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Universe
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Progress in Group Field Theory and Related Quantum Gravity Formalisms
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Progress in Group Field Theory and Related Quantum Gravity Formalisms

A special issue of Universe (ISSN 2218-1997).

Deadline for manuscript submissions: closed (30 June 2019) | Viewed by 46923

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Dr. Steffen Gielen

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Guest Editor
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK
Interests: quantum gravity; quantum cosmology; group field theory
Dr. Sylvain Carrozza

E-Mail Website
Guest Editor
Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada
Interests: quantum gravity; quantum field theory; mathematical physics; quantum foundations
Dr. Daniele Oriti

E-Mail Website
Guest Editor
Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Golm, Germany
Interests: theoretical physics; quantum gravity; fundamental cosmology; foundations of physics and philosophy of science

Special Issue Information

Dear Colleagues,

In recent years, the Group Field Theory (GFT) formalism for quantum gravity has seen significant progress in different directions: Our understanding of its quantum geometric degrees of freedom has been deepened, new techniques have been developed towards GFT renormalization, and the possibility of establishing a continuum limit; new connections between quantum gravity, condensed matter physics and quantum information have been explored; applications of GFT to the physics of cosmology and black holes point to possible novel resolutions of puzzles of classical gravity and represent promising avenues for connecting quantum gravity with observations. The many (conceptual and mathematical) relations and structural overlaps with other quantum gravity formalisms, such as random tensor models, loop quantum gravity, and lattice gravity, allow also to take advantage of the many developments occurring in them, which include both interesting results of direct implementation within GFT as well as inspiring advances towards shared goals. This stimulates mutual and joint progress.

This Special Issue focuses on GFT, and more broadly on related formalisms which share the same methods and foundations. Contributions from tensor models, loop quantum gravity, spin foam models, or other discrete and combinatorial approaches to quantum gravity which have direct implications for GFT will be particularly welcomed. We also encourage submissions from research fields with direct relevance to more specific aspects of GFT research, including for instance fundamental cosmology, quantum information, or condensed matter theory, but also mathematical and formal aspects.

Research papers or review articles are welcome.

No Article Processing Charges (APC) will be applied to submitted manuscripts.  If the Special Issue reaches more than 10 published papers, it will be printed in book form, with an ISBN number.

Dr. Steffen Gielen
Dr. Sylvain Carrozza
Dr. Daniele Oriti
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Universe is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • quantum gravity
  • group field theory
  • tensor models
  • spin foam models
  • loop quantum gravity
  • quantum cosmology

Published Papers (15 papers)

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Editorial

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8 pages, 244 KiB  
Editorial
Editorial for the Special Issue “Progress in Group Field Theory and Related Quantum Gravity Formalisms”
by Sylvain Carrozza, Steffen Gielen and Daniele Oriti
Universe 2020, 6(1), 19; https://0-doi-org.brum.beds.ac.uk/10.3390/universe6010019 - 20 Jan 2020
Cited by 6 | Viewed by 1993
Abstract
This editorial introduces the Special Issue “Progress in Group Field Theory and Related Quantum Gravity Formalisms” which includes a number of research and review articles covering results in the group field theory (GFT) formalism for quantum gravity and in various neighbouring areas of [...] Read more.
This editorial introduces the Special Issue “Progress in Group Field Theory and Related Quantum Gravity Formalisms” which includes a number of research and review articles covering results in the group field theory (GFT) formalism for quantum gravity and in various neighbouring areas of quantum gravity research. We give a brief overview of the basic ideas of the GFT formalism, list some of its connections to other fields, and then summarise all contributions to the Special Issue. Full article
(This article belongs to the Special Issue Progress in Group Field Theory and Related Quantum Gravity Formalisms)

Research

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22 pages, 656 KiB  
Article
Thermal Quantum Spacetime
by Isha Kotecha
Universe 2019, 5(8), 187; https://0-doi-org.brum.beds.ac.uk/10.3390/universe5080187 - 12 Aug 2019
Cited by 8 | Viewed by 2534
Abstract
The intersection of thermodynamics, quantum theory and gravity has revealed many profound insights, all the while posing new puzzles. In this article, we discuss an extension of equilibrium statistical mechanics and thermodynamics potentially compatible with a key feature of general relativity, background independence; [...] Read more.
The intersection of thermodynamics, quantum theory and gravity has revealed many profound insights, all the while posing new puzzles. In this article, we discuss an extension of equilibrium statistical mechanics and thermodynamics potentially compatible with a key feature of general relativity, background independence; and we subsequently use it in a candidate quantum gravity system, thus providing a preliminary formulation of a thermal quantum spacetime. Specifically, we emphasise an information-theoretic characterisation of generalised Gibbs equilibrium that is shown to be particularly suited to background independent settings, and in which the status of entropy is elevated to being more fundamental than energy. We also shed light on its intimate connections with the thermal time hypothesis. Based on this, we outline a framework for statistical mechanics of quantum gravity degrees of freedom of combinatorial and algebraic type, and apply it in several examples. In particular, we provide a quantum statistical basis for the origin of covariant group field theories, shown to arise as effective statistical field theories of the underlying quanta of space in a certain class of generalised Gibbs states. Full article
(This article belongs to the Special Issue Progress in Group Field Theory and Related Quantum Gravity Formalisms)
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21 pages, 1204 KiB  
Article
Spin Foam Vertex Amplitudes on Quantum Computer—Preliminary Results
by Jakub Mielczarek
Universe 2019, 5(8), 179; https://0-doi-org.brum.beds.ac.uk/10.3390/universe5080179 - 26 Jul 2019
Cited by 19 | Viewed by 4259
Abstract
Vertex amplitudes are elementary contributions to the transition amplitudes in the spin foam models of quantum gravity. The purpose of this article is to make the first step towards computing vertex amplitudes with the use of quantum algorithms. In our studies we are [...] Read more.
Vertex amplitudes are elementary contributions to the transition amplitudes in the spin foam models of quantum gravity. The purpose of this article is to make the first step towards computing vertex amplitudes with the use of quantum algorithms. In our studies we are focused on a vertex amplitude of 3+1 D gravity, associated with a pentagram spin network. Furthermore, all spin labels of the spin network are assumed to be equal j = 1 / 2 , which is crucial for the introduction of the intertwiner qubits. A procedure of determining modulus squares of vertex amplitudes on universal quantum computers is proposed. Utility of the approach is tested with the use of: IBM’s ibmqx4 5-qubit quantum computer, simulator of quantum computer provided by the same company and QX quantum computer simulator. Finally, values of the vertex probability are determined employing both the QX and the IBM simulators with 20-qubit quantum register and compared with analytical predictions. Full article
(This article belongs to the Special Issue Progress in Group Field Theory and Related Quantum Gravity Formalisms)
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40 pages, 552 KiB  
Article
Dynamical Properties of the Mukhanov-Sasaki Hamiltonian in the Context of Adiabatic Vacua and the Lewis-Riesenfeld Invariant
by Max Joseph Fahn, Kristina Giesel and Michael Kobler
Universe 2019, 5(7), 170; https://0-doi-org.brum.beds.ac.uk/10.3390/universe5070170 - 13 Jul 2019
Cited by 8 | Viewed by 3284
Abstract
We use the method of the Lewis-Riesenfeld invariant to analyze the dynamical properties of the Mukhanov-Sasaki Hamiltonian and, following this approach, investigate whether we can obtain possible candidates for initial states in the context of inflation considering a quasi-de Sitter spacetime. Our main [...] Read more.
We use the method of the Lewis-Riesenfeld invariant to analyze the dynamical properties of the Mukhanov-Sasaki Hamiltonian and, following this approach, investigate whether we can obtain possible candidates for initial states in the context of inflation considering a quasi-de Sitter spacetime. Our main interest lies in the question of to which extent these already well-established methods at the classical and quantum level for finitely many degrees of freedom can be generalized to field theory. As our results show, a straightforward generalization does in general not lead to a unitary operator on Fock space that implements the corresponding time-dependent canonical transformation associated with the Lewis-Riesenfeld invariant. The action of this operator can be rewritten as a time-dependent Bogoliubov transformation, where we also compare our results to already existing ones in the literature. We show that its generalization to Fock space has to be chosen appropriately in order to not violate the Shale-Stinespring condition. Furthermore, our analysis relates the Ermakov differential equation that plays the role of an auxiliary equation, whose solution is necessary to construct the Lewis-Riesenfeld invariant, as well as the corresponding time-dependent canonical transformation, to the defining differential equation for adiabatic vacua. Therefore, a given solution of the Ermakov equation directly yields a full solution of the differential equation for adiabatic vacua involving no truncation at some adiabatic order. As a consequence, we can interpret our result obtained here as a kind of non-squeezed Bunch-Davies mode, where the term non-squeezed refers to a possible residual squeezing that can be involved in the unitary operator for certain choices of the Bogoliubov map. Full article
(This article belongs to the Special Issue Progress in Group Field Theory and Related Quantum Gravity Formalisms)
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21 pages, 549 KiB  
Article
Switching Internal Times and a New Perspective on the ‘Wave Function of the Universe’
by Philipp A. Höhn
Universe 2019, 5(5), 116; https://0-doi-org.brum.beds.ac.uk/10.3390/universe5050116 - 14 May 2019
Cited by 60 | Viewed by 4704
Abstract
Despite its importance in general relativity, a quantum notion of general covariance has not yet been established in quantum gravity and cosmology, where, given the a priori absence of coordinates, it is necessary to replace classical frames with dynamical quantum reference systems. As [...] Read more.
Despite its importance in general relativity, a quantum notion of general covariance has not yet been established in quantum gravity and cosmology, where, given the a priori absence of coordinates, it is necessary to replace classical frames with dynamical quantum reference systems. As such, quantum general covariance bears on the ability to consistently switch between the descriptions of the same physics relative to arbitrary choices of quantum reference system. Recently, a systematic approach for such switches has been developed. It links the descriptions relative to different choices of quantum reference system, identified as the correspondingly reduced quantum theories, via the reference-system-neutral Dirac quantization, in analogy to coordinate changes on a manifold. In this work, we apply this method to a simple cosmological model to demonstrate how to consistently switch between different internal time choices in quantum cosmology. We substantiate the argument that the conjunction of Dirac and reduced quantized versions of the theory defines a complete relational quantum theory that not only admits a quantum general covariance, but, we argue, also suggests a new perspective on the ‘wave function of the universe’. It assumes the role of a perspective-neutral global state, without immediate physical interpretation that, however, encodes all the descriptions of the universe relative to all possible choices of reference system at once and constitutes the crucial link between these internal perspectives. While, for simplicity, we use the Wheeler-DeWitt formulation, the method and arguments might be also adaptable to loop quantum cosmology. Full article
(This article belongs to the Special Issue Progress in Group Field Theory and Related Quantum Gravity Formalisms)
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20 pages, 439 KiB  
Article
Reconstruction of Mimetic Gravity in a Non-Singular Bouncing Universe from Quantum Gravity
by Marco de Cesare
Universe 2019, 5(5), 107; https://0-doi-org.brum.beds.ac.uk/10.3390/universe5050107 - 07 May 2019
Cited by 16 | Viewed by 2698
Abstract
We illustrate a general reconstruction procedure for mimetic gravity. Focusing on a bouncing cosmological background, we derive general properties that must be satisfied by the function f(□ϕ) implementing the limiting curvature hypothesis. We show how relevant physical information can be [...] Read more.
We illustrate a general reconstruction procedure for mimetic gravity. Focusing on a bouncing cosmological background, we derive general properties that must be satisfied by the function f(□ϕ) implementing the limiting curvature hypothesis. We show how relevant physical information can be extracted from power-law expansions of f in different regimes, corresponding e.g., to the very early universe or to late times. Our results are then applied to two specific models reproducing the cosmological background dynamics obtained in group field theory and in loop quantum cosmology, and we discuss the possibility of using this framework as providing an effective field theory description of quantum gravity. We study the evolution of anisotropies near the bounce, and discuss instabilities of scalar perturbations. Furthermore, we provide two equivalent formulations of mimetic gravity: one in terms of an effective fluid with exotic properties, the other featuring two distinct time-varying gravitational “constants” in the cosmological equations. Full article
(This article belongs to the Special Issue Progress in Group Field Theory and Related Quantum Gravity Formalisms)
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36 pages, 499 KiB  
Article
Status of Background-Independent Coarse Graining in Tensor Models for Quantum Gravity
by Astrid Eichhorn, Tim Koslowski and Antonio D. Pereira
Universe 2019, 5(2), 53; https://0-doi-org.brum.beds.ac.uk/10.3390/universe5020053 - 05 Feb 2019
Cited by 49 | Viewed by 3146
Abstract
A background-independent route towards a universal continuum limit in discrete models of quantum gravity proceeds through a background-independent form of coarse graining. This review provides a pedagogical introduction to the conceptual ideas underlying the use of the number of degrees of freedom as [...] Read more.
A background-independent route towards a universal continuum limit in discrete models of quantum gravity proceeds through a background-independent form of coarse graining. This review provides a pedagogical introduction to the conceptual ideas underlying the use of the number of degrees of freedom as a scale for a Renormalization Group flow. We focus on tensor models, for which we explain how the tensor size serves as the scale for a background-independent coarse-graining flow. This flow provides a new probe of a universal continuum limit in tensor models. We review the development and setup of this tool and summarize results in the two- and three-dimensional case. Moreover, we provide a step-by-step guide to the practical implementation of these ideas and tools by deriving the flow of couplings in a rank-4-tensor model. We discuss the phenomenon of dimensional reduction in these models and find tentative first hints for an interacting fixed point with potential relevance for the continuum limit in four-dimensional quantum gravity. Full article
(This article belongs to the Special Issue Progress in Group Field Theory and Related Quantum Gravity Formalisms)
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8 pages, 231 KiB  
Article
Equivalence of Models in Loop Quantum Cosmology and Group Field Theory
by Bekir Baytaş, Martin Bojowald and Sean Crowe
Universe 2019, 5(2), 41; https://0-doi-org.brum.beds.ac.uk/10.3390/universe5020041 - 23 Jan 2019
Cited by 10 | Viewed by 2690
Abstract
The paradigmatic models often used to highlight cosmological features of loop quantum gravity and group field theory are shown to be equivalent, in the sense that they are different realizations of the same model given by harmonic cosmology. The loop version of harmonic [...] Read more.
The paradigmatic models often used to highlight cosmological features of loop quantum gravity and group field theory are shown to be equivalent, in the sense that they are different realizations of the same model given by harmonic cosmology. The loop version of harmonic cosmology is a canonical realization, while the group-field version is a bosonic realization. The existence of a large number of bosonic realizations suggests generalizations of models in group field cosmology. Full article
(This article belongs to the Special Issue Progress in Group Field Theory and Related Quantum Gravity Formalisms)
17 pages, 361 KiB  
Article
On the Geometry of No-Boundary Instantons in Loop Quantum Cosmology
by Suddhasattwa Brahma and Dong-han Yeom
Universe 2019, 5(1), 22; https://0-doi-org.brum.beds.ac.uk/10.3390/universe5010022 - 10 Jan 2019
Cited by 11 | Viewed by 3375
Abstract
We study the geometry of Euclidean instantons in loop quantum cosmology (LQC) such as those relevant for the no-boundary proposal. Confining ourselves to the simplest case of a cosmological constant in minisuperspace cosmologies, we analyze solutions of the semiclassical (Euclidean) path integral in [...] Read more.
We study the geometry of Euclidean instantons in loop quantum cosmology (LQC) such as those relevant for the no-boundary proposal. Confining ourselves to the simplest case of a cosmological constant in minisuperspace cosmologies, we analyze solutions of the semiclassical (Euclidean) path integral in LQC. We find that the geometry of LQC instantons have the peculiar feature of an infinite tail which distinguishes them from Einstein gravity. Moreover, due to quantum-geometry corrections, the small-a behaviour of these instantons seem to naturally favor a closing-off of the geometry in a regular fashion, as was originally proposed for the no-boundary wavefunction. Full article
(This article belongs to the Special Issue Progress in Group Field Theory and Related Quantum Gravity Formalisms)
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16 pages, 1912 KiB  
Article
Primordial Power Spectra from an Emergent Universe: Basic Results and Clarifications
by Killian Martineau and Aurélien Barrau
Universe 2018, 4(12), 149; https://0-doi-org.brum.beds.ac.uk/10.3390/universe4120149 - 18 Dec 2018
Cited by 10 | Viewed by 3142
Abstract
Emergent cosmological models, together with the Big Bang and bouncing scenarios, are among the possible descriptions of the early Universe. This work aims at clarifying some general features of the primordial tensor power spectrum in this specific framework. In particular, some naive beliefs [...] Read more.
Emergent cosmological models, together with the Big Bang and bouncing scenarios, are among the possible descriptions of the early Universe. This work aims at clarifying some general features of the primordial tensor power spectrum in this specific framework. In particular, some naive beliefs are corrected. Using a toy model, we investigate the conditions required to produce a scale-invariant spectrum and show to what extent this spectrum can exhibit local features sensitive to the details of the scale factor evolution near the transition time. Full article
(This article belongs to the Special Issue Progress in Group Field Theory and Related Quantum Gravity Formalisms)
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18 pages, 341 KiB  
Article
Rainbow-Like Black-Hole Metric from Loop Quantum Gravity
by Iarley P. Lobo and Michele Ronco
Universe 2018, 4(12), 139; https://0-doi-org.brum.beds.ac.uk/10.3390/universe4120139 - 01 Dec 2018
Cited by 13 | Viewed by 2368
Abstract
Hypersurface deformation algebra consists of a fruitful approach to derive deformed solutions of general relativity based on symmetry considerations with quantum-gravity effects, of which the linearization has been recently demonstrated to be connected to the DSR program by κ -Poincaré symmetry. Based on [...] Read more.
Hypersurface deformation algebra consists of a fruitful approach to derive deformed solutions of general relativity based on symmetry considerations with quantum-gravity effects, of which the linearization has been recently demonstrated to be connected to the DSR program by κ -Poincaré symmetry. Based on this approach, we analyzed the solution derived for the interior of a black hole and we found similarities with the so-called rainbow metrics, like a momentum-dependence of the metric functions. Moreover, we derived an effective, time-dependent Planck length and compared different regularization schemes. Full article
(This article belongs to the Special Issue Progress in Group Field Theory and Related Quantum Gravity Formalisms)

Review

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25 pages, 2307 KiB  
Review
Holographic Entanglement in Group Field Theory
by Goffredo Chirco
Universe 2019, 5(10), 211; https://0-doi-org.brum.beds.ac.uk/10.3390/universe5100211 - 09 Oct 2019
Cited by 2 | Viewed by 2567
Abstract
This work is meant as a review summary of a series of recent results concerning the derivation of a holographic entanglement entropy formula for generic open spin network states in the group field theory (GFT) approach to quantum gravity. The statistical group-field computation [...] Read more.
This work is meant as a review summary of a series of recent results concerning the derivation of a holographic entanglement entropy formula for generic open spin network states in the group field theory (GFT) approach to quantum gravity. The statistical group-field computation of the Rényi entropy for a bipartite network state for a simple interacting GFT is reviewed, within a recently proposed dictionary between group field theories and random tensor networks, and with an emphasis on the problem of a consistent characterisation of the entanglement entropy in the GFT second quantisation formalism. Full article
(This article belongs to the Special Issue Progress in Group Field Theory and Related Quantum Gravity Formalisms)
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23 pages, 950 KiB  
Review
Group Field Theory Condensate Cosmology: An Appetizer
by Andreas G. A. Pithis and Mairi Sakellariadou
Universe 2019, 5(6), 147; https://0-doi-org.brum.beds.ac.uk/10.3390/universe5060147 - 13 Jun 2019
Cited by 36 | Viewed by 3091
Abstract
This contribution is an appetizer to the relatively young and fast-evolving approach to quantum cosmology based on group field theory condensate states. We summarize the main assumptions and pillars of this approach which has revealed new perspectives on the long-standing question of how [...] Read more.
This contribution is an appetizer to the relatively young and fast-evolving approach to quantum cosmology based on group field theory condensate states. We summarize the main assumptions and pillars of this approach which has revealed new perspectives on the long-standing question of how to recover the continuum from discrete geometric building blocks. Among others, we give a snapshot of recent work on isotropic cosmological solutions exhibiting an accelerated expansion, a bounce where anisotropies are shown to be under control, and inhomogeneities with an approximately scale-invariant power spectrum. Finally, we point to open issues in the condensate cosmology approach. Full article
(This article belongs to the Special Issue Progress in Group Field Theory and Related Quantum Gravity Formalisms)
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31 pages, 992 KiB  
Review
Progress in Solving the Nonperturbative Renormalization Group for Tensorial Group Field Theory
by Vincent Lahoche and Dine Ousmane Samary
Universe 2019, 5(3), 86; https://0-doi-org.brum.beds.ac.uk/10.3390/universe5030086 - 26 Mar 2019
Cited by 21 | Viewed by 2917
Abstract
This manuscript aims at giving new advances on the functional renormalization group applied to the tensorial group field theory. It is based on the series of our three papers (Lahoche, et al., Class. Quantum Gravity 2018, 35, 19), (Lahoche, et al., Phys. Rev. [...] Read more.
This manuscript aims at giving new advances on the functional renormalization group applied to the tensorial group field theory. It is based on the series of our three papers (Lahoche, et al., Class. Quantum Gravity 2018, 35, 19), (Lahoche, et al., Phys. Rev. D 2018, 98, 126010) and (Lahoche, et al., Nucl. Phys. B, 2019, 940, 190–213). We consider the polynomial Abelian U ( 1 ) d models without the closure constraint. More specifically, we discuss the case of the quartic melonic interaction. We present a new approach, namely the effective vertex expansion method, to solve the exact Wetterich flow equation and investigate the resulting flow equations, especially regarding the existence of non-Gaussian fixed points for their connection with phase transitions. To complete this method, we consider a non-trivial constraint arising from the Ward–Takahashi identities and discuss the disappearance of the global non-trivial fixed points taking into account this constraint. Finally, we argue in favor of an alternative scenario involving a first order phase transition into the reduced phase space given by the Ward constraint. Full article
(This article belongs to the Special Issue Progress in Group Field Theory and Related Quantum Gravity Formalisms)
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22 pages, 483 KiB  
Review
Quantum Gravity on the Computer: Impressions of a Workshop
by Lisa Glaser and Sebastian Steinhaus
Universe 2019, 5(1), 35; https://0-doi-org.brum.beds.ac.uk/10.3390/universe5010035 - 18 Jan 2019
Cited by 6 | Viewed by 2600
Abstract
Computer simulations allow us to explore non-perturbative phenomena in physics. This has the potential to help us understand quantum gravity. Finding a theory of quantum gravity is a hard problem, but, in the last several decades, many promising and intriguing approaches that utilize [...] Read more.
Computer simulations allow us to explore non-perturbative phenomena in physics. This has the potential to help us understand quantum gravity. Finding a theory of quantum gravity is a hard problem, but, in the last several decades, many promising and intriguing approaches that utilize or might benefit from using numerical methods were developed. These approaches are based on very different ideas and assumptions, yet they face the common challenge to derive predictions and compare them to data. In March 2018, we held a workshop at the Nordic Institute for Theoretical Physics (NORDITA) in Stockholm gathering experts in many different approaches to quantum gravity for a workshop on “Quantum gravity on the computer”. In this article, we try to encapsulate some of the discussions held and talks given during this workshop and combine them with our own thoughts on why and how numerical approaches will play an important role in pushing quantum gravity forward. The last section of the article is a road map providing an outlook of the field and some intentions and goalposts that were debated in the closing session of the workshop. We hope that it will help to build a strong numerical community reaching beyond single approaches to combine our efforts in the search for quantum gravity. Full article
(This article belongs to the Special Issue Progress in Group Field Theory and Related Quantum Gravity Formalisms)
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