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Arguably, the most important issue in physics today is trying to unify the twin pillars of modern physics, quantum theory and general relativity, into a single theory known as quantum gravity. This introductory presentation will discuss various aspects of quantum gravity.

Graphity models are characterized by configuration spaces in which states correspond to graphs and Hamiltonians that depend on local properties of graphs such as degrees of vertices and numbers of shortcycles. It has been argued that such models can be useful in studying how an extended geometry might emerge from a background independent dynamical system. As statistical systems, graphity models can be studied analytically by estimating their partition functions or numerically by Monte Carlo simulations. In this talk I will present recent results obtained using both of these approaches. In particular, I will describe the transition between the high and low temperature regimes and arguethat matter degrees of freedom must play an important role in order for the graph states dominating in the low temperature regime to resemble interesting extended geometries.

This talk is concerned with the existence of spectral triples in quantum gravity. I will review the construction of a spectral triple over a functional space of connections. Here, the *-algebra is generated by holonomy loops and the Dirac type operator has the form of a global functional derivation operator. The spectral triple encodes the Poisson structure of General Relativity when formulated in terms of Ashtekars variables. Finally I will argue that the Hamiltonian of General Relativity may emerge from the construction via the requirement that inner automorphisms vanish on the vacuum sector.

For quantum gravity, the requirement of metric positivity suggests the use of noncanonical, affine kinematical field operators. In view of gravity\'s set of open classical first class constraints, quantization before reduction is appropriate, leading to affine commutation relations and affine coherent states. The anomaly in the quantized constraints may be accommodated within the projection operator approach, which treats first and second class quantum constraints in an equal fashion. Functional integral representations are derived for expressions both with and without constraint imposition. As with all coherent state formulations, close contact between the classical and quantum theories is maintained throughout. Perturbative nonrenormalizability is understood as a partial hard-core behavior of the interaction, which as soluble models suggest, leads to a perturbative formulation, not about the traditional free theory, but rather about a suitable pseudofree theory that properly incorporates the essence of the hard core.

This talk presents some recent results in renormalizable noncommutative quantum field theory. After introducing the renormalization group approach in the commutative setting I will procede to its generalization to the simplest noncommutative model, $phi_4^{star 4}$ on the Moyal space. The well known phenomenon of ultraviolet/infrared mixing is cured by adding a harmonic potential term to the free action. Under the new renormalization group, adapted to the noncommutative geometry, this model turns out to be renormalizable to all orders in perturbation theory. Moreover it is {f asymptotically safe} at all orders in perturbation theory. The consequences of this results are discussed.

The theory of Quantum Mechanics requires \'completeness\', that is, we need to know the complete set of physically allowed states before we can reliably compute quantum mechanical amplitudes. Among these possible states are microscopic black holes, since they are valid solutions to Einstein\'s equations for the gravitational force. However, a quantum description of black holes requires a drastic revision of our notions of space and time, in particular if we were to accept the interpretation of their microstates as given by superstring theories. The logical foundations of our physical world view are touched upon here. A natural sounding solution of our problems here could come from a re-interpretation of what quantum mechanics really is. The first thing to dispose of should be all references to \'magic\' and \'mystery\' when dealing with quantum mechanics or string theory.

We analyze the trans-Planckian problem and its formulation in the context of cosmology, black-hole physics, and analogue models of gravity. In particular, we discuss the phenomenological approach to the trans-Planckian problem based on modified, locally Lorentz-breaking, dispersion relations (MDR). The main question is whether MDR leave an detectable imprint on macroscopic physics. In the framework of the semi-classical theory of gravity, this question can be unambiguously answered only through a rigorous formulation of quantum field theory on curved space with MDR. In this context, we propose a momentum-space analysis of the Green\'s function, which will hopefully lead to the correct renormalization of the stress tensor.

The problem of time is studied in a toy model for quantum gravity: Barbour and Bertotti\'s timeless formulation of non-relativistic mechanics. We quantize this timeless theory using path integrals and compare it to the path integral quantization of parameterized Newtonian mechanics, which contains absolute time. In general, we find that the solutions to the timeless theory are energy eigenstates, as predicted by the usual canonical quantization. Nevertheless, the path integral formalism brings new insight as it allows us to precisely determine the difference between the theory with and without time. This difference is found to lie in the form of the constraints imposed on the gauge fixing functions by the boundary conditions. In the stationary phase approximation, the constraints of both theories are equivalent. This suggests that a notion of time can emerge in systems for which the stationary phase approximation is either good or exact. As there are many similarities between this model of classical mechanics and general relativity, these results could provide insight to how time might be emergent in a theory of quantum gravity.

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.