scholarly journals Temporal Lorentzian spectral triples

2014 ◽  
Vol 26 (08) ◽  
pp. 1430007 ◽  
Author(s):  
Nicolas Franco
Keyword(s):  
Noncommutative Geometry ◽  
Minkowski Spacetime ◽  
Spectral Triple ◽  
Spectral Triples ◽  
Lorentzian Space ◽  
Pure States ◽  
Global Time ◽  
Distance Formula

We present the notion of temporal Lorentzian spectral triple which is an extension of the notion of pseudo-Riemannian spectral triple with a way to ensure that the signature of the metric is Lorentzian. A temporal Lorentzian spectral triple corresponds to a specific 3 + 1 decomposition of a possibly noncommutative Lorentzian space. This structure introduces a notion of global time in noncommutative geometry. As an example, we construct a temporal Lorentzian spectral triple over a Moyal–Minkowski spacetime. We show that, when time is commutative, the algebra can be extended to unbounded elements. Using such an extension, it is possible to define a Lorentzian distance formula between pure states with a well-defined noncommutative formulation.

scholarly journals Spectral distance on Lorentzian Moyal plane

2020 ◽  
Vol 17 (06) ◽  
pp. 2050089
Author(s):  
Anwesha Chakraborty ◽  
Biswajit Chakraborty
Keyword(s):  
Noncommutative Geometry ◽  
Coherent States ◽  
Euclidean Case ◽  
Spectral Triples ◽  
Spectral Distance ◽  
Moyal Plane ◽  
Pure States ◽  
Math Phys ◽  
Distance Formula

We present here a completely operatorial approach, using Hilbert–Schmidt operators, to compute spectral distances between time-like separated “events”, associated with the pure states of the algebra describing the Lorentzian Moyal plane, using the axiomatic framework given by [N. Franco, The Lorentzian distance formula in noncommutative geometry, J. Phys. Conf. Ser. 968(1) (2018) 012005; N. Franco, Temporal Lorentzian spectral triples, Rev. Math. Phys. 26(8) (2014) 1430007]. The result shows no deformations of non-commutative origin, as in the Euclidean case, if the pure states are constructed out of Glauber–Sudarshan coherent states.

scholarly journals TWISTED ENTIRE CYCLIC COHOMOLOGY, J-L-O COCYCLES AND EQUIVARIANT SPECTRAL TRIPLES

2004 ◽  
Vol 16 (05) ◽  
pp. 583-602 ◽  
Author(s):  
DEBASHISH GOSWAMI
Keyword(s):  
Spectral Data ◽  
Noncommutative Geometry ◽  
Positive Operator ◽  
Chern Character ◽  
Cyclic Cohomology ◽  
Spectral Triple ◽  
Usual Sense ◽  
Spectral Triples ◽  
Definition Of ◽  
Made In

We study the "quantized calculus" corresponding to the algebraic ideas related to "twisted cyclic cohomology" introduced in [12]. With very similar definitions and techniques as those used in [9], we define and study "twisted entire cyclic cohomology" and the "twisted Chern character" associated with an appropriate operator theoretic data called "twisted spectral data", which consists of a spectral triple in the conventional sense of noncommutative geometry [1] and an additional positive operator having some specified properties. Furthermore, it is shown that given a spectral triple (in the conventional sense) which is equivariant under the (co-) action of a compact matrix pseudogroup, it is possible to obtain a canonical twisted spectral data and hence the corresponding (twisted) Chern character, which will be invariant (in the usual sense) under the (co-)action of the pseudogroup, in contrast to the fact that the Chern character coming from the conventional noncommutative geometry need not to be invariant. In the last section, we also try to detail out some remarks made in [3], in the context of a new definition of invariance satisfied by the conventional (untwisted) cyclic cocycles when lifted to an appropriate larger algebra.

scholarly journals Multiplicativity of Connes’ calculus

2019 ◽  
Vol 31 (09) ◽  
pp. 1950033
Author(s):  
Partha Sarathi Chakraborty ◽  
Satyajit Guin
Keyword(s):  
Noncommutative Geometry ◽  
Spectral Triple ◽  
Graded Algebra ◽  
Spectral Triples ◽  
Differential Graded Algebra

In his book on noncommutative geometry, Connes constructed a differential graded algebra out of a spectral triple. Lack of monoidality of this construction is investigated. We identify a suitable monoidal subcategory of the category of spectral triples and show that when restricted to this subcategory the construction of Connes is monoidal. Richness of this subcategory is exhibited by establishing a faithful endofunctor to this subcategory.

scholarly journals Missing the point in noncommutative geometry

Synthese ◽  
2021 ◽  
Author(s):  
Nick Huggett ◽  
Fedele Lizzi ◽  
Tushar Menon
Keyword(s):  
Scalar Field ◽  
Noncommutative Geometry ◽  
Smooth Manifold ◽  
Operational Definition ◽  
Spectral Triple ◽  
Fundamental Quantity

AbstractNoncommutative geometries generalize standard smooth geometries, parametrizing the noncommutativity of dimensions with a fundamental quantity with the dimensions of area. The question arises then of whether the concept of a region smaller than the scale—and ultimately the concept of a point—makes sense in such a theory. We argue that it does not, in two interrelated ways. In the context of Connes’ spectral triple approach, we show that arbitrarily small regions are not definable in the formal sense. While in the scalar field Moyal–Weyl approach, we show that they cannot be given an operational definition. We conclude that points do not exist in such geometries. We therefore investigate (a) the metaphysics of such a geometry, and (b) how the appearance of smooth manifold might be recovered as an approximation to a fundamental noncommutative geometry.

scholarly journals On the Koszul formula in noncommutative geometry

2020 ◽  
Vol 32 (10) ◽  
pp. 2050032 ◽  
Author(s):  
Jyotishman Bhowmick ◽  
Debashish Goswami ◽  
Giovanni Landi
Keyword(s):  
Scalar Curvature ◽  
Noncommutative Geometry ◽  
Spectral Triple ◽  
Fuzzy Sphere ◽  
Civita Connection ◽  
Canonical Metric ◽  
Levi Civita Connection

We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul formula, we show that our Levi-Civita connection is a bimodule connection. We construct a spectral triple on a fuzzy sphere and compute the scalar curvature for the Levi-Civita connection associated to a canonical metric.

scholarly journals NONCOMMUTATIVE GEOMETRY SPECTRAL ACTION AS A FRAMEWORK FOR UNIFICATION: INTRODUCTION AND PHENOMENOLOGICAL/COSMOLOGICAL CONSEQUENCES

2011 ◽  
Vol 20 (05) ◽  
pp. 785-804 ◽  
Author(s):  
MAIRI SAKELLARIADOU
Keyword(s):  
Standard Model ◽  
Noncommutative Geometry ◽  
Particle Physics ◽  
Geometrical Model ◽  
Energy Scale ◽  
Noncommutative Space ◽  
Geometrical Approach ◽  
Neutrino Mixing ◽  
Spectral Triples ◽  
Spectral Action

I will summarize Noncommutative Geometry Spectral Action, an elegant geometrical model valid at unification scale, which offers a purely gravitational explanation of the Standard Model, the most successful phenomenological model of particle physics. Noncommutative geometry states that close to the Planck energy scale, spacetime has a fine structure and proposes that it is given as the product of a four-dimensional continuum compact Riemaniann manifold by a tiny discrete finite noncommutative space. The spectral action principle, a universal action functional on spectral triples which depends only on the spectrum of the Dirac operator, applied to this almost commutative product geometry, leads to the full Standard Model, including neutrino mixing which has Majorana mass terms and a see-saw mechanism, minimally coupled to gravity. It also makes various predictions at unification scale. I will review some of the phenomenological and cosmological consequences of this beautiful and purely geometrical approach to unification.

scholarly journals Spectral triples on irreversible C∗-dynamical systems

2021 ◽  
Author(s):  
Valeriano Aiello ◽  
Daniele Guido ◽  
Tommaso Isola
Keyword(s):  
Dynamical Systems ◽  
Product Formula ◽  
Crossed Product ◽  
Spectral Triple ◽  
Crossed Products ◽  
Spectral Triples ◽  
Main Assumption ◽  
Noncommutative Spaces ◽  
Covering Projection ◽  
Injective Endomorphism

Given a spectral triple on a [Formula: see text]-algebra [Formula: see text] together with a unital injective endomorphism [Formula: see text], the problem of defining a suitable crossed product [Formula: see text]-algebra endowed with a spectral triple is addressed. The proposed construction is mainly based on the works of Cuntz and [A. Hawkins, A. Skalski, S. White and J. Zacharias, On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113(2) (2013) 262–291], and on our previous papers [V. Aiello, D. Guido and T. Isola, Spectral triples for noncommutative solenoidal spaces from self-coverings, J. Math. Anal. Appl. 448(2) (2017) 1378–1412; V. Aiello, D. Guido and T. Isola, A spectral triple for a solenoid based on the Sierpinski gasket, SIGMA Symmetry Integrability Geom. Methods Appl. 17(20) (2021) 21]. The embedding of [Formula: see text] in [Formula: see text] can be considered as the dual form of a covering projection between noncommutative spaces. A main assumption is the expansiveness of the endomorphism, which takes the form of the local isometricity of the covering projection, and is expressed via the compatibility of the Lip-norms on [Formula: see text] and [Formula: see text].

scholarly journals Loop groups and noncommutative geometry

2017 ◽  
Vol 29 (09) ◽  
pp. 1750029
Author(s):  
Sebastiano Carpi ◽  
Robin Hillier
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Representation Theory ◽  
Noncommutative Geometry ◽  
Loop Group ◽  
Positive Energy ◽  
Loop Groups ◽  
Spectral Triples ◽  
Conformal Field ◽  
Coset Models

We describe the representation theory of loop groups in terms of K-theory and noncommutative geometry. This is done by constructing suitable spectral triples associated with the level [Formula: see text] projective unitary positive-energy representations of any given loop group [Formula: see text]. The construction is based on certain supersymmetric conformal field theory models associated with [Formula: see text] in the setting of conformal nets. We then generalize the construction to many other rational chiral conformal field theory models including coset models and the moonshine conformal net.

scholarly journals CATEGORIFIED NONCOMMUTATIVE MANIFOLDS

2009 ◽  
Vol 24 (15) ◽  
pp. 2802-2819 ◽  
Author(s):  
R. A. DAWE MARTINS
Keyword(s):  
Dirac Operator ◽  
Noncommutative Geometry ◽  
Tangent Bundle ◽  
Mass Matrix ◽  
Fermion Mass ◽  
Spectral Triples ◽  
Noncommutative Manifolds ◽  
Real Spectral Triples ◽  
Fermion Mass Matrix

We construct a noncommutative geometry with generalised 'tangent bundle' from Fell bundle C*-categories (E) beginning by replacing pair groupoid objects (points) with objects in E. This provides a categorification of a certain class of real spectral triples where the Dirac operator D is constructed from morphisms in a category. Applications for physics include quantization via the tangent groupoid and new constraints on D finite (the fermion mass matrix).

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