Quantum Gravity

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.

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.

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.

There is a deep relation between Loop Quantum Gravity and notions from category theory, which have been pointed out by many researchers, such as Baez or Velhinho. Concepts like holonomies, connections and gauge transformations can be naturally formulated in that language. In this formulation, the (spatial) diffeomorphisms appear as the path grouopid automorphisms. We investigate the effect of extending the diffeomorphisms to all such automorphisms, which can be viewed as \"distributional diffeomorphisms\". We also give a notion of \"categorial holonomy-flux-algebra\", and present the construction of the automorphism-invariant Hilbert space for abelian gauge groups, which will be entirely combinatorial.