Quantum Gravity

I will review some problems of the black hole paradigm and explore other
possibilities for the final state of stellar collapse other than an evaporating
black hole. In particular I will use the so-called transplanckian problem as a
guide in this search for a compelling scenario for the evaporation of
ultracompact objects.

I will recall the main motivations for considering spin foam models in their Group Field Theory (GFT) versions, which are quantum field theories defined on group manifolds. As for any other quantum field theory, a fully consistent definition of the latter must involve renormalization. I will briefly review a specific class of GFTs, called tensorial, for which progress in this direction has recently been possible. A new just-renormalizable model, in three dimensions and on the SU(2) group, will be presented. Interestingly, it includes the geometric constraint of the Boulatov model, and might as such be related to Euclidean quantum gravity in three dimensions. Furthermore, this opens the way to a similar analysis of current 4d gravity spin foam models.

Tensor models are generalization of matrix models, and are studied as discrete models for quantum gravity for more than two-dimensions. Among them, the rank-three tensor models can be interpreted as theories of dynamical fuzzy spaces, and they generally have the feature of
respecting symmetries. In this talk, after briefly reviewing some results on Euclidean models such as spontaneous generation of fuzzy spaces and Euclidean general relativity respecting the diffeomorphism symmetry on them, I will present a way to introduce “local” time to the rank-three tensor models by constructing “local” Hamiltonians. The consistency among the multiple ways of local time evolutions is guaranteed by the on-shell closure of the constraint algebra among the
local Hamiltonians and the symmetry generators of the tensor models. The constraint algebra is shown to approach the DeWitt algebra in a formal continuum limit. I will also discuss the two-fold uniqueness of the local Hamiltonians, and will briefly show some preliminary results on the quantization.

We describe of the evaporation process as driven by the dynamical evolution of the quantum gravitational degrees of freedom resident at the horizon, as identified by the Loop Quantum Gravity kinematics. Using a parallel with the Brownian motion, we interpret the first law of quantum dynamical horizon in terms of a fluctuation-dissipation relation applied to this fundamental discrete structure. In this way, the horizon evolution is described in terms of relaxation to an equilibrium state balanced by the excitation of Planck scale constituents of the horizon. We show how from this setting the emergence of several conservative scenarios for the final stage of the evaporation process can be microscopically derived. Namely, the leakage of part of the horizon quantum geometry information prior to the Planckian phase and the stabilization of the hole surface shrinkage forming a massive remnant, which can eventually decay, are shown to take place.

It is known that the entanglement entropy of quantum fields on the black hole background contributes to the Bekenstein-Hawking entropy,and that its divergences can be absorbed into the renormalization of gravitational couplings. By introducing a Wilsonian cutoff scale and the concepts of the renormalization group, we can expand this observation into a broader framework for understanding black hole entropy. At a given RG scale, two contributions to the black hole entropy can be identified: the "gravitational" contribution coming from the running effective gravitational action, and the entanglement entropy of the quantum degrees of freedom below the cutoff scale. At different RG scales the balance is different, though the total black hole entropy is invariant. I will describe this picture for free fields, considering both minimal and non-mininal coupling, and discuss the extension to interacting fields and the difficulties it raises.

The combinatorial problems associated with the counting of black hole states in loop quantum gravity can be analyzed by using suitable generating functions. These not only provide very useful tools for exact computations, but can also be used to perform an statistical analysis of the black hole degeneracy spectrum, study the interesting substructure found in the entropy of microscopic black holes and its asymptotic behavior for large horizon areas. The methods that will be described are relevant for the discussion of the thermodynamic limit for black holes in the area canonical ensemble. This is an important issue in order to understand sub-leading corrections to the Bekenstein-Hawking law.

There are several fundamental predictions of quantum field theory, such as Hawking radiation (i.e., black hole evaporation) or the Sauter-Schwinger effect (i.e., electron-positron pair creation out of the quantum vacuum by a strong electric field), which have so far eluded direct experimental verification.
However, it should be possible to gain some experimental access to these effects via suitable condensed matter analogues. In this talk, some possibilities for reproducing such fundamental
quantum effects in the laboratory are discussed.

The space of convex polyhedra can be given a dynamical structure. Exploiting this dynamics we have performed a Bohr-Sommerfeld quantization of the volume of a tetrahedral grain of space, which is in excellent agreement with loop gravity. Here we present investigations of the volume of a 5-faced convex polyhedron. We give for the first time a constructive method for finding these polyhedra given their face areas and normals to the faces and find an explicit formula for the volume. This results
in new information about cylindrical consistency in loop gravity and a couple of surprises about polyhedra. In particular, we are interested in discovering whether the evolution generated by this volume is chaotic or integrable as this will impact the interpretation of the spin network basis in loop gravity. 

In spacetime physics any set C of events—a causal set—is taken to be partially ordered by the relation £ of possible causation: for p, q Î C,   p £ q means that q is in p’s future light cone. Fotini Markopoulou has proposed that the causal structure of spacetime itself be represented by “sets evolving over C” —that is, in essence, by the topos Set C of presheaves on Cop.  In this talk I am going to show how Set  C may be effectively replaced by a certain model V(H) of intuitionistic set theory—frame-valued set theory—with (I hope) illuminating results. In particular, Markopoulou’s idea of viewing the universe “from the inside” will amount to placing oneself “inside” V(H). I will also sketch the role played in this framework by covering schemes and sheaves.

Recent progress in the quantization of nonrenormalizable scalar fields has found that a suitable non-classical modification of the ground state wave function leads to a result that eliminates term-by-term divergences that arise in a conventional perturbation analysis. After a brief review of both the scalar field story and the affine quantum gravity program, examination of the procedures used in the latter surprisingly shows an analogous formulation which already implies that affine quantum gravity is not plagued by divergences that arise in a standard perturbation study. Additionally, guided by the projection operator method to deal with quantum constraints, trial reproducing kernels are introduced that satisfy the diffeomorphism constraints. Furthermore, it is argued that the trial reproducing kernels for the diffeomorphism constraints also satisfy the Hamiltonian constraint as well. 
This talk is based on arXiv:1203.0691 .

We propose a new method of unifying gravity and the Yang-Mills fields by introducing a spin-foam model. We realize a unification between an SU(2) Yang-Mills interaction and 3D general relativity by considering a constrained Spin(4) ~SO(4) Plebanski action. The theory is quantized a la spin-foam by implementing the analogue of the simplicial constraints for the Spin(4) symmetry, providing a way to couple Yang-Mills fields to spin-foams. We also present a way to recover 2-point correlation functions between the connections as a first way to implement scattering amplitudes  between particle states. We conclude with speculations about extension of the model to 4D and  incorporate a newly developed model of Dark Energy.

I will review the construction of lattice theories which maintain one or more exact supersymmetries for non zero lattice spacing concentrating in particular on the case of N=4 super Yang-Mills. Such lattice theories may be studied using Monte Carlo techniques borrowed from lattice QCD and can be used to explore issues in holography. In three dimensions the same constructions can be used to formulate a topological theory of gravity which we argue is equivalent to Witten's Chern Simons theory.