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Journal articles on the topic 'Connes' noncommutative geometry'

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Published: 10 December 2022

Last updated: 29 January 2023

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1

Viet, Nguyen Ai. "A New Solution to the Structure Equation in Noncommutative Spacetime." Communications in Physics 24, no. 1 (March 12, 2014): 21. http://dx.doi.org/10.15625/0868-3166/24/1/3606.

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In this paper, starting from the common foundation of Connes' noncommutative geometry ( NCG)\cite{Connes1, Connes2, CoLo, Connes3}, various possible alternatives in the formulation of atheory of gravity in noncommutative spacetime are discussed indetails. The diversity in the final physical content of the theory is shown to be the consequence of the arbitrary choices in each construction steps. As an alternative in the last step, when the structure equations are to be solved, a minimal set of constraints on the torsion and connection is found to determine all the geometric notions in terms of the metric. In the Connes-Lott model of noncommutative spacetime, in order to keep the full spectrum of the discretized Kaluza-Klein theory \cite{VW2}, it is necessary to include the torsion in the generalized Einstein-Hilbert-Cartan action.

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2

AASTRUP, JOHANNES, and JESPER MØLLER GRIMSTRUP. "INTERSECTING CONNES NONCOMMUTATIVE GEOMETRY WITH QUANTUM GRAVITY." International Journal of Modern Physics A 22, no. 08n09 (April 10, 2007): 1589–603. http://dx.doi.org/10.1142/s0217751x07035306.

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An intersection of noncommutative geometry and loop quantum gravity is proposed. Alain Connes' noncommutative geometry provides a framework in which the Standard Model of particle physics coupled to general relativity is formulated as a unified, gravitational theory. However, to this day no quantization procedure compatible with this framework is known. In this paper we consider the noncommutative algebra of holonomy loops on a functional space of certain spin-connections. The construction of a spectral triple is outlined and ideas on interpretation and classical limit are presented.

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3

SCHÜCKER, THOMAS. "NONCOMMUTATIVE GEOMETRY AND THE STANDARD MODEL." International Journal of Modern Physics A 20, no. 11 (April 30, 2005): 2471–80. http://dx.doi.org/10.1142/s0217751x05024791.

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4

LIZZI, F., G. MANGANO, G. MIELE, and G. SPARANO. "MIRROR FERMIONS IN NONCOMMUTATIVE GEOMETRY." Modern Physics Letters A 13, no. 03 (January 30, 1998): 231–37. http://dx.doi.org/10.1142/s0217732398000292.

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In a recent paper we pointed out the presence of extra fermionic degrees of freedom in a chiral gauge theory based on Connes' noncommutative geometry. Here we propose a mechanism which provides a high mass to these mirror states, so that they decouple from low energy physics.

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5

Sergeev, A. G. "Spin geometry of Dirac and noncommutative geometry of Connes." Proceedings of the Steklov Institute of Mathematics 298, no. 1 (August 2017): 256–93. http://dx.doi.org/10.1134/s0081543817060177.

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6

Landi, Giovanni. "BOOK REVIEW: Noncommutative Geometry, by Alain Connes." General Relativity and Gravitation 30, no. 10 (October 1998): 1543–48. http://dx.doi.org/10.1023/a:1018821310333.

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7

VAN DEN DUNGEN, KOEN, and WALTER D. VAN SUIJLEKOM. "PARTICLE PHYSICS FROM ALMOST-COMMUTATIVE SPACETIMES." Reviews in Mathematical Physics 24, no. 09 (October 2012): 1230004. http://dx.doi.org/10.1142/s0129055x1230004x.

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Our aim in this review paper is to present the applications of Connes' noncommutative geometry to elementary particle physics. Whereas the existing literature is mostly focused on a mathematical audience, in this paper we introduce the ideas and concepts from noncommutative geometry using physicists' terminology, gearing towards the predictions that can be derived from the noncommutative description. Focusing on a light package of noncommutative geometry (so-called "almost-commutative manifolds"), we shall introduce in steps: electrodynamics, the electroweak model, culminating in the full Standard Model. We hope that our approach helps in understanding the role noncommutative geometry could play in describing particle physics models, eventually unifying them with Einstein's (geometrical) theory of gravity.

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8

LIZZI, F., G. MANGANO, G. MIELE, and G. SPARANO. "CONSTRAINTS ON UNIFIED GAUGE THEORIES FROM NONCOMMUTATIVE GEOMETRY." Modern Physics Letters A 11, no. 32n33 (October 30, 1996): 2561–72. http://dx.doi.org/10.1142/s0217732396002575.

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We analyze the possibility to extend the Connes and Lott reformulation of the standard model to larger unified gauge groups. Noncommutative geometry imposes very stringent constraints on the possible theories, and remarkably, the analysis seems to suggest that no larger gauge groups are compatible with the noncommutative structure, unless one enlarges the fermionic degrees of freedom, namely the number of particles.

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9

RENNIE, A. "COMMUTATIVE GEOMETRIES ARE SPIN MANIFOLDS." Reviews in Mathematical Physics 13, no. 04 (April 2001): 409–64. http://dx.doi.org/10.1142/s0129055x01000673.

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In [1], Connes presented axioms governing noncommutative geometry. He went on to claim that when specialised to the commutative case, these axioms recover spin or spinc geometry depending on whether the geometry is "real" or not. We attempt to flesh out the details of Connes' ideas. As an illustration we present a proof of his claim, partly extending the validity of the result to pseudo-Riemannian spin manifolds. Throughout we are as explicit and elementary as possible.

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10

Várilly, Joseph C., and JoséM Gracia-Bondía. "Connes' noncommutative differential geometry and the standard model." Journal of Geometry and Physics 12, no. 4 (November 1993): 223–301. http://dx.doi.org/10.1016/0393-0440(93)90038-g.

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11

REUTER, M. "NONCOMMUTATIVE GEOMETRY ON QUANTUM PHASE SPACE." International Journal of Modern Physics A 11, no. 07 (March 20, 1996): 1253–78. http://dx.doi.org/10.1142/s0217751x96000560.

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A noncommutative analog of the classical differential forms is constructed on the phase space of an arbitrary quantum system. The noncommutative forms are universal and are related to the quantum-mechanical dynamics in the same way as the classical forms are related to classical dynamics. They are constructed by applying the Weyl-Wigner symbol map to the differential envelope of the linear operators on the quantum-mechanical Hilbert space. This leads to a representation of the noncommutative forms considered by A. Connes in terms of multiscalar functions on the classical phase space. In an appropriate coincidence limit they define a quantum deformation of the classical tensor fields and both commutative and noncommutative forms can be studied in a unified framework. We interpret the quantum differential forms in physical terms and comment on their possible applications.

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12

Chakraborty, Partha Sarathi, and Satyajit Guin. "Multiplicativity of Connes’ calculus." Reviews in Mathematical Physics 31, no. 09 (October 2019): 1950033. http://dx.doi.org/10.1142/s0129055x19500338.

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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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13

LIZZI, F., G. MIELE, G. SPARANO, and G. MANGANO. "INFLATIONARY COSMOLOGY FROM NONCOMMUTATIVE GEOMETRY." International Journal of Modern Physics A 11, no. 16 (June 30, 1996): 2907–29. http://dx.doi.org/10.1142/s0217751x96001413.

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In the framework of the Connes-Lott model based on noncommutative geometry, the basic features of a gauge theory in the presence of gravity are reviewed, in order to show the possible physical relevance of this scheme for inflationary cosmology. These models naturally contain at least two scalar fields, interacting with each other whenever more than one fermion generation is assumed. In this paper we propose to investigate the behavior of these two fields (one of which represents the distance between the copies of a two-sheeted space-time) in the early stages of the universe evolution. In particular the simplest Abelian model, which preserves the main characteristics of more complicate gauge theories, is considered and the corresponding inflationary dynamics is studied. We find that a chaotic inflation is naturally favored, leading to a field configuration in which no symmetry breaking occurs and the final distance between the two sheets of space-time is smaller the greater the number of e fold in each sheet.

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De Nittis, Giuseppe, and Maximiliano Sandoval. "The noncommutative geometry of the Landau Hamiltonian: differential aspects." Journal of Physics A: Mathematical and Theoretical 55, no. 2 (December 16, 2021): 024002. http://dx.doi.org/10.1088/1751-8121/ac3da4.

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Abstract In this work we study the differential aspects of the noncommutative geometry for the magnetic C*-algebra which is a 2-cocycle deformation of the group C*-algebra of R 2 . This algebra is intimately related to the study of the quantum Hall effect in the continuous, and our results aim to provide a new geometric interpretation of the related Kubo’s formula. Taking inspiration from the ideas developed by Bellissard during the 80s, we build an appropriate Fredholm module for the magnetic C*-algebra based on the magnetic Dirac operator which is the square root (à la Dirac) of the quantum harmonic oscillator. Our main result consist of establishing an important piece of Bellissard’s theory, the so-called second Connes’ formula. In order to do so, we establish the equality of three cyclic 2-cocycles defined on a dense subalgebra of the magnetic C*-algebra. Two of these 2-cocycles are new in the literature and are defined by Connes’ quantized differential calculus, with the use of the Dixmier trace and the magnetic Dirac operator.

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15

MORETTI, VALTER. "ASPECTS OF NONCOMMUTATIVE LORENTZIAN GEOMETRY FOR GLOBALLY HYPERBOLIC SPACETIMES." Reviews in Mathematical Physics 15, no. 10 (December 2003): 1171–217. http://dx.doi.org/10.1142/s0129055x03001886.

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Connes' functional formula of the Riemannian distance is generalized to the Lorentzian case using the so-called Lorentzian distance, the d'Alembert operator and the causal functions of a globally-hyperbolic spacetime. As a step of the presented machinery, a proof of the almost-everywhere smoothness of the Lorentzian distance considered as a function of one of the two arguments is given. Afterwards, using a C*-algebra approach, the spacetime causal structure and the Lorentzian distance are generalized into noncommutative structures giving rise to a Lorentzian version of part of Connes' noncommutative geometry. The generalized noncommutative spacetime consists of a direct set of Hilbert spaces and a related class of C*-algebras of operators. In each algebra a convex cone made of self-adjoint elements is selected which generalizes the class of causal functions. The generalized events, called loci, are realized as the elements of the inductive limit of the spaces of the algebraic states on the C*-algebras. A partial-ordering relation between pairs of loci generalizes the causal order relation in spacetime. A generalized Lorentz distance of loci is defined by means of a class of densely-defined operators which play the role of a Lorentzian metric. Specializing back the formalism to the usual globally-hyperbolic spacetime, it is found that compactly-supported probability measures give rise to a non-pointwise extension of the concept of events.

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16

MARTINETTI, PIERRE. "LINE ELEMENT IN QUANTUM GRAVITY: THE EXAMPLE OF DSR AND NONCOMMUTATIVE GEOMETRY." International Journal of Modern Physics A 24, no. 15 (June 20, 2009): 2792–801. http://dx.doi.org/10.1142/s0217751x09046242.

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We question the notion of line element in some quantum spaces that are expected to play a role in quantum gravity, namely noncommutative deformations of Minkowski spaces. We recall how the implementation of the Leibniz rule forbids to see some of the infinitesimal deformed Poincaré transformations as good candidates for Noether symmetries. Then we recall the more fundamental view on the line element proposed in noncommutative geometry, and re-interprete at this light some previous results on Connes' distance formula.

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17

Shojaei-Fard, Ali. "Application of Deformed Lie Algebras to Non-Perturbative Quantum Field Theory." Journal of the Indian Mathematical Society 84, no. 1-2 (January 2, 2017): 109. http://dx.doi.org/10.18311/jims/2017/5839.

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The manuscript implements Connes-Kreimer Hopf algebraic renormalization of Feynman diagrams and Dubois-Violette type noncommutative differential geometry to discover a new class of differential calculi with respect to infinite formal expansions of Feynman diagrams which are generated by Dyson-Schwinger equations.

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18

BRAIN, SIMON, and WALTER D. VAN SUIJLEKOM. "THE ADHM CONSTRUCTION OF INSTANTONS ON NONCOMMUTATIVE SPACES." Reviews in Mathematical Physics 23, no. 03 (April 2011): 261–307. http://dx.doi.org/10.1142/s0129055x1100428x.

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We present an account of the ADHM construction of instantons on Euclidean space-time ℝ4 from the point of view of noncommutative geometry. We recall the main ingredients of the classical construction in a coordinate algebra format, which we then deform using a cocycle twisting procedure to obtain a method for constructing families of instantons on noncommutative space-time, parametrized by solutions to an appropriate set of ADHM equations. We illustrate the noncommutative construction in two special cases: the Moyal–Groenewold plane [Formula: see text] and the Connes–Landi plane [Formula: see text].

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19

Cuntz, J. "Noncommutative simplicial complexes and the Baum--Connes conjecture." Geometric And Functional Analysis 12, no. 2 (June 1, 2002): 307–29. http://dx.doi.org/10.1007/s00039-002-8248-6.

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20

Liu, Yang. "Scalar curvature in conformal geometry of Connes–Landi noncommutative manifolds." Journal of Geometry and Physics 121 (November 2017): 138–65. http://dx.doi.org/10.1016/j.geomphys.2017.07.012.

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21

LAI, ALAN. "ON THE JLO COCYCLE AND ITS TRANSGRESSION IN ENTIRE CYCLIC COHOMOLOGY." International Journal of Geometric Methods in Modern Physics 10, no. 07 (June 10, 2013): 1350037. http://dx.doi.org/10.1142/s0219887813500370.

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The JLO character formula due to Jaffe–Lesniewski–Osterwalder [Quantum K-theory: the Chern character, Commun. Math. Phys.112 (1988) 75–88] assigns to each Fredholm module a cocycle in entire cyclic cohomology. It descends to define a cohomological Chern character on K-homology. This paper extends the definition of the JLO character formula for Breuer–Fredholm modules, the modules that represent type II noncommutative geometry; and shows that the JLO character formula coincides with the Connes character formula [see M. Benameur and T. Fack, Type II noncommutative geometry. I. Dixmier trace in von Neumann algebras, Adv. Math.199 (2006) 29–87] at the level of entire cyclic cohomology.

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22

Singh, Tejinder P. "Proposal for a New Quantum Theory of Gravity II." Zeitschrift für Naturforschung A 74, no. 11 (November 26, 2019): 989–92. http://dx.doi.org/10.1515/zna-2019-0211.

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AbstractIn the first article of this series, we have introduced the concept of an atom of space-time-matter (STM), which is described by the spectral action of noncommutative geometry, corresponding to a classical theory of gravity. In the present work, we use the Connes time parameter, along with the spectral action, to incorporate gravity into trace dynamics. We then derive the spectral equation of motion for the STM atom, which turns out to be the Dirac equation on a noncommutative space.

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23

Devastato, Agostino, Manuele Filaci, Pierre Martinetti, and Devashish Singh. "Actions for twisted spectral triple and the transition from the Euclidean to the Lorentzian." International Journal of Geometric Methods in Modern Physics 17, supp01 (May 6, 2020): 2030001. http://dx.doi.org/10.1142/s0219887820300019.

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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.

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24

MARCOLLI, MATILDE. "BUILDING COSMOLOGICAL MODELS VIA NONCOMMUTATIVE GEOMETRY." International Journal of Geometric Methods in Modern Physics 08, no. 05 (August 2011): 1131–68. http://dx.doi.org/10.1142/s0219887811005592.

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This is an overview of new and ongoing research developments aimed at constructing cosmological models based on noncommutative geometry, via the spectral action functional, thought of as a modified gravity action, which includes the coupling with matter when computed on an almost commutative geometry. This survey is mostly based on recent results obtained in collaboration with Elena Pierpaoli and Kevin Teh. We describe various aspects of cosmological models of the very early universe, developed by the author and Pierpaoli, based on the asymptotic expansion of the spectral action functional and on renormalization group analysis of the associated particle physics model (an extension of the standard model with right-handed neutrinos and Majorana mass terms previously developed in collaboration with Chamseddine and Connes). We also describe nonperturbative results, more recently obtained by Pierpaoli, Teh, and the author, which extend to the more modern universe, which show that, for different candidate cosmic topologies, the form of the slow-roll inflation potentials obtained from the nonperturbative calculation of the spectral action are strongly coupled to the underlying geometry and topology. We discuss some ongoing directions of research and open questions in this new field of "noncommutative cosmology". The paper is based on the talk given by the author at the conference "Geometry and Quantum Field Theory" at the MPI, in honor of Alan Carey.

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25

Golse, François, and Eric Leichtnam. "Applications of Connes' Geodesic Flow to Trace Formulas in Noncommutative Geometry." Journal of Functional Analysis 160, no. 2 (December 1998): 408–36. http://dx.doi.org/10.1006/jfan.1998.3305.

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26

Rozenblum, Grigori. "Eigenvalues of singular measures and Connes’ noncommutative integration." Journal of Spectral Theory 12, no. 1 (March 24, 2022): 259–300. http://dx.doi.org/10.4171/jst/401.

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27

KASTLER, DANIEL. "A DETAILED ACCOUNT OF ALAIN CONNES’ VERSION OF THE STANDARD MODEL IN NON-COMMUTATIVE DIFFERENTIAL GEOMETRY III." Reviews in Mathematical Physics 08, no. 01 (January 1996): 103–65. http://dx.doi.org/10.1142/s0129055x96000056.

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We describe in detail Alain Connes’ last presentation of the (classical level of the) standard model in noncommutative differential geometry, now free of the cumbersome adynamical fields which parasited the initial treatment. Accessorily, the theory is presented in a more transparent way by systematic use of the skew tensor-product structure, and of 2×2 matrices with 2×2 matrix-entries instead of the previous 4×4 matrices.

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Singh, Tejinder P. "From quantum foundations to spontaneous quantum gravity – An overview of the new theory." Zeitschrift für Naturforschung A 75, no. 10 (October 25, 2020): 833–53. http://dx.doi.org/10.1515/zna-2020-0073.

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AbstractSpontaneous localisation is a falsifiable dynamical mechanism which modifies quantum mechanics and explains the absence of position superpositions in the macroscopic world. However, this is an ad hoc phenomenological proposal. Adler’s theory of trace dynamics, working on a flat Minkowski space-time, derives quantum (field) theory and spontaneous localisation, as a thermodynamic approximation to an underlying noncommutative matrix dynamics. We describe how to incorporate gravity into trace dynamics, by using ideas from Connes’ noncommutative geometry programme. This leads us to a new quantum theory of gravity, from which we can predict spontaneous localisation and give an estimate of the Bekenstein-Hawking entropy of a Schwarzschild black hole.

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29

de Albuquerque, Luiz C., Jorge L. deLyra, and Paulo Teotonio-Sobrinho. "Fluctuating Commutative Geometry." Modern Physics Letters A 18, no. 33n35 (November 20, 2003): 2517–24. http://dx.doi.org/10.1142/s0217732303012763.

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We use the framework of noncommutative geometry to define a discrete model for fluctuating geometry. Instead of considering ordinary geometry and its metric fluctuations, we consider generalized geometries where topology and dimension can also fluctuate. The model describes the geometry of spaces with a countable number n of points. The spectral principle of Connes and Chamseddine is used to define dynamics. We show that this simple model has two phases. The expectation value 〈n〉, the average number of points in the universe, is finite in one phase and diverges in the other. Moreover, the dimension δ is a dynamical observable in our model, and plays the role of an order parameter. The computation of 〈δ〉 is discussed and an upper bound is found, 〈δ〉 < 2. We also address another discrete model defined on a fixed d = 1 dimension, where topology fluctuates. We comment on a possible spontaneous localization of topology.

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Ponge, Raphaël, and Hang Wang. "Noncommutative geometry and conformal geometry, II. Connes–Chern character and the local equivariant index theorem." Journal of Noncommutative Geometry 10, no. 1 (2016): 303–74. http://dx.doi.org/10.4171/jncg/235.

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31

Guin, Satyajit. "The tensor product of supersymmetric N = (1,1) spectral data." International Journal of Geometric Methods in Modern Physics 15, no. 12 (December 2018): 1850207. http://dx.doi.org/10.1142/s0219887818502079.

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We define the notion of tensor product of supersymmetric [Formula: see text] spectral data in the context of supersymmetric quantum theory and noncommutative geometry. We explain in which sense our definition is canonical and also establish its compatibility with the tensor product of [Formula: see text] spectral data defined earlier by Connes. As an application, we show that the unitary connections on the individual [Formula: see text] spectral data give rise to a unitary connection on the product [Formula: see text] spectral data.

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HWANG, DAE SUNG, and TAEHOON LEE. "GAUGED SU(2)L×SU(2)R/SU(2)L+Rσ MODEL IN THE FRAMEWORK OF NONCOMMUTATIVE GEOMETRY." International Journal of Modern Physics A 09, no. 31 (December 20, 1994): 5531–39. http://dx.doi.org/10.1142/s0217751x94002259.

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We study the gauged SU(2) L× SU(2) Rσ model in the SU(2|2) superalgebra formalism. The superconnection is taken to have one-form vector fields as its even part and zero-form scalar fields as its odd part. Incorporating the matrix derivative of noncommutative geometry proposed by Connes and Coquereaux et al., we naturally obtain the spontaneously symmetry broken SU(2) L× SU(2) Rσ model. The masses of the axial vector gauge fields and the Higgs fields are obtained.

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COQUEREAUX, R., R. HÄUβLING, and F. SCHECK. "ALGEBRAIC CONNECTIONS ON PARALLEL UNIVERSES." International Journal of Modern Physics A 10, no. 01 (January 10, 1995): 89–98. http://dx.doi.org/10.1142/s0217751x95000048.

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For any manifold M we introduce a ℤ-graded differential algebra Ξ, which, in particular, is a bimodule over the associative algebra C(M⋃M). We then introduce the corresponding covariant differentials and show how this construction can be interpreted in terms of Yang-Mills and Higgs fields. This is a particular example of noncommutative geometry. It differs from the prescription of Connes in the following way: the definition of Ξ does not rely on a given Dirac-Yukawa operator acting on a space of spinors.

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MARTINETTI, PIERRE, FLAVIO MERCATI, and LUCA TOMASSINI. "MINIMAL LENGTH IN QUANTUM SPACE AND INTEGRATIONS OF THE LINE ELEMENT IN NONCOMMUTATIVE GEOMETRY." Reviews in Mathematical Physics 24, no. 05 (May 24, 2012): 1250010. http://dx.doi.org/10.1142/s0129055x12500109.

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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 Roberts (DFR) quantum spacetime, as well as in the canonical noncommutative spacetime (θ-Minkowski); on the other side, Connes' spectral distance in noncommutative geometry. Although in the Euclidean space the two notions merge into the one of geodesic distance, they yield distinct results in the noncommutative framework. In particular, in the Moyal plane, the quantum length is bounded above from zero while the spectral distance can take any real positive value, including infinity. We show how to solve this discrepancy by doubling the spectral triple. This leads us to introduce a modified quantum length d′L, which coincides exactly with the spectral distance dD on the set of states of optimal localization. On the set of eigenstates of the quantum harmonic oscillator — together with their translations — d′L and dD coincide asymptotically, both in the high energy and large translation limits. At small energy, we interpret the discrepancy between d′L and dD as two distinct ways of integrating the line element on a quantum space. This leads us to propose an equation for a geodesic on the Moyal plane.

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35

Martinetti, Pierre, and Luca Tomassini. "Noncommutative Geometry of the Moyal Plane: Translation Isometries, Connes’ Distance on Coherent States, Pythagoras Equality." Communications in Mathematical Physics 323, no. 1 (July 20, 2013): 107–41. http://dx.doi.org/10.1007/s00220-013-1760-8.

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Singh, Tejinder P. "Proposal for a New Quantum Theory of Gravity." Zeitschrift für Naturforschung A 74, no. 7 (July 26, 2019): 617–33. http://dx.doi.org/10.1515/zna-2019-0079.

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AbstractWe recall a classical theory of torsion gravity with an asymmetric metric, sourced by a Nambu–Goto + Kalb–Ramond string [R. T. Hammond, Rep. Prog. Phys. 65, 599 (2002)]. We explain why this is a significant gravitational theory and in what sense classical general relativity is an approximation to it. We propose that a noncommutative generalisation of this theory (in the sense of Connes’ noncommutative geometry and Adler’s trace dynamics) is a “quantum theory of gravity.” The theory is in fact a classical matrix dynamics with only two fundamental constants – the square of the Planck length and the speed of light, along with the two string tensions as parameters. The guiding symmetry principle is that the theory should be covariant under general coordinate transformations of noncommuting coordinates. The action for this noncommutative torsion gravity can be elegantly expressed as an invariant area integral and represents an atom of space–time–matter. The statistical thermodynamics of a large number of such atoms yields the laws of quantum gravity and quantum field theory, at thermodynamic equilibrium. Spontaneous localisation caused by large fluctuations away from equilibrium is responsible for the emergence of classical space–time and the field equations of classical general relativity. The resolution of the quantum measurement problem by spontaneous collapse is an inevitable consequence of this process. Quantum theory and general relativity are both seen as emergent phenomena, resulting from coarse graining of the underlying noncommutative geometry. We explain the profound role played by entanglement in this theory: entanglement describes interaction between the atoms of space–time–matter, and indeed entanglement appears to be more fundamental than quantum theory or space–time. We also comment on possible implications for black hole entropy and evaporation and for cosmology. We list the intermediate mathematical analysis that remains to be done to complete this programme.

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EL-NABULSI, RAMI AHMAD. "FRACTIONAL DIRAC OPERATORS AND LEFT-RIGHT FRACTIONAL CHAMSEDDINE–CONNES SPECTRAL BOSONIC ACTION PRINCIPLE IN NONCOMMUTATIVE GEOMETRY." International Journal of Geometric Methods in Modern Physics 07, no. 01 (February 2010): 95–134. http://dx.doi.org/10.1142/s0219887810003951.

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The generalization of the Chamseddine–Connes spectral triples action to its (left and right) fractional counterpart is constructed within the context of the Riemann–Liouville and Erdelyi–Kober (left and right) fractional operators. In the fractional approach, the Dirac operators [Formula: see text] is approximated by [Formula: see text] and the spectral triple [Formula: see text] is replaced by its fractional equivalent [Formula: see text], [Formula: see text], [Formula: see text], 0 < α < 1. When the (left) fractional action is applied to the noncommutative space defined by the spectrum of the Standard Model, one obtains many attractive characteristics including time-dependent gauge couplings constants ([Formula: see text]), a time-dependent cosmological constant (Λ cos ), a time-dependent scalar Ricci curvature (R), a time-dependent Newton's coupling constant, and a time-dependent Higgs square mass [Formula: see text]. Furthermore, [Formula: see text], Λ cos , R, and [Formula: see text] were found to be nonsingulars at the Planck's time. When the (left and right) fractional bosonic action is taken into account, all the previous functions are found to be complexified, including gravity. Many additional interesting features are discussed and explored in some details.

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38

Singh, Tejinder P. "Octonions, trace dynamics and noncommutative geometry—A case for unification in spontaneous quantum gravity." Zeitschrift für Naturforschung A 75, no. 12 (November 18, 2020): 1051–62. http://dx.doi.org/10.1515/zna-2020-0196.

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AbstractWe have recently proposed a new matrix dynamics at the Planck scale, building on the theory of trace dynamics and Connes noncommutative geometry program. This is a Lagrangian dynamics in which the matrix degrees of freedom are made from Grassmann numbers, and the Lagrangian is trace of a matrix polynomial. Matrices made from even grade elements of the Grassmann algebra are called bosonic, and those made from odd grade elements are called fermionic—together they describe an ‘aikyon’. The Lagrangian of the theory is invariant under global unitary transformations and describes gravity and Yang–Mills fields coupled to fermions. In the present article, we provide a basic definition of spin angular momentum in this matrix dynamics and introduce a bosonic(fermionic) configuration variable conjugate to the spin of a boson(fermion). We then show that at energies below Planck scale, where the matrix dynamics reduces to quantum theory, fermions have half-integer spin (in multiples of Planck’s constant), and bosons have integral spin. We also show that this definition of spin agrees with the conventional understanding of spin in relativistic quantum mechanics. Consequently, we obtain an elementary proof for the spin-statistics connection. We then motivate why an octonionic space is the natural space in which an aikyon evolves. The group of automorphisms in this space is the exceptional Lie group G2 which has 14 generators [could they stand for the 12 vector bosons and two degrees of freedom of the graviton?]. The aikyon also resembles a closed string, and it has been suggested in the literature that 10-D string theory can be represented as a 2-D string in the 8-D octonionic space. From the work of Cohl Furey and others it is known that the Dixon algebra made from the four division algebras [real numbers, complex numbers, quaternions and octonions] can possibly describe the symmetries of the standard model. In the present paper we outline how in our work the Dixon algebra arises naturally and could lead to a unification of gravity with the standard model. From this matrix dynamics, local quantum field theory arises as a low energy limit of this Planck scale dynamics of aikyons, and classical general relativity arises as a consequence of spontaneous localisation of a large number of entangled aikyons. We propose that classical curved space–time and Yang–Mills fields arise from an effective gauging which results from the collection of symmetry groups of the spontaneously localised fermions. Our work suggests that we live in an eight-dimensional octonionic universe, four of these dimensions constitute space–time and the other four constitute the octonionic internal directions on which the standard model forces live.

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39

Hochs, P., and N. P. Landsman. "The Guillemin–Sternberg conjecture for noncompact groups and spaces." Journal of K-theory 1, no. 3 (February 11, 2008): 473–533. http://dx.doi.org/10.1017/is008001002jkt022.

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AbstractThe Guillemin–Sternberg conjecture states that “quantisation commutes with reduction” in a specific technical setting. So far, this conjecture has almost exclusively been stated and proved for compact Lie groups G acting on compact symplectic manifolds, and, largely due to the use of Spinc Dirac operator techniques, has reached a high degree of perfection under these compactness assumptions. In this paper we formulate an appropriate Guillemin–Sternberg conjecture in the general case, under the main assumptions that the Lie group action is proper and cocompact. This formulation is motivated by our interpretation of the “quantisation commuates with reduction” phenomenon as a special case of the functoriality of quantisation, and uses equivariant K-homology and the K-theory of the group C*-algebra C*(G) in a crucial way. For example, the equivariant index – which in the compact case takes values in the representation ring R(G) – is replaced by the analytic assembly map – which takes values in K0(C*(G)) – familiar from the Baum–Connes conjecture in noncommutative geometry. Under the usual freeness assumption on the action, we prove our conjecture for all Lie groups G having a discrete normal subgroup Γ with compact quotient G/Γ, but we believe it is valid for all unimodular Lie groups.

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40

Oussi, Lahcen, and Janusz Wysoczański. "bm-Central Limit Theorems associated with non-symmetric positive cones." Probability and Mathematical Statistics 39, no. 1 (June 10, 2019): 183–97. http://dx.doi.org/10.19195/0208-4147.39.1.12.

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Analogues of the classical Central Limit Theorem are proved in the noncommutative setting of random variables which are bmindependent and indexed by elements of positive non-symmetric cones, such as the circular cone, sectors in Euclidean spaces and the Vinberg cone. The geometry of the cones is shown to play a crucial role and the related volume characteristics of the cones is shown.

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41

Brzeziński, Tomasz, and Andrzej Sitarz. "Smooth geometry of the noncommutative pillow, cones and lens spaces." Journal of Noncommutative Geometry 11, no. 2 (2017): 413–49. http://dx.doi.org/10.4171/jncg/11-2-1.

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42

Huggett, Nick, Fedele Lizzi, and Tushar Menon. "Missing the point in noncommutative geometry." Synthese, January 18, 2021. http://dx.doi.org/10.1007/s11229-020-02998-1.

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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.

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43

De Nittis, Giuseppe, and Maximiliano Sandoval. "The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects." Symmetry, Integrability and Geometry: Methods and Applications, December 28, 2020. http://dx.doi.org/10.3842/sigma.2020.146.

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This work provides a first step towards the construction of a noncommutative geometry for the quantum Hall effect in the continuum. Taking inspiration from the ideas developed by Bellissard during the 80's we build a spectral triple for the C∗-algebra of continuous magnetic operators based on a Dirac operator with compact resolvent. The metric aspects of this spectral triple are studied, and an important piece of Bellissard's theory (the so-called first Connes' formula) is proved.

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44

GIMPERLEIN, HEIKO, and MAGNUS GOFFENG. "NONCLASSICAL SPECTRAL ASYMPTOTICS AND DIXMIER TRACES: FROM CIRCLES TO CONTACT MANIFOLDS." Forum of Mathematics, Sigma 5 (2017). http://dx.doi.org/10.1017/fms.2016.33.

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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 Hölder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudodifferential calculus is available, variations of Connes’ residue trace theorem and related integral formulas continue to hold. On the circle, a large class of nonmeasurable Hankel operators is obtained from Hölder continuous functions$f$, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.

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45

Foissy, Loïc. "Algebraic Structures on Typed Decorated Rooted Trees." Symmetry, Integrability and Geometry: Methods and Applications, September 21, 2021. http://dx.doi.org/10.3842/sigma.2021.086.

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Typed decorated trees are used by Bruned, Hairer and Zambotti to give a description of a renormalisation process on stochastic PDEs. We here study the algebraic structures on these objects: multiple pre-Lie algebras and related operads (generalizing a result by Chapoton and Livernet), noncommutative and cocommutative Hopf algebras (generalizing Grossman and Larson's construction), commutative and noncocommutative Hopf algebras (generalizing Connes and Kreimer's construction), bialgebras in cointeraction (generalizing Calaque, Ebrahimi-Fard and Manchon's result). We also define families of morphisms and in particular we prove that any Connes-Kreimer Hopf algebra of typed and decorated trees is isomorphic to a Connes-Kreimer Hopf algebra of non-typed and decorated trees (the set of decorations of vertices being bigger), through a contraction process, and finally obtain the Bruned-Hairer-Zambotti construction as a subquotient.

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46

Szczesny, Matt. "Colored Trees and Noncommutative Symmetric Functions." Electronic Journal of Combinatorics 17, no. 1 (April 5, 2010). http://dx.doi.org/10.37236/468.

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Let ${\cal CRF}_S$ denote the category of $S$-colored rooted forests, and H$_{{\cal CRF}_S}$ denote its Ringel-Hall algebra as introduced by Kremnizer and Szczesny. We construct a homomorphism from a $K^+_0({\cal CRF}_S)$–graded version of the Hopf algebra of noncommutative symmetric functions to H$_{{\cal CRF}_S}$. Dualizing, we obtain a homomorphism from the Connes-Kreimer Hopf algebra to a $K^+_0({\cal CRF}_S)$–graded version of the algebra of quasisymmetric functions. This homomorphism is a refinement of one considered by W. Zhao.

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47

Thürigen, Johannes. "Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme." Symmetry, Integrability and Geometry: Methods and Applications, October 27, 2021. http://dx.doi.org/10.3842/sigma.2021.094.

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Various combinatorially non-local field theories are known to be renormalizable. Still, explicit calculations of amplitudes are very rare and restricted to matrix field theory. In this contribution I want to demonstrate how the BPHZ momentum scheme in terms of the Connes-Kreimer Hopf algebra applies to any combinatorially non-local field theory which is renormalizable. This algebraic method improves the understanding of known results in noncommutative field theory in its matrix formulation. Furthermore, I use it to provide new explicit perturbative calculations of amplitudes in tensorial field theories of rank r>2.

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48

Perez-Sanchez, Carlos I. "On Multimatrix Models Motivated by Random Noncommutative Geometry II: A Yang-Mills-Higgs Matrix Model." Annales Henri Poincaré, April 23, 2022. http://dx.doi.org/10.1007/s00023-021-01138-w.

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AbstractWe continue the study of fuzzy geometries inside Connes’ spectral formalism and their relation to multimatrix models. In this companion paper to Pérez-Sánchez (Ann Henri Poincaré 22:3095–3148, 2021, arXiv:2007.10914), we propose a gauge theory setting based on noncommutative geometry, which—just as the traditional formulation in terms of almost-commutative manifolds—has the ability to also accommodate a Higgs field. However, in contrast to ‘almost-commutative manifolds’, the present framework, which we call gauge matrix spectral triples, employs only finite-dimensional algebras. In a path-integral quantization approach to the Spectral Action, this allows to state Yang–Mills–Higgs theory (on four-dimensional Euclidean fuzzy space) as an explicit random multimatrix model obtained here, whose matrix fields exactly mirror those of the Yang–Mills–Higgs theory on a smooth manifold.

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49

González-Pérez, Adrían, Marius Junge, and Javier Parcet. "Singular integrals in quantum Euclidean spaces." Memoirs of the American Mathematical Society 272, no. 1334 (July 2021). http://dx.doi.org/10.1090/memo/1334.

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We shall establish the core of singular integral theory and pseudodifferential calculus over the archetypal algebras of noncommutative geometry: quantum forms of Euclidean spaces and tori. Our results go beyond Connes’ pseudodifferential calculus for rotation algebras, thanks to a new form of Calderón-Zygmund theory over these spaces which crucially incorporates nonconvolution kernels. We deduce L p L_p -boundedness and Sobolev p p -estimates for regular, exotic and forbidden symbols in the expected ranks. In the L 2 L_2 level both Calderón-Vaillancourt and Bourdaud theorems for exotic and forbidden symbols are also generalized to the quantum setting. As a basic application of our methods, we prove L p L_p -regularity of solutions for elliptic PDEs.

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50

Besnard, Fabien. "On symmetry breaking in the B-L extended spectral Standard Model." Journal of Physics A: Mathematical and Theoretical, May 25, 2022. http://dx.doi.org/10.1088/1751-8121/ac7368.

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Abstract We apply Connes-Chamseddine spectral action to the U(1)_B-L- extension of the Standard Model. We show that in order for the scalar potential to reach its minimum for a non-zero value of the new complex scalar ﬁeld, thus triggering the breaking of B-L symmetry, a constraint on the quartic coupling constants must be satisﬁed at uniﬁcation scale. We then explore the renormalization ﬂow of this model in two opposite scenarios for the neutrino sector, and show that this constraint is not compatible with the pole masses of the top quark and SM Higgs boson. We also show that the model suﬀers from a mass-splitting problem similar to the doublet-triplet splitting problem of Grand Uniﬁed Theories. We discuss potential implications for the Noncommutative Geometry program.

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