  The quantum entanglement of many states of matter can be represented by
electric and magnetic fields, much like those found in Maxwell's theory. These
fields "emerge" from the quantum structure of the many-electron state, rather
than being fundamental degrees of freedom of the vacuum. I review basic aspects
of the theory of emergent gauge fields in insulators in an intuitive manner. In
metals, Fermi liquid theory relies on adiabatic continuity from the free
electron state, and its central consequence is the existence of long-lived
electron-like quasiparticles around a Fermi surface enclosing a volume
determined by the total density of electrons, via the Luttinger theorem.
However long-range entanglement and emergent gauge fields can also be present
in metals. I focus on the "fractionalized Fermi liquid" (FL*) state, which also
has long-lived electron-like quasiparticles around a Fermi surface; however the
Luttinger theorem on the Fermi volume is violated, and this requires the
presence of emergent gauge fields, and the associated loss of adiabatic
continuity to the free electron state. Finally, I present a brief survey of
some recent experiments in the hole-doped cuprate superconductors, and
interpret the properties of the pseudogap regime in the framework of the FL*
theory.
