Note, that these incorrect for heterogeneous, polyscale media averaging procedures dominated the first half of XX century.

Lorentz (1923) on the Future Modifications of Maxwell Equations

Lorentz himself in his "Clerk Maxwell's Electromagnetic Theory. The Rede Lecture for 1923, Cambridge" (1923) used to say that:

"Will it be possible to maintain these equations? I am not thinking here of the comparatively slight modifications that have been necessary in the theory of relativity;.....

A greater and really serious danger is threatening from the side of the quantum theory, for the existence of amounts of energy that remain concentrated in small spaces during their propagation, to which several phenomena seem to point, is in to Maxwell's equations.

However this may be, even if further development should require profound alterations, Maxwell's theory will always remain a step of the highest importance in the progress of physics."

where he was citing Maxwell himself:

"It appears to me that while we derive great advantage from the recognition of the many analogies between the electrical current and a current of a material fluid, we must carefully avoid making any assumption not warranted by experimental evidence,......

A knowledge of these things would amount to at least the beginning of a complete dynamical theory of electricity, in which we should regard electrical action, not, as in this treatise, as a phenomenon due to an unknown cause, subject only to the general laws of dynamics, but as the result of known motions of known portions of matter, in which not only the total effects and final results, but the whole intermediate mechanism and details of the motion, are taken as the object of study."

Our comments:

Note that Maxwell said these words long before an electron, a nucleus, other particles, and atomic structures of matter were discovered.

It would be relevant to cite here what else Lorentz (1923) writes in p. 32:

"Maxwell did not always express himself so cautiously: at other times he did not shrink from imagining an elaborate mechanical model.

All physicists know its principal features. The magnetic energy is considered as a true vis viva, the magnetic field being the seat of invisible motions, rotations of small particles about the lines of force.

The system of these particles may be compared to a wheelwork, and Maxwell has to explain how it can be that all the wheels in an element of volume are rotating in the same direction.

This shows that the motion is not transmitted directly from one wheel to the next. So Maxwell is led to assume that between these wheels of the magnetic field and in contact with them, there are smaller ones which transmit the motion in the manner of friction wheels.

What I called wheels might in reality be spheres capable of turning about axes in any direction and, as Maxwell showed, a system of this kind is amenable to mathematical analysis. In his image the friction wheels represent what we call electricity; in a conductor we must conceive them to be freely movable, ....."

Our comments:

Yes, that might be atoms, molecules, electrons, other particles involved into magnetic fields of local and averaged (bulk) significance.

The words of Lorentz above are pretty on target regarding what was derived with the HSP-VAT application to the sub-atomic Lower scale wheelwork to get to the bulk properties and mathematical dependencies in - "Incompatibility of Maxwell-Lorentz Electrodynamics Equations at Atomic and Continuum Scales"

and

".... The outcome of these various attempts is, in my opinion, this: that we must admit the possibility of mechanical representations, a possibility that is already shown by the fact that the formulae of the electromagnetic field can be given the form of the general equations of dynamics.

On the other hand it cannot be denied that, when we desire them to be applicable to a comparatively wide range of phenomena, the theories that have proposed become so complicated that they give us but little satisfaction......"

Our comments:

What would Lorentz to say today when watching the ITER and LHC projects activities? He might's turning upside down.

Landau and Lifshits on Averaging of Atomic Scale Maxwell-Lorentz Electromagnetism Equations

Landau and Lifshits (1957, 1960) wrote on the Macroscopic form of Maxwell set of atomic scale equations.

In the very first page 11 (Russian publication) they wrote:

"That is, instead of real "microscopic" value of electric field strength we will be considering its averaged value, determine it as

Basic equations of continuum mechanics electrodynamics usually obtained by averaging of equations of electromagnetic field in the vacuum. This transformation from micro- to macroscopic equations was firstly performed by G.A.Lorentz."

Our comments:

In p. 12 authors just boldly saying that averaging of microscopic equations and straightly bring the forms Averaging has been made mainly verbally.

In the page 55 a reader can find:

"The second is made with the averaging of equation

Assuming that inside dielectric substance there is none externally inserted charges; this is the most usual and important case. Then the full charge in the whole volume of dielectric is set equal to zero and after introduction of it into electrical field:

besides, outside of the body In fact, having integration over the volume that is surrounded by surface having it inside and going everywhere outside, we will have

Further in the next page 12, authors derive:

"By definition of dipole moment, this is the integral

Making substitution from (6,3) and taking integral over the volume, that is having the body within it, get

The surface integral is disappearing, and in the second one we have that means

That is why the polarization vector presents itself the dipole moment ( or the electric moment) of dielectric unity volume."

Our comments:

The same derivation as in Levich (1969 we can see, only Landau and Lifshits (1957, 1960) took that very simple with the outright suggestion that the averaged

***************************************

Then, in the page 304 reader can find excerpts about the averaging process regarding the whole set of microscopic Maxwell-Lorentz equations:

"Equations

can be obtained immediately by changing and in exact microscopic Maxwell equations with their averaged variables and That is why these equations by no means are not needed any change. What related to equation

then this equation obtained with averaging of exact microscopic equation , besides here is used the fact that the full charge of the body equal to zero."

"One more equation must be produced by averaging of the exact equation

Immediate averaging gives

Meanwhile, when the macroscopic field is time dependent establishing the connection of the mean value of with the earlier introduced variables is pretty complicated task. It is easier to make the needed averaging not directly, but using the following more normal way."

Then, authors make few transformations of like already "averaged" (in reality not averaged, just declared as averaged) other equations from the Maxwell set and bring to readers the following equation for magnetic field as the "averaged" one

From this set (unaveraged)

Landau and Lifshits immediately obtained the fundamental expression on the speed of electromagnetic field in the dielectrics - as the following equality

Our comments:

These exercises in formal interplaying of three Maxwell equations as of already averaged ones - but in reality just declared as "averaged," can not be considered as the averaging of microscale, atomic scale equations found from that set of electrodynamics equations.

Interesting to note that the immediate transformation of field into the is not considered as the fault, error, etc.?

*******************************************************************

Then, at last the final text describing the no-one approach to averaging of microscopic equations by Landau and Lifshits (1957, 1960) we might cite from page 466 that:

"Electromagnetic quantities present in macroscopic electrodynamics, having obtained as the result of averaging that can be imagined as the collection of two operations. If to accept, for graphical picture, from the classic point of view then that could be distinguished the averaging over the physically infinitely small volume when the locations of all particles are assigned and then by averaging of that value over the particles movements."

Our comments:

What does it mean - "averaging over the physically infinitely small volume" of a heterogeneous multiphase substance? Over the infinitely scaled down morphology of the same kind? Each part, each phase should be infinitely scaled down?

That was not a question in all over this book.

Levich's "Averaging" of Atomic Scale Maxwell-Lorentz Equations

Levich (1969) wrote on the Macroscopic form of Maxwell set of equations-

before averaging

for vacuum with charges-particles

for vacuum in the CGSE system.

In pgs. 684-5 (Vol. 1) one can read that for microscale, atomic scale Maxwell-Lorentz equations we should use the following set:

Our comments:

This set of local electrodynamics equations in a medium which consists of "vacuum" and point-particles - nuclei, electrons, we can find in Lorentz' equations - that Lorents meant in writing these kind of equations that the electrons and aether constitute the same medium just with the space depending properties with a smooth transition boundaries in between. And the electrons had a volumetrical character in his model.

Here in the set of post QM times - the electrons are considered as points (just no volume and boundaries!) and in the matter is just the "vacuum" in between.

Further in p. 685 we can read that:

"Providing averaging of the Maxwell-Lorentz equations over the physically infinitely small volume and time interval , we determine averaged values via

Then we have

Let's introduce the following definitions

and will name the mean value of strength of electric field by strength of the field but the mean value of magnetic field strength in the medium by magnetic induction (this name of the mean magnetic field is connected exclusively to the historic tradition)."

Our comments:

Well, in this way of averaging when the that is averaged incorrectly and declared as averaged volumetrically - Oh, yes, it is pretty convenient to do this!

At the same time - this is incorrect averaging. The equality is the great streching in atomic physics.

Because in CGSE in the already averaged induction equation; but assuming that is not the same as and making instantly just in the induction equation alone?! How is it by pure averaging of this equation, let this explain to us someone with clarity?

Also, there is no description on - What is the averaging volume? and

How it exactly will be averaged - if the bounding averaging volume surface is cutting through the particles, charges, atoms, molecules, etc.?

They did not need the form, shape and bounding surface of the REV, because they did not describe that REV at all!! Can we imagine this now?

And there is no correct way of saying that this surface will be moving all the time with the closest (and not even closest?) to it particles, atoms, etc., just to include all the previous particles within the

More on that - they just do the integration! In the first part of XX century they did not bother at all - how to do that averaging? They don't do averaging in most cases at all, a mean integral value, no - they do just the unspecified volume integration! That's it, no deletion on the volume of REV?! Unbelievable, but people are so accustomed to this incorrect 'averaging," that do not note - What is this mathematics all about?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Then in p. 686 (Vol. 1) one can continue to read that:

"Then, the equations (1.7) - (1.11) take forms

Our comments:

What a simple averaging !

That is why nobody wanted throughout these 40 years after 1967, when firstly appeared the Heterogeneous Gauss-Ostrogradsky and other heterogeneous media theorems, to recognize the correct averaging over the heterogeneous subjects and media. Because of the averaging mode that had established itself in atomic physics, particle physics as that has being thought that in the 1900th-50s of XX century the good procedures, mathematical methods for averaging the points with nothing in between in an unspecified unknown volume to average the atomic, sub-atomic matters all over the same unspecified volumes.

Also, the last equation ((1.18) should loose the to get to the correct (free charge flow) conservation final form. Let we see how it can be performed in homogeneous volumeless particles atomic physics.

Figure 1: Simplified draft of Bottom-Up consecutive series of Representative Elementary Volumes (REVs) at three scales from a molecular (one of the REVs is in an aqueous solution) to continuum scales.

In Homogeneous physics - like in studied in this sub-section textbooks, there are no specifics, theorems to do averaging over these REVs, and even there are no definitions or wrong definitions (as in Continuum Homogeneous Mechanics) of a REV.

Levich's "Averaging" of Terms and in Atomic Scale Maxwell-Lorentz Equations

Then in pg. 686 (Vol. 1) one can continue to read that:

"Further transformation of equations is based on the finding of the mean values of and . For finding these mean variables there are some assumptions regarding a matter's sctructure should be made."

Our comments:

In p. 687 Levich starts to develop formulae for averaged charge that is all physicists need to do to justify the established conventional formulae for polarization and time dependent (transient) bound current within the substance of interest (matter).

" Dipole moment, obtained by bulk substance, equal, by definition

"The subscript (bound) defines that the mean charge caused by displacement of charges, that bound within atoms of a substance. This appearance of bound or induced charge named electrostatic induction or polarization.

To find out let's use the definition (2.1) and transform the integral to where - is the some function of polarization. Then, comparing with the one can find Using the formula (I,53) and denote in it , , find that it is used also the extension of homogeneous GO theorem as

based on use of

where is the dyadic.

Then we can get (in Levich, 1969)

as soon as

See for reference - Korn, G.A., and Korn, T.M. (1961, Russian edition) - p. 169-170.

Our comments:

We need to note that here in the above mathematics is used the notation of the polarization field as if it is At the Upper scale. That is - the substitution of the microscale values (functions) under integral by the like ALREADY Averaged Macroscale function!

Then Levich (1969, V.1) writes the same trick conventional physicists did while averaging over the unspecified REV and even for the whole body under investigation.

He writes - "Choosing the surface of integration outside of volume occupied by body (subject matter), we have

as soon as outside body polarization in equal zero."

Well, but this is the , , and at the same time only for the bulk, the total body assessment is valid procedure. How physicists are going to agree with this for 60-80? years? The matter is about the "point" or the "dot" of physical matter estimation. And by this way they suggest us to integrate over the whole body, even outside of it?

"That is why

Comparing with

Our comments:

Meanwhile, this should be an averaged value! What a trick?

We would like to point out again - that here used the notation of the polarization field as if it is And is at the Upper scale. That is - the substitution of the microscale values (functions) under integral by the like ALREADY Averaged Macroscale function !

Because really should be

but we can not get functions out of integral sign BEFORE integration.

Nevertheless, in Levich (1969) presented the designation of the following equalities (for bound charges)

which is looks like in the domain - artificially, for this equation, then we can write

Meanwhile, even in brackets of assumed bulk outside and acceptance that correct will be the expression

Going further in p. 689-695 one can find the mathematical shifts used for derivation of needed up-front formulae for averaged product

Starting in the p. 689:

"A bit more complicated is the task of calculation of the mean density of current in the medium This calculation might be performed on the basis of some moleding ideas, or, that is more formal, having general representations on electromagnetic properties of a medium.

We will choose the second path, inasmuch as the models of atoms and molecules that being used usually along presentations of macroscopic electrodynamics not only far from reality,..."

Starting in the p. 690:

"The mean current density in a medium in each point of a body is the function of strength of electric and magnetic fields. Apart of that, if these fields are changing in the space and in time, the mean current density will depend of speed of changing of vectors and in time and of spatial derivatives of these variables, i.e.

As soon as the fields apparently assumed to be weak, it is possible to expand function into the series over powers of the variables and restrain the expansion by the first power terms."

In the p. 691:

"Then to the accuracy of the first order terms

where , and - are scalar values, dependent on the medium properties. The reason for introduction into the last additive term the factor of light speed , will be obvious from the further text.

The nil term in the expansion is absent, because the absence of field should correspond the current equal to zero."

Our comments:

Author say straight in the explanations of the approach that the whole procedure and idea is of approximate nature, even the light speed is taken to fit the future substitution into the like "averaged" Amper-Maxwell law equation.

In the further derivation Levich takes course to introduce a few coefficients and heavy use of the same unrealistic bulk assumption of surface integration over the outside of studied piece of material (pgs. 693 in , 694) just as of the frivolous measure to get to the needed form of final mathematical expressions, as we respond above in comments.

In the p. 694:

"If the surface of integration passes outside of the volume occupied by the body, then at this surface .."

Then, at last, in the p. 695 one can find the resulting expression for :

"Exploring values , , , the mean density of current in a substance can be finally written in the following form

Our comments:

In a couple of pages author starts to warn reader on the domains of validity of derived in this way equations of Maxwell-Heavisde-Lorentz for a continuum mechanics applications.

In the p. 697:

"We would like to stress out that in comparison to the equations of Maxwell-Lorentz (for microscopic atomic scale electromagnetism- our note), that is among the most accurate and universal nature laws, the system of Maxwell equations has a restricted area of applicability due to limitations of domain of constitutive equations usage. "

In the p. 701:

"Let start investigation of assumptions, that were made for Maxwell equations derivation. Th emain assumption is the expansion (3.2)."

"....The more complicated is the issue applicability of expansion (3.2) in cases of high frequency and spatially-heterogeneous systems."

Our comments:

We will extend our analysis of theoretical procedures used by Levich to assure students that the averaging and communication between atomic and continuum scales equations can be correct - which we saw that was not true any more after heterogeneous scaled procedures (HSP-VAT) have been created in 60s - 90s.

Electromagnetism Modeling and Maxwell-Lorentz Equations in Vlasov's Theory of Long-Range Collective Phenomena (Averaging)

The Vlasov equation widely is known in circles of plasma physics and statistical mechanics professionals along with mathematical physics specialists.

Among many worthwhile arguments Vlasov had put forward regarding processes of collective character by physics itself, not by the theoreticians wish the few are most appealing and sound even today when researchers, people still are not getting through to the issue:

1) Theory of pair collisions formally not applicable to Coulomb interaction.

2) Theory of pair collisions is the Local events describer.

3) Long-range Coulomb interaction existence in many-many physical fields and processes (not only of plasma physics, as it is written on his intentions), is the strong reason to seek for another model, than the Boltzmann equation.

Well enough, it is known meanwhile, that Vlasov was run after even for that his arguments list above.

To this list we can add many issues, but of the only these two right now - And not only of Coulomb interaction. What about other known types of long-range collective interactions?

Another one point is that as it was written relatively recently and is not in a wide debating mode that - the Boltzmann Equation is Invalid in the Derivation. More than 100 Years of Misled Research?

The Vlasov equation can be seen as the first swallow in a series of theories, modeling schemes for a Long-Range interaction phenomena finalized with the new type of modeling equations - the integro-differential ones. In physics first of all, as well as in other sciences. This theory must be analyzed with regarding of its motivations, physical model, methods of mathematical approach and performance, and of its results so far.

Vlasov's set of equations is being presented mainly as the kinetic equation describing the behavior of a single(?) particle distribution functions of a plasma under the influence of electric and magnetic fields.

Just in case of questions - short summary is this - the Vlasov's theory and equations while are in the right direction and with good techniques, they are still of an insufficient base for a collective interaction description by physical and mathematical models.

In line with the statistical mechanics approach and equations for point-particles movement the Maxwell-Lorentz (ML) equations had been brought to the "collective self-consistent" electromagnetic fields array as the full standard set of ML equations where electric field E(r,t) and magnetic induction field B(r,t) are named as the fields of collective interaction (quasi-averaged). That form means that the Maxwell-Lorentz equations were not averaged neither in their atomic format nor in a continuum mechanics appearance. The effect of collective behavior is introduced into ML equations, model implicitly via the Vlasov equations themselves.

The techniques are still not averaging, but taking into account the collective phenomena - that we call quasi-averaging.

This piece of analysis to be continued with some mathematics.

Jackson (1999) Pseudo-Averaging of Atomic Scale Maxwell-Lorentz Equations

In p. 248 Jackson (1999) starts the attempt to connect the microscale - atomic actually, and the continuum mechanics model of Maxwell-Lorentz equations, starting writing the macroscale Maxwell equations.

"6.6 Derivation of the Equations of Macroscopic Electromagnetism

The discussion of electromagnetism in the preceding chapters has been based on the macroscopic Maxwell equations,

where and are the macroscopic electric and magnetic field quantities, and are corresponding derived fields, related to and through the polarization and the magnitization of the material medium by

Our comments:

Which is the artificial move with regard of . If one could attentively follow the further transformations of atomic scale Maxwell-Lorentz equations to the macroscopic scale equations one would easily locate the spot where the magnitization field when substituted within the Ampere-Maxwell for induced magnetic field equation

will forcibly return the formulation with field in the left hand side of the above equation to the original notation with field that should be held just before averaging

Well, it would be naturally writing that

instead of

Further Jackson writes in the same page 248:

"Similarly, and are the macroscopic (free) charge density and current density, respectively. Although these equations are familiar and totally acceptable, we have yet to present a serious derivation of them from a microscopic starting point.

This deficiency is removed in the present section. The derivation remains within a classical framework even though atoms must be described quantum-mechanically. The excuse for this apparent inadequacy is that the quantum-mechanical discussion closely parallels the classical one,...."

"We consider a microscopic world made up of electrons and nuclei. For dimensions large compared to , the nuclei can be treated as point systems, as can the electrons. We assume that the equations governing electromagnetic phenomena for these point charges are the microscopic Maxwell equations,

Our comments:

Well, comparing to the of really vacuum with point charges specified Maxwell-Lorentz equations as for homogenized medium, for example, these

and where the Faraday law of induction equation is

;

or as above

one can notice that there is difference of the three equations - first, second and the fourth.

In the Gauss law for magnetic field equation which can be written as while in this and in other two equations scaling treatment dramatically change the averaging output for a poly-"phase" atomic system, as we will sea below.

Jackson specifically hides here the real initial form of atomic scale equations and writes equations in , quantities (why not in al , , not in because this would bring the pseudo-averaging done for homogeneous fields closer and easier to the Lorentz' writing of macroscale EM equations.

In p. 249 we can read the explanations for this form of microscale equations as of before averaging, specific for Homogeneous electrodynamics reasoning:

"where and are the microscopic electric and magnetic fields and and are the microscopic charge and current densities. There are no corresponding fields and because all the charges are included in and "

Our comments:

What if the charges are included into equations (3) and (4)?

The charge presence is not a justification or reason for substitution of averaging mathematics; the charges are included because the particles are artificial - meaning the particles are smeared all over the volume.

Meanwhile, there is no atomic scale magnetic induction dot-point field unless we either making a "SPECIAL" theory related to a separate particle (electron, nucleus) and only, or have been averaged the field over the some specified volume (that would include the "vacuum" and the particles, but not a homogenized unknown one "phase" which is called vacuum (and which is the mix of "vacuum" and of smeared particles presented as volumeless "dot-points") - as of an atom or of larger dimension as the REV and have already received that field as a result of averaging with all components of this field, but obtained as by averaging and shown as averaged either over the atom or the REV.

Otherwise, the instant unjustified change of with is just the "fixing" to the Lorentz form of continuum scale equations.

Also, why not having the same substitution with regard of the ? Because anyway there is no polarization field so far in the equation?

Well, the same should be said and about - there is no still magnetization in the equations at atomic scale, right? Why to change with in this case?

Further in p. 249 we can read:

"A macroscopic amount of matter at rest contains of the order of electrons and nuclei, all in incessant motion because of thermal agitation, zero point vibration, or orbital motion. The microscopic electromagnetic fields produced by these charges vary extremely rapidly in space and in time.

The spatial variationsoccur over distances of the order of or less, and the temporal fluctuations occur with periods ranging from for nuclear vibrations to for electronic orbital motion. Macroscopic measuring devices generally average over intervals in space and time much larger that these.

All the microscopic fluctuations are therefore averaged out, giving relatively smooth and slowly varying macroscopic quantities, such as appear in the macroscopic Maxwell equations."

Further in p. 249 we can read:

"The question of what type of averaging is appropriate must be examined with some care. At first glance one might think averages over both space and time are necessary. But this is not true. Only a spatial averaging is necessary.

(Parenthetically, we note that a time averaging alone would certainly not be sufficient, as can be seen by considering an ionic crystal whose ions have small zero point vibrations around well-defined and separated lattice sites.)

To delimit the domain where we expect a macroscopic description of electromagnetic phenomena to work, we observe that the reflection and refraction of visible light are adequately described by the Maxwell equations with a continuous dielectric constant, where as x-ray diffraction clearly exposes the atomistic nature of matter. It is plausible therefore to take the length Å as the absolute lower limit to the macroscopic domain.

The period of oscillation associated with light of this wavelength is s. In a volume of there are, in ordinary matter, still of the order of nuclei and electrons. Thus in any region of macroscopic interest with there are so many nuclei and electrons that the fluctuations will be completely washed out by a spatial averaging.

On the other hand, because the time scale associated with is actually in the range of atomic and molecular motions, a time-averaging would not be appropriate. There is, nevertheless, no evidence after the spatial averaging of the microscopic time fluctuations of the medium."

Our comments:

Well, we can comment with assessments for close by the sub-volumes REVs for continuum media averaging.

If in a REV we have then in REV

then in a REV

That is, in all these two macroscale REVs simulated above - of and of there are tremendous number of particles, atoms, small molecules.

In p. 249 we can observe the firstly introduced the concept of averaging in Homogeneous electrodynamics:

"The spatial average of function with respect to a test function is defined as

where is real, nonzero in some neighborhood of and normalized to unity over all space. It is simplest, though not necessary, to imagine to be nonnegative. To preserve without bias directional characteristics of averaged physical properties, we make isotropic in space. Two examples are

In p. 250 we can observe

"Since space and time derivatives enter the Maxwell equations, we must consider these operations with respect to averaging according to (6.65). Evidently, we have

and

The operations of space and time differentiation thus commute with the averaging operation."

Our comments:

We ought to declare the complete ignorance also and of this rather well known author regarding the averaging mathematics as it was developed since 1967.

The matter of fact is that in the (6.67) both equations are incorrect and the following wrong statement about the commutation of averaging and operators is the base of the hundred years falsification in atomic and macroscopic as they say physics.

We need to write here regarding these pseudo-averaging rules the very important two notes and observe the following two issues:

1) The first is that the real averaging volume - Representative Elementary Volume (REV) has never been described or even briefly illustrated in homogeneous physics.

Even there is no figure that students can locate in textbooks (if anyone find and point me out the place of publication, I will comment on that).

And this is common in orthodox homogeneous physics - researcher, reader or student can be easily driven into a variable false constructions in need at any moment. Mostly in atomic and particle physics this is the frivolous REV - most often the REV that is so large that it embraces the whole subject, piece of matter, material that is under investigation. This is done to declare many surficial integrals equal to zero.

Figure 2: A computer generated picture of simulation Volume at molecular scale with the rigid for data reduction bounding surfaces. No one commercial software package can do the HSVAT for Heterogeneous and Atomic scale simulation.

The very important reason to hide and not specify the REV for this kind of averaging is obvious from the averaging construction (by Lorentz) of Atomic scale (microscale) Maxwell-Lorentz equations when for the key averaged terms and at the end needs to be fixed (fitted) the mathematical expressions for surficial integral in a way that nullify the very important terms that we need to retain for correct integration.

In this way the equations are brought to the homogeneous continuum media form obtained by Lorentz and that form is incorrect.

All subsequent effort in atomic physics should followed these simplifications by Lorentz to get the same bulk continuum media electrodynamics equations - as we know them for more than 110 years.

This misfortunate situation lasts for so long because the proper mathematical tools for heterogeneous media averaging were not at disposal of Lorentz.

2) The formulae hide the important features of averaging for multiphase heterogeneous media.

Generally, these formulae are not correct for Heterogeneous media. That is why the orthodox conventional physics for so many years since 1967 and during the following in the 80-90s developments in the HSP-VAT tried to hide, suppress and silence the truthful physics and mathematics of multiphase microscale theory, modeling with averaging and scaleportation, presented in the HSP-VAT methods and math.

Because in this way all these constructions of averaging in atomic, particle physics, and continuum mechanics as given above for only this example of averaging Maxwell-Heaviside-Lorentz equations, are falsified.

And they have been falsified anyway, because the correct physical and mathematical methods of HSP-VAT already have proved for many problems that homogeneous methods just bring approximate and often misleading incorrect results..

The thing is that the averaging for heterogeneous and scaled matter is the quite different operations comparing to what was accepted at the edge of XX century in physics and math, see basics in our -

and in

At the same time - it would be useful to remind here in more detail what is in many sciences including homogeneous electrodynamics understood under the term "averaging."

It is just

for whatever function or operator one might has.

Meanwhile, if the medium consists of at least two constituents then we need to derive the REV averaged function over for example, by integration

with the all heterogeneous, scaled consequences of that for derivatives, operators, and equations.

Further in p. 250 we can observe:

"We can now consider the averaging of the microscopic Maxwell equations (6.64). The macroscopic electric and magnetic field quantities and are defined as the averages of the microscopic fields and

Then the averages of the two homogeneous equations in (6.64) become the corresponding macroscopic equations,

Our comments:

We must state again that are the wrong averaging expressions and equations for our polyphase media in the REV.

In p. 251 we can continue to observe the path with which author tries to pseudo-average the other two microscale equations:

"The averaged inhomogeneous equations from (6.64) become

Comparison with the inhomogeneous pair of macroscopic equations in indicates the already known fact that the derived fields and are introduced by the extraction from and of certain contributions that can be identified with the bulk properties of the medium. The examination of and is therefore the next task."

Our comments:

These averaged equations (6.70) are also pseudo-averaged. This is the incorrect averaging for heterogeneous media.

We are witnessing a trick in this path - first in there were just the substitution of instead of , then now in the same like "averaged" quantity get back substituted to in the following one-two steps in derivation, while the pseudo-averaged is declared as that it is in the final "averaged" equations the quantity will appear to be the "derived" field, along with the

Schwinger et al. (1998) Pseudo-Averaging of Microscale Maxwell-Lorentz Equations

In the textbook by Schwinger et al. (1998) given the good example of reasoning, methods used in homogeneous physics to derive (and by this to validate, justify) the Maxwell-Lorentz electrodynamics equations for a continuum. Everyone was able to analyze these rather complicated mathematically ideas and methods used for "homogenized" mixing of atomic scale particles and of the "vacuum" to get Up to the continuum media electromagnetism governing equations.

We would extend our consideration of this theory in the pairing sub-section -

where the contribution to the multi"phase" Scaleportation theory for electrodynamics equations, models between atomic and continuum descriptions have been worked out.

Finally authors bring all the formulae working for the idea to get the Lorentz averaged electrodynamics equations.

In p. 42:

" The final form of the historical, macroscopic Maxwell equations is

We are approaching this interesting and deep text on atomic to continuum mechanics Maxwell-Lorentz transformation basics with the open and adverse analysis in our accompanying sub-section -

R.Mills' Theory of Classical Quantum Mechanics (CQM) and Maxwell Equations as a Communicator Between CQM and the Rest of Physics

One can read from the abstract of Mills (2003) paper:

"Despite its successes, quantum mechanics (QM) has remained mysterious to all who have encountered it. Starting with Bohr and progressing into the present, the departure from intuitive, physical reality has widened. The connection between QM and reality is more than just a "philosophical" issue. It reveals that QM is not a correct or complete theory of the physical world and that inescapable internal inconsistencies and incongruities arise when attempts are made to treat it as physical as opposed to a purely mathematical "tool.""

"In an attempt to provide some physical insight into atomic problems and starting with the same essential physics as Bohr of moving in the Coulombic field of the proton and the wave equation as modified by Schrodinger, a classical approach is explored that yields a remarkably accurate model and provides insight into physics on the atomic level."

"Physical laws and intuition may be restored when dealing with the wave equation and quantum-mechanical problems. Specifically, a theory of classical quantum mechanics (CQM) is derived from first principles that successfully applies physical laws on all scales. Rather than using the postulated Schrodinger boundary condition " as r ," which leads to a purely mathematical model of the electron, the constraint is based on experimental observation. Using Maxwell's equations, the classical wave equation is solved with the constraint that the bound (n = 1)-state electron cannot radiate energy."

Our comments:

Many good things are versed here in the abstract. The promise is great.

"The result gives a natural relationship among Maxwell's equations, special relativity, and general relativity. CQM holds over a scale of space-time of 85 orders of magnitude --- it correctly predicts the nature of the universe from the scale of the quarks to that of the cosmos."

Our comments:

That is seemed of too much of scales and of too much of physics included within these scales to make this connection all happens. Also, and the main drawback of this statement, that the different scales physics, objects, phenomena are of heterogeneous nature - so, it can not be described with the OSFA (One Scale For All) governing, modeling equations.

About 85 orders of magnitudes it is an excessive claim.

In p. 436 author writes on peculiarities of atomic physics:

"2. CLASSICAL QUANTUM THEORY OF THE ATOM BASED ON MAXWELL'S EQUATIONS THAT HOLDS OVER ALL SCALES

In this paper the old view that the electron is a zero or one-dimensional point in an all-space probability wave-function Y(x) is not taken for granted. The theory of CQM, derived from first principles, must successfully and consistently apply physical laws on all scales. Historically, the point at which QM broke with classical laws can be traced to the issue of nonradiation of the one-electron atom that was addressed by Bohr with a postulate of stable orbits in defiance of the physics represented by Maxwell's equations.

Later, physics was replaced by "pure mathematics" based on the notion of the inexplicable wave-particle duality nature of electrons, which lead to the Schrodinger equation, where the consequences of radiation predicted by Maxwell's equations were ignored. Ironically, both Bohr and Schrodinger used the electrostatic Coulomb potential of Maxwell's equations but abandoned the electrodynamic laws."

In p. 437 author continues:

"The mass energy and angular momentum of the electron are constant; this requires that the equation of motion of the electron be temporally and spatially harmonic. Thus the classical wave equation applies and

where is the time-dependent charge density function of the electron in time and space. In general, the wave equation has an infinite number of solutions. To arrive at the solution that represents the electron a suitable boundary condition must be imposed. It is well known from experiments that each single atomic electron of a given isotope radiates to the same stable state. Thus the physical boundary condition of nonradiation of the bound electron was imposed on the solution of the wave equation for the time-dependent charge density function of the electron. The condition for radiation by a moving point charge given by Haus is that its space-time Fourier transform possess components that are synchronous with waves traveling at the speed of light."

"Conversely, it is proposed that the condition for nonradiation by an ensemble of moving point charges that makes up a current density function is as follows: For nonradiative states, the current density function must not possess space-time Fourier components that are synchronous with waves traveling at the speed of light."

"Thus an electron is a spinning, two-dimensional spherical surface (zero thickness1), called an electron orbitsphere, that can exist in a bound state at only specified distances from the nucleus, as shown in Fig. 1. The corresponding current function shown in Fig. 2, which gives rise to the phenomenon of spin, is derived in Section 4."

Our comments:

No matter what is in the CQM theory is right and what is wrong, the acute issues, questions raised by other interested physicists, mathematicians should be either overthrown or explained. In line with critics from participants of http://tech.groups.yahoo.com/group/hydrino/message/11316

this is a partial list of serious questions toward the theory of CQM:

1. The claimed function f(r) for the radial component of the solution to the wave equation does not satisfy the differential equation.

2. The computation of the orbitsphere radius is wrong, with the consequence that the computation of the potential energy is also wrong.

3. The computation of angular kinetic energy is wrong.

4. The two computations of angular momentum are both wrong.

5. The computed energy of trapped photons is wrong.

6. The computation of mass/charge density of the orbitsphere

is wrong.

7. The computed moment of inertia is wrong.

8. There is no coherent explanation for resonant transfers of energy.

9. The 'free' electron configuration is not a solution to the 'wave equation'.

10. The computations of spectral lines are done incorrectly.

Our further comments:

The number of issues in CQM also are of serious value for understanding of physical realities, not modeled with SQM, or modeled without connection to reality.

Among them what is welcomed have a long rather list of things that describe at least the attempts to model phenomena with the nature recognized tools. We would enumerate some as:

1) The claim to present particles, at least electron, photon as the objects in 3D space with behavior partially, at least, complied with the Maxwell-Lorentz modeling equations. There are issues remained about an orbitsphere and other.

2) The attempts to describe phenomena with help of classical deterministic tools. It is known that the deterministic picture of the events, phenomena in physics or elsewhere always are in change of statistical (statistical mechanics) methods when the amount of knowledge about that phenomena of objects is accumulated above the certain level when it is preferable or even obligatory to turn to more elucidative and accurate deterministic (causal) methods, tools.

What is objectionable in the CQM Theory regarding of what is being suggested:

1) The presentation of particles still as the point objects.

This contradicts to the idea of particles as the physical objects - the points are not the physical objects. Those are just the points with the assigned BY US PROPERTIES.

2) Directly related to the above is the issue of using the Maxwell-Lorentz equations for modeling of the POINTS?

3) The attempts to deal with the collective behavior of electrons, other particles in a surrounding media as things that could be described as at the same scale and with the homogeneous theoretical approach.

CONCLUSIONS:

1) All the effort spent by physicists in XX century when for justifying the appearance of Maxwell-Heaviside-Lorentz electrodynamics continuum scales equations was to find out the appropriate or approximate mathematical procedures which can be called "fixing" for making way to pseudo-average the atomic scale Lorentz electromagnetism equations and present them looking as the macroscale equations also by Lorentz, but for which Lorentz could not use the correct averaging (scaled) mathematics and physics that appeared at the end of XX century.

Up to now physicists had not used the correct scaled averaging methods for atomic scale Maxwell-Heaviside-Lorentz electrodynamics equations.

2) Because the problem of averaging of the array of moving atoms, molecules, free electrons embedded in a medium that can be called vacuum (and is not really empty space) is the problem of scaled heterogeneous physics, it should be treated with the tools of that physics including first of all the various Volume-Surface integration theorems, developed for Heterogeneous media.

That is why the methods used in homogeneous physics must fail and have been failing for >130 years to develop the correct macroscale medium electrodynamics governing equations.

3) What is used right now in physics as the macroscale continuum medium, even homogeneous, Maxwell-Heaviside-Lorentz electrodynamics equations is the statement that adjusted (and we have shown how it is usually done) to the form developed by Lorentz and that is the incomplete governing equations set for electromagnetism phenomena.

4) Generally, these averaging formulae and pseudo-averaged EM governing equations used up to now in homogeneous microscale electrodynamics, are not correct for atomic scale, for the Upper scales averaging, for Heterogeneous media. That is why the orthodox conventional physics for so many years since 1967 and during the following in the 80-90s developments in the HSP-VAT tried to ignore, suppress and silence the truthful physics and mathematics of multiphase microscale electrodynamics theory, modeling with averaging and scaleportation, presented in the HSP-VAT methods and math.

Because of this way of homogeneous averaging in atomic and particle physics as given above for averaging of Maxwell-Heaviside-Lorentz equations, the Upper scale (continuum mechanics) equations have been falsified and incomplete.

All conventional textbooks on electrodynamics and materials science are showing the same type of incorrect mathematical procedures for heterogeneous averaging.

Meanwhile this falsified electrodynamics that is being adjusted for every case, lies in the very core of the Conventional Orthodox Homogeneous physics (COHP) - from particle and atomic physics up to astrophysics.

5) Lorentz himself in his "Clerk Maxwell's Electromagnetic Theory. The Rede Lecture for 1923, Cambridge" (1923) used to say that:

"Will it be possible to maintain these equations? I am not thinking here of the comparatively slight modifications that have been necessary in the theory of relativity;.....

A greater and really serious danger is threatening from the side of the quantum theory, for the existence of amounts of energy that remain concentrated in small spaces during their propagation, to which several phenomena seem to point, is in absolute contradiction to Maxwell's equations.

However this may be, even if further development should require profound alterations, Maxwell's theory will always remain a step of the highest importance in the progress of physics."

6) There would be many formulations in electrodynamics for different matters and for different scales. Not as of the case that used right now which is the application of the same kind of Lorentz' Continuum Mechanics electrodynamics equations for any scale and any medium. Just using changed coefficients that are the adjusting parameters right now.

7) We have found the ways to obtain the Upper macroscale of Continuum Mechanics modeling electrodynamics equations all based on correct heterogeneous mathematics applied to the atomic scale Equations of Maxwell-Heaviside-Lorentz -

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