Electromagnetism as an emergent phenomenon: a step-by-step guide

We give a detailed description of electrodynamics as an emergent theory from condensed-matter-like structures, not only {\it per se} but also as a warm-up for the study of the much more complex case of gravity. We will concentrate on two scenarios that, although qualitatively different, share some important features, with the idea of extracting the basic generic ingredients that give rise to emergent electrodynamics and, more generally, to gauge theories. We start with Maxwell's mechanical model for electrodynamics, where Maxwell's equations appear as dynamical consistency conditions. We next take a superfluid $^3$He-like system as representative of a broad class of fermionic quantum systems whose low-energy physics reproduces classical electrodynamics (Dirac and Maxwell equations as dynamical low-energy laws). An important lesson that can be derived from both analyses is that the vector potential has a microscopic physical reality and that it is only in the low-energy regime that this physical reality is blurred in favour of gauge invariance, which in addition turns out to be secondary to effective Lorentz invariance.