Re: Renormalization
From: Eugene Stefanovich (eugenev_at_synopsys.com)
Date: 03/16/05
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Date: Wed, 16 Mar 2005 16:19:04 +0000 (UTC)
Arnold Neumaier wrote:
> Eugene Stefanovich wrote:
>
>>Chris Oakley wrote:
>>
>>
>>>>What bothers me more than renormalization is the process of
>>>>regularization. Renormalization is the process of removing the
>>>>divergences by absorbing them in physical constants such as mass and
>>>>charge. Renormalization, on the other hand, is a different process.
>>>>Renormalization is required because, when standard perturbation theory
>>>>is used, it is found that the photon has a divergent mass. However,
>>>>unlike the electron, there is no mass term to absorb this divergence.
>>>>Renormalizatoin is the mathematical process by which the divergence
>>>>associated with photon mass is removed. The problem I have with this
>>>>is that the process is very "ad hoc" (in my opinion). The
>>>>renormalization of the electron mass has a physical bases, i.e, the
>>>>bare mass is infinite which partially cancels the divergence that comes
>>>>about in perturbation theory. This, then, results in a finite
>>>>experimental mass. However the bare mass of the photon is presumed to
>>>>be zero so that there is no way to remove the divergent term.
>>>
>>>
>>>Underlying renormalization theory is the belief that one can selectively
>>>violate basic laws of mathematics and still have enough left to make a
>>>meaningful theory.
>>>
>>>I take the (minority) view that this is not possible. Even when one is
>>>using approximations, one can use mathematics to determine the extent of
>>>their validity. But when one starts to deal in quantities that are,
>>>quite simply, meaningless, then the whole edifice collapses. Infinity
>>>minus infinity is indeterminate. That is all one needs to know.
>>>
>>
>>
>>I tend to agree with you, and I offer a solution: forget about the
>>quantization + renormalization + dressing nightmare and
>>"infinity minus infinity" tricks involved there.
>>Just take the RQD dressed particle Hamiltonian as a starting point
>>for doing physics. (I have a few low order terms of this Hamiltonian
>>presented in the book; if you give me some time I'll derive higher
>>order terms that should be more than enough to do practical
>>calculations). With this Hamiltonian you'll never meet divergent
>>loop integrals, there is no need for regularization,
>>renormalization, and "infinity minus infinity" subtractions.
>>As long as you don't ask me where I got this Hamiltonian,
>
>
> and what happens to the zero photon mass,
I believe, this can be dealt with by using standard techniques
for cancelling the infrared divergences. I haven't done that,
you are right.
> and whether the loop
> expansion converges, etc...
As far as I know, convergence is an unsolved problem in
all realistic QFT theories. It will stay unsolved for a
while, I guess.
>
>
>>your sense of mathematical purity will not be offended.
>
>
> There is a lot missing before your approach is mathematically
> acceptable, and you know it!
>
Yes, there are many pieces missing, but at least the ugliest
of them all - the ultraviolet divergences - is in a good shape now.
Eugene Stefanovich.
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