Second Quantization and the Spectral Action

Dong, Rui; Khalkhali, Masoud; Suijlekom, Walter D. van

doi:10.48550/arxiv.1903.09624

Cited by 5 publications

(4 citation statements)

References 8 publications

(13 reference statements)

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“…An interpretation of the spectral action as the von Neumann entropy of a second-quantized spectral triple has been found recently in [20] (cf. [42]).…”

Section: The Spectral Action Principlementioning

confidence: 99%

A survey of spectral models of gravity coupled to matter

Chamseddine

Suijlekom

2019

Advances in Noncommutative Geometry

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This is a survey of the historical development of the Spectral Standard Model and beyond, starting with the ground breaking paper of Alain Connes in 1988 where he observed that there is a link between Higgs fields and finite noncommutative spaces. We present the important contributions that helped in the search and identification of the noncommutative space that characterizes the fine structure of spacetime. The nature and properties of the noncommutative space are arrived at by independent routes and show the uniqueness of the Spectral Standard Model at low energies and the Pati-Salam unification model at high energies.2 Early days of the spectral Standard ModelThe first serious attempt to utilize the ideas of noncommutative geometry in particle physic was made by Alain Connes in 1988 in his paper "Essay on physics and noncommutative geometry" [28]. He observed that it is possible to change the structure of the (Euclidean) space-time so that the action functional gives the Weinberg-Salam model. The main emphasis was on the conceptual understanding of the Higgs field, which he calls, the black box of the standard model. The qualitative picture was taken to be of a twosheeted Euclidean space-time separated by a distance of the order of 10 −16 cm. In order to simplify the presentation, and to easily follow the historical development, we will use a uniform notation, representing old results in a new format. It is therefore more efficient to start with the basic definitions.

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“…An interpretation of the spectral action as the von Neumann entropy of a second-quantized spectral triple has been found recently in [20] (cf. [42]).…”

Section: The Spectral Action Principlementioning

confidence: 99%

A survey of spectral models of gravity coupled to matter

Chamseddine

Suijlekom

2019

Advances in Noncommutative Geometry

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“…This allows one to make precise sense of path integrals over noncommutative geometries. Although this formulation is valid at the moment only for a small class of geometries, the present method might shed light on the general problem of quantization of NCG, already tackled using von Neumann's information theoretic entropy in [CCvS19] and [DKvS19], by fermionic and bosonic-fermionic second quantization, respectively.…”

Section: Introductionmentioning

confidence: 99%

On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model

Perez-Sanchez

2021

Preprint

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We continue the study of fuzzy geometries inside Connes' spectral formalism and their relation to multimatrix models. In this companion paper to [arXiv:2007:10914, Ann. Henri Poincaré, 2021 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 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.

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“…The proposals on presenting noncommutative geometries in an inherently quantum setting are diverse: A spin network approach led to the concept of gauge networks, along with a blueprint for spin foams in NCG, as a quanta of NCG [MvS14]; therein, from the spectral action (for Dirac operators) on gauge networks, the Wilson action for Higgs-gauge lattice theories and the Kogut-Susskind Hamiltonian (for a 3-dimensional lattice) were derived, as an interesting result of the interplay among lattice gauge theory, spin networks and NCG. Also, significant progress on the matter of fermionic second quantization of the spectral action, relating it to the von Neumann entropy, has been proposed in [CCvS18]; and a bosonic second quantization was undertaken more recently in [DK19]. The context of this paper is a different, random geometrical approach motivated by the path-integral quantization of noncommutative geometries…”

Section: Introductionmentioning

confidence: 99%

Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models

Perez-Sanchez¹

2019

Preprint

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A fuzzy geometry is a certain type of spectral triple whose Dirac operator crucially turns out to be a finite matrix. This notion was introduced in [J. Barrett, J. Math. Phys. 56, 082301 (2015)] and accommodates familiar fuzzy spaces like spheres and tori. In the framework of random noncommutative geometry, we use Barrett's characterization of Dirac operators of fuzzy geometries in order to systematically compute the spectral action S(D) = Trf (D) for 2ndimensional fuzzy geometries. In contrast to the original Chamseddine-Connes spectral action, we take a polynomial f with f (x) → ∞ as |x| → ∞ in order to obtain a well-defined path integral that can be stated as a random matrix model with action of the type S(D) = N • tr F + i tr Ai • tr Bi, being F, Ai and Bi noncommutative polynomials in 2 2n−1 complex N × N matrices that parametrize the Dirac operator D. For arbitrary signature-thus for any admissible KO-dimension-formulas for 2-dimensional fuzzy geometries are given up to a sextic polynomial, and up to a quartic polynomial for 4-dimensional ones, with focus on the octo-matrix models for Lorentzian and Riemannian signatures. The noncommutative polynomials F, Ai and Bi are obtained via chord diagrams and satisfy: independence of N ; self-adjointness of the main polynomial F (modulo cyclic reordering of each monomial); also up to cyclicity, either self-adjointness or anti-self-adjointness of Ai and Bi simultaneously, for fixed i. Collectively, this favors a free probabilistic perspective for the large-N limit we elaborate on.

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Second Quantization and the Spectral Action

Cited by 5 publications

References 8 publications

A survey of spectral models of gravity coupled to matter

A survey of spectral models of gravity coupled to matter

On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model

Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models

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