Fractional Charge and Spectral Asymmetry in 1-d: a Closer Look*

The physics of charge fractionalization is studied using a simple and physical approach. The normal ordered charge is related to the Atiyah-Patodi-Singer invariant, and the physical interpretation of the spectral asymmetry is clarified in the presence of a continuous spectrum. By introducing the quantity B(E) which is a ratio of Jost-type determinants we relate the asymmetry to the phase and zeros or poles of B(E). The fractional part of the charge is determined by the high energy behavior of the phase and the integer part is related to the spectral flow. We give simple examples showing that only the fractional part of the charge is a topological invariant; the integer part is determined by local properties of the background fields. Submitted to Physical Review D * Work eupported by the Department of Energy, contract DE AC03 76SF00515