Consistent gauge interaction involving dynamical coupling and anomalous current

We show a possible way to construct a consistent formalism where the effective electric charge can change with space and time without destroying the gauge invariance. In the previous work^1,2 we took the gauge coupling to be of the form g(Φ)j_μ(A^μ + ∂^μB) where B is an auxiliary field, Φ is a scalar field and the current j_μ is the Dirac current. This term produces a constraint (∂_μΦ)j^μ = 0 which can be related to MIT bag model by boundary condition. In this paper, we show that when we use the term g(Φ)j_μ(A^μ - ∂^μ(1/〉∂_ρA^ρ)), instead of the auxiliary field B, there is a possibility to produce a theory with dynamical coupling constant, which does not produce any constraint or confinement. The coupling j_μ^A(A^μ - ∂^μ(1/〉∂_ρA^ρ)) where j_μ^A is an anomalous current is also discussed.