Quantization of non-Lagrangian systems and noncommutative quantum mechanics

Author(s)

Vladislav Kupriyanov

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Física (IF/SBI)
• Defense date: 2009-03-23

Examining board members

Dmitri Maximovitch Guitman; Joao Carlos Alves Barata; Marcelo Otavio Caminha Gomes; Aleksandr Nikolaievich Pinzul; Francisco Toppan

• Advisor: Dmitri Maximovitch Guitman

Abstract

We present here three interrelated problems: quantization of non-Lagrangian theories, noncommutative quantum mechanics (NCQM) and a constructions of the star product trough the the Weyl ordering. In the context of the first problem an approach to the canonical quantization of systems with non-Lagrangian equations of motion is proposed. We construct an action principle for an equivalent first-order equations of motion. There exists an ambiguity (not reducible to a total time derivative) in associating a Lagrange function with the given set of equations. We give a complete description of this ambiguity. The proposed scheme is applied to quantization of a general quadratic theory. Also the quantization of a damped oscillator and a radiating point-like charge is constructed. In the context of NCQM we propose a path integral formulation of relativistic NCQM and construct a noncommutative generalization of superparticle action. After quantization, the proposed action reproduces the Klein-Gordon and Dirac equations in the noncommutative field theories. In the context of the third problem we develop an approach to the deformation quantization on the real plane with an arbitrary Poisson structure which based on Weyl symmetrically ordered operator products. A simple and effective iterative procedure of the construction of star products is formulated. This procedure allowed us to calculate the third and the fourth order star products. Modulo some cohomology issues which we do not consider here, the method gives an explicit and physics-friendly description of the star products. (AU)