Motion in Quantum Gravity

The black hole information puzzle can be resolved if two conditions are met. The first is that the information about what falls inside a black hole remains encoded in degrees of freedom that persist after the black hole completely evaporates. These degrees of freedom should be capable of purifying the information. The second is if these purifying degrees of freedom do not significantly contribute to the system's energy, as the macroscopic mass of the initial black hole has been radiated away as Hawking radiation to infinity. The presence of microscopic degrees of freedom at the Planck scale provides a natural mechanism for achieving these two conditions without running into the problem of the large pair-creation probabilities of standard remnant scenarios. In the context of Hawking radiation, the first condition implies that correlations between the in and out Hawking partner particles need to be transferred to correlations between the microscopic degrees of freedom and the out partners in the radiation. This transfer occurs dynamically when the in partners reach the singularity inside the black hole, entering the UV regime of quantum gravity where the interaction with the microscopic degrees of freedom becomes strong. The second condition suggests that the conventional notion of the vacuum's uniqueness in quantum field theory should fail when considering the full quantum gravity degrees of freedom. In this paper, we demonstrate both key aspects of this mechanism using a solvable toy model of a quantum black hole inspired by loop quantum gravity.

Canonical quantum gravity provides insights into the quantum dynamics as well\nas quantum geometry of space-time by its implications for constraints. Loop\nquantum gravity in particular requires specific corrections due to its\nquantization procedure, which also results in a discrete picture of space. The\ncorresponding changes compared to the classical behavior can most easily be\nanalyzed in isotropic models, but perturbations around them are more involved.\nFor one type of corrections, consistent equations have been found which shed\nlight on the underlying space-time structure at the Planck scale: not just\nquantum dynamics but also the concept of space-time manifolds changes in\nquantum gravity. Effective line elements provide indications for possible\nrelationships to other frameworks, such as non-commutative geometry.\n

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.

Modern Canonical Quantum General Relativity

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,

It is well known that the quantum double structure plays an important role in three-dimensional quantum gravity coupled to matter field. In this paper, we show how this algebraic structure emerges in the context of three-dimensional Riemannian loop quantum gravity (LQG) coupled to a finite number of massive spinless point particles. In LQG, physical states are usually constructed from the notion of SU(2) cylindrical functions on a Riemann surfaced Σ and the Hilbert structure is defined by the Ashtekar-Lewandowski measure. In the case where Σ is the sphere S2, we show that the physical Hilbert space is in fact isomorphic to a tensor product of simple unitary representations of the Drinfeld double DSU(2): the masses of the particles label the simple representations, the physical states are tensor products of vectors of simple representations, and the physical scalar product is given by intertwining coefficients between simple representations. This result is generalized to the case of any Riemann surface Σ.

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 Karolyhazy uncertainty relation states that if a device is used to measure a length l, there will be a minimum uncertainty \(\delta l\) in the measurement, given by \((\delta l)^3 \sim L_{\mathrm {P}}^2 l\). This is a consequence of combining the principles of quantum mechanics and general relativity. In this letter we show how this relation arises in our approach to quantum gravity, in a bottom-up fashion, from the matrix dynamics of atoms of space–time–matter. We use this relation to define a space–time–matter (STM) foam at the Planck scale, and to argue that our theory is holographic. By coarse graining over time-scales larger than Planck time, one obtains the laws of quantum gravity. Quantum gravity is not a Planck scale phenomenon; rather it comes into play whenever classical space–time background is not available to describe a quantum system. Space–time and classical general relativity arise from spontaneous localisation in a highly entangled quantum gravitational system. The Karolyhazy relation continues to hold in the emergent theory. An experimental confirmation of this relation will constitute a definitive test of the quantum nature of gravity.

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.

<p><strong>The future of theoretical physics is unclear. Two large areas that fall under the umbrella of theoretical physics are cosmology and quantum gravity. Modern cosmology is relatively a much younger field than quantum gravity, and both of these fields require further developments of general relativity. In this thesis we do not hope to resolve the problems facing modern cosmology or theories of quantum gravity. Rather, we will conduct original research into aspects of general relativity that may be used in the future to aid the development and testing of theories of cosmology and quantum gravity.</strong></p><p>It is our view that the largest problem facing astrophysics and cosmology stem from the existence of the dark sector of the Universe. The implication here being that more than ninety percent of the energy density of the Universe is “missing in action” and seemingly consists of dark energy and dark matter. Furthermore, it is apparent that there exist conceptual flaws in our understanding of observational concepts such as expansion versus motion and observer biases. To this end, we investigate the standard spacetime metric used in cosmology, the Friedmann–Lemaˆıtre-Roberston–Walker (FLRW) metric in a peculiar coordinate system — the Painlev´e–Gullstrand coordinates. In this coordinate system (slicing), space is no longer expanding, rather, the galaxies are receding from each other. We hope this will aid in the understanding of expansion, motion, curvature, and observer bias with future work. We further investigate the possibility of black holes in cosmology being directly coupled to the accelerated expansion of the Universe — in other words, black holes as a source for dark energy. However, we show that this is highly implausible.</p><p>Relatively recently it has been postulated that the near black hole horizon limit may be a regime where quantum gravity effects become relevant i.e., quantum gravity may not be restricted to near the Planck scale. We investigate a curious model of black and white holes that shows how one may transition into the other over a finite period of time. This is research conducted in the near horizon limit of the Schwarzschild black hole. We introduce a time dependent function into the usual Schwarzschild black hole spacetime (leaving this new spacetime not a simple coordinate transformed version of the original). This function allows the black hole to transition into a white hole. Importantly, the action for this transition can be shown to be zero, meaning it can be added to the Feynman path integral at no cost.</p><p>Finally, we move to investigating the black hole memory effect. During the last decade, there has been an interesting connection made between the Bondi– Metzner–Sachs (BMS) group — an infinite dimensional group of symmetries found at null infinity — and the gravitational memory effect. In particular, it was shown that the passage of a gravitational wave that alters a Schwarzschild black hole is seen as a supertranslation of the spacetime at null infinity. We extend these calculations to the Kerr and Kerr–Newman black holes. Hence, showing that there may be a way to verify the abstract mathematical ideas predicated on the BMS group by detection of the memory effect in future observations.</p><p>It is our hope that when future gravitational wave detectors such as the laser-interferometer-space-antenna (LISA) are launched, research conducted in this thesis may shed light on how the memory may relate to black holes in their asymptotic & near horizon limits to aid our understanding of the nature of quantum gravity.</p>

The loop quantum gravity theory is a background-independent and non-perturbative quantization of general relativity. It has been developed both in the canonical and covariant formalisms. As for all tentative theories of quantum gravity, one of the key question is to produce clear experimental or observational predictions. As the discrete nature of spacetime at the Planck scale is extremely difficult to probe directly, phenomenological attempts do focus mostly on the cosmological sector, on black holes and on possible high energy astrophysics effects. The main prediction of loop quantum gravity in the cosmological sector is that the Big Bang is replaced by a Big Bounce: a contracting phase should have taken place before the expanding phase we are now living in. In the black hole sector, the Hawking evaporation spectrum should receive substantial corrections and the very existence of an event horizon can be questioned, potentially leading to black holes bouncing into white holes. Finally, on the astroparticle physics side, the quantum discreteness of spacetime could lead to interesting effects for high energy cosmic rays.

The Karolyhazy uncertainty relation is the statement that if a device is used to measure a length $l$, there will be a minimum uncertainty $\delta l$ in the measurement, given by $(\delta l)^3 \sim L_P^2\; l$. This is a consequence of combining the principles of quantum mechanics and general relativity. In this note we show how this relation arises in our approach to quantum gravity, in a bottom-up fashion, from the matrix dynamics of atoms of space-time-matter. We use this relation to define a space-time-matter foam at the Planck scale, and to argue that our theory is holographic. By coarse graining over time scales larger than Planck time, one obtains the laws of quantum gravity. Quantum gravity is not a Planck scale phenomenon; rather it comes into play whenever no classical space-time background is available to describe a quantum system. Space-time and classical general relativity arise from spontaneous localisation in a highly entangled quantum gravitational system. The Karolyhazy relation continues to hold in the emergent theory. An experimental confirmation of this relation will constitute a definitive test of the quantum nature of gravity.

IS A GRAVITON DETECTABLE?

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.

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.