Dirac on renormalization (was: Is general relativity incompatible...)

Eugene Stefanovich wrote:

> "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.
>
> 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.

The concept of bare particles (and infinite masses and coupling
constants) does not figure at all in modern settings of renormalization
based on the renormalization group. (See, e.g., Salmhofer's book).

Thus these worries are no longer relevant.

The only persisting worries are those about the meaning of the
scattering matrix which so far exists only as an asymptotic series
rather than as a mathematically well-defined operator.

Arnold Neumaier

.

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