Composite system in noncommutative space and the equivalence principle

The motion of a composite system made of N particles is examined in a space with a canonical noncommutative algebra of coordinates. It is found that the coordinates of the center-of-mass position satisfy noncommutative algebra with effective parameter. Therefore, the upper bound of the parameter of noncommutativity is re-examined. We conclude that the weak equivalence principle is violated in the case of a non-uniform gravitational field and propose the condition for the recovery of this principle in noncommutative space. Furthermore, the same condition is derived from the independence of kinetic energy on the composition.     

System of interacting harmonic oscillators in rotationally invariant noncommutative phase space

Rotationally invariant space with noncommutativity of coordinates and noncommutativity of momenta of canonical type is considered. A system of N interacting harmonic oscillators in uniform field and a system of N particles with harmonic oscillator interaction are studied. We analyze effect of noncommutativity on the energy levels of these systems. It is found that influence of coordinates noncommutativity on the energy levels of the systems increases with increasing of the number of particles. The spectrum of N free particles in uniform field in rotationally invariant noncommutative phase space is also analyzed. It is shown that the spectrum corresponds to the spectrum of a system of N harmonic oscillators with frequency determined by the parameter of momentum noncommutativity.     

Spin Hall effect and Berry phase of spinning particles

We consider the adiabatic evolution of the Dirac equation in order to compute its Berry curvature in momentum space. It is found that the position operator acquires an anomalous contribution due to the non-Abelian Berry gauge connection making the quantum mechanical algebra noncommutative. A generalization to any known spinning particles is possible by using the Bargmann–Wigner equation of motions. The noncommutativity of the coordinates is responsible for the topological spin transport of spinning particles similarly to the spin Hall effect in spintronic physics or the Magnus effect in optics. As an application we predict new dynamics for nonrelativistic particles in an electric field and for photons in a gravitational field.     

非对易相空间Dirac oscillator的研究

在极坐标系中研究了非对易相空间中的Dirac oscillator问题.研究显示:系统的波函数可以表示为合流超几何函数,而非对易相空间Dirac oscillator的量子行为类似于朗道问题.最后,对η=0和对易极限两种特殊情况进行了简单讨论.     

非对易相空间中谐振子体系热力学性质的探讨

2000年以来, 有关非对易空间的各种物理问题一直是研究的热点, 并在量子力学、场论、凝聚态物理、天体物理等各领域中已被广泛地探讨. 采用统计物理方法讨论非对易效应对谐振子体系热力学性质的影响. 先以对易相空间中确定二维和三维谐振子的配分函数求出谐振子体系的热力学函数; 非对易相空间中的坐标和动量通过坐标-坐标和动量-动量之间的线性变换而以对易相空间中的坐标和动量来表示; 最终以非对易相空间中求出配分函数来讨论非对易效应对谐振子体系热力学性质的影响. 结果显示, 在非对易相空间中谐振子体系的配分函数和熵表达式均包含因非对易引起的修正项. 从分析结果得出如下结论: 非对易效应对谐振子的配分函数和熵函数等微观状态函数有一定的影响, 但对谐振子体系的内能、热容量等宏观热力学函数没有影响. 研究结果只是对应于满足玻尔兹曼统计的经典体系, 对于满足费米-狄拉克和玻色-爱因斯坦统计的量子体系需进一步推广研究.     

A New Type of Seiberg-Witten Map and Its Application

Deformation quantization is a powerful tool to deal with systems in noncommutative space to get their energy spectra and corresponding Wigner functions, especially for the case of both coordinates and momenta being noncommutative. In order to simplify solutions of the relevant *-genvalue equation, we introduce a new kind of Seiberg-Witten-like map to change the variables of the noncommutative phase space into ones of a commutative phase space, and demonstrate its role via an example of two-dimensional oscillator with both kinetic and elastic couplings in the noncommutative phase space.     

Mass Defect in the Noncommutative Schwarzschild Space-Time

In this paper we investigate the mass defect and other gravitational effects in noncommutative Schwarzschild space-time obtained by considering particles as smeared objects. The effects of space-time noncommutativity on mass defect of a test particle and a homogeneous spherical shell are calculated. The NC corrections to gravitational redshift, and light-speed in Schwarzschild field are briefly discussed. The results show that the NC corrections have weakening action on these gravitational effects comparing with those in commutative cases.     

Nontraditional role of interparticle forces in quantum mechanics

A quantum-mechanical model for systems of interacting bodies that takes into account noncommutativity of the operators of coordinates and momenta of different particles and correlations of the uncertainties of these quantities is considered. Here, the noncommutativity of these operators is due to the effect of interparticle forces and arises as a natural generalization of the usual commutation relation between the coordinate and momentum operators for a single particle. The efficiency of the model is proven by specific calculations for systems known in atomic and nuclear physics.     

The Geometry of Noncommutative Space-Time

Stabilization, by deformation, of the Poincaré-Heisenberg algebra requires both the introduction of a fundamental lentgh and the noncommutativity of translations which is associated to the gravitational field. The noncommutative geometry structure that follows from the deformed algebra is studied both for the non-commutative tangent space and the full space with gravity. The contact points of this approach with the work of David Finkelstein are emphasized.     

Noncommutativity effects in FRW scalar field cosmology

We study effects of noncommutativity on the phase space generated by a non-minimal scalar field which is conformally coupled to the background curvature in an isotropic and homogeneous FRW cosmology. These effects are considered in two cases, when the potential of scalar field has zero and nonzero constant values. The investigation is carried out by means of a comparative detailed analysis of mathematical features of the evolution of universe and the most probable universe wave functions in classically commutative and noncommutative frames and quantum counterparts. The influence of noncommutativity is explored by the two noncommutative parameters of space and momentum sectors with a relative focus on the role of the noncommutative parameter of momentum sector. The solutions are presented with some of their numerical diagrams, in the commutative and noncommutative scenarios, and their properties are compared. We find that impose of noncommutativity in the momentum sector causes more ability in tuning time solutions of variables in classical level, and has more probable states of universe in quantum level. We also demonstrate that special solutions in classical and allowed wave functions in quantum models impose bounds on the values of noncommutative parameters.     

Deformations of Fluid Dynamics and Friedmann Equation on Noncommutative Phase Space

Noncommutative phase space is one of the widely studied extensions of ordinary phase space, and has profound implications in cosmological physics. In this paper we study the dynamics of perfect fluid on noncommutative phase space, as well as deformations of the Friedmann equation. The Lagrangian formalism is used to take into account of the phase space noncommutativities. Then a map from canonical Lagrangian variables to Eulerian variables is employed to derive the equations of motion of the mass and current densities. We find that both these two equations receive noncommutative corrections that are linear in the noncommutative parameters. However, we also find that in the approximation of vanishing comoving velocity the leading order noncommutative correction due to momentum noncommutativity on the Friedmann equation is zero.     

Classical mechanics on noncommutative space with Lie-algebraic structure

We investigate the kinetics of a nonrelativistic particle interacting with a constant external force on a Lie-algebraic noncommutative space. The structure constants of a Lie algebra, also called noncommutative parameters, are constrained in general due to some algebraic properties, such as the antisymmetry and Jacobi identity. Through solving the constraint equations the structure constants satisfy, we obtain two new sorts of algebraic structures, each of which corresponds to one type of noncommutative spaces. Based on such types of noncommutative spaces as the starting point, we analyze the classical motion of the particle interacting with a constant external force by means of the Hamiltonian formalism on a Poisson manifold. Our results not only include that of a recent work as our special cases, but also provide new trajectories of motion governed mainly by marvelous extra forces. The extra forces with the unimaginable -, and -dependence besides with the usual t-, x-, and -dependence, originating from a variety of noncommutativity between different spatial coordinates and between spatial coordinates and momenta as well, deform greatly the particle’s ordinary trajectories we are quite familiar with on the Euclidean (commutative) space.     

A complete $\mathcal{O}(\alpha_{\mathrm{S}}^{2})$ calculation of the signal–background interference for the Higgs diphoton decay channel

We analyze a noncommutative model of BTZ spacetime based on deformation of the standard symplectic structure of phase space, i.e., a modification of the standard commutation relations among coordinates and momenta in phase space. We find a BTZ-like solution that is nonperturbative in the non-trivial noncommutative structure. It is shown that the use of deformed commutation relations in the modified non-canonical phase space eliminates the horizons of the standard metric.     

DKP Oscillator with Spin-0 in Three-dimensional Noncommutative Phase Space

The DKP equation with Dirac oscillator potential for spin-0 particles has been studied when both space-space noncommutativity and momentum-momentum noncommutativity are considered. The exact wave functions and corresponding energy levels have been found. Due to the noncommutative effect, the energy spectrum is not degenerate.     

非对易相空间中各向同性谐振子的能级分裂

非对易空间的效应是出现在弦尺度下的一种物理效应. 本文介绍了量子力学非对易空间的代数关系; 讨论了非对易相空间中服从玻色-爱因斯坦统计的粒子的连续性条件, 最后给出了非对易相平面和非对易相空间中的线性谐振子的能级分裂.     

An alternative formulation of Hall effect and quantum phases in noncommutative space

A recent method of constructing quantum mechanics in noncommutative coordinates, alternative to implying noncommutativity by means of star product is discussed. Within this approach we study Hall effect as well as quantum phases in noncommutative coordinates. The θ-deformed phases which we obtain are velocity independent.     

Doubly special relativity in de Sitter spacetime

We discuss the generalization of Doubly Special Relativity to a curved de Sitter background. The model has three fundamental observer‐independent scales, the velocity of light c, the de Sitter radius α, and the Planck energy κ, and can be realized through a nonlinear action of the de Sitter group on a noncommutative position space. We consider different choices of coordinates on the de Sitter hyperboloid that, although equivalent, may be more suitable for treating different problems. Also the momentum space can be described as a hyperboloid embedded in a five‐dimensional space, but in this case different choices of coordinates lead to inequivalent models. We investigate the kinematics and the Hamiltonian dynamics of some specific models and describe some of their phenomenological consequences. Finally, we show that it is possible to construct a model exhibiting a duality for the interchange of positions and momenta together with the interchange of α and κ.     

An integrable noncommutative version of the sine-Gordon system

Using the bicomplex approach we discuss an integrable noncommutative system in two-dimensional Euclidean space. It is described by an equation of motion which reduces to the ordinary sine-Gordon equation when the noncommutation parameter is removed, plus a constraint equation which is nontrivial only in the noncommutative case. The implications of this constraint, which is required by integrability but seems to reduce the space of classical solutions, remain to be understood. We show that the system has an infinite number of conserved currents and we give the general recursive relation for constructing them. For the particular cases of lower spin nontrivial currents we work out the explicit expressions and perform a direct check of their conservation. These currents reduce to the usual sine-Gordon currents in the commutative limit. We find classical “localized” solutions to first order in the noncommutativity parameter and describe the Backlund transformations for our system. Finally, we comment on the relation of our noncommutative system to the commutative sine-Gordon system.     

Quantum Hall effect in noncommutative quantum mechanics

We study the quantum Hall (QH) effect for an electron moving in a plane whose coordinates and momenta are noncommuting under the influence of uniform external magnetic and electric fields. After solving the time independent Schrödinger equation both on a noncommutative space (NCS) and a noncommutative phase space (NCPS), we obtain the energy eigenvalues and eigenfunctions of the relevant Hamiltonian. We derive the electric current whose expectation value gives the QH effect both on a NCS and a NCPS.