Reformulation instead of Renormalizations

[Bob_for_short]

It is certainly interesting that the Quantum Field Theories like QED can be reformulated so that no divergences appear at all. Usually the ultraviolet divergences are "cured" with renormalizations. But recently I posted a paper (http://arxiv.org/abs/0811.4416) with the following abstract [1]:

In this article I show why the fundamental constants obtain perturbative corrections in higher orders, why the renormalizations work and how to reformulate the theory in order to avoid these technical and conceptual complications. I demonstrate that the perturbative mass and charge corrections are caused exclusively with the kinetic nature of the interaction Lagrangian. As soon as it is not purely quantum mechanical (or QFT) specific feature, the problem can be demonstrated on a classical two-body problem. The latter can be solved in different ways, one of them being correct and good for applying the perturbation theory (if necessary) and another one being tricky and awkward. The first one is physically and technically natural - it is a center-of-inertia-and-relative-variable formulation. The second one - a mixed variable formulation - is unnecessarily complicated and leads to the mass and "charge" corrections even in the Newtonian mechanics of two bound bodies. The perturbation theory in QFT is factually formulated in the mixed variables - that is why it brings corrections to the fundamental constants. This understanding opens a way of correctly formulating the QFT equations and thus to simplify the QFT calculations technically and conceptually. For example, in scattering problems in QED it means accounting exactly the quantized electromagnetic field influence in the free in and out states of charged particles so no infrared and ultraviolet problems arise. In bound states it means obtaining the energy corrections (the Lamb shift, the anomalous magnetic moment) quite straightforwardly and without renormalizations.

The most important findings are:

1. The energy-momentum conservation law can be preserved in QED in a physically and mathematically natural way rather than in the frame of self-action ansatz with inevitable renormalizations.

2. The Novel QED has the correct classical limit where the radiation is unavoidably taken into account (the inclusive picture) rather than neglected.

3. The electron ( or more generally, a charge) and the quantized electromagnetic field form a compound, in a certain sense "welded" rather than "mountable-dismountable" system. I call it an electronium. Its quantum mechanical description is quite similar to the atomic one [2]. In particular, photons are just excited electronium states. No constant renormalizations are necessary in such an approach, no divergences appear.

4. I propose to construct the other "gauge" theories in the same spirit - as theories where compound systems (fermioniums) interact with possible exciting their internal (relative) degrees of freedom ("gauge" bosons). The simplest physical analogy to that is the fast atom1-atom2 scattering at large angles when only nucleus1-nucleus2 (Coulomb or not) potential is important and the final atomic states are excited [2].

I hope this approach deserves attention and a further development.

Vladimir Kalitvianski.


[1] Reformulation instead of Renormalizations, http://arxiv.org/abs/0811.4416.

[2] Atom as a "Dressed" Nucleus, Central European Journal of Physics, V. 7, N. 1, pp. 1-11, (2009) by Vladimir Kalitvianski, (available also at http://arxiv.org/abs/0806.2635).

 

[Bob_for_short]

The journal version of "Atom..." is available as an attachment to my posts #28 at https://www.physicsforums.com/showthread.php?t=309473&page=2.

Bob.

 

[Bob_for_short]

It is also available online from the journal site: http://www.springerlink.com/content/h3414375681x8635/?p=309428ad758845479b8aeb522c6adfdd&pi=0

 

[Bob_for_short]

One more article on removing divergent corrections by reformulation of the original problem in better terms. It is available at http://arxiv.org/abs/0906.3504.

 

[Fra]

Hello Bob,

A comment on your comment in the other thread.

So this all is about making somehow the QG renormalizable?

To me it's much more. The renormalization problem of GR, is a problem phrased in the mainstream framework. In the new framework I am seeking, most certainly the "problem" would not appear.

So to me, the renormalization issue, is a symptom of a flawed reasoning, not the prime problem per see. Once the right reasoning is in place, which solves also other problem, this is expected to be a non-issue.

In my opinion, the QED renormalizability plays a bad role - it makes an impression that the renormalizations are a "good soluiton" to mathematical and conceptual difficulties. "Cowboy's attacs" to manage divergences in QG fail and all these strings and superstrings, loops, dimension reductions are just attempts to get something meaningfull, at least mathematically.

At the same time, there is another approach that contains really natural (physical) regularizators or cut-offs and thus is free from divergences.

I share your sentiment here. But IMO it's not ONLY about renormalization problem. It's a lot more. It's about unification and understanding the whole. Many things. Fine tuning problems, cosmological constant problem. The relation between spacetime and matter. QM foundations.

I skimmed your paper and to me your aim to adress specifically the renormalization alone.

I think the physical natural regulators you seek, are implicit in the "inside view", which is again the key to understanding how matter "sees", or relates to spacetime.

In your paper you seem to make use of spacetime as if it were a platform?

So my first amateurish impression is that,

1) I share your sentiment of renormalization beeing a lucky trick, and that in a proper formulation, those tricks should not be necessary. The formulation should come out right from start.

2) But I think that there are more problems than this, and you doesn't acknowledge them explicity. Which worries me that you try to isolate this problem from other problems that might be related.

/Fredrk

 

[Fra]

Bob, what is your spontaneous impression of the ideas of Olaf Dreyer?

He is aiming towards a new research program, he calls it "internal relativity", where one of the core ideas is the emphasis the inside view of things. However, there are to my knowledge not a lot of papers yet.

But here is a brief description of his ideas from fqxi.
http://www.fqxi.org/data/articles/Dreyer_Olaf.pdf

/Fredrik

 

[Fra]

About my conceptual opinon of "perturbation theory" in general, the only physical basis of a perturbation theory is where you perturb a prior information, with speculations, then there is a physical rational behind the ordering implicit in the specific perturbation expansion - it corresponds to a principle of minimum speculation. The speculation is truncated along with the perturbation expansion, and at some point, from the inside perspective the higer order speculations simply aren't distinguishable from the inside perspective. so there is a physical motivation for the cutoff which leads to the point that - the ACTION of the system actually behaves exactly AS IF the perturbvation is truncated. So the "behaviour of the system" should reveal this.

But I think to make sense out of that, we need a reconstruction of the information models, from the intrinsic perspective. The wrong basis of information (ie an externa one) implies that we get the wrong "physical perturbation" and thus there is no physical truncation.

All this could be built-in, into an intrinsic measurement theory coupled with action on reaction. Ie. the self-interaction would always be limited by the inside-resources, which I think of as the coherent degrees of freedom, relative to the inside. What is beyong that is action, upton the environments reaction, and this must contain a logic of correction - how to revise your information, when the prior and the new info are not consistent. It's unfortunately not as simple as a plain bayesian update or static Maxent. I think we need something more clever. I think the RULE of the information update, on which bayesian update is a natural but yet SPECIAL case, is subject to evolution an selection. And it's the context that must determine which rule of information update that is more "fit".

This is the type of inside view I seek.I'm not sure if this makes sense.

/Fredrik

 

[Bob_for_short]

It's about time to develop this approach in details, solve concrete problems and work out calculation technique.

 

[zetafunction]

renormalization of UV can be 'cured' partially by zeta regularization .. (see Elizalde Book or the free paper http://wbabin.net/science/moreta23.pdf as an intro )

 

[Bob_for_short]

So what? I propose reformulation on a physical basis rather than renormalizations.

 

[Bob_for_short]

Dear JustinLevy,

Thank you for your assessment of my approach. I really appreciate any constructive feedback. I really need improvement of my writing style as well as the way of result presenting.

... Before even getting into details, the broad strokes of your approach don't seem motivated well at all...

I will think about it. Maybe I will add "maybes" to those phrases to make them less striking.

Your writing style also, unfortunately, comes off as crackpottish due to the ratio of complaints against mainstream theory to actual content, and also the absolutism of the phrasing. All in all, cleaning up your paper so that someone can skim it and understand what you are claiming would be great.

Roger. I will smooth my complaints and add some uncertainty to my phrasing.

All you seem to really be doing is proposing a different Lagrangian

A Hamiltonian, to be exact...

... put that front and center. Many physicists can read the majority of the physical content off of a Lagrangian themselves. If your trial lagrangian has the correct classical limits and interesting features, they will be much more inclined to read the intro and conclusion (and maybe even skim or read the whole paper).

I will do it, thanks for advising.

Since all you are really doing is just proposing a different Lagrangian and claiming it is _exact_ instead of approximate and therefore doesn't need regulating,...

No, it's a trial one, it is clearly stated.

... I can't help but ask: Do you really think something as fundamental as electrodynamics has been using the wrong lagrangian all these years and you were the one that came up with the correct one?

You exaggerate here. Most of problems are using approximate Lagrangians. When the charge-current is a known function of space-time (not an unknown variable), the field is easily found. No physical and mathematical problems arise. Similarly, when the external filed is given (not an unknown variable) the charge motion is well defined. These two extreme cases cover the majority of practical problems. The only problematic case is to build a self-consistent theory where both charge-current and field variables are unknown. H. Lorentz found nothing better than a self-action ansatz. I found another way (interaction without self-action).

The first thing people will ask for is, at the very least, experimental post-diction.
If your theory is better than QED, can you derive the anomalous magnetic moment of the electron with your theory?

Yes, they ask for it. At this stage (a trial Hamiltonian proposal with some non-relativistic estimations) it is too early to report the fourth-order calculation of (g-2). Preliminary estimations show however that it is possible.

Considering your theory doesn't even reduce to Maxwell's equations, I think that would be very very unlikely.

My trial Hamiltonian contains the quantum oscillator Hamiltonians like any other QED Hamiltonian in the Coulomb gauge. What are you speaking about?

 

[Bob_for_short]

...But I think is has to be the the problem of the advocate of the competing approach to show that it can outperform the main approach. Probably all the one working on the main approaches do so because it's what they find most promising, in which case it's still rational...

Can you aim at some of the generally acknowledge open question in physics?...

Thank you, Fredrik, for your thoughts and suggestions.

Yes, I address the main problem in physics since electrodynamics invention. I speak of self-action. It is not only “theoretical” but also practical questions. Namely, difficulties with non-renormalizable theories block practical calculations.

What I found is a quite physical possibility to build a theory with interaction and without self-action. Many say that the self-action is necessary for predicting some experimental data. But they are cheating. Self-action ansatz alone introduces fundamental difficulties, and only renormalizations, introduced later, remove (perturbatively) the unnecessary self-action contributions. So a renormalized result is free from self-action effect. (It is the only purpose of renormalizations.)

Instead of carrying out renormalizations perturbatively in self-acting theories, I propose to start from a Hamiltonian without self-action. As simple as that. This simplifies tremendously the calculations and makes everything clear. I hope to be heard by respected particle physicists because there is indeed a way of preserving the energy-momentum conservation laws without self-action.

 

[Fra]

Thank you, Fredrik, for your thoughts and suggestions.

Yes, I address the main problem in physics since electrodynamics invention. I speak of self-action. It is not only “theoretical” but also practical questions. Namely, difficulties with non-renormalizable theories block practical calculations.

What I found is a quite physical possibility to build a theory with interaction and without self-action. Many say that the self-action is necessary for predicting some experimental data. But they are cheating. Self-action alone introduces fundamental difficulties, and only renormalizations remove (perturbatively) the unnecessary self-action contributions. So a renormalized result is free from self-action effect.

Instead of carrying out renormalizations perturbatively, I propose to start from a Hamiltonian without self-action. As simple as that. This simplifies tremendously the calculations and makes everything clear. I hope to be heard by respected particle physicists.

Just regarding the standard renormalization formalism, and the fact that it's invented as a somewhat ambigous to remove non-physical and ambigous degrees of freedom that shouldn't be there in the first place - I share you objection. I am not defending current standards, except that it's at least until we have something better, they best we have so to speak.

So in a sense I'm probably just as far off the main roads as you are.

Still your reasoning and objection about this quite is different than mine. I'll try to think another round on your point and maybe see if I can find a more constructive than earlier in the thread.

/Fredrik

 

[Bob_for_short]

Just regarding the standard renormalization formalism, and the fact that it's invented as a somewhat ambigous to remove non-physical and ambigous degrees of freedom that shouldn't be there in the first place - I share you objection...

It is not even non-physical degrees of freedom. It is non-physical corrections to the fundamentals constants. For example, when you solve a heat conduction equation by the perturbation theory with a known heat conductivity, the latter should not acquire "divergent perturbative corrections", it is a nonsense. For the heat conduction equation I managed to reformulate the equation (with a simple variable change) and obtained immediately good, convergent series. Starting from that time, I tried to make a similar thing with QED and finally I found how to do this. Now I need funding to carry out calculations with my ne Hamiltonian.

 

[Fra]

It's quite clear that we attack this very differently and I'm still trying to see the main point but I'll just throw out something here and see if it makes sense to you...I'm trying to understand

It is not even non-physical degrees of freedom. It is non-physical corrections to the fundamentals constants. For example, when you solve a heat conduction equation by the perturbation theory with a known heat conductivity, the latter should not acquire "divergent perturbative corrections", it is a nonsense. For the heat conduction equation I managed to reformulate the equation (with a simple variable change) and obtained immediately good, convergent series. Starting from that time, I tried to make a similar thing with QED and finally I found how to do this. Now I need funding to carry out calculations with my ne Hamiltonian.

Suppose we consider this "action space".

It seems to be you are effectively suggesting that the divergences can be cured, but choosing another starting point for the perturbation? Mathematicall this makes good sense of course, but I'm not sure I see the physical idea here, keep in mind that my strange perspective is that of operational inference.

What is the physical significance of "perturbation" anyway, in your view? MAthematically one can see it as trying to find a solution to something, but perturbing another solution, and it's intuitively clear that for a particular perturbation/expansion techique, there might be a sensible perturbation series only for certain starting points. But IMO that has very little physical significance unless the physical meaning of the notion of perturbation si clear.

To me, the choice of starting poitn for a perturbation can't be chosen at will, it's defined by the physical context (the observer, or measurement setup), isn't it? that's how I see it.

But maybe you suggest that the wrong physical starting point is used in the first place, if so, I can connect to that. But where does the observer come in? doesn't the observer define the observational scale?

or is your main point that there is an alternative formualtion of the SM actions, including the fundamental constants so that all perturbations can work w/o divergences??

/Fredrik

 

[Bob_for_short]

Dear Fredrik,

I started answering your questions here but finally decided to refer to my publications where they all have already been answered. Start from reading my weblog, please.

Regards,

Bob.

 

[JustinLevy]

If you change the Hamiltonian in the classical limit, then how can it possibly reproduce Maxwell's equations?

Regardless, you have not avoided the "infinities" anyway. You still have an infinite degrees of freedom (the field), which you represent as oscillators and state you will quantize these oscillators. Therefore your ground state energy is infinite. To talk about any measurements (energy needed to excite a state, etc.) you will indeed need to renormalize to get a finite value out of the differences of two infinite values.

 

[Bob_for_short]

If you change the Hamiltonian in the classical limit, then how can it possibly reproduce Maxwell's equations?

Regardless, you have not avoided the "infinities" anyway. You still have an infinite degrees of freedom (the field), which you represent as oscillators and state you will quantize these oscillators. Therefore your ground state energy is infinite. To talk about any measurements (energy needed to excite a state, etc.) you will indeed need to renormalize to get a finite value out of the differences of two infinite values.

If you take the Maxwell equations, you can represent them as a set of independent equations for canonical coordinates and momenta. The corresponding Hamiltonian is a sum of oscillator Hamiltonians. So the both representations are equivalent.

The ground state energy has never been a problem. It does lead to the mass and charge renormalizations.

 

[JustinLevy]

If you take the Maxwell equations, you can represent them as a set of independent equations for canonical coordinates and momenta. The corresponding Hamiltonian is a sum of oscillator Hamiltonians. So the both representations are equivalent.

All of (classical) electrodynamics is covered by Maxwell's equations and the Lorentz force law. Furthermore, these are not independent equations; they are coupled equations.

You propose a different Hamiltonian, and thus the physics will be different. So your theory does not reproduce the correct classical limit.

If instead, your theory is indeed equivalent as you claim, than there are no experimental differences at all and thus you have proposed nothing new.

The ground state energy has never been a problem. It does lead to the mass and charge renormalizations.

You claim your theory avoids all the "infinities" and "mathematical difficulties". Yet the infinities are still there, and now you seem to be saying 'yes' you do need renormalization.

 

[Bob_for_short]

All of (classical) electrodynamics is covered by Maxwell's equations and the Lorentz force law.

Correct.

Furthermore, these are not independent equations; they are coupled equations.

Correct. The textbook coupling, though, includes a self-action term. I couple them without self-action.

You propose a different Hamiltonian, and thus the physics will be different. So your theory does not reproduce the correct classical limit.

Correct. In my approach there is no correction to the electron mass and run-away solution.

If instead, your theory is indeed equivalent as you claim, than there are no experimental differences at all and thus you have proposed nothing new.

Wrong. New are notions and equations that correctly describe the known experiments.

You claim your theory avoids all the "infinities" and "mathematical difficulties". Yet the infinities are still there, and now you seem to be saying 'yes' you do need renormalization.

It is infinities in your imagination that you try to impose to my theory.

 

[Evo]

Dear Fredrik,

I started answering your questions here but finally decided to refer to my publications where they all have already been answered. Start from reading my weblog, please.

Regards,

Bob.

Sorry Bob, that's not sufficient. If you wish to have a discussion here, your information must be posted here. If you do not wish to post answers here, then the thread will be closed.

 

[JustinLevy]

Correct. The textbook coupling, though, includes a self-action term. I couple them without self-action.

You keep saying that, but I see no evidence you have removed any so called "self-action". Charged particles still are a source of $A^\mu$ and particles are still affected by $A^\mu$. You act as though you subtracted some interaction that makes every particle only see the fields from every other particle ... you cannot separate the fields this way, if only for the simple reason that now physics doesn't depend on just the state of the system, but the entire history of the system and therefore we lose a great deal of predictive power.

You propose a different Hamiltonian, and thus the physics will be different. So your theory does not reproduce the correct classical limit.

Correct. In my approach there is no correction to the electron mass and run-away solution.

So if you agree that the classical limit is different, then you must agree that your theory does not agree classically with Maxwell's equations and the Lorentz force law.

What is your new "version" of Maxwell's equations and the Lorentz force law?

Wrong. New are notions and equations that correctly describe the known experiments.

Since your equations disagree with Maxwell's equations and the Lorentz force law, I fail to see how you can possibly reproduce even most classical experiments.

It is infinities in your imagination that you try to impose to my theory.

You still have an infinite degrees of freedom (the field), which you represent as oscillators and state you will quantize these oscillators. Therefore your ground state energy is infinite. To talk about any measurements (energy needed to excite a state, scattering, etc.) you will indeed need to renormalize to get a finite value out of the differences of two infinite values.

 

[Bob_for_short]

Yesterday I got sick (a flu?), so it is very difficult to me to answer, I am sorry.

You keep saying that, but I see no evidence you have removed any so called "self-action". Charged particles still are a source of $A^\mu$ and particles are still affected by $A^\mu$. You act as though you subtracted some interaction that makes every particle only see the fields from every other particle ... you cannot separate the fields this way, if only for the simple reason that now physics doesn't depend on just the state of the system, but the entire history of the system and therefore we lose a great deal of predictive power.

Dear Justin,

Self-action means action of the proper field, including radiated, on the particle. My theory "lost" the following predictive power: it does not predict infinite and unnecessary correction to the electron mass and it does not predict non physical run-away solutions. The rest remains quite the same as in the usual CED.

Frankly, Fra and you are asking questions that directly and clearly addressed in my publications, which are available on internet for free. I do not see any reason why I should retype them in this thread. I wrote quite detailed articles with concrete examples on purpose. Your statements, Justin, testify that you do not want to read and understand my motivation and my results.

Sorry Bob, that's not sufficient. If you wish to have a discussion here, your information must be posted here. If you do not wish to post answers here, then the thread will be closed.

Dear Evo,

You may, if you like so, close this thread. Since its opening no real discussion has started yet, unfortunately.

Regards,

 

[Fra]

Yesterday I got sick (a flu?), so it is very difficult to me to answer, I am sorry.

I hope you get better soon.

Frankly, Fra and you ask questions that directly and clearly addressed in my publications, which are available on internet for free. I do not see any reason why I should retype them in this thread. I wrote quite detailed articles with concrete examples on purpose.

I did skim your articles, and I guess on first skimming I didn't see any major vision. Now I am certainly not everybody or even representative to those you address, but perhaps your message could be clearer.

I think a reader needs to, in a few paragraphs get motivated, why they should read the rest in detail. And finally that should provide motivation to think further. Some of your phrasings is what caught my first interest, such as "natural cutoffs" and reformulation instead of renormalization.

But your change of variables, and thus transforming the hamiltonians or langragian forms seem somewhat ambigous still to me, and I don't see how it addresses how it adresses some of the problems that at least I think a new approach should.

Is there a proportion to the benefit of this, and the effort of reformulating everything?

I'm just trying to get you to motivate your approach more.

I only speak for myself, and I do not read a lot of papers. Some authors tend to write short and brief paper, making the essential motivating points without going into details. And then refer to other papers for details.

Maybe this sounds silly but I like papers no more than 5 pages :) I rather read 5 papers with 5 pages each, than one paper with 25 pages.

Everyone is lazy, and tend to use their own time economically. I want very good reasons to read a 50 page random paper. But I do not require as strong reasons to read a random 2-page paper.

I do not see any reason why I should retype them in this thread.

I could think of one reason: google is very fast and efficient at indexing all threads on physicsforum, and it has great ranking. Explaining your reasoning here, even if it means retyping, might even increase your visibility.

If I for example google "reformulation renormalization", the two top rankings are physicsforums threads. I can't see your own arxiv papers anywhere on the first page with the same search words.

Just some sincere ideas.

/Fredrik

 

[JustinLevy]

Frankly, Fra and you are asking questions that directly and clearly addressed in my publications, which are available on internet for free.

You do not address these things.
Nowhere do I see proof of many of your claims. Sure, you may think that because your paper has your talking points repeated time and again and interspersed with equations that you have demonstrated all your claims. You unfortunately have not. Please don't fall back on the crackpot defense of "it is obvious so I will repeat my talking points as if that answered it".

While it may be "your theory/framework" in the sense that you are presenting it. A theory/framework has to be 'everyone's' in the sense that it has to be clearly explained well enough that everyone can agree on what the theory/framework even IS so that experimental predictions can be made.


In particular, I will ask once again:

What is your new "version" of Maxwell's equations and the Lorentz force law?
Please write them as the 5 coupled equations so we can see explicitly and completely unambiguously how you are suggesting they be changed.

What is the ground state energy of your infinite number of oscillators / field degrees of freedom?


If you really feel that your paper clearly explains the theory enough that I should know the answers to those (and agree with you on those answers), then try a little experiment. Have a colleague you work with read your paper and try to answer those.

At least attempt discussion and answer some basic questions for us.

 

[Bob_for_short]

Justin,

1) which of my articles did you read?,

2) tell me more about your level of education, please. This information is absent in your public profile.

Depending on this information I will decide on what level of physics I can reply.

 

[strangerep]

In a thread in the quantum physics forum, i.e.,
https://www.physicsforums.com/showthread.php?t=348911&page=5
Bob asked:

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

I'll answer here, since any answer I might give leads inevitably to more discussion of
your theory.

It is far too strong to say that standard QED is a "complete" failure.
I am reasonably happy if the predictions of a theory become better and better
when more accurate calculations are performed.

However, I also have no problem if other people consider alternate starting points
which might give faster "convergence" to (all aspects of) the observed physics.

I am sorry, but the above is the best answer I can give to your question at this time.

Since you mentioned bremstrahlung, perhaps you will now accept a question from me...

Have you read, or do you have access to, this paper:
----------------------------------
V. Chung, "Infrared Divergence in QED",
Phys Rev, vol 10, no 4B, (1965), pp857-869.

Abstract: The infrared divergences of quantum electrodynamics are eliminated to all
orders of perturbation theory in the matrix elements by an appropriate choice of initial
and final soft photon states. The condition for this cancellation restricts these states
to representations of the canonical commutation rules which are unitarily inequivalent
to the usual Fock representation.
----------------------------------

If you've read it, can you comment on any relation (or lack thereof) between it
and your electronium construction?

 

[JustinLevy]

Sorry for the late reply. I am still interested in a discussion, but I had family visiting for the last few days and did not have time to devote to this.

1) which of my articles did you read?

The one you started this thread on:
http://arxiv.org/abs/0811.4416

2) tell me more about your level of education, please. This information is absent in your public profile.

I am a physics graduate student. I feel I know classical EM and quantum well, and while I have learned the basics of QED for classes I do not use it on a regular basis.

To help set the starting 'level' of this discussion, let me begin it in more detail.
First, let's build a common language so that terminology issues don't plague this discussion.

There are some definitions,
$$\vec{E} = - \nabla \phi - \frac{\partial}{\partial t} \vec{A}$$
$$\vec{B} = \nabla \times \vec{A}$$
$$A^\mu = (\phi,\vec{A})$$
$$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$
and some sign conventions:
$$g^{00} = -1$$
such that
$$F_{\mu\nu} = \begin{bmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & B_z & -B_y \\ E_y/c & -B_z & 0 & B_x \\ E_z/c & B_y & -B_x & 0 \end{bmatrix}$$

Do you agree we can use these definitions for this discussion?The standard approach to classical electrodynamics (which in your paper you call CED, so let's use that acronym here as well):
All of CED is included in these five equations
$$\nabla \times \vec{E} = -\frac{\partial \vec{B}} {\partial t}$$
$$\nabla \cdot \vec{B} = 0$$
$$\nabla \cdot \vec{E} = \frac {\rho} {\varepsilon_0}$$
$$\nabla \times \vec{B} = \mu_0\vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}} {\partial t}$$
$$\vec{F} = q (\vec{E} + \vec{v}\times\vec{B})$$
Notice that the first two of these come directly from the definitions of the potentials.

Do you agree this encompasses CED?

You claim these equations of CED lead to infinites (they do, when considering the energy in an electromagnetic field) and run-away solutions (which they do not, naive application of the Abraham-Lorentz formula lead to this, but even the wiki article cites resolutions to this when considered appropriately http://en.wikipedia.org/wiki/Abraham–Lorentz–Dirac_force ). I don't want to get into an argument over that, since I really just want to understand what your theory predicts so let's leave the discussion of CED pathologies for some other time.For a point particle, a (non-relativistic) Lagrangian that gives CED is:
$$\mathcal{L} = \frac{1}{2}m\dot{\vec{x}}^2 - q\phi + q\dot{\vec{x}} \cdot \vec{A} - \frac{1}{4\mu_0} \int F_{\mu\nu} F^{\mu\nu} \ d^3r$$
where the 'coordinates' are the particle coordinates and the field coordinates $A^\mu$. Varying the particle coordinates gives the force equation, and varying the field coordinates gives the two source equations (and again the remaining two equations come from the definitions).

While the usual momentum of the particle is of course
$$\vec{p} = m \vec{v}$$
the canonical momentum is (using the capital symbol to distinguish it):
$$\vec{P} = \frac{\partial \mathcal{L}}{\partial \dot{\vec{x}}} = \vec{p} + q \vec{A}$$

The Hamiltonian is in general (summing over all pairs of coordinates and their conjugate momentum):
$$\mathcal{H} = \sum \dot{x} P_x - \mathcal{L}$$
which should be rewritten in terms of just the coordinates and conjugate momenta so that we can apply the evolution equations later. Here we get:
$$\mathcal{H} = \frac{(\vec{P} - q \vec{A})^2}{2m} + q \phi + \frac{1}{2}\int (\epsilon_0 E^2 + \frac{1}{\mu_0} B^2) \ d^3r$$

Given the Hamiltonian, the equations of motion are then
$$\frac{d}{dt} P_x = - \frac{\partial \mathcal{H}}{\partial x}$$
$$\frac{d}{dt} x = \frac{\partial \mathcal{H}}{\partial P_x}$$
for each pair of coordinate $x$ and conjugate momentum $P_x$.

Notice the p - qA only shows up because the Hamiltonian is referring to the conjugate momentum. In terms of 'normal' momentum $p$, the Hamiltonian is:
$$\mathcal{H} = \frac{p^2}{2m} + q \phi + \frac{1}{2}\int (\epsilon_0 E^2 + \frac{1}{\mu_0} B^2) \ d^3r$$
This is still the same Hamiltonian, but can be misleading since it is not written in terms of the conjugate momentum and if one accidentally used the evolution equations using p instead ... it would not give the correct evolution. So it is extremely important to be precise on what the Hamiltonian and coordinates are. If the Lagrangian is given, the momentum of course can be readily seen.Since you complain about the qA term alot, and claim to remove it, it is not clear to me at all:
1] precisely what Lagrangian and coordinates you are using
(you build up a lot with two particles and then appear to abruntly start treating parts as fields and present a Hamiltonian in eq. 54 without it even being possible to tell if the momentum are indeed conjugate to the coordinates)
Please state precisely what your Lagrangian is, and what you consider the coordinates.

2] how you can claim this reproduces CED at ALL
(since again, I need to know the conjugate momenta. furthermore, you don't seem to have any spatial derivatives of the spatial components of A_mu ... so you can't possibly have a current create a magnetic field, or have any chance of reproducing the Lorentz force equation)
Please state what your new "version" of the five CED equations I listed above are.Please at least respond to the four bolded items.

 

[Bob_for_short]

There are some definitions,
$$\vec{E} = - \nabla \phi - \frac{\partial}{\partial t} \vec{A}$$
$$\vec{B} = \nabla \times \vec{A}$$
$$A^\mu = (\phi,\vec{A})$$
$$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$
and some sign conventions:
$$g^{00} = -1$$
such that
$$F_{\mu\nu} = \begin{bmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & B_z & -B_y \\ E_y/c & -B_z & 0 & B_x \\ E_z/c & B_y & -B_x & 0 \end{bmatrix}$$

Do you agree we can use these definitions for this discussion?

Yes, I do. (I did not verified them though, but it does not matter for instance.)

All of CED is included in these five equations
$$\nabla \times \vec{E} = -\frac{\partial \vec{B}} {\partial t}$$
$$\nabla \cdot \vec{B} = 0$$
$$\nabla \cdot \vec{E} = \frac {\rho} {\varepsilon_0}$$
$$\nabla \times \vec{B} = \mu_0\vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}} {\partial t}$$
$$\vec{F} = q (\vec{E} + \vec{v}\times\vec{B})$$
Notice that the first two of these come directly from the definitions of the potentials. Do you agree this encompasses CED?

Yes, I do but with some reservations. Here they are.

1) 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. But this is inessential for instance.

You claim these equations of CED lead to infinites (they do, when considering the energy in an electromagnetic field) and run-away solutions (which they do not, naive application of the Abraham-Lorentz formula lead to this, but even the wiki article cites resolutions to this when considered appropriately http://en.wikipedia.org/wiki/Abraham–Lorentz–Dirac_force ). I don't want to get into an argument over that, since I really just want to understand what your theory predicts so let's leave the discussion of CED pathologies for some other time.]

2) My second reservations is just about equations. If you look in my article, it starts from self-action discussion. 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. 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. So he decided to develop further the particle equations and introduced new terms. Let us note that the notions of particle mass and charge had already been introduced in the theory as external (phenomenological or fundamental) constants before the Lorentz self-action ansatz.

It is not true that the problem of non-physical solutions has already been resolved in the frame of CED, as you formulated it above. For example, F. Rohrlich wrote a lot of papers on this subject. The last one is at http://arxiv.org/abs/0804.4614. Its disproval is given at http://arxiv.org/abs/0904.2377.

Anyway, any attempts to get rid of non-physical solutions were reduced to modification of CED. You and me will speak of another way of preserving the conservation laws, OK? Factually my approach joins the two limiting cases outlined above but without self-action.

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.

 

[Bob_for_short]

...Have you read, or do you have access to, this paper:
----------------------------------
V. Chung, "Infrared Divergence in QED",
Phys Rev, vol 10, no 4B, (1965), pp857-869.

Abstract: The infrared divergences of quantum electrodynamics are eliminated to all
orders of perturbation theory in the matrix elements by an appropriate choice of initial
and final soft photon states. The condition for this cancellation restricts these states
to representations of the canonical commutation rules which are unitarily inequivalent
to the usual Fock representation.
----------------------------------

If you've read it, can you comment on any relation (or lack thereof) between it
and your electronium construction?

No, I have not read it and I have no access to it. But I read some time ago a conference presentation (slides) where the autors stated that the finite T imposed a finite energy for soft modes and thus this resolved the IR problem. Their annonce was accompanied with many exclamation signs (!).

I think the paper of V. Chung is different since he mentiones "appropriate choice of initial and final states". I think that imposing appropriate initial and final states is not legitimate since it is not really physically motivated. In my approach you may have any initial distribution of the oscillator populations (in pure or mixed states) and calculate the inclusive quantities summing up over all possible (rather than "appropriate") final states.

 

[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.

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

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
$$\vec{E} = \vec{E}_{total /\ 'actual'} +\nabla \phi_{ext} + \frac{\partial}{\partial t} \vec{A}_{ext}$$
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.