A Feynman loop diagrams and Dyson series for anomalous magnetic moment

[Meir Achuz]

Well, the discussion is about the question, whether there's a theoretical explanation for the value of the fine structure constant. This has been asked since Sommerfeld introduced it when treating the fine structure of the hydrogen spectrum using Bohr-Sommerfeld quantization. It's indeed an ironic coincidence that he gets the fine structure right without introducing spin (which is pretty impossible within Bohr-Sommerfeld quantization anyway) by just solving the relativistic equations of motion of a classical point particle in a Coulomb field.

Now the fine structure constant is, in the SI, simply
$$\alpha=\frac{e^2}{4 \pi \epsilon_0 \hbar c}.$$
With the new SI the question about the status of its understanding is very easily answered: in the above formula $e$ ("elementary electric charge", $\hbar=h/(2 \pi)$ ("modified Planck's action quantum"), and $c$ ("speed of light in vacuo") are all defined constants fixing our system of units, but $\epsilon_0$ has to be measured, i.e., it cannot be derived somehow from any theory. So the value of $\alpha$ is an empirical input in our contemporary best theory we have (in this case the Standard Model of elementary particle physics).

But, it is $\alpha$, not $\epsilon_0$, that is actually measured.

 

[PeterDonis]

I thought that someone put a 'skeptical' imogi on my original post.

Probably because they had no idea what you were trying to say in that post. Neither did I.

 

[Adrian59]

Maybe, but the papers reporting these calculations generally explain, at a conceptual level, all of the terms that go into those calculations. Like any Feynman diagram calculation, it includes all possible paths from the starting point to the ending point and usually sorted by the number of loops involved.

So, since we are agreed that α cannot be derived from first principles, it is only the coefficients of the power series that connects the anomalous magnetic moment to the fine structure constant (α) that are derived. I can see how one could do this by a straightforward numerical method. However, it appears that these authors are using Feynman diagrams to derive these coefficients, and depending on the order being calculated, one can use either 1, 7, 72, 891, or 12 672 for the lowest order terms. Even at order 5 there are far too many loops to do this without a computer.

This table is based on the same kinds of derivations we have already discussed: they are not derivations of α from first principles, they are basically consistency checks on the perturbation models using the comparisons of model predictions of actual observables with the experimental values.

So, I have two remaining questions:
1) how does one explicitly derive this from a single Feynman loop, especially what values are ascribed to the momenta carried by the loop;
2) how are the more complex numbers of loops dealt with numerically.

 

[vanhees71]

But, it is $\alpha$, not $\epsilon_0$, that is actually measured.

True, but $\epsilon_0$ then follows through simple algebra from that measured value.

 

[vanhees71]

So, since we are agreed that α cannot be derived from first principles, it is only the coefficients of the power series that connects the anomalous magnetic moment to the fine structure constant (α) that are derived. I can see how one could do this by a straightforward numerical method. However, it appears that these authors are using Feynman diagrams to derive these coefficients, and depending on the order being calculated, one can use either 1, 7, 72, 891, or 12 672 for the lowest order terms. Even at order 5 there are far too many loops to do this without a computer.

It can not (yet?) derived from first principles but has to be measured (see my posting above).

So, I have two remaining questions:
1) how does one explicitly derive this from a single Feynman loop, especially what values are ascribed to the momenta carried by the loop;
2) how are the more complex numbers of loops dealt with numerically.

I've no idea to which calculations you refer to, i.e., Feynman diagrams for which processes you are talking about.

 

[Adrian59]

An example is from the Aoyama et al paper referenced in the OP. Equation 31 appears to be written from a loop diagram. Later on the paper the authors derive the 4th order coefficient with α to the power 8.

 

[vanhees71]

Yes, it's a two-loop contribution to the electron self-energy. What has this to do with a derivation of the numerical value of $\alpha$? In QED $\alpha$ is an input parameter, not something that's calculated.

 

[Vanadium 50]

Let.s step back.

The fine structure constant is the electric charge (squared). It is the one element of its definition present in all systems of units, and if you were to somehow define a sert of units without an e², in those units it would still be 4x bigger when discussing a (hypothetical) Q=+2 proton and Q=-2 electron atom.

The mathematical structure of QED permits any value of elementary charge.

Therefore, QED alone cannot predict α. No matter what you do. Sure, it is possible that some larger theory can predict it, but QED cannot. Because the mathematical structure of QED permits any value of elementary charge.

The answer tp the question posed in this thread is "no". If something does allow it to be predicted in the future, that something will not be QED.

 

[Adrian59]

Yes, it's a two-loop contribution to the electron self-energy. What has this to do with a derivation of the numerical value of α? In QED α is an input parameter, not something that's calculated.

That is my point entirely. So more specifically, what are Aoyama et al up to when they appear to start from such a loop diagram in section 4 of the referenced paper!

 

[vanhees71]

They are up to what's said in the title of the paper, i.e., to review the theory of the anomalous magnetic moment of the electron, i.e., the electron's Lande factor including higher-order loop corrections.

 

[Adrian59]

The answer to the question posed in this thread is "no". If something does allow it to be predicted in the future, that something will not be QED.

Thanks for your input. Have you any opinion on the issue I raised in #59 with vanhees71.

 

[Adrian59]

They are up to what's said in the title of the paper, i.e., to review the theory of the anomalous magnetic moment of the electron, i.e., the electron's Lande factor including higher-order loop corrections.

Still with you on this. However, my issue is how they go from Feynman loop calculations to the power series coefficients. They appear to jump from setting up the calculation. They state at the top of page 22 that they wish to 'explicitly work out the fourth order case', and by the middle of the page they quote a numerical result of -0.334166. How do they do this?

 

[PeterDonis]

But, it is $\alpha$, not $\epsilon_0$, that is actually measured.

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.

 

[Vanadium 50]

Have you any opinion on the issue I raised in #59 with vanhees71.

I am starting to conclude that a) the paper does what it says it does and does not calculate α, and b) nothing we write will convince you that α is not something derivable from QED. Other issues belong in a separate thread.

 

[Adrian59]

I am starting to conclude that a) the paper does what it says it does and does not calculate α, and b) nothing we write will convince you that α is not something derivable from QED. Other issues belong in a separate thread

Why is that despite agreeing with you, you keep attributing the wrong idea to me? I agree that α cannot be derived from QED. I think it is pertinent to this thread to explore the issues raised by the Aoyama et al paper. It is deeply mathematical and that may be problematic to some contributors to this thread. However, I did assign this as advanced, so I would hope it would attract suitably proficient responses. So far so good, but I feel I am missing something. I have already stated that my OP could have been worded better: more like, many authors appear to use Feynman loop diagrams to assist in the derivation of the fine structure constant from the anomalous magnetic moment or vice versa. Clearly this is what Aoyama et al are doing. However, they appear to get so far then jump to an answer as I said in #62.

I think their language might be confusing. Normally I would expect an "analytic" result to mean one computed from a closed-form formula instead of numerically. If the formula is an integral, as I would expect for terms in a perturbation series, it would have to be an integral that was solvable using explicit functions, rather than numerically integrated. It is not clear to me that all of the results that this paper says are "analytic" were obtained in this way.

It would appear we have arrived at an agreement on two points:
1) that α cannot be derived from QED;
2) we are not quite sure what Aoyama et al are doing in their so called analytic calculation.

 

[Vanadium 50]

Why is that despite agreeing with you, you keep attributing the wrong idea to me?

Because the titel of this thread is "QED derivation of the fine structure constant?" and the answer "no" has been given by multiple people at multiple times for multiple reasons. Yet the thread continues. It;s natural for people to conclude that you aren't really convinced.

 

[Adrian59]

Because the titel of this thread is "QED derivation of the fine structure constant?" and the answer "no" has been given by multiple people at multiple times for multiple reasons. Yet the thread continues. It;s natural for people to conclude that you aren't really convinced.

I am, can we move on to the other issues I've mentioned.

 

[PeterDonis]

can we move on to the other issues I've mentioned.

What should we change the thread title to to make it clear that it is those other issues that you are really interested in?

 

[Adrian59]

What should we change the thread title to to make it clear that it is those other issues that you are really interested in?

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?

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

 

[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?

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

That's way too long for a thread title.

 

[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.