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.     

Exact Solution of D-Dimensional Klein-Gordon Oscillator with Minimal Length

Specific modifications of the usual canonical commutation relations between position and momentum operators have been proposed in the literature in order to implement the idea of the existence of a minimal observable length. Here, we study a consequence of this proposal in relativistic quantum mechanics by solving in the momentum space representation the Klein-Gordon oscillator in arbitrary dimensions. The exact bound states spectrum and the corresponding momentum space wave function are obtained. Following Chang et al, （Phys. Rev. D 65 （2002） 125027）, we discuss constraint that can be placed on the minimal length by measuring the energy levels of an electron in a penning trap.     

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.     

Quantization of Space in the Presence of a Minimal Length

In this article,we apply the Generalized Uncertainty Principle(GUP),which is consistent with quantum gravity theories to an elementary particle in a finite potential well,and study the quantum behavior in this system.The generalized Hamiltonian contains two additional terms,which are proportional to αp~3(the result of the maximum momentum assumption) and α~2p~4(the result of the minimum length assumption),where α ~ 1/M_(PIC) is the GUP parameter.On the basis of the work by Ali et al.,we solve the generalized Schrodinger equation which is extended to include the α~2 correction term,and find that the length L of the finite potential well must be quantized.Then a generalization to the double-square-well potential is discussed.The result shows that all the measurable lengths especially the distance between the two potential wells are quantized in units of α_0l_(PI) in GUP scenario.     

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.     

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.     

Dynamics of Two-Level Trapped Ion in a Standing Wave Laser in Noncommutative Space

We study the dynamics of a two-level trapped ion in a standing wave electromagnetic field in two-dimensional （2D） noncommutative spaces in the Lamb-Dicke regime under the rotating wave approximation. We obtain the explicit analytical expressions for the energy spectra, energy eigenstates, unitary time evolution operator, atomic inversion, and phonon number operators. The Rabi oscillations, the collapse, and revivals in the average atomic inversion and the average phonon number are explicitly shown to contain the information of the parameter of the space noncommutativity, which sheds light on proposing new schemes based on the dynamics of trapped ion to test the noncommutativity.     

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.     

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.     

Factorizations of Polynomials over Noncommutative Algebras and Sufficient Sets of Edges in Directed Graphs

To directed graphs with unique sink and source we associate a noncommutative associative algebra and a polynomial over this algebra. Edges of the graph correspond to pseudo-roots of the polynomial. We give a sufficient condition when coefficients of the polynomial can be rationally expressed via elements of a given set of pseudo-roots (edges). Our results are based on a new theorem for directed graphs also proved in this paper. To the memory of Felix Alexandrovich Berezin. Vladimir Retakh was partially supported by NSA     

The Klein-Gordon and the Dirac Oscillators in a Noncommutative Space

We study the Dirac and the Klein-Gordon oscillators in a noncommutative space. It is shown that the Klein-Gordon oscillator in a noncommutative space has a similar behaviour to the dynamics ofa particle in a commutative space and in a constant magnetic field. The Dirac oscillator in a noncommutative space has a similar equation to the equation of motion for a relativistic fermion in a commutative space and in a magnetic field, however a new exotic term appears, which implies that a charged fermion in a noncommutative space has an electric dipole moment.     

The Klein-Gordon and the Dirac Oscillators in a Noncommutative Space

We study the Dirac and the Klein-Gordon oscillators in a noncommutative space. It is shown that the Klein-Gordon oscillator in a noncommutative space has a similar behaviour to the dynamics of a particle in a commutative space and in a constant magnetic field. The Dirac oscillator in a noncommutative space has a similar equation to the equation of motion for a relativistic fermion in a commutative space and in a magnetic field, however a new exotic term appears, which implies that a charged fermion in a noncommutative space has an electric dipole moment.     

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.     

Constraints via the Event Horizon Telescope for Black Hole Solutions with Dark Matter under the Generalized Uncertainty Principle Minimal Length Scale Effect

Four spherically symmetric but non-asymptotically flat black hole solutions surrounded with spherical dark matter distribution perceived under the minimal length scale effect is derived via the generalized uncertainty principle. Here, the effect of this quantum correction, described by the parameter $\gamma$, is considered on a toy model galaxy with dark matter and the three well-known dark matter distributions: the cold dark matter, scalar field dark matter, and the universal rotation curve. The aim is to find constraints to $\gamma$ by applying these solutions to the known supermassive black holes: Sagittarius A (Sgr. A*) and Messier 87* (M87*), in conjunction with the available Event Horizon telescope. The effect of $\gamma$ is then examined on the event horizon, photonsphere, and shadow radii, where unique deviations from the Schwarzschild case are observed. As for the shadow radii, bounds are obtained for the values of $\gamma$ on each black hole solution at $3\sigma$ confidence level. The results revealed that under minimal length scale effect, black holes can give positive (larger shadow) and negative values (smaller shadow) of $\gamma$, which are supported indirectly by laboratory experiments and astrophysical or cosmological observations, respectively.     

Solutions of Laplace Equation in n-Dimensional Spaces

The solutions of the Laplace equation inn-dimensional space are studied. The angular eigenfunctions have the form of associated Jacobi polynomials. The radial solution of the Helmholtz equation is derived.     

Differential Forms and the Noncommutative Residue for Manifolds with Boundary in the Non-product Case

In this Letter, for an even-dimensional compact manifold with boundary which has the non-product metric near the boundary, we use the noncommutative residue to define a conformal invariant pair. For a four-dimensional manifold, we compute this conformal invariant pair under some conditions and point out the way of computations in the general.     

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.     

电磁场中带电粒子在非对易相空间的能级

非对易空间效应是出现在弦的尺度下的一种物理效应。 首先扼要介绍了非对易相空间中的量子力学代数、 Moyal Weyl乘法和广义Bopp变换, 然后讨论了电磁场中带电粒子的Hamiltonian算符, 最后给出了其在非对易相空间中的能级情况。