[FOM] Question about theoretical physics

Arnold Neumaier Arnold.Neumaier at univie.ac.at
Tue Feb 26 12:11:32 EST 2013

On 02/26/2013 04:56 AM, Joe Shipman wrote:
> After seeing all the replies here and on a physics discussion forum where this was cross-posted, I find no reason to change my views, which are the same as they were 20 years ago. The complete failure of "constructive quantum field theory" to provide anything that is recognizably an effective algorithm to predict results of real experiments is a scandal, and the lack of awareness of this in all of the articles and books addressed to non-physicists is another scandal. None of the literature I have checked on CQFT gives any good reason to expect that they are on the right track, and all of it tangles the physics and math together in the frustrating way almost all theoretical physicists always do, instead of describing things in a rigorous way.

The purpose of constructive quantum field theory was never to find an 
effective algorithm to predict results of real experiments, just as the 
purpose of the theory of partial differential equations was never to
find an effective algorithm to solve partial differential equations.
Finding algorithms is the realm of numerical analysis, and there it is 
done with various degrees of rigor, depending on the problem to be solved.

> Is it really too much to ask that the actual recipes that are officially supposed to be used in practice to calculate prefictions of experimental results be described in a precise algorithmic fashion, so that mathematicians who aren't theoretical physics can have a crack at them?

To complain with some measure of justice requires to know of a way how 
to do it.

That no one has been able to see a way just shows how hard the problem 
is. It is on par with proving or disproving the Riemann hypothesis; both 
are among the seven Clay millenium problems.

> Even if a proof of convergence is too much to ask of the physicists, they ought to at least be able to describe precisely what they actually DO to generate the successive digits of, say, the prediction of the anomalous magnetic moment of the electron.

How to obtain the first order Taylor expansion is discussed in many 
textbooks on quantum field theory.

How to get (in principle) the values of all coefficients of the 
asymptotic series is described with full rigor, e.g., in Kreiner's work 
on renormalization in terms of Hopf algebras. Finding the values
of higher and higher coefficients is essentially reduced to evaluating 
lots of integrals in higher and higher dimensions. Evaluating the latter 
reliably to high accuracy is itself a very difficult and time-consuming 

You may as well complain about numerical analysis not being able to 
evaluate complicated integrals in 100 dimensions to 12 digits of accuracy.

Arnold Neumaier

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