[FOM] Question about theoretical physics

Arnold Neumaier Arnold.Neumaier at univie.ac.at
Fri Mar 1 06:25:17 EST 2013

On 02/28/2013 01:41 AM, Joe Shipman wrote:
> The difference is that I can figure out from textbooks of numerical analysis how to write a program that does specific calculations to desired degrees of accuracy, while textbooks of quantum theory fail to describe the calculations precisely enough that I can do this.

It just takes more effort. Just as high energy physics experiments are 
very time-consuming and cannot be checked by most people unless they 
work on it full time, so high energy physics predicitions are very 
time-consuming and cannot be checked by most people unless they work on 
it full time.

We are no longer in a time where a scientist can easily check in detail 
what is going on in a discipline. People may work for half a year full 
time on figuring out how to calculate a 5th order term for a single 
prediction. this is the nature of high precision tests. It is not a 
simple task.

> Can you give a reference to the description of the calculation of the prediction of the anomalous magnetic moment of the electron that, in your opinion, does the best job of giving a good programmer what he needs to actually implement the calculation?

I have given in my FAQ detailed references to the relevant publications. 
    Is QED consistent? (from Chapter B5)
They give an amount of details considered to be adequate by the physics 
community. To repeat the calculations based on these publications should 
be possible for every physicist trained in quantum field theory, 
prepared to study the background literature, and diligent enough in high 
performance programming. But it is a lot of work, and I never tried. It 
is highly specialized work, and checked only by highly specialized people.

Related material from my FAQ:
    QED and relativistic quantum chemistry (from Chapter B6)
    The usefulness of QED (from Chapter B6)
    Summing divergent series (from Chapter B5)

> For example, such a description might include:
> 1) a method of enumerating the relevant Feynman diagrams of each order, and testing whether they are equivalent
> 2) a method of numerically approximating the relevant integral for each diagram to a prescribed accuracy
> 3) a scheme for managing the summation of the infinite series by specifying, for a desired accuracy in the result, and a given precision in the fine-structure parameter alpha, what degree of diagram to go out to, how many Feynman diagrams of each degree to estimate the integral for, and how far to carry the numerical approximation of each Feynman diagram.

Each of these topics could easily fill a PhD thesis, and each such 
thesis would probably do it quite differently. (Point 1 has become a bit 
easier through Kreiner's work mentioned earlier in this thread.)

But there is no scheme for reaching a desired accuracy, as lacking a 
mathematical definition  of QED, it is impossible to give error bounds.

Even in areas where the conceptual basis is clear, error bounds are 
often unknown. For example, the Navier-Stokes equations are (unlike QED) 
mathematicaslly well-defined, but its rigorous solvability (a 
prerequisite for error bounds) under realistic conditions is still 
unsolved (another Clay millenium problem).

> The theoretical physicists publish "theoretical" numbers which must be the output of SOME program, but I have not seen any explanations which are precise enough to qualify as specifications for writing such a program. What's the best source you've got?

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

Arnold Neumaier

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