Casimir Energy for a Coupled Fermion-Soliton System

TL;DR

This paper calculates the Casimir energy for a solvable fermion-pseudoscalar system with adjustable parameters, revealing its behavior across different regimes and its relation to topological charge and bound states.

Contribution

It provides an exact computation of Casimir energy in a coupled fermion-soliton model with variable parameters, exploring its dependence on topological charge and adiabaticity.

Findings

Casimir energy approaches zero in the adiabatic limit.

Casimir energy is generally positive and increases with .

Configurations with zero modes are energetically unfavorable.

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 \textit{static} and \textit{prescribed} with two adjustable parameters. These parameters determine the values of the field at infinity ($\pm \theta_0$) and its scale of variation ($\mu$). One can build up a field configuration with arbitrary topological charge by changing $\theta_0$, and interpolate between the extreme adiabatic and non-adiabatic regimes by changing $\mu$. 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 $\theta_0$ 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.