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One of the tests of Quantum Electrodynamics is the value of the "Anomalous magnetic dipole moment".

The theoretical value is:

$$a_e = 0.001\ 159\ 652\ 181....$$

We say that QED "predicts" this value. But as far as I'm aware, the series which predicts this actually diverges. (Or at least not proven to converge?) Hence the full series does not predict this value it predicts infinity.

So in order that the theory of QED is predictive, it means that we must state two things:

(1) How many terms of the series are we to use.

(2) After discarding the rest of the series, how close to the real value is this theoretical value.

e.g. We might say, QED is accurate up to 10 terms in the series and this gives a value which will be within 0.001% of the experimental value.

If we don't state these two things, and the series diverges, all we can say is that coincidentally the first few terms off the series matches the experimental value. And that adding more terms MIGHT or MIGHT NOT give a closer value.

So assuming the series in the coupling constant diverges, has anyone worked out how many terms are we supposed to take, how close to the experimental value we should expect to get, and at what point will the series begin to get less accurate?

Alternatively we might say that we don't know when the series will begin to get less accurate and we can only add more terms and compare it with experiment. In which case this is not very predictive.

A third possiblility is that the number of terms we can use would be proportional to the cut-off we are using (as in renormalisation), but if this is the case we would need a formula relating the number of terms to use with the cut of parameter.

Qmechanic
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    I like that you talk not about the fact that without renormalisation every loop order on its own would already diverge, but also that it is said that the whole series in the coupling constant is not a divergent, but an asymptotic series, which does diverge from the actual value if you include higher and higher orders. This always puzzles me: Both in canonical quantisation as well as in the path integral method this series is an expansion of an exponential function... but why should this be an asymptotic series at all (Since an exponential is defined by its series)? Any references for that?? – Koschi May 31 '21 at 18:41
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    I'm looking at this: http://jakobschwichtenberg.com/divergence-perturbation-series-qft/ which seems to suggest that we can trust the series up to 137 terms if the coupling constant is 1/137. But it doesn't say how close to the experimental value we should eexpect to get. –  May 31 '21 at 18:57
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    A minor quibble. You're asking a good question about the series (+1), but QED's predictive power doesn't rely on the series. Perfectly well-defined finite functions often have divergent small-parameter expansions. QED is like that. It has a perfectly legit, finite definition in discrete spacetime. The discreteness is artificial, but practical experiments can't tell the difference. One of QED's key properties is that changing the discretization scale doesn't noticeably effect its predictions for practical experiments. (That's roughly what "renormalizable" means.) – Chiral Anomaly May 31 '21 at 20:33
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    Related: How can an asymptotic expansion give an extremely accurate prediction, as in QED? – Chiral Anomaly May 31 '21 at 20:52
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    @Koschi The paper Asymptotic behavior of the QED perturbation series is a place to start. – Chiral Anomaly May 31 '21 at 20:52
  • @ChiralAnomaly Could you not expand your comment to an answer? Looking through the references I vaguely understood that simple expansions (as used in fourier and taylor ones) cannot include the perturbative expansion illustrated with Feynman diagrams, that an asymptotic series is a different type of expansion. May be you should make it clear for the non theorists of us. – anna v Jun 01 '21 at 04:48
  • @annav I wish I knew the answer. The usual Feynman-diagram series comes from expanding a scattering amplitude in powers of a small coupling constant. As we include higher powers (more diagrams), the sum gets closer to the correct answer, but only up to a point. After that, including more diagrams just makes it worse. We can verify this in simpler examples, but the question asks quantitatively when that happens in QED (how many Feynman diagrams we can include before it starts to make the approximation worse). I don't know when that happens in QED, so I'm not qualified to write an answer. – Chiral Anomaly Jun 01 '21 at 13:30
  • @ChiralAnomaly Lubos Motl, , who is no longer interactive here, has this relevant to the question blog entry https://motls.blogspot.com/2010/11/quantum-field-theory-has-no-problems.html#more . I have learned a lot from his contributions here and in his blog, and I trust him as a theoretical physicist. Note that he calls the perturbative expansion in Feynman diagrams a Tailor expansion. ( I suspect that this proposal that asymptotic expansion is something different than the Tailor expansion for me means to be classified together with all the navel gazing about entanglement and interpretation – anna v Jun 01 '21 at 14:22
  • etc ...) Quote :"The actual thing that will be compared with the experiments is actually the cross section of an observable process." This is the gauge he uses on whether divergences lead to infinities. – anna v Jun 01 '21 at 14:24

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