S.S. Gousheh, S.S. Mousavi, L. Shahkarami
Phys. Rev. D 90, 045027 (2014)
Publication year: 2014

Abstract

We investigate the vacuum polarization and Casimir energy of a Dirac field coupled to a scalar potential in one spatial dimension. Both of these effects have a common cause, which is the distortion of the spectrum of the Dirac field due to its coupling with the background field. Choosing the potential to be a symmetrical square well renders the problem exactly solvable, and we can obtain the whole spectrum of the system analytically. We show that the total number of states and the total density remain unchanged as compared with the free case, as one expects. Furthermore, since there is a reflection symmetry between positive- and negative-energy eigenstates of the fermion, the total density and the total number of negative and positive states remain unchanged, separately. This, along with the fact that there is no zero mode, mandate that the vacuum polarization in this model is zero for any choice of the parameters of the potential. It is important to note that although the vacuum polarization is zero due to the symmetries of the model, the Casimir energy of the system is not zero in general. In the graph of the Casimir energy as a function of the depth of the well, there is a maximum approximately when the bound energy levels change direction and move back towards their continuum of origin. The Casimir energy for a fixed value of the depth is an almost linear increasing function of the width. Moreover, the Casimir energy density (the energy density of all the negative-energy states) and the energy density of all the positive-energy states are exactly the mirror images of each other. Finally, we compute the total energy of a valence fermion present in the lowest positive-energy fermionic bound state. We find that taking into account the Casimir energy does not result in the appearance of any local minima in the graphs of the total energy as a function of the parameters of the model, and this is in sharp contrast to the cases where there are levels crossing the line E=0.