Global Gauge Symmetry Breaking in the Abelian Higgs Mechanism

Abstract: 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. The symmetries that are spontaneously broken by the Higgs mechanism are then the global ones. Second, we explain how the dressing field method singles out the Coulomb gauge as a preferred gauge for a gauge-invariant account of the Abelian Higgs mechanism. Based on the existence of this group of global gauge symmetries that are physical, 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.