Fermion Determinants in Static, Inhomogeneous Magnetic Fields

The renormalized fermionic determinant of QED in 3 + 1 dimensions, $\mbox{det}_{{ren}}$, in a static, unidirectional, inhomogeneous magnetic field with finite flux can be calculated from the massive Euclidean Schwinger model's determinant, $\mbox{det}_{{Sch}}$, in the same field by integrating $\mbox{det}_{{Sch}}$, over the fermion's mass. Since $\mbox{det}_{{ren}}$ for general fields is central to QED, it is desirable to have nonperturbative information on this determinant, even for the restricted magnetic fields considered here. To this end we continue our study of the physically relevant determinant $\mbox{det}_{{Sch}}$. It is shown that the contribution of the massless Schwinger model to $\mbox{det}_{{Sch}}$ is cancelled by a contribution from the massive sector of QED in 1 + 1 dimensions and that zero modes are suppressed in $\mbox{det}_{{Sch}}$. We then calculate $\mbox{det}_{{Sch}}$ analytically in the presence of a finite flux, cylindrical magnetic field. Its behaviour for large flux and small fermion mass suggests that the zero-energy bound states of the two-dimensional Pauli Hamiltonian are the controlling factor in the growth of $\ln \mbox{det}_{{Sch}}$. Evidence is presented that $\mbox{det}_{{Sch}}$ does not converge to the determinant of the massless Schwinger model in the small mass limit for finite, nonzero flux magnetic fields.