Bound state problem for a Dirac particle in an external static charge distribution in (1+1)-dimensions

TL;DR

This paper investigates the self-interaction effects of a Dirac particle in a static external charge distribution in (1+1)-dimensions, demonstrating that self-interaction can stabilize bound states in a system with zero total charge.

Contribution

It introduces a nonlinear Dirac equation including self-potential effects and shows that self-interaction cancels the linearly rising potential, enabling stable bound states.

Findings

Self-interaction cancels the linearly rising potential.

Stable bound states are possible with zero total charge.

The system exhibits confinement of the Dirac particle.

Abstract

We study the self-interaction effects for the Dirac particle moving in an external field created by static charges in (1+1)-dimensions. Assuming that the total electric charge of the system vanishes, we show that the asymptotically linearly rising part of the external potential responsible for nonexistence of bound states in the external field problem without self-interaction is cancelled by the self-potential of the zero mode of the Dirac particle charge density. We derive the Dirac equation which includes the self-potential of the non-zero modes and is nonlinear. We solve the spectrum problem in the case of two external positive charges of the same value and prove that the Dirac particle and external charges are confined in a stable system.