Fractional charge, a sharp quantum observable

Abstract

The magnitude of quantum fluctuations of the charge of a fractionally charged soliton is calculated. The soliton charge operator is defined as ${\hat{Q}}_s={\hat{Q}}_f-〈0\left|{\hat{Q}}_f\right|0\mathrm{}〉$, where ${\hat{Q}}_f$ is the integral of the charge-density operator sampled by a function $f$ peaked at the position of the soliton, and falling smoothly to zero on a scale $L$. $|0\mathrm{}〉$ is the ground state of the system in the absence of solitons. It is shown that the mean-square fluctuation of ${\hat{Q}}_s$ taken about its fractional average value $Q_s$ vanishes as $O\left(\frac{{\xi }_0}{L}\right)$ for $L>>{\xi }_0$, where ${\xi }_0$ is the width of the soliton. Thus, as $L\to \infty$, the soliton is an eigenfunction of the charge operator with fractional eigenvalue. We also show that the portion of the charge fluctuations that are due to the soliton falls as $\exp \left(\frac{-L}{{\xi }_0}\right)$ as $L\to \infty$. Nonetheless, the charge of the entire system, including all solitons, is integral.