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Reformulation instead of Renormalizations

  1. Apr 15, 2009 #1
    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).
     

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  3. May 25, 2009 #2
    Last edited: May 25, 2009
  4. Jun 5, 2009 #3
  5. Jun 18, 2009 #4
    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.
     
    Last edited: Sep 5, 2009
  6. Jul 9, 2009 #5

    Fra

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    Hello Bob,

    A comment on your comment in the other thread.
    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.

    I share your sentiment here. But IMO it's not ONLY about renormalization problem. It's alot 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
     
  7. Jul 9, 2009 #6

    Fra

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    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 alot of papers yet.

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

    /Fredrik
     
  8. Jul 9, 2009 #7

    Fra

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    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
     
  9. Sep 17, 2009 #8
    It's about time to develop this approach in details, solve concrete problems and work out calculation technique.
     
  10. Oct 19, 2009 #9
  11. Oct 19, 2009 #10
    So what? I propose reformulation on a physical basis rather than renormalizations.
     
    Last edited: Oct 19, 2009
  12. Nov 4, 2009 #11
    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.
    I will think about it. Maybe I will add "maybes" to those phrases to make them less striking.
    Roger. I will smooth my complaints and add some uncertainty to my phrasing.
    A Hamiltonian, to be exact...
    I will do it, thanks for advising.
    No, it's a trial one, it is clearly stated.
    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).
    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.
    My trial Hamiltonian contains the quantum oscillator Hamiltonians like any other QED Hamiltonian in the Coulomb gauge. What are you speaking about?
     
  13. Nov 4, 2009 #12
    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.
     
    Last edited: Nov 4, 2009
  14. Nov 4, 2009 #13

    Fra

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    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
     
  15. Nov 4, 2009 #14
    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.
     
  16. Nov 4, 2009 #15

    Fra

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    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

    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
     
  17. Nov 4, 2009 #16
    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.
     
    Last edited: Nov 5, 2009
  18. Nov 5, 2009 #17
    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.
     
  19. Nov 5, 2009 #18
    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.
     
  20. Nov 5, 2009 #19
    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.

    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.
     
  21. Nov 5, 2009 #20
    Correct.
    Correct. The textbook coupling, though, includes a self-action term. I couple them without self-action.
    Correct. In my approach there is no correction to the electron mass and run-away solution.
    Wrong. New are notions and equations that correctly describe the known experiments.
    It is infinities in your imagination that you try to impose to my theory.
     
    Last edited: Nov 5, 2009
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