Fermion determinant for general background gauge fields

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.

Worldline instantons and pair production in inhomogenous fields

“…Indeed, they are related by the analytic continuation γ → iγ, as can be seen in the expressions for the instanton paths and in the expressions for the stationary actions. This analytic continuation is similar to one connecting time dependent electric fields with spatially inhomogeneous magnetic backgrounds [37,38].…”

“…, as a function of the eE , in (38) for the time-dependent electric field E(t) = E sech 2 (ωt), plotted as a function of the adiabaticity parameter γ. Contrast this plot with the behavior in Figure 7 for a spatial inhomogeneity of the same form.…”

Heisenberg–euler Effective Lagrangians: Basics and Extensions

“…This approach has been developed recently by M. Fry [76], and also has great potential to yield important information about the behavior of fermion determinants in general backgrounds. For example, in Euclidean 2-dimensions, for a unidirectional abelian field strength F ( x), the fermion determinant is bounded below by [76] …”

Fermion Determinants: Some Recent Analytic Results

“…The analysis in Ref. 13 is not sufficient to rule out the (ma) 2 |eΦ| ln |eΦ| term in the remainder in (9); it may only be (ma) 2 |eΦ|. This result is therefore valid for a whole class of background gauge fields in the nonperturbative region of small mass and strong coupling.…”