Wigner Function for Klein-Gordon Landau Problem

First we calculate the Wigner phase-space distribution function for the Klein-Gordan Landau problem on a commmutative space. Then we study the modifications introduced by the coordinate-coordinate noncommuting and momentum-momentum noncommuting, namely, by using a generalized Bopp＇s shift method we construct the Wigner function for the Klein-Gordan Landau problem both on a noncommutative space （NCS） and a noncommutative phase space （NCPS）.     

Landau problem in noncommutative quantum mechanics

The Landau problem in non-commutative quantum mechanics (NCQM) is studied. First by solving the Schrodinger equations on noncommutative (NC) space we obtain the Landau energy levels and the energy correction that is caused by space-space noncommutativity. Then we discuss the noncommutative phase space case, namely, space-space and momentum-momentum non-commutative case, and we get the explicit expression of the Hamiltonian as well as the corresponding eigenfunctions and eigenvalues.     

非对易相空间中的狄拉克振子的能级和Wigner函数

非对易几何、弦论和圈量子引力理论的发展,使非对易空间受到越来越多的关注.非对易量子理论不同于平常的量子理论,它是弦尺度下的特殊的物理效应,处理非对易量子力学问题需要特殊方法.本文首先介绍了Moyal方程与Wigner函数,利用Moyal-Weyl乘法与Bopp变换将H(x,p)变换成^H(^x,^p),考虑坐标—坐标、动量—动量的非对易性,实现对非对易相空间中星乘本征方程的求解.并利用非对易相空间量子力学的代数关系,讨论了非对易相空间中狄拉克振子的Wigner函数和能级,研究结果发现非对易相空间中狄拉克振子的能级明显依赖于非对易参数.     

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.     

Landau problem in noncommutative quantum mechanics

The Landau problem in non-commutative quantum mechanics (NCQM) is studied.First by solving the Schr(o)dinger equations on noncommutative (NC) space we obtain the Landau energy levels and the energy correction that is caused by space-space noncommutativity.Then we discuss the noncommutative phase space case,namely,space-space and momentum-momentum non-commutative case,and we get the explicit expression of the Hamfltonian as well as the corresponding eigenfunctions and eigenvalues.     

二维模糊球上的量子霍尔流体

在回顾了Haldane对量子Hall效应在二维球面S2上的描述后,本文构造了二维模糊球S2上的非对易代数及其Hilbert空间的Moyal结构.通过构造模糊球上不可压缩量子霍尔流体的非对易Chern-Simons理论,求解具有准粒子源的Gaussian约束,找出模糊球上的Calogero矩阵及最低Landau能级Laughlin波函数的完全集,此Laughlin波函数由旋量坐标推广的Jack多项式表示.     

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.     

自旋1/2非对易朗道问题的Wigner函数(英文)

Wigner函数在对量子体系状态的描述方面具有重要的意义。 讨论了自旋1/2非对易朗道问题的Wigner函数。首先回顾了对易空间中Wigner函数所服从的星本征方程, 然后给出了非对易相空间中自旋1/2朗道问题的Hamiltonian, 最后利用星本征方程（Moyal 方程）计算了非对易相空间中自旋1/2朗道问题具有矩阵表示形式的Wigner函数及其能级。With great significance in describing the state of quantum system, the Wigner function of the spin half non commutative Landau problem is studied in this paper. On the basis of the review of the Wigner function in the commutative space, which is subject to the *eigenvalue equation, Hamiltonian of the spin half Landau problem in the non commutative phase space is given. Then, energy levels and Wigner functions in the form of a matrix of the spin half Landau problem in the non commutative phase space are obtained by means of the _*-eigenvalue equation (or Moyal equation).     

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.     

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.     

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.     

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.     

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.     

Relativistic Oscillators in a Noncommutative Space and in a Magnetic Field

In this work, we study the relativistic oscillators in a noncommutative space and in a magnetic field. It is shown that the effect of the magnetic field may compete withthat of the noncommutative space and that is able to vanish the effect of the noncommutative space.     

Quantum phase for an electric quadrupole moment in noncommutative quantum mechanics

We study the noncoInmutative nonrelativistic quantum dynamics of a neutral particle, which possesses an electric qaudrupole moment, in the presence of an external magnetic field. First, by intro ducing a shift for the magnetic field, we give the Schrodinger equations in the presence of an external magnetic field both on a noncommutative space and a noncomlnutative phase space, respectively. Then by solving the SchrSdinger equations both on a noneommutative space and a noncommutative phase space, we obtain quantum phases of the electric quadrupole moment, respectively. Wc demonstrate that these phases are geometric and dispersive.     

带电线性谐振子在非对易空间中的Wigner函数

在利用Wigner函数性质的基础上, 考虑到空间变量的对易关系中包含了坐标 坐标的非对易性, 得到了带电线性谐振子在非对易空间中的Wigner函数。 Based on the property of wigner function, the Wigner function of charged Linear Harmonic Oscillator in non commutative space was obtained by considering the noncommutative of the coordinate coordinate in the relation of space variable.     

Deformed Schrödinger symmetry on noncommutative space

We construct the deformed generators of Schrödinger symmetry consistent with noncommutative space. The examples of the free particle and the harmonic oscillator, both of which admit Schrödinger symmetry, are discussed in detail. We construct a generalised Galilean algebra where the second central extension exists in all dimensions. This algebra also follows from the Inonu–Wigner contraction of a generalised Poincaré algebra in noncommuting space.     

The Noncommutative Harmonic Oscillator Based on Symplectic Representation of Galilei Group

We study symplectic unitary representations for the Galilei group and derive the Schrödinger equation in phase space. Our formalism is based on the noncommutative structure of the star product. Guided by group theoretical concepts, we construct a physically consistent phase-space theory in which each state is described by a quasi-probability amplitude associated with the Wigner function. As applications, we derive the Wigner functions for the 3D harmonic oscillator and the noncommutative oscillator in phase space.     

The Geometric Construction of WZW Effective Action in Noncommutative Manifold

By constructing close-one-cochain density Ω^12n in the gauge group space we get the Wess-Zumino-Witten (WZW) effective Lagrangian on high-dimensional noncommutative space.Especially consistent anomalies derived from this WZW effective action in noncommutative four-dimensional space coincide with those obtained by L.Bonora etc.(het-th/0002210).     

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.