An All-Orders Derivative Expansion

TL;DR

This paper derives an exact effective action for fermions in 2+1 dimensional QED with inhomogeneous magnetic fields, revealing that the derivative expansion is asymptotic and not convergent.

Contribution

It provides the first exact all-orders derivative expansion of the QED effective action in inhomogeneous magnetic fields, highlighting its asymptotic nature.

Findings

The effective action is obtained exactly for various magnetic profiles.

The derivative expansion is shown to be asymptotic, not convergent.

Implications for quantum field theory calculations in inhomogeneous backgrounds.

Abstract

We evaluate the exact $QED_{2+1}$ effective action for fermions in the presence of a family of static but spatially inhomogeneous magnetic field profiles. This exact result yields an all-orders derivative expansion of the effective action, and indicates that the derivative expansion is an asymptotic, rather than a convergent, expansion.