I am very interested in the fundamental problem of making quantum theory and relativity compatible in what is known as quantum gravity. One of the most conservative approaches to this problem is the implementation of `conventional' quantization methods to the gravitational field. This canonical quantization program has become in recent years a serious candidate for a quantum theory of geometry (without any other matter field), thanks to the introduction of connection and loop variables by Ashtekar, Rovelli and Smolin. One of the most remarkable results coming from this approach is the particular picture of the geometry at the Planck scale. Excitations of geometry are one dimensional, polymer-like, and the spectra of geometric operators turn out to be discrete. Furthermore, the resulting quantum geometry happens to be intrinsically non-commutative: the operators associated, for instance, to the area of two intersecting surfaces do not commute. This fact posses very interesting challenges: How is one to recover a commutative, smooth geometry in certain limit?, What are the implications of this non-commutativity in, say, the definition of `coordinates' on the manifold? Thus, my current research is in the direction of answering such questions. In particular, I would like to understand the role that the non-commutativity plays in the whole formalism. The other question I am interested in is the macroscopic/semi-classical limit of the theory. Historically, the first attempts were the so called weave states, where the state was defined over a complicated loop that was woven to reproduce a smooth geometry at certain scales. In this direction I have proposed some simple and preliminary ``Gaussian'' weaves, that might approximate flat space. It is important to understand the relation between this states and other proposals such as the coherent states of Thiemann and Winkler and the Statistical Geometry approach of Ashtekar and Bombelli. Still, the basic question of what a semi-classical state is and how one should construct them is still open. New proposals, in particular in the direction of linking non-perturbative states with Fock excitations have recently appeared. It is important to understand its relevance and interconnection with the other approaches. In particular it is very important to understand how one is going to address the issue of constructing dynamical semi-classical states. I have been studying the problem of constructing dynamical semi-classical states and comparing them to kinematical `coherent' states, for simple systems. I plan to to continue working on these problems in the near future.
In order to gain a full understanding of the conceptual problems that are common to the quantum theory of gravity, it is important to consider simpler models. Usually, this implies performing a symmetry reduction or considering theories in less than four dimensions. In this spirit, I have been involved in the study of the quantum collapse of a spherically symmetric dust shell, and in the midi-superspace quantization of Gowdy models. The Gowdy model is the simplest, inhomogeneous, closed cosmological model. It has been extensively studied at the quantum level. We have shown that a very natural quantization is not unitarily implementable. This Gowdy model has turned out to be of interest to the formulation of the Schrodinger representation on a curved spacetime and has lead to an example of quantization ambiguities in field theory that were not explicitly constructed in the literature.
Classical General Relativity and Black Holes.
The quantum geometry formalism has also been applied to the horizons of black holes in order to compute the Bekenstein-Hawking entropy. In order to do this, it was necessary to restrict attention to the `black hole' sector of phase space. This lead to what has become known as Isolated Horizons. The idea is to give geometrically motivated boundary conditions on the theory which allow for a well defined action principle and Hamiltonian description. This formalism has in particular, generalized the laws of black hole mechanics to situations in which the horizon is isolated, but the exterior region is allowed to be dynamical. I have been interested in the formalism when non-linear matter fields are considered. This has lead to an understanding of the rich interplay between the solitons in, say, EYM system and the colored black holes. The formalism is also well suited for posing uniqueness (`no-hair') conjectures for the existence of solutions. More recently, I have considered isolated horizons for theories where the matter is non-minimally coupled to gravity, where the treatment based on Noether charges tells us that the Black Hole entropy is not proportional to the area but it also depends on the matter fields at the horizon . The IH formalism is able to incorporate this situations. I have also been studying isolated horizons when there are two horizons present such as a Schwarzschild-de Sitter spacetime where there is a BH and a cosmological horizon.
Open problems I am interested in include the relation to the Brown-York formalism and Euclidean Techniques both in 2+1 and 3+1 dimensions. One would also like to have a precise definition of what are called `horizon charges' and of the uniqueness and completeness conjectures. These is especially important in view of recent numerical calculations that contradict an uniqueness conjecture as originally stated. This has important consequences in the IH formalism and should be explored.
I am also interested in other related problems involving geometry and quantum theory. Among these, I have been working in the geometrical formulation of quantum mechanics, and, in particular, in the incorporation of the superposition principle into the formalism. This is my view a very important issue that should be addressed, specially since the superposition principle it at the forefront of the quantum theory and is naively excluded from the geometric formulation of the theory where the space of states is a non-linear manifold.
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