Noncommutative Quantum Mechanics and Quantum Cosmology

Abstract

We present a phase-space noncommutative version of quantum mechanics and apply this extension to Quantum Cosmology. We motivate this type of noncommutative algebra through the gravitational quantum well (GQW) where the noncommutativity between momenta is shown to be relevant. We also discuss some qualitative features of the GQW such as the Berry phase. In the context of quantum cosmology we consider a Kantowski-Sachs cosmological model and obtain the Wheeler-DeWitt (WDW) equation for the noncommutative system through the ADM formalism and a suitable Seiberg-Witten (SW) map. The WDW equation is explicitly dependent on the noncommutative parameters, $\theta$ and $\eta$. We obtain numerical solutions of the noncommutative WDW equation for different values of the noncommutative parameters. We conclude that the noncommutativity in the momenta sector leads to a damped wave function implying that this type of noncommmutativity can be relevant for a selection of possible initial states for the universe.

based on a talk given in Second Workshop on Quantum Gravity and Noncommutative Geometry, 22nd-24th September 2008, Universidade Lusofona, Lisboa

References in corpus (3)

• Weyl-Wigner Formulation of Noncommutative Quantum Mechanics
• Noncommutative Gravity
• Berry Phase in the Gravitational Quantum Well and the Seiberg-Witten map

Cited by in corpus (22)

• A comparative review of four formulations of noncommutative quantum mechanics
• New bounds for Tsallis parameter in a noncommutative phase-space entropic gravity and nonextensive Friedmann equations
• Black Holes and Phase Space Noncommutativity
• Entropic Gravity, Phase-Space Noncommutativity and the Equivalence Principle
• Probing phase-space noncommutativity through quantum beating, missing information and the thermodynamic limit
• Noncommutative Graphene
• Non-Canonical Phase-Space Noncommutativity and the Kantowski-Sachs singularity for Black Holes
• The singularity problem and phase-space noncanonical noncommutativity
• Probing phase-space noncommutativity through quantum mechanics and thermodynamics of free particles and quantum rotors
• Exact solution of two dimensional Dunkl harmonic oscillator in Non-Commutative phase-space
• Non-commutative black holes of various genera in the connection formalism
• Modified entropies, their corresponding Newtonian forces, potentials, and temperatures
• Quantum engines and the range of the second law of thermodynamics in the noncommutative phase-space
• Classical and quantum bicosmology with noncommutativity
• Superbradyons and some possible dark matter signatures
• Preon models, relativity, quantum mechanics and cosmology (I)
• Generalized phase-space description of non-linear Hamiltonian systems and the Harper-like dynamics
• Gaussian fidelity distorted by external fields
• Classical and statistical limits of the quantum singular oscillator
• Noncommutative Black Holes
• Accelerating cosmologies in an integrable model with noncommutative minisuperspace variables
• Emergent time crystals from phase-space noncommutative quantum mechanics