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This thread contains discussion concerning these topic that occured on other threads, where they would have lead too far away from the original topic. Relevant earlier discussion: http://www.physicsoverflow.org/20116/advice-about-renormalizations-in-physics?show=20116#q20116
Discussion between Vladimir Kalitvianski and Ron Maimon on the submission, Living with infinities.
@VladimirKalitviansky: I say "ultra high-energy" because the unitarity issues don't appear if you make the cutoff less than some ridiculous energy like exp(1/alpha)m_e, and consider perturbative calculations at less than this scale. You can also define a regularized theory using Pauli-Villars regularization, with two additional massive fermions at least one of which contributes with negative norm, and so long as the mass of the subtraction fermions is nowhere near the absurd mass, you will define a perfectly good subtracted theory, with perfectly reasonable scattering behavior at energies lower than that where you start producing negative norm fermions.
The analogy with the Ising model just gives a mathematical example of an analogous breakdown of perturbative renormalization, so that it is clear that it isn't that the method is wrong, just that it is revealing new physics. This example needs to be understood in depth, because here you know everything mathematically, and you can simulate it on a computer. All modern renormalization theory starts from this starting point, the Ising model. Not because it is realistic, but because it is simple, and it shows that any kind of high-energy nonsense can give rise to a renormalizable theory at long distances, it's not some special property, it's natural.
But of course we don't have a lattice at short distances! It conflicts with Lorentz invariance. But we (most probably) have string theory at short distances, or something else, but it doesn't matter, because the lessons of the Ising model are generic. It really doesn't matter what is going on at short distances, so long as it is new physics.
The reason I went to great length about this is because I suspected that you have some idea that these problems are still open or controversial. They aren't, they're solved as far as physics is concerned. That doesn't mean that all the issues of electron self-field are solved, just that the particular meaning of short-distance renormalization is no longer an issue.
There is no notion of "bare electron" in a Pauli-Villars regulated theory, by the way, you can just compute the correlation functions, and they are whatever they are, you get finite answers at all orders of perturbation theory, at the expense of negative norm states at some enormous energy which can be shoved up to the Landau pole. So the only issues that this paper is discussing is Landau pole issues, not the low-energy issues which involve the long-distance self-field, which are the interesting research questions left unresolved in electrodynamics.
I explained why the classical self-field problem is misleading in another answer--- the issues with the self-field in classical electromagnetism is that the mass correction diverges too fast, linearly, as opposed to logarithmically. This means that trying to take a pointlike limit you get inconsistencies in the classical limit when you hit the classical electron radius, with all the attendant sicknesses, but these problems don't occur until a much higher scale in quantum electrodynamics, because of positrons.
What this means is that it is possible to analyze the short-distance renormalization properties of QEDwithout solving or even considering the classical problem of motion in response to the self-field. This problem is still interesting and important, but it is an infrared problem, it is relevant for describing electron motion at distances larger than the Compton wavelength, it has very little to do with the renormalization program as understood today.
Thank you, Ron, very much. I will read your answers later, I have to rush at work now.
The electron is still pointlike in QED, but the divergence is slowed down from the classical 1/R divergence to log(R), because the electron world-line is fluctuating into positrons half the time at short distances, making the field less divergent. This is the log divergence, and it makes QED more consistent than CED, the perturbation theory doesn't break down until a much higher scale.
I don't understand why you say that the Lamb-shift calculation (which I never went through) involves a "smearing" of the electron charge. The smearing of the electron at high energies is about the divergence near-field, the field right next to the electron only. When you talk about "smearing to large distances" you are talking about far-field, the infrared divergences at long distances.
It is really annoying that you refuse to separate the two issues. There are two completely different physical things here, the slow radiated field of an electron moving at frequency omega, which involves photons of order omega or less, which is infrared divergent in QED, and for which a proper description is open, and the near-field, involving the local log-divergences in the renormalization prescription of perturbative QED, and they are two separate things, with completely separate physics. The physics of the local divergences doesn't care about the environment at all, it's about the microscopic structure of the electron. The far-field involves the long-distance radiation from the electron, and is nothing to do with the other thing.
Thank you, Ron, for taking your time and efforts to answer my questions. Yes, I know that we can construct a theory without bare particles, - a Pauli-Villars, a supersymmetric theory with very heavy masses, or just with counter-terms subtracting unnecessary "effects". As far as we are subtracting them, I wonder why not "subtract" them in the physical construction, in the equations? I mentioned the effect of self-induction elsewhere; it comes from the total field whereas we need a radiation field solely (no Plank scale physics is responsible for that). Why not take into account the radiated field solely in CED? It has not been done yet. We physicists are creative, so it is within our power/duty. We are not obliged to stick to a beaten track.
Models like Ising's one or molecular dynamics are not really relevant here. In them you performcalculations, you discard (subtract) nothing. Let us concentrate on CED, because including the self-induction (near field) effect and subtracting it looks somewhat silly. Maybe without it in our construction we will arrive at good CED equations with exact energy-momentum conservation and the point-likeness of the electron will not be an issue at all. That would be a replacement of the Ising and other nice, but irrelevant, model "ideology". Am I reasonable in my reasoning?
Ron, you wrote: "One physical intepretation is that the electron's charge is not concentrated at a point on any time-slice in quantum electrodynamics, but the back-and-forth path in time of the electron means that the charge on any one time slice has both electron and positron intersections, and the charge ends up fractally smeared out over a region comparable to the Compton wavelength, partially smoothing the charge and mollifying the divergence."
I would like to note that any finite charge distribution gives a finite energy. In case of QED you describe the electron is still pointlike or with a somewhat singular charge concentration at the center for the energy to be divergent.
But there is another description of the smearing effect in QED. If we look at the Lamb shift calculation (an estimation by T. Welton), we will see an "additional" electron smearing in its orbit due to coupling to the electromagnetic degrees of freedom. The size of the smearing region depends on $n$, i.e., it is different for different orbits. When we push $n\to\infty$, i.e., when we consider a nearly free and still electron in atom, this smearing size grows to infinity. It is not as small as the Compton length, but much larger. One can find the $n$-dependence of this "additional size" in some textbooks. (The energy shift decreases, but the smearing region increases.) In other words, a "free" electron (no external field) is smeared to infinity and this is an exact QED result.
This is an aside discussion to the thread What is the status of the existence problem for scalar QFT and QED?, started as a response to this comment.
@RonMaimon, you give a physical meaning to the field energy (field-mass, self-mass, electromagnetic mass), that's the problem. CED works fine with any value of charge and without this energy. That is why the latter is always discarded. It has no application at all.
Connections of bare constants with cutoff proceeds from non physical idea of running constants like $m_0(\Lambda)$. Note, the bare mass $m_0$ is a mass of non interacting charge and it cannot (by its definition) depend on the cutoff determined with interaction. As well, if you think a self-field is responsible for the particle mass, then our way of taking into account the radiation reaction force takes into account the self-mass once again. This shows that our way of reasoning is wrong and only discarding this unnecessary mass correction "helps". Similarly for the charge corrections. They are discarded with all their "deep physics". So Landau pole is not a problem at all since it concerns the "bare" charge. You see, the idea of bare constants only worsens understanding. It is much better to think of "self-interaction" as of $j\cdot A + L_{CT}$, i.e., staffed with the conter-terms. This is a "true" interaction, and then no corrections to the physical constants arise (no need in a silly idea of pre-exisiting bare but running constants).
CED interacting with point particles is simply not consistent for any mass of the point particle and any charge, it is only consistent if you make the particle have a charge density of some smeared out kind of size larger than the classical radius for that charge.
When you make the radius of the charged particle smaller, it starts getting exponentially runaway solutions, and if you get rid of them artificially, you still have a pre-acceleration that prevents a causal formulation of any kind, not Hamiltonian, not local Lagrangian.
@RonMaimon: Ron, CED does not only consist of self-action effects. CED is about interactions and as such it works fine (to some precision). As long as interactions are considered ("inter" means two and more charges), there is no conceptual problem. And if this self-action is added and then discarded, it does not make CED inonsistent. It just demonstrates that the self-action is not modeled correctly. One has just to exclude the self-induction force from the total self-force. It is awkwardly done by subtraction or mass renormalization rather than by advancing a better self-force. We need to replace $\dot{F}_{ext}$ with another, radiation field-dependent term to make ends meet in CED.
A better self-force is a force of interaction with the radiated field, which carries away some energy-momentum. This physical development is still missing in CED, but CED is consistent apart from the self-induction term. There is no need to take into account the self-induction part because the near field does not propagate and does not carry away anything. We just have to recognize that this idea does not provide the energy-momentum conservation and there is no need in this term.
The near-field self-energy is strongly model-dependent and is always removed from practical calculations. We can safely say it does not exist. And the radiation reaction force $\propto\ddot{v}$ is not used in practice, but replaced with $\dot{F}_{ext}$, so CED works to some precision. It exists although it is not completely developed yet, in my opinion.
Dilaton does not leave me alone and accuses me in abusing something. I am already very disappointed with my inability to get people discuss friendly my stuff and I am not sure that this my explanation will be understood correctly. The last try.
David Gross, speaking of QFT, mentions that the Wilsonian point of view does not provide a good understanding at short distances. In particular, String Theory gives a UV completion to some QFTs since it has some specific model at short distances.
My model can also be understood as providing some specific physics at short distances, some sort of UV completion. There is nothing bad in my try. I do not deny the usefulness of the Wilsonian approach, I just know it is incomplete and needs a completion at least. Besides, "Reformulation" is equivalent to a great extent (in my toy model - absolutely equivalent) to exactly renormalized theory. So, when dimension10 writes: "Reformulation is plain unnecessary, yet you have made inaccurate claims to justify it", I think he exaggerates or does not understand. Recently he wrote to me: "I've read your paper, but I just don't understand it's significance, motivation, or purpose, which makes it pointless in my eyes. I consider all metrics besides originality to go into accuracy, so I downvoted the submission for accuracy."
Misunderstanding is a reason of mistreating me here. So I propose first to study my bullshit and then extract what is useful. Remember, physicists extract good results from bullshit since long ago ;-)
@RonMaimon: you wrote: "The inconsistency of quantum electrodynamics in the deep ultraviolet is Landau's triviality, it is not controversial. It also is not a mistake of bad mathematics, of leaving out a term or writing the equations wrong, or something like this, it is an actual physical change in the theory at short distances, when the coupling becomes large." I wonder how can you be so sure, when the results and the physical picture depend entirely on equations? What if I write down some other equations? Everything is model-dependent rather than exists in Nature objectively. Landau pole is a feature of this specific model and a specific renormalization framework. In the same model, but with discarding (instead of absorbing by bare constants), there is no bare stuff nor Landau pole. If you do not like discarding, use a "causal approach" by Scharf where no bare stuff is present and no need in a "new physics" exists nor in excuses. Do not impose a certain model with a certain framework as a unique and ultimate physical description. My toy model is intended, amongst other things, to break this frozen dogmatic ice.
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