Magnetic permeability of constrained scalar QED vacuum

TL;DR

This paper investigates how boundary conditions affect the magnetic properties of the scalar QED vacuum, revealing new boundary-dependent contributions to magnetic susceptibility and permeability.

Contribution

It introduces a novel analysis of boundary effects on the scalar QED vacuum, deriving explicit boundary-dependent modifications to magnetic properties using the Euler-Heisenberg Lagrangian.

Findings

Boundary conditions induce new contributions to magnetic susceptibility.

Existence of a boundary-dependent vacuum permeability in the linear regime.

The confined vacuum exhibits expected diamagnetic behavior.

Abstract

We compute the influence of boundary conditions on the Euler-Heisenberg effective Lagrangian scalar QED scalar for the case of a pure magnetic field. The boundary conditions constrain the quantum scalar field to vanish on two parallel planes separeted by a distance $a$ and the magnetic field is assumed to be constant, uniform and perpendicular to the planes. The effective Lagrangian is obtained using Schwinger's proper-time representation and exhibits new contributions generated by the boundary condition much in the same way as a material pressed between two plates exhibits new magnetic properties. The confined bosonic vacuum presents the expected diamagnetic properties and besides the new non-linear $a$-dependent contributions to the susceptibility we show that there exists also a new $a$-dependent contribution for the vacuum permeability in the linear approximation.