Fractional total-charge Eigenvalues for a fermion in a finite one-dimensional box

Abstract

We show that the Jackiw-Rebbi and Goldstone-Wilczek hamiltonians describing a two-component Dirac fermion in bounded background field and confined to a finite one-dimensional box has self-adjoint extensions which are labelled by a 2×2 unitary matrix U. The fractional part of the eigenvalue of the total vacuum charge N₀ is found to be given by α/π where det U=−exp(21α). The (non-periodic) boundary condition U=1 models the Jackiw-Rebbi infinite space solitonic situation leading to half-odd integral N₀.