Influence of Noncommutativity on the Motion of Sun-Earth-Moon System and the Weak Equivalence Principle

Features of motion of macroscopic body in gravitational field in a space with noncommutativity of coordinates and noncommutativity of momenta are considered in general case when coordinates and momenta of different particles satisfy noncommutative algebra with different parameters of noncommutativity. Influence of noncommutativity on the motion of three-body Sun-Earth-Moon system is examined. We show that because of noncommutativity the free fall accelerations of the Moon and the Earth toward the Sun in the case when the Moon and the Earth are at the same distance to the source of gravity are not the same even if gravitational and inertial masses of the bodies are equal. Therefore, the Eotvos-parameter is not equal to zero and the weak equivalence principle is violated in noncommutative phase space. We estimate the corrections to the Eotvos-parameter caused by noncommutativity on the basis of Lunar laser ranging experiment results. We obtain that with high precision the ratio of parameter of momentum noncommutativity to mass is the same for different particles.     

Associated production evidence against Higgs impostors and anomalous couplings

We analyze bicovariant differential calculus on κ-Minkowski spacetime. It is shown that corresponding Lorentz generators and noncommutative coordinates compatible with bicovariant calculus cannot be realized in terms of commutative coordinates and momenta. Furthermore, κ-Minkowski space and NC forms are constructed by twist related to a bicrossproduct basis. It is pointed out that the consistency condition is not satisfied. We present the construction of κ-deformed coordinates and forms (super-Heisenberg algebra) using extended twist. It is compatible with bicovariant differential calculus with κ-deformed $\mathfrak{igl}(4)$ -Hopf algebra. The extended twist leading to κ-Poincaré-Hopf algebra is also discussed.     

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.     

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.     

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.     

System of interacting harmonic oscillators in rotationally invariant noncommutative phase space

Rotationally invariant space with noncommutativity of coordinates and noncommutativity of momenta of canonical type is considered. A system of N interacting harmonic oscillators in uniform field and a system of N particles with harmonic oscillator interaction are studied. We analyze effect of noncommutativity on the energy levels of these systems. It is found that influence of coordinates noncommutativity on the energy levels of the systems increases with increasing of the number of particles. The spectrum of N free particles in uniform field in rotationally invariant noncommutative phase space is also analyzed. It is shown that the spectrum corresponds to the spectrum of a system of N harmonic oscillators with frequency determined by the parameter of momentum noncommutativity.     

Distances in finite spaces from noncommutative geometry

Following the general principles of noncommutative geometry, it is possible to define a metric on the space of pure states of the noncommutative algebra generated by the coordinates. This metric generalizes the usual Riemannian one. We investigate some general properties of this metric in finite commutative cases corresponding to a metric on a finite set, and also compute explicitly some distances associated to commutative or noncommutative algebras.     

Non-commutative Euclidean and Minkowski structures

A noncommutative *-algebra that generalizes the canonical commutation relations and that is covariant under the quantum groups SO _q (3) or SO _q(1, 3) is introduced. The generating elements of this algebra are hermitean and can be identified with coordinates, momenta and angular momenta. In addition a unitary scaling operator is part of the algebra.     

Lie algebra type noncommutative phase spaces are Hopf algebroids

For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite-dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way; therefore, obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here, we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.     

Duality between coordinates and wave functions on noncommutative space

The relation between coordinates and the solutions of the stationary Schrödinger equation in the noncommutative algebra of functions on R^2N is discussed. We derive this relation for a certain class of wave functions for which the quantum prepotentials depend linearly on the coordinates similarly to the commutative case. Also, the differential equation satisfied by the prepotentials is given.     

Lagrangian fractional mechanics — a noncommutative approach

The extension of coordinate-velocity space with noncommutative algebra structure is proposed. For action of fractional mechanics considered on such a space the respective Euler-Lagrange equations are derived via minimum action principle. It appears that equations of motion in the noncommutative framework do not mix left and right derivatives thus being simple to solve at least in the linear case. As an example, two models of oscillator with fractional derivatives are studied. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Noncommutative lattices as finite approximations

Lattice discretizations of continuous manifolds are common tools used in a variety of physical contexts. Conventional discrete approximations, however, cannot capture all aspects of the original manifold, notably its topology. In this paper we discuss an approximation scheme due to Sorkin (1991) which correctly reproduces important topological aspects of continuum physics. The approximating topological spaces are partially ordered sets (posets), the partial order encoding the topology. Now, the topology of a manifold M can be reconstructed from the commutativè C^*algebra C(M) of continuous functions defined on it. In turn, this algebra is generated by continuous probability densities in ordinary quantum physics on M. The latter also serves to specify the domains of observables like the Hamiltonian. For a poset, the role of this algebra is assumed by a noncommutative C^*-algebra A. This fact makes any poset a genuine ‘noncommutative’ (‘quantum’) space, in the sense that the algebra of its ‘continuous functions’ is a noncommutative C^*-algebra. We therefore also have a remarkable connection between finite approximations to quantum physics and noncommutative geometries. We use this connection to develop various approximation methods for doing quantum physics using A.     

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.     

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.     

Thermodynamics of a Charged Particle in a Noncommutative Plane in a Background Magnetic Field

Landau system in noncommutative space has been considered. To take into account the issue of gauge invariance in noncommutative space, we incorporate the Seiberg-Witten map in our analysis. Generalised Bopp-shift transformation is then used to map the noncommutative system to its commutative equivalent system. In particular we have computed the partition function of the system and from this we obtained the susceptibility of the Landau system and found that the result gets modified by the spatial noncommutative parameter θ. We also investigate the de Hass–van Alphen effect in noncommutative space and observe that the oscillation of the magnetization and the susceptibility gets noncommutative corrections. Interestingly, the susceptibility in the noncommutative scenario is non-zero in the range of the magnetic field greater than the threshold value which is in contrast to its commutative counterpart. The results obtained are valid upto all orders in the noncommutative parameter θ.     

Moduli Space and Structure of Noncommutative 3-Spheres

We analyze the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and prove a general algebraic result which considerably refines the classical homomorphism from a quadratic algebra to a cross-product algebra associated to the characteristic variety and lands in a richer cross-product. It allows to control the C*-norm on involutive quadratic algebras and to construct the differential calculus in the desired generality. The moduli space of noncommutative 3-spheres is identified with equivalence classes of pairs of points in a symmetric space of unitary unimodular symmetric matrices. The scaling foliation of the moduli space is identified to the gradient flow of the character of a virtual representation of SO(6). Its generic orbits are connected components of real parts of elliptic curves which form a net of biquadratic curves with eight points in common. We show that generically these curves are the same as the characteristic variety of the associated quadratic algebra. We then apply the general theory of central quadratic forms to show that the noncommutative 3-spheres admit a natural ramified covering π by a noncommutative three-dimensional nilmanifold. This yields the differential calculus. We then compute the Jacobian of the ramified covering π by pairing the direct image of the fundamental class of the noncommutative three-dimensional nilmanifold with the Chern character of the defining unitary and obtain the answer as the product of a period (of an elliptic integral) by a rational function. Finally, we show that the hyperfinite factor of type II₁ appears as cross-product of the field K _q of meromorphic functions on an elliptic curve by a subgroup of its Galois group ${\text{Aut}}_\mathbb{C} \left( {K_q } \right)$ .     

Commutation superoperator of a state and its applications to the noncommutative statistics

For a normal state on a von Neumann algebra the space of square-integrable operators is introduced. As distinct from the L² space in the classical probability theory, it possesses an additional skew-symmetric form and the associated superoperator, which is a convenient tool to describe commutation properties in L². It is shown that a state on the algebra of canonical commutation relations is Gaussian (quasi-free) iff the space of canonical observables is an invariant subspace of the corresponding commutation superoperator. Basing on these ideas a new approach to some problems in the noncommutative statistic is developed.     

Spin Hall effect in noncommutative coordinates

A semiclassical constrained Hamiltonian system which was established to study dynamical systems of matrix valued non-Abelian gauge fields is employed to formulate spin Hall effect in noncommuting coordinates at the first order in the constant noncommutativity parameter θ. The method is first illustrated by studying the Hall effect on the noncommutative plane in a gauge independent fashion. Then, the Drude model type and the Hall effect type formulations of spin Hall effect are considered in noncommuting coordinates and θ deformed spin Hall conductivities which they provide are acquired. It is shown that by adjusting θ different formulations of spin Hall conductivity are accomplished. Hence, the noncommutative theory can be envisaged as an effective theory which unifies different approaches to similar physical phenomena.     

Field theory amplitudes in a space with SU(2) fuzziness

The structure of transition amplitudes in field theory in a three-dimensional space whose spatial coordinates are noncommutative and satisfy the SU(2) Lie algebra commutation relations is examined. In particular, the basic notions for constructing the observables of the theory as well as subtleties related to the proper treatment of δ distributions (corresponding to conservation laws) are introduced. Explicit examples are given for scalar field theory amplitudes in the lowest order of perturbation.     

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.