Re: Is general relativity incompatible with the Newtonian limit?

"Arnold Neumaier" <Arnold.Neumaier@xxxxxxxxxxxx> wrote in message
news:43785F95.6060407@xxxxxxxxxxxxxxx
> > However great a theorist was Dirac, he was wrong about this assessment
> > of renormalization. Perturbative renormalization is based on sound
> > mathematics *and* is capable of produce correct veriable (and verified)
> > predictions.
>
> True and not true. It produces verifiable predictions if you restrict
> attention to the first few terms of a (most probably divergent)
> asymptotic series, but it has no way to make sense of the whole
> series. This is what Dirac found deficient in the foundations.
>
> An asymptotic series is a series such as
> f(x) = sum_{k=0:inf} k! x^k
> with radius of convergence zero. For small enough x, the first few
> terms give seemingly good approximations, but if one includes for
> any fixed nonzero x enough terms, the series diverges. Thus, as Dirac
> asserts, one neglects arbitrarily large terms to get the approximations
> which work so well in QED.

I think that Dirac was frustrated not by the zero radius of convergence
of the perturbation series for the renormalized S-matrix. I think his
frustration was directed at the basic renormalization algorithm that is
at work in each given perturbation order. This algorithm has two
flavors. One is to simply throw away some infinite terms on the pretext
that they are "physically unacceptable", e.g., they violate the gauge
invariance or they give infinite contributions to the electron mass or
charge. Another, slightly more agreeable approach, is to add certain
infinite counterterms to the interaction Hamiltonian, so that
contributions from these counterterms exactly cancel other infinite
contributions to the S-matrix. I think Dirac disliked both these
approaches. I agree with him completely.

There is another quote from a leading theoretician that seems to confirm
my interpretation:

"Thus, present quantum electrodynamics is one of the strangest
achievements of the human mind. No theory has been confirmed by
experiment to higher precision; and no theory has been plagued by
greater mathematical difficulties which have withstood repeated attempts
at their elimination. There can be no doubt that the present agreement
with experiments is not fortuitous. Nevertheless, the renormalization
procedure can only be regarded as a temporary crutch which holds up the
present framework. It should be noted that, even if the renormalization
constants were not infinite, the theory would still be unsatisfactory,
as long as the unphysical concept of "bare particle" plays a dominant
role. If one considers quantum electrodynamics as a phenomenological
theory with respect to the mass and charge of the interacting particles,
and if one consequently condones the necessity of infinite mass and
charge renormalizations, one is tempted to consider quantum
electrodynamics as a pretty satisfactory theory..." F. Rohrlich in
http://www.philsoc.org/1962Spring/1526transcript.html

It is clear that Rohrlich was worrying about the difference between bare
and physical particles, rather than about the convergence of the
perturbation series. Eugene.

.

Relevant Pages

  • Dirac on renormalization (was: Is general relativity incompatible...)
    ... Perturbative renormalization is based on sound ... This is what Dirac found deficient in the foundations. ... > of the perturbation series for the renormalized S-matrix. ... One is to simply throw away some infinite terms on the pretext ...
    (sci.physics.research)
  • Re: When do we need renormalization
    ... >>parameters and renormalization. ... >>phases of the theory while the cutoff is still finite. ... >>the parameters in the Hamiltonian are infinite, ...
    (sci.physics.research)
  • Re: A version of QED without ultraviolet divergences
    ... field theory is defined fundamentally with a distance cutoff D that has some ... The renormalization and it infinites is really nothing to worry about at ... If we use normal QED Hamiltonian with finite ... > is to use INFINITE masses M and charges E in the ...
    (sci.physics.relativity)
  • Re: Connes & Marcolli paper on renormalization
    ... how renormalization of formally infinite quantities can be done ... treatment produces in these cases the same kind of infinities ... is fine - this corresponds exactly to renormalized perturbation theory ... Since nonrelativistic and 2D or 3D relativistic theories also have ...
    (sci.physics.research)
  • Re: When do we need renormalization
    ... >>The infinities you encounter in QM are different from infinities in QFT. ... >>that the binding energy is infinite. ... And it gives (after renormalization) completely reasonable ... But one can study a nonrelativistic, particle number ...
    (sci.physics.research)