General Relativity & Quantum Cosmology

There are two notions that play a central role in the mathematical theory of computation. One is that of a computable problem, i.e., of a problem that can, in principle, be solved by an (idealized) computer. It is known that there exist problems that \'have answers\', but for which those answers are not computable. The other is that of the difficulty of a computation, i.e. of the number of (idealized) steps required actually to carry out that computation. It is known that, given any appropriate \'degree of difficulty\', there exists a problem that, while computable, is at least that difficult. These two notions, while purely mathematical, are designed to reflect, in some broad sense, the physics of the computation process. But there are indications that physics may have something further to say about them. Indeed, it has been suggested that, by using general relativity, some problems that are (mathematically) non-computable may become computable; and that, by using quantum mechanics, some problems that are (mathematically) difficult may become less so. Are there, in principle, any limitations on what physics can do for us in this area?

The principles of Quantum Mechanics and of Classical General Relativity imply Uncertainty Relations between the different spacetime coordinates of the events, which yield to a basic model of Quantum Minkowski Space, having the full (classical) Poincare\' group as group of symmetries. The four dimensional Euclidean distance is a positive operator bounded below by a constant of order one in Planck units; the area operator and the four volume operator are normal operators - the latter being a Lorentz invariant operator with pure point spectrum - whose moduli are also bounded below by a constant of order one. While the spectrum of the 3 volume operator includes zero. These findings are in perfect agreement with the physical intuition suggested by the Spacetime Uncertainty Relations which are implemented by the Algebra of Quantum Spacetime. The formulations of interactions between quantum fields on Quantum Spacetime will be discussed. The various approaches to interactions, equivalent to one another on the Minkowski background, yield to different schemes on Quantum Spacetime, with the common feature of a breakdown of Lorentz invariance due to interactions. In particular one of these schemes will be discussed and motivated, which leads to fully Ultraviolet-Finite theories. Quantum fields will depend on the quantum coordinates, but, in presence of Gravity, the commutators of the coordinates might in turn depend on the quantum fields, giving rise to a quantum texture where fields and spacetime coordinates cannot be separated. Possible deep physical consequences will be outlined.

Loop Quantum Gravity and Deformation Quantization Abstract: We propose a unified approach to loop quantum gravity and Fedosov quantization of gravity following the geometry of double spacetime fibrations and their quantum deformations. There are considered pseudo--Riemannian manifolds enabled with 1) a nonholonomic 2+2 distribution defining a nonlinear connection (N--connection) structure and 2) an ADM 3+1 decomposition. The Ashtekar-Barbero variables are generalized and adapted to the N-connection structure which allows us to write the general relativity theory equivalently in terms of Lagrange-Finsler variables and related canonical almost symplectic forms and connections. The Fedosov results are re-defined for gravitational gauge like connections and there are analyzed the conditions when the star product for deformation quantization is computed in terms of geometric objects in loop quantum gravity. We speculate on equivalence of quantum gravity theories with 3+1 and 2+2 splitting and quantum analogs of the Einstein equations.

There has been a dream that matter and gravity can be unified in a fundamental theory of quantum gravity. One of the main philosophies to realize this dream is that matter may be emergent degrees of freedom of a quantum theory of gravity. We study the propagation and interactions of braid-like chiral states in models of quantum gravity in which the states are (framed) four-valent spin networks embedded in a topological three manifold and the evolution moves are given by the dual Pachner moves. There are results for both the framed and unframed case. We study simple braids made up of two nodes which share three edges, which are possibly braided and twisted. We find three classes of such braids, those which both interact and propagate, those that only propagate, and the majority that do neither. These braids may serve as fundamental matter content.

We show that the matrix-model for noncommutative U(n) gauge theory actually describes SU(n) gauge theory coupled to gravity. The nonabelian gauge fields as well as additional scalar fields couple to a dynamical metric G_ab, which is given in terms of a Poisson structure. This leads to a gravity theory which is naturally related to noncommutativity, encoding those degrees of freedom which are usually interpreted as U(1) gauge fields. Essential features such as gravitational waves and the Newtonian limit are reproduced correctly. UV/IR mixing is understood in terms of an induced gravity action. The framework appears suitable for quantizing gravity.

After a brief motivation for studying 3d gravity, a review of Strominger argument will be given, showing that 3d quantum gravity with negative cosmological constant is a conformal field theory. Classical phase space tools will be introduced to develop semi-classical analyses of gravity with zero cosmological constant, and, with negative cosmological constant and closed timelike curves.

There has been much interest, in the past few years, in the kappa-Poincare\'/kappa-Minkowski framework as a possible scenario for a deformation of Poincare\' symmetries at Planck scale. I will show how it is possible to give a physical characterization of the concept of quantum symmetries described by a nontrivial Hopf algebra. In particular, I will discuss the generalization of the Noether analysis for a scalar field in kappa-Minkowski space-time and derive conserved charges associated with each generator of the kappa-Poincare\' Hopf-algebra. Then I will report on a recent proposal for the quantization of a scalar field enjoying kappa-Poincare\' symmetries, which consists in a construction of the Fock-space of the theory consistent with the structure of deformed symmetries. Finally I will comment on possible applications of deformed symmetries scenarios in cosmology.

Relational particle mechanics are theories of relative angles and relative (ratios of) separations only. These bear a number of resemblances to the geometrodynamical formulation of general relativity and as such are useful analogues for at least some approaches to the notorious problem of time in quantum gravity. I have recently provided a fairly complete study of the configuration spaces of these theories in spatial dimension 1 and 2, am subsequently studying the redused forms of these theories at the quantum level, and this shall provide a number of useful examples for the conceptual discussion of various problem of time strategies

In this talk we propose a Reduced Phase Space Quantization approach to Loop Quantum Gravity. The idea is to combine the relational formalism introduced by Rovelli in the extended form developed by Dittrich and the Brown-Kuchar-Mechanism. The relational formalism can be used to construct gauge invariant observables for constrained systems such as General Relativity, while the Brown-Kuchar-Mechanism is a particular application of the relational formalism in which pressureless dust is taken as the clock of the system. By combining these two we obtain a framework in which the constraints of General Relativity deparametrize such that the algebra of observables has a very simple structure and furthermore we obtain a so called physical Hamiltonian generating the evolution of those observables. The quantization of the reduced phase space and the physical Hamiltonian can be obtained by using standard LQG techniques and gives a direct access to the physical Hilbert space, which is much harder to achieve in the standard Dirac quantization. Additionally we will analyze the quantization in the Algebraic Quantum Gravity context and discuss the differences that occur. Finally we will present recent results where this framework has been applied to cosmological perturbation theory.

In the first part of the talk we introduce a technique to compute large scale correlations in LQG and spinfoam models. Using this formalism we calculate some components of the graviton propagator and of the n-points function. In the second part we apply the ideas suggested by LQG to the black hole interior . The result of the quantum analysis is that the classical singularity disappears in loop quantum gravity. Solving the semiclassical Einstein equation of motion we obtain a metric regular and singularity free in contrast to the classical one. By using the new metric we calculate the Hawking temperature, entropy and we study the mass evaporation process.