My current research interests are centered around several aspects of classical and quantum gauge theories, in particular, theories which are invariant under the group of diffeomorphisms, such as topological field theories, BF-like models and gravity. In the classical setting, the approaches I am most familiar with are the constrained Hamiltonian analysis, the canonical covariant methods and multisymplectic geometry. In the quantum aspect, I have been involved in different studies of the framework known as the deformation quantization formalism or phase space quantum mechanics, and with non-perturbative canonical quantization methods, such as Loop Quantum Gravity (LQG). Related to this, I am very interested in the fundamental problem of making general relativity and quantum theory compatible in what is called as quantum gravity. One of the most interesting results obtained from LQG, is that the geometry behaves in a quantum way, and the spectra of geometric operators turn out to be discrete and non-commutative. In addition, we have been studying the LQG approach under the deformation quantization formalism. Under this formalism, the Dirac quantization rules are achieved by replacing the usual product of the algebra of smooth functions on the classical phase space with an associative non-commutative product, such that the resulting commutator is a deformation of the Poisson structure. Finally these quantization frameworks allow us to analyze quantum phenomena related to Relativistic Quantum Information, Quantum Field Theory on curved spacetimes and algorithms in quantum computing.

1. Quantum Algorithms and Quantum Computing
2. Deformation quantization
3. Quantum gravity
4. Relativistic Quantum Information

1. Application an Development of Computer Tehnologies in Mathematics and Teaching Research
2. Geometry of Dynamical Systems and Applications to Physics
3. Theoretical Physics and Quantization

1. La representación de lazos en el formalismo de cuantización por deformación
   21/05/2018 - 01/06/2021
   CONACyT-SEP de Ciencia Básica