Feynman loop diagrams and Dyson series for anomalous magnetic moment

[PeterDonis]

Have complex QED calculations involving Feynman loops with say 891 (4th order) or 12672 (5th order) anything to do with calculating either the fine structure constant or the anomalous magnetic moment, accepting we need one of these to be experimentally measured?

Yes. In fact I think calculations now have to be carried to 6th order to get sufficient accuracy to match experiments.

If so how does one deal with these large number of loops at a conceptual level?

I'm not sure what you mean by "at a conceptual level".

 

[Adrian59]

I'm not sure what you mean by "at a conceptual level".

I meant more than 'Oh, we use a computer algorithm'. That is, I was asking for more of an explanation of the mathematical processes.

 

[PeterDonis]

I was asking for more of an explanation of the mathematical processes.

AFAIK they're just grinding through numerical evaluation of the Feynman integrals corresponding to each Feynman diagram, order by order, using standard numerical techniques for evaluating integrals.

 

[Adrian59]

That's way too long for a thread title.

How about, 'using Feynman loop diagrams to calculate the Dyson series for the anomalous magnetic moment.'

 

[PeterDonis]

How about, 'using Feynman loop diagrams to calculate the Dyson series for the anomalous magnetic moment.'

That works. Title change done (I abbreviated it just a little bit).

 

[Adrian59]

AFAIK they're just grinding through numerical evaluation of the Feynman integrals corresponding to each Feynman diagram, order by order, using standard numerical techniques for evaluating integrals.

Hooray, actually this is exactly my thought from the start. Apologies if the OP has mislead anyone.

Also, as you have stated before I cannot see any justification for Aoyama et al claiming there derivation is analytic when it looks like a pure numerical technique.

So if that is the case, the inclusion of Feynman loop propagators is superfluous.

 

[PeterDonis]

if that is the case, the inclusion of Feynman loop propagators is superfluous

Why? They're part of the calculation. Saying it's "numerical" doesn't mean you throw away the equations. It means you evaluate the equations numerically instead of as a closed form answer.

 

[Haborix]

On the analytic vs. numeric thing, I think the distinction isn't really about using numerics or not using numerics. The distinction is how much work do you do starting from the fundamental governing equations before you decide to use numerics. Usually doing something analytically means you gain insight or information you otherwise wouldn't. For example, I might just solve Newton's equations for a pendulum numerically right off the bat. Alternatively, I can discover the energy conservation and what it tells me about solutions, even though in the end I will be reduced to writing Elliptic integrals that will be solved numerically. Similarly, using perturbation series usually tells you something about the strength/significance of various terms/effects, whereas if I just put QFT on a lattice I might not learn as much (or I learn something different). Random aside done.

 

[Meir Achuz]

That depends on the measurement. If you measure the force between two stationary charges and you are using SI units, you are measuring $\epsilon_0$, since that's the only quantity in the equation for the force that does not have a fixed value in SI units.

Yes, you are right. $\epsilon_0$ could now be measured from Coulomb's law. Is there a reference for such an experiment? In any event, the measured value of $\epsilon$ would be limited in accuracy by the present value of $\alpha$. Before the improvement in SI, $\epsilon _0$ was only known as a conversion constant, with no error.

 

[vanhees71]

I guess that's a very inaccurate way of measuring $\epsilon_0$. Rather one measures $\alpha$ very accurately via the quantum Hall effect, the Josephson effect, or determining the gyrofactor of the electron in a Penning trap, which afaik is the most accurate one:

https://arxiv.org/abs/2209.13084

Having $\alpha$ you also have $\epsilon_0$, which is the only constant that's not defined in the SI:
$$\alpha=\frac{e^2}{4 \pi \epsilon_0 \hbar c}=\frac{e^2}{2 \epsilon_0 h c}.$$
Both $h$ (Planck's quantum of action) and $c$ (vacuum speed of light) are used to define the SI units.

 

[Vanadium 50]

Why? They're part of the calculation. Saying it's "numerical" doesn't mean you throw away the equations.

And what is "numerical"? If I have a thousand terms and I calculate them all analytically and add them up, is it analytic? Is it numeric? Does it even matter?

These days, even "analytic" calculations are often done by computer. It reduces the chance for a inconsistency in sign convention (which has historically been a problem with this calculation). Does that make it "numeric"? And does it matter?

I sense this thread is changing direction yet again. And I am not sue that deciding what the right words are to use is going to shed much light on things.