Spectral Distances: Results for Moyal Plane and Noncommutative Torus

The spectral distance for noncommutative Moyal planes is considered in the framework of a non compact spectral triple recently proposed as a possible noncommutative analog of non compact Riemannian spin manifold. An explicit formula for the distance between any two elements of a particular class of pure states can be determined. The corresponding result is discussed. The existence of some pure states at infinite distance signals that the topology of the spectral distance on the space of states is not the weak * topology. The case of the noncommutative torus is also considered and a formula for the spectral distance between some states is also obtained

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

The spectral distance on the Moyal plane

We study the noncommutative geometry of the Moyal plane from a metric point of view. Starting from a non compact spectral triple based on the Moyal deformation A of the algebra of Schwartz functions on R^2, we explicitly compute Connes' spectral distance between the pure states of A corresponding to eigenfunctions of the quantum harmonic oscillator. For other pure states, we provide a lower bound to the spectral distance, and show that the latest is not always finite. As a consequence, we show that the spectral triple [20] is not a spectral metric space in the sense of [5]. This motivates the study of truncations of the spectral triple, based on M_n(C) with arbitrary integer n, which turn out to be compact quantum metric spaces in the sense of Rieffel. Finally the distance is explicitly computed for n=2.Comment: Published version. Misprints corrected and references updated; Journal of Geometry and Physics (2011

Aspects of the metric and differential noncommutative geometry : application to physics

La géométrie non commutative, du fait qu'elle permet de généraliser des objets géométriques sous forme algébrique, offre des perspectives intéressantes pour réunir la théorie quantique des champs et la relativité générale dans un seul cadre. Elle peut être abordée selon différents points de vue et deux d'entre eux sont présentés dans cette thèse. Le premier, le calcul différentiel basé sur les dérivations, nous a permis de construire une action de Yang-Mills-Higgs dans laquelle apparait des champs pouvant être interprétés comme des champs de Higgs. Avec le second, les triplets spectraux, on peut généraliser la notion de distance entre état et calculer des formules de distance. C'est ce que nous avons fait dans le cas de l'espace de Moyal et du tore non commutatif.Noncommutative geometry offers interesting prospects to gather the quantum field theory and relativity in one general framework because it allows one to generalize geometric objects algebraically. It can be approached from different points of view and two of them are presented in this PhD. The first, calculus based on derivations, allowed us to construct a Yang-Mills-Higgs action which appears in fields that can be interpreted as Higgs fields. With the second, spectral triples, we can generalize the notion of distance between states. We calculated the distance formulas in the case of the Moyal space and the noncommutative torus

Aspects différentiels et métriques de la géométrie non commutative. Application à la physique.

La géométrie non commutative, du fait qu'elle permet de généraliser des objets géométriques sous forme algébrique, offre des perspectives intéressantes pour réunir la théorie quantique des champs et la relativité générale dans un seul cadre. Elle peut être abordée selon différents points de vue et deux d'entre eux sont présentés dans cette thèse. Le premier, le calcul différentiel basé sur les dérivations, nous a permis de construire une action de Yang-Mills-Higgs dans laquelle apparait des champs pouvant être interprétés comme des champs de Higgs. Avec le second, les triplets spectraux, on peut généraliser la notion de distance entre état et calculer des formules de distance. C'est ce que nous avons fait dans le cas de l'espace de Moyal et du tore non commutatif.Noncommutative geometry offers interesting prospects to gather the quantum field theory and relativity in one general framework because it allows one to generalize geometric objects algebraically. It can be approached from different points of view and two of them are presented in this PhD. The first, calculus based on derivations, allowed us to construct a Yang-Mills-Higgs action which appears in fields that can be interpreted as Higgs fields. With the second, spectral triples, we can generalize the notion of distance between states. We calculated the distance formulas in the case of the Moyal space and the noncommutative torus.PARIS11-SCD-Bib. électronique (914719901) / SudocSudocFranceF

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

Abstract. 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 M. We show that the differential calculus, generated by the maximal subalgebra of the derivation algebra of M that can be related to infinitesimal symplectomorphisms, gives rise to a natural construction of Yang-Mills-Higgs models on 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 .C/ and the algebra of matrix valued functions C 1 .M /˝M n .C/. The UV/IR mixing problem of the resultingYang-MillsHiggs models is also discussed. (2010). 81T75, 81T13, 81T15. Mathematics Subject Classificatio