On spectral triples in quantum gravity II

Aastrup, Johannes; Grimstrup, Jesper M.; Nest, Ryszard

doi:10.4171/jncg/30

Cited by 30 publications

(113 citation statements)

References 12 publications

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“…This result shows that this construction encodes information about the differential structure of the manifold M . This is in stark contrast to the measures used in for example loop quantum gravity and previously by ourselves, where the smooth connections have zero measure.…”

Section: A Hilbert Space Representation Of Qhd(m)contrasting

confidence: 73%

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On Representations of the Quantum Holonomy Diffeomorphism Algebra

Aastrup

Grimstrup²

2019

Fortschritte der Physik

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In this paper we establish the existence of the non‐perturbative theory of quantum gravity known as quantum holonomy theory by showing that a Hilbert space representation of the QHD(M) algebra, which is an algebra generated by holonomy‐diffeomorphisms and by translation operators on an underlying configuration space of connections, exist. We construct operators, which correspond to the Hamiltonian of general relativity and the Dirac Hamiltonian, and show that they give rise to their classical counterparts in a classical limit. We also find that the structure of an almost‐commutative spectral triple emerge in the same limit. The Hilbert space representation, that we find, is non‐local, which appears to rule out spacial singularities such as the big bang and black hole singularities. Finally, the framework also permits an interpretation in terms of non‐perturbative Yang‐Mills theory as well as other non‐perturbative quantum field theories. This paper is the first of two, where the second paper contains mathematical details and proofs.

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Section: A Hilbert Space Representation Of Qhd(m)contrasting

confidence: 73%

“…This algebra, which was first introduced and studied in [] can be shown to be metric independent. For details on the

HD (M)

algebra we refer the reader to .…”

Section: The Quantum Holonomy‐diffeomorphism Algebramentioning

confidence: 99%

On Representations of the Quantum Holonomy Diffeomorphism Algebra

Aastrup

Grimstrup²

2019

Fortschritte der Physik

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“…It is the Dirac operator used as a basic construction block of the spectral triple of Aastrup, Grimstrup, and Nest in loop quantum gravity to model tetrads. 1,10 The Dirac operator, D 1/3 , with parameter t = 1/3 is the cubic Dirac operator of Kostant, 9 whose square, we will see in a moment, has the simple property of consisting only of its degree two term and its degree zero term. Therefore, the family of Dirac operators D t interpolates the most important Dirac operators one considers on a Lie group.…”

Section: One-parameter Family Of Dirac Operators D Tmentioning

confidence: 99%

Spectral action for a one-parameter family of Dirac operators on $\bm{ SU}\textbf {(2)}$SU(2) and $\bm{SU}\textbf {(3)}$SU(3)

Lai

Teh

2013

Journal of Mathematical Physics

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The one-parameter family of Dirac operators containing the Levi-Civita, cubic, and the trivial Dirac operators on a compact Lie group is analyzed. The spectra for the one-parameter family of Dirac Laplacians on SU(2) and SU(3) are computed by considering a diagonally embedded Casimir operator. Then the asymptotic expansions of the spectral actions for SU(2) and SU(3) are computed, using the Poisson summation formula and the two-dimensional Euler-Maclaurin formula, respectively. The inflation potential and slow-roll parameters for the corresponding pure gravity inflationary theory are generated, using the full asymptotic expansion of the spectral action on SU(2). C 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4790484] I. INTRODUCTIONOn a compact Lie group, one may naturally associate a one-parameter family of Dirac operators by interpolating the torsion of a connection given by the Lie bracket. This family of Dirac operators, is parametrized by a real parameter, t. For t = 1/2 one obtains the geometric spin Dirac operator, algebraic cubic Dirac operator of Kostant when t = 1/3, 8 and the trivial Dirac operator when t = 0. The trivial Dirac operator is used in a model of loop quantum gravity (LQG) to model tetrads. 1 While such a generalized family of operators is mathematically interesting, the authors' original motivation for considering this family was to attempt to calculate the spectral action of the LQG spectral triple of Aastrup, Grimstrup, and Nest. 1, 10 The results obtained in this paper are the asymptotic expansions of the spectral actions for SU(2), SU(3), and U(1) × SU(2). Given an energy scale , the spectral action counts the number of eigenstates with energy below . As grows to a large scale, the coefficients of the -asymptotic of the spectral action capture the residues of the noncommutative Zeta function. 4 The spectral action serves as an action functional in noncommutative geometry. These results may serve as a step toward a LQG spectral action computation (for t = 0), also coupled to matter.The paper is organized in the following manner. Section I defines the one-parameter family of Dirac operators that we are considering. In Sec. II, we express the Dirac Laplacian action on any Clifford module in terms of the action of the Lie algebra's Casimir element on finite-dimensional irreducible representations of the Lie group. This interpretation reduces the problem of finding the spectrum on a general compact Lie group to the specific Clebsch-Gordan decomposition of the tensor products of an irreducible representation with the Weyl representation. We calculate the spectrum of the one-parameter family of Dirac Laplacians on SU(2) and the asymptotic expansion of their spectral actions on in Sec. III. In Sec. IV, we apply the developed machinery to compute the Dirac Laplacian spectrum for SU(3), and in Sec. V, we compute the asymptotic expansion of the spectral action for SU(3) using the two-dimensional Euler-Maclaurin formula, to constant order in . In Sec. VI, we consider a toy infl...

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“…As a step to quantizing gravity in the Ashtekar framework, there are recent developments of describing such an extended space of connections using a spectral triple in noncommutative geometry [1,8], which captures the geometry of the space of generalized connections as operators on a Hilbert space. While the geometries of the base manifold and the space of G-connections on it are in theory retained, the construction of the spectral triple is considered too discrete to practically allow one to reconstruct the base manifold and its G-connections.…”

Section: Introductionmentioning

confidence: 99%

“…In another description, one uses a finite set of curves to probe the space of G-connections to obtain a finite dimensional manifold that depend on the sets of curves. By successively refining the finite sets of curves, one obtains a pro-manifold that extends the original space of connections to the so-called space of generalized connections [4].As a step to quantizing gravity in the Ashtekar framework, there are recent developments of describing such an extended space of connections using a spectral triple in noncommutative geometry [1,8], which captures the geometry of the space of generalized connections as operators on a Hilbert space. While the geometries of the base manifold and the space of G-connections on it are in theory retained, the construction of the spectral triple is considered too discrete to practically allow one to reconstruct the base manifold and its G-connections.…”

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confidence: 99%

Quantization of the G-Connections via the Tangent Groupoid

Lai

2013

Int. J. Geom. Methods Mod. Phys.

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A description of the space of G-connections using the tangent groupoid is given. As the tangent groupoid parameter is away from zero, the G-connections act as convolution operators on a Hilbert space. The gauge action is examined in the tangent groupoid description of the G-connections. Tetrads are formulated as Dirac type operators. The connection variables and tetrad variables in Ashtekar's gravity are presented as operators on a Hilbert space. IntroductionIn Ashtekar's theory of gravity, a SU (2)-connection captures the extrinsic curvature of a space-like leaf in a time-transversal foliation, and the intrinsic geometry of the leaf is given by a tetrad [2]. It is shown that the Einstein-Hilbert functional and Einstein equations can be written in terms of SU (2)-connections and tetrads, thus these variables together recast Einstein's theory of gravity. By rewriting Einstein's gravity with the connection variables and tetrad variables, one could attempt to quantize gravity via the Hamiltonian formalism, and obtain a theory of quantum gravity [3]. The reader may refer to Thiemann's introductory [9]. This article offers an alternative view of the space of connections to ones that appear in other loop quantum gravity literatures.To take the connections as dynamic variables in a quantum theory, one studies wave functions on the space of connections. Unfortunately, the lack of a measure on such a space already poses the first challenge. One typical solution to obtain a measure on the connection space comes from spaces of progressively refined cylindrical functions. In another description, one uses a finite set of curves to probe the space of G-connections to obtain a finite dimensional manifold that depend on the sets of curves. By successively refining the finite sets of curves, one obtains a pro-manifold that extends the original space of connections to the so-called space of generalized connections [4].As a step to quantizing gravity in the Ashtekar framework, there are recent developments of describing such an extended space of connections using a spectral triple in noncommutative geometry [1,8], which captures the geometry of the space of generalized connections as operators on a Hilbert space. While the geometries of the base manifold and the space of G-connections on it are in theory retained, the construction of the spectral triple is considered too discrete to practically allow one to reconstruct the base manifold and its G-connections. To rid the discrete description of using finite sets of embedded curves, this article proposes an alternative way to smoothly probe the space of connections using the tangent groupoid. The purposes of this article are to present an alternative idea to the studies *

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On spectral triples in quantum gravity II

Cited by 30 publications

References 12 publications

On Representations of the Quantum Holonomy Diffeomorphism Algebra

On Representations of the Quantum Holonomy Diffeomorphism Algebra

Spectral action for a one-parameter family of Dirac operators on $\bm{ SU}\textbf {(2)}$SU(2) and $\bm{SU}\textbf {(3)}$SU(3)

Quantization of the G-Connections via the Tangent Groupoid

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