Remarks on propagating waves in non-linear vacuum electrodynamics

Using the quadratic expansion in the photon fields of Euler-Heisenberg (EH) non-linear electrodynamics (NLED) Lagrangian model we study relevant vacuum properties in a scenario involving the propagation of a photon probe in the presence of a background constant and static magnetic field, ${\bf B_e}$. We compute the gauge invariant, symmetric and conserved energy-momentum tensor (EMT) and angular momentum tensor (AMT) for arbitrary magnetic field strength using the Hilbert method under the soft-photon approximation. We discuss how the presence of magneto-electric terms in the EH Lagrangian is a source of anisotropy, induce the non-zero trace in the EMT and leads to differences between EMT calculated by the Hilbert or Noether method. From the Hilbert EMT we analyze some quantities of interest such as the energy density, pressures, Poynting vector, and angular momentum vector, comparing and discussing the differences with respect to the improved Noether method. The magnetized vacuum properties are also studied showing that a photon effective magnetic moment can be defined for different polarization modes. The calculations are done in terms of derivatives of the two scalar invariants of electrodynamics, hence, extension to other NLED Lagrangian is straightforward. We discuss further physical implications and experimental strategies to test magnetization, photon pressure, and effective magnetic moment.