A energia de Casimir para o campo eletromagnético à temperatura finita com quebra de simetria de Lorentz

Abstract

The Casimir effect, theoretically predicted in 1948 by the Dutch physicist Hendrik Brugt Gerhard Casimir, demonstrates that two perfectly conducting parallel plates undergo the action of an attractive force when introduced into a vacuum. In the calculation, Casimir considered the fluctuations of the quantum vacuum, and concluded that the effect comes from the imposition of boundary conditions to the electromagnetic vacuum. Since its publication, other geometries and other fields have been calculated, and new results have been obtained, in spherical geometry there is a repulsive force, unlike parallel plates which present attractive force, the massive field has a smaller effect than the zero-mass scalar field, in fact, the theory can still be tested in new areas. To study the Casimir effect at finite temperature, consider the approach presented by Takahashi and Umezawa known as Thermal Field Dynamics, or simply TFD formalism. To introduce temperature effects through this formalism, two fundamental elements are needed, the doubling of the Hilbert space and the Bogoliubov transformations. Doubling is done using the tilde conjugation rules. A Bogoliubov transformation consists of rotating between the operators of the two spaces, the usual Hilbert space and the tilde space, to introduce the effects of temperature. With the advent of contemporary physics, it is almost a consensus among physicists that the Lorentz symmetry fundamental, in relativity, needs to be broken at the energy upper scale of the standard model, the breaking of the Lorentz symmetry in this case is due to the coupling inserted in the Lagrangian scale. Here, we coupled an ether field to the electromagnetic field and observed that the ether interacts with the electromagnetic field distorting its metric and violating the Lorentz symmetry, in our calculations the modified electromagnetic field showed symmetry between the spatial directions ( x, y, z) by the Stefan-Boltzmann law violating symmetry in the direction of time (t). Due to the symmetries involved in this system, the Casimir effect for the modified electromagnetic field at zero temperature and finite temperature does not depend on the constant (κ) for the spatial directions (x, y) and also demonstrated symmetry between these directions, showing violation for temporal (t) and spatial (z) direction.