Reformulation instead of Renormalizations

[JustinLevy]

Thank you for your responses.

Historically the Maxwell equations were written in terms of field tensions. Very soon the vector-potential was introduced for some conveniences. So it is the properties of the vector potential that follow from the Maxwell equations, not on the contrary.

Yes, that is the historical order.

However, since we will be working with Lagrangians and Hamiltonians in CED, where it is the potentials that are given higher status (as coordinates, where as the E and B fields are not coordinates and are just defined in terms of the potentials), in the context that Maxwell's equations and the Lorentz force law are derived from the Lagrangian or Hamiltonian, I hope we can agree that my wording there are least makes sense.

As you say though, quibbling about such wording is pointless. The mathematical definition relating the fields to potentials is a definition either way.

Thus, in order to answer properly your questions I have to note the following: The Maxwell (field) equations together with particle equations work fine in two limiting cases:

I. The fields in the particle equation are known functions of space-time, so the Lorentz force is known (case of external fileds). Then we look for trajectories.

II. The charge-current distribution is known function of space-time, i.e., the source terms for fields are known (external sources). Then we look for the field solution.

In temrs of Lagrangian of interaction it corresponds to two cases: jA = jA_ext and j_extA, where j and A are four-vectors. This covers practically all CED applications.

I am really hesitant here. I do not want to derail this discussion by arguing over pathologies in CED. But I do want you to at least understand where I am coming from.

So for now, we can agree to disagree, but here is my stance on these:

The theoretical question raised by H. Lorentz was to make ends meet with the energy-momentum conservation for a radiated particle. As long as its equations contained only an external filed (magnetic, for example), no particle energy losses were taken into account.

This comment makes no sense. The equations do NOT "only" contain an external field. The equations very clearly contain source equations for the fields (or in the Lagrangian context, j.A not only provides a term in the force equation, but also for the evolution of the fields).

Said an even more concretely way:
The Lagrangian has time translation symmetry and space translation symmetry. Via Noether's theorem, this very clearly has energy-momentum conservation in any situation. Nothing needs to be added.

So he decided to develop further the particle equations and introduced new terms.

We just agreed on what the equations of CED are.
To then complain that a different set of equations containing the Abraham-Lorentz force can cause problems is immaterial.

Because the Lagrangian has energy-momentum conservation, there CANNOT be any run-off solutions. Any such solutions must be due to mathematical error.

Anyway, any attempts to get rid of non-physical solutions were reduced to modification of CED.

No, it is the other way around. Non-physical solutions were the result of modification of CED as listed above. CED expressly forbids run-away solutions as shown by Noether's theorem.

Okay. That is my stance on CED pathologies.
We can agree to disagree. I don't want to argue about those; I just want you to understand where I am coming from. I also don't want to argue about these because the motivations of your theory are not important for this discussion, just the details of what your theory is.

So... moving on.

You and me will speak of another way of preserving the conservation laws, OK?

Yes, let us move onto your approach now.

If you are agree, I will continue answering tonight. By the way, "skimming" my article is not sufficient. I tried to explain everything in details including canonical coordinates and momenta.

I'm sorry, but the article really wasn't clear enough for me to extract the answer to those bolded requests myself. I was hoping someone else would jump in and help since your paper is online for everyone, but despite the "read counts" for the thread going up, no such luck. Maybe no one is willing to answer for fear of stepping on your toes if they misunderstood the paper as well.

Anyway, if you could just answer the last two bolded requests, then the content of your theory will be clear enough to me so that I can start playing with it myself. It is not even necessary to make any comments beyond those, for once I have that it will allow me to work through everything myself so that the results in the paper will hopefully make much more sense to me.

So yes, let's please move on to the meat of the discussion.
Thank you.

 

[Bob_for_short]

Please state precisely what your Lagrangian is, and what you consider the coordinates.

Let me consider your CED Hamiltonian where everything is clear to you.

<br /> \mathcal{H} = \frac{(\vec{P} - q \vec{A})^2}{2m} + q \phi + \frac{1}{2}\int (\epsilon_0 E^2 + \frac{1}{\mu_0} B^2) \ d^3r<br />

Here, as you say, there are not only external but also the radiated fields. Let me consider the simplest case - without external magnetic filed and with a constant and uniform external electric field E_ext. Then φ_ext = E_extr. The external electric field is not present in the integral since it is a known function. The unknown variables of the radiated field E_rad and B_rad can be decomposed into independent oscillators with conjugated coordinates Q_k and momenta P_k.

Although this problem is simple, you will never find its solution. You will try later, OK?

Now, what I propose is

1) to omit the radiated field A_rad from the first term. Otherwise it brings self-action which is a bad idea,

2) to consider the momentum P as a momentum of the center of inertia of the entire system (electron+radiated filed).

3) to consider field oscillators as internal degrees of freedom of a compound system,

4) to express the electron coordinates r in this compound system via the CI coordinates R (conjugated to P) and the "relative" or "internal" coordinates Q_k. I write it in a free way like r = R + ∑_kε_kQ_k where ε_k are "coupling constants". You can yourself write down my Hamiltonian now.

Then derive the equations of motions. You will see that in this simplest case the CI momentum equation will include only the external field and no proper radiated field will be involved. So the CI equation is solved exactly.

The oscillator equations will contain a pumping term proportional to the known external force (rather that to the unknown electron acceleration). So the oscillator problems are solved exactly too. The external filed accelerates the system as a whole and pumps the internal degrees of freedom due to acting on the electron. Both works of the external force are additive.

Please make the equation derivation yourself in order to feel how this works.

I have a remark to make about the Noether's theorem failure in CED but maybe later on, if you are interested.

 

[JustinLevy]

You did not give me a Lagrangian.
Since you seem to know what the conjugate momentum is and its relation to the ordinary momentum as well as what the coordinates are, if you don't have a Lagrangian, can you PLEASE work the Hamiltonian backwards to present a Lagrangian. I do not feel confident that your manipulations of the Hamiltonian directly, and on top of that changing what coordinates you want to use after the Hamiltonian was written from the original Lagrangian, are valid mathematical manipulations. So I'd really like to see the Lagrangian for all the reasons I stated earlier.

Also, as I already mentioned, if you remove Arad like that, the particle cannot source the usual fields. Yes indeed, that term causes a force on the particle. Yet that same term also causes the particle to source the usual fields.

Remove it and you don't even correctly describe radiation anymore! Heck, you don't even correctly describe things like self-induction anymore.

Since this appears to be such a huge hole in your theory, I am still willing to give the benefit of the doubt that I am misunderstanding your theory. So again I request:
Please state precisely what your Lagrangian is, and what you consider the coordinates.

From this I can directly see the conjugate momentum. I can directly see any conservation laws. I can derive the Hamiltonian myself. I can derive the new "version" of the five CED equations. So please. Even if you don't normally work with the Lagrangian, please work backwards to obtain it. Once you give us all the Lagrangian and what you consider the coordinates, there can be no shred of confusion left.

I do not feel this is an unreasonable request. I feel it would also help many others that are reading this (the read count keeps shooting up) understand your theory more clearly as well. So please, please,
Please state precisely what your Lagrangian is, and what you consider the coordinates.

 

[Bob_for_short]

You did not give me a Lagrangian.

See formula (49b) in my article.

I do not feel confident that your manipulations of the Hamiltonian directly, and on top of that changing what coordinates you want to use after the Hamiltonian was written from the original Lagrangian, are valid mathematical manipulations.

My manipulations are not just variable changes in the frame of CED. In this sense they are not "valid mathematically transformations". It is an ansatz - how to construct Lagrangian and Hamiltonian from physical reasoning. It's an act of creation.

In the discussed formulation (with only one charged particle + an external field + radiated field) there is no Coulomb and magnetic fields created by the charge itself. They are present in the interaction Lagrangian/Hamiltonian of two and more charges (like L_int = ∫j_µ(x)D(x-y)j_µ(y)dxdy, see inter-charge interaction term in the Coulomb gauge)). These quasi-static fields are not physical degrees of freedom that take or give away some energy unlike oscillators and CI. They are absent here because there is no other charge where they could serve as external fields.

One charge does not have a self-induction.

In the Lagrangian formulation there are velocities and coordinates, kinetic and potential energies, as usual.

The Noether's theorem fails in "your" CED because CED equations do not have self-consistent solutions.

 

[JustinLevy]

See formula (49b) in my article.

That doesn't answer the question.

Fine, I will try to "interpret" it myself based on your paper and comments.
After all the substitutions, is the Lagrangian this?
\mathcal{L} = \frac{1}{2}m (\dot{\vec{r}}_{CI})^2 - \phi_{ext}(\vec{r}}_{CI} + \sum_k \epsilon_k \vec{E}_k) + \sum_k [\frac{1}{2} \mu_k (\dot{\vec{E}}_k)^2 - \phi_{ext}(\vec{E_k})]

Where the coordinates are: \vec{r}_{CI} and \vec{E}_k
The values \mu_k and \epsilon_k are experimental constants.
Also, noting that E only refers to the non-external field such that
\vec{E} = \vec{E}_{total /\ &#039;actual&#039;} +\nabla \phi_{ext} + \frac{\partial}{\partial t} \vec{A}_{ext}
And that the actual particle position is
\vec{r} = \vec{r}_{CI} - \sum_k \epsilon_k \vec{E}_k


If, as you suggest, we restrict ourselves to a constant electric field such that V(r) = a r, then this becomes:
\mathcal{L} = \frac{1}{2}m (\dot{\vec{r}}_{CI})^2 - a(\vec{r}}_{CI} + \sum_k \epsilon_k \vec{E}_k) + \sum_k [\frac{1}{2} \mu_k (\dot{\vec{E}}_k)^2 - a(\vec{E_k})]

Solving, I get the following equations of motion:
m \ddot{\vec{r}}_{CI} = -a

\mu_k \ddot{\vec{E}}_k = -a(1+\epsilon_k)

So, ALL frequencies are radiated, and in all directions, and with the power growing unbounded.

And how does the particle move?
We have:
\ddot{\vec{r}} = - \frac{a}{m} + \sum_k \frac{a}{\mu_k}(1+\epsilon_k)


The radiation clearly doesn't match experiment.
Furthermore, it is unclear how to add more particles and magnetic fields.

Please provide the general Lagrangian (multiple particles, arbitrary external field, B_rad field, etc.) for your theory along with what you consider the coordinates to be.
If you restrict yourself to single particle with electrostatic fields ... no one will never be able to calculate your new 'version' of the five CED equations for comparison.

 

[Bob_for_short]

It seems to me you made a typos in the first equation: the field Lagrangian does not contain phi (the last term). The last term should be E².

The second Lagrangian contains the letter a at two places. So your oscillator equation is wrong: there is still no E_k in it. The oscillator equation should be a free oscillator with its proper frequency and a right-hand-side source, proportional to a. Such an oscillator does not radiate with power unbounded. You need initial conditions, for example E_k(0)=0. Then each oscillator will take only a finite part of energy, although each take will last different time (depending on proper frequency), so in total the EMF power grows.

 

[JustinLevy]

So the Lagrangian is this?
\mathcal{L} = \frac{1}{2}m (\dot{\vec{r}}_{CI})^2 - \phi_{ext}(\vec{r}}_{CI} + \sum_k \epsilon_k \vec{E}_k) + \sum_k [\frac{1}{2} \mu_k (\dot{\vec{E}}_k)^2 -(\vec{E_k})^2]

Now the equation of motion in a constant external electric field is
\mu_k \ddot{\vec{E}}_k = -a \epsilon_k - 2 \vec{E}_k

This cannot be correct.

I wish you'd just provide me the full Lagrangian to play with. If you don't want to include multiple particles, fine, but at least include arbitrary external field as well as sourcing of magnetic field. Then I can calculate your new 'version' of the five CED equations for comparison.

 

[Bob_for_short]

You forgot that a is a vector, it is E_ext. The external potential <br /> \mathcal\phi_{ext} is a scalar product proportional to

\mathcal ( \vec{r}_{CI} - \sum_k \epsilon_k \vec{E}_k) \vec{E}_{ext}

There is no run-away solutions. An oscillator with a constant driving force has a physically reasonable solution.

Take your time. Do not try to catch me. Better think of physics.

 

[JustinLevy]

I think you may have seen a previous version of my post. The sign error is corrected. You are correct that 'a' should be a vector. Neither of these change the fact that the result is wrong compared to experiment.

I don't understand why you won't just provide the full Lagrangian for me to play with.
Please?

 

[Bob_for_short]

Now the equation of motion in a constant external electric field is
\mu_k \ddot{\vec{E}}_k = -a \epsilon_k - 2 \vec{E}_k

This cannot be correct...

I think you may have seen a previous version of my post. The sign error is corrected. You are correct that 'a' should be a vector. Neither of these change the fact that the result is wrong compared to experiment.

I don't understand why you won't just provide the full Lagrangian for me to play with.
Please?

Dear Justin,

In each your post you state that "this cannot be correct" or so. Meanwhile you hurry and make errors in elementary derivations the main part of which is contained in my paper. I do not understand why you are interested in the "full Lagrangian" if my approach is wrong in advance in your eyes.

Please explain why "the result is wrong compared to experiment." I feel uneasy to continue without clarifying this question.

 

[Bob_for_short]

Strange results of QFT

There are several threads discussing "rigorous results" of some QFT. For example, it is states that once the quantum fields are distributions, the renormalizations are "necessary" and inevitable. I completely agree with the following reservation: in the present QFTs with self-action, no physical results can be obtained without removing the self-action contributions. It is known since long-long ago, and many-many books have been written on this subject rewording this problem in different ways. The main conclusion of "happy with renormalizations" researchers is the following: the renormalization is not an issue.

At the same time I see a crying problem in such QFTs (QED for certainty): whatever momentum q is transferred to the electron, no soft radiation appears on the tree level in charge-from-charge scattering (Rutherford, Mott, Moeller elastic cross sections). Is it physical? Don't you view this as a problem?

So, in the first non-vanishing order the standard QED predicts events that never happen elastic processes). And it does not predict the phenomenon that happen always (soft radiation, inelastic processes). Don’t you consider this theory "feature" as a complete failure in the physics description? Isn’t it a too bad start for the perturbation theory?

QM predicts probabilities (or cross sections). If the probability of some event is equal to 1 and the theory predicts 0 in the first non-vanishing order, that means a complete missing the point in physics description by physicists in their trial theory. Proving in these conditions that there are no issues, everything is fine and well understood is fooling oneself and others.
No wonder such a practice fails on most cases including QG. When I attract attention to this in the appropriate threads, I obtain infractions. It seems I am the sole person who worries about it.

So I have a sole question to you: why you do not see such a mismatch in the probability prediction as a severe problem of theory formulation?

 

[JustinLevy]

In each your post you state that "this cannot be correct" or so. Meanwhile you hurry and make errors in elementary derivations the main part of which is contained in my paper.

I have been practically begging you to clearly state your full Lagrangian and what the coordinates are for many posts now. Instead you try to lead me around and complain when I interpret you wrong. Yes, I am 'hurrying' because I want to get to my original questions and you seem to be forcing me through an obstacle course before you won't just once and for all clearly lay out what the lagrangian and coordinates for your theory are.

In the calculations I made a sign error that I noticed immediately and fixed myself before you even finished writing your post. And yes, I accidentally didn't mark a constant as a vector. Is this really reason to continue to ignore my simple request?

Please provide the general Lagrangian (multiple particles, arbitrary external field, B_rad field, etc.) for your theory along with what you consider the coordinates to be.
If you don't want to include multiple particles, fine, but at least include arbitrary external field as well as sourcing of magnetic field. Then I can calculate your new 'version' of the five CED equations for comparison.

I do not understand why you are interested in the "full Lagrangian" if my approach is wrong in advance in your eyes.

I am trying to give you the benefit of the doubt, in that there is a possibility that I have a 'straw-man' view of your theory. I want to make absolutely sure I understand what your theory IS before I get into detailed discussion of it with you.

Please explain why "the result is wrong compared to experiment." I feel uneasy to continue without clarifying this question.

If the particle is accelerating in the x direction, are you really claiming there is no electric field in the y or z direction?

Infinite hyperbolic motion (well, parabolic here) can be confusing to imagine, so let me use a much more familiar problem involving radiation.

Imagine a charged particle at the origin, with the external magnetic field on the z axis being zero, and the external electric field E_x=E_y=0 and E_z = a \cos(\omega t).

So we can use \phi_{ext}(x,y,z) = -a z \cos(\omega t).
The charged particle should oscillate along the z axis.

For the x and y components of the electric field, we get the equations of motion:
\mu_k \ddot{E}_k^{(x)} = - 2 E_k^{(x)}
\mu_k \ddot{E}_k^{(y)} = - 2 E_k^{(y)}
while for the z components we get:
\mu_k \ddot{E}_k^{(z)} = a \epsilon_k \cos(\omega t) - 2 E_k^{(z)}

Nothing couples the x and y components of the electric field to the charged particle. This is not correct.
Furthermore, we don't see the correct distribution of radiation (this lagrangian gives only radiation with E_z non-zero). Experiment shows dipole radiation giving a field with radiation going like sin^2(theta) from the z-axis.Either that Lagrangian and set of coordinates is not your actual theory, or your theory is just plain wrong. I am giving you the benefit of the doubt, and assuming I just don't have the correct Lagrangian and set of coordinates. So please, provide the full lagrangian and state explicitly what your coordinates are so that everyone here may learn what exactly your theory IS.

 

[Bob_for_short]

Imagine a charged particle at the origin, with the external magnetic field on the z axis being zero, and the external electric field E_x=E_y=0 and E_z = a \cos(\omega t).

So we can use \phi_{ext}(x,y,z) = -a z \cos(\omega t).
The charged particle should oscillate along the z axis.

For the x and y components of the electric field, we get the equations of motion:
\mu_k \ddot{E}_k^{(x)} = - 2 E_k^{(x)}
\mu_k \ddot{E}_k^{(y)} = - 2 E_k^{(y)}
while for the z components we get:
\mu_k \ddot{E}_k^{(z)} = a \epsilon_k \cos(\omega t) - 2 E_k^{(z)}

Nothing couples the x and y components of the electric field to the charged particle. This is not correct. Furthermore, we don't see the correct distribution of radiation (this lagrangian gives only radiation with E_z non-zero). Experiment shows dipole radiation giving a field with radiation going like sin^2(theta) from the z-axis.

I want to point out that the pumped z-component of oscillator filed propagates mostly along X and Y axes. Do you agree?

Either that Lagrangian and set of coordinates is not your actual theory, or your theory is just plain wrong. I am giving you the benefit of the doubt, and assuming I just don't have the correct Lagrangian and set of coordinates. So please, provide the full lagrangian and state explicitly what your coordinates are so that everyone here may learn what exactly your theory IS.

OK, let me put it in this way:

If you take the classical CED equations and neglect the radiative friction effect (which is really small) you will obtain the same equations as mines, just instead of the known external force it will be the known charge acceleration - they are proportional so it is the same radiation field source. So my results do not differ in this sense from the CED ones if in the latter the radiative friction term is neglected. There is nothing to criticize in my theory. In the standard CED one has to neglect this term because with it the solution is at least difficult to find (and it is actually wrong - run-away solution). In my pet theory the radiative friction term is just absent in the "particle" equation so I obtain easily the right solution for CI and radiation. The electron coordinate r is highly fluctuating but on average it behaves smoothly, like R(t). Now you see, my theory gives nearly the same results as CED. (In fact, I advanced it for QED, not for CED, where the electron fluctuations are quantum rather than classical.)

You may safely use CED equations without self-action (with the radiative friction term neglected) and you will obtain my theory results for fields and averaged results for the electron coordinate. Estimate the relative contribution of the radiative friction in CED and you will see that it is an extremely small value. So the standard CED equations without radiative friction term is the answer to your question about my Lagrangian and charge equations in general, multi-particle case.

Now you see that my purpose was to exclude small but non-physical self-action from the theory and preserve the energy-momentum conservation laws in a more physical way.

Concerning the Noether's theorem. You know, in physics the equations came first and the least action (LA) principle came later. It is OK if the equations have physically meaningful solutions. Then the conserving quantities are well defined. But even in this case the artificial character of the least action (LA) principle is seen easily: after obtaining the equations of motion we never use the future coordinates x(t₂). We use the initial coordinates and velocities. The latter are absent in the least action principle. And x(t₂) are absent as "boundary" conditions since it is non-physical situation - to know future.

So the Noether's theorem derived from LA principle, may formally "work" even when the equation solutions do not exist in the physical sector. This is just the case with CED. An this is what I just fixed by advancing my own approach.

 

[JustinLevy]

I want to point out that the pumped z-component of oscillator filed propagates mostly along X and Y axes. Do you agree?

That Lagrangian only predicts radiation parallel to the x-y plane.
As mentioned, this disagrees with experiment. The fact that sin(theta) is maximum at theta=90 degrees does not mean you can ignore the rest of the distribution.

That Lagrangian disagrees with experiment.

So please, provide the full lagrangian and state explicitly what your coordinates are so that everyone here may learn what exactly your theory IS.

If you take the classical CED equations and neglect the radiative friction effect (which is really small) you will obtain the same equations as mine, just instead of the known external force it will be the known (!) charge acceleration (they are proportional so it is the same field source). So my results do not differ in this sense from the CED ones if in the latter the radiative friction term is neglected. There is nothing to criticize in my theory. In the standard CED one has to neglect this term because with it the solution is at least difficult to find (and it is actually wrong - run-away solution). In my pet theory the radiative friction term is just absent in the "particle" equation so I obtain easily the right solution for CI and radiation. The electron coordinate r is highly fluctuating but on average it behaves smoothly, like R(t). Now you see, my theory gives nearly the same results as CED. (In fact, I advanced it for QED, not for CED, where the electron fluctuations are quantum rather than classical.)

This is getting incredibly frustrating.
You keep repeating your talking points instead of ever telling me specifically what your theory is. If your theory is that lagrangian and coordinates then the simple example above already shows that your theory does not match CED or experiment closely at all.

So I request yet again:
Please, provide the full lagrangian and state explicitly what your coordinates are so that everyone here may learn what exactly your theory IS.

So the standard CED equations without radiative friction term is the answer to your question about my Lagrangian and charge equations in general, multi-particle case.

PLEASE! You know that is not an answer. We already agreed on what the five equations of CED are. There is no explicit "radiative friction" term. Furthermore, repeating your talking point here in no way is explicit enough for me to "guess" your Lagrangian and coordinates.

Please stop giving the run around.
Please just directly answer the question with as much specific math as possible.
Please provide the general Lagrangian (multiple particles, arbitrary external field, B_rad field, etc.) for your theory along with what you consider the coordinates to be.

This thread has shot up to > 3000 views since this conversation started. If you want people to learn your theory, please state the lagrangian and coordinates here so that everyone knows explicitly what your theory is. You keep claiming your theory is almost equivalent to CED, but I see absolutely no indication of that from what math I've seen.

 

[Bob_for_short]

That Lagrangian only predicts radiation parallel to the x-y plane. As mentioned, this disagrees with experiment. The fact that sin(theta) is maximum at theta=90 degrees does not mean you can ignore the rest of the distribution.

That Lagrangian disagrees with experiment.

Take the CED formulation in terms of E_k from post 28 and obtain the field equations as in post 42, please. I bet they are the same as mine.

Then we will move farther. Each thing in its time.

 

[JustinLevy]

Take the CED formulation in terms of E_k from post 28 and obtain the field equations as in post 42, please. I bet they are the same as mine.

Then we will move farther. Each thing in its time.

No, they will not be close at all.
First of all because there is a SOURCE TERM coupling the fields to the particle which the lagrangian above is missing and also because the Lagrangian of CED uses different coordinates than your theory.

There is no need to even do the calculations to see how much a difference these make.
For instance, the Lagrangian of CED leads to Maxwell's equations and the Lorentz force law. The Lagrangian from above does not.

Stop stalling me with more hoops to jump through "first" before you tell me what your theory is.
Please, provide the full lagrangian and state explicitly what your coordinates are so that everyone here may learn what exactly your theory IS.

This is a reasonable request and I've requested it in 9 different posts now. If you wish to discuss your theory in the independent research forum, you should be willing at a minimum to explain what your theory is ... in this case just providing a single equation along with explanation of what the coordinates are.

 

[Bob_for_short]

No, they will not be close at all.
First of all because there is a SOURCE TERM coupling the fields to the particle which the lagrangian above is missing and also because the Lagrangian of CED uses different coordinates than your theory. ... There is no need to even do the calculations to see how much a difference these make.

I still would like to see the CED equations for E_k, please.

This is a reasonable request and I've requested it in 9 different posts now.

Even more times you have declared my approach wrong whereas it was your misunderstanding.

If you wish to discuss your theory in the independent research forum, you should be willing at a minimum to explain what your theory is ... in this case just providing a single equation along with explanation of what the coordinates are.

I want to be understood, I want you to understand my approach. Each time I gave "a single equation", you hurried to declare it wrong physically. With difficulties we advance nevertheless. Now derive the equation for E_k in CED and let us see what is the difference. Without it we cannot advance. If you do not want to follow my advice, I will not be able to prove you anything. Then you may consider my attempt as failed, whatever, I will not care about your groundless opinion.

 

[JustinLevy]

I want to be understood, I want you to understand my approach.

Then give me the full lagrangian and state the coordinates for your theory already.

Even more times you have declared my approach wrong whereas it was your misunderstanding.

That is patently false.
I have been trying to ask you what your theory is for several pages now. That you won't tell me specifically what your theory IS doesn't mean you can blame me when I try to make guesses based on your talking points.

NONE of this would be a problem if you just gave me the Lagrangian and coordinates like I keep asking. Then it would be abundantly clear precisely what your theory is.

I still would like to see the CED equations for E_k, please.

Fine.
This is the last hoop though. I expect you to finally respond in kind to my simple request then.

I don't feel like showing every step, so a google search gave this which provides a similar path with more details
http://www.oberlin.edu/physics/dstyer/AppliedQM/photon.pdf

Let us look at the free field term in the Lagrangian:
F_{\mu\nu} F^{\mu\nu} = 2(B^2 - \frac{1}{c^2}E^2)
In the free field Coulomb gauge, Maxwell's equations in terms of potentials become:
\partial_\nu \partial^\nu A^\mu = 0
these four equations are each individually a coupled set of equations for the field 'coordinates' at each point in space. To make them a set of decoupled equations, we can go into the momentum space instead
\vec{A}(\vec{r}) = \int \vec{A}(\vec{k})e^{+i\vec{k}\cdot\vec{r}} d^3r
-\frac{1}{c^2}\ddot{A}^\mu(\vec{k}) + k^2 A^\mu(\vec{k}) = 0
In the free field, these modes are completely decoupled and appear as harmonic oscillators.

\vec{E}(\vec{k}) = - \dot{\vec{A}}(\vec{k})
\vec{B}(\vec{k}) = i \vec{k} \times \vec{A}(\vec{k})

So the free field terms become:
\frac{1}{2}F_{\mu\nu} F^{\mu\nu} = -k^2 \vec{A}(\vec{k})^2 + \frac{1}{c^2}\dot{\vec{A}}(\vec{k})^2

Previously the CED Lagrangian was written as:
\mathcal{L} = \frac{1}{2}m\dot{\vec{x}}^2 - q\phi + q\dot{\vec{x}} \cdot \vec{A} - \frac{1}{4\mu_0} \int F_{\mu\nu} F^{\mu\nu} \ d^3r
where the 'coordinates' are the particle coordinates and the field coordinates A^\mu(x,y,z)
Now rewriting it in terms of field coordinates at each point in momentum space, we have:
\mathcal{L} = \frac{1}{2}m\dot{\vec{x}}^2 - q\int \phi(\vec{k})e^{+i\vec{k}\cdot\vec{x}} \frac{d^3k}{(2\pi)^3} + q\dot{\vec{x}} \cdot \int \vec{A}(\vec{k})e^{+i\vec{k}\cdot\vec{x}} \frac{d^3k}{(2\pi)^3} - \frac{1}{2} \int [ <br /> \epsilon_0 \dot{\vec{A}}(\vec{k})^2 - \frac{k^2}{\mu_0} \vec{A}(\vec{k})^2] \frac{d^3k}{(2\pi)^3}


Unlike your Lagrangian, where the radiated field doesn't even couple with the particle if the external potential is linear in position, this clearly couples the radiated fields to the particle. Here, the particle can actually source some fields.

Also notice, unlike your lagrangian, this coupling in CED will allow the radiated electric field x and y components to be non-zero.

Each time I gave "a single equation", you hurried to declare it wrong physically.

The ONLY equation you have written here in this entire thread is when you copy-pasted an equation I wrote earlier (a Hamiltonian), then repeated many of your talking points as if that somehow clearly gave me the Lagrangian and coordinates.

I have spent a great deal of my time to satisfy you so that you'd just answer a single question of mine. Will you please, please, finally grant my request:
Please provide the general Lagrangian (multiple particles, arbitrary external field, B_rad field, etc.) for your theory along with what you consider the coordinates to be.

 

[Bob_for_short]

Now rewriting it in terms of field coordinates at each point in momentum space, we have:
\mathcal{L} = \frac{1}{2}m\dot{\vec{x}}^2 - q\int \phi(\vec{k})e^{+i\vec{k}\cdot\vec{x}} \frac{d^3k}{(2\pi)^3} + q\dot{\vec{x}} \cdot \int \vec{A}(\vec{k})e^{+i\vec{k}\cdot\vec{x}} \frac{d^3k}{(2\pi)^3} - \frac{1}{2} \int [ <br /> \epsilon_0 \dot{\vec{A}}(\vec{k})^2 - \frac{k^2}{\mu_0} \vec{A}(\vec{k})^2] \frac{d^3k}{(2\pi)^3}

Unlike your Lagrangian, where the radiated field doesn't even couple with the particle if the external potential is linear in position, this clearly couples the radiated fields to the particle. Here, the particle can actually source some fields.

The radiated field should be caused with the particle acceleration determined in turn with an external force. And a particle in my approach is a part of oscillator. Push the particle and the oscillator gets excited. Very simple and physical mechanism of coupling. You see, you still do not understand or do not know what my approach tells about the relationship (coupling mechanism) of electron and the field. And you want "the total Lagrangian"! For whom I wrote my detailed articles?

Also notice, unlike your lagrangian, this coupling in CED will allow the radiated electric field x and y components to be non-zero.

Each radiated mode propagates along k. You hurry again to judge. I am afraid that you need "the full Lagrangian" solely in order to declare it "obviously wrong", in your understanding. And I want you to open your eyes. You are already close. Derive the equations for E_k. It should be an ordinary oscillator equation with a pumping source. Get it. You will achieve much more than you think.

I have spent a great deal of my time to satisfy you so that you'd just answer a single question of mine.

Me too. We should speak the same language first. What I am trying to get is your comprehension of the coupling mechanism in my theory. There is a conceptual difference to overcome. Then you will be able to judge and appreciate. There is no hoops, we are advancing strait ahead, thanks to my patient efforts.

 

[JustinLevy]

You continue to complain that I don't know what your theory is. Yet you refuse to state in clear math what your theory is. Complaining more to me and repeating talking points is not going to somehow magically impart any clear math about your theory.

If you stated the general lagrangian clearly and what you consider the coordinates, there would be absolutely no room for confusion. Anyone could work out the equations of motion.

I have asked for three pages for you to answer ONE REQUEST.
You are clearly not interested in discussing your theory if you are not willing to write a single equation that would have prevented three pages of "discussion".

Do you deny that if you wrote the general lagrangian for your theory and stated clearly what you consider the coordinates, that the theory would be precisely laid out in just that one equation?

As you have already been warned by one moderator, if you are going to have a thread in the Independent Research forum, you must respond to questions about your theory.

If you want someone to understand your theory, stop complaining and mathematically specify your theory. Please provide the general Lagrangian (multiple particles, arbitrary external field, B_rad field, etc.) for your theory along with what you consider the coordinates to be.

 

[Bob_for_short]

You continue to complain that I don't know what your theory is.

I do not "complain" but attract your attention. We were speaking of the simplest case of one charge. It is a correct methodological approach - explain the mechanism in a simple case and then generalize to many-particle case. We are still there.

Yet you refuse to state in clear math what your theory is.

It is not true. In my articles I clearly and repeatedly introduce this "math". It is a usual math for a compound system. I use the CI and relative coordinates with the corresponding conjugated momenta in the Hamiltonian formulation or velocities in the Lagrangian formulation.

I have asked for three pages for you to answer ONE REQUEST.

I have already answered it. It is a pity you missed it.

As a matter of fact, I wrote a general Hamiltonian (60) for QED; not for CED. It may describe as many particles as you like. CED is obtained as the inclusive result of QED.

You asked for a CED Lagrangian although "my CED" is obtained as the inclusive QED result. Yet I agreed to explain you what is what in principle in an elementary CED case. Even such an elementary CED case looks ridiculous from a classical point of view because the electron coordinate r(t) is highly fluctuating: it contains a smooth part R(t) and oscillating part because in my model the electron is a part of oscillators. On average one obtains R(t). In QED it corresponds to the inclusive picture which is more physically correct than just averaging the classical trajectory.

I do not have the most general CED Lagrangian. Lagrangian serves to obtain equations of motion. They are more important. We have them already, fortunately. Let us start from mechanical equations.

From practical point of view my approach corresponds to neglecting the radiative friction (jerk) in the charge equations of the usual CED and considering the charge positions as electronium's CI positions. The charge equations may contain only external fields - as the Lorentz force (i.e., in a usual way). This is a "mechanical part" of "my CED". So you have these equation already.

The radiated energy or power is entirely contained in the Maxwell equations since, according to my model, they are equations of the "internal degrees of freedom". The energy-conservation laws are already preserved perfectly in this model. We should not, unlike H. Lorentz, add a radiative friction term like jerk (2e²/3c³)da/dt in the charge equations because they are the CI equation in my model. So I removed the "uneasiness" in practising CED without radiative friction term. According to my model, the mechanical equations are more correct without it than with it.

So you have the Maxwell equations already. Together with mechanical equations they are "my CED", if you like. Of course, such a description is valid only in case of small quantum effects.

When you look for a charge trajectory in an external field, the Lagrangian contains the term L_int = jA_ext.

When you look for a field evolution with given sources, the Lagrangian contains the term L_int = j_extA. By the way, in this case the field equations can be formally solved and their solutions can be put in the mechanical equations of another charge, thus one excludes the field variables from consideration. This is clearly seen from the Hamiltonian (60) (four-fermion trem ∫∫jDj).

The self-induction is contained in this current-current term. It is a mutual effect of several charges, not a self-action.

For a self-consistent description in CED it is sufficient to use the ordinary equations without the radiative friction (jerk term) in the mechanical equations. You can use L_int = ∑{j_extA + jA_ext}, where the sum is done over all elementary charges and fields. Is it OK with you?

You see, there is a conceptual gap between your understanding of CED and my theory. It is not reduced just to different math. CED equations contain already the necessary math but in my model we have different physical meaning of variables given just above.

So take the CED Lagrangian and use the corresponding equations without radiative self-action (jerk contribution). That is my answer to your demand.

Now, derive the oscillator equations in case of CED, please. What is a source of radiation in "your CED"? I want to compare it with my theory. You said it is quite different. Show me that.

Regards,

Vladimir.

 

[JustinLevy]

It is not true. In my articles I clearly and repeatedly introduce this "math".

You do not state the lagrangian for this simple case, let alone for the general case in your paper. You forced me to guess based on your statements.

I have asked for three pages for you to answer ONE REQUEST.

I have already answered it. It is a pity you missed it.

NO YOU HAVE NOT! HOW DARE YOU MAKE ME WRITE PAGES WORTH AND THEN LIE STRAIGHT TO MY FACE.

I do not have the most general CED Lagrangian.

So you finally admit why you refuse to answer my question.

As a matter of fact, I wrote a general Hamiltonian (60) for QED; not for CED. It may describe as many particles as you like. CED is obtained as the inclusive result of QED.

I told you multiple times why presenting just the Hamiltonian is not enough. The Hamiltonian equations of motion are useless unless it is precisely clear mathematically what the conjugate momenta are ... if you feel you know these, then just take them and obtain your lagrangian.

It is starting to appear that you are either:
A) dragging me along without intent to EVER answer my question
or
B) don't have your theory mathematically well defined enough to even be ABLE to answer my question

I have spent much time and wrote many equations here.
The least you can do is try to answer my one simple request.
Once your theory is mathematically specified, then there can be no "confusion". Heck, at that point we don't even need to "interpret" anything. We just work the equations and if we do the correct math we will HAVE to agree on the answers.

So please finally answer my request.

From practical point of view my approach corresponds to neglecting the radiative friction (jerk) in the charge equations of the usual CED and considering the charge positions as electronium's CI positions.

I am getting very sick of your talking points.
Especially this one. First of all you already agreed to what the equations of CED were. There is no 'radiative jerk' in those equations. That is not a fundamental part of CED. It is misapplication of Abraham-Lorentz that cause many of these problems.

PLEASE, LET'S FOCUS ON WHAT THE HECK YOUR THEORY IS INSTEAD OF YOUR COMPLAINTS ON CED. We can return to that once we agree what your theory even is.

You see, there is a conceptual gap between your understanding of CED and my theory. It is not reduced just to different math.

NO! It does reduce to math.
If you mathematically state what your Lagrangian and what you consider the coordinates, THEN THERE IS NO ROOM FOR "CONFUSION". It is precisely defined. The answers follow by calculation and the "philosophy"/interpretation of the equations in this sense are meaningless metaphysics. It is good to have a mental picture, but the math must be first.

So take the CED Lagrangian and use the corresponding equations without radiative self-action (jerk contribution). That is my answer to your demand.

THAT IS NOT IN THE CED LAGRANGIAN!
DAMN IT. Please state MATHEMATICALLY what your lagrangian is.

Now, derive the oscillator equations in case of CED, please. What is a source of radiation in "your CED"? I want to compare it with my theory. You said it is quite different. Show me that.

You already agreed, for a point particle a (non-relativistic) Lagrangian that gives CED is:
\mathcal{L} = \frac{1}{2}m\dot{\vec{x}}^2 - q\phi + q\dot{\vec{x}} \cdot \vec{A} - \frac{1}{4\mu_0} \int F_{\mu\nu} F^{\mu\nu} \ d^3r
The coordinates are x and A^\mu, with the fields being a function of position.

I was sloppy in 'deriving' the momentum space version of the fields, as I was talking too much about the free field. This lead to me accidentally dropping one part. Here is the correct one
\mathcal{L} = \frac{1}{2}m\dot{\vec{x}}^2 - q\int \phi(\vec{k})e^{+i\vec{k}\cdot\vec{x}} \frac{d^3k}{(2\pi)^3} + q\dot{\vec{x}} \cdot \int \vec{A}(\vec{k})e^{+i\vec{k}\cdot\vec{x}} \frac{d^3k}{(2\pi)^3} - \frac{1}{2} \int [ \frac{k^2}{\mu_0} \vec{A}(\vec{k})^2 - \epsilon_0 \dot{\vec{A}}(\vec{k})^2 -<br /> \epsilon_0 k^2 \phi(\vec{k})^2 + \frac{1}{\mu_0} \dot{\phi}(\vec{k}^2 <br /> ] \frac{d^3k}{(2\pi)^3}
Now the coordinates are x and A^u, with the fields being a function of momentum space.
You can derive:
(-k^2 -\frac{1}{c^2}\frac{\partial^2}{\partial t^2})\phi(k) = -\rho/\epsilon_0
(-k^2 -\frac{1}{c^2}\frac{\partial^2}{\partial t^2})\vec{A}(k) = -\mu_0 \vec{j}
Which is maxwell's equations in terms of the potentials in the Lorenz gauge.

And do you agree that your Lagrangian is:
\mathcal{L} = \frac{1}{2}m (\dot{\vec{r}}_{CI})^2 - \phi_{ext}(\vec{r}}_{CI} + \sum_k \epsilon_k \vec{E}_k) + \frac{1}{2}\sum_k [\mu_k (\dot{\vec{E}}_k)^2 -(\vec{E_k})^2]
With the coordinates being r_CI and E_k.

If we can agree on these things, I will agree to work out more math for you.

 

[Bob_for_short]

...It is starting to appear that you are either:
A) dragging me along without intent to EVER answer my question, or
B) don't have your theory mathematically well defined enough to even be ABLE to answer my question.

Calm down, take it easy, I have no bad intentions.

We see the CED differently, it is obvious. For example, you do not find there the jerk contribution. Let me tell you that here you are alone.

The whole point of my research and my model is to get rid of this jerk. It has severe consequences in QED. Look at my title. You have to understand that I was motivated by this problem. I don't hide my general Lagrangian from you. I work with QED, not with CED.
I am coming from QED reformulation, if you like this vision better. CED was not my concern because even for one particle it has ridiculous features. If you accept fluctuating electron coordinate - it is OK, we can advance in CED. But it is much better accepted in QED (quantum mechanical charge smearing), so I worked and work with QED actually.

Concerning classical things, I gave a quite detailed mechanical analogy in my paper where the oscillator is mechanical. Being a part of oscillator resolves the energy-momentum conservation problems in interactions. This is my fundamental result which I apply in QED just as in Classical Mechanics.

Concerning "my Lagrangian", the particle part is OK and the oscillator part is like yours but in the Coulomb gauge. It is nearly the same. The only difference is that the field is radiated one - the vector potential A(k) is transversal (orthogonal to k). It represents physical degrees of freedom that take and give away (exchange) the energy-momentum. (In other gauges there are non-physical degrees of freedom decoupled from matter.)

If you take a time derivative of your vector potential equation, you will obtain an equation for the electric field expressed via particle acceleration. The latter is proportional to the external force from the particle equation so they are interchangeable if there is nothing but an external force. My theory gives naturally the known external force as a field source (rather than an unknown particle acceleration in case of taking into account self-action). This excludes the self-action of the radiated field on "particle" motion in my model. This was my primary concern in my research.

 

[JustinLevy]

Calm down, take it easy, I have no bad intentions.

Okay. I was just very very upset that you still won't answer my question, but yet claimed you had answered it despite saying in the same post that you don't currently have an answer.

Concerning "my Lagrangian", the particle part is OK and the oscillator part is like yours but in the Coulomb gauge.

Is what I wrote for your Lagrangian and coordinates in the single particle case with no external magnetic field correct?
If not, then please write the correct equation and coordinates here.
Also, please specify here in this thread what r_CI is in terms of the radiated fields and particle position.
Then we will have everything that mathematically specifies your theory all here in one place for unambiguous discussion.

It is nearly the same. The only difference is that the field is radiated one - the vector potential A(k) is transversal (orthogonal to k).

I disagree strongly with your statement that it is nearly the same.

We need to be precise enough with the math that we can discuss this precisely. Once you commit to stating what your Lagrangian and coordinates are, then there can be absolutely no room for confusion of the consequences of your changes ... I want to finally move on to discussing these consequences and predictions of your theory.

As explained above, it appears to me that your theory does not give the correct radiation distribution for dipole radiation. If you disagree with me, fine. But let's agree on what the math of your theory is so that we can work on the calculations and come to an agreement.


As for my request, if you believe you know the Hamiltonian and conjugate momenta, please work this backwards to get the Lagrangian. I do not feel this is an unreasonable request considering how little work it should be for you.

 

[Bob_for_short]

OK, Justin, I will do it tomorrow. It's late now in Grenoble.

Regards,

Vladimir.

 

[Redbelly98]

As for my request, if you believe you know the Hamiltonian and conjugate momenta, please work this backwards to get the Lagrangian. I do not feel this is an unreasonable request considering how little work it should be for you.

Bob_for_short, please satisfactorily address JustinLevy's request as your next response. Otherwise, this thread will be "locked pending moderation". This has gone on too long.

OK, Justin, I will do it tomorrow. It's late now in Grenoble.

Regards,

Vladimir.

Okay.

 

[JustinLevy]

After your PM's, I decided I'd be willing to post ONE more post before your response. But I am not willing to be stringed along any further. Please finally respond to my request, stating here precisely and mathematically, the lagrangian and coordinates and any supporting mathematical definitions needed to define your theory. Then there can be no room for confusion about what your theory actually is, and we can move on to discussing predictions.

------------

Since you wish to work in the Couloumb gauge and in reciprocal space, for comparison, here is the Lagrangian for classical electrodynamics (CED) written that way.

Gauge condition:
\nabla \cdot \vec{A}(\vec{r}) = 0
in reciprocal space this is
\vec{k}\cdot \vec{A}(\vec{k})=0
so the vector field is purely transverse, and thus only has two free components.Non-relativistic, since the discussion has been non-relativistic so far, and for an arbitrary number of particles:
\mathcal{L} = \sum_\alpha \frac{1}{2}m (\dot{\vec{r}}_\alpha)^2<br /> - \frac{1}{2}\int d^3k [\phi^*(\vec{k}) \rho(\vec{k})+\rho^*(\vec{k})\phi(\vec{k})] <br /> + \frac{1}{2}\int d^3k [\vec{j}^*(\vec{k})\cdot\vec{A}(\vec{k})+\vec{A}^*(\vec{k})\cdot\vec{j}(\vec{k})]
\ \ \ <br /> + \frac{\epsilon_0}{2}\int d^3k [k^2 \phi^*(\vec{k})\phi(\vec{k}) + \dot{\vec{A}}^*(\vec{k})\cdot\dot{\vec{A}}(\vec{k}) -c^2k^2 \vec{A}^*(\vec{k})\cdot\vec{A}(\vec{k})]
where
\vec{\rho}(\vec{r}) = \sum_\alpha q_\alpha \delta^3(\vec{r} -\vec{r}_\alpha)
\vec{j}(\vec{r}) = \sum_\alpha q_\alpha \dot{\vec{r}}_\alpha \delta^3(\vec{r} -\vec{r}_\alpha)

Since the vector \vec{A}(\vec{r}) is real, \vec{A}(\vec{k}) = \vec{A}^*(-\vec{k}), and similarly for the scalar potential. So the generalized coordinates are:
only half the reciprocal space k \phi(\vec{k}),\phi^*(\vec{k}),\vec{A}(\vec{k}),\vec{A}^*(\vec{k}) (only the transverse components for A), and \vec{r}_\alpha

1] Do you agree the above gives CED?

2] Please finally respond to my request, stating here precisely and mathematically, the lagrangian and coordinates and any supporting mathematical definitions needed to define your theory.

 

[Bob_for_short]

... in the Couloumb gauge and in reciprocal space, for comparison, here is the Lagrangian for classical electrodynamics (CED) written that way.

Non-relativistic, since the discussion has been non-relativistic so far, and for an arbitrary number of particles:
\mathcal{L} = \sum_\alpha \frac{1}{2}m (\dot{\vec{r}}_\alpha)^2<br /> - \frac{1}{2}\int d^3k [\phi^*(\vec{k}) \rho(\vec{k})+\rho^*(\vec{k})\phi(\vec{k})] <br /> + \frac{1}{2}\int d^3k [\vec{j}^*(\vec{k})\cdot\vec{A}(\vec{k})+\vec{A}^*(\vec{k})\cdot\vec{j}(\vec{k})]
\ \ \ <br /> + \frac{\epsilon_0}{2}\int d^3k [k^2 \phi^*(\vec{k})\phi(\vec{k}) + \dot{\vec{A}}^*(\vec{k})\cdot\dot{\vec{A}}(\vec{k}) -c^2k^2 \vec{A}^*(\vec{k})\cdot\vec{A}(\vec{k})]
where
\vec{\rho}(\vec{r}) = \sum_\alpha q_\alpha \delta^3(\vec{r} -\vec{r}_\alpha)
\vec{j}(\vec{r}) = \sum_\alpha q_\alpha \dot{\vec{r}}_\alpha \delta^3(\vec{r} -\vec{r}_\alpha)

Since the vector \vec{A}(\vec{r}) is real, \vec{A}(\vec{k}) = \vec{A}^*(-\vec{k}), and similarly for the scalar potential. So the generalized coordinates are:
only half the reciprocal space k \phi(\vec{k}),\phi^*(\vec{k}),\vec{A}(\vec{k}),\vec{A}^*(\vec{k}) (only the transverse components for A), and \vec{r}_\alpha

1] Do you agree the above gives CED?

Not really. φ should not be involved in the filed Lagrangian, it's a mistake. φ has an explicit solution (∆φ ∝ ρ) in this gauge so it can and should be excluded. The inter-charge electrostatic interaction - the second term in your expression - is then described with the following sum: ∑_(α>β) q_αq_β/|r_α - r_β|. (There was no need to make a Fourier transform.) In case of one charge in an external filed Φ_ext(r_e), the latter is present in the Lagrangian as a potential energy. Similarly, there may be the term jA_ext describing an external magnetic field, for example. Both Φ_ext and A_ext are given function of space-time, not the dynamics variables.

2] Please finally respond to my request, stating here precisely and mathematically, the lagrangian and coordinates and any supporting mathematical definitions needed to define your theory.

Do you imagine me to be a boy to run your errands? Whose confusion we are trying to resolve? Exclude φ from the dynamics, please, and add external filed contributions. Then the dynamic variables are particle variables and the vector potential ones in the same sense as given in all textbooks. This is the standard CED. As soon as you get it ready yourself and understand what is what in it, we will be able to compare it with my theory.

 

[JustinLevy]

Do you imagine me to be a boy to run your errands? Whose confusion we are trying to resolve?

YOU decided to promote your theory here. And this is YOUR theory.
As explained to you multiple times now by forum moderators, you are expected to answer questions about your theory.

Yes there is confusion, because your theory is currently not mathematically explained well enough. When I or others state there is something wrong, you instead just claim we don't know the theory. When we try to understand better, you instead just give talking points. YOU NEED TO GIVE SPECIFIC MATH HERE, so that there can be no room for confusion.

You have promised me multiple times now that you would give me this SIMPLE response.

Not really. φ should not be involved in the filed Lagrangian, it's a mistake.

Then you are wrong.
Vary the coordinates and you will get the Maxwell's and Lorentz force law. Do you deny that? If so, vary them and prove it to me mathematically.

Yes, φ can be removed since there is no dependence on its time derivative in the Lagrangian. So you can define φ in terms of its 'equation of motion' in the Coulomb gauge and then plug that into the Lagrangian to remove dependence on φ explicitly. This is important when changing to the Hamiltonian, but we are not discussing this yet. The Lagrangian I gave IS in the Coulomb gauge, and it DOES give classical electrodynamics.

Since your Lagrangian looks like it involves placing a different coordinate in the evaluation of the scalar potential, I thought it better to leave it explicitly in for comparison. Either way, leaving it in does not violate the gauge condition nor the CED equations of motion.

please, and add external filed contributions. Then the dynamic variables are particle variables and the vector potential ones in the same sense as given in all textbooks.

To add external field contributions just replace \phi and A with \phi + \phi_{ext} and A + A_{ext} respectively in the Lagrangian. The generalized coordinates remain the same.1] If you still disagree that what I wrote in the previous post gives CED, please prove it mathematically.

2] Please finally respond to my request, stating here precisely and mathematically, the lagrangian and coordinates and any supporting mathematical definitions needed to define your theory.

 

[Bob_for_short]

OK, we are nearly here. I have got to go to work right now and you, please, just think of use of φ if it is not involved in radiation. It determines the instant Coulomb interaction and can be written explicitly. So the searched variables are in fact the particle ones and the radiated filed A. The rest is known. If you agree, I will write the Lagrangian without φ as a variable.