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Let us take the standard QED ($e^-, e^+, \gamma$) as a model of QFT and ask what is its "short-distance" physics?

They say the UV infinities appear because we do not know the real physics of short distances and initially we introduce it wrong. OK, but after renormalizations, what physics does remain? Do we replace the unknown/wrong physics with certain/right one? Can anybody describe it without appealing to unphysical bare particles? Have we an idea about the real electron from QED? If so, why we cannot use it as the input to construct a reasonable theory from the very beginning?

P.S. Moderators, please do not close my questions before they are answered, let people answer.

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5 Answers 5

Let us take standard fluid dynamics as a model of continuum media physics. It is obvious what is its "short-distance" physics isn't it? These are atoms and molecules. Let's apply your questions to this case -- suppose that we know nothing about atoms and molecules. in that case we can still use hydrodynamics, don't we?

Do we replace the unknown/wrong physics with certain/right one?

What is "wrong" and "right" physics in a first place? If your hydrodynamics predicts physical phenomena in it's range of validity then I'm free to call it "right". Even if it says nothings about atoms. Anything else is a philosophy.

Can anybody describe it without appealing to unphysical bare particles?

Can anybody describe hydrodynamics without appealing to "unphysical" continuum medium? Yes -- if one knows about atoms. If one doesn't know then one can try to speculate about underlying theory. But it doesn't render hydrodynamics useless and "wrong".

Have we an idea about the real electron from QED? Why we cannot use it as the input to construct a reasonable theory from the very beginning?

Have we an idea about "real" stream of water? It is made of atoms, right? So in this particular case there is no "real" stream of water at all. That's why we cannot use it as the input to construct a reasonable theory from the very beginning.

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Thank you, Kostya, for your answer. I hope you will be able to say something about QED now. –  Vladimir Kalitvianski Feb 20 '11 at 20:47
    
Hydrodynamics as a physical model does not have severe conceptual and mathematical difficulties. It has its region of applicability, as any theory, but it is a self-consistent model. In other words, we do not rebuild its solutions "on go" with referring to unknown physics at short distances because the initial model has no physical (or reasonable) solutions. –  Vladimir Kalitvianski Feb 20 '11 at 20:49
    
I see often people ascribing me weird statements I never made. In particular, I did not blame hydrodynamics and never called it "useless" and "wrong". As a model, hydrodynamics works fine analytically and numerically within its validity region. –  Vladimir Kalitvianski Feb 20 '11 at 20:54
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@Vladimir: Yes, we in fact do have to reformulate these models 'on the go'--For instance, in order to make the thermodynamics of the Van der Walls equation of state consistent in the neighborhood of a phase transition, it's necessary to introduce a Maxwell construction. The mathematics of renormalization in stat mech is identical to the mathematics of renormalization in paritcle theory. –  Jerry Schirmer Feb 22 '11 at 16:31
    
I deleted some comments that were veering off topic. –  David Z Feb 23 '11 at 6:10

Lets start from experimental data. What is an electron, what do we know about an electron? It is too small to touch or see or smell. Everything we know about an electron comes from several levels of proxies. We end up measuring a track circle in a magnetic field and get e/m, consistently for different "electrons" and we do the millikan oil drop and get e and then we can assign a mass to these manifestations consistently.

That is all we have for the electron, it has a mass m_e measured and a charge e, measured.

Nature has been good to us and a working theory exists for QED. Mathematics is a tool, it can describe and predict measurements but it is not something that creates reality. Reality is what one measures . If the theory predicts, it does not matter if it goes into a yoga position to do so, as long as it can predict consistently. They want to call them bare and dressed mass? Fine. Who can measure anything more than that the measured mass is m_e and and the measured charge e?

Better theories/computations may come up, but to be better they should describe existing measurements and predict more and different ones, that QED cannot explain, for anybody to pay any attention. Or be as overwhelmingly economic and elegant as the heliocentric is to the geocentric pov. QED works.

added: I want to give an example from real physics history that I heard from the horse's mouth back in the 1980s, of how succesful new methods of computation overwhelm tradition and sweep over reluctances once shown to successfully predict faster and accurately.

Back in the Manhattan day project, a physicist think tank had been set up with the best brains of the time to calculate crossections needed for making the bomb. Feynman was a junior member of the team. They gave the group a problem and a week later people reported the result of their independent calculations, parallel processing. Feynman said that one afternoon he was lying on his bed with his feet on the wall, when the Feynman diagram method came to him, whole ( he had eidetic memory so he probably saw it). He calculated the current problem and waited impatiently for the report of the others. When he gained confidence that his method was as good as the long drawn out s matrix calculations he started playing games with the team. He would get the result in an evening, tell them the next day what they would find, and it would take them the rest of the week to confirm.

Of course Feynman diagrams were universally accepted after that.

I was reminded of this story when I listened to the talk of Nima Arkani-Hamed which he gave on the twistor revolution. He finds extremely cumbersome the Feynman diagrams method and is exploring a new one that gives the same results as the thousands of summed QCD feynman diagrams. I was amused, and am sure that Feynman would have been too, if he were still alive.

If a new computational method is faster, sleeker and as predictive, it will be adopted as surely as God made little cabbages.

In my experimentalist's opinion of course.

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The comment thread was becoming quite a mess. I deleted them all (this seems to be a preferred method of dealing with such situations). Let's not start up a discussion in any further comments. –  David Z Feb 24 '11 at 18:22

If by "short distance physics" you mean "arbitrarily short distances" then the answer to your question is that your question is ill-defined. You cannot consistently describe arbitrarily small distances within QED.

Here is why.

QED (as opposed to QCD) has a Landau pole and thus is inconsistent if viewed as a "fundamental" theory. Like most other quantum field theories QED should be viewed as an effective field theory with its range of validity. You should not trust QED as an appropriate description of Nature at (or beyond) the Planck scale (or even at the GUT scale or at whichever new physics scale may lie between TeV and the GUT scale).

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I agree that QED as a model is incomplete but the question is not here. QED as a model is renormalizable and therefore it has some certain physics of short distances (implied, implemented, incorporated, whatever). I just want to know that (model) physics. –  Vladimir Kalitvianski Feb 24 '11 at 15:54
    
QED is renormalizable but INCONSISTENT. You cannot escape the Landau pole. –  Daniel Grumiller Feb 24 '11 at 19:29
    
I know and I agree but the Landau pole appears in connection of "bare" charge with the real charge solely. As soon as nobody cares about bare charge value and uses only real charge in calculations, it does not make a problem for calculation results which are quite certain. –  Vladimir Kalitvianski Feb 24 '11 at 20:34
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If you know and agree then the issue is settled: QED cannot make a statement about what happens at arbitrarily small distances, simply because it is not applicable there. In the fluid analogy by Kostya questions like "what is the temperature of a single atom" are meaningless. You have to go to larger distance scales to talk about temperature. Your question is of the same kind, at least if by "short distance physics" you mean "arbitrarily short distances". –  Daniel Grumiller Feb 25 '11 at 13:10
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@Vladimir Your assertions about thermodynamics are just wrong. The temperature of one molecule is just undefined, not demanded to be equal to the liquid temperature. Second, molecules (and atoms) are the containers of degrees of freedom of a system in thermodynamics; volume doesn't do this. You lost a lot of credibility for me in a thread about advanced physics when you revealed that you don't know much about the very basics... –  kharybdis Mar 6 '11 at 19:33

Before attempting an Answer to this Question, I up-voted Anna V's and Daniel Grumiller's Answers. If we consider that "QED" includes all the calculational apparatus of an expansion as a series of divergent Feynman integrals, their regularization, resummation, and renormalization, then Daniel Grumiller's criticism of the question seems unavoidable.

If we consider "QED" to be defined more minimally, however, just by a deformed free field Hamiltonian or Lagrangian, without commitment to a particular elaboration of that definition, then it seems possible that there is a reasonable, analytic answer, but we have not so far constructed any such mathematical approach. In such a case, however, I suppose that any calculational treatment would nonetheless have to make contact with Anna V's request for engineering utility, which I further suppose requires a convergent series expansion of some sort on some finite domain from some different starting point. It seems unlikely, therefore, that we should start from the "bare particle" free field, since we already know that constructing a series centered on that starting point that is convergent in some sense requires extraordinary, time-consuming measures. To my knowledge, no-one has proved that every possible expansion of QED is everywhere divergent, but functional mathematics is not the relatively elementary playground of complex analytic functions, so such a proof may be possible.

With this prelude, I feel myself in a position to give partial answers to the sequence of subsidiary questions:

Do we replace the unknown/wrong physics with certain/right one? Maybe, maybe not. Can anybody describe it without appealing to unphysical bare particles? I think the bare particles thing is a problematic starting point, I think we have to start somewhere else. Have we an idea about the real electron from QED? If QED has an alternative series expansion that has some engineering utility, then the current expansion presumably has some relationship to the new expansion, in which case, yes. If so, why we cannot use it as the input to construct a reasonable theory from the very beginning? It's not necessarily the theory that's problematic, it may be the calculational approach. Just as one example, all quantized free fields already have something similar to the scalar field's property that a 2-point covariance function $C(x-y)$ diverges for $x\rightarrow y$, which makes free fields a questionable starting point.

Here, however, I acknowledge a fundamental difference from your approach as I understand it, Vladimir, that, to me, algebras of operator-valued distributions are a "better" starting point than particles, so that I have had to contort my answers to get near your starting point, a process which may have left the above near to nonsense, though I find the process instructive for my own rethinking of my ideas. For a long time I have thought that QED was itself a poor starting point for constructing mathematically consistent models, whether physical or not, but (for reasons unspoken here) I find myself more open to the idea as a result of attempting to answer this question.

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The benefit of this genre of posts, is that they are challenging. There will be no progress if we speak only of what we were taught. Wisdom is the result of testing what we know, and everyone has misconceptions, including myself ;) Honestly:

on wikipedia: Renormalization: ... was first developed in quantum electrodynamics (QED) to make sense of infinite integrals in perturbation theory. ...

What infinities? I can not conceive of any infinite amount that can be attributed to any object in this universe. And why not? Because it would take an infinite time or infinite energy or a null or infinite length.

For example what is the logic of integrating the EM field from r = 0 to infinity? Can be done mathematically, but physically is nonsensical because the short total lifetime of the particle (and universe). The wavelength of a photon is limited by the size of the oscillator where it originated. Enormous (or miserably small) wavelengths imply correspondingly long (or short) time of interaction.

Thus the application of cutoffs is natural and the right response.

As for perturbative methods, the calculation of a quantity by an infinite sum (not of energy, wavelength, etc) is one approach to another still unknown function. A situation that could raise such representation is the existence of feedback between two components. Any physical process responds within the short interaction time and do not involve computers nor have endless time to do the calculation.


Can someone explain, please, because I am troubled with Vladimir comment: 'At least, Classical Electrodynamics predicts EMW when the electron is accelerated. So this phenomenon (radiation) occurs always. Now look at the Mott, Bhabha, Klein-Nishina formulas - they are the first Born approximations describing elastic processes. No radiation is obtained at'

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Whatever the transferred momentum q is in particle collisions (whatever particle acceleration is while colliding), the radiation does not happen! It is so because in this approximation the field is decoupled from charges. The most essential charge feature (capacity to create fields) is not taken into account. None of participants of this discussion recognizes it as a flaw of the initial approximation. In the next perturbative order, instead of small corrections, they obtain explosion of everything: radiative corrections are infinite. So no calculation is possible without "repairing" results. –  Vladimir Kalitvianski Feb 22 '11 at 23:28
    
I saw your site and I agree that you are by far nearer to a better explanation. A point particle is absolute nonsence. A point is only a math abstraction and has no extension at all. You must read Haus, Goedecke, Pinnow (Our Ressonant Universe). The non-radiation condition. Wave mechanics. The Corpuscule notion is a construct of our senses. Danger. The longest time of interaction must limit the shortest wavelength, is'nt it? and the length scale must limit the upper frequency, IMO. I wonder I have no comments, only bad votes. You understand me, it is very diff to be against. –  Helder Velez Feb 26 '11 at 23:58
    
In a linear problem with unrelated vars we can add solutions, else we must do the calcs altogether. A runaway solution is IMO a ill defined playground. Limits of integration, etc... The distance from my home to Lisbon is +-10km, but I will not try to measure from here to Moon and subtract from Moon to Lisbon. It is not the same! ;-) –  Helder Velez Feb 27 '11 at 1:20
    
Phonons are photons -» a wave. The quantization is due to the finite possible states that emitters and receivers can possess (they are resonators with distributed charges). QM do correct results but I think that a different toolbox.. Wave mech... Good luck. I'm risking many points, gime100 ;-) –  Helder Velez Feb 27 '11 at 1:22

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