Emergent gauge fields and the high temperature superconductors

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.