theory
• Minimal momentum estimation in noncommutative phase space of canonical type with preserved rotational and time reversal symmetries
• Field Theory with Coordinate Dependent Noncommutativity
• Harmonic oscillator chain in noncommutative phase space with rotational symmetry
• The 1/N Expansion in Noncommutative Quantum Mechanics
• Probing Planck scale effects on absolute mass limit in neutrino flavor evolution
• Quantization of Length in Spaces with Position-Dependent Noncommutativity
• Refractive index profiles for a $\mathcal{P}\mathcal{T}$-symmetric optical structure