A free particle in noncommutative space-time

By examining some known wave equations it is shown that the dynamics of a free quantum particle in a slightly noncommutative space-time is equivalent to that of a charged particle moving in a self-generated (weak) electromagnetic field.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.     

A noncommutative view on topology and order

In this paper we put forward the definition of particular subsets on a unital C^∗$C^{\ast }$-algebra, that we call isocones, and which reduce in the commutative case to the set of continuous non-decreasing functions with real values for a partial order relation defined on the spectrum of the algebra, which satisfies a compatibility condition with the topology (complete separateness). We prove that this space/algebra correspondence is a dual equivalence of categories, which generalizes the Gelfand–Naimark duality. Thus we can expect that general isocones could serve to define a notion of “noncommutative ordered spaces”. We also explore some basic algebraic constructions involving isocones, and classify those which are defined in _M2(C)$M_2\left(\mathbb{C}\right)$.     

A noncommutative deformation of topological field theory

Cohomological Yang–Mills theory is formulated on a noncommutative differentiable four manifold through the -deformation of its corresponding BRST algebra. The resulting noncommutative field theory is a natural setting to define the -deformation of Donaldson invariants and they are interpreted as a mapping between the Chevalley–Eilenberg homology of noncommutative spacetime and the Chevalley–Eilenberg cohomology of noncommutative moduli of instantons. In the process we find that in the weak coupling limit the quantum theory is localized at the moduli space of noncommutative instantons.     

A quantum-group-like structure on noncommutative 2-tori

We show that on noncommutative 2-tori, there are natural structures which have analogous formal properties as Hopf algebra structures, but where the comultiplication has values in a deformation of the tensor product.Supported by Project P 7724 PHY of Fonds zur Förderung der wissenschaftlichen Forschung.     

On A. Connes' noncommutative integration theory

This paper is a comment (written in a self-contained way) upon A. Connes' noncommutative integration theory. A simple observation (Lemma 9) allows the direct definition of transverse measures as functionals on the general positive random functions. Connes' first definition is then recovered by considering the transverse functions as a separating subset of the positive random functions (to which they are related by explicit natural transformations).This work is supported in part by the National Science Foundation, Grant MCS 79-03041     

Three-dimensional noncommutative bosonization

We consider the extension of the (2+1)-dimensional$(2+1)\text{-dimensional}$ bosonization process in noncommutative (NC) spacetime. We show that the large mass limit of the effective action obtained by integrating out the fermionic fields in NC spacetime leads to the NC Chern–Simons action. The present result is valid to all orders in the noncommutative parameter θ. We also discuss how the NC Yang–Mills action is induced in the next to leading order.     

A noncommutative geometric approach to left-right symmetric weak interactions

By combining the generalized exterior algebra of forms over a noncommutative algebra with the gauging of discrete directions and the associated Higgs fields, we consider the construction of the bosonic sector of left-right symmetric models of the form SU(2) _L SU(2) _R U(1). We see that within this formalism maximal use can be made of the gauge connection associated with the noncommutative graded algebra.     

A new kind of representations on noncommutative phase space

We introduce new representations to formulate quantum mechanics on noncommutative phase space, in which both coordinate-coordinate and momentum-momentum are noncommutative. These representations explicitly display entanglement properties between degrees of freedom of different coordinate and momentum components. To show their potential applications, we derive explicit expressions of Wigner function and Wigner operator in the new representations, as well as solve exactly a two-dimensional harmonic oscillator on the noncommutative phase plane with both kinetic coupling and elastic coupling.     

Two-dimensional noncommutative quantum dynamics

This paper gives the two-dimensional extension for the noncommutative quantum dynamics of Rembielinski and Smolinski.     

A note on noncommutative Chern–Simons theories

The three-dimensional Chern–Simons theory on is studied. Considering the gauge transformations under the group elements which are going to one at infinity, we show that under arbitrary (finite) gauge transformations action changes with an integer multiple of 2π if, the level of noncommutative Chern–Simons is quantized. We also briefly discuss the case of the noncommutative torus and some other possible extensions.     

Hard noncommutative loops resummation

The noncommutative version of the Euclidean g2phi4 theory is considered. By using Wilsonian flow equations the ultraviolet renormalizability can be proved to all orders in perturbation theory. On the other hand, the infrared sector cannot be treated perturbatively and requires a resummation of the leading divergences in the two-point function. This is analogous to what is done in the hard thermal loops resummation of finite temperature field theory. Next-to-leading order corrections to the self-energy are computed, resulting in O(g3) contributions in the massless case, and O(g6logg2) in the massive one.     

Quantum noncommutative multidimensional cosmology

We present exact quantum solutions for a noncommutative, multidimensional cosmological model and show that stabilization of extra dimensions sets in with the introduction of noncommutativity between the scale factors. An interpretation is offered to accommodate the notion of time, rendering comparison with the classical solutions possible.     

Riemannian manifolds in noncommutative geometry

We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spin^c${}^c$ manifolds; and conversely, in the presence of a spin^c${}^c$ structure. We also show how to obtain an analogue of Kasparov’s fundamental class for a Riemannian manifold, and the associated notion of Poincaré duality. Along the way we clarify the bimodule and first-order conditions for spectral triples.     

Plane waves in noncommutative fluids

We study the dynamics of the noncommutative fluid in the Snyder space perturbatively at the first order in powers of the noncommutative parameter. The linearized noncommutative fluid dynamics is described by a system of coupled linear partial differential equations in which the variables are the fluid density and the fluid potentials. We show that these equations admit a set of solutions that are monochromatic plane waves for the fluid density and two of the potentials and a linear function for the third potential. The energy–momentum tensor of the plane waves is calculated.     

Renormalizable noncommutative quantum field theory

We discuss the Euclidean noncommutative f⁴₄{\phi^4_4}-quantum field theory as an example of a renormalizable field theory. Using a Ward identity, Disertori, Gurau, Magnen and Rivasseau were able to prove the vanishing of the beta function for the coupling constant to all orders in perturbation theory. We extend this work and obtain from the Schwinger–Dyson equation a non-linear integral equation for the renormalised two-point function alone. The non-trivial renormalised four-point function fulfils a linear integral equation with the inhomogeneity determined by the two-point function. These integral equations might be the starting point of a nonperturbative construction of a Euclidean quantum field theory on a noncommutative space. We expect to learn about renormalisation from this almost solvable model.     

Causality in noncommutative two-sheeted space-times

We investigate the causal structure of two-sheeted space-times using the tools of Lorentzian spectral triples. We show that the noncommutative geometry of these spaces allows for causal relations between the two sheets. The computation is given in detail when the sheet is a 2- or 4-dimensional globally hyperbolic spin manifold. The conclusions are then generalised to a point-dependent distance between the two sheets resulting from the fluctuations of the Dirac operator.     

Integrable mappings for noncommutative objects

The method of integrable mappings is generalized to the noncommutative case. Hierarchies of integrable systems corresponding to the noncommutative Darboux-Toda substitution in the two-dimensional spaces and superspaces are constructed.