Confining boundary conditions from dynamical coupling constants

It is shown that it is possible to consistently and gauge invariantly formulate models where the coupling constant is a non-trivial function of a scalar field. In the U(1) case, the coupling to the gauge field contains a term of the form g(φ)_jμ(^Aμ + ^∂μB) where B is an auxiliary field and _jμ is the Dirac current. The scalar field φ determines the local value of the coupling of the gauge field to the Dirac particle. The consistency of the equations determines the condition ^∂μφ_jμ = 0 which implies that the Dirac current cannot have a component in the direction of the gradient of the scalar field. As a consequence, if φ has a soliton behaviour, like defining a bubble that connects two vacua, we obtain that the Dirac current cannot have a flux through the wall of the bubble, defining a confinement mechanism where the fermions are kept inside those bags. Consistent models with time dependent fine structure constant can be also constructed.