Group-theoretical quantization of non-linear systems and dissipative systems

Abstract

A group-theoretical analysis of the non-Abelian Stueckelberg mass-generating mechanism leads to a quantum theory for electro-weak interactions, formulated in the framework of a Group Approach to Quantization, which departs from standard perturbation theory, and in which there is no need for a Higgs boson. Non-canonical basic commutation relations are proposed, and the Goldstone-like boson sector, described by a partial-trace non-linear sigma model, is carefully analyzed. A Hamiltonian operator, respecting the Hilbert space of states, is given in terms of the basic operators. In the second part, we develop a quantum version of the classical Arnold transformation, which establishes the correspondence among the quantum theories of classically linear theories. It is used, on the one hand, to construct Hermite-Gauss and Laguerre-Gauss states for the free particle in one, two and three dimensions, and, on the other hand, to study the Caldirola-Kanai model for the damped harmonic oscillator on symmetry grounds. The inclusion of the time symmetry in this system, algebraically, leads to the Bateman dual system, for which a group law is constructed and corresponding quantum representations, with first and second order Schrödinger equations, are given.