0 electrodynamics Proca equation

The Lagrangian (850) shows that 0(3) electrodynamics is consistent with the Proca equation. The inhomogeneous field equation (32) of 0(3) electrodynamics is a form of the Proca equation where the photon mass is identified with a vacuum charge-current density. To see this, rewrite the Lagrangian (850) in vector form as follows ... 

The 0(3) Proca equation (856) does not have this artificial constraint on the potentials, which are regarded as physical in this chapter. This overall conclusion is self-consistent with the inference by Barrett [104] that the Aharonov-Bohm effect is self-consistent only in 0(3) electrodynamics, where the potentials are, accordingly, physical. 

There is rub to this construction. This Proca equation is really only applicable on a scale that approaches high-energy physics where the A 3 boson has appreciable influence. This will be only at a range of 10 17 cm. On the scale of atomic physics 10 3 cm, where quantum optics is applicable, this influence will be insignificant. In effect on a scale where the Al 3 does not exist, as it has decayed into pion pairs, the duality is established and there is no Lagrangian for the B 3 field. This puts us back to square one, where we must consider non-Abelian electrodynamics as effectively U(l) electrodynamics plus additional nonLagrangian and nonHamiltonian symmetries. 

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