Massive Schwinger model and four-dimensional QED: The connection.

Abstract

We state the connection between the fermion determinant in four-dimensional QED (${\mathrm{QED}}_{4}$) and the massive Schwinger model, ${\mathrm{QED}}_{2}$, for the case of smooth, polynomial-bounded, unidirectional magnetic fields. Using the diamagnetic bound on the fermion determinant in ${\mathrm{QED}}_{2}$, we obtain an upper bound on the fermion determinant in ${\mathrm{QED}}_{4}$ for this class of fields. Using Kato's inequality, we obtain an upper bound on the one-loop effective action in scalar ${\mathrm{QED}}_{4}$ for smooth, polynomial-bounded but otherwise general fields with fast decrease at infinity.