Uma introdução ao efeito Casimir no contexto do Campo de Dirac

Abstract

In order to study the Casimir effect for the fermionic field, it was first necessary to understand the Dirac equation: its motivation, mathematical and physical aspects. We solve such an equation in Cartesian coordinates, and obtain its normalized general solutions, that is, the free solutions. Furthermore, we consider the Lagrangian density referring to those free solutions and obtain the corresponding energy-momentum tensor of the Dirac field. Once these solutions are known, we promote the field to an operator with the so-called second quantization. Once this was done, it became possible to investigate some fundamental quantities about nature, arising from fluctuations in the quantum vacuum of the field. In this sense, our first approach consisted of analyzing the spin field subjected to the bag boundary condition, the results of which manifest our interest phenomenon, the Casimir effect. In our second approach, we subject the field to the anti-helix condition and make sure of the influence that topology has on the fluctuations of the fermionic quantum vacuum, by once again seeing the Casimir effect manifestation, this time known as the topological Casimir effect. Finally, it is worth emphasizing that the entire study contained in this work took place in a flat space-time, in (3 + 1) dimensions. To be more specific, the mathematical scenario considered is the Minkowski spacetime.