One-loop renormalization in quantum gravity

Modern Canonical Quantum General Relativity

Loop quantum gravity is a physical theory that aims at unifying general relativity and quantum mechanics. It takes general relativity very seriously and modifies it via quantisation. General relativity describes gravity in terms of geometry; therefore, quantising such theory must be equivalent to quantising geometry and that is what loop quantum gravity does. This sounds like a mathematical task as well, thus, in this paper we will present the mathematics of loop quantum gravity. We will do it from a mathematician’s point of view. This paper is intended to be an introduction to loop quantum gravity for postgraduate students of physics and mathematics. In this work we will restrict ourselves to the three-dimensional case.

The unification of quantum principles with Einstein's geometric conception of spacetime and gravity into a consistent theory of quantum gravity is one of the main open challenges at the foundations of modern theoretical physics. Among several approaches developed over the years, the two main candidates are string theory and loop quantum gravity (LQG). Both theories are characterized by their own achievements and open issues so that the solution to the problem of quantum gravity remains still elusive. In lack of experimental guidance, in order to make progress, it becomes important to single out common features shared by different approaches allowing to merge tools and ideas, thus providing indirect tests to potentially overcome current limitations. In this respect, one of the major recent developments is the so-called holographic principle and its string theory-based realization within the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, a conjectured duality between string theory in asymptotically AdS spacetime and a SU(N) gauge theory thought of as living on its conformal boundary. As such it provides a promising arena to look for possible connections between non-perturbative LQG and string theory non-perturbatively defined via its dual field theory. With this motivation in mind, in this thesis we focus on effective symmetry-reduced models for resolved cosmological and black hole singularities to which lot of effort has been devoted within the LQG community and their possible embedding within the AdS/CFT framework. First of all, we focus on the field theory signatures of the singularity resolution in the bulk of Kasner-AdS spacetimes and show via several examples of quantum corrected effective geometries that the finite-distance pole in the boundary two-point correlator previously interpreted as the holographic signature of the bulk singularity is smoothed out by quantum gravity effects. The time dependence of the boundary spacetime however prevents us from setting up an independent field theory computation to compare with the bulk gravity results. We move then to black hole (BH) singularities whose asymptotic boundary is Minkowski or AdS. Since no fully satisfactory effective LQG model is available already for the simplest static spherically symmetric case with zero cosmological constant, we consider the case of a 4-dimensional Schwarzschild BH as a necessary preparation for higher dimensional extensions to AdS BHs. New models based on new sets of canonical variables directly related to the Kretschmann scalar are thus introduced to take the onset of quantum effects under control, and the resulting quantum corrected spacetime structure is discussed in detail. A key ingredient of our analysis is the study of Dirac observables for the asymptotic ADM masses and their relation with admissible initial conditions for the effective dynamics compatibly with the requirement of a unique upper bound on curvature invariants resolving the central singularity. Finally, quantum corrections to thermodynamic quantities are also analyzed and, coming back to the original motivation of using LQG techniques for AdS/CFT, a possible extension to arbitrary dimensions and negative cosmological constant is sketched.

Of the different approaches to quantum gravity, the best developed, from the point of view of addressing the key theoretical questions a quantum theory of gravity must answer, is loop quantum gravity. While string theory appears to better address the issue of unification, at least so far, it fails to provide a background independent approach to the quantum mechanics of spacetime geometry-a necessary condition for any quantum theory of gravity. Moreover, many key conjectures remain unproven, including perturbative finiteness and consistency, which have not been demonstrated for any string theory past second order in perturbation theory. By contrast, loop quantum gravity appears to provide a consistent and finite background independent approach to quantum gravity. There is recent progress on several issues, including accounting for the black hole entropy [8], and giving a precise quantum mechanical description of the earliest phases of the evolution of the universe [9, 10]. Furthermore, it gives unique predictions of novel quantum gravitational phenomena, such as the discreteness of area, volume and other observables. However, in this new era of quantum gravity phenomenology, a quantum theory of gravity must pass a stricter test to be taken as a serious candidate: it must make unambiguous predictions for the upcoming experiments which probe the Planck scale. One way such predictions may arise is by modifications of the energymomentum relations of low energy physics, of the form,

In many quantum gravity approaches, the cosmological constant is introduced by deforming the gauge group into a quantum group. In three dimensions, the quantization of the Chern-Simons formulation of gravity provided the first example of such a deformation. The Turaev-Viro model, which is an example of a spin-foam model, is also defined in terms of a quantum group. By extension, it is believed that in four dimensions, a quantum group structure could encode the presence of $\mathrm{\ensuremath{\Lambda}}\ensuremath{\ne}0$. In this article, we introduce by hand the quantum group ${\mathcal{U}}_{q}(\mathfrak{s}u(2))$ into the loop quantum gravity (LQG) framework; that is, we deal with ${\mathcal{U}}_{q}(\mathfrak{s}u(2))$-spin networks. We explore some of the consequences, focusing in particular on the structure of the observables. Our fundamental tools are tensor operators for ${\mathcal{U}}_{q}(\mathfrak{s}u(2))$. We review their properties and give an explicit realization of the spinorial and vectorial ones. We construct the generalization of the $\mathrm{U}(N)$ formalism in this deformed case, which is given by the quantum group ${\mathcal{U}}_{q}(\mathfrak{u}(N))$. We are then able to build geometrical observables, such as the length, area or angle operators, etc. We show that these operators characterize a quantum discrete hyperbolic geometry in the three-dimensional LQG case. Our results confirm that a quantum group structure in LQG can be a tool to introduce a nonzero cosmological constant into the theory. Our construction is both relevant for three-dimensional Euclidian quantum gravity with a negative cosmological constant and four-dimensional Lorentzian quantum gravity with a positive cosmological constant.

Spin foam models, loop quantum gravity, and group field theory are discussed as quantum gravity candidate theories and usually involve a continuous Lie group. We advocate here to consider quantum gravity-inspired models with finite groups, firstly as a test bed for the full theory and secondly as a class of new lattice theories possibly featuring an analogue diffeomorphism symmetry. To make these notes accessible to readers outside the quantum gravity community, we provide an introduction to some essential concepts in the loop quantum gravity, spin foam, and group field theory approach and point out the many connections to the lattice field theory and the condensed-matter systems.

Modified dispersion relations(MDRs) as a manifestation of Lorentz invariance violation, have been appeared in alternative approaches to quantum gravity problem. Loop quantum gravity is one of these approaches which evidently requires modification of dispersion relations. These MDRs will affect the usual formulation of the Compton effect. The purpose of this paper is to incorporate the effects of loop quantum gravity MDRs on the formulation of Compton scattering. Using limitations imposed on MDRs parameters from Ultra High Energy Cosmic Rays(UHECR), we estimate the quantum gravity-induced wavelength shift of scattered photons in a typical Compton process. Possible experimental detection of this wavelength shift will provide strong support for underlying quantum gravity proposal.

There are several approaches to quantum gravitational corrections of black hole thermodynamics. String theory and loop quantum gravity, by direct analysis on the basis of quantum properties of black holes, show that in the entropy–area relation the leading order correction should be of log-area type. On the other hand, generalized uncertainty principle (GUP) and modified dispersion relations (MDRs) provide perturbational framework for such modifications. Although both GUP and MDRs are common features of all quantum gravity scenarios, their functional forms are quantum gravity model dependent. Since both string theory and loop quantum gravity give more reliable solution of the black hole thermodynamics, one can use their results to test approximate results of GUP and MDRs. In this Letter, we find quantum corrected black hole thermodynamics in the framework of GUP and MDR and then we compare our results with string theory solutions. This comparison suggests severe constraints on the functional form of GUP and MDRs. These constraints may reflect characteristic features of ultimate quantum gravity theory.

Quantum gravity aims to describe gravity in quantum mechanical terms. How exactly this needs to be done remains an open question. Various proposals have been put on the table, such as canonical quantum gravity, loop quantum gravity, string theory, etc. These proposals often encounter technical and conceptual problems. In this chapter, we focus on canonical quantum gravity and discuss how many conceptual problems, such as the measurement problem and the problem of time, can be overcome by adopting a Bohmian point of view. In a Bohmian theory (also called pilot-wave theory or de Broglie-Bohm theory, after its originators de Broglie and Bohm), a system is described by certain variables in space-time such as particles or fields or something else, whose dynamics depends on the wave function. In the context of quantum gravity, these variables are a space-time metric and suitable variable for the matter fields (e.g., particles or fields). In addition to solving the conceptual problems, the Bohmian approach yields new applications and predictions in quantum cosmology. These include space-time singularity resolution, new types of semi-classical approximations to quantum gravity, and approximations for quantum perturbations moving in a quantum background.

IS A GRAVITON DETECTABLE?

We review aspects of loop quantum gravity and spin foam models at an introductory level, with special attention to questions frequently asked by non-specialists.

We have searched for a universal property of quantum gravity, where "universal" means to be independent of any existing model of quantum gravity (such as the super string theory, loop quantum gravity, causal dynamical triangulation, and so on). To do so, we have investigated carefully the basis of black hole thermodynamics. The uniqueness of Bekenstein-Hawking entropy lets us extract a reasonable universal property of quantum gravity from the conditions justifying Boltzmann formula. The universal property indicates a repulsive gravity at Planck length scale, otherwise stationary black holes can not be regarded as thermal equilibrium states of gravity.

This chapter is the first chapter of the book dedicated to the study of the combinatorics of various quantum gravity approaches. After a brief introductory section to quantum gravity, we shortly mention the main candidates for a quantum theory of gravity: string theory, loop quantum gravity, and group field theory (GFT), causal dynamical triangulations, matrix models. The next sections introduce some GFT models such as the Boulatov model, the colourable and the multi-orientable model. The saddle point method for some specific GFT Feynman integrals is presented in the fifth section. Finally, some algebraic combinatorics results are presented: definition of an appropriate Conne–Kreimer Hopf algebra describing the combinatorics of the renormalization of a certain tensor GFT model (the so-called Ben Geloun–Rivasseau model) and the use of its Hochschild cohomology for the study of the combinatorial Dyson–Schwinger equation of this specific model.

We tackle the question of motion in Quantum Gravity: what does motion mean at the Planck scale? Although we are still far from a complete answer we consider here a toy model in which the problem can be formulated and resolved precisely. The setting of the toy model is a three-dimensional Euclidean gravity. Before studying the model in detail, we argue that Loop Quantum Gravity may provide a very useful approach when discussing the question of motion in Quantum Gravity.

The conference Conceptual and Technical Challenges for Quantum Gravity at Sapienza University of Rome, from 8 to 12 September 2014, has provided a beautiful opportunity for an encounterbetween different approaches and different perspectives on the quantum-gravity problem. It contributedto a higher level of shared knowledge among the quantum-gravity communities pursuing each specific research program.There were plenary talks on many different approaches, including in particular string theory, loop quantum gravity, spacetime noncommutativity, causal dynamical triangulations, asymptotic safety and causal sets. Contributions from the perspective of philosophy of science were also welcomed.In addition several parallel sessions were organized. The present volume collects contributions from the Noncommutative Geometry and Quantum Gravity parallel session4, with additional invited contributions from specialists in the field.Noncommutative geometry in its many incarnations appears at the crossroad of many researches in theoretical and mathematical physics:• from models of quantum space-time (with or without breaking of Lorentz symmetry) to loop gravity and string theory,• from early considerations on UV-divergencies in quantum field theory to recent models of gauge theories on noncommutative spacetime,• from Connes description of the standard model of elementary particles to recent Pati-Salam like extensions.This volume provides an overview of these various topics, interesting for the specialist as well as accessibleto the newcomer. 4partially funded by CNRS PEPS /PTI ‘‘Metric aspect of noncommutative geometry: from Monge to Higgs’’