The renormalized fermionic determinant of QED in 3 + 1 dimensions, det ren , in a static, unidirectional, inhomogeneous magnetic field with finite flux can be calculated from the massive Euclidean Schwinger model's determinant, det Sch , in the same field by integrating det Sch , over the fermion's mass.Since 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 det Sch . It is shown that the contribution of the massless Schwinger model to det Sch is cancelled by a contribution from the massive sector of QED in 1 + 1 dimensions and that zero modes are suppressed in det Sch . We then calculate 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 det Sch . Evidence is presented that det Sch does not converge to the determinant of the massless Schwinger model in the small mass limit for finite, nonzero flux magnetic fields.

A representation for the fermionic determinant of the massive Schwinger model, or QED 2 , is obtained that makes a clean separation between the Schwinger model and its massive counterpart. From this it is shown that the index theorem for QED 2 follows from gauge invariance, that the Schwinger model's contribution to the determinant is canceled in the weak field limit, and that the determinant vanishes when the field strength is sufficiently strong to form a zero-energy bound state.

We state the connection between the fermion determinant in four-dimensional QED (QED,) and the massive Schwinger model, QED,, for the case of smooth, polynomial-bounded, unidirectional magnetic fields. Using the diamagnetic bound on the fermion determinant in QED,, we obtain an upper bound on the fermion determinant in QED, for this class of fields. Using Kato's inequality, we obtain an upper bound on the one-loop effective action in scalar QED, for smooth, polynomial-bounded but otherwise general fields with fast decrease at infinity. PACS number(s): 12.20.D~

No abstract

An exact representation of the Euclidean fermion determinant in two dimensions for centrally symmetric, finite-ranged Abelian background fields is derived. Input data are the wave function inside the field's range and the scattering phase shift with their momenta rotated to the positive imaginary axis and fixed at the fermion mass for each partial-wave. The determinant's asymptotic limit for strong coupling and small fermion mass for square-integrable, unidirecitonal magnetic fields is shown to depend only on the chiral anomaly. The concept of duality is extended from one to twovariable fields, thereby relating the two-dimensional Euclidean determinant for a class of background magnetic fields to the pair production probability in four dimensions for a related class of electric pulses. Additionally, the "diamagnetic" bound on the two-dimensional Euclidean determinant is related to the negative sign of ∂ImS eff /∂m 2 in four dimensions in the strong coupling, small mass limit, where S eff is the one-loop effective action.