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Pierre Martinetti

Publications and source records attributed to Pierre Martinetti.

48 records · Page 3Linked to original sources
arXiv2008

Spectral distance on the circle

A building block of noncommutative geometry is the observation that most of the geometric information of a compact riemannian spin manifold M is encoded within its Dirac operator D. Especially via Connes' distance formula one is able to extract from the spectral properties of D the geodesic distance on M. In this paper we investigate the distance d encoded within a covariant Dirac operator on a trivial U(n)-fiber bundle over the circle with arbitrary connection. It turns out that the connected components of d are tori whose dimension is given by the holonomy of the connection. For n=2 we explicitly compute d on all the connected components. For n>2 we restrict to a given fiber and find that the distance is given by the trace of the module of a matrix. The latest is defined by the holonomy and the coordinates of the points under consideration. This paper extends to arbitrary n and arbitrary connection the results obtained in hep-th/0506147 for U(2)-bundle with constant connection. It confirms interesting properties of the spectral distance with respect to another distance naturally associated to connection, namely the horizontal or Carnot-Caratheodory distance d_H. Especially in case the connection has irrational components, the connected components for d are the closure of the connected components of d_H within the euclidean topology on the torus.

math.OA↗
arXiv2008

Wilsonian renormalization, differential equations and Hopf algebras

In this paper, we present an algebraic formalism inspired by Butcher's B-series in numerical analysis and the Connes-Kreimer approach to perturbative renormalization. We first define power series of non linear operators and propose several applications, among which the perturbative solution of a fixed point equation using the non linear geometric series. Then, following Polchinski, we show how perturbative renormalization works for a non linear perturbation of a linear differential equation that governs the flow of effective actions. Then, we define a general Hopf algebra of Feynman diagrams adapted to iterations of background field effective action computations. As a simple combinatorial illustration, we show how these techniques can be used to recover the universality of the Tutte polynomial and its relation to the $q$-state Potts model. As a more sophisticated example, we use ordered diagrams with decorations and external structures to solve the Polchinski's exact renormalization group equation. Finally, we work out an analogous construction for the Schwinger-Dyson equations, which yields a bijection between planar $ϕ^{3}$ diagrams and a certain class of decorated rooted trees.

hep-th↗
arXiv2007

Twisted Hopf symmetries of canonical noncommutative spacetimes and the no-pure-boost principle

We study the twisted-Hopf-algebra symmetries of observer-independent canonical spacetime noncommutativity, for which the commutators of the spacetime coordinates take the form [x^{mu},x^{nu}]=i theta^{mu nu} with observer-independent (and coordinate-independent) theta^{mu nu}. We find that it is necessary to introduce nontrivial commutators between transformation parameters and spacetime coordinates, and that the form of these commutators implies that all symmetry transformations must include a translation component. We show that with our noncommutative transformation parameters the Noether analysis of the symmetries is straightforward, and we compare our canonical-noncommutativity results with the structure of the conserved charges and the "no-pure-boost" requirement derived in a previous study of kappa-Minkowski noncommutativity. We also verify that, while at intermediate stages of the analysis we do find terms that depend on the ordering convention adopted in setting up the Weyl map, the final result for the conserved charges is reassuringly independent of the choice of Weyl map and (the corresponding choice of) star product.

hep-th↗
arXiv2007

Is life a thermal horizon ?

This talk aims at questioning the vanishing of Unruh temperature for an inertial observer in Minkovski spacetime with finite lifetime, arguing that in the non eternal case the existence of a causal horizon is not linked to the non-vanishing of the acceleration. This is illustrated by a previous result, the diamonds temperature, that adapts the algebraic approach of Unruh effect to the finite case.

gr-qc↗
arXiv2007

A no-pure-boost uncertainty principle from spacetime noncommutativity

We study boost and space-rotation transformations in kappa-Minkowski noncommutative spacetime, using the techniques that some of us had previously developed (hep-th/0607221) for a description of translations in kappa-Minkowski, which in particular led to the introduction of translation transformation parameters that do not commute with the spacetime coordinates. We find a similar description of boosts and space rotations, which allows us to identify some associated conserved charges, but the form of the commutators between transformation parameters and spacetime coordinates is incompatible with the possibility of a pure boost.

hep-th↗
arXiv2007

Carnot-Caratheodory metric and gauge fluctuation in Noncommutative Geometry

Gauge fields have a natural metric interpretation in terms of horizontal distance. The latest, also called Carnot-Caratheodory or subriemannian distance, is by definition the length of the shortest horizontal path between points, that is to say the shortest path whose tangent vector is everywhere horizontal with respect to the gauge connection. In noncommutative geometry all the metric information is encoded within the Dirac operator D. In the classical case, i.e. commutative, Connes's distance formula allows to extract from D the geodesic distance on a riemannian spin manifold. In the case of a gauge theory with a gauge field A, the geometry of the associated U(n)-vector bundle is described by the covariant Dirac operator D+A. What is the distance encoded within this operator ? It was expected that the noncommutative geometry distance d defined by a covariant Dirac operator was intimately linked to the Carnot-Caratheodory distance dh defined by A. In this paper we precise this link, showing that the equality of d and dh strongly depends on the holonomy of the connection. Quite interestingly we exhibit an elementary example, based on a 2 torus, in which the noncommutative distance has a very simple expression and simultaneously avoids the main drawbacks of the riemannian metric (no discontinuity of the derivative of the distance function at the cut-locus) and of the subriemannian one (memory of the structure of the fiber).

hep-th↗
arXiv2006

Smoother than a circle, or How non commutative geometry provides the torus with an egocentred metric

We give an overview on the metric aspect of noncommutative geometry, especially the metric interpretation of gauge fields via the process of "fluctuation of the metric". Connes' distance formula associates to a gauge field on a bundle P equipped with a connection H a metric. When the holonomy is trivial, this distance coincides with the horizontal distance defined by the connection. When the holonomy is non trivial, the noncommutative distance has rather surprising properties. Specifically we exhibit an elementary example on a 2-torus in which the noncommutative metric d is somehow more interesting than the horizontal one since d preserves the S^1-structure of the fiber and also guarantees the smoothness of the length function at the cut-locus. In this sense the fiber appears as an object "smoother than a circle". As a consequence, from a intrinsic metric point of view developed here, any observer whatever his position on the fiber can equally pretend to be "the center of the world".

hep-th↗
arXiv2003

A brief introduction to the noncommutative geometry description of particle physics standard model

Notes from a course given at Oujda university, Morocco, october 2002 - march 2003 within the support of a fellowship from the Agence Universitaire de la Francophonie. These notes present a brief introduction to Connes' non commutative geometry description of the standard model of particle physics. The notion of distance is emphasized, especially the possible interpretation of the Higgs field as the component of a discrete internal dimension. These notes are in french and are taken from the author's phD thesis.

math-ph↗
arXiv2001

Distances en geometrie non commutative

Cette these etudie l'aspect metrique de la geometrie non commutative a travers la formulation de Connes de la distance entre etats d'une algebre. Sont etudies des exemples d'espaces finis et le modele standard ou le champs de Higgs s'interprete comme coefficient d'une metrique dans une dimension interne discrete.

math-ph↗