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 Actions for twisted spectral triple and the transition from the Euclidean to the Lorentzian

2020 ◽  
Vol 17 (supp01) ◽  
pp. 2030001
Author(s):  
Agostino Devastato ◽  
Manuele Filaci ◽  
Pierre Martinetti ◽  
Devashish Singh
Keyword(s):  
Scalar Field ◽  
Standard Model ◽  
Noncommutative Geometry ◽  
Spectral Triple ◽  
Spectral Action ◽  
Lorentzian Signature ◽  
The Standard Model ◽  
Form Field

This is a review of recent results regarding the application of Connes’ noncommutative geometry to the Standard Model, and beyond. By twisting (in the sense of Connes-Moscovici) the spectral triple of the Standard Model, one does not only get an extra scalar field which stabilises the electroweak vacuum, but also an unexpected [Formula: see text]-form field. By computing the fermionic action, we show how this field induces a transition from the Euclidean to the Lorentzian signature. Hints on a twisted version of the spectral action are also briefly mentioned.

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 DIFFERENTIAL CALCULUS ON FUZZY SPHERE AND SCALAR FIELD

1998 ◽  
Vol 13 (19) ◽  
pp. 3235-3243 ◽  
Author(s):  
URSULA CAROW-WATAMURA ◽  
SATOSHI WATAMURA
Keyword(s):  
Scalar Field ◽  
Dirac Operator ◽  
Differential Calculus ◽  
Spectral Triple ◽  
Fuzzy Sphere ◽  
Alternative Possibility ◽  
Operator Ordering

We find that there is an alternative possibility to define the chirality operator on the fuzzy sphere, due to the ambiguity of the operator ordering. Adopting this new chirality operator and the corresponding Dirac operator, we define Connes' spectral triple on the fuzzy sphere and the differential calculus. The differential calculus based on this new spectral triple is simplified considerably. Using this formulation the action of the scalar field is derived.

scholarly journals INEQUALITIES FOR GRADIENT EINSTEIN AND RICCI SOLITONS

2020 ◽  
pp. 351
Author(s):  
Adara-Monica Blaga ◽  
Mircea Crasmareanu
Keyword(s):  
Scalar Field ◽  
Smooth Manifold ◽  
Short Note ◽  
Riemannian Metric ◽  
Ricci Solitons ◽  
Quadratic Equations ◽  
Gradient Ricci Solitons

This short note concerns with two inequalities in the geo\-me\-try of gradient Einstein solitons $(g, f, \lambda )$ on a smooth manifold $M$. These inequalities provide some relationships between the curvature of the Riemannian metric $g$ and the behavior of the scalar field $f$ through two quadratic equations satisfied by the scalar $\lambda $. The similarity with gradient Ricci solitons and a slightly generalization involving a $g$-symmetric endomorphism $A$ are provided.

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 Noncommutative geometry, Grand Symmetry and twisted spectral triple

2015 ◽  
Vol 634 ◽  
pp. 012008 ◽  
Author(s):  
Agostino Devastato
Keyword(s):  
Noncommutative Geometry ◽  
Spectral Triple

scholarly journals The 750 GeV diphoton excess in unified SU(2)L ×SU(2)R ×SU(4) models from noncommutative geometry

2016 ◽  
Vol 31 (18) ◽  
pp. 1650101 ◽  
Author(s):  
Ufuk Aydemir ◽  
Djordje Minic ◽  
Chen Sun ◽  
Tatsu Takeuchi
Keyword(s):  
Scalar Field ◽  
Large Hadron Collider ◽  
Standard Model ◽  
Noncommutative Geometry ◽  
Cross Sections ◽  
Hadron Collider ◽  
Gauge Coupling ◽  
The Standard Model

We discuss a possible interpretation of the 750 GeV diphoton resonance, recently reported at the large hadron collider (LHC), within a class of [Formula: see text] models with gauge coupling unification. The unification is imposed by the underlying noncommutative geometry (NCG), which in these models is extended to a left–right symmetric completion of the Standard Model (SM). Within such unified [Formula: see text] models the Higgs content is restrictively determined from the underlying NCG, instead of being arbitrarily selected. We show that the observed cross-sections involving the 750 GeV diphoton resonance could be realized through a SM singlet scalar field accompanied by colored scalars, present in these unified models. In view of this result, we discuss the underlying rigidity of these models in the NCG framework and the wider implications of the NCG approach for physics beyond the SM.

scholarly journals NONCOMMUTATIVE GEOMETRY, STRINGS AND DUALITY

2000 ◽  
Vol 14 (22n23) ◽  
pp. 2383-2396
Author(s):  
FEDELE LIZZI
Keyword(s):  
String Theory ◽  
Noncommutative Geometry ◽  
Spectral Triple ◽  
Target Space ◽  
Vertex Operators ◽  
Work Done

In this talk, based on work done in collaboration with G. Landi and R. J. Szabo, I will review how string theory can be considered as a noncommutative geometry based on an algebra of vertex operators. The spectral triple of strings is introduced, and some of the string symmetries, notably target space duality, are discussed in this framework.

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 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.

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