Effective Lagrangian at cubic order in electromagnetic fields and vacuum birefringence

The effective Lagrangian of electromagnetic fields at the cubic order in field strength is considered. This generalized Lagrangian is motivated by electrodynamics on non-commutative spaces. We find the canonical and symmetrical energy-momentum tensors and show that the vacuum in the model behaves like an anisotropic medium. The propagation of a linearly polarized laser beam in the external transverse magnetic field is investigated. We obtain the dispersion relation and refraction indexes for two polarizations. From experimental values of the induced ellipticity, reported by PVLAS Collaboration, the constraint on parameters in the effective Lagrangian is evaluated.