Proca equation

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]. 

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]. 

This program is not consistent with the Proca equation on the U(l) level. If the Proca equation... 

A condition is imposed on one of the four components of so that there are only three free components. However, the Lagrangian leading to the Proca equation is not gauge invariant due to the presence of a mass term [15]... 

It is well known that the Proca equation [6], Eq. (809), for a massive photon is not gauge-invariant because the Lagrangian (827) corresponding to it is not gauge-invariant. In SI units, this Lagrangian is... 

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 photon with mass has three degrees of freedom, so the 0(3) procedure is again self-consistent. The key advantage of the 0(3) procedure is that it produces a Proca equation that does not indicate the necessity for the Lorenz condition. 

The U(l) Proca equation (819) implies that the Lorenz condition always holds, because Eq. (819) leads to... 

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. 

Having derived the Proca equation in gauge-invariant form on the U(l) and 0(3) levels, canonical quantization can be attempted. Defining the photon mass in reduced units as... 

So A can act as a wave function and the Proca equation can be regarded as a quantum equation if A is a wave function in configuration space, and as a classical equation in momentum space. 

It is customary to develop the Proca equation in terms of the vacuum charge current density... 

The potential A1 therefore has a physical meaning in the Proca equation because it is directly proportional to / (vac). The Proca equations in the vacuum are therefore... 

The problem with the Proca equation, as derived originally, is that it is not gauge-invariant because, under the U(l) gauge transform [46]... 

In order to show that the Proca equation from gauge theory is gauge-invariant, it is convenient to consider the Jacobi identity... 

On the U(l) level, for example, the structure of the Lehnert [45] and gauge-invariant Proca equations is obtained as follows ... 

On the 0(3) level, one can write the Proca equation in the following form... 

Therefore the Proca equation can be recovered on the 0(3) level from the special solution (236) as the operator equation ... 

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. 

However, there remains the problem of how to obtain a locally gauge-invariant Proca equation. To address this problem rigorously, it is necessary to use a non-Abelian Higgs mechanism applied within gauge theory. 

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

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