[FOM] Question about theoretical physics

Kreinovich, Vladik vladik at utep.edu
Thu Feb 21 17:18:53 EST 2013

In QED volume of Landau and Lifchitz series of book (this one was written after Landau's death), it is stated very clearly that the corresponding series diverge, so they are only asymptotic. 

-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of Joe Shipman
Sent: Wednesday, February 20, 2013 9:19 PM
To: fom at cs.nyu.edu
Subject: [FOM] Question about theoretical physics

I want to focus on a point which arose in the discussion about physical theories. The question is, when we measure a very precisely known physical quantity like the (dimensionless) anomalous magnetic moment of the electron, "g", and compare it with a theoretical prediction from Quantum Electrodynamics, what kind of mathematical object is the predicted value?

My understanding is that it is the sum of a power series in the fine-structure constant "alpha", where the coefficients of the power series are computable numbers, but no computable modulus of convergence is involved and no proof of convergence is known. This makes the predicted value Pi^0_3 in the parameter alpha. 

Furthermore, alpha is measured by the same experimental technique that is used to measure g; there is a range of values of  alpha which is consistent with the measured value for g. QED would be considered falsified if it were shown that for all alpha, the predicted value of g would lie outside the experimentally measured range of possible values for g. In other words, an experimentally observed range (r1,r2) with rational endpoints for g, and the assumption that QED is correct, gives rise to a parameter-free mathematical sentence Phi of type Sigma^0_4, that looks like the following:

Phi: ***There exists a rational value alpha_r, such that for all rational epsilon>0, there exists an integer N, such that for all integers M>N,

the calculation of the Mth partial sum of the power series with alpha=alpha_r lies within the rational interval (r1-epsilon, r2+epsilon).***

Note that the coefficients of the power series are computable real numbers with known convergence moduli, without which we would need a statement of even higher type.

If, after choosing rationals r1 and r2 bracketing the current experimental range, the resulting purely mathematical statement Phi were to be disproven, QED (in its usual form where the Nth coefficient of the power series comes from summing Feynman integrals across all the degree-N Feynman diagrams) would be falsified. If, on the other hand, Phi were to be proven, QED would be shown "consistent with the experimental measurement of g".

It is my impression that neither of these has occurred. Instead, the power series has been treated as an asymptotic series where the error is assumed to be smaller than the next term, so that proofs of convergence can be avoided, and the resulting modification is what has been calculated to be consistent with experiment. It is also my impression that nobody expects to be able to prove Phi, but some mathematical physicists believe that Phi may be refutable, because the number of Feynman diagrams involved in the sequence of coefficients grows factorially, eventually overwhelming the term-by-term decrease coming from the powers of alpha.

I hereby beseech any theoretical physicists, or mathematicians who know more about QED than I do, to confirm or correct the account I have given. If my account is correct, then even the supposedly well-established theory of Quantum Electrodynamics does not have a valid mathematical foundation, because it unjustifiably assumes convergence of the relevant series.

-- JS
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