_{1}

^{*}

Prevailing and conventional wisdom holds that intermediate gauge Bosons for long range interactions such as the gravitational and electromagnetic interactions must be massless as is assumed to be the case for the photon which mediates the electromagnetic interaction. We have argued in a different reading that it should in-principle be possible to have massive photons. The problem of whether or not these photons will lead to short or long range interactions has not been answered. Naturally, because these photons are massive, one would without much pondering and excogitation on the matter assume that these photons can only take part in short range interactions. Contrary to this and to conventional wisdom; via a subtlety—namely, the foregoing of the Lorenz gauge and in line with ideas set out in out proposed Unified Field Theory, the introduction of a vector potential whose components are 4 × 4 Hermitian matrices; we show within the confines of Proca Electrodynamics under the said modifications that massive photons should be long lived ( i.e., stable) and be able to take part in long range interactions without any problem.

“We make our world significant, by the courage of our questions and, by the depth of our answers.”

Carl Sagan (1934-1996).

Despite the dearth of solid experimental proof [cf.] [

where

If (1) and (2) are both applicable to the photon with all the identical symbols holding the same meaning, then, it follows directly that

The hidden assumption in all the reasoning leading to the fact that for photons

The idea of a zero-mass particle usually presents a challenge to freshman students encountering this for the first time [

In the present reading, first we show that the idea founded in the reading [

The remainder of this reading is organised as follows. In Section 2, we give an exposition of PED and thereafter in Section 3, we give the theory or set of ideas leading to a gauge invariant theory of massive long range photons. With respect to experimental and observational philosophy, in Section 4, we discuss the meaning of the new theory and thereafter in Section 5, we give a general discussion and the conclusion drawn thereof.

As is well known, Maxwellian Electrodynamics (MED) is based on the hypothesis of a massless photon. If at all, what evidence there is for this, experience is yet to furnish us with a solid answer. As regards the quintessence of a zero-mass photon is the resulting gauge invariance of MED i.e., to those that seek beauty in a physical theory, one appealing feature of MED is that it quantum mechanical version i.e. Quantum Electrodynamics (QED) is constructed from a gauge invariant Lagrangian. Gauge invariance was first introduced by Professor Herman Weyl [

where

is the electromagnetic field tensor. As usual, the Greek indices

The term

pected normalization condition for the four vector

The equations of motion associated with the Proca Lagrangian

and the resulting Equation is:

Since

The over-brace has been inserted in the above Equation and this has been done for latter purposes so as to make it easy to reference.

Now, if we are to institute the Lorentz gauge condition

where:

is the usual D’Alembert operator.

To see why is is said that mass photons must lead to only short range interaction we have to solve (8) and, for simplicity, we shall assume that spatially,

and in empty space, the electric potential

where

For a minute, let us pause a question. What if we can show that within the framework of the same theory just laid down above (i.e. PED), that, one can obtain the desired long range Coulomb potential for a non-vanishing photon mass? Would the above reason for assuming a non-zero photon mass still hold? We think not. This is what we shall do in the next section; we shall present a trivially simple condition for attaining the said.

In the present section, we will address the two issues that make a massive photon non-desirable and these are the issue of gauge invariance and the issue that these massive photons can only be short range. In Section 3.1, we show that if the components of the four potential

If the following anti-commutator relations are to hold true, i.e.:

then, it is crystal clear that the following gauge transformations:

will leave the Maxwell-Proca Lagrangian (3) completely invariant. The anti-commutator relations (12) will require that

If the reader accepts the idea that

where

then, the relations (12) will be possible, thus, leading to the gauge transformations (13) to leave the Maxwell-Proca Lagrangian (3) completely invariant as desired. In this way, we solve once and for altime this nagging problem of obtaining a gauge invariant Maxwell-Procca Lagrangian. This comes at the cost of:

1) Making the current

2) Making the current

Before we close this section, we need to say something about

In massless MED, the function

The reasons why the photon is considered massless have been discussed in the previous sections. Of particular concern here is the fact that because the electromagnetic interaction is a long range interaction, therefore, the intermediate vector Boson―the photon; must be massless for this to be so. If we can demonstrate that even a massive photon can mediate long range interactions, will the above reason for vanishing photon mass still hold? We think not.

Notice that if we set equal to zero the over-braced terms in (7), then, we would obtain the same Equations that are obtained in the case of a Lorenz gauge invariant massless photon i.e. Equation (10) with

requires that we abandon the Lorenz gauge i.e. set

new gauge condition that replaced the Lorenz gauge condition

Since

so that:

where

With the over-brace terms set to zero, (7) reduces to:

Equation (20) is the same Equation obtained in the case of massless photon under the Lorenz gauge. We know that Equation (20) leads to long range and long lived photon. The difference now is that, these photons are now massive instead of massless.

Now, in-order that electric charge and current are conserved

Therefore, the condition

Under the usual or the traditional Proca Electrodynamics, a massive photon is expected to decay on a time scale

Tu et al. [

1) Laboratory Experiments e.g., [

2) Large Scale Observations e.g., [

In these measurements, derivations from MED are sought because the addition of the Proca term invariably modifies two of Maxwell’s four Equations i.e., the source coupled field Equations. These two Equations are:

where

where

In the large scale observations, the deviations are sought from Solar and planetary magnetic fields while in laboratory experiments these deviations are sought from usual laboratory magnetic fields and electric circuits. In both the laboratory and large scale observational measurements, the mass of the photon is not measured directly but limits to the photon mass are derived. Laboratory measurements find upper limits in the range

Clearly, the equivalent of Equations (22) and (23) to be derived from (20) will contain no mass-term hence, it follows that if one where to try and use Equations (22) and (23) as happens in all effort to measure the mass of the photon, they must obtain values of the photon mass that are compatible with zero because the actual final Equations (20) under the proposed scheme, this term does not appear in Equations (22) and (23), implying that this term is identically equal to zero, hence a mass compatible with zero is expected to be detected if one where to assume Equations (22) and (23) for

Using the idea from the reading [

In comparison to the Stückelberg mechanism and its descendants, the present method (theory) does not need the addition of a new assortment of particles to achieve gauge invariance. According to Occam’s Razor which requires a minimal addition of parameters into a theory, this fact that in the present theory we do not include nor need a menagerie of scalars as is the with the Stückelberg mechanism, this fact alone may be considered to be a significant improvement insofar a massive photon theory is concerned.

Another problem associated with massive photons is that such photons will have three extra degrees of freedom [

of rest. The total energy

to the equipartition theorem, adding three degrees of freedom will result in the photon gaining an extra amount

of energy

into account in deriving Planck’s radiation law―then, in complete contradiction with results from experimental philosophy, this would alter the Planck’s radiation law by a factor of

The above stated problem associated with the massive photon comes about because the Proca photon is capable of being at rest. That is to say, there exists a Lorentz frame in which the photon is at rest and in the this rest frame, it has three degrees of freedom. This problem is solved by making the mass of the photon to be dependant on the photon’s frequency (wavelength). As shown in [

tude of its group velocity

the present gauge invariant massive long range and long lived photon has three and not six degrees of freedom.

In-closing, allow us to say that in our most modest view, it is with great confidence that we have say that the present reading, together with [

Assuming the correctness (or acceptability) of the ideas presented herein, we hereby make the following conclusions:

1) It is in principle possible to have gauge invariant massive long lived photons that take part in long range interactions.

2) As suggested in [

3) Due to the new gauge condition (18)’s concealment of the mass of the photon from the corridors of Maxwell’s Equations for a massive photon, experiments and observations based on the Proca Equations (22) and (23) are not expected to yield a mass of the photon that is compatible with zero because this term does not appear in the final Equation of a massive photon as presented herein. This non-appearance of the mass-terms implies that measurements which make use of Equations (22) and (23) for a massive photon must fail to detect a non-zero mass for the photon.

We are grateful to the National University of Science & Technology (NUST)’s Research & Innovation Department and Research Board for their unremitting support rendered toward our research endeavours; of particular mention, Prof. Dr. P. Mundy, Dr. P. Makoni and Prof. Y. S. Naik’s unwavering support. This reading is dedicated to my mother Setmore Nyambuya and to the memory of departed father Nicholas Nyambuya. (27.10.1947- 23.09.1999).