[FOM] Question about theoretical physics

Arnold Neumaier Arnold.Neumaier at univie.ac.at
Fri Feb 22 04:45:50 EST 2013

On 02/21/2013 05:19 AM, Joe Shipman wrote:

> 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?

It is a number obtained by an uncontrolled approximation of a
mathematically not yet well-defined theory. See Chapter B5: Divergences
and renormalization of my theoretical physics FAQ at

> 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.

No. The sum of a power series with convergence radius zero is logically

> 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.

No. By convention, falsification in high energy physics requires a
discrepancy of 5 times the standard deviation of the measurement errors
(which are realizations of a random variable).
But since the predicted value is itself an uncontrolled approximation,
it is not 100% clear when precisely QED would count as being falsified.

> 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.

This can be (and is) safely be assumed as in QED the initial
corrections are rapidly decaying (most likely until over 100 terms).

> even the supposedly well-established  theory of Quantum Electrodynamics
> does not have a valid mathematical  foundation,

This is well-known since around 1950. See the FAQ cited above.

> because it unjustifiably > assumes convergence of the relevant series.

This is not the case since every physicist is aware of the divergence of 
the series. However, it is also known that one can work rigorously with 
many asymptotic series in the manner described, for sufficiently
small values of the expansion parameter. Here the meaning of 
sufficiently small depends on the number of terms used and on the
function to which the series is asymptotic. Thus only unverified 
assumption is that the value of alpha is small enough.

Arnold Neumaier

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