On-shell effective field theory and quantum transport for hard photons

We develop an effective field theory for the description of high energetic or hard photons, the on-shell effective theory (OSEFT). The OSEFT describes the so called eikonal or semi-classical optical limit, allowing for corrections organized in a systematic expansion on inverse powers of the photon energy. We derive the OSEFT from the Maxwell Lagrangian, and study its different properties, such as the gauge symmetry and reparametrization invariance. The theory can be finally formulated in terms of a gauge invariant vector gauge field, without the need to introduce gauge-fixing. We then use the OSEFT to compute corrections to the Wigner photon function, and derive its associated side jump effect from reparametrization invariance. Finally, we discuss how to properly define the Stokes parameters from transport theory once quantum effects are considered, so as to preserve their well-defined properties under Lorentz transformations.