VACUUM POLARIZATION BY SOLITONS IN (1+1)-DIMENSIONS

We have computed the vacuum polarization of fermions by solitons of any topological charge in (1+1) dimensions by a modification of the adiabatic method of Goldstone and Wilczek. We have obtained solutions which continuously interpolate between the adiabatic and nonadiabatic regimes and have investigated the effect of the scale of variation of the solitons on the vacuum polarization (VP) that they induce and have verified that energy level crossing occurs only for sharply varying solitons. However we have shown explicitly that in all cases, every time the topological charge (TC) of the system increases by one, one unit of fermion number escapes from the system, thereby showing that the adiabatic results of Goldstone and Wilczek hold even in the nonadiabatic case. Hence we have verified that in the adiabatic case VP = -TC for all TC. However, we have obtained the surprising result that even in the nonadiabatic case, the solitons will polarize the vacuum provided that they have a high enough TC (although in this case \VP\ < \TC\) only the infinitely sharp solitons can never polarize the vacuum. We have also shown that, during the entire process of building up a soliton of any TC, the total ''number'' of eigenfunctions of the Hamiltonian is conserved, the total spatial density of the spectrum remains uniform and the eigenfunctions remain complete.