Proca field equations

It should finally be mentioned that the basic equations (l)-(8) have been derived from gauge theory in the vacuum, using the concept of covariant derivative and Feynman s universal influence [38]. These equations and the Proca field equations are shown to be interrelated to the well-known de Broglie theorem, in which the photon rest mass m can be interpreted as nonzero and be related to a frequency v = moc2/h. A gauge-invariant Proca equation is suggested by this analysis and relations (l)-(8). It is also consistent with the earlier conclusion that gauge invariance does not require the photon rest mass to be zero [20,38]. 

In this connection it should finally be mentioned that Comay [63] and Hunter [64] have discussed the field concept by Evans and Vigier on the basis of conventional electromagnetic theory where the 4-current of Eq. (22) vanishes in the vacuum. Their analysis leads to the obvious conclusion that the field vanishes in such a case. This does, however, not rule out the existence of B(3) when there is a nonzero 4-current of the type (26) introduced by de Broglie, Vigier and Evans. Thus, without a Proca-type equation (22), no steady-state magnetic spin field can exist in a rest frame K. ... 

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 ... 

In order to derive field equations in the vacuum that are self-consistent, cause must precede effect and the classical current of the Proca current must be gauge-invariant. The starting point for the development is the concept of scalar field... 

The locally gauge-invariant Lehnert field equation corresponding to Eq. (374) was derived as Eq. (350). The photon picks up mass from the vacuum itself, and having derived a locally gauge-invariant Proca equation, canonical quantization can be applied to produce a photon with mass with three space dimensions. 

At the Higgs minimum, this field equation reduces to the locally gauge-invariant Proca equation... 

With a nonzero rest mass one would at a first glance expect a photon gas to have three degrees of freedom two transverse and one longitudinal. This would alter Planck s radiation law by a factor of, in contradiction with experience [20]. A detailed analysis based on the Proca equation shows, however, that the B3 spin field cannot be involved in a process of light absorbtion [5]. This is also made plausible by the present model of Sections VII and VIII, where the spin field is carried away by the pilot field. As a result, Planck s law is recovered in all practical cases [20]. In this connection it has also to be observed that transverse photons cannot penetrate the walls of a cavity, whereas this is the case for longitudinal photons which would then not contribute to the thermal equilibrium [43]. 

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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