[FOM] Question about theoretical physics

Joe Shipman JoeShipman at aol.com
Wed Feb 27 19:41:23 EST 2013

Neumaier wrote:
How to obtain the first order Taylor expansion is discussed in many textbooks on quantum field theory.

How to get (in principle) the values of all coefficients of the asymptotic series is described with full rigor, e.g., in Kreiner's work on renormalization in terms of Hopf algebras. Finding the values
of higher and higher coefficients is essentially reduced to evaluating lots of integrals in higher and higher dimensions. Evaluating the latter reliably to high accuracy is itself a very difficult and time-consuming problem.

You may as well complain about numerical analysis not being able to evaluate complicated integrals in 100 dimensions to 12 digits of accuracy.

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.

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?

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.

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?

-- JS

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