Macroscopic Electric Polarization as a Geometric Quantum Phase

I outline a modern theory of the macroscopic polarization P in crystalline dielectrics. While P itself is not an observable, the polarization difference ΔP between any two crystal states can be measured, defined and calculated as a bulk material property: the root of this fact lies in a geometric quantum phase. In general, ΔP cannot be recovered from periodic charge densities, where any phase information is deleted: dielectric polarization is in fact a quantum phenomenon, governed by a gauge-invariant phase feature of the crystal wave functions.