Quantum Gravity

During the last couple of years Dupuis, Freidel, Livine, Speziale and Tambornino developed a twistorial formulation for loop quantum gravity.
Constructed from Ashtekar--Barbero variables, the formalism is restricted to SU(2) gauge transformations.
In this talk, I perform the generalisation to the full Lorentzian case, that is the group SL(2,C).
The phase space of SL(2,C) (i.e. complex or selfdual) Ashtekar variables on a spinnetwork graph is decomposed in terms of twistorial variables. To every link there are two twistors---one to each boundary point---attached. The formalism provides a clean derivation of the solution space of the reality conditions of loop quantum gravity.
Key features of the EPRL spinfoam model are perfectly recovered.
If there is still time, I'll scatch my current project concerning a twistorial path integral for spinfoam gravity as well. 

The emergence of fractal features in the microscopic structure of space-time is a common theme in many approaches to quantum gravity. In particular the spectral dimension, which measures the return probability of a fictitious diffusion process on space-time, provides a valuable probe which is easily accessible both in the continuum functional renormalization group and discrete Monte Carlo simulations of the gravitational action. In this talk, I will give a detailed exposition of the fractal properties associated with the effective space-times of asymptotically safe Quantum Einstein Gravity (QEG). Comparing these continuum results to three-dimensional Monte Carlo simulations, we demonstrate that the resulting spectral dimensions are in very good agreement. This comparison also provides a natural explanation for the apparent conflicts between the short distance behavior of the spectral dimension reported from Causal Dynamical Triangulations (CDT), Euclidean Dynamical Triangulations (EDT), and Asymptotic Safety.

We relate the discrete classical phase space of loop gravity to the continuous phase space of general relativity. Our construction shows that the flux variables do not label a unique geometry, but rather a class of gauge-equivalent geometries. We resolve the tension between the loop gravity geometrical interpretation in terms of singular geometry, and the spin foam interpretation in terms of piecewise-flat geometry, showing that both geometries belong to the same equivalence class. We also establish a clear relationship between Regge geometries and the piecewise-flat spin foam geometries. All of this is based on arXiv:1110.4833.

Tensor models appear as the higher dimensional extension of the so-called matrix models describing 2D quantum gravity through the sum over triangulations of surfaces. In the light of the recent $1/N$ expansion for these tensor models, we uncover a new class of tensor models for 4D and 3D gravity which are renormalizable at all orders of perturbation theory. An overview of two papers, [arXiv:1111.4997 [hep-th]] and [arXiv:1201.0176 [hep-th]], on the renormalization of these tensor models and their beta function will be given.

Group field theories show up as a higher dimensional generalization of matrix models in background independent approaches to quantum gravity.
Their Feynman expansion generates simplicial complexes of all topologies weighted by spin foam amplitudes.
In this talk, we will present a dual formulation of these theories as non-commutative quantum fields theories, whose variables have a clear interpretation in terms of simplicial geometry. We will show that it gives a geometrically clear ways to define spin foam models for gravity which can be cast as

To study the continuum limit of a microscopic model of gravity we need microscopic observables that have a clear interpretation in terms of continuum geometry. In general the construction of such observables is notoriously difficult. In the model of causal dynamical triangulations (CDT) it is clear what the microscopic observables are, but at present the only known well-behaved observables with a continuum interpretation are spatial volumes. In this talk I will demonstrate what it takes to go beyond these by introducing the moduli as observables for CDT in 2+1 dimensions with spatial topology of the torus. Measurements of these observables using computer simulations provide valuable clues concerning the effective action describing CDT in the continuum. In particular I will present numerical evidence indicating that the effective kinetic term is well described by a modified Wheeler-De Witt metric like the one appearing in Horava-Lifshitz gravity.

Entanglement is a paradigmatic example of quantum correlations, a presumed reason for the superior performance of quantum computation and an obvious divider of states and processes into classical and quantum. In the last decade all these notions were challenged. Entanglement does not capture the totality of non-classical behavior. Quantum discord (in its different versions) is a more general measure of quantum correlations. It can be related to the advantage in some tasks like the extraction of work from a Szilrad heat engine using Maxwell's demons with various resources. The discord turns out to be the difference between the work extracted from a given bipartite system using a a global and a local strategy. Different strategies relate to different definitions of discord, but all definitions agree on zero, so "classical" systems are universal to all heat engines. One way of identifying a task as quantum or classical is by examining the quantum resources required to implement it. This can be done by examining the entanglement required for a LOCC implementation of the task. Creation (or non-creation) of entanglement as a result of its implementation is not enough to identify these resources. An example is a bi-local implementation of an entangling quantum gate (C-NOT) by LOCC with unentagled input and output states. This lack of entanglement does not guarantee the LOCC implementability. A method to determine if entanglement resources are required is to track the change in quantum discord during the process.

We construct the q-deformed spinfoam vertex amplitude using Chern-Simons theory on the boundary 3-sphere of the 4-simplex. The rigorous definition involves the construction of Vassiliev-Kontsevich invariant for trivalent knot graph. Under the semiclassical asymptotics, the q-deformed spinfoam amplitude reproduce Regge gravity with cosmological constant at nondegenerate critical configurations.

In general relativity, the fields on a black hole horizon are obtained from those in the bulk by pullback and restriction. In quantum gravity, it would be natural to obtain them in the same manner. This is not fully realized in the quantum theory of isolated horizons in loop quantum gravity, in which a Chern-Simons phase space on the horizon is quantized separately from the bulk. I will outline an approach in which the quantum horizon degrees of freedom are simply components of the quantized bulk degrees of freedom. A condition is imposed on the quantum states to encode the existence of a horizon. I will present evidence that solutions to this condition have properties on the horizon that are remarkably similar to those of Chern-Simons theory. Instrumental in formulating the horizon condition are novel flux operators that use the Duflo isomorphism and seem to represent some type of quantum deformed SU(2). I will review their definition and summarize what I know about their properties.

I briefly introduce the recently introduced idea of relativity of locality, which is a consequence of a non-flat geometry of momentum space. Momentum space can acquire nontrivial geometrical properties due to quantum gravity effects. I study the relation of this framework with noncommutative geometry, and the Quantum Group approach to noncommutative spaces. In particular I'm interested in kappa-Poincaré, which is a Quantum Group that, as shown by Freidel and Livine, in the 1+1D case emerges as the symmetry of effective field theory coupled with quantum gravity, once that the gravitational degrees of freedom are integrated out. I'm interested in particular in the Lorentz covariance of this model which is present, but is realized in a nontrivial way. If I still have time, I'll then speak about an under-course general study of the Lorentz covariance of Relative Locality models.