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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 $g(ϕ)j_μ(A^μ +\partial^μB)$ where $B$ is an auxiliary field and $j_μ$ 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 determine the condition $\partial^μϕj_μ= 0$ 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 vacuua, 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