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Renormalizable quantum field theories

  1. Jun 1, 2009 #1
    In Quantum Field Theory, there are 'infinity' problems:

    At extremely short distances the energy quanta increase so
    much that infinities would occur. In order to overcome this
    a cut off is imposed that postulates that quanta cannot possess
    energy above some arbitary high value.
    This works well for low energy calculations but not for high
    energy interactions.

    1) At what length does this cut off happen - approximately
    2) How is this justified?

    A reference if you don't know what I am talking about:
    http://en.wikipedia.org/wiki/Quantum_field_theory#Renormalization
     
  2. jcsd
  3. Jun 1, 2009 #2
    The necessity to make a cut-off, let us say, in QED for certainty, originates from the fact that without it the integrals diverge.
    The cut of is an artificial trick, it is not related to some physics. Introducing a cut-off is called a regularization. There are many of them. Then, in the "renormalizable" theories, the divergent terms are grouped so that they are perturbative corrections to the initial mass and charge.
    After that they say: "Let us discard these corrections", and the corrections are discarded. So no dependence of the cut-off parameter remains. Others say: "Our original mass and charge are not observable but with these big corrections they are observable, therefore they are equal to the finite m and e." Both ideologies give the same final "renormalized" finite expressions. They are cut-off independent.

    This discarding prescription is not legitimate mathematically and reflects a too bad (too distant) initial approximation and bad interaction (self-action) used in the perturbation theory. The "renormalized" solutions are not solutions of the original theory but those of another one that I call a Novel QED.

    Practically the renormalization justification is done by comparison with the experimental data. It is a good luck that sometimes this prescription works (see my "Reformulation instead of Renormalizations" by Vladimir Kalitvianski). In fact there is a possibility to reformulate QED in better terms and obtain convergent series automatically.

    Bob_for_short.
     
    Last edited: Jun 1, 2009
  4. Jun 1, 2009 #3
    I basically agree with Bob_for_short's assessment. Just wanted to add a couple of points.

    In QFT (really, I have QED in mind) we meet problems when calculating the S-matrix (e.g., scattering amplitudes). Some momentum integrals in these calculations appear to be divergent when the integration momentum tends to infinity. The fix suggested by the renormalization theory is two-fold:

    1. Introduce momentum cutoff, so that integrals are forced to be finite.
    2. Add certain (momentum-dependent) "counterterms" to the Hamiltonian of QED.

    If you do these steps carefully, then you can find that calculated scattering amplitudes

    1. Become finite and cutoff-independent
    2. Agree perfectly well with experiment when involved particle momenta are below the cutoff.

    All this is nice and good. The question is what if we want to study interactions at momenta higher than the cutoff? The problem is that we cannot take the cutoff momentum to infinity, because "counterterms" in the Hamiltonian are divergent in this limit and we would obtain an ill-defined Hamiltonian. The prevailing attitude is that we should not take the infinite cutoff limit, because this would mean probing interactions at such small distances that our ideas about smooth space-time are no longer applicable. It is assumed that at small distances (on the order of Planck length or whatever), some new effects take place (like space-time granularity or string-like nature of particles,...) which invalidate the use of standard QED. So, the suggested solution is to pick the momentum cutoff to be above the characteristic momentum in the considered physical problem and below the inverse Planck length.

    There is also an alternative point of view, which is called the "dressed particle" approach. It says that the QED Hamiltonian with countertems is badly screwed up. It suggests to fix the divergences in counterterms by applying an unitary "dressing" transformation to the Hamiltonian. As a result of this we obtain:

    1. New (cutoff-independent and finite) Hamiltonian of QED.
    2. All scattering amplitudes computed with this Hamiltonian are exactly the same as in the traditional renormalized QED, i.e., agree with experiment very well.
     
  5. Jun 1, 2009 #4
    In the classical field theory of electromagnetism it is necessary to do the same renormalization procedure in order to account for the radiation backreaction.

    I just see renormalization as a 'problem' with Lorentz invariant theories involving point particles, the impossibility of having a finite energy lorentz-invariant near-field of an accelerating point particle being similar but obviously more subtle than the problem of having a lorentz invariant rigid body. But my point is that this issue does not only arise in QFT.

    By the way, for those who do not like renormalization, one of the awesome properties about strings as opposed to particles is that their interactions are finite to all orders in perturbation theory, no renormalization required, the integrals converge!
     
  6. Jun 2, 2009 #5
    This is not unique to strings. The same is true in the "dressed particle" approach to QFT. All loop integrals are convergent. No regularization/renormalization required.

    E. V. Stefanovich, "Quantum field theory without infinities", Ann. Phys. (NY), 292 (2001), 139
     
  7. Jun 2, 2009 #6
    A unitary clothing transformation in which the Hamiltonian remains finite
    in the limit of removed regularization because infinities present in theta
    exactly cancel infinities in V (where a transformation 'e to the i theta'
    -sorry LATEX is too hard for me - has been applied).


    E. V. Stefanovich,
    Quantum Field Theory without Infinities
    http://www.geocities.com/meopemuk/AOPpaper.html
     
  8. Jun 2, 2009 #7
    As soon as Eugene starts from the same QED Hamiltonian and from the same initial approximation (free particles), he is bound to reproduce the usual renormalized QED expansion however it is called - dressing or just renormalizations, whatever. In both approaches occurs the perturbative "dressing". By the way, this name was invented in the standard QED, before "dressing" transformation. Nobody has been able to explain how finally a dressed electron looks like. (At best, one speaks of infinite "vacuum polarization" around a point-like charge that screens it original (infinite) "bare" charge.)

    In Eugene's approach the same banal infra-red difficulties arise. It means too distant initial approximation (the strong photon-electron coupling neglected).

    As I said in my first reply, the fundamental constant renormalizations are equivalent to discarding the correction to the known, phenomenological constants. This reveals their non legitimate mathematical character. Renormalizations remove perturbatively the self-action interaction term. What does remain after renormalization? A potential interaction of compound systems, not self-action. That is why in the Novel QED I start directly from another Hamiltonian - without the self-interaction term and with electroniums as the initial approximations. No wonder I obtain immediately mathematically finite and physically correct results.

    In order to show that there is no problem at short distances, let as consider an atom. It is practically unknown but true that the positive charge in atoms is quantum mechanically smeared, just like the negative (electron) charge. The positive charge cloud size is much smaller than the negative charge cloud size, but it is finite and is of the same nature (turning aroung the atomic center of inertia). So the effective ("dressed" if you like) potential is not as singular as the Coulomb one (1/r) but is much softer, for example, Ueff(r) ≈ 1/{r2+[(me/MA)a0]2}1/2 for Hydrogen. This potential gives correct amplitude of elastic scattering at large angles (it gives the positive charge elastic form-factor that serves as a natural regularization factor in integrals). This potential does not tend to infinity when r →0 but remains constant. An of course, the cut-off size is much much larger than the Plank's length.

    The same is valid for an electron permanently coupled to the quantized electromagnetic field: it charge is quantum mechanically smeared, so no singularity of the effective potential appears in calculations if one takes the coupling exactly in the initial approximation. Perturbative taking into account leads to the infra-red divergence. It is easy to understand: the potential (1/r) is infinitely "far" from Ueff(r) at short distances and the corresponding integrals diverge. That is why it is necessary to start from better initial approximation where photon-electron coupling is taken into account exactly.

    Pushing the bound nucleus in atom or pushing the bound electron in electronium excites the internal degrees of freedom of the corresponding compound system, so the inelastic processes (atom exciting or photon oscillator exciting) happens automatically in the first Born approximation. So no infra-red divergence arises. The inclusive cross section give well known classical results - the Rutherford cross section, as if the target charge were point-like ans situated at the center of inertia of the compound system.

    Details of physics and mathematics of the Novel QED are reported in "Atom as a "Dressed" Nucleus" and in "Reformulation instead of Renormalizations" by Vladimir Kalitvianski (available on arXiv).

    Bob_for_short.
     
    Last edited: Jun 2, 2009
  9. Jun 2, 2009 #8
    bose condensate & renormalizable quantum field theories

    1) Are we assuming that the wavefunction region is a limit of energy quanta following the 1/r laws? So, as we approach the centre of a 'wavefunction region' the energy quanta diminish to zero instead of diverging to infinity?

    2) How does this cut off affect an Einstein Bose condensate where wavefunctions are merged?
     
  10. Jun 2, 2009 #9
    I could not understand you first question. Apparently we think differently.

    I could not understand your second question: Bose-Einstein condensate of what? Of electrons? I am sorry, I cannot reply.

    Bob.
     
  11. Jun 2, 2009 #10
    I start my article "Reformulation instead of Renormalizations" from analysis of H. Lorentz ansatz about self-action and point-likeness of the electron. I show that the mass renormalization is just discarding corrections (perturbative or exact) to the phenomenological electron mass. In other words, it is postulating new equation for the electron dynamics.
    The same discarding is made in QED. It is good luck that such a prescription works. Normally it does not work - the number of non-renormalizable theories is much larger.

    I show also how the energy-momentum conservation law can be preserved without self-action ansatz.

    Finally, I show that in compound systems there is always a natural cut-off mechanism so no necessity to invent strings or other "grained space-time" appear.

    Starting from compound systems gives naturally soft radiation which is not case in Eugene dressing transformation or in the standard QED approaches. I propose physically and mathematically justified approach. I do not rely on good luck.

    Bob_for_short.
     
    Last edited: Jun 2, 2009
  12. Jun 2, 2009 #11

    malawi_glenn

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    bob, read the forum ruels; dont adress answers to "problems" from your own, non-published, ideas/theories.

     
  13. Jun 2, 2009 #12
    My works were published. For example, "Atom as a "dressed" nucleus" has been published in the Central European Journal of Physics, V. 7, N. 1, pp. 1-11 (2009). I consider there the same problem. As well it was published long ago in the USSR (1990-93). My "RiR" is available on arXiv as a prepublication and I refer to it since it answers the questions of this thread. They are not problems of my own but the "eternal" problems of interacting fields. I, as many of us, tried to resolve them. None has found an error in my works so far. What is my own is my opinion based on my results. It is what this forum is made for - an exchange of opinions. Those who have no their own results refer to somebody else's opinions.

    Bob.
     
    Last edited: Jun 2, 2009
  14. Jun 2, 2009 #13

    malawi_glenn

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    oh it has been published recently? That is better :-)

    Well we tend to discuss accepted opinions here, if you read the forum rules you will understand.

    have fune
     
  15. Jun 2, 2009 #14
    No, I published it long ago. I propose you to take part in my poll about positive charge atomic form-factor, please.

    I read somewhere your opinion about people who are not happy with renormalizations. You consider them to be crackpots.

    I am sure you are happy with renormalizations and I am sure you know the whole universe history from the Big Bang till the end. You feel so high, you even touch the sky. That is why you take a liberty to lecture me.

    Learn hard and be open-minded.

    Regards,

    Bob_for_short.
     
    Last edited: Jun 3, 2009
  16. Jun 2, 2009 #15
    Bob, I got this from a physics blog (I believe its yours??):
    http://vladimirkalitvianski.wordpress.com/

    “It is also described with an atomic (positive charge or “second”) form-factor, so the positive charge in an atom is not “point-like”. The positive charge “cloud” in atoms is small but finite. It gives a natural “cut-off” or regularization factor in calculations
    also in blog:
    "It is practically unknown but true that the positive (nucleus electric) charge in an atom is quantum mechanically smeared, just like the negative (electron) charge. "

    I assume by +ve 'charge cloud' you are referring to some sort of wavefunction region?
    Its interesting to me to know what's going on here - a wavefunction region seems a good reason for capping energies - or maybe I am on the wrong path here?
     
  17. Jun 2, 2009 #16

    George Jones

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    Let me remind everyone that Physics Forums rules to which everyone who registers agrees,

    https://www.physicsforums.com/showthread.php?t=5374,

    in part, state
    I'm locking this thread. Any discussion of Bob_for_short's ideas should take place in the appropriate thread in the Independent Research Forum,

    https://www.physicsforums.com/showthread.php?t=307642.
     
  18. Jun 3, 2009 #17
    Is there a discussion of renormalization and dressing anywhere else?
    I am interested in this topic, but cannot get a discussion going or join
    one.
     
  19. Jun 3, 2009 #18
    I think you may discuss the generally accepted issues here. I will not participate any more in order not to have the thread locked.

    If you are interested in my personal findings, you can read my articles available on arXiv and discus them in the independent research group, the thread https://www.physicsforums.com/showthread.php?t=307642.

    Bob.
     
  20. Jun 3, 2009 #19

    DarMM

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    I just wanted to mention that this is not correct. The addition of counterterms to the Hamiltonian makes it finite as the cutoff is removed. There are several field theories where this has proven to be the case, see Glimm and Jaffe's book.
    In fact to be totally accurate without renormalizations the Green's functions of theories are zero, not infinite. So the addition of infinite counterterms makes the theory non-zero and finite.
     
  21. Jun 3, 2009 #20
    Dear DarMM and George Jones,

    Don't you think that starting from "bare" (non physical, wrong) Hamiltonian furnished with non physical counter-terms is less overt crackpottery than starting from physical (well defined, everything is known experimentally) Hamiltonian with physical interaction?

    Could you also participate in my poll on the positive charge atomic form-factor, please?

    Bob (for short).
     
    Last edited: Jun 3, 2009
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