SLh(2)-Symmetric Torsionless Connections

The most general SL _h (2)–symmetric torsionless linear connection is constructed. This is done based on a recently proposed definition of a linear connection in noncommutative geometry. Part of the results can be obtained by using the singular map which relates the q-plane to the h–plane. There is also a part in the covariant derivative, linearconnection, and curvature which does not have any q-analogue. It is seen that the covariant derivative of the h-plane is more classical or less quantized than that of the q-plane.     

Dirac Operators on Quantum Flag Manifolds

A Dirac operator D on quantized irreducible generalized flag manifolds is defined. This yields a Hilbert space realization of the covariant first-order differential calculi constructed by I. Heckenberger and S. Kolb. All differentials df=i[D,f] are bounded operators. In the simplest case of Podle' quantum sphere one obtains the spectral triple found by L. Dabrowski and A. Sitarz.     

Commutator Representations of Covariant Differential Calculi on Quantum Groups

Let (, d) be a first-order differential *-calculus on a *-algebra . We say that a pair (, F) of a *-representation of on a dense domain of a Hilbert space and a symmetric operator F on gives a commutator representation of if there exists a linear mapping : L( ) such that (adb) = (a)i[F, (b) ], a, b . Among others, it is shown that each left-covariant *-calculus of a compact quantum group Hopf *-algebra has a faithful commutator representation. For a class of bicovariant *-calculi on , there is a commutator representation such that F is the image of a central element of the quantum tangent space. If is the Hopf *-algebra of the compact form of one of the quantum groups SL _q (n+1), O _q (n), Sp _q (2n) with real trancendental q, then this commutator representation is faithful.     

Field Theory on a q = −1 Quantum Plane

We build a q = –1 deformation of a plane on a product of two copies of algebras of functions on the plane. This algebra contains a subalgebra of functions on the plane. We present a general scheme (which could be also used to construct a quaternion from pairs of complex numbers) and we use it to derive differential structures and metrics, and discuss sample field-theoretical models.Mathematics Subject Classifications (1991):46L87, 81T13, 17B37.     

Noncommutative Unification of General Relativity and Quantum Mechanics. A Finite Model

We construct a model unifying general relativity and quantum mechanics in a broader structure of noncommutative geometry. The geometry in question is that of a transformation groupoid given by the action of a finite group on a space E. We define the algebra of smooth complex valued functions on , with convolution as multiplication, in terms of which the groupoid geometry is developed. Owing to the fact that the group G is finite the model can be computed in full details. We show that by suitable averaging of noncommutative geometric quantities one recovers the standard space-time geometry. The quantum sector of the model is explored in terms of the regular representation of the algebra , and its correspondence with the standard quantum mechanics is established.     

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=ε_ij^kθ_k and a momentum noncommutativity matrix parameter β=ε_ij^kβ_k, is shown to be equivalent to Quantum Mechanics (QM) on a suitable transformed Quantum Phase Space (QPS). Imposing some constraintson this particular transformation, we firstly find that the product of the two parameters θ and β possesses alower 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 thephysical 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. Amongthe 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$vec{B}=q^{-1}vec{beta}$. For the other examples, additionalcorrection 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 noncommutativity is effectively feeled with opposite sign.     

Quantum electrodynamics with arbitrary charge on a noncommutative space

Using the Seiberg-Witten map,we obtain a quantum electrodynamics on a noncommutative space,which has arbitrary charge and keep the gauge invariance to at the leading order in theta.The one-loop divergence and Compton scattering are reinvestigated.The uoncommutative effects are larger than those in ordinary noncommutative quantum electrodynamics.     

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

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

Lessons from Quantum Field Theory: Hopf Algebras and Spacetime Geometries

We discuss the prominence of Hopf algebras in recent progress in Quantum Field Theory. In particular, we will consider the Hopf algebra of renormalization, whose antipode turned out to be the key to a conceptual understanding of the subtraction procedure. We shall then describe several occurrences of this, or closely related Hopf algebras, in other mathematical domains, such as foliations, Runge-Kutta methods, iterated integrals and multiple zeta values. We emphasize the unifying role which the Butcher group, discovered in the study of numerical integration of ordinary differential equations, plays in QFT.     

Noncommutative 4-Spheres Based on all Podleś 2-Spheres and Beyond

A wide class of noncommutative spaces, including 4-spheres based on all the quantum 2-spheres and suspensions of matrix quantum groups is described. For each such space a noncommutative vector bundle is constructed. This generalises and clarifies various recent constructions of noncommutative 4-spheres.     

Functional Integrals and Quantum Fluctuations on Two-DimensionalNoncommutative Space-Time

The generalized Thirring model with impurity coupling is defined ontwo-dimensional noncommutative space-time, a modified propagator and freeenergy are derived by means of functional integrals method. Moreover,quantum fluctuations and excitation energies are calculated on two-dimensional black hole and soliton background.     

Quantum electrodynamics with arbitrary charge on a noncommutative space

Using the Seiberg-Witten map, we obtain a quantum electrodynamics on a noncommutative space, which has arbitrary charge and keep the gauge invariance to at the leading order in theta. The one-loop divergence and Compton scattering are reinvestigated. The noncommutative effects are larger than those in ordinary noncommutative quantum electrodynamics.     

Geodesic Curves on Quantized Manifolds

A general definition of curves and geodesics associated with a given connection on a quantized manifold is given. In the particular case of functional quantization, we define geodesics in the same way as in the classical case and we will show that the two definitions are compatible. As an example, we examine our results for the quantum Manin plane.     

Noncommutative Quantum Hall Effect of Bilayer Systems

In this paper we study the bilayer quantum Hall （QH） effect on a noncommutative phase space （NCPS）. By using perturbation theory, we calculate the energy spectrum, eigenfunction, Hall current, and Hall conductivity of the bilayer QH system, and express them in terms of noncommutative parameters θ and θ^-, respectively. In our calculation, we assume that these parameters vary from laver to laver.     

Noncommutative Quantum Hall Effect of Bilayer Systems

In this paper we study the bilayer quantum Hall (QH) effect on a noncommutative phase space (NCPS). By using perturbation theory, we calculate the energy spectrum, eigenfunction, Hall current, and Hall conductivity of the bilayer QH system, and express them in terms of noncommutative parameters θ and \bar{θ}, respectively. In our calculation, we assume that these parameters vary from layer to layer.     

One-loop renormalizability of noncommutative U(1) gaugetheory with scalar fields

This paper uses the background field method to calculate one-loop divergent corrections to the gauge field propagators in noncommutative U(1) gauge theory with scalar fields. It shows that for a massless scalar field, the gauge field propagators are renormalizable to θ²-order, but for a massive scalar field they are renormalizable only to θ-order.     

On Spin Structures and Dirac Operators on the Noncommutative Torus

We find and classify possible equivariant spin structures with Dirac operators on the noncommutative torus, proving that, similarly as in the classical case, the spectrum of the Dirac operator depends on the spin structure.     

Differential and Complex Geometry of Two-Dimensional Noncommutative Tori

We analyze in detail projective modules over two-dimensional noncommutative tori and complex structures on these modules. We concentrate our attention on properties of holomorphic vectors in these modules; the theory of these vectors generalizes the theory of theta-functions. The paper is self-contained; it can be used also as an introduction to the theory of noncommutative spaces with simplest space of this kind thoroughly analyzed as a basic example.     

Realization of the N(odd)-Dimensional Quantum Euclidean Space by Differential Operators

The quantum Euclidean space R_q^N is a kind ofnoncommutative space that is obtained from ordinary Euclidean spaceR^N by deformation with parameter q. When N is odd, thestructure of this space is similar to R_q³. Motivated byrealization of R_q³ by differential operators inR³, we give such realization for R_q⁵ andR_q⁷ cases and generalize our results toR_q^N (N odd) in this paper, that is, we show that thealgebra of R_q^N can be realized by differential operatorsacting on C^∝ functions on undeformed spaceR^N.