Implementing quantum Ricci curvature

Klitgaard, N.F.; Loll, R.

doi:10.1103/physrevd.97.106017

Cited by 39 publications

(59 citation statements)

References 18 publications

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“…Moreover, the Ollivier curvature of an edge is discrete, bounded and local in the sense that the curvature of an edge only depends on short cycles supported by that edge, where a cycle is short if its length is at most 5. This makes it an attractive model of a quantised gravitational field in some lattice regularisation; indeed, a somewhat similar proposal for a 'quantum Ricci curvature' closely related to the Ollivier curvature in its basic intuition is made in [52,53] from a dynamical triangulations perspective.…”

Section: The Ollivier Curvature In Graphsmentioning

confidence: 90%

Self-assembly of geometric space from random graphs

Kelly

Trugenberger

Biancalana

2019

Class. Quantum Grav.

117

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We present a Euclidean quantum gravity model in which random graphs dynamically self-assemble into discrete manifold structures. Concretely, we consider a statistical model driven by a discretisation of the Euclidean Einstein-Hilbert action; contrary to previous approaches based on simplicial complexes and Regge calculus our discretisation is based on the Ollivier curvature, a coarse analogue of the manifold Ricci curvature defined for generic graphs. The Ollivier curvature is generally difficult to evaluate due to its definition in terms of optimal transport theory, but we present a new exact expression for the Ollivier curvature in a wide class of relevant graphs purely in terms of the numbers of short cycles at an edge. This result should be of independent intrinsic interest to network theorists. Action minimising configurations prove to be cubic complexes up to defects; there are indications that such defects are dynamically suppressed in the macroscopic limit. Closer examination of a defect free model shows that certain classical configurations have a geometric interpretation and discretely approximate vacuum solutions to the Euclidean Einstein-Hilbert action. Working in a configuration space where the geometric configurations are stable vacua of the theory, we obtain direct numerical evidence for the existence of a continuous phase transition; this makes the model a UV completion of Euclidean Einstein gravity. Notably, this phase transition implies an area-law for the entropy of emerging geometric space. Certain vacua of the theory can be interpreted as baby universes; we find that these configurations appear as stable vacua in a mean field approximation of our model, but are excluded dynamically whenever the action is exact indicating the dynamical stability of geometric space. The model is intended as a setting for subsequent studies of emergent time mechanisms.

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Section: The Ollivier Curvature In Graphsmentioning

confidence: 90%

Self-assembly of geometric space from random graphs

Kelly

Trugenberger

Biancalana

2019

Class. Quantum Grav.

117

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“…The Wilson loop investigation just described provides another motivation for finding more useful curvature observables with better averaging properties. The recent suggestion of defining a scalable notion of quantum Ricci curvature [100] without referring to deficit angles is a promising step in this direction, which has already been tested for dynamical triangulations in lower dimensions [101].…”

Section: Curvature Observablesmentioning

confidence: 99%

“…22). A first indication of the robustness of the quantum Ricci curvature comes from evaluating it on the ensemble of Euclidean dynamical triangulations in two dimensions [101], which is known to have a highly nonclassical and fractal geometry. The measured expectation value of its curvature can be mapped best to that of a five-dimensional continuum sphere, where it should be recalled that Euclidean quantum gravity in two dimensions has a Hausdorff dimension of four (which at least comes close).…”

Section: Curvature Observablesmentioning

confidence: 99%

Quantum gravity from causal dynamical triangulations: a review

Loll

2019

Class. Quantum Grav.

302

205

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This topical review gives a comprehensive overview and assessment of recent results in Causal Dynamical Triangulations (CDT), a modern formulation of lattice gravity, whose aim is to obtain a theory of quantum gravity nonperturbatively from a scaling limit of the lattice-regularized theory. In this manifestly diffeomorphism-invariant approach one has direct, computational access to a Planckian spacetime regime, which is explored with the help of invariant quantum observables. During the last few years, there have been numerous new and important developments and insights concerning the theory's phase structure, the roles of time, causality, diffeomorphisms and global topology, the application of renormalization group methods and new observables. We will focus on these new results, primarily in four spacetime dimensions, and discuss some of their geometric and physical implications. 1 arXiv:1905.08669v1 [hep-th]

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“…As with other situations where non-perturbative physics is involved, one could try to cross-check results obtained with continuum QFT methods with lattice studies. It is worth mentioning that also in lattice approaches to quantum gravity, observables are very hard to define and especially to implement in the simulations, see, e.g., [322] for encouraging recent results. This is in stark contrast to the large number of observables that can be defined in the presence of an asymptotically flat background.…”

Section: Remarksmentioning

confidence: 99%

Critical Reflections on Asymptotically Safe Gravity

et al. 2020

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Asymptotic safety is a theoretical proposal for the ultraviolet completion of quantum field theories, in particular for quantum gravity. Significant progress on this program has led to a first characterization of the Reuter fixed point. Further advancement in our understanding of the nature of quantum spacetime requires addressing a number of open questions and challenges. Here, we aim at providing a critical reflection on the state of the art in the asymptotic safety program, specifying and elaborating on open questions of both technical and conceptual nature. We also point out systematic pathways, in various stages of practical implementation, toward answering them. Finally, we also take the opportunity to clarify some common misunderstandings regarding the program.

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Implementing quantum Ricci curvature

Cited by 39 publications

References 18 publications

Self-assembly of geometric space from random graphs

Self-assembly of geometric space from random graphs

Quantum gravity from causal dynamical triangulations: a review

Critical Reflections on Asymptotically Safe Gravity

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