Noncommutative Classical Mechanics

In this work, I investigate the noncommutative Poisson algebra of classical observables corresponding to a proposed general noncommutative quantum mechanics, Djemai, A. E. F. and Smail, H. (2003). I treat some classical systems with various potentials and some physical interpretations are given concerning the presence of noncommutativity at large scales (celestial mechanics) directly tied to the one present at small scales (quantum mechanics) and its possible relation with UV/IR mixing.     

非对易相空间中角动量的分裂

非对易空间效应是一种在弦尺度下出现的物理效应. 本文首先介绍了在Schwinger表象中角动量的3个分量用产生--消灭算符的表示形式, 接着讨论了非对易相空间的量子力学代数; 然后用对易空间谐振子的产生-消灭算符表示出了在非对易情况下的角动量; 最后讨论了非对易相空间中角动量的分裂.     

The Phase Space Model of Nonrelativistic Quantum Mechanics

We focus on several questions arising during the modelling of quantum systems on a phase space. First, we discuss the choice of phase space and its structure. We include an interesting case of discrete phase space. Then, we introduce the respective algebras of functions containing quantum observables. We also consider the possibility of performing strict calculations and indicate cases where only formal considerations can be performed. We analyse alternative realisations of strict and formal calculi, which are determined by different kernels. Finally, two classes of Wigner functions as representations of states are investigated.     

Quantum Field Theory with a Minimal Length Induced from Noncommutative Space

From the inspection of noncommutative quantum mechanics, we obtain an approximate equivalent relation for the energy dependence of the Planck constant in the noncommutative space, which means a minimal length of the space. We find that this relation is reasonable and it can inherit the main properties of the noncommutative space. Based on this relation, we derive the modified Klein-Gordon equation and Dirac equation. We investigate the scalar field and φ4 model and then quantum electrodynamics in our theory, and derive the corresponding Feynman rules. These results may be considered as reasonable approximations to those of noncommutative quantum field theory. Our theory also shows a connection between the space with a minimal length and the noncommutative space.     

Noncommutative Quantum Hall Effect of Bilayer Systems

In this paper we study the bilayer quantum Hall （QH） effect on a noncommutative phase space （NCPS）. By using perturbation theory, we calculate the energy spectrum, eigenfunction, Hall current, and Hall conductivity of the bilayer QH system, and express them in terms of noncommutative parameters θ and θ^-, respectively. In our calculation, we assume that these parameters vary from laver to laver.     

An Algebra of Effects in the Formalism of Quantum Mechanics on Phase Space

Defining an addition of the effects in the formalism of quantum mechanics on phase space, we obtain a new effect algebra that is strictly contained in the effect algebra of all effects. A new property of the phase space formalism comes to light, namely that the new effect algebra does not contain any pair of noncommuting projections. In fact, in this formalism, there are no nontrivial projections at all. We illustrate this with the spin-1/2 algebra and the momentum/position algebra. Next, we equip this algebra of effects with the sequential product and get an interpretation of why certain properties fail to hold. PACS: 02.10.Gd, 03.65.Bz. This paper was a submission to the Fifth International Quantum Structure Association Conference (QS5), which took place in Cesena, Italy, March 31–April 5, 2001.     

Noncommutative Unification of General Relativity and Quantum Mechanics. A Finite Model

We construct a model unifying general relativity and quantum mechanics in a broader structure of noncommutative geometry. The geometry in question is that of a transformation groupoid given by the action of a finite group on a space E. We define the algebra of smooth complex valued functions on , with convolution as multiplication, in terms of which the groupoid geometry is developed. Owing to the fact that the group G is finite the model can be computed in full details. We show that by suitable averaging of noncommutative geometric quantities one recovers the standard space-time geometry. The quantum sector of the model is explored in terms of the regular representation of the algebra , and its correspondence with the standard quantum mechanics is established.     

Noncommutative Quantum Hall Effect of Bilayer Systems

In this paper we study the bilayer quantum Hall (QH) effect on a noncommutative phase space (NCPS). By using perturbation theory, we calculate the energy spectrum, eigenfunction, Hall current, and Hall conductivity of the bilayer QH system, and express them in terms of noncommutative parameters θ and \bar{θ}, respectively. In our calculation, we assume that these parameters vary from layer to layer.     

Quantum Mechanics in Curved Configuration Space

Different approaches are compared to formulation of quantum mechanics of a particle on the curved spaces. At first, the canonical, quasiclassical, and path integration formalisms are considered for quantization of geodesic motion on the Riemannian configuration spaces. A unique rule of ordering of operators in the canonical formalism and a unique definition of the path integral are established and, thus, a part of ambiguities in the quantum counterpart of geodesic motion is removed. A geometric interpretation is proposed for noninvariance of the quantum mechanics on coordinate transformations. An approach alternative to the quantization of geodesic motion is surveyed, which starts with the quantum theory of a neutral scalar field. Consequences of this alternative approach and the three formalisms of quantization are compared. In particular, the field theoretical approach generates a deformation of the canonical commutation relations between operators of coordinates and momenta of a particle. A cosmological consequence of the deformation is presented in short.     

Coordinate Eigenstate Representation in Noncommutative Space as a New Entangled State in Quantum Mechanics

For the first time we construct the eigenstate ｜τ〉 of noncommutatlve coordinate. It turns out that｜τ〉 is an entangled state in the ordinary space. Remarkably, its Schmidt decomposition has definite expression in the coordinate representation and the momentum representation. The 〈τ｜ representation can simplify some calculations for obtaining energy level of two-dimensional oscillator in noncommutative space.     

Functional Integrals and Quantum Fluctuations on Two-Dimensional Noncommutative Space-Time

The generalized Thirring model with impurity coupling is defined on two-dimensional noncommutative space-time, a modified propagator and free energy are derived by means of functional integrals method. Moreover, quantum fluctuations and excitation energies are calculated on two-dimensional black hole and soliton background.     

Gauge Theory on a Discrete Noncommutative Space

In this paper, we apply Connes' noncommutative geometry and the Seiberg—Witten map to a discrete noncommutative space consisting of n copies of a given noncommutative space R ^m . The explicit action functional of gauge fields on this discrete noncommutative space is obtained.     

The Phase Space Formalism for Quantum Mechanics and C* Axioms

On any quantum mechanical Hilbert space, the phase space localization operators form a set of operators that are both physically motivated and form the groundwork for a C* algebra. This set is shown to be informationally complete in the original Hilbert space. We also revisit the relation between having a complete set of eigenvectors, commutability and compatibility. Dedicated to G.G. Emch.     

Classical Mechanics and Quantum Mechanics

The Newton equation of motion is derived from quantum mechanics.     

Realization of the N(odd)-Dimensional Quantum Euclidean Space by Differential Operators

The quantum Euclidean space R_q^N is a kind of noncommutative space that is obtained from ordinary Euclidean space R^N by deformation with parameter q. When N is odd, the structure of this space is similar to R_q³. Motivated by realization of R_q³ by differential operators in R³, we give such realization for R_q⁵ and R_q⁷ cases and generalize our results to R_q^N (N odd) in this paper, that is, we show that the algebra of R_q^N can be realized by differential operators acting on C^∝ functions on undeformed space R^N.     

Klein Gordon Oscillators in Commutative and Noncommutative Phase Space with Psudoharmonic Potential in the Presence and Absence Magnetic Field

We study the Klein-Gordon oscillator in commutative, noncommutative space, and phase space with psudoharmonic potential under magnetic field hence the other choice is studying the Klein-Gordon equation oscillator in the absence of magnetic field. In this work, we obtain energy spectrum and wave function in different situations by NU method so we show our results in tables.     

Klein Gordon Oscillators in Commutative and Noncommutative Phase Space with Psudoharmonic Potential in the Presence and Absence Magnetic Field

We study the Klein-Gordon oscillator in commutative, noncommutative space, and phase space with psudoharmonic potential under magnetic field hence the other choice is studying the Klein-Gordon equation oscillator in the absence of magnetic field. In this work, we obtain energy spectrum and wave function in different situations by NU method so we show our results in tables.     

State-Vector Space and Canonical Coherent States in Noncommutative Plane

The structure of the state-vector space of identical bosons in noncommutative spaces is investigated. To maintain Bose-Einstein statistics the commutation relations of phase space variables should simultaneously include coordinate-coordinate non-commutativity and momentum-momentum non-commutativity, which leads to a kind of deformed Heisenberg-Weyl algebra. Although there is no ordinary number representation in this state-vector space, several set of orthogonal and complete state-vectors can be derived which are common eigenvectors of corresponding pairs of commuting Hermitian operators. As a simple application of this state-vector space, an explicit form of two-dimensional canonical coherent state is constructed and its properties are discussed.     

Time Flow in a Noncommutative Regime

We develop an approach to dynamical and probabilistic properties of the model unifying general relativity and quantum mechanics, initiated in the paper (Heller et al. (2005) International Journal Theoretical Physics 44, 671). We construct the von Neumann algebra of random operators on a groupoid, which now is not related to a finite group G, but is the pair groupoid of the Lorentzian frame bundle E over spacetime M. We consider the time flow on depending on the state . The state defining the noncommutative dynamics is assumed to be normal and faithful. Then the pair is a noncommutative probabilistic space and can also be interpreted as an equilibrium thermal state, satisfying the Kubo-Martin-Schwinger condition. We argue that both the “time flow” and thermodynamics have their common roots in the noncommutative unification of dynamics and probability.