Framework Of Loop Quantum Gravity Research Articles

Coherent states, quantum gravity, and the Born-Oppenheimer approximation. I. General considerations

This article, as the first of three, aims at establishing the (time-dependent) Born-Oppenheimer approximation, in the sense of space adiabatic perturbation theory, for quantum systems constructed by techniques of the loop quantum gravity framework, especially the canonical formulation of the latter. The analysis presented here fits into a rather general framework and offers a solution to the problem of applying the usual Born-Oppenheimer ansatz for molecular (or structurally analogous) systems to more general quantum systems (e.g., spin-orbit models) by means of space adiabatic perturbation theory. The proposed solution is applied to a simple, finite dimensional model of interacting spin systems, which serves as a non-trivial, minimal model of the aforesaid problem. Furthermore, it is explained how the content of this article and its companion affect the possible extraction of quantum field theory on curved spacetime from loop quantum gravity (including matter fields).

Invariant Connections in Loop Quantum Gravity

Given a group $G$ and an abelian $C^*$-algebra $\mathfrak{A}$, the antihomomorphisms $\Theta\colon G\rightarrow \mathrm{Aut}(\mathfrak{A})$ are in one-to-one with those left actions $\Phi\colon G\times \mathrm{Spec}(\mathfrak{A})\rightarrow \mathrm{Spec}(\mathfrak{A})$ whose translation maps $\Phi_g$ are continuous; whereby continuities of $\Theta$ and $\Phi$ turn out to be equivalent if $\mathfrak{A}$ is unital. In particular, a left action $\phi\colon G \times X\rightarrow X$ can be uniquely extended to the spectrum of a $C^*$-subalgebra $\mathfrak{A}$ of the bounded functions on $X$ if $\phi_g^*(\mathfrak{A})\subseteq \mathfrak{A}$ holds for each $g\in G$. In the present paper, we apply this to the framework of loop quantum gravity. We show that, on the level of the configuration spaces, quantization and reduction in general do not commute, i.e., that the symmetry-reduced quantum configuration space is (strictly) larger than the quantized configuration space of the reduced classical theory. Here, the quantum-reduced space has the advantage to be completely characterized by a simple algebraic relation, whereby the quantized reduced classical space is usually hard to compute.

Black hole spectroscopy from loop quantum gravity models

Using Monte Carlo simulations, we compute the integrated emission spectra of black holes in the framework of Loop Quantum Gravity (LQG). The black hole emission rates are governed by the entropy whose value, in recent holographic loop quantum gravity models, was shown to agree at leading order with the Bekenstein-Hawking entropy. Quantum corrections depend on the Barbero-Immirzi parameter $\gamma$. Starting with black holes of initial horizon area $A \sim 10^2$ in Planck units, we present the spectra for different values of $\gamma$. Each spectrum clearly decomposes in two distinct parts: a continuous background which corresponds to the semi-classical stages of the evaporation and a series of discrete peaks which constitutes a signature of the deep quantum structure of the black hole. We show that $\gamma$ has an effect on both parts that we analyze in details. Finally, we estimate the number of black holes and the instrumental resolution required to experimentally distinguish between the considered models.

No fermion doubling in quantum geometry

In loop quantum gravity the discrete nature of quantum geometry acts as a natural regulator for matter theories. Studies of quantum field theory in quantum space–times in spherical symmetry in the canonical approach have shown that the main effect of the quantum geometry is to discretize the equations of matter fields. This raises the possibility that in the case of fermion fields one could confront the usual fermion doubling problem that arises in lattice gauge theories. We suggest, again based on recent results on spherical symmetry, that since the background space–times will generically involve superpositions of states associated with different discretizations the phenomenon may not arise. This opens a possibility of incorporating chiral fermions in the framework of loop quantum gravity.

Horizon entropy with loop quantum gravity methods

We show that the spherically symmetric isolated horizon can be described in terms of an SU(2) connection and an su(2)-valued one-form, obeying certain constraints. The horizon symplectic structure is precisely the one of 3d gravity in a first order formulation. We quantize the horizon degrees of freedom in the framework of loop quantum gravity, with methods recently developed for 3d gravity with non-vanishing cosmological constant. Bulk excitations ending on the horizon act very similarly to particles in 3d gravity. The Bekenstein–Hawking law is recovered in the limit of imaginary Barbero–Immirzi parameter. Alternative methods of quantization are also discussed.

Projective structures in loop quantum cosmology

Projective structures have successfully been used for the construction of measures in the framework of loop quantum gravity. In the present paper, we establish such structures for the configuration space R⊔RBohr, recently introduced in the context of homogeneous isotropic loop quantum cosmology. In contrast to the traditional space RBohr, the first one is canonically embedded into the quantum configuration space of the full theory. In particular, for the embedding of states into a corresponding symmetric sector of loop quantum gravity, this is advantageous. However, in contrast to the traditional space, there is no Haar measure on R⊔RBohr defining a canonical kinematical L2-Hilbert space on which operators can be represented. The introduced projective structures allow to construct a family of natural measures on R⊔RBohr whose corresponding L2-Hilbert spaces we finally investigate.

Symmetric scalar constraint for loop quantum gravity

In the framework of loop quantum gravity, we define a new Hilbert space of states which are solutions of a large number of components of the diffeomorphism constraint. On this Hilbert space, using the methods of Thiemann, we obtain a family of gravitational scalar constraints. They preserve the Hilbert space for every choice of lapse function. Thus adjointness and commutator properties of the constraint can be investigated in a straightforward manner. We show how the space of solutions of the symmetrized constraint can be defined by spectral decomposition, and the Hilbert space of physical states by subsequently fully implementing the diffeomorphism constraint. The relationship of the solutions to those resulting from a proposal for a symmetric constraint operator by Thiemann remains to be elucidated.

A note on entanglement entropy and quantum geometry

It has been argued that the entropy computed in the isolated horizon framework of loop quantum gravity is closely related to the entanglement entropy of the gravitational field, and that the calculation performed is not restricted to horizons. We recall existing work on this issue and explain how recent work on generalizing these computations to arbitrary spacetime dimensions supports this point of view and makes the duality between entanglement entropy and the entropy computed from counting boundary states manifest. In a certain semiclassical regime in dimensions, this entropy is given by the Bekenstein–Hawking formula.

BF theory explanation of the entropy for nonrotating isolated horizons

We consider the nonrotating isolated horizon as an inner boundary of a four-dimensional asymptotically flat spacetime region. Due to the symmetry of the isolated horizon, it turns out that the boundary degrees of freedom can be described by a SO(1,1) BF theory with sources. This provides a new alternative approach to the usual one using Chern-Simons theory to study the black hole entropy. To count the microscopical degrees of freedom with the boundary BF theory, the entropy of the isolated horizon can also be calculated in the framework of loop quantum gravity. The leading-order contribution to the entropy coincides with the Bekenstein-Hawking area law only for a particular choice of the Barbero-Immirzi parameter, which is different from its value in the usual approach using Chern-Simons theory. Moreover, the quantum correction to the entropy formula is a constant term rather than a logarithmic term.

Quantum-reduced loop gravity: Cosmology

We introduce a new framework for loop quantum gravity: mimicking the spinfoam quantization procedure we propose to study the symmetric sectors of the theory imposing the reduction weakly on the full kinematical Hilbert space of the canonical theory. As a first application of Quantum-Reduced Loop Gravity we study the inhomogeneous Bianchi I model. The emerging quantum cosmological model represents a simplified arena on which the complete canonical quantization program can be tested. The achievements of this analysis could elucidate the relationship between Loop Quantum Cosmology and the full theory.

Isolated Horizons and Black Hole Entropy in Loop Quantum Gravity

We review the black hole entropy calculation in the framework of Loop Quantum Gravity based on the quasi-local definition of a black hole encoded in the isolated horizon formalism. We show, by means of the covariant phase space framework, the appearance in the conserved symplectic structure of a boundary term corresponding to a Chern-Simons theory on the horizon and present its quantization both in the U(1) gauge fixed version and in the fully SU(2) invariant one. We then describe the boundary degrees of freedom counting techniques developed for an infinite value of the Chern-Simons level case and, less rigorously, for the case of a finite value. This allows us to perform a comparison between the U(1) and SU(2) approaches and provide a state of the art analysis of their common features and different implications for the entropy calculations. In particular, we comment on different points of view regarding the nature of the horizon degrees of freedom and the role played by the Barbero-Immirzi parameter. We conclude by presenting some of the most recent results concerning possible observational tests for theory.

Revisiting the quantum scalar field in spherically symmetric quantum gravity

We extend previous results in spherically symmetric gravitational systems coupled with a massless scalar field within the loop quantum gravity framework. As a starting point, we take the Schwarzschild spacetime. The results presented here rely on the uniform discretization method. We are able to minimize the associated discrete master constraint using a variational method. The trial state for the vacuum consists of a direct product of a Fock vacuum for the matter part and a Gaussian centered around the classical Schwarzschild solution. This paper follows the line of research presented by Gambini et al (2009 Class. Quantum Grav. 26 215011 (arXiv:0906.1774v1)) and a comparison between their result and the one given in this work is made.

Intersecting Quantum Gravity with Noncommutative Geometry - a Review

We review applications of noncommutative geometry in canonical quantum gravity. First, we show that the framework of loop quantum gravity includes natural noncommutative structures which have, hitherto, not been explored. Next, we present the construction of a spectral triple over an algebra of holonomy loops. The spectral triple, which encodes the kinematics of quantum gravity, gives rise to a natural class of semiclassical states which entail emerging fermionic degrees of freedom. In the particular semiclassical approximation where all gravitational degrees of freedom are turned off, a free fermionic quantum field theory emerges. We end the paper with an extended outlook section.

Feynman diagrammatic approach to spinfoams

The spinfoams for people without the 3D/4D imagination could be an alternative title of our work. We derive spinfoams from operator spin-network diagrams. Our diagrams are the spin-network analog of the Feynman diagrams. Their framework is compatible with the framework of loop quantum gravity (LQG). For every operator spin-network diagram, we construct a corresponding operator spinfoam. Admitting all the spin-networks of LQG and all possible diagrams leads to a clearly defined large class of operator spinfoams. This way our framework provides a proposal for a class of 2-cell complexes that should be used in the spinfoam theories of LQG. Within this class, our diagrams are just equivalent to the spinfoams. The advantage, however, in the diagram framework is that it is self contained and all the amplitudes can be calculated directly from the diagrams without explicit visualization of the corresponding spinfoams. The spin-network diagram operators and amplitudes are consistently defined on their own. Each diagram encodes all the combinatorial information. We illustrate the applications of our diagrams: we introduce a diagram definition of Rovelli’s surface amplitudes as well as of the canonical transition amplitudes. Importantly, our operator spin-network diagrams are defined in a way general enough to accommodate all the versions of the Engle–Pereira–Rovelli–Livine or the Freidel–Krasnov model, as well as other possible models. The diagrams are also compatible with the structure of the LQG Hamiltonian operators, which is an additional advantage. Finally, a scheme for a complete definition of a spinfoam theory by declaring a set of interaction vertices emerges from the examples presented at the end of the paper.

Canonical quantization of non-commutative holonomies in 2 + 1 loop quantum gravity

In this work we investigate the canonical quantization of 2 + 1 gravity with cosmological constant Λ > 0 in the canonical framework of loop quantum gravity. The unconstrained phase space of gravity in 2 + 1 dimensions is coordinatized by an SU(2) connection A and the canonically conjugate triad field e. A natural regularization of the constraints of 2 + 1 gravity can be defined in terms of the holonomies of $ {A_\pm } = A\pm \sqrt {{\Lambda e}} $ . As a first step towards the quantization of these constraints we study the canonical quantization of the holonomy of the connection A λ = A + λe (for λ ∈ $ \mathbb{R} $ ) on the kinematical Hilbert space of loop quantum gravity. The holonomy operator associated to a given path acts non trivially on spin network links that are transversal to the path (a crossing). We provide an explicit construction of the quantum holonomy operator. In particular, we exhibit a close relationship between the action of the quantum holonomy at a crossing and Kauffman’s q-deformed crossing identity (with $ q = \exp \left( {i\hbar \lambda /2} \right) $ ). The crucial difference is that (being an operator acting on the kinematical Hilbert space of LQG) the result is completely described in terms of standard SU(2) spin network states (in contrast to q-deformed spin networks in Kauffman’s identity). We discuss the possible implications of our result.

Black hole entropy: lessons from loop quantum gravity

The purpose of this contribution is is to discuss black hole entropy in the loop quantum gravity framework. Special attention is paid to the description of the microscopic degrees of freedom responsible for the entropy, the statement of the combinatorial problems that must be solved in order to count them, and the behaviour of the entropy as a function of the horizon area. In particular I will review the derivation of the Bekenstein-Hawking law and its logarithmic corrections. I end with a comparison between the results derived within loop quantum gravity and the ones obtained by other approaches.

Dynamics for a 2-vertex quantum gravity model

We use the recently introduced U(N) framework for loop quantum gravity to study the dynamics of spin network states on the simplest class of graphs: two vertices linked with an arbitrary number N of edges. Such graphs represent two regions, in and out, separated by a boundary surface. We study the algebraic structure of the Hilbert space of spin networks from the U(N) perspective. In particular, we describe the algebra of operators acting on that space and discuss their relation to the standard holonomy operator of loop quantum gravity. Furthermore, we show that it is possible to make the restriction to the isotropic/homogeneous sector of the model by imposing the invariance under a global U(N) symmetry. We then propose a U(N)-invariant Hamiltonian operator and study the induced dynamics. Finally, we explore the analogies between this model and loop quantum cosmology and sketch some possible generalizations of it.

Interacting Entropy-Corrected Holographic Chaplygin Gas Model

Holographic dark energy (HDE), presents a dynamical view of dark energy which is consistent with the observational data and has a solid theoretical background. Its definition follows from the entropy-area relation $S(A)$, where $S$ and $A$ are entropy and area respectively. In the framework of loop quantum gravity, a modified definition of HDE called "entropy-corrected holographic dark energy" (ECHDE) has been proposed recently to explain dark energy with the help of quantum corrections to the entropy-area relation. Using this new definition, we establish a correspondence between modified variable Chaplygin gas, new modified Chaplygin gas and the viscous generalized Chaplygin gas with the entropy corrected holographic dark energy and reconstruct the corresponding scalar potentials which describe the dynamics of the scalar field.

On the regularization of the constraint algebra of quantum gravity in 2 + 1 dimensions with a nonvanishing cosmological constant

We use the mathematical framework of loop quantum gravity (LQG) to study the quantization of three-dimensional (Riemannian) gravity with a positive cosmological constant (Λ > 0). We show that the usual regularization techniques (successful in the Λ = 0 case and widely applied in four-dimensional LQG) led to a deformation of the classical constraint algebra (or anomaly) proportional to the local strength of the curvature squared. We argue that this is an unavoidable consequence of the nonlocal nature of generalized connections.

Spin-foams for all loop quantum gravity

The simplicial framework of Engle–Pereira–Rovelli–Livine spin-foam models is generalized to match the diffeomorphism invariant framework of loop quantum gravity. The simplicial spin-foams are generalized to arbitrary linear 2-cell spin-foams. The resulting framework admits all the spin-network states of loop quantum gravity, not only those defined by triangulations (or cubulations). In particular, the notion of embedded spin-foam we use allows us to consider knotting or linking spin-foam histories. Also the main tools, the vertex structure and the vertex amplitude, are naturally generalized to an arbitrary valency case. The correspondence between all the SU(2) intertwiners and the SU(2)×SU(2) EPRL intertwiners is proved to be 1–1 in the case of the Barbero–Immirzi parameter |γ| ⩾ 1.