Classical electrodynamics (CED) and quantum electrodynamics (QED) are well established
theories and their predictions have been confirmed experimentally in various regimes.
However, there are still areas of CED and of QED that deserve theoretical and experimental
investigation, especially when processes occur in the presence of "strong" background
electromagnetic fields, i.e., of the order of the so-called critical field of QED [1]. In view of
the increasingly stronger available laser fields it is becoming feasible to employ them to test
CED and QED under the extreme conditions supplied by intense fields [1]. After a broad
introduction on the typical scales of CED and QED, we will describe different regimes of
laser-matter interaction at ultra-high laser intensities and experimental efforts to test the
two theories under such extreme conditions. As a prominent theoretical example of open
problems which can be addressed also experimentally, we focus on the so-called "radiation
reaction" problem: classically, when a charged particle is accelerated by an external field,
it emits radiation and this emission alters the motion of the charge itself. The problem of
finding a self-consistent equation of motion in the realm of CED has resulted in unphysical
equations, which have raised a long-standing, still open debate on the subject already at a
classical level. Also, the quantum origin of radiation reaction will be discussed, by relating
radiation reaction itself to the emission by a charge of multiple photons.
[1] A. Di Piazza, C. Muller, K. Z. Hatsagortsyan, and C. H. Keitel, Rev. Mod. Phys. 84, 1177 (2012).
