Magnetic permeability of constrained scalar QED vacuum

We compute the influence of boundary conditions on the effective Lagrangian of scalar QED for the case of a pure magnetic field. The boundary conditions assumed here constrain the quantum scalar field but not the radiation field, so that the effect to be considered already appears at the one-loop level. The boundary conditions generate new contributions to the effective Lagrangian much in the same way as a material pressed between two plates exhibits new magnetic properties. For the case of a periodic boundary condition we show that the bosonic vacuum behaves as if it were a diamagnetic medium. Our results show that the vacuum permeability has a linear term that is completely absent when there is no boundary condition.