Related papers: The spectral distance on the Moyal plane
We question the emergence of a minimal length in quantum spacetime, comparing two notions that appeared at various points in the literature: on the one side, the quantum length as the spectrum of an operator L in the Doplicher Fredenhagen…
We investigate whether the identification between Connes' spectral distance in noncommutative geometry and the Monge-Kantorovich distance of order 1 in the theory of optimal transport - that has been pointed out by Rieffel in the…
We discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes' distance, between pure states, in the product of commutative spectral triples. We investigate the…
We study the spectral metric aspects of the standard Podles sphere, which is a homogeneous space for quantum SU(2). The point of departure is the real equivariant spectral triple investigated by Dabrowski and Sitarz. The Dirac operator of…
Examples of noncommutative self-coverings are described, and spectral triples on the base space are extended to spectral triples on the inductive family of coverings, in such a way that the covering projections are locally isometric. Such…
We study the canonical quantization of the theory given by Chamseddine-Connes spectral action on a particular finite spectral triple with algebra $M_2(\Cset)\oplus\Cset$. We define a quantization of the natural distance associated with this…
We study the convergence aspects of the metric on spectral truncations of geometry. We find general conditions on sequences of operator system spectral triples that allows one to prove a result on Gromov-Hausdorff convergence of the…
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…
In noncommutative geometry, Connes's spectral distance is an extended metric on the state space of a C*-algebra generalizing Kantorovich's dual formula of the Wasserstein distance of order 1 from optimal transport. It is expressed as a…
We show that Connes' metric on the state space associated with a spectral triple is nowhere infinite exactly when it is globally bounded. Moreover, we produce a family of simple examples showing that this is not automatically the case.
We develop a systematic study about the spectrality of measures supported on piecewise smooth curves by studying the support of the tempered distributions arising from the tiling equation of some singular spectral measures. In doing so, we…
We present a first numerical investigation of a non-commutative gauge theory defined via the spectral action for Moyal space with harmonic propagation. This action is approximated by finite matrices. Using Monte Carlo simulation we study…
This is a review of explicit computations of Connes distance in noncommutative geometry, covering finite dimensional spectral triples, almost-commutative geometries, and spectral triples on the algebra of compact operators. Several…
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…
To any spectral triple (A,D,H) a dimension d is associated, in analogy with the Hausdorff dimension for metric spaces. Indeed d is the unique number, if any, such that |D|^-d has non trivial logarithmic Dixmier trace. Moreover, when d is…
Modulo some natural generalizations to noncompact spaces, we show in this letter that Moyal planes are nonunital spectral triples in the sense of Connes. The action functional of these triples is computed, and we obtain the expected result,…
We consider the spectral behavior and noncommutative geometry of commutators $[P,f]$, where $P$ is an operator of order $0$ with geometric origin and $f$ a multiplication operator by a function. When $f$ is H\"{o}lder continuous, the…
We study the Connes distance of quantum states of $2D$ harmonic oscillators in phase space. Using the Hilbert-Schmidt operatorial formulation, we construct a boson Fock space and a quantum Hilbert space, and obtain the Dirac operator and a…
This work prolongs, using an operator method, the investigations started in our recent paper J. Math. Phys. 51., 102108 on the spectrum and states of the harmonic oscillator on twisted Moyal plane, where rather a Moyal-star-algebraic…
A fundamental tool in noncommutative geometry is Connes' character formula. This formula is used in an essential way in the applications of noncommutative geometry to index theory and to the spectral characterisation of manifolds. A…