^{*}

It is largely believed (or strongly assumed) that photons are massless particles and the most compelling evidence there—it is said—is found in the manifestations of photons being long ranged and long lived particles. As we have done before, albeit, with a much better and clear insight in the present than before; we argue herein that massive photons can still enjoy the special and rare privilege of travelling at the speed of light c while being long ranged, long lived and most of all, obeying the much desired gauge symmetry. This we achieve by breaking the traditional Lorenz gauge and in its place, we introduce a new Special Gauge Condition (SGC) that does the work of assuring the photon its longevity, long range-ness and that it [photon] propagates at the speed c. However, the most melancholic outcome of our investigation is that if the present scheme is what subtle Nature has chosen to endow the photon a non-zero mass so that it [photon] still obeys gauge invariance, is long ranged, long lived and travels at the sacred speed c; then, this non-zero photon mass may be very difficult, if not impossible to measure. We use the equations developed to investigate Lorentz violation in
*?*-ray bursts.

Despite the inconclusive experimental evidence (see e.g., Refs. [

In our earlier works (presented in Refs. [

Apart from the aforesaid introduction of the Stueckelberg [

In its classical form, Maxwellian Electrodynamics (MED) is described by two pristine and condensed tensor equations-namely, the source coupled and source free field equations. The source free field equations are given by the equation:

where:

is the MED field tensor,

potential and the magnetic vector potential respectively. In the present reading, the electromagnetic four vector potential

The source free field equations are given by the equation:

where

where

Now, substituting (2) into (1), we will have:

where

it follows that Equation (8) will reduce to:

Given the Lorenz gauge (9) and using the following definitions

By applying the methods of least action, the source coupled Equation (1) is derived from the Maxwellian Electrodynamic Lagrangian

by submitting this Lagrangian (11) into the Lagrangian equation of motion, namely:

The Lagrangian (11) is invariant under the following gauge transformation:

where

Now, it may not be immediately clear that the Lagrangian (11) is invariant under the gauge transformation (13) because this transformation leads to this Lagrangian (11) now becoming:

As is well known, the MED tensor field

For example, Srednicki [

ing in (14) vanishes identically. In this way, the Lagrangian (11) is clearly invariant under the gauge transforma- tion (13). So far, so good, let us proceed to Maxwell-Stückelberg Electrodymanics and let us call the photon(s) described by MED as presented here-MED photon(s).

Another way to make the Lagrangian (11) invariant under the gauge transformation (13) without having to relay on a judicious process and selection of boundary conditions is to add a Stückelberg scalar field

The Stückelberg scalar field

This MSED Lagrangian (15) is invariant under the following pair of gauge transformations [

To obtain the MSED field equations from this Lagrangian (15), via the method of least action, we substitute this MSED Lagrangian into the following Lagrangian equation of motion:

from which one obtains the usual massless photon MED field Equation (1) and the Stückelberg scalar field disappears completely from the midst of the resulting field equations. Its job [Stückelberg scalar field] here is the restoration of the much desired gauge symmetry. In the Standard Model, it can also be used to get ride of the Higgs boson [

As is well known, Maxwellian Electrodynamics (MED) is based on the hypothesis of a massless photon. If at all, what evidence there is to 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 which does not need either the Stückelberg scalar field nor the special boundary conditions to attain gauge invariance. From an Ockham Razor’s stand-point the QED would be the most preferred Lagrangian.

Gauge invariance was first introduced by the great German mathematician, mathematical physicist and philosopher-Professor Herman Klaus Hugo Weyl (1885-1955) [

The resulting source coupled Maxwell-Proca field equations are:

This Equation (19) is the classical Maxwellian Electrodynamic field equation with it added the Proca mass term,

In one of the early Solar system investigations of Maxwell-Proca electrodynamics, the great Austrian physicist-Erwin Rudolf Josef Alexander Schrödinger (1887-1961) was perhaps the first [

and:

In seeking a photon mass, both laboratory and astronomical observations focus on the two extra terms

usual Maxwell’s source coupled equations with a vanishing mass term. What this means is that if once tried to use MPED to decipher a non-zero mass for the photon, they will not detect a non-zero mass but a vanishing mass because it has been `whipped' away despite it being non-zero. Therefore, the special gauge condition that we shall introduce renders it difficult if not impossible to detect a non-vanishing photon mass-this put the question of a non-zero mass into a serious anti-juxtaposition because, in the end, laboratory and astronomical observations that employ MPED to detect a non-vanishing photon mass via the terms

Contrary to physical and natural reality, Proca’s massive photon theory leads to short-lived and short ranged photons and this is one of its greatest setbacks insofar as Electrodynamics of massive photons is concerned. Short-lived and short ranged photons would do well for the Standard Model, but not for the seemingly long- lived and longed ranged electromagnetic interaction. Additionally, the resulting Lagrangian of massive photons is not readily gauge invariant. In a very simple and trivial manner-as did Stueckelberg [

a special gauge condition. The modified MPED Lagrangian, which we shall call the Maxwell-Proca-Stückelberg (MPSED) Lagrangian

This Lagrangian is the same MPED Lagrangian albeit with the four electrodynamic

Lagrangian (22) is also invariant under the same gauge transformations. So, the issue of gauge invariance is no issue at all. The real issue is that the resulting MPED equations requires that photons must be short-lived and short ranged because the mass-term exists in the resulting equations. We will see this in the subsequent subsection.

Special Gauge ConditionTo obtain the resulting field equations from the Lagrangian

Because of the existence of the mass-term

The Special Gauge Condition (SGC) that we shall introduce so as to attain the desired theory of massive photons that are gauge invariant, long ranged and long lived is to set the terms in the under-brace [in (24)] so that these terms are identically equal to zero, i.e.:

The resulting equation after the introduction of the SGC, is:

and this equation is the same as Equation (10) for a massless photons in MED theory. At this point we are done with what we intend to achieve insofar as long range, long lived and gauge invariant massive photons are concerned. However, if we are to leave it at this point, a lot of questions about the Stückelberg scalar field will remain unanswered. To avoid clamping too many issues in one reading, we shall try and answer these questions but not with the level depth that is required because this would require a separate reading (or maybe readings) altogether, therefore, the reader with a strong taste for thoroughness must surely forgive us in this instance. In-closing, let us call the photon(s) described by MPSED under the proposed SGC (25) as presented here-MPSED photon(s).

We have shown that massive photons can exist without the three major problems that are normally associated with massive photons, that is, the problems of them being short ranged, short lived and non-gauge invariant. This achievement has come at a severe cost, namely that the Lorenz gauge

where

then we are back to the usual Lorenz gauge [

and given (27), it follows that the resulting equation is an equation for the field

which is the Klein-Gordon equation for the Stückelburg scalar. If

then, despite it having a non-zero mass, the Stückelburg scalar will either move at a speed less than, or, equal to the speed of light. We need to explain the latter-for the present purposes, we will ignore the tachyon solution

If

problems with such a particle propagating at the speed of light since it is massive-for massive particles, are, according to the STR incarcerated to travel only at speeds less than the speed of light. Against this background, we have argued in an earlier reading [

Now, since the massive Stückelburg scalar is coupled to the photon, it must propagate at the same speed with the photon-it must propagate at the speed c. This is the assumption that we shall adopt, the meaning of which is that we shall assume for the Stückelburg scalar that

The Stückelburg scalar is not-in general-expected to have the same energy and momentum as the photon, its energy and momentum can be different and we shall take this assumption that the photon and Stückelburg scalar

have different energies and momentum. In general, the wavefunction of a free Stückelburg scalar is

where

It turns out that the rate of change of orbital angular momentum of the Stückelberg scalar can be fixed and this fixture is so as to conform with observations. To do this, we have to re-look at the way the electrical field is defined. The magnetic field will remain defined as

where a new electrical field,

to take the

and this can be rewritten as:

From the new Lorenz gauge (27), we have:

and given that:

where

and further:

hence:

and from this

Notice that the Stückelberg field

this field must be present in all matter, including the electrical neutral neutrino. Because no significant electrical field has been measured around a neutrino, this field must be extremely small. It must be so small that it is only significant on cosmic scales. If this field exists in all matter, then, the Universe must be filled with a tiny all- pervading and permeating cosmic electrical field.

Now, given that

Further, taking the curl of

Evaluating-in rectangular Cartesian coordinates-the expression

From (41), it is clear that in-order for us to obtain the complete set of the usual equations that we are used to know, i.e., Maxwell’s Equations (4, 5, 6 & 7), the Stückelberg scalar field will have to have a non-changing orbital angular momentum i.e.

Clearly, if

One of our real motives [

As is well known, a cornerstone of Einstein’s STR [

A great effort is currently under-way to investigate possibilities of Lorentz violations [

these quantum gravity efforts seek to find deviations in the Einstein energy-momentum relation

One such deviation that has been proposed and continues to enjoy a great deal of attention is the proposal by the Italian physicist-Professor Amelino-Camelia et al. [

where

Against this background, if the outcome of the present investigations are to accepted, then, it is no longer tenable (for us-at least) to ascribe this time delay in the arrival times of these g-rays of different energies to the mass of the photon as the present ideas are translucently clear in that massive photons will-in a vacuum, travel at the speed of light, c. That is to say, in a vacuum, the present ideas predict that,

Photons will propagate at the speed c only in a perfect vacuum where the refractive index (n) is unity

Actually, the IGM is known to be a rarefied plasma [

If the SEF

Assuming for

Given that the group velocity

Further, if

We can rewrite (47) as:

where

From this Formula (48), if

High-energy

At this point, let us digress a little and ask the question “What does a positive (or negative) conductance?” To answer this question, we need to realize that in the derivation of Equations (43) and (44), we assumed Ohms Law

Now, after the above digression, let us proceed with the main business of the present section. We want to derive a formula that will allow us to compute the time lag per unity energy difference. For this, we need first to compute

Let D be the actual physical distance from Earth where with GRB has occurred and let t be the time taken for this flush to travel from the moment of emission to when it is observed on Earth. We know that

which it follows that

so that:

The distance D is the light-travel distance or the look-back time

i.e., the expanding Universe within the framework of the standard Cosmological-Constant Cold-Dark-Matter (LCDM)-model; and that the redshift

where

The result (51) is an exact result which can be put to the test if all the parameters are known. What is required are four parameters, namely the time lag

The majority of known quasars have very high redshifts

Because of the above said, it is our strong opinion that there is need for one to seriously consider the issue of how to interpret high redshifts because distances are very important in astronomy and cosmology. If we get them wrong, all our beautiful results that we currently obtaining are like castles built in the air high above the clouds, for soon and very soon, when truth finally catches up with us, our castles will not be spared by the ever present “gravitational force”, these beautiful castles will just crash on touching the ground. We must say; we are only comfortable working with low redshifts. With these low redshifts, the error in the distance estimation is obviously much smaller than in the case of very large redshifts. The high redshifts of GRBs may just suffer the same fate as quasar redshifts. There is need to verify beyond most doubt whether or not high redshift objects-in this case high redshift GRB; are at their cosmological distances.

Therefore, in the crude calculation that we are going to make below, we shall consider two low redshift GRBs. In this case of low redshifts GRBs, we know that to first order approximation D is such that

Now, by considering two low redshift GRBs, we shall use Equation (53) to compute a “rough” estimate of the conductance of the cosmic plasma. From the data of the GRB PKS 2155-304 [

whose magnitude is of the order

If the idea that we have just proposed of an all-pervading and permeating rarefied cosmic plasma and the SEF is acceptable, and the time delays of g-rays of different energies are to be measured accurately for most if not all the GRBs, then, a sky-map of the conductance of this plasma can-in-principle, be made. A test of the correctness of this idea would be if the same value of

For good reasons largely to do with aestheticism, physicists have nursed a “phobia” against massive photons. They have laid three very strong charges against them; the first of which is that, such photons will have be short ranged-the meaning of which is that they would not be able to traverse―as they do―the vast expanse of the observable Universe; the second being that such photons should not be able to live long and last and most of all, against the desideratum of the purest soul of the theoretical physicist, such photons would sacrilegiously violate the sacrosanct and embellished symmetry of gauge invariance.

Of these three serious charges laid against them, we have demonstrated herein that massive photons (i.e., MPSED massive photons) can be acquitted―there really is no case against them. These charges come about if we assume MPED and as-well that the electromagnetic four vector potential obeys the Lorenz gauge [

In the framework of MPED (which is our current best model for massive photons), if photons really did have a non-zero mass

limit of

Insofar as the present findings are concerned regarding these measurements in all their range of diversity, novelty and ingenuity, what is deeply disturbing about the present acquittal is that, if photons really did have a non-zero mass and these photons are MPSED photons, then, this mass may be concealed from any kind of revelation by Nature to such an extent that scientific experiments may not be capable of positively detecting this non-zero mass. Anything that is not measurable surely is outside of the realm of science since science concerns itself with matters that can be put before the Grand Jury of Science via the inviolable and embellished methods of scientific experimentation and enquiry.

We can not help but express our greatest and deepest disappointment at the result that we have obtained herein. Yes the result maybe very important-and perhaps beautiful; but the very fact that it can not be put to the test is not only disturbing and ugly but repugnant to the purest soul of the searching theoretical physicist. How are we to take of it? That the issue of the mass of the photon is a metaphysical issue? How does a scientist accept this? This result is akin to that of the great Austrian mathematician, logician and philosopher-Kurt Gödel (1906-1978)’s incompleteness theory of 1931. That is, to have an idea that tells you of the shorting comings, in this case, we have this idea that is telling us that yes, the photon may or may not have a mass, but, whatever the case, it will be very difficult is not impossible to discern. Surely, how does one handle such matters in experi- mental or even abstract philosophy?

Apart from the the said three charges, physicists have held another diabolic charge against massive photons- namely, that they can not possibly travel at the speed of light c as this would lead to a serious and chronic problems with Einstein’s widely accepted STR [

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

If one where to take this 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 3/2. Such a paradigm shift would be measurable [

On that note, i.e. existence of extra degrees of free for a massive photon, one will have to ask how these extra degrees of freedom come about? The answer is that they come about because these massive MPED photons do have a rest frame. They have a rest frame because of the very fact that-their having a non-zero mass, allows them to be brought to rest or at least to have in principle a rest frame. Normal massless MED photons have no rest frame whether in-principle or in practice, they always are travelling. The SGC (25) that we have introduced assures massive MPSED photons of this special state of not having a rest frame as they will always travel at the speed of light c in vacuum. They do not have a rest frame as is the case with massive MPED photons. Therefore, these massive MPSED photons are not longitudinal photons, they will result in the Planck's radiation law that we are always used to know.

The scheme that we have proposed herein to endow MPED photons with mass while at the sametime concealing it, this mechanism can not in-principle applied it to the Standard Model without any new modifications; it can not possibly be used in its bare form to endow the quantum gauge bosons of the Standard Model with mass, since these quantum gauge bosons of the Standard Model are short lived and short ranged because the SGC that we have introduced will-against physical and natural reality―make them long lived, long ranged.

For example, Sonoda & Tsai [

ranged and long lived

In closing, allow us to say that the idea that we have presented here, i.e., the idea of massive long range, long lived and gauge invariant photons is an exact idea, while the consequences thereof are only exploratory in nature. These consequences are this idea’s implications on GRB with regard to the observed time delays in the arrival times of g-rays of different energies. There certainly is need for a detailed exploration on this matter. All we can say is that these ideas surely look promising and worthwhile for further investigations if we are to understand Nature at a much deeper level than at present.

Assuming the acceptability of the thesis presented herein, we hereby lay down the following as our inescapable conclusion.

1. As has been demonstrated herein, the misgivings physicists have of massive photons―i.e., misgivings derived from the assumption of Proca Electrodynamics and the Lorenz gauge [?], namely that against physical and natural reality as we experience it, massive photons are expected to be short ranged, short lived and in contempt of the sacrosanct symmetry of gauge invariance; these misgivings can easily be “whisked” away by (a) the introduction of a Stückelberg scalar field

2. Apart from the pleasant outcome mentioned above, the most melancholic outcome of our achievement is that if the present scheme is what Nature has chosen to endow the photon with a non-zero mass so that this massive photon obeys gauge invariance, is long ranged and long lived; then, this photon mass may be very difficult if not impossible to measure because the terms involving its mass are concealed from the dynamic equations by the SGC. There seems to be no way to detect this non-zero mass except perhaps by directly detect- ing the Stückelberg scalar field which may very well be inseparable from the massive photon field thus making it impossible to detect. Separation of the massive photon from the Stückelberg scalar field must result in the decay of the photon into a stable particle-antiparticle pair (i.e.

3. The scheme that we have proposed herein to endow the photon with mass while in principle one can apply it to the Standard Model, albeit, with new modifications; it can not possibly be used in its bare form to endow the quantum gauge bosons of the Standard Model with mass, since these quantum gauge bosons of the Standard Model are short lived and short ranged because the SGC that we have introduced, despite the preservation of the much desired gauge symmetry, this will―against physical and natural reality―make them long lived, long ranged.

4. If Lorentz Invariance is an exact and inviolable symmetry of Nature, then, as we have suggested herein, namely that, the apparent Lorentz violation measured in GRB events, this can in principle be explained (amongst a set of possibilities) if the great expanse of cosmological space is filled with a rarefied cosmic plasma which is itself dominated by non-random currents due to negative electrical charges. These negative charge electric currents lead to

Golden GadzirayiNyambuya, (2016) Long Range, Long Lived and Gauge Invariant Massive Photons via Stückelberg Scalar Coupling. Journal of Modern Physics,07,7-24. doi: 10.4236/jmp.2016.71002