Quantum Gravity

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.

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.

I discuss the status of Quantum Gravity Phenomenology, focusing separately on the 3 key areas: ability to discover, ability to constrain, and ability to falsify. And I stress the importance of adopting carefully taylored test theories as a remedy to difficulties encountered when comparing experimental evidence to theory evidence.

I discuss how physics beyond the Planck scale and before inflation might leave an imprint on the primordial spectrum. There are interesting limitations connected with the information paradox that suggests unexpected new ways to test ideas on quantum gravity.

We use the example of inflationary physics to discuss the possibility that short distance physics might be imprinted on long-distance observables. In particular, we focus on issues involving decoupling in field theory.

If string theory is the correct theory which unifies gravity with the other forces of nature at a quantum level, it should determine the evolution of the earliest stages of the universe. I will discuss how stringy signatures of this early phase may be visible in current cosmological observations.