Quantum diffusion of magnetic fields in a numerical worldline approach

We propose a numerical technique for calculating effective actions of electromagnetic backgrounds based on the worldline formalism. As a conceptually simple example, we consider scalar electrodynamics in three dimensions to one-loop order. Beyond the constant-magnetic-field case, serving as a benchmark test, we analyze the effective action of a step-function-like magnetic field - a configuration that is inaccessible to derivative expansions. We observe magnetic-field diffusion, i.e., nonvanishing magnetic action density at space points near the magnetic step where the classical field vanishes.