  The elastic properties of solids are described in close analogy with General
Relativity, by locally gauging the translational group of space-time. Electron
interactions with the crystal lattice are thus generated by enforcing full
gauge invariance, with the introduction of a gauge field. Elementary
excitations are associated with the local gauge, contrasting to the usual
interpretation as being Goldstone bosons emerging from global symmetry
breaking. In the linear limit of the theory, the gauge field displays elastic
waves, that we identify with acoustic phonons, when the field is quantized.
Coupling with the electronic part of the system yields the standard
electron-phonon interaction. If spin-orbit effects are included, unusual
couplings emerge between the strain field and the electronic spin current,
leading to novel physics that may be relevant for spintronic applications.
