Cusp Interaction in Minimal Length Quantum Mechanics

The modified Schrödinger equation with a minimal length is considered under a Cusp potential which includes the exponential interaction. Next, exact analytical solutions of the problem are reported and thereby the scattering states as well as the corresponding transmission and reflection coefficients are reported.     

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.     

Maximum Momentum,Minimal Length and Quantum Gravity Effects of Compact Star Cores

Based on the generalized uncertainty principle with maximum momentum and minimal length,we discuss the equation of state of ideal ultra-relativistic Fermi gases at zero temperature.Maximum momentum avoids the problem that the Fermi degenerate pressure blows up since the increase of the Fermi energy is not limited.Applying this equation of state to the Tolman-Oppenheimer-Volkoff(TOV) equation,the quantum gravitational effects on the cores of compact stars are discussed.In the center of compact stars,we obtain the singularity-free solution of the metric component,g_(tt) ~-(1 + 0.2185 x r~2).By numerically solving the TOV equation,we find that quantum gravity plays an important role in the region r ~10~4a_0(△x)_(min).Current observed masses of neutron stars indicate that the dimensionless parameter z_0 cannot exceed 10~(19).     

Quantum Gravitational Corrections to the Real Klein-Gordon Field in the Presence of a Minimal Length

The (D+1)-dimensional (β,β′)-two-parameter Lorentz-covariant deformed algebra introduced by Quesne and Tkachuk (J. Phys., A Math. Gen. 39, 10909, 2006), leads to a nonzero minimal uncertainty in position (minimal length). The Klein-Gordon equation in a (3+1)-dimensional space-time described by Quesne-Tkachuk Lorentz-covariant deformed algebra is studied in the case where β′=2β up to first order over deformation parameter β. It is shown that the modified Klein-Gordon equation which contains fourth-order derivative of the wave function describes two massive particles with different masses. We have shown that physically acceptable mass states can only exist for b < \frac18m²c²\beta<\frac{1}{8m^{2}c^{2}} which leads to an isotropic minimal length in the interval 10⁻¹⁷ m<(ΔX ⁱ )₀<10⁻¹⁵ m. Finally, we have shown that the above estimation of minimal length is in good agreement with the results obtained in previous investigations.     

A Minimal Framework for Non-Commutative Quantum Mechanics

Deformation quantisation is applied to ordinary Quantum Mechanics by introducing the star product in a configuration space combining a Riemannian structure with a Poisson one. A Hilbert space compatible with such a configuration space is designed. The dynamics is expressed by a Hermitian Hamiltonian containing a scalar potential and a one-form potential. As a simple illustration, it is shown how a particular type of non-commutativity of the star product is interpretable as generating the Zeeman effect of ordinary Quantum Mechanics.     

The Minimal Length Uncertainty and the Nonextensive Thermodynamics

In this paper, we study the thermodynamics of quantum harmonic oscillator in the Tsallis framework and in the presence of a minimal length uncertainty. The existence of the minimal length is motivated by various theories such as string theory, loop quantum gravity, and black-hole physics. We analytically obtain the partition function, probability function, internal energy, and the specific heat capacity of the vibrational quantum system for \(1<q<\frac {3}{2}\) and compare the results with those of Tsallis and Boltzmann-Gibbs statistics without the minimal length scale.     

Minimal Irreversible Quantum Mechanics: An Axiomatic Formalism

An axiomatic formalism for a minimalirreversible quantum mechanics is introduced. It isshown that a quantum equilibrium and the decoherencephenomenon are consequences of the axioms and thatLyapunov variables, exponential survival probabilities,and a classical conditional never-decreasing entropy canbe defined.     

Minimal Length, Maximal Momentum and the Entropic Force Law

Different candidates of quantum gravity proposal such as string theory, noncommutative geometry, loop quantum gravity and doubly special relativity, all predict the existence of a minimum observable length and/or a maximal momentum which modify the standard Heisenberg uncertainty principle. In this paper, we study the effects of minimal length and maximal momentum on the entropic force law formulated recently by E. Verlinde.     

Minimal Irreversible Quantum Mechanics: Decoherence and Classical Equilibrium Limit

Using the Minimal Irreversible Quantum Mechanicsformalism, it is demonstrated that the quantum regimecan be considered as the transient phase while the finalclassical equilibrium regime is the permanent state. A basis where exact matrix decoherenceappears for these final states is found. The appearanceof a classical universe in quantum gravity models is thecosmological version of this problem.     

Classical Dynamics Based on the Minimal Length Uncertainty Principle

In this paper we consider the quadratic modification of the Heisenberg algebra and its classical limit version which we call the β-deformed Poisson bracket for corresponding classical variables. We use the β-deformed Poisson bracket to discuss some physical problems in the β-deformed classical dynamics. Finally, we consider the (α,β)- deformed classical dynamics in which minimal length uncertainty principle is given by \( [ \hat {x} , \hat {p}] = i \hbar (1 + \alpha \hat {x}^{2} + \beta \hat {p}^{2} ) \). For two small parameters α,β, we discuss the free fall of particle and a composite system in a uniform gravitational field.     

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.     

对单胶子交换势的非微扰修正研究

对单胶子交换势的非微扰修正进行了系统研究,结果表明,要获得规范不变的修正形式,QCD真空胶子场必须取为协变规范.在协变规范下,给出了单胶子交换势的非微扰修正的新形式.     

Discrete Time and Intrinsic Length in Quantum Mechanics

We consider the possibility that simultaneously time and intrinsic length can be regarded as discrete real parameters. We study the dynamics of the free particle. For both scattering and bound states there are configurations where the energy is bounded from above and from below even for positive wave-function solutions. For the case of continuous evolution we show that the wave equation with a linear scalar coupling describes an oscillator that has built-in hidden supersymmetry.     

Relativistic Approach to the Hydrogen Atom in a Minimal Length Scenario

We show that relativistic contributions to the ground-state energy of the hydrogen atom from a minimal length introduced by a Lorentz-covariant algebra are more important than non-relativistic contributions; the non-relativistic approach is therefore unsuitable. We compare our result with experimental data to estimate an upper bound of the order 10^?20m for the minimal length.     

Solution of the Dipoles in Noncommutative Space with Minimal Length *

The single neutral spin-half particle with electric dipole moment and magnetic dipole moment moving in an external electromagnetic field is studied. The Aharonov-Casher effect and He-McKellar-Wilkens effect are emphatically discussed in noncommutative (NC) space with minimal length. The energy eigenvalues of the systems are obtained exactly in terms of the Jacobi polynomials. Additionally, a special case is discussed and the related energy spectra are plotted.     

A Nonperturbative Regularization of the Supersymmetric Schwinger Model

It is shown that noncommutative geometry is a nonperturbative regulator which can manifestly preserve a space supersymmetry and a supergauge symmetry while keeping only finite number of degrees of freedom in the theory. The simplest N= 1 case of the U(1) supergauge theory on the sphere is worked out in detail. Received: 15 March 1999 / Accepted: 8 April 1999     

椭圆量子群极小表示的系数代数

给出了对应于A(1)n–1面模型的玻尔兹曼权的L矩阵及FelderAn系列椭圆量子群的极小表示(所有矩阵元均为c数)的系数代数.这个代数满足杨–Baxter方程.同时,对此代数还给出了一组PBW基.