Casimir energy for a coupled fermion-soliton system

Abstract

In this paper we compute the Casimir energy for a coupled fermion-pseudoscalar field system. In the model considered in this paper the pseudoscalar field is static and prescribed with two adjustable parameters. These parameters determine the values of the field at infinity (±θ0) and its scale of variation (µ). One can build up a field configuration with arbitrary topological charge by changing θ0, and interpolate between the extreme adiabatic and non-adiabatic regimes by changing µ. This system is exactly solvable and therefore we compute the Casimir energy exactly and unambiguously by using an energy density subtraction scheme. We show that in general the Casimir energy goes to zero in the extreme adiabatic limit, and in the extreme non-adiabatic limit when the asymptotic values of the pseudoscalar field properly correspond to a configuration with an arbitrary topological charge. Moreover, in general the Casimir energy is always positive and on the average an increasing function of θ0 and always has local maxima when there is a zero mode, showing that these configurations are energetically unfavorable. We also compute and display the energy densities associated with the spectral deficiencies in both of the continua, and those of the bound states. We show that the energy densities associated with the distortion of the spectrum of the states with E > 0 and E < 0 are mirror images of each other. We also compute and display the Casimir energy density. Finally we compute the energy of a system consisting of a soliton and a valance electron and show that the Casimir energy of the system is comparable with the binding energy.

An investigation of the Casimir energy for a fermion coupled to the sine-Gordon soliton with parity decomposition

“…(2) only contain the negative-energy states. However, in [61] we showed that the total negativeand positive-energy densities are exactly the mirror images of each other for the SESM. Therefore, the Casimir energy for that model can be computed only from the negative states or only the positive states, or the average of all of the states and the results are exactly the same in all cases.…”

“…The Casimir energy in the standard approach is obtained by computing the change in the zero-point energy of the field due to the presence of the boundary conditions or the background field. This energy for a Fermi field can be written as follows (see for example [61,62]):…”

“…Some authors compute the Casimir energy as the lowestorder quantum correction to the mass of the solitons including supersymmetric solitons [49][50][51][52][53][54][55][56][57][58][59][60][61]. Most of the models with solitons are not exactly solvable and the Casimir energy cannot be calculated directly.…”

Fermions on wobbling kinks: normal versus quasinormal modes

“…The system composed of a fermion interacting with background φ 4 kink is analytically solvable and exhibits bound states as well as scattering states [6][7][8]. The properties of fermion-kink systems, such as the fermion energy spectrum and Casimir energy, have been studied for many different models [9][10][11][12][13][14]. In these systems, it is also possible to consider the back-reaction of the fermion on the soliton, which modifies the kink profile, especially for strong coupling constants [15][16][17][18].…”

Fermions on wobbling kinks: normal versus quasinormal modes

“…The system composed of a fermion interacting with a background φ 4 kink is analytically solvable and exhibits bound states as well as scattering states [6][7][8]. The properties of fermion-kink systems, such as the fermion energy spectrum and Casimir energy, have been studied for many different models [9][10][11][12][13][14]. In these systems, it is also possible to consider the back-reaction of the fermion on the soliton, which modifies the kink profile, especially for strong coupling constants [15][16][17][18].…”