Magnetic permeability of constrained scalar QED vacuum

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. ∗ e-mail:[email protected] † e-mail:[email protected] ‡ e-mail:[email protected] § e-mail:[email protected]