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

TL;DR

This paper explores how classical electrodynamics can emerge from condensed matter systems, illustrating the microscopic reality of the vector potential and the conditions under which gauge invariance and Lorentz invariance arise as effective phenomena.

Contribution

It provides a detailed step-by-step analysis of emergent electrodynamics from condensed matter models, including Maxwell's mechanical model and superfluid helium-3-like systems.

Findings

Vector potential has a microscopic physical reality.

Gauge invariance emerges as a low-energy effective symmetry.

Effective Lorentz invariance arises in the low-energy regime.

Abstract

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.