Cuantización por Deformación de la 2-Brana Bosónica

Abstract

In this thesis we address the problem of applying the deformation quantization formalism to treat extended objects, in particular, we analize the bosonic 2-brane case with spherical symmetry. The thesis is divided in two parts. The first one consists in a review of the formalism treatment of systems with a finite number of degrees of freedom, where the Stratonovich-Weyl quantizer, the Moyal ?M product and the Wigner function were obtained. Subsequently, we study its extension to systems with an infinite number of degrees of freedom (fields), finding the analog of the Stratonovich-Weyl quantizer, the Moyal ?M product and the corresponding Wigner functional in order to describe the quantum behaviour of these systems. In the second part, we dealt with the bosonic string in the canonical quantization and then with the deformation quantization method. These results are used to work out the p-dimensional extended objects embedded in D-dimensional Minkowski spacetime. We have also found the Stratonovich-Weyl quantizer, the Moyal ? product and the integral form of the Wigner functional for these systems. As an application, we treat the bosonic 2-brane under this method. Because in this problem is hard to fix a gauge, we consider the symmetrically spherical case. For this problem, we obtain the Wigner function using the Hartle-Hawking, Linde and Vilenkin boundary conditions by means of numerical methods and we analyze each case. The results of this thesis are the first steps in the treatment of high dimensional extended objects as well as for more complicated gauges using the deformation quantization formalism.