Noncommutative-Geometry Wormholes Based on the Casimir Effect

While wormholes are as good a prediction of Einstein’s theory as black holes, they are subject to severe restrictions from quantum field theory. In particular, holding a wormhole open requires a violation of the null energy condition, calling for the existence of exotic matter. The Casimir effect has shown that this physical requirement can be met on a small scale, thereby solving a key conceptual problem. The Casimir effect does not, however, guarantee that the small-scale violation is sufficient for supporting a macroscopic wormhole. The purpose of this paper is to connect the Casimir effect to noncommutative geometry, which also aims to accommodate small-scale effects, the difference being that these can now be viewed as intrinsic properties of spacetime. As a result, the noncommutative effects can be implemented by modifying only the energy momentum tensor in the Einstein field equations, while leaving the Einstein tensor unchanged. The wormhole can therefore be macroscopic in spite of the small Casimir effect.

CONVERGENCE OF WEIGHTED AVERAGES OF MARTINGALES IN NONCOMMUTATIVE BANACH FUNCTION SPACES

Let x （xn）≥1 be a martingale on a noncommutative probability space n （M, r） and （wn）n≥1 a sequence of positive numbers such that Wn = ∑ k=1^n wk →∞ as n →∞ We prove that x = （x.）n≥1 converges in E（M） if and only if （σn（x）n≥1 converges in E（.hd）, where E（A//） is a noncommutative rearrangement invariant Banach function space with the Fatou property and σn（x） is given by σn（x） = 1/Wn ∑k=1^n wkxk, n=1, 2, .If in addition, E（Ad） has absolutely continuous norm, then, （an（x））≥1 converges in E（.M） if and only if x = （Xn）n≥1 is uniformly integrable and its limit in measure topology x∞∈ E（M）.

NONCOMMUTATIVE ORLICZ-HARDY SPACES

Let （Φ,Ψ） be a pair of complementary N-functions and HΦ（A） and HΨ（A） be the associated noncommutative Orlicz-Hardy spaces. We extend the Riesz, Szeg＆#168;o and inner-outer type factorization theorems of Hp（A） to this case.

Noncommutative Differential Calculus and Its Application on the Lattice

By introducing the noncommutative differential calculus on the function space of the infinite/finite set and construct a homotopy operator, one prove the analogue of the Poincare lemma for the difference complex. As an application of the differential calculus, a two dimensional integral model can be derived from the noncommutative differential calculus.

The Foldy-Wouthuysen Transformation of the Dirac Equation in Noncommutative Phase-Space

A method of Foldy-Wouthuysen transformation for relativistic spin-1/2 particles in external fields is proposed;in the present work the basic properties of the Dirac hamiltonian in the FW representation in the noncommutative phase-space are investigated and the Schrödinger-Pauli equation is found, knowing that the used methods for extracting the full phase-space noncommutative Dirac equation are, the Bopp-shift linear translation method, and the Moyal-Weyl product (*-product).

SZEG TYPE FACTORIZATION THEOREM FOR NONCOMMUTATIVE HARDY-LORENTZ SPACES

We introduce noncommutative Hardy-Lorentz spaces and give the Szegō and inner-outer type factorizations of these spaces.

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

The generalized Thirring model with impurity coupling is defined on two-dimensional noncommutativespace-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.

Noncommutative Differential Calculus and Its Application on Discrete Spaces

We present the noncommutative differential calculus on the function space of the infinite set and construct a homotopy operator to prove the analogue of the Poincare lemma for the difference complex. Then the horizontal and vertical complexes are introduced with the total differential map and vertical exterior derivative. As the application of the differential calculus, we derive the schemes with the conservation of symplecticity and energy for Hamiltonian system and a two-dimensional integral models with infinite sequence of conserved currents. Then an Euler-Lagrange cohomology with symplectic structure-preserving is given in the discrete classical mechanics.

Parikh-Wilczek Tunneling as Massive Particles from Noncommutative Schwarzschild Black Hole

In this paper, we apply the tunneling of massive particle through the quantum horizon of a Schwarzschild black hole in noncommutative spaeetime. The tunneling effects lead to modified Hawking radiation due to inclusion of back-reaction effects. Our calculations show also that noncommutativity effects cause the further modifications to the thermodynamical relations in black hole. We calculate the emission rate of the massive particles＇ tunneling from a Schwarzschild black hole which is modified on account of noncommutativity influences. The issues of information loss and possible correlations between emitted particles are discussed. Unfortunately even by considering noneommutativity view point, there is no correlation between different modes of evaporation at least at late-time. Nevertheless, as a result of spacetime noncommutativity, information may be conserved by a stable black hole remnant.

On Quantum Mechanics on Noncommutative Quantum Phase Space

In this work,we develop a general framework in which Noncommutative Quantum Mechanics (NCQM), characterized by a space noncommutativity matrix parameter θ=∈_(ji)~kθ_k and a momentum noncommutativity matrix parameter β_(ij)=∈_(ij)~kβ_k,is shown to be equivalent to Quantum Mechanics (QM) on a suitable transformed Quantum Phase Space (QPS).Imposing some constraints on this particular transformation,we firstly find that the product of the two parameters θ and β possesses a lower bound in direct relation with Heisenberg incertitude relations,and secondly that the two parameters are equivalent but with opposite sign,up to a dimension factor depending on the physical system under study.This means that noncommutativity is represented by a unique parameter which may play the role of a fundamental constant characterizing the whole NCQPS.Within our framework,we treat some physical systems on NCQPS:free particle,harmonic oscillator,system of two-charged particles,Hydrogen atom.Among the obtained results, we discover a new phenomenon which consists of a free particle on NCQPS viewed as equivalent to a harmonic oscillator with Larmor frequency depending on β,representing the same particle in presence of a magnetic field=q~(-1).For the other examples,additional correction terms depending on β appear in the expression of the energy spectrum.Finally,in the two-particle system case,we emphasize the fact that for two opposite charges noncornmutativity is effectively feeled with opposite sign.

Quasideterminant Solutions of a Noncommutative Nonisospectral Kadomtsev-Petviashvili Equation

Solutions of a noncommutative nonisospectral Kadomtsev-Petviashvili equation are given in terms of quasiwronskian and quasigrammian respectively. These solutions are verified by direct substitutions. Dynamics of some obtained solutions are illustrated.

On Symmetry Flows of Noncommutative Kadomtsev-Petviashvili Hierarchy

We discuss symmetry flows of noncommutative Kadomtsev-Petviashvili (NCKP) hierarchy. An operatorbased formulation, alternative to the star-product approach of extended symmetry flows is presented. Noncommutative additional symmetry flows of the NCKP hierarchy are formulated. A rescaling symmetry flow which is associated with the rescaling of whole coordinates is introduced.

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.

Accounting for Some Aspects of Dark Matter and Dark Energy via Noncommutative Geometry

The purpose of this paper is to seek a connection between noncommutative geometry, an offshoot of string theory, and certain aspects of dark matter and dark energy. The former case is based on a simple mathematical argument showing that the main manifestation of dark matter in connection with flat galactic rotation curves is also a consequence of noncommutative geometry. The latter case requires an examination of the local effect of noncommutative geometry and the subsequent extension to the global phenomenon of an accelerating Universe.

A Relation of the Noncommutative Parameters in Generalized Noncommutative Phase Space

We introduce the deformed boson operators which satisfy a deformed boson algebra in some special types of generalized noncommutative phase space. Based on the deformed boson algebra, we construct coherent state representations. We calculate the variances of the coordinate operators on the coherent states and investigate the corresponding Heisenberg uncertainty relations. It is found that there are some restriction relations of the noncommutative parameters in these special types of noncommutative phase space.

COMPLEX INTERPOLATION OF NONCOMMUTATIVE HARDY SPACES ASSOCIATED WITH SEMIFINITE VON NEUMANN ALGEBRAS

We proved a complex interpolation theorem of noncommutative Hardy spaces associated with semi-finite von Neumann algebras and extend the Riesz type factorization to the semi-finite case.

CONDITIONAL EXPECTATIONS AND MARTINGALES IN NONCOMMUTATIVE Lp-SPACES ASSOCIATED WITH CENTER-VALUED TRACES

In this paper we prove the existence of conditional expectations in the noncom- mutative Lp（M, Ф）-spaces associated with center-valued traces. Moreover, their description is also provided. As an application of the obtained results, we establish the norm convergence of weighted averages of martingales in noncommutative Lp（M, Ф）-spaces.

Vacuum Solutions of Classical Gravity on Cyclic Groups from Noncommutative Geometry

Based on the observation that the moduli of a link variable on a cyclic group modify Connes' distance on this group, we construct several action functionals for this link variable within the framework of noncommutative geometry. After solving the equations of motion, we find that one type of action gives nontrivial vacuum solution for gravity on this cyclic group in a broad range of coupling constants and that such a solution can be expressed with Chebyshev's polynomials.

LENTICULAR NONCOMMUTATIVE TORI

All C*-algebras of sections of locally trivial C* -algebra bundles over ∏i=1sLki(ni) with fibres Aw Mc(C) are constructed, under the assumption that every completely irrational noncommutative torus Aw is realized as an inductive limit of circle algebras, where Lki (ni) are lens spaces. Let Lcd be a cd-homogeneous C*-algebra over whose cd-homogeneous C*-subalgebra restricted to the subspace Tr × T2 is realized as C(Tr) A1/d Mc(C), and of which no non-trivial matrix algebra can be factored out.The lenticular noncommutative torus Lpcd is defined by twisting in by a totally skew multiplier p on Tr+2 × Zm-2. It is shown that is isomorphic to if and only if the set of prime factors of cd is a subset of the set of prime factors of p, and that Lpcd is not stablyisomorphic to if the cd-homogeneous C*-subalgebra of Lpcd restricted to some subspace LkiLki (ni) is realized as the crossed product by the obvious non-trivial action of Zki on a cd/ki-homogeneous C*-algebra over S2ni+1 for ki an integer greater than 1.

Minimal Length Quantum Mechanics of Dirac Particles in Noncommutative Space

We study the two-dimensional harmonic oscillator in commutative and noncommutative space within the framework of minimal length quantum mechanics for spin-l^2 particles. The energy spectra and the eigenfunction are obtained in both cases. Special cases are also deduced.