Mapeamento de sistemas quânticos invariantes de forma via deformação de campos clássicos.

Abstract

In our work, we study systems of ordinary real scalar fields. We analyze the static configurations, which lead us to nonlinear second order differential equations. The solutions of these equations, called defects, may have a topological character, leading to two types of defect: topological and non-topological. Next, we talk about the Bogomol'nyi method, which, for non-negative potentials, enables us to solve the equations of motion by means of a first-order differential equation. We study the stability of the solution through perturbations to the linear order around it, which makes it possible to separate the equation of motion from the perturbation field, in temporal part and spatial part. From this second part we arrive at what we call the fluctuation operator, which governs the quantum oscillations around the classical solution. If the procedure can be inverted, that is, it is possible to construct classical models of fields, starting from the stability equation, which is of paramount importance for the purpose of our research. In our work we reconstructed the classical models for several quantum systems, identifying their Hamiltonian as flotation operators. Another point discussed was the deformation method. Through it it is possible to perform the mapping between scalar field potentials via a deformation function. This new potential, the deformed potential, is described in terms of the initial potential and its solutions. The method allows to generate a multitude of new models with characteristics different from the originals, but allows, to some extent, to control characteristics such as energy and width of defects. We then review supersymmetric quantum mechanics and their potential supersymmetric partners. These potentials are obtained through the factorization of the Hamiltonians, being possible to create an entire hierarchy of Hamiltonians, which presents a degeneration of the energy spectrum. Certain potential partners have a special feature, called form invariance. Form invariance is obtained by making a change in the potential parameters, which leads to a potential of the same shape, but with an increase of energy. This property is a condition of integrability, which ensures that quantum systems based on form invariant potentials are solvable. In the present work we seek to establish a new type of mapping between quantum potentials with form invariance, using classical field deformation. Another very interesting feature of the shape invariant potentials, which was the starting point for the present work, is that all known invariants can be mapped to each other by means of canonical transformations, parameters and point coordinates. For this, we mapped the supersymmetric potential, reconstructed a field model for each potential, and mapped the field models using the deformation method. In this way, we achieve a closed relationship involving a classical and a quantum part.