Vacuum charge fractionlization re-examined

TL;DR

This paper re-examines vacuum charge fractionalization in a one-dimensional Dirac model, clarifying that total charge conservation is maintained unless zero modes are present, and discusses related conceptual issues.

Contribution

It provides a detailed analysis showing vacuum charge fractionalization requires zero modes and clarifies misconceptions about charge conservation and regularization.

Findings

Vacuum charge fractionalization occurs only with zero modes.

Total charge remains conserved despite fractionalization.

Regularization and continuum limit issues are critical in the analysis.

Abstract

We consider a model of a quantized fermion field that is based on the Dirac equation in one dimensional space and re-examine how the fermion number of the vacuum, or the vacuum charge, varies when an external potential is switched on. With this model, fractionization of the vacuum charge has been illustrated in the literature by showing that the external potential can change the vacuum charge from zero to a fractional number. Charge conservation then appears violated in this process. This is because the charge that has been examined in this context is only a part of the total charge of the vacuum. The total charge is conserved. It is not fractionalized unless the Dirac equation has a zero mode. Two other confusing aspects are discussed. One is concerned with the usage of the continuum limit and the other with the regularization of the current operator. Implications of these aspects of the vacuum problem are explored.