How can an asymptotic expansion give an extremely accurate prediction, as in QED?

What is the meaning of "twenty digits accuracy" of certain QED calculations? If I take too little loops, or too many of them, the result won't be as accurate, so do people stop adding loops when the result of their calculation best agrees with experiment? I must be missing something.

• May 1 '13 at 14:38

Suppose you're interested in computing some quantity $F(\alpha)$, like the excess magnetic moment $g-2$, which depends on the fine structure constant $\alpha \simeq .007$.

Perturbation theory gives a recipe for the coefficients $F_i$ of an infinite series $\sum_{i \geq 0} F_i \alpha^i$, which is expected to be asymptotic to $F$. This means that, if you add up the first few terms, you should get a decent approximation to $F$.

$F \simeq F_0 + F_1 \alpha + F_2 \alpha^2 + F_3 \alpha^3 + ... + F_N \alpha^N$

If you keep adding higher order corrections, making $N$ bigger, the approximation will get better for a while, and then, eventually, it will start to get worse and diverge in the limit $N \to \infty$. In QED, we don't know exactly where the approximation gets worse. However, one expects on general grounds that the series approximation will start to get worse when $N \simeq 1/\alpha \simeq 137$. (And this is indeed what we see in toy models, where we can make everything completely rigorous.)

In principle, then, we should add up all the Feynman diagrams of order $\leq 137$. In practice, we don't have the ability to do this. Computing all those diagrams is very time-consuming. Even with computer assistance, we have difficulty going beyond 4 or 5 loops.

• hm... this describes what an asymptotic series is but does not explain why it behaves that way or what properties it has. granted, that'll make it just purely mathematical, but given that one never encounters asymptotic series in undergrad mathematics (or at least I have not) I'd be happy to be given a quite intro to them. May 1 '13 at 14:28
• @nervxxx: I think it answers the original question, which I took to be "how do we choose when to stop adding loop corrections, and why is this reasonable?" Understanding the behavior of asymptotic series is a worthwhile goal. I'd begin by looking at Marino's notes laces.web.cern.ch/laces/LACES10/notes/instlargen.pdf May 1 '13 at 14:37
• @nervxxx We did series convergence when I was in high school, though people on a slower math track didn't get it until early in college. Generally with easier to compute coefficients, but still. I think the class was often called "Precalculus" and covered the first pass of analytic geometry, basic trigonometry and various bits of foundational stuff on sequences and series to prepare you for calculus. May 1 '13 at 14:50
• @dmckee I'm pretty sure what you covered was on regular series, not asymptotic series. They are quite different things. Of course you do the usual series stuff in high school and college, but I don't think anyone formally does asymptotic series anywhere. the fact that you're assuming I'm talking about regular series goes to show that fact... May 1 '13 at 17:14
• @nervxxx I wouldn't be at all surprised to learn that dmckee learned the basics of asymptotic series in high school. That's where I first learned it. May 1 '13 at 17:24