Cutoff and asymptotic series in QED

Question

From the thread http://www.physicsoverflow.org/26751/physics-of-asymptotic-series-and-resummation?show=29617#c29617

...note that QED is valid only with some large but finite cutoff $\Lambda$.

What you call "valid" means "incomplete above $\Lambda$" rather than "obligatorily divergent". QED is $\Lambda$-divergent for another reason. I tired to explain it (again, on a simple toy model) here.

Comments

''valid'' means that QED is an approximation to a more complete theory that changes the UV behavior for large energies is a significant way, so that too large values of the cutoff $\Lambda$ are not really physical.

If one uses QED with a finite cutoff of the size of the masses of the weak interaction bosons, and a tiny photon mass compatible with current experiment then its predictions are finite (i.e., there are nowhere divergences, all resummations are done exacly in the spirit of your renormalization scheme in the other thread), and the results agree with experiment to 12 decimal digits, provided the constants in the bare interaction are chosen to fit the experimental data (which means, large enough, as specified by the renormalization calculations).

This shows that the bare interaction is an excellent approximation to reality, much better than anything we have. However, the bare constants in the interaction have no physical meaning, since the physics is in the resulting S-matrix, not in the coefficients. One cannot measure the bare constants, only the long-distance behavior. 

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Too large values of $\Lambda$ may not contribute in a reasonably formulated theory. The short-distance physics of QED is incomplete, but it may not be harmful because the transferred momentum/energy is well below "new" physics. Even out of its validity, a reasonably formulated QED would be numerically wrong, but with no catastrophes. You imply that $\Lambda$ matters and cannot be sent to infinity. I object to this narrow point of view as it only is valid for a badly formulated theory.

Arnold, it would be nice if you could write a review to my "Short-distance" paper (positive or negative, no matter). Because you are implicitly proceeding from the idea that QED formulation is unique (the standard one), and I believe in the opposite.

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What I don't like about your papers is that you solve a little problem only and then assert boldly ''It is proposed to build QFT in the same manner.'' This makes it impossible to take you seriously. You first need to have understood QFT properly before you are qualified to propose to rebuild it. Lacking this understanding you sound to me like my little son building a Lego house and then proposing to build a skyscraper in the same way. It is funny and it indicates his level of understanding, but it not to be taken seriously.

Your belief that QED can have other formulations is meaningless unless you can at least make it plausible how your ideas are compatible with Lorentz invariance and with gauge invariance.

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Actually, the value of $\Lambda$ in QED is largely irrelevant If it is large enough. It just means that for each value of the cutoff, the other parameters must be refitted to the experimental data, giving them a cutoff dependence. Thus there is a 1-parameter family of accurate QED formulations, all corresponding to essentially the same renormalized coupling constants.

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Yes, I know this "effect" of toy models. But to a great extent it is a matter of confidence in the author. If S. Weinberg would propose such a toy model, you would take it more seriously than from me. If my examples make it impossible to take me seriously, then do impossible - search for a rational in my stuff despite lack of confidence in me.

Thus there is a 1-parameter family of accurate QED formulations, all corresponding to essentially the same renormalized coupling constants.

Yes, I know that. But we do not converge - each of us repeats his own phrases. What can we do to converge? Imagine, I have already learned QED and know it better than you. But I am not happy with it, just like P. Dirac. Because it is not a calculations when one fits the results rather than equations. For some reason you think I do not know QFT and its problems, probably because I am not discussing non interesting to me aspects of QFT.

Another thing is that you are completely happy with your understanding of QFT, so you do not see any need in reformulating it. All this makes you consider me as an annoying person fighting with shadows.

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Actually, I am not happy with QED either, like Dirac. And like you I am dreaming since many, many years ago that my ideas may be sufficient to produce a better approach to QED. I created one toy theory after another, smaller and bigger ones, and always found that it fell short from what the real QED must satisfy. Through my trials and errors I learnt more and more about QED as actually practiced. I also learnt to appreciate the power of QED as it currently is, and the reasons why it is so difficult to create better foundations. It is nearly impossible...  I also saw many people trying it and failing; I tried to learn from all attempts I could find in the literature (including yours). Maybe I now understand QED better than anyone else in the world; at least my powers to explain it well have grown tremendously, compared to what I have seen in the literature. But all techniques combined weren't enough to reach the mountain top - a rigorous foundation for QED that has the same power and accuracy as the existing QED but can be reasonably explained without any awkward trickery.

Like in many past years, I am still dreaming to be successful one day - no, not one day but until the coming summer! But one summer after the other passes, and I am slowly getting old. Nevertheless, my insight into the matter still increases year after year, so maybe this summer I'll succeed!!?

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You see, we are not so different although our motivations and approaches are different. In the last paper I just showed that a reasonably formulated theory is $\Lambda$-insensitive by construction and without fitting game involving bare constants. It is more physical and the toy model itself is quite realistic atomic example. No stretches.

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One big difference is that I no longer announce my dreams as proposals, but publicize my little steps (if I make them public at all) as just being good solutions to small problems, and leave the speculative part to my readers and to my heart. 

The stretch in your submission is your summarizing sentence, '' It is proposed to build QFT in the same manner.''

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The refitting needed in traditional QED is not that bad, as you only need to refit two constants, the electron mass and the electron charge, to the renormalization conditions. Everything else can then be calculated from the fit. Thus one has a highly predictive theory. 

Whatever you'll be doing, you also need to match these two data points. Your fit is slightly easier, since you simply take them as parameters - but as long as you ignore all its mathematical structure and replace it by a heuristic ansatz,  you'll have a lot of trouble to make your alternative theory be as predictive as the real QED. This is why I believe your approach to be doomed.

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... as long as you ignore all its mathematical structure and replace it by a heuristic ansatz,  you'll have a lot of trouble to make your alternative theory be as predictive as the real QED. This is why I believe your approach to be doomed.

Those toy models (mechanical and atomic) are supposed to convince you in the opposite.

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''convince you in the opposite''??

Electrodynamics is characterized by Lorentz invariance and gauge invariance, both in the classical and in the quantum domain. I see nothing in your toy models that give the slightest hint about how to combine your ideas with these two basic properties of electrodynamics. Only this would convince me, since this is the main difficulty to be overcome in reaching a better QED. 

Note that experiment is in full agreement with both principles, and Dirac never renounced these!

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You and Ron are blaming me for the lack of relativistic and gauge invariance as if I were against these properties. It is very strange! I never said that these properties must be abandoned in a future theory. I said that calling them "guiding principles" is too much since they are simple properties and dictate nothing. You cannot imagine how many relativistic- and gauge-invariant expressions can be written down, but the main difficulty is in correctly encoding physics in the new equations. For example, the correct radiation reaction term in CED should be a radiation reaction term rather than a self-induction, self-acceleration, or an ersatz term.

Answer 1

Standard QED correctly encodes all finite time physics about electromagnetic processes in its renormalized equations, consistent with relativistic and gauge invariance; and almost all in the asymptotic domain, where the infrared problems are partially but not fully resolved. 

So any proposal to reform QED must be measured against these achievements.

I am not blaming that your models are not relativistic and gauge invariance. But I am blaming that you seriously consider your toy models to be a big step towards a novel QED though neither you nor anyone else has the slightest idea how to make the step from the toy to the real thing. The path you propose is covered with unsurmountable barriers called relativistic and gauge invariance. But you don't care and happily repeat your proposals.

If QED were a mountain top in the sea of relativistic and gauge invariance, your toy model is like a small rocky hill on the other side of the shore, that you discovered how to climb. And, knowing that the famous climber Dirac thought that the current difficult path to the top of QED could not be the final answer, you shout, ''Hey, I finally found what Dirac was missing - the principle by means of which one can climb the QED top'', without having the slightest idea of how to cross the sea. Why should anyone take you seriously?

One must begin at the right side of the shore - assuming relativistic and gauge invariance -, and then see how to find from there a better path to the top. But this cannot be done without getting from the literature the best charts that already exist, and studying them in more detail than the others, so that one can discover the uncharted territory there that might hide a better way to the top. And yes, if one then has better equipment than the Elders, one might be more successful. 

So if you say,

You cannot imagine how many relativistic- and gauge-invariant expressions can be written down, but the main difficulty is in correctly encoding physics in the new equations,

I fully agree, this is indeed the essential problem. But instead of suggesting a new way how to approach this problem (which would count as a step towards a novel QED) you throw away the first, essential half - you don't have QED unless you have these! You replace it by another problem where the essential difficulty is absent, solve the resulting easy problem, but claim you are close to a novel QED.

The essential difficulty is not to cope with short-distance physics or with the photon dressing. Here your toy models indeed propose techniques for handling these. Not the only known techniques by the way; for example, you might want to look at the work by Derezinski, who spent most of his academic career with various aspects of such problems. 

The essential difficulty is rather to find a way to formulate a relativistic and gauge-invariant theory in which the short- and long-distance problems can be handled better than with the current, successful but somewhat awkward renormalization techniques. In approximations without relativity or gauge invariance, the problem has been already solved by people like Derezinski, with models far closer to real QED than your simple toy models.

Thus there is no shortage of methods for handling short- and long-distance problems, but only of methods for combining these with relativistic and gauge invariance. You contributed nothing to the latter, hence nothing towards a novel QED.

Comments

... there is no shortage of methods for handling short- and long-distance problems ...

If you read my "Short-distance" paper, you will see that no methods are necessary since in a reasonably formulated theory there are no IR and UV problems.

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 in a reasonably formulated theory there are no IR and UV problems.

The essential problem is then to find a (covariant and gauge invariant) formulation of QED that is a reasonably formulated theory in this sense. You brought us no step closer to this goal.

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The essential problem is then to find a (covariant and gauge invariant) formulation of QED that is a reasonably formulated theory in this sense. You brought us no step closer to this goal.

It is the next step. First we should make sure that not having those problems is possible in principle.

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I am looking forward to you taking the next step. (Your first step doesn't count in my eyes, as one can always make difficulties disappear by simplifying the problem enough. Things start to get interesting only when you approach the real obstacle.)

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Arnold, by now you should know for sure that the real obstacle for me is the lack of funding.

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This is a poor excuse. My work in physics is also not funded. I do it in my spare time, besides a full-time job as a professor for Computational Mathematics that costs me more than 40 hours a week. Lack of funding still allows me to read the relevant literature, to learn what is needed to understand QED properly, and to be modest in my claims about the possible relevance of my work.

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I am not you. If I say I need funding, it means I cannot do this work otherwise.

Answer 2

''What can we do to converge?''

I don't want to converge, I want to reach the top! Your toys are not suitable for this, or I would have used them.

But if you want to converge with me, walk my ways and we'll reach the top together. If you trod a different path than I we'll never converge.

Comments

:-)