Question

Is the vacuum degenerate in the Abelian Higgs model?

Answer 1

Consider the theory with Lagrangian

L=−14FμνFμν+(Dμϕ)∗(Dμϕ)−U(ϕ),L=−14FμνFμν+(Dμϕ)∗(Dμϕ)−U(ϕ),

where U(ϕ)U(ϕ) breaks the U(1)U(1) symmetry of the system. If we were working in an ungauged theory, in which the U(1)U(1) symmetry is global, we would have no problem claiming that there are infinitely many distinct vacuum states, all with the same energy, and labelled by an angle θθ.