processes in strong laser field quantum electrodynamics
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QED in strong laser fields, vacuum polarization effects

Introduction
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Nonlinear processes have always played a fundamental role in different areas of physics, spanning from hydrodynamics, atomic and laser physics to plasma and high-energy physics [@Scott_b_2005]. From a theoretical point of view the description of such nonlinear processes, though attractive, is also particularly challenging. Since the invention of the laser, it was manifest that one of its unique features, the coherence, would allow for the experimental investigation of nonlinear phenomena. In a laser beam, in fact, a large number of photons propagate in phase and, depending on the laser intensity and on the process at hand, they may act cooperatively. One example is atomic high-order harmonic generation (HHG), in which a large number of laser photons is absorbed by a single atom and only one high-energy photon is emitted (see the reviews [@Agostini_2004; @Midorikawa_2011]). When laser-driven electrons (mass $m$ and charge $e<0$) are bound in atoms, nonlinear phenomena start at laser field amplitudes $E_0$ of the order of the typical atomic binding field $E_{\text{at}}=m^2|e|^5$, which corresponds to a laser intensity of $I_{\text{at}}=E_{\text{at}}^2/4\pi=7.0\times 10^{16}\;\text{W/cm$^2$}$ (units with $\hbar=c=1$ are employed throughout). In this case 