[FOM] Question about theoretical physics

[Joe Shipman]

Both my concerns are serious, but they are independent. 

It is clear that QED does not have a sound mathematical foundation, and I objected to the lack of progress in over 60 years of attempting to give it one and the false impression given that it was some kind of model for a mathematized physical theory. This is not the "scandalous" part, since you can't force theoretical progress and the amount of grant money that has been spent on this is not scandalously large, being a minute fraction of the money spent on the experimental side--I'd say that the grant money was scandalously small in comparison except I wouldn't know who to give additional money to to significantly improve the chances of solving this problem.

The poor documentation of the calculations that have actually been done is less serious from a theoretical point of view, but more embarrassing from a professional point of view. The assertion that the theories agree with experiment to N decimal places is the kind of thing that must be checkable and reproducible. 

Tim's analogies to mathematics are on the right track but there are distinctions his examples need to reflect. The Appel-Haken-Koch proof was objectionable only because the calculations were too large to be verified by hand, but the documentation was always excellent so that it was easy both to verify the proof that the algorithm was correct, and to reproduce the calculation. 

The Hales-Ferguson proof is a better example, but the issues are very clear and the authors are involved in a project (the "Flyspeck" project) to create a completely formalized and machine-verifiable proof comparable to the proof of the Four-Color Theorem in its transparency. 

This is not something that the QED theorists are likely to do because QED is not the final "Theory of Everything" anyway, and there is reason to believe a formal disproof of convergence would be the result rather than a formal proof of convergence. 

However, there must still exist actual source code that was used to generate the published theoretical predictions, and publishing and documenting this source code in order to allow verification and reproducibility is a fair request and the physicists deserve criticism for not having done this. 

Can Professor Neumaier, or any other theoretical physicist, provide a link the following much simpler pieces of documentation?

1) a method for enumerating Feynman diagrams relevant to the calculation of the magnetic moment of the electron

2) a high-level description of the process by which the value of a Feynman integral for *one specific nontrivial Feynman diagram* is numerically approximated

This seems like a bare minimum that ought to be reducible to terms understandable by a mathematician who does not know advanced physics. The first item is purely combinatorial. For the second item I am not asking how the function to be integrated was arrived at, merely how, GIVEN the function and the diagram, its Feynman integral can be numerically approximated.  This is again a purely mathematical question--I am willing to assume that the Feynman diagram and the function to be used in the Feynman integral have been GIVEN by "physics", and just want to know what the numerical integration process is.

-- JS

Sent from my iPhone

On Mar 2, 2013, at 12:23 AM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:

Arnold Neumaier wrote:

> See the citations from ''Is QED consistent?'' mentioned above.

I took a brief look at a couple of these.  Now that Joe Shipman has clarified that his objection is simpler than the "caricature" that I painted in previous posts, I think that his complaint is analogous to the complaints that have been leveled against the Appel-Haken-Koch proof of the four-color theorem and against the Hales-Ferguson proof of the Kepler conjecture.  Namely, the excessively detailed computations involved have not been documented in a fully satisfactory manner.  For example, Passera
(hep-ph/0606174) writes:

The exact expressions for these coefficients are rather complicated,
containing hundreds of polylogarithmic functions up to fifth degree (for
the light-by-light diagrams) and complex arguments (for the vacuum
polarization ones). Indeed, they were too long to be listed in [19, 20]
(but were kindly provided by their authors), although series expansions
were given for the cases of physical relevance.

I interpret "kindly provided by their authors" to mean a private communication with Passera.

The fact that the computations involve non-rigorous mathematics arouses additional anxiety in a mathematician, who may wonder if a non-rigorous calculation is independently reproducible.  But it seems to me that the lack of rigorous mathematical foundation for QED is something of a red herring, if Shipman's main concern is being able to reproduce the calculation.  The real problem is that both the mathematics and the physics communities haven't completely solved the problem of adequately documenting extremely complicated calculations.  I think this is a legitimate concern, though Shipman's use of the term "scandal" in a way that implies that QFT is a particularly egregious offender seems a bit over-sensationalized to me.

Tim
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