[FOM] Question about theoretical physics

[joeshipman at aol.com]

Tim, what you are saying would be perfectly acceptable, and I would have no problem with the statement that physicists do not have algorithms to calculate even the finite-order approximations to QED predictions, except that the physicists are clearly giving the opposite impression, except for Professor Neumaier.


Let me try to be very careful and precise here. For background, I am using these two articles


http://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment


http://en.wikipedia.org/wiki/Precision_tests_of_QED


The second of these articles covers many experiments, the first covers a single experiment in more detail.

(1) Experimental predictions are expressed in terms of a power series expansion in the fine-structure constant /alpha, where the coefficients for each term come from a sum of contributions from individual "Feynman diagrams" of order corresponding to the exponent of /alpha, and the calculation is cut off after a finite number of terms.

(2) In general the first (linear) term dominates to such an extent that the prediction is a monotonic function of /alpha, so that any given experiment can merely provide a range of possible values for alpha rather than directly falsify QED, where the uncertainty in the interval containing alpha comes from both experimental error and error due to finite approximation. To falsify QED we would need two different experiments whose corresponding ranges of values for /alpha did not overlap. However, from the mathematical point of view what matters is the COEFFICIENTS, which ought to be defined from the theory without reference to experiment. 


(3) The first article states: "As of 2009, the coefficients of the QED formula for the anomalous magnetic moment of the electron have been calculated through order /alpha^4 and are known analytically up to /alpha^3.".



I interpret "known analytically" to mean to arbitrary precision, that they are recursive real numbers. I interpret "have been calculated" to mean that interval estimates are available. I understand most physicists to be saying that, in principle, an arbitrary precision calculation for the coefficients exist, but that for fourth order and higher this calculation has not been reduced to an analytic formula.
To first order, the "one-loop contribution", there is a simple analytic formula, first found by Schwinger, namely 1/2pi. To third order, the expression is given in a 1996 paper by Laporta and Remiddi that can be found here
 http://arxiv.org/pdf/hep-ph/9602417v1.pdf 
and it is a rational polynomial in pi, ln(2), RiemannZeta(3), and RiemannZeta(5) which is ten terms long and for which the most complicated rational coefficient is 28259/5184, and which numerically evaluates to (1/(pi^3))*1.18124...
My issues for Professor Neumaier and for Tim Chow can now be boiled down to these extremely specific and well-defined questions: 


(1) Can the result of Laporta and Remiddi be regarded as a rigorous theorem of mathematics?
(2) considering this experiment only (anomalous magnetic moment of the electron), does the fourth-order coefficient that has been calculated come with a rigorous upper and lower numerical bound?
(3) Is there any reason to believe that an analytic expression of the type that Laporta and Remiddi established for the third-order coefficient does not exist for higher-order coefficients?
(4) Is there a method in principle to calculate arbitrarily precise bounds for the fourth-order and higher coefficients, which we may regard as algorithmic by applying the Church-Turing thesis to the mind of either Professor Neumaier, or of some more expert specialist in the field whom Professor Neumaier could name?


-- JS


-----Original Message-----
From: Timothy Y. Chow <tchow at alum.mit.edu>
To: Foundations of Mathematics <fom at cs.nyu.edu>
Sent: Wed, Jul 3, 2013 5:43 pm
Subject: Re: [FOM] Question about theoretical physics


On Tue, 2 Jul 2013, Joe Shipman wrote:
> That's a good distinction, but what I said, and what Professor Neumaier 
> said, and what Lubos Motl said, all apply to the most fundamental and 
> canonical experiments of all, such as the measurements of the Lamb shift 
> and the magnetic moment of the electron, so the issues I raise are 
> unaffected by this point.

But I think they are affected, because the mere act of asking for an 
*algorithm* means that you're asking for a precise specification of an 
infinite family of inputs and a systematic procedure for treating all 
those inputs.  I don't think that physics operates that way.

The analogy in mathematics might be asking for an "algorithm" for 
generating a computer-verifiable proof of, say, the Poincare conjecture, 
from Perelman's preprints.  There isn't any such algorithm.  What we do is 
to hope that a bunch of experts will get together and study the proof and 
then tell everyone else that they understand how it works.  When this 
happens, we generally declare the problem solved.  It's nice if there is a 
beautifully written exposition that any random mathematician can use to 
confirm the result independently, without the need to talk to any experts 
in order to pick up the tricks of the trade that are needed to fill in the 
"obvious" steps of the proof that the experts don't bother to write down. 
But it's hardly a "scandal" if no such exposition exists.  We trust that 
the experts know what they're doing and that if someone were to take the 
time to become an expert, they would be able to reproduce the same 
results.  That's the only "algorithm" in most cases.

Tim
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