Luis, although I think that the Maxwell equations have to be enhanced by additional terms (see for example

https://www.researchgate.net/publication/310994179_On_Divergence_in_Radiation_Fields_Toward_highly_focused_neuron_stimulation_for_neurology_and_HCI

), I do not agree with your 7 points.  I feel that you are looking - in a more or less abstract way - only at parts of the picture, and I would suggest to calculate all aspects of some distinct examples.

1.: Imagine a simple plane wave incident on a charged particle.  If you would neglect the effect of the B field you would arrive at your point 1; by taking the B field into account, the transfer of momentum appears without introduction of further quantities.  BTW, that the behavior of a single photon or a small number of photons cannot be described by classical theory (e. g. Compton effect) is of course well known and one of the main reasons why quantum theory cannot be replaced by simply modifying classical theory.

2.: But these fields in phase comply perfectly with the Maxwell equations.

3. seems to be missing.

4.: I do not understand this point.  Give me the power flow density of a monochromatic, homogeneous, plane wave, and I'll give you the intensities of the E and B fields.

5.: Why "... without a volume"?  The unit of power flow density is power flow per m^2, so if we know the propagation time per meter (about 3.3 ns in vacuum), and if we assume that the energy moves in a continuous way, we can calculate the energy contained in 1 m^3.

6.: Without changing any physical content, you can get rid of complex numbers. As penalties, you have to replace each complex number by two real numbers, and you have to introduce branches into your calculations, e. g. for the cases "argument of square root positive" and "argument of square root negative".

7.: In a propagating wave, the E and B fields in a certain point of space become zero momentarily (and, in the framework of classical theory, so does the energy), but the energy is not disappearing and reappearing again; it is, well,  propagating.

8.: If the two waves are collinear within some volume of space, then they are collinear everywhere. Consequently, at least one wave is incident on the source of the other wave. If you calculate the energy exchanged between sources and incident waves, you will get a perfect energy balance. Example: A wave incident on an electrically conducting layer thick enough to "shield" the wave completely: The charges near the surface of the layer produce a wave collinear to the incident wave with a phase difference of 180°, and both waves cancel each other resulting in zero energy transmitted through the layer (extinction theorem).  But the incident energy equals the sum of reflected energy and EM energy turned into thermal energy.

Hope my comments do not sound too destructive but are helpful.

Working Paper On Divergence in Radiation Fields Toward highly focused neur...

The energy equation of EM field and Lorentz force equation?  The electric field energy do transfer energy to the charged particle in Thomson scattering (which is classical case of Compton effects) and then the charged particle would emit radiation when accelerated by the EM field. 

Maxwell theory does not explain the electromagnetic wave in full, because the photons include the gravitational component. 

Certainly, photons can be expected to have a gravitational component, but I think that

a) the reason is not one of the 7 points mentioned above but simply Newton's third law (plus Special and General Relativity), and

b) the gravitational field of a visible photon (600 nm) is weaker than that of a resting electron by a factor of about 4*10^-5. So I guess it might be hard or even impossible to measure any effect based on the gravitational field of light.

Mr.  Fricke I disagree with you completely. Let us start:

"I do not agree with your 7 points.  I feel that you are looking - in a more or less abstract way - only at parts of the picture, and I would suggest to calculate all aspects of some distinct examples.

1.: Imagine a simple plane wave incident on a charged particle.  If you would neglect the effect of the B field you would arrive at your point 1; by taking the B field into account, the transfer of momentum appears without introduction of further quantities.  BTW, that the behavior of a single photon or a small number of photons cannot be described by classical theory (e. g. Compton effect) is of course well known and one of the main reasons why quantum theory cannot be replaced by simply modifying classical theory."

Remember what the Compton Effect is. It is not an interaction among charged particles. Or the behavior of a charged particle in a field. This behavior will be different for a positive or negative charge, and will be nul for a neutral particle. This effect is about the transmission of momentum independent of their charge. It is the collision effect among particles, and the transmission of energy. The photon hit an electron, a proton, a neutron, an atom, and the behavior of the photon and the particle hit will depend only of the vector speed and mass of the particles, not on their charge. In the case of a photon. Of what mass we are talking about? Diogenes Aybar in his article the EMG theory of the Photon proposes that the light wave have a gravitational component that goes in direction of the movement, that is responsible of the Compton Effect.  This theory can be read at:

  http://www.journaloftheoretics.com/Links/Papers/EMG%20III.pdf

"2.: But these fields in phase comply perfectly with the Maxwell equations."

The problem with the compliance of the Maxwell equations is that in this way the law of conservation of energy is violated. See below

"3. seems to be missing." OK

"4.: I do not understand this point.  Give me the power flow density of a monochromatic, homogeneous, plane wave, and I'll give you the intensities of the E and B fields."

Well, show me how you calculate the electric and magnetic fields. Their values.

"5.: Why "... without a volume"?  The unit of power flow density is power flow per m^2, so if we know the propagation time per meter (about 3.3 ns in vacuum), and if we assume that the energy moves in a continuous way, we can calculate the energy contained in 1 m^3."

Not so easy. First, you do not have to assume that the energy moves in a continuous way. It cannot be otherwise.  If you have the frequency you can have the energy of the wave. I am not asking about the energy of a given volume, that you have restricted to 1 m3, but to the EM wave; you do not have a wave distributed in a space with a transversal area of 1 meter, but following a sinusoid, with an amplitude and a frequency. You cannot integrate over an area and multiply for a distance in the direction of the displacement.  But, well, link the two energy parameters, make a formula relating the total energy and the energy density, and show me.

"6.: Without changing any physical content, you can get rid of complex numbers. As penalties, you have to replace each complex number by two real numbers, and you have to introduce branches into your calculations, e. g. for the cases "argument of square root positive" and "argument of square root negative".

This is your position, but not of everybody, especially not mine. Let us follow George P. Shpenkov.

“Thus, ‘imaginary’ numbers are not actually imaginary. All conjugate (“real” and “imaginary”) numbers are real. In particular, the wave function, called in modern physics as a “complex” function comprising real and imaginary terms, in actual fact, is contained only real components, reflecting thus the potential-kinetic essence of rest-motion. So, summing up, we emphasize once again that  the "imaginary" unit i is a symbol for a qualitatively polar opposite essence (property, number, parameter) obeying to the polar opposite (“negative”) algebra of signs relative to the conventional (“positive”) algebra.” http://shpenkov.janmax.com/ImaginUnitEng.pdf

"7.: In a propagating wave, the E and B fields in a certain point of space become zero momentarily (and, in the framework of classical theory, so does the energy), but the energy is not disappearing and reappearing again; it is, well,  propagating."

This is not true. The energy can never become zero, even for a microsecond. The propagation of energy does mean that you have, at any time, an energy value different from zero; if the energy is conserved this value must be constant.  

Following Maxwell, in the case of an electromagnetic wave travelling in empty space:

E = Em sin (kx - ωt);  B = Bm sin (kx - ωt)

Where E is the electric field and B the magnetic field; k is the wave number, ω, the angular frequency, x the position, and t is time. Be Em the maximum value of E and Bm the maximum value of B.

From this equations it is clear that E is changing in phase with B. Besides, the relation ∇ E = - B/ , requires that E and B be in phase. (∇ E = -partial derivative B/t).

Then, they will reach their maxima at the same time, and will keep changing until a time in the cycle when they will become zero. The energy density of the wave is contained exclusively in these two fields, and its value is a function of them, and, since they are changing with time, so do the energy: UT = UE + UB;

UT = ½ εo E2 + ½ µo B2 

The above equation describe an energy density that fluctuates in both the time and space coordinates. It does so in such a way that at certain moment there is no energy while at a predictable later moment the energy density is maximal, with a continuous variation from zero energy to a maximum. Because the moments and the positions are perfectly known, it is not possible to use the uncertainty principle to explain the energy fluctuations. Nor can it be explained in terms of variations of the energy density with constant total energy, because this presupposition implies that the volume of the photon varies with the oscillation from zero to infinity, nor by saying that the energy balance has to be taken in a volume that encompasses an integer number of whole cycles of the wave. Therefore the system behaves as if it had an energy reservoir (outside of this space-time?) from which it can extract and pour in energy. This assertion is incompatible with the principle of conservation of energy.

Normally the energy density is averaged. May I ask something? If the energy of the electromagnetic wave is constant, why it needs to be averaged? The Pointing  Vector is a very used concept in the study of  electromagnetic waves and is defined as the velocity  of the energy flux per unit of area, S = 1/µO EmxBm,  which implies again a variation of the energy. To face the problem of the changing energy, another concept is utilized, the Mean Pointing Vector (Sm = 1/2µo EmxBm).  The Mean Pointing vector is defined as the mean of the velocity of the energy flux per unit of area, and is calculated when the electric and magnetic fields reach their maxims. With the Mean Pointing Vector the values of the Pointing Vector are leveled, hiding the fact that the energy of the wave is changing with time.

 Let us cite Resnick and Halliday, only because they, better than anybody, put in evidence this situation. “This relation S = 1/µO EmxBm, refers to S, E, and B at any instant.  Ordinarily we are more interested in the mean value of S. An observer making many measurements of the intensity of the wave passing for it, would measure this value, Sm = 1/2 µo EmxBm

Please, read again. “Ordinarily we are more interested in the mean value of the Pointing Vector”. Why? What is the rationale of calculating a mean of an energy that have and must have the same value? It will be because with this mathematical subterfuge is hided the evident fact of a changing energy?

8.: If the two waves are collinear within some volume of space, then they are collinear everywhere. Consequently, at least one wave is incident on the source of the other wave. If you calculate the energy exchanged between sources and incident waves, you will get a perfect energy balance. Example: A wave incident on an electrically conducting layer thick enough to "shield" the wave completely: The charges near the surface of the layer produce a wave collinear to the incident wave with a phase difference of 180°, and both waves cancel each other resulting in zero energy transmitted through the layer (extinction theorem).  But the incident energy equals the sum of reflected energy and EM energy turned into thermal energy.

Hope my comments do not sound too destructive but are helpful.

Excuse me, but you did not understand the situation. There is not the case when a light beam travelling in one medium enters in another that have different density. This is the Ewald-Oseen theorem, where there is extinction of the incident wave and the collinear opposing wave, but the refracted wave plus the thermal energy had the energy of the incident wave.  I am talking about the complete destruction of the waves in the phenomenon called Destructive Interference.

Let us have these two waves:

E1 = Em sin (kx - ωt) E2 = Em sin (kx - ωt + p)

B1 = Bm sin (kx - ωt) B2 = Bm sin (kx - ωt + p)

E = E1 + E2 and B = B1 + B2

E = Em sin (kx - ωt) + Em sin (kx - ωt + p)

B =Bm sin (kx - ωt) + Bm sin (kx - ωt + p)

But, sin (kx - ωt + p) = - sin (kx – ωt) ,

Then, E = 0 and B = 0 and,

UT = UE + UB

UT = ½ εo E2 + ½ o B2

UT = 0

According to Maxwell equations, in destructive interference of two collinear beams out of phase 180 degree, the energy density, which is a function of the electric and magnetic fields, is destroyed too. In this way, the principle of conservation of energy is violated.

What happens with the wave if the electric and magnetic fields have vanished through interference? The EMG theory of the photon predicts that a new wave will continue traveling, this time as a gravitational wave with spin two.

This is in agreement with Robert Stirniman who stated: “Notably, a spin one photon, or circularly polarized EM wave (spin one), could be interposed with the same wave but 180 degrees out-of-phase. Conventionally the electric and magnetic fields vanish -- but the spin must superpose additively. Now a spin 2 wave, but with no transverse electric and magnetic field.”

Mr,  C.Y. LO,  As your research have led you to conclude that the EM radiation has a gravitational component, I would be pleased you read the EMG theory of the Photon to see its relevance. I also would like that everybody interested in EM and Gravitation read the theory in order to ascertain if it has any value.

The EMG theory of the photon postulates that light is an electromagnetic-gravitational entity. The electro-magnetic- gravitational wave contains an electric, a magnetic and a gravitational field; the electric and magnetic fields changing in phase and oscillating transversally, and the gravitational field, 90 degrees out of phase, oscillating longitudinally. This configuration makes the photon an interactive entity capable of gaining and losing energy through its interaction.

The gravitational redshift can be explained through this interaction, where the gravitational field associated with light interacts with the gravitational field of a massive body.  If the lines of force of the

gravitational field produced by the massive body are collinear with the trajectory of the photon, it can

happens two things: the light redshifts, if the photon goes against the gravitational field; or blueshifts, if

the photon goes with it. If the lines of force of the gravitational field are not collinear with the

direction of light, then the beam is deflected. These phenomena can be explained by an interactive

photon, which have within three fields.

This theory explains the Compton Effect and the transmission of momentum by the wave, badly understood when one thinks in a transverse electromagnetic wave. Also it explains, the necessary conditions to produce interference in waves, conditions well known empirically, but not predicted by any known theory. The EMG theory establishes precisely where the energy is when a total destructive interference of two waves happens. Besides, a new prediction is that the bending of light must produce chromatic aberration. This would give an explanation to the galaxy lensing.

In the other hand, new concepts are derived from the theory, like the volume of the photon, that

establishes the characteristics the wave must have to observe a Compton Effect; the indissolubility and

discreteness of space and time; the translocation or the way that the photon moves. And, with its GEM induction equations (two equations added to Maxwell’s four) opens a door to get an advance in the awaited unification of electromagnetism and gravity.

The theory, beautiful, elegant and economic, is a superstructure that lies over a supposed violation of the application of the electromagnetism of Maxwell to the light wave. If the theory is validated it will means an earthquake in the edifice of modern physic. But, even if it is only a beautiful creation of the human mind, I think Dr. Aybar has put the finger over the injury, about the problem of conservation of energy in the EM wave theory.

 I would like you examine the theory, check the mathematics and gives an opinion of it, because you are one who thinks the EM theory is incomplete.

The EMG of the photon can be found at:

http://www.journaloftheoretics.com/Links/links-papers.htm

The photons have a gravitational wave component is necessary. Otherwise, there would be conflict between the notion of photons and the static Einstein equation.

There are some problems finding the EMG theory of the Photon, in the Journal of Theorethics, so I will add an attachment.

Mr. Fondeur, thanks for your reply! I will take the liberty to give my answers in several parts due to time limitations; I'll start with 1. and 4.:

1. I didn't know that the Compton effect appears between photons and electrically neutral particles.  Could you please provide a link to a publication on the experimental verification?

4. Prerequisites: a) The average power flow density Sav is known.

b) Since the wave is harmonic (monochromatic) the maximum power flow density is Smax = 2 Sav.

c) The power flow density S (Poynting vector) is given by E x H.

d) For a TEM wave, E x H reduces to E * H.

e) Because the wave propagates in vacuum, H = E / Z0 with the wave impedance of vacuum Z0 (about 377 Ohm).

Smax = 2 Sav = Emax * Hmax = Emax^2 / Z0  

Emax = sqrt(2 * Sav * Z0)

Hmax = Emax / Z0

Now we know the amplitudes of both E field and H field. Frequency and the plane of polarization are unknown but can be measured.

Dear Thierry de Mees, I'm glad you took the time to have a look at the paper on divergence.  Indeed, I mixed Maxwellian electromagnetism with special relativity; I felt justified by the facts that E-M theory not only "survived" SR unchanged but was one of the starting points, and that, for example, Heaviside discovered the deformation of a spherical object at high speed into a spheroidal one even before the advent of SR.

But it is always rewarding to look at the same thing from different angles, so: Many thanks for your suggestion to formulate sections III A and B taking Jefimenko's work into account. I guess rereading chapters 1 to 4 of "Causality ..." will be sufficient for this purpose. I hope the approach will be successful, and I'm curious about the results!

Best regards.

Maxwell equations explain the continuum aspects of light but not the quantum aspects. Example E = hf is not derived from Maxwell. Also it is not assumed that all the natures of light are known.

 Maxwell Equations lead to a set of sine waves or cosine waves with amplitudes E or D and B or H, resulting in a Poynting Vector which is power flow per unit cross section area. With a few other assumptions Planck's Law can be derived for one complete cycle of the wave, but one of the assumptions is the effective cross section of a wave.

Conversely with Planck Law given, the effective cross section can be calculated from the Maxwell Equations.

The key to it all lies in establishing the deeper physical meaning behind the displacement current, which in modern textbooks is disguised as the magnetic vector potential A. With div A = 0 we can ascertain that EM waves are transverse to the polar origin used in the analysis. It further follows that curl A = B places B and H into the realms of vorticity and angular momentum. A is therefore the fine-grained circumferential momentum in Maxwell's sea of molecular vortices, and EM waves are therefore a propagation of angular acceleration through the sea. As regards the issue of these waves carrying linear momentum, we need to resort to Maxwell's equation for radiation pressure which is not part of the set of equations known as Maxwell's equations. It can then be reasoned that EM waves involve pressurized aether spilling over from one molecular vortex to the next during angular acceleration (including precession).  Maxwell incorporated the result of the 1856 Weber/Kohlrausch experiment into his analysis hence exposing the speed of EM waves to be equal to the speed of light.

I invite you to take knowledge of the book Essays on the Formal Aspects of Electromagnetic Theory (editor Akhlesh Lakhtakia, World Scientific, 1993) and of an older interesting book Relativity Reexamined by L. Brillouin (Academic Press, 1970).

Antoine J.H. Acke’s original and interesting work, which I have just learned about, deserves to be studied more thoroughly.

Dear Noel Kallas,

Thank you for your interest in my work. 

One can find my vision on EM on RG: https://www.researchgate.net/publication/305774508_ELECTROMAGNETISM_EXPLAINED_BY_THE_THEORY_OF_INFORMATONS-2

Article ELECTROMAGNETISM EXPLAINED BY THE THEORY OF INFORMATONS-2

Maxwell theory cannot explain photons.