General Relativity & Quantum Cosmology

According to general relativity, space-time ends at singularities and classical physics just stops. In particular, the big bang is regarded as The Beginning. However, general relativity is incomplete because it ignores quantum effects. Through simple models, I will illustrate how the quantum nature of space-time geometry resolves the big bang singularity. Quantum physics does not stop there. Indeed, quantum space-times can be vastly larger than what general relativity had us believe, with unforeseen physical effects in the deep Planck regime.

It has long been thought that theories based on equations of motion possessing derivatives of order higher than second are not unitary. Specifically, they are thought to possess unphysical ghost states with negative norm. However, it turns out that the appropriate Hilbert space for such theories had not been correctly constructed, and when the theory is formulated properly [Bender and Mannheim, PRL 100, 110402 (2008). (arXiv:0706.0207 [hep-th]] there are no ghost states at all and time evolution is fully unitary. Unitarity can be established for theories based on both second and fourth order derivatives, and for theories based on fourth order derivatives alone. In this latter case the Hamiltonian is a non-diagonalizable, Jordan-block operator which possesses fewer eigenstates than eigenvalues. Despite the lack of completeness of the energy eigenstates, a consistent, unitary quantum mechanics for the theory can still be formulated [Bender and Mannheim, PRD 78, 025022 (2008). (arXiv:0807.2607 [hep-th]).] The implications of these results for the construction of a consistent theory of quantum gravity in four spacetime dimensions will be briefly discussed.

As well known, cosmic ray experiments can put strong constraints on possible Lorentz Invariance Violations. In particular, the presence of the so called GZK \'cut-off\' may indicate that protons do propagate in the Universe as expected from relativistic invariance. The presence of this feature in the spectrum has been convincingly indicated by the HiRes and Auger experiments, while the Auger Observatory has given indication on the correlation of Ultra High Energy Cosmic particles with nearby sources, as predicted by the GZK feature. I review the experimental results and discuss in particular both the theoretical and experimental intricacies and the limits on LIV parameters that can be deduced.

Motivated by the analogy proposed by Witten between Chern-Simons theories and CFT-Wess-Zumino-Witten models, we explore a new way of computing the entropy of a black hole starting from the isolated horizon framework in Loop Quantum Gravity. The results seem to indicate that this analogy can work in this particular case. This could be a good starting point for the search of a deeper connection between the description of black holes in LQG and a conformal field theory.

I discuss a class of compact objects (\'monsters\') with more entropy than a black hole of the same ADM mass. Such objects are problematic for AdS/CFT duality and the conventional interpretation of black hole entropy as counting of microstates. Nevertheless, monster initial data can be constructed in semi-classical general relativity without requiring large curvatures or energy densities.

The semiclassical-quantum correspondence (SQC) is a new principle which has enabled the explicit solution of the quantum constraints of GR in the full theory in the Ashtekar variables for gravity coupled to matter. The solutions, which constitute the physical space of states implementing the quantum dynamics of GR in the Dirac procedure, include a special class of states known as the generalized Kodama states (GKod). The GKodS can be seen as an analogue of the pure Kodama state (Kod) when quantum gravity (QGRA) is coupled to matter fields quantized on the same footing. The criterion for finiteness stems from a precise cancellation of the ultraviolet singularities stemming from the quantum Hamiltonian constraint, allowing for an exact solution. This signifies the following developments for 4D QGRA: (i) Equivalence among the Dirac, reduced phase, geometric and path integration approaches to quantization for GKods; (ii) A generalization of topological field theory to include matter fields via the instanton representation of GKod; (iii) A possible mechanism to establish 4D QGRA, via tree networks, as a renormalizable theory (iv) A direct link from QGRA to Minkoswki spacetime physics, which would enable tests of 4D QGRA without the necessity to access the Planck scale (v) A third-quantized analogy to second quantized spin network states implementing the quantum dynamics of GR. The aforementioned algorithm is designed to construct explicit solutions to the constraints of the full theory by inspection, while implementing any desired ‘boundary’ conditions on the states necessary to reduce to the appropriate semiclassical limit. Conversely, the finite states of 4D QGRA can place severe restrictions on phenomena occurring in the weak gravitational limit below the Planck scale. While we demonstrate this for the GKodS in this talk, the procedure can be applied to obtain a family of states labeled by two arbitrary functions of position, which possess the requisite Hilbert space structure in the limit where the matter fields are turned off. Remaining areas of research in progress include the illumination of the Hilbert space structure of the GKodS, analysis of various models for which the SQC can produce tractable solutions, in the full theory and in minisuperspace, and the addressal of any issues of interest regarding the mathematical rigor of the states.

We discuss the possibility that spacetime geometry may be an emergent phenomenon. This idea has been motivated by the Analogue Gravity programme. An \'effective gravitational field\' dominates the kinematics of small perturbations in an Analogue Model. In these models there is no obvious connection between the \'gravitational\' field tensor and the Einstein equations, as the emergent spacetime geometry arises as a consequence of linearising around some classical field. After a brief introduction on this topic, we present our recent contributions to the field.

After reviewing Wilson\'s picture of renormalization, and the associated Exact Renormalization Group, I will show that no (physically acceptable) non-trivial fixed points exist for scalar field theory in D>=4. Consequently, an asymptotic safety scenario is ruled out, and the triviality of the theory is confirmed.

In an asymptotically anti-de Sitter space, three-dimensional topologically massive gravity has some remarkable properties, which suggest interesting applications to quantum gravity. Unfortunately, though, the theory appears to be unstable, even at the special \'chiral\' value of the coupling. I will discuss recent work, and recent controversies, in this field.

A modified version of the double potential formalism for the electrodynamics of dyons is constructed. Besides the two vector potentials, this manifestly duality invariant formulation involves four additional potentials, scalar potentials which appear as Lagrange multipliers for the electric and magnetic Gauss constraints and potentials for the longitudinal electric and magnetic fields. In this framework, a static dyon appears as a Coulomb-like solution without string singularities. Dirac strings are needed only for the Lorentz force law, not for Maxwell\'s equations. The magnetic charge no longer appears as a topological conservation law but as a surface integral on a par with electric charge. The theory is generalized to curved space. As in flat space, the string singularities of dyonic black holes are resolved. As a consequence all singularities are protected by the horizon and the thermodynamics is shown to follow from standard arguments in the grand canonical ensemble.

The general boundary state formulation is a key tool for extracting the semiclassical limit of nonpertubative theories of quantum gravity. In this talk I will discuss how this formalism works in the context of four-dimensional quantum Regge calculus with a general triangulation. A Gaussian boundary state selects a classical internal solution and peaks the path integral on it. As a result boundary observables, in particular the two-point function, can be computed order by order in a semiclassical asymptotic expansion. When the same methods are applied to a modified Regge theory that substitutes the exponential of the action by its cosine at each simplex in the triangulation, as conjectured from the semiclassical limit of spin foam models, the contributions from the sign-reversed terms are suppressed and the results match those of conventional Regge calculus. This talk is based on the results published in arXiv:0808.1107 [gr-qc].

The standard Hamiltonian formulation of (first order) gravity breaks manifest covariance both in its retention of the Lorentz group as a local gauge group and in its discrepant treatment of spacelike and timelike diffeomorphisms. Here we promote more covariant alternatives for canonical quantum gravity that address each of these problems, and discuss the implications for both the classical and the quantum theory of gravity. By retaining the full local Lorentz group, one gains significant insight into the geometric and algebraic properties of the Hamiltonian dynamics. As an example, we discuss the possibility of computing the internal spin angular momentum of asymptotically flat spacetimes, which may lead to insight into the nature of spin in quantum gravity. By treating the spacelike and timelike diffeomorphisms on equal footing, using techniques from geometric quantization we find a new representation of the quantum constraints where the total Hamiltonian is kinematical in the same vein as the Gauss and diffeomorphism constraints. Finally, we discuss the possibility of a manifestly 4-dimensional symplectic form on the Lagrangian phase space.