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 ${\stackrel{^}{Q}}_{s}={\stackrel{^}{Q}}_{f}\ensuremath{-}〈0|{\stackrel{^}{Q}}_{f}|0〉$, where ${\stackrel{^}{Q}}_{f}$ 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 $L$. $|0〉$ is the ground state of the system in the absence of solitons. It is shown that the mean-square fluctuation of ${\stackrel{^}{Q}}_{s}$ taken about its fractional average value ${Q}_{s}$ vanishes as $O(\frac{{\ensuremath{\xi}}_{0}}{L})$ for $Lgg{\ensuremath{\xi}}_{0}$, where ${\ensuremath{\xi}}_{0}$ is the width of the soliton. Thus, as $L\ensuremath{\rightarrow}\ensuremath{\infty}$, 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 $\mathrm{exp}(\frac{\ensuremath{-}L}{{\ensuremath{\xi}}_{0}})$ as $L\ensuremath{\rightarrow}\ensuremath{\infty}$. Nonetheless, the charge of the entire system, including all solitons, is integral.