Noncommutative field theories on Rλ3R^3_\lambda: Towards UV/IR mixing freedom

We consider the noncommutative space $\mathbb{R}^3_\lambda$, a deformation of the algebra of functions on $\mathbb{R}^3$ which yields a "foliation" of $\mathbb{R}^3$ into fuzzy spheres. We first construct a natural matrix base adapted to $\mathbb{R}^3_\lambda$. We then apply this general framework to the one-loop study of a two-parameter family of real-valued scalar noncommutative field theories with quartic polynomial interaction, which becomes a non-local matrix model when expressed in the above matrix base. The kinetic operator involves a part related to dynamics on the fuzzy sphere supplemented by a term reproducing radial dynamics. We then compute the planar and non-planar 1-loop contributions to the 2-point correlation function. We find that these diagrams are both finite in the matrix base. We find no singularity of IR type, which signals very likely the absence of UV/IR mixing. We also consider the case of a kinetic operator with only the radial part. We find that the resulting theory is finite to all orders in perturbation expansion.Comment: 31 pages, 4 figures. Improved version. Sections 5.1 and 5.2 have been clarified. A minor error corrected. References adde

Anyonic Excitations in Fast Rotating Bose Gases

The role of anyonic excitations in fast rotating harmonically trapped Bose gases in a fractional Quantum Hall State is examined. Standard Chern-Simons anyons as well as "non standard" anyons obtained from a statistical interaction having Maxwell-Chern-Simons dynamics and suitable non minimal coupling to matter are considered. Their respective ability to stabilize attractive Bose gases under fast rotation in the thermodynamical limit is studied. Stability can be obtained for standard anyons while for non standard anyons, stability requires that the range of the corresponding statistical interaction does not exceed the typical wavelength for the atoms.Comment: 6 pages. Presented at the "School on Quantum Phase Transitions and Non-Equilibrium Phenomena in Cold Atomic Gases", ICTP Trieste, 11-22 july 200

The role of proximity relations in regional and territorial development processes

Proximity analyses have nowadays turned out to be a part of the toolbox of regional scientists and this notion recently became very popular in the position of politics, and private or public stakeholders. Proximity is an argument for selling food or financial products, as well as a good slogan for local networks or social devices or even for policymakers. In parallel, the notion of proximity spread in the academic literature and is now commonly used by scholars in regional science, geography or spatial economics. The use of the word proximity increased and grown in importance, in particular for authors interested in the question of milieus, districts, distance analyses, or in recent advances in economic geography or evolutionary geography. Interest is affecting now the works dedicated to innovation process, links between science and industry, relations between users and producers or sub-contractors, national systems of innovation, innovative milieus, about also local labour markets or urban policies. Indeed, the use of the concept of proximity, plural by nature by its spatial as well as non-spatial dimensions, is the key for overcoming the apparent opposition between the reaffirmation of the importance of the local and the death of distance and for escaping the sterile confinement in one or the other extreme positions. But despite the substantial literature on proximity processes and relations, only a few academic works have been devoted to studying the link between regional development and proximity relations. This paper intends to fill this gap and to pave the way for future research in this field. We consider that the integration of the notion of proximity into the framework of regional development analysis provides interesting input due to its plasticity and ability to draw connections between spatial, economic and social dimensions; but also suggests ways of possible changes for regional and territorial policies. The main outline is to try to assess the importance of proximity relations (or obstacles led by proximity relations) in regional development processes, and discuss approaches of different disciplines

Involutive representations of coordinate algebras and quantum spaces

We show that $\frak{su}(2)$ Lie algebras of coordinate operators related to quantum spaces with $\frak{su}(2)$ noncommutativity can be conveniently represented by $SO(3)$-covariant poly-differential involutive representations. We show that the quantized plane waves obtained from the quantization map action on the usual exponential functions are determined by polar decomposition of operators combined with constraint stemming from the Wigner theorem for $SU(2)$. Selecting a subfamily of $^*$-representations, we show that the resulting star-product is equivalent to the Kontsevich product for the Poisson manifold dual to the finite dimensional Lie algebra $\mathfrak{su}(2)$. We discuss the results, indicating a way to extend the construction to any semi-simple non simply connected Lie group and present noncommutative scalar field theories which are free from perturbative UV/IR mixing.Comment: 29 pages, several paragraphs added, published in JHE

Noncommutative Yang-Mills-Higgs actions from derivation-based differential calculus

Derivations of a noncommutative algebra can be used to construct differential calculi, the so-called derivation-based differential calculi. We apply this framework to a version of the Moyal algebra ${\cal{M}}$. We show that the differential calculus, generated by the maximal subalgebra of the derivation algebra of ${\cal{M}}$ that can be related to infinitesimal symplectomorphisms, gives rise to a natural construction of Yang-Mills-Higgs models on ${\cal{M}}$ and a natural interpretation of the covariant coordinates as Higgs fields. We also compare in detail the main mathematical properties characterizing the present situation to those specific of two other noncommutative geometries, namely the finite dimensional matrix algebra $M_n({\mathbb{C}})$ and the algebra of matrix valued functions $C^\infty(M)\otimes M_n({\mathbb{C}})$. The UV/IR mixing problem of the resulting Yang-Mills-Higgs models is also discussed.Comment: 23 pages, 2 figures. Improved and enlarged version. Some references have been added and updated. Two subsections and a discussion on the appearence of Higgs fiels in noncommutative gauge theories have been adde

Derivations of the Moyal Algebra and Noncommutative Gauge Theories

The differential calculus based on the derivations of an associative algebra underlies most of the noncommutative field theories considered so far. We review the essential properties of this framework and the main features of noncommutative connections in the case of non graded associative unital algebras with involution. We extend this framework to the case of ${\mathbb{Z}}_2$-graded unital involutive algebras. We show, in the case of the Moyal algebra or some related ${\mathbb{Z}}_2$-graded version of it, that the derivation based differential calculus is a suitable framework to construct Yang-Mills-Higgs type models on Moyal (or related) algebras, the covariant coordinates having in particular a natural interpretation as Higgs fields. We also exhibit, in one situation, a link between the renormalisable NC $\phi^4$-model with harmonic term and a gauge theory model. Some possible consequences of this are briefly discussed.Comment: 25 pages, 1 figure. Based on a talk given at the XVIIth International Colloquium on Integrable Systems and Quantum Symmetries, June 19-22, 2008, Pragu

Vacuum energy and the cosmological constant problem in κ\kappa-Poincar\'e invariant field theories

We investigate the vacuum energy in $\kappa$-Poincar\'e invariant field theories. It is shown that for the equivariant Dirac operator one obtains an improvement in UV behavior of the vacuum energy and therefore the cosmological constant problem has to be revised.Comment: improved version, 15 page

Spectral theorem in noncommutative field theories: Jacobi dynamics

Jacobi operators appear as kinetic operators of several classes of noncommutative field theories (NCFT) considered recently. This paper deals with the case of bounded Jacobi operators. A set of tools mainly issued from operator and spectral theory is given in a way applicable to the study of NCFT. As an illustration, this is applied to a gauge-fixed version of the induced gauge theory on the Moyal plane expanded around a symmetric vacuum. The characterization of the spectrum of the kinetic operator is given, showing a behavior somewhat similar to a massless theory. An attempt to characterize the noncommutative geometry related to the gauge fixed action is presented. Using a Dirac operator obtained from the kinetic operator, it is shown that one can construct an even, regular, weakly real spectral triple. This spectral triple does not define a noncommutative metric space for the Connes spectral distance.Comment: 31 pages. Improved version to be published. Section 4 modified. Various misprints correcte

A Remark on the Spontaneous Symmetry Breaking Mechanism in the Standard Model

In this paper we consider the Spontaneous Symmetry Breaking Mechanism (SSBM) in the Standard Model of particles in the unitary gauge. We show that the computation usually presented of this mechanism can be conveniently performed in a slightly different manner. As an outcome, the computation we present can change the interpretation of the SSBM in the Standard Model, in that it decouples the SU(2)-gauge symmetry in the final Lagrangian instead of breaking it.Comment: 16 page