Fractional charge, a sharp quantum observable

The magnitude of quantum fluctuations of the charge of a fractionally charged soliton is calculated. The soliton charge operator is defined as ^Qₛ=^Qf-〈0|^Qf|0〉, where ^Qf 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 ^Qₛ taken about its fractional average value Qₛ vanishes as O(ξ₀L) for L>>ξ₀, where ξ₀ is the width of the soliton. Thus, as L→∞, 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(-Lξ₀) as L→∞. Nonetheless, the charge of the entire system, including all solitons, is integral.