I discussed Berry phase back in February, (Berry or Pancharatnam-Berry or Quantum phase) from a vector / density matrix point of view, but I thought it would be nice to describe Berry phase from the point of view of the U(1) gauge symmetry of quantum mechanics. From a density matrix point of view, the U(1) gauge symmetry is what arises in the state vector formalism, from the requirement that all physical observables be capable of being written in the density matrix formalism.
Let’s consider functions of three normalized spin-1/2 SU(2) spinors: |a), |b), and |c) in the bra-ket notation. (And I’m using round brackets to avoid a WordPress screwup.) Since the state vectors of quantum mechanics are unchanged by phase, we can also choose three arbitrary fixed real numbers, to respectively multiply these spinors. The bras and kets take the complex conjugate phases so the arbitrary phase transformation on these objects is as follows:
Quantum mechanic’s physical predictions will be unchanged by the substitutions. This symmetry is the U(1) symmetry of the SU(3)xSU(2)xU(1) symmetry of elementary particles and it was the first gauge symmetry. In short, letting the phase depend on position, and requiring that the theory of electrons be unchanged, defines a new field which turns out to be the photon.