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 per se but also as a warm-up for the study of the much more complex case of gravity. We 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 Maxwellʼs mechanical model for electrodynamics, where Maxwellʼs equations appear as dynamical consistency conditions. We next take a superfluid 3He-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 only in the low-energy regime is this physical reality blurred in favor of gauge invariance, which in addition turns out to be secondary to effective Lorentz invariance.