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27 pages, 662 KiB

Open AccessArticle

by Eugenio Bianchi and Pierre Martin-Dussaud

Universe 2024, 10(4), 181; https://doi.org/10.3390/universe10040181 - 14 Apr 2024

Viewed by 515

The metric field of general relativity is almost fully determined by its causal structure. Yet, in spin foam models of quantum gravity, the role played by the causal structure is still largely unexplored. The goal of this paper is to clarify how causality [...] Read more.

The metric field of general relativity is almost fully determined by its causal structure. Yet, in spin foam models of quantum gravity, the role played by the causal structure is still largely unexplored. The goal of this paper is to clarify how causality is encoded in such models. The quest unveils the physical meaning of the orientation of the two-complex and its role as a dynamical variable. We propose a causal version of the EPRL spin foam model and discuss the role of the causal structure in the reconstruction of a semiclassical space–time geometry. Full article

(This article belongs to the Special Issue Loop Quantum Gravity: A Themed Issue in Honor of Prof. Abhay Ashtekar)

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22 pages, 359 KiB

Open AccessArticle

Diffeomorphism Covariance and the Quantum Schwarzschild Interior

by I. W. Bornhoeft, R. G. Dias and J. S. Engle

Universe 2024, 10(2), 89; https://doi.org/10.3390/universe10020089 - 13 Feb 2024

Viewed by 1031

Abstract

We introduce a notion of residual diffeomorphism covariance in quantum Kantowski–Sachs (KS) describing the interior of a Schwarzschild black hole. We solve for the family of Hamiltonian constraint operators satisfying the associated covariance condition, as well as parity invariance, preservation of the Bohr [...] Read more.

We introduce a notion of residual diffeomorphism covariance in quantum Kantowski–Sachs (KS) describing the interior of a Schwarzschild black hole. We solve for the family of Hamiltonian constraint operators satisfying the associated covariance condition, as well as parity invariance, preservation of the Bohr Hilbert space of the Loop Quantum KS and a correct (naïve) classical limit. We further explore the imposition of minimality for the number of terms and compare the solution with those of other Hamiltonian constraints proposed for the Loop Quantum KS in the literature. In particular, we discuss a lapse that was recently commonly chosen due to the resulting decoupling of the evolution of the two degrees of freedom and the exact solubility of the model. We show that such a choice of lapse can indeed be quantized as an operator that is densely defined on the Bohr Hilbert space and that any such operator must include an infinite number of shift operators. Full article

(This article belongs to the Special Issue Loop Quantum Gravity: A Themed Issue in Honor of Prof. Abhay Ashtekar)

18 pages, 975 KiB

Open AccessArticle

On Covariant and Canonical Hamiltonian Formalisms for Gauge Theories

by Alejandro Corichi, Juan D. Reyes and Tatjana Vukašinac

Universe 2024, 10(2), 60; https://doi.org/10.3390/universe10020060 - 29 Jan 2024

Cited by 1 | Viewed by 907

Abstract

The Hamiltonian description of classical gauge theories is a very well studied subject. The two best known approaches, namely the covariant and canonical Hamiltonian formalisms, have received a lot of attention in the literature. However, a full understanding of the relation between them [...] Read more.

The Hamiltonian description of classical gauge theories is a very well studied subject. The two best known approaches, namely the covariant and canonical Hamiltonian formalisms, have received a lot of attention in the literature. However, a full understanding of the relation between them is not available, especially when the gauge theories are defined over regions with boundaries. Here, we consider this issue, by first making it precise what we mean by equivalence between the two formalisms. Then, we explore several first-order gauge theories and assess whether their corresponding descriptions satisfy the notion of equivalence. We shall show that, even when in several cases the two formalisms are indeed equivalent, there are counterexamples that signal that this is not always the case. Thus, non-equivalence is a generic feature of gauge field theories. These results call for a deeper understanding of the subject. Full article

(This article belongs to the Special Issue Loop Quantum Gravity: A Themed Issue in Honor of Prof. Abhay Ashtekar)

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13 pages, 338 KiB

Open AccessArticle

Chiral Loop Quantum Supergravity and Black Hole Entropy

by Konstantin Eder and Hanno Sahlmann

Universe 2023, 9(7), 303; https://doi.org/10.3390/universe9070303 - 23 Jun 2023

Cited by 2 | Viewed by 647

Abstract

Recent work has shown that local supersymmetry on a spacetime boundary in

N

-extended AdS supergravity in chiral variables implies coupling to a boundary

OSp {(N | 2)}_{C}

super Chern–Simons theory. Consequently there has been a proposal to define and [...] Read more.

Recent work has shown that local supersymmetry on a spacetime boundary in

N

-extended AdS supergravity in chiral variables implies coupling to a boundary

OSp {(N | 2)}_{C}

super Chern–Simons theory. Consequently there has been a proposal to define and calculate the entropy S for the boundary, in the supersymmetric version of loop quantum gravity, for the minimal case

N = 1

, via this super Chern–Simons theory. We give an overview of how supergravity can be treated in loop quantum gravity. We review the calculation of the dimensions of the quantum state spaces of

UOSp (1 | 2)

super Chern–Simons theory with punctures, and its analytical continuation, for the fixed quantum super area of the surface, to

OSp {(1 | 2)}_{C}

. The result is

S = a_{H} / 4

for large (super) areas. Lower order corrections can also be determined. We begin also a discussion of the statistical mechanics of the surface degrees of freedom by calculating the grand canonical partition function at zero chemical potential. This is a new result. Full article

(This article belongs to the Special Issue Loop Quantum Gravity: A Themed Issue in Honor of Prof. Abhay Ashtekar)

24 pages, 389 KiB

Open AccessArticle

Dynamically Implementing the

\bar{μ}

-Scheme in Cosmological and Spherically Symmetric Models in an Extended Phase Space Model

by Kristina Giesel and Hongguang Liu

Universe 2023, 9(4), 176; https://doi.org/10.3390/universe9040176 - 03 Apr 2023

Cited by 1 | Viewed by 1001

Abstract

We consider an extended phase space formulation for cosmological and spherically symmetric models in which the choice of a given

\bar{μ}

-scheme can be implemented dynamically. These models are constructed in the context of the relational formalism by using a canonical transformation [...] Read more.

We consider an extended phase space formulation for cosmological and spherically symmetric models in which the choice of a given

\bar{μ}

-scheme can be implemented dynamically. These models are constructed in the context of the relational formalism by using a canonical transformation on the extended phase space, which provides a Kuchař decomposition of the extended phase space. The resulting model can be understood as a gauge-unfixed model of a given

\bar{μ}

-scheme. We use this formalism to investigate the restrictions to the allowed

\bar{μ}

-scheme from this perspective and discuss the differences in the cosmological and spherically symmetric case. This method can be useful, for example, to obtain a

\bar{μ}

-scheme in a top-down derivation from full LQG to symmetry-reduced effective models, where, for some models, only the

μ_{0}

-scheme has been obtained thus far. Full article

(This article belongs to the Special Issue Loop Quantum Gravity: A Themed Issue in Honor of Prof. Abhay Ashtekar)

23 pages, 561 KiB

Open AccessArticle

Modeling Quantum Particles Falling into a Black Hole: The Deep Interior Limit

by Alejandro Perez, Salvatore Ribisi and Sami Viollet

Universe 2023, 9(2), 75; https://doi.org/10.3390/universe9020075 - 31 Jan 2023

Cited by 2 | Viewed by 898

Abstract

In this paper, we construct a solvable toy model of the quantum dynamics of the interior of a spherical black hole with falling spherical scalar field excitations. We first argue about how some aspects of the quantum gravity dynamics of realistic black holes [...] Read more.

In this paper, we construct a solvable toy model of the quantum dynamics of the interior of a spherical black hole with falling spherical scalar field excitations. We first argue about how some aspects of the quantum gravity dynamics of realistic black holes emitting Hawking radiation can be modeled using Kantowski–Sachs solutions with a massless scalar field when one focuses on the deep interior region

r ≪ M

(including the singularity). Further, we show that in the

r ≪ M

regime, and in suitable variables, the KS model becomes exactly solvable at both the classical and quantum levels. The quantum dynamics inspired by loop quantum gravity is revisited. We propose a natural polymer quantization where the area a of the orbits of the rotation group is quantized. The polymer (or loop) dynamics is closely related to the Schroedinger dynamics away from the singularity with a form of continuum limit naturally emerging from the polymer treatment. The Dirac observable associated with the mass is quantized and shown to have an infinite degeneracy associated with the so-called

ϵ

-sectors. Suitable continuum superpositions of these are well-defined distributions in the fundamental Hilbert space and satisfy the continuum Schroedinger dynamics. Full article

(This article belongs to the Special Issue Loop Quantum Gravity: A Themed Issue in Honor of Prof. Abhay Ashtekar)

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36 pages, 871 KiB

Open AccessFeature PaperArticle

On Propagation in Loop Quantum Gravity

by Thomas Thiemann and Madhavan Varadarajan

Universe 2022, 8(12), 615; https://doi.org/10.3390/universe8120615 - 23 Nov 2022

Cited by 5 | Viewed by 1012

Abstract

A rigorous implementation of the Wheeler–Dewitt equations was derived in the context of Loop Quantum Gravity (LQG) and was coined Quantum Spin Dynamics (QSD). The Hamiltonian constraint of QSD was criticised as being too local and to prevent “propagation” in canonical LQG. That [...] Read more.

A rigorous implementation of the Wheeler–Dewitt equations was derived in the context of Loop Quantum Gravity (LQG) and was coined Quantum Spin Dynamics (QSD). The Hamiltonian constraint of QSD was criticised as being too local and to prevent “propagation” in canonical LQG. That criticism was based on an algorithm developed for QSD for generating solutions to the Wheeler–DeWitt equations. The fine details of that algorithm could not be worked out because the QSD Hamiltonian constraint makes crucial use of the volume operator, which cannot be diagonalised analytically. In this paper, we consider the U(1)

^{3}

model for Euclidean vacuum LQG which consists of replacing the structure group SU(2) by U(1)

^{3}

and otherwise keeps all properties of the SU(2) theory intact. This enables analytical calculations and the fine details of the algorithm ingto be worked out. By considering one of the simplest possible non-trivial classes of solutions based on very small graphs, we show that (1) an infinite number of solutions ingexist which are (2) generically not normalisable with respect to the inner product on the space of spatially diffeomorphism invariant distributions and (3) generically display propagation. Due to the closeness of the U(1)

^{3}

model to Euclidean LQG, it is extremely likely that all three properties hold also in the SU(2) case and even more so in physical Lorentzian LQG. These arguments can in principle be made water tight using modern numerical (e.g., ML or QC) methods combined with the techniques developed in this paper which we reserve for future work. Full article

(This article belongs to the Special Issue Loop Quantum Gravity: A Themed Issue in Honor of Prof. Abhay Ashtekar)

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31 pages, 626 KiB

Open AccessArticle

Dirac Observables in the 4-Dimensional Phase Space of Ashtekar’s Variables and Spherically Symmetric Loop Quantum Black Holes

by Geeth Ongole, Hongchao Zhang, Tao Zhu, Anzhong Wang and Bin Wang

Universe 2022, 8(10), 543; https://doi.org/10.3390/universe8100543 - 19 Oct 2022

Cited by 5 | Viewed by 1013

Abstract

In this paper, we study a proposal put forward recently by Bodendorfer, Mele and Münch and García-Quismondo and Marugán, in which the two polymerization parameters of spherically symmetric black hole spacetimes are the Dirac observables of the four-dimensional Ashtekar’s variables. In this model, [...] Read more.

In this paper, we study a proposal put forward recently by Bodendorfer, Mele and Münch and García-Quismondo and Marugán, in which the two polymerization parameters of spherically symmetric black hole spacetimes are the Dirac observables of the four-dimensional Ashtekar’s variables. In this model, black and white hole horizons in general exist and naturally divide the spacetime into the external and internal regions. In the external region, the spacetime can be made asymptotically flat by properly choosing the dependence of the two polymerization parameters on the Ashtekar variables. Then, we find that the asymptotical behavior of the spacetime is universal, and, to the leading order, the curvature invariants are independent of the mass parameter m. For example, the Kretschmann scalar approaches zero as

K ≃ A_{0} r^{- 4}

asymptotically, where

A_{0}

is generally a non-zero constant and independent of m, and r the geometric radius of the two-spheres. In the internal region, all the physical quantities are finite, and the Schwarzschild black hole singularity is replaced by a transition surface whose radius is always finite and non-zero. The quantum gravitational effects are negligible near the black hole horizon for very massive black holes. However, the behavior of the spacetime across the transition surface is significantly different from all loop quantum black holes studied so far. In particular, the location of the maximum amplitude of the curvature scalars is displaced from the transition surface and depends on m; so does the maximum amplitude. In addition, the radius of the white hole is much smaller than that of the black hole, and its exact value sensitively depends on m, too. Full article

(This article belongs to the Special Issue Loop Quantum Gravity: A Themed Issue in Honor of Prof. Abhay Ashtekar)

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7 pages, 254 KiB

Open AccessArticle

A Covariant Polymerized Scalar Field in Semi-Classical Loop Quantum Gravity

by Rodolfo Gambini, Florencia Benítez and Jorge Pullin

Universe 2022, 8(10), 526; https://doi.org/10.3390/universe8100526 - 10 Oct 2022

Cited by 7 | Viewed by 1005

Abstract

We propose a new polymerization scheme for scalar fields coupled to gravity. It has the advantage of being a (non-bijective) canonical transformation of the fields, and therefore ensures the covariance of the theory. We study it in detail in spherically symmetric situations and [...] Read more.

We propose a new polymerization scheme for scalar fields coupled to gravity. It has the advantage of being a (non-bijective) canonical transformation of the fields, and therefore ensures the covariance of the theory. We study it in detail in spherically symmetric situations and compare to other approaches. Full article

(This article belongs to the Special Issue Loop Quantum Gravity: A Themed Issue in Honor of Prof. Abhay Ashtekar)

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15 pages, 326 KiB

Open AccessArticle

Mass of Cosmological Perturbations in the Hybrid and Dressed Metric Formalisms of Loop Quantum Cosmology for the Starobinsky and Exponential Potentials

by Simon Iteanu and Guillermo A. Mena Marugán

Universe 2022, 8(9), 463; https://doi.org/10.3390/universe8090463 - 07 Sep 2022

Cited by 4 | Viewed by 1132

Abstract

The hybrid and the dressed metric formalisms for the study of primordial perturbations in Loop Quantum Cosmology lead to dynamical equations for the modes of these perturbations that are of a generalized harmonic-oscillator type, with a mass that depends on the background but [...] Read more.

The hybrid and the dressed metric formalisms for the study of primordial perturbations in Loop Quantum Cosmology lead to dynamical equations for the modes of these perturbations that are of a generalized harmonic-oscillator type, with a mass that depends on the background but is the same for all modes. For quantum background states that are peaked on trajectories of the effective description of Loop Quantum Cosmology, the main difference between the two considered formalisms is found in the expression of this mass. The value of the mass at the bounce is especially important, since it is only in a short interval around this event that the quantum geometry effects on the perturbations are relevant. In a previous article, the properties of this mass were discussed for an inflaton potential of quadratic form, or with similar characteristics. In the present work, we extend this study to other interesting potentials in cosmology, namely the Starobinsky and the exponential potentials. We prove that there exists a finite interval of values of the potential (which includes the zero but typically goes beyond the sector of kinetically dominated inflaton energy density) for which the hybrid mass is positive at the bounce whereas the dressed metric mass is negative. Full article

(This article belongs to the Special Issue Loop Quantum Gravity: A Themed Issue in Honor of Prof. Abhay Ashtekar)

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