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.     

Deformed squeezed states in noncommutative phase space

A deformed boson algebra is naturally introduced from studying quantum mechanics on noncommutative phase space in which both positions and momenta are noncommuting each other. Based on this algebra, corresponding intrinsic noncommutative coherent and squeezed state representations are constructed, and variances of single- and two-mode quadrature operators on these states are evaluated. The result indicates that in order to maintain Heisenberg's uncertainty relations, a restriction between the noncommutative parameters is required.     

Differential calculus and gauge transformations on a deformed space

We consider a formalism by which gauge theories can be constructed on noncommutative space time structures. The coordinates are supposed to form an algebra, restricted by certain requirements that allow us to realise the algebra in terms of star products. In this formulation it is useful to define derivatives and to extend the algebra of coordinates by these derivatives. The elements of this extended algebra are deformed differential operators. We then show that there is a morphism between these deformed differential operators and the usual higher order differential operators acting on functions of commuting coordinates. In this way we obtain deformed gauge transformations and a deformed version of the algebra of diffeomorphisms. The deformation of these algebras can be clearly seen in the category of Hopf algebras. The comultiplication will be twisted. These twisted algebras can be realised on noncommutative spaces and allow the construction of deformed gauge theories and deformed gravity theory. Dedicated to the 60th birthday of Prof. Obregon.     

Noncommutative coordinate picture of the quantum phase space

We illustrate an isomorphic representation of the observable algebra for quantum mechanics in terms of the functions on the projective Hilbert space, and its Hilbert space analog, with a noncommutative product in terms of explicit coordinates and discuss the physical and dynamical picture. The isomorphism is then used as a base for the translation of the differential symplectic geometry of the infinite dimensional manifolds onto the observable algebra as a noncommutative geometry. Hence, we obtain the latter from the physical theory itself. We have essentially an extended formalism of the Schr̎odinger versus Heisenberg picture which we describe mathematically as like a coordinate map from the phase space, for which we have presented argument to be seen as the quantum model of the physical space, to the noncommutative geometry coordinated by the six position and momentum operators. The observable algebra is taken essentially as an algebra of formal functions on the latter operators. The work formulates the intuitive idea that the noncommutative geometry can be seen as an alternative, noncommutative coordinate, picture of familiar quantum phase space, at least so long as the symplectic geometry is concerned.     

经典q变形谐振子及其?量子化

通常所谓的量子包络代数可以在经典Poisson括号的意义下实现,以谐振子系统为例,给出U_q(SU(2))的这种经典实现,指出这种q变形伴随着相空间复坐标的Beltrami变形。并讨论了这种经典q变形谐振子系统的?量子化,得到在Lie括号意义下的U_q(SU(2))代数。 关键词：     

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.     

New realizations of Lie algebra kappa-deformed Euclidean space

We study Lie algebra κ-deformed Euclidean space with undeformed rotation algebra SO_a(n) and commuting vectorlike derivatives. Infinitely many realizations in terms of commuting coordinates are constructed and a corresponding star product is found for each of them. The κ-deformed noncommutative space of the Lie algebra type with undeformed Poincaré algebra and with the corresponding deformed coalgebra is constructed in a unified way.     

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 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.     

Dynamical symmetries and conservation laws for Korteweg-de Vries equation

The KdV-equation in two space time dimensions with the set of rapidly decreasing test functions as initial conditions is treated in the setting of nonlinear group and Lie algebra representations. The topological properties of the direct and inverse scattering mappings are discussed in detail.The algebra of continuous constants of motion turns out to be generated as in the linear case by three constants of motion and an extension of a representation of the e₂ Lie algebra on space-time symmetries to its enveloping algebra. The integrability of these representations is studied.It is further proved that the “moment problem” does not have a unique solution in this setting.The existence of noncommutative algebras of smooth time independent constants of motion is pointed out.     

Non-𝒜belian Geometry

 Spatial noncommutativity is similar and can even be related to the non- Abelian nature of multiple D-branes. But they have so far seemed independent of each other. Reflecting this decoupling, the algebra of matrix valued fields on noncommutative space is thought to be the simple tensor product of constant matrix algebra and the Moyal-Weyl deformation. We propose scenarios in which the two become intertwined and inseparable. Therefore the usual separation of ordinary or noncommutative space from the internal discrete space responsible for non-Abelian symmetry is really the exceptional case of an unified structure. We call it non-Abelian geometry. This general structure emerges when multiple D-branes are configured suitably in a flat but varying B field background, or in the presence of non-Abelian gauge field background. It can also occur in connection with Taub-NUT geometry. We compute the deformed product of matrix valued functions using the lattice string quantum mechanical model developed earlier. The result is a new type of associative algebra defining non-Abelian geometry. A possible supergravity dual is also discussed. Received: 13 December 2000 / Accepted: 24 October 2002 Published online: 24 January 2003 Communicated by R. H. Dijkgraaf     

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.     

Deformation Quantization of Nonregular Orbits of Compact Lie Groups

In this paper we construct a deformation quantization of the algebra of polynomials of an arbitrary (regular and nonregular) coadjoint orbit of a compact semisimple Lie group. The deformed algebra is given as a quotient of the enveloping algebra by a suitable ideal.     

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。     

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.     

Noncommutative parameters of quantum symmetries and star products

The star product technique translates the framework of local fields on noncommutative spacetime into nonlocal fields on standard spacetime. We consider the example of fields on κ-deformed Minkowski space, transforming under κ-deformed Poincaré group, with noncommutative parameters. By extending the star product to the tensor product of functions on κ-deformed Minkowski space and κ-deformed Poincaré group we represent the algebra of noncommutative parameters of deformed relativistic symmetries by functions on classical Poincaré group.     

Some Graded Bialgebras Related to Borcherds Superalgebras

In present paper we define a new kind of weak quantized enveloping algebra of Borcherds superalgebras. We denote this algebra by wU_q^t(G)wU_{q}^{\tau}(\mathcal{G}). It is a noncommutative and noncocommutative weak graded Hopf algebra under some additional condition. It has a homomorphic image which is isomorphic to the usual quantum enveloping algebra U_q(G)U_{q}(\mathcal{G}) of G\mathcal{G}.     

Deformed “commutative” Chern–Simons’ system

Noncommutative Chern–Simons’ system is non-perturbatively investigated at a full deformed level. A deformed “commutative” phase space is found by a non-canonical change between two sets of deformed variables of noncommutative space. It is explored that in the “commutative” phase space all calculations are similar to the case in commutative space. Spectra of its energy and angular momentum of the Chern–Simons’ system are obtained at the full deformed level. The noncommutative–commutative correspondence is clearly showed. Formalism for the general dynamical system is briefly presented. Some subtle points are clarified.     

Noncommutative lattices and their continuum limits

We consider finite approximations of a topological space M by noncommutative lattices of points. These lattices are structure spaces of noncommutative C^*-algebras which in turn approximate the algebra C(M) of continuous functions on M. We show how to recover the space M and the algebra C(M) from a projective system of noncommutative lattices and an inductive system of noncommutative C^*-algebras, respectively.     

QFT on homothetic Killing twist deformed curved spacetimes

We study the quantum field theory (QFT) of a free, real, massless and curvature coupled scalar field on self-similar symmetric spacetimes, which are deformed by an abelian Drinfel’d twist constructed from a Killing and a homothetic Killing vector field. In contrast to deformations solely by Killing vector fields, such as the Moyal-Weyl Minkowski spacetime, the equation of motion and Green’s operators are deformed. We show that there is a *-algebra isomorphism between the QFT on the deformed and the formal power series extension of the QFT on the undeformed spacetime. We study the convergent implementation of our deformations for toy-models. For these models it is found that there is a *-isomorphism between the deformed Weyl algebra and a reduced undeformed Weyl algebra, where certain strongly localized observables are excluded. Thus, our models realize the intuitive physical picture that noncommutative geometry prevents arbitrary localization in spacetime.