The quantum vacuum in electromagnetic fields: From the Heisenberg-Euler   effective action to vacuum birefringence
  The focus of these lectures is on the quantum vacuum subjected to classical electromagnetic fields. To this end we explicitly derive the renowned Heisenberg-Euler effective action in constant electromagnetic fields in a rather pedagogical and easy to conceive way. As an application, we use it to study vacuum birefringence constituting one of the most promising optical signatures of quantum vacuum nonlinearity. 
Felix Karbstein
hep-th hep-ph