Global Gauge Symmetry Breaking in the Abelian Higgs Mechanism

This paper aims to resolve the incompatibility between two extant gauge-invariant accounts of the Abelian Higgs mechanism: the first account uses global gauge symmetry breaking, and the second eliminates spontaneous symmetry breaking entirely. We resolve this incompatibility by using the constrained Hamiltonian formalism in symplectic geometry. First we argue that, unlike their local counterparts, global gauge symmetries are physical in the presence of boundary conditions. The symmetry that is spontaneously broken by the Higgs mechanism is this global one. Second, we explain how the Coulomb gauge is the preferred gauge for a gauge-invariant account of the Abelian Higgs mechanism. Based on the existence of the physical global gauge symmetry, we resolve the incompatibility between the two accounts by arguing that the correct way to carry out the second method is to eliminate only the redundant gauge symmetries, i.e. those local gauge symmetries which are not global. We extend our analysis to quantum field theory, where we show that the Abelian Higgs mechanism can be understood as spontaneous global $U(1)$ symmetry breaking in the $C^*$-algebraic sense.