A Feynman loop diagrams and Dyson series for anomalous magnetic moment

[Adrian59]

TL;DR Summary

Using Feynman loop diagrams to calculate the Dyson series for the anomalous magnetic moment

Many authors appear to assert that they can derive the fine structure constant from a Feynman loop diagram(s) from first principles. Most recently a team from Japan lead by Aoyama (Aoyama, T., Kinoshita, T. and Nio , M. (2019). Atoms; vol. 7, (issue 1): pg 28) have attempted this. I have not seen a convincing derivation anywhere. Many commentators state that the fine structure constant can only be measured experimentally either directly or via the anomalous magnetic moment. Who is right?

 

[PeterDonis]

Many authors appear to assert that they can derive the fine structure constant from a Feynman loop diagram(s) from first principles. Most recently a team from Japan lead by Aoyama (Aoyama, T., Kinoshita, T. and Nio , M. (2019). Atoms; vol. 7, (issue 1): pg 28) have attempted this.

Where in that paper do you see them deriving the fine structure constant from first principles?

 

[Adrian59]

Starting from equation 31 which appears to be from a loop diagram with Feynman propagators; they then in equation 32 introduce the Feynman integration parameters. After a lot of algebra we get to equation 67.They then deal with the UV and IR divergences to get to equation 96, and the fourth order co-efficient of -0.344131(48). They then write that this 'is in agreement with the analytical result -0.344166'. I understand that they still have α in their power series, but they don't appear to robustly derive the co-efficient that they quote. It just appears to drop in their paper from nowhere.

 

[PeterDonis]

Starting from equation 31 which appears to be from a loop diagram with Feynman propagators; they then in equation 32 introduce the Feynman integration parameters. After a lot of algebra we get to equation 67.They then deal with the UV and IR divergences to get to equation 96, and the fourth order co-efficient of -0.344131(48). They then write that this 'is in agreement with the analytical result -0.344166'. I understand that they still have α in their power series, but they don't appear to robustly derive the co-efficient that they quote. It just appears to drop in their paper from nowhere.

Even given all this, it still isn't anything like deriving $\alpha$ from first principles. So I still don't see any basis for your claim that that is what this paper is doing, or attempting to do.

 

[PeterDonis]

They then write that this 'is in agreement with the analytical result -0.344166'. I understand that they still have α in their power series, but they don't appear to robustly derive the co-efficient that they quote. It just appears to drop in their paper from nowhere.

The "analytic results" given in the paper appear to be taken from other papers listed in their references.

 

[Adrian59]

I have checked some of the references, and they do not appear any more enlightening. Did you have any particular reference in mind when you made your comment?

Of course this power of series would need an experimentally derived anomalous magnetic moment to solve for alpha. So, I accept that my original post could have been worded better. However, I still cannot see how they get these coefficients for the power series. They appear random.

I could see how one would derive these coefficients numerically, but the authors suggest that they get this coefficient both analytically and numerically. How do they get the analytically derived result?

 

[PeterDonis]

Did you have any particular reference in mind when you made your comment?

Not any one in particular, no. A number of them are mentioned at various points in the paper in connection with "analytic" or "near-analytic" results.

I could see how one would derive these coefficients numerically, but the authors suggest that they get this coefficient both analytically and numerically. How do they get the analytically derived result?

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.

But in any case, I don't see anywhere where they are computing a value for $\alpha$ without any experimental input. They are just making various uses of various experimentally obtained values of $\alpha$ to try to come up with self-consistent estimates of terms in perturbation series ("self-consistent" meaning that the value of $\alpha$ that you get back out of the computation is within the range of possible values of $\alpha$ that you put in to the computation).

 

[Adrian59]

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.

You appear to arrive at the same conclusion as I have regarding this paper. I also agree that this paper does not derive alpha directly, but from a Dyson power series, so my original use of this reference was slightly misleading.

Does this mean that you do not think that Schwinger did derive alpha as he appears to have claimed though I believe the paper in which he was going to show this never materialised.

 

[PeterDonis]

Does this mean that you do not think that Schwinger did derive alpha as he appears to have claimed

I wasn't aware that Schwinger had made such a claim. Do you have a reference?

AFAIK nobody has derived $\alpha$ from first principles independent of any experimental input. Plenty of physicists hope that when we find a more comprehensive theory that has our current Standard Model as a special case, that theory will predict the values of constants like $\alpha$ that we currently have to measure in experiments. But nobody has actually come up with such a theory yet.

 

[PeterDonis]

this paper does not derive alpha directly, but from a Dyson power series

A Dyson power series in powers of alpha. Which means you can't "derive" alpha from such a series at all. The best you can do is, as I said before, to check that the result of the series gives you a value of alpha that is consistent with the value that you put in to compute the series.

 

[Adrian59]

I wasn't aware that Schwinger had made such a claim. Do you have a reference?

AFAIK nobody has derived $\alpha$ from first principles independent of any experimental input. Plenty of physicists hope that when we find a more comprehensive theory that has our current Standard Model as a special case, that theory will predict the values of constants like $\alpha$ that we currently have to measure in experiments. But nobody has actually come up with such a theory yet.

Schwinger, (1947). Physical Reviews; 73(1): pg 416-417.

As to the rest of your entry #10, I am still in agreement that AFAIK no one has produced such a derivation. The primary purpose of my opening this thread was to test this claim in a wider community.

 

[PeterDonis]

Schwinger, (1947). Physical Reviews; 73(1): pg 416-417.

I assume you mean this classic paper?

https://journals.aps.org/pr/abstract/10.1103/PhysRev.73.416

If so, I don't see anywhere in it where Schwinger claims to have derived $\alpha$ from first principles. The paper is his classic calculation of the magnetic moment of the electron including radiative corrections from QED, assuming the best experimental value of $\alpha$ that was then known.

 

[Adrian59]

A Dyson power series in powers of alpha. Which means you can't "derive" alpha from such a series at all. The best you can do is, as I said before, to check that the result of the series gives you a value of alpha that is consistent with the value that you put in to compute the series.

Returning to your point in entry #7, I cannot see why Aoyama et al go to the trouble of setting up an integral from a Feynman loop as if they are pursuing an analytical approach, if in the end they appear to just numerically solve a Dyson series equation. This whole section of the paper appears redundant and as you say confusing.

 

[Adrian59]

I assume you mean this classic paper?

https://journals.aps.org/pr/abstract/10.1103/PhysRev.73.416

If so, I don't see anywhere in it where Schwinger claims to have derived $\alpha$ from first principles. The paper is his classic calculation of the magnetic moment of the electron including radiative corrections from QED, assuming the best experimental value of $\alpha$ that was then known.

Yes, I do. This reference was only a letter though he appears to make this claim cf third paragraph second sentence. Of note that this letter predates the Pocono conference when he made his slightly infamous presentation lasting hours that Oppenheimer was so derogatory about.

However, the historical facts are not my main issue, but what Aoyama et al are up to. Are they trying to complicate things in the hope that readers may think they have achieved an analytical result when in fact they have not?

 

[PeterDonis]

he appears to make this claim cf third paragraph second sentence.

Do you mean this sentence?

The detailed application of the theory shows that the radiative correction to the magnetic interaction energy
corresponds to an additional magnetic moment associated with the electron spin, of magnitude $\delta \mu / \mu = \left( \frac{1}{2} \pi \right) e^2 / \hbar c = 0.001162$

If so, this is not making any such claim. It is talking about the same sort of thing the paper you referenced in the OP is doing: summing terms in a Dyson series. Of course the sum referred to by Schwinger here is only the next order correction after the base value of the magnetic moment, since that was all that could be computed at the time. But the "detailed application of the theory" referred to is summing a Dyson series with $\alpha$ as an input. It is not any kind of theoretical derivation of the value of $\alpha$ as an output.

 

[dx]

I think one must consider the possibility that the fine structure constant is exactly 1/137. This is because the effective fine structure constant at high energy is 1/127.

 

[renormalize]

Even given all this, it still isn't anything like deriving $\alpha$ from first principles.

One genuine (numerological?) attempt to "derive" the fine structure constant was made by mathematician Armand Wyler in 1969. My vague understanding is that it emerges from ratios of the volumes of certain compact subgroups of the invariance group of a massless wave equation. The result is Wyler's Constant:$$\alpha_{W}\equiv\frac{9}{8\pi^{4}}\left(\frac{\pi^{5}}{2^{4}\,5!}\right)^{1/4}=0.007297348130\ldots=\frac{1}{137.0360824\ldots}$$

 

[PeterDonis]

I think one must consider the possibility that the fine structure constant is exactly 1/137. This is because the effective fine structure constant at high energy is 1/127.

You're contradicting yourself. If the effective fine structure constant changes with energy scale (which it does), it can't have a single exact value. (Also, I don't think the high energy limiting value is exactly 1/127. You need to give a reference if you think that is an exact value.)

If you mean that the limiting value of the fine structure constant at zero energy (i.e., the infrared limit of the running coupling in the renormalization group flow) is 1/137, this is known to be experimentally false.

 

[Adrian59]

One genuine (numerological?) attempt to "derive" the fine structure constant was made by mathematician Armand Wyler in 1969. My vague understanding is that it emerges from ratios of the volumes of certain compact subgroups of the invariance group of a massless wave equation. The result is Wyler's Constant:$$\alpha_{W}\equiv\frac{9}{8\pi^{4}}\left(\frac{\pi^{5}}{2^{4}\,5!}\right)^{1/4}=0.007297348130\ldots=\frac{1}{137.0360824\ldots}$$

I was trying to avoid discussion on numerological arguments. They seem to only get near the correct answer but not spot on. Currently the best value for alpha is 1/137.035999.

 

[Adrian59]

Do you mean this sentence?

If so, this is not making any such claim. It is talking about the same sort of thing the paper you referenced in the OP is doing: summing terms in a Dyson series. Of course the sum referred to by Schwinger here is only the next order correction after the base value of the magnetic moment, since that was all that could be computed at the time. But the "detailed application of the theory" referred to is summing a Dyson series with $\alpha$ as an input. It is not any kind of theoretical derivation of the value of $\alpha$ as an output.

Yes, I do but the next sentence says, “It is indeed gratifying that recently acquired experimental data confirm this prediction”. This suggests that Schwinger is making a theoretical prediction.

I agree that this is only to the second order. However, this Schwinger paper predates Dyson’s introduction of a power series.

On reflection, the Dyson series has the anomalous magnetic moment expanded as a power series in alpha with coefficients. So in effect one can find any one of these knowing the other two. Aoyama et al seem to be deriving the coefficients, so I would suggest they needs to insert the experimental values of the anomalous magnetic moment and the fine structure constant.

 

[ohwilleke]

TL;DR Summary: Deriving the fine structure constant from a Feynman loop diagram(s) from first principles.

Many authors appear to assert that they can derive the fine structure constant from a Feynman loop diagram(s) from first principles. Most recently a team from Japan lead by Aoyama (Aoyama, T., Kinoshita, T. and Nio , M. (2019). Atoms; vol. 7, (issue 1): pg 28) have attempted this. I have not seen a convincing derivation anywhere. Many commentators state that the fine structure constant can only be measured experimentally either directly or via the anomalous magnetic moment. Who is right?

These terms are using first principles to convert experimental measurements made at high precision in the form measured to the theoretical QED coupling constant as a coupling constant of the theory.

To do this you express the theoretical value of your precision experiment's expected result in terms of the fine structure constant, and then solve for the fine structure constant given the experimental result.

It isn't a first principles determination of the fine structure constant, it is a first principles conversion of an experimental result's value to a fine structure constant value.

As the saga of the muon g-2 experiment illustrates, for extremely high precision measurements, this conversion is no back of napkin calculation and you can no longer neglect the non-tree level contributions of QCD and the weak force to the precision expected value calculated in terms of the fine structure constant, which imposes some limitations on how precisely the fine structure constant can be determined in isolation.

For example, in the muon g-2 calculation, the different contributions to the experimental value break down approximately as follows:

The bit that my screenshot obscured is HLbL = "92(18) × 10⁻¹¹" which is a 20% relative uncertainty.

The screenshot above is from an April 7, 2021 Zoom press conference from Fermilab announcing a new muon g-2 measurement at Fermilab that was derived from T. Aoyama, et al., "The anomalous magnetic moment of the muon in the Standard Model" arXiv (June 8, 2020), with the same lead author of the paper in question.

You can use the muon g-2 measurement to measure the QED coupling constant to a precision of 1 part per million (or less precisely), since it is a pure QED theoretical calculation in which the QCD and weak force contributions can be ignored as negligible up to that level of precision. But even then, you need to do the QED calculation to something like five or six loops to convert the experimentally measured value of muon g-2 to the QED coupling constant's value, which is a non-trivial calculation (although far simpler than the parallel QCD calculations).

If you want to determine the QED coupling constant from experiments more precisely than you can with muon g-2 (the current experimentally measured value is roughly 100 times more precise than that), you need to find an experiment with less QCD and weak force noise than muon g-2 and do a theoretical calculation of its value in terms of the QED coupling constant for that experiment.

The currently preferred experiments to measure the fine structure constant are experimental measurements of electron anomalous magnetic moments and of photon recoil in atom interferometry, in part, because they have smaller weak force and QCD contributions to the theoretically predicted values of these experiments than muon g-2 does, so the experimental result can be more precisely expressed as a conversion from the experimental result to the fine structure constant.

The electron g-2 calculation is, in form, almost identical to the muon g-2 calculation (literally the only difference is the electron mass v. the muon mass in the equation), but because the electron has a mass of just 0.511 MeV while the muon has a mass of about 105 MeV, the impact of the weak force (which is driven, in part, by the mass of the W boson to relative to the electron) and the impact of the strong force of QCD (which is driven, in part, by the mass of light hadrons relative to the electron mass) is a much smaller share of the overall predicted experimental value of electron g-2 than they are in muon g-2. The magnitude of the greater precision of the fine structure constant from electron g-2 relative to what is possible from muon g-2 is roughly proportional to the ratio of the muon mass to the electron mass.

For reference purposes the g factors of the electron and muon measured in experiments (muon g-2 = ((-muon g)-2)/2):

Another consideration that makes the conversion non-trivial

The fine structure constant runs with energy scale, so any quotation of its value has to be linked to an energy scale (which as an aside is a dimensionful quantity which means that any given quoted value of the fine structure constant is not truly dimensionless, the dimensions are just hidden away in the definitions and footnotes).

So, in some experiments, you'll need to include the beta functions of the QED, QCD, and weak force coupling constants (and perhaps other quantities involved in the experiment) to get the right result for the running of these experimentally determined physical constants with energy scale.

The beta functions can be calculated from first principles in the Standard Model without experimental input. These calculations are also highly non-trivial. Each involves months of work by a small team of mathematical physicists with multiple high end PCs, or a supercomputer, to do and confirm that the calculation done was done correctly.

You can use these beta functions to pick the optimal energy-scale of the experiment you want to use to determine the fine structure constant with, to minimize the noise from QCD and weak force interactions in the theoretically predicted value of your experiment.

 

[ohwilleke]

One genuine (numerological?) attempt to "derive" the fine structure constant was made by mathematician Armand Wyler in 1969. My vague understanding is that it emerges from ratios of the volumes of certain compact subgroups of the invariance group of a massless wave equation. The result is Wyler's Constant:$$\alpha_{W}\equiv\frac{9}{8\pi^{4}}\left(\frac{\pi^{5}}{2^{4}\,5!}\right)^{1/4}=0.007297348130\ldots=\frac{1}{137.0360824\ldots}$$

When proposed, Wyler's constant was within 1.5 sigma of the experimentally measured value, but at current levels of precision measurement Wyler's constant is off by about 1300 sigma from the experimental result (although some of this may be due to errors in Wyler's calculation). From the link in the quoted material above:

While it appears to have a connection with the invariance group of a relativistic quantum theoretical wave equation, a number of errors in Wyler's papers are cited in Robertson (1971). Robertson (1971) also notes, "It is appealing to think that it [the fine-structure constant] might be derivable theoretically. Wyler's number ... appears to have better chances to be derived from a theory than any of the other numbers that also agree with experiment. It may be that even though the expression (8) [from Wyler's paper] is not correct, the number (12) [Wyler's constant] somehow is correct." Adler (1972) terms the constant, "a number in search of a theory" and notes that "whether the agreement of Eq. (23) [Wyler's constant] with experiment has a basis in physics, or is purely fortuitous, remains at present a completely open question."

 

[PeterDonis]

the next sentence says, “It is indeed gratifying that recently acquired experimental data confirm this prediction”. This suggests that Schwinger is making a theoretical prediction.

Sure, but it's not a theoretical prediction of the value of $\alpha$. It's a theoretical prediction of the electron's magnetic moment, making use of the best then current experimental value of $\alpha$.

I agree that this is only to the second order.

Irrelevant. See above.

the Dyson series has the anomalous magnetic moment expanded as a power series in alpha with coefficients.

Exactly.

So in effect one can find any one of these knowing the other two.

There are only two measurable quantities involved here: $\alpha$ and the electron's magnetic moment. The experimental confirmation of Schwinger's calculation amounts to a consistency check on the model. It does not amount to anything like a first principles theoretical derivation of either $\alpha$ or the electron's magnetic moment.

Aoyama et al seem to be deriving the coefficients

Only as intermediate values in the calculation. The coefficients are not measurable. They are theoretical artifacts of the particular calculation method being used.

I would suggest they needs to insert the experimental values of the anomalous magnetic moment and the fine structure constant.

Why would they need experimental values for the magnetic moment in their calculation? That does not appear in the Dyson series they are calculating. Obviously they will need to compare their calculation with experimental measurements, and they do.

For $\alpha$, they already say they are using experimental data.

 

[Adrian59]

Irrelevant. See above.

I was agreeing with your previous comment so I can’t see how this is irrelevant unless your initial comment was itself irrelevant. I actually think your point was correct, so there is nothing irrelevant in my agreeing with that.

 

[PeterDonis]

I was agreeing with your previous comment so I can’t see how this is irrelevant unless your initial comment was itself irrelevant. I actually think your point was correct, so there is nothing irrelevant in my agreeing with that.

It's irrelevant to the larger point that I was making in my previous post, about which we have already exchanged several posts in this thread, and which was the original question in both the title and the OP of this thread. The thread seems to have widened in scope to just a general discussion of QED calculations of measurable quantities; I agree your comment was not irrelevant in that wider context.

 

[Vanadium 50]

First, as said before α = π(g-2) (plus corrections, if you want) is in no way a prediction of alpha.

Second, α varies with distance. At a distance of about 1/40 of a femtometer, it's up to about 1/125. If you're predicting alpha, at what scale?

Finally, in the Standard Model, alpha is entirely predictable from other numbers: you tell me M(W), M(Z) and Γ(Z) and I can tell you alpha. But that just pushes it back a level. If you look under the hood, alpha is essentially the charge of the electron, and in the kinds of theories similar to electromagnetism (called "U(1) gauge theories") any particle can have any charge. So if alpha is predictable at all, it must be from some other theory.

 

[vanhees71]

Even given all this, it still isn't anything like deriving $\alpha$ from first principles. So I still don't see any basis for your claim that that is what this paper is doing, or attempting to do.

Of course, $\alpha=4 \pi e^2$ (in natural Heaviside-Lorentz units), is an input parameter in the Standard model. Nowadays it's defined in the International System of Units (SI) to define the unit of charge, C.

What's calculated in a decade-long effort is the anomalous magnetic moment of the electron (and also of the muon) within the Standard Model. Particularly the value for the electron can be determined so accurately (as well as the theoretical prediction) by experiment that it's an important test for "physics beyond the standard model".

 

[Adrian59]

These terms are using first principles to convert experimental measurements made at high precision in the form measured to the theoretical QED coupling constant as a coupling constant of the theory.

To do this you express the theoretical value of your precision experiment's expected result in terms of the fine structure constant, and then solve for the fine structure constant given the experimental result.

It isn't a first principles determination of the fine structure constant, it is a first principles conversion of an experimental result's value to a fine structure constant value.

Hello Ohwilleke, it is always a pleasure to have you contributing to a thread.

I presume you meant team and not term in the quote; so if I read you correctly, and noting my comment in #20, Aoyama et al are deriving the coefficients to derive an expression linking the anomalous magnetic moment to a power series in the fine structure constant.

So, in some experiments, you'll need to include the beta functions of the QED, QCD, and weak force coupling constants (and perhaps other quantities involved in the experiment) to get the right result for the running of these experimentally determined physical constants with energy scale.

My understanding of these beta functions is that they are just the coefficients in another power series, that for the coupling parameters. So, that they would be closely connected to alpha.

The beta functions can be calculated from first principles in the Standard Model without experimental input. These calculations are also highly non-trivial. Each involves months of work by a small team of mathematical physicists with multiple high end PCs, or a supercomputer, to do and confirm that the calculation done was done correctly.

The problem with understanding these complex calculations is that they are hidden deep inside some computer, and we are then handed a result. Furthermore, I have not seen a convincing derivation even at the one loop level despite reading several good texts (Weinberg, Peskin and Schroeder, Zee, Srednicki, Maggiore). They all start with a loop diagram, and then use Feynman propagators to set up an integral; then the integral is manipulated using the Feynman integral trick. Some authors even go and solve the parameter integrations, but no one seems to explicitly solve the final momentum integral!

 

[Meir Achuz]

In realms of atoms, where electrons dance,
A constant governs, subtle yet so grand,
The fine structure constant, nature's stance,
A guiding force, held by an unseen hand.

Alpha, its name, a symbol of finesse,
Defines the strength of light's enchanting touch,
Unveiling secrets, bringing darkness less,
A cosmic whisper, oh, so very much.

From spectra's hues to atoms' intricate schemes,
Alpha's influence permeates the scene,
Shaping the cosmos, weaving its own dreams,
A constant marvel, nature's masterpiece.

Oh, fine structure constant, ever true,
A testament to harmony, a cosmic clue.

 

[ohwilleke]

Hello Ohwilleke, it is always a pleasure to have you contributing to a thread.

Thanks.

I presume you meant team and not term in the quote;

I do. I am innocent of all typos. Those are caused by evil Internet gremlins after the fact.

so if I read you correctly, and noting my comment in #20, Aoyama et al are deriving the coefficients to derive an expression linking the anomalous magnetic moment to a power series in the fine structure constant.

Correct.

My understanding of these beta functions is that they are just the coefficients in another power series, that for the coupling parameters. So, that they would be closely connected to alpha.

They are closely connected to alpha, but the coefficients of the beta function aren't functions of alpha.

The problem with understanding these complex calculations is that they are hidden deep inside some computer, and we are then handed a result.

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.

You can look at the QED coupling constant beta function in this open access paper explaining how it is calculated.

 

[ohwilleke]

Second, α varies with distance. At a distance of about 1/40 of a femtometer, it's up to about 1/125. If you're predicting alpha, at what scale?

α varies with the momentum transfer in the interaction in question, commonly called "energy scale", and often identified with the variable "s", not with distance.

See Section 10.2.2 of "10. Electroweak Model and Constraints on New PhysicsRevised March 2022" by J. Erler and A. Freitas in R.L. Workmanet al.(Particle Data Group), Prog. Theor. Exp. Phys. 2022, 083C01 (August 11, 2022) at https://pdg.lbl.gov/2023/reviews/contents_sports.html

Specifically:

In most EW renormalization schemes, it is convenient to define a running α dependent on the energy scale of the process, with α⁻¹≈137.036 appropriate at very low energy, i.e. close to the Thomson limit. The OPAL and L3 collaborations at LEP could also observe the running directly in small and large angle Bhabha scattering, respectively. For scales above a few hundred MeV the low energy hadronic contribution to vacuum polarization introduces a theoretical uncertainty in α.

 

[Vanadium 50]

α varies with the momentum transfer in the interaction in question, commonly called "energy scale", and often identified with the variable "s", not with distance.

Wrong.

The source of the variation in alpha is charge screening by vacuum polarization. Shorter distance means less polarization. As a practical matter, short distance also means high momentum transfer (not s in general), which is more experimentally accessible, but the fundamental cause is that you get "inside" the region of charge screening by the vacuum.

 

[Adrian59]

As I said, the primary purpose of opening this thread was to was to explore the origin of claims made to the effect that α can be theoretically derived, and that its value when compared with experimentally derived values is the origin of the assertion that QED is one of the most accurate physical theories . This certainly is the claim made by many popular science books. It appears in more academic circles that there is no absolute derivation of α without some experimental value as an input.

Sure, but it's not a theoretical prediction of the value of α. It's a theoretical prediction of the electron's magnetic moment, making use of the best then current experimental value of α.

However, this is an indictment on academic physics that this is not made clear, and the myth of a pure theoretical derivation is allowed to permeate popular science books. Certainly twenty years ago that was what I thought, having read such claims.

You can look at the QED coupling constant beta function in this open access paper explaining how it is calculated.

The paper you linked here was an extremely good summary of the basics of quantum field theory (QFT) which the earlier sections were all very familiar to me having studied QFT. However, there it is again in table 1.1 a comparison of the theoretical and the experimental of α. Also, the reference is back to our old friend Aoyama, see my opening reference.

Then the first sentence of chapter 6, the conclusions explicitly makes the claim, 'quantum electrodynamics’ unparalleled agreement with experiment has made it one of the most successful theories in the physical sciences.'

There needs to be some clarity as to what is being claimed and by what method.

 

[renormalize]

As I said, the primary purpose of opening this thread was to was to explore the origin of claims made to the effect that α can be theoretically derived, and that its value when compared with experimentally derived values is the origin of the assertion that QED is one of the most accurate physical theories . This certainly is the claim made by many popular science books.

Can you cite some specific references from "popular science books" that claim $\alpha$ can be theoretically derived?

 

[PeterDonis]

the myth of a pure theoretical derivation is allowed to permeate popular science books

Pop science books are not good sources for learning actual science, so it would be no surprise to me (unfortunately) if they made claims about this that are not made in textbooks or peer-reviewed papers. This problem is certainly not limited to claims about theoretical derivations of $\alpha$.

The paper

It's not a paper, it's a thesis. The rules for those are somewhat different.

there it is again in table 1.1 a comparison of the theoretical and the experimental of α.

This table is based on the same kinds of derivations we have already discussed: they are not derivations of $\alpha$ 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.

the conclusions explicitly makes the claim, 'quantum electrodynamics’ unparalleled agreement with experiment has made it one of the most successful theories in the physical sciences.'

There needs to be some clarity as to what is being claimed and by what method.

It seems to me that you are the only one who is (mistakenly) claiming that any of these papers are giving a derivation of $\alpha$ solely from first principles, or that "agreement with experiment" requires every single quantity in the theory to be derived solely from first principles. Nobody else seems to be misunderstanding them that way.

 

[Adrian59]

It seems to me that you are the only one who is (mistakenly) claiming that any of these papers are giving a derivation of $\alpha$ solely from first principles, or that "agreement with experiment" requires every single quantity in the theory to be derived solely from first principles. Nobody else seems to be misunderstanding them that way.

Despite my best efforts you have repeatedly got the wrong end of the stick. I am in no way mistaken on this issue as I have said repeatedly in this thread. If you doubt this then you need to re-read the whole thread. In fact I have agreed with you consistently through the thread. My stated aim was to explore the origin of such a claim which I have no doubt has been made by OTHERS as I have referenced.

The only issue may be that the terminology is poorly applied, so that when an author says theoretically derived they do not mean completely theoretically derived right from first principles. However, my point is that these authors should be more careful with their wording.

 

[PeterDonis]

In fact I have agreed with you consistently through the thread.

Not when you make claims like this:

this is an indictment on academic physics

There needs to be some clarity as to what is being claimed and by what method.

my point is that these authors should be more careful with their wording.

These are the kinds of claims I am pushing back on, because I do not agree with them, and I don't see where there is any basis for them in any of the references you have given.

 

[PeterDonis]

when an author says theoretically derived they do not mean completely theoretically derived right from first principles

We do not have anything in physics that is "completely theoretically derived right from first principles". Every single physical theory we have and have ever had has quantities in it that have no first principles derivation but must be input from experimental results. Every physicist knows this, and expects anyone reading what they write to know it too, so they don't have to laboriously restate it every time they write a paper.

 

[Adrian59]

We do not have anything in physics that is "completely theoretically derived right from first principles". Every single physical theory we have and have ever had has quantities in it that have no first principles derivation but must be input from experimental results. Every physicist knows this, and expects anyone reading what they write to know it too, so they don't have to laboriously restate it every time they write a paper.

So we do actually agree, as I thought all along. I will have to re-read the Aoyama et al paper to try and find what they are really up to. I can see if it is purely a numerical derivation how it works; but as I have said, and something you agreed on in #7 their terminology does appear rather confusing.

 

[Meir Achuz]

Whether were not its size can be derived (It has not yet been derived, although I have tried.), its size is remarkable. If it were larger or smaller by enough, life as we know it would be impossible.

 

[PeterDonis]

we do actually agree

If we do, then why did you post the things I quoted in post #37? Those are all direct quotes from you, and as I said in that post, I disagree with all of them. Are you now retracting those statements?

 

[PeterDonis]

its size

The size of what? Did you possibly intend this post for a different thread?

 

[PeterDonis]

something you agreed on in #7 their terminology does appear rather confusing

It's confusing with regard to their use of the term "analytic" as opposed to "numerical", as I described in that post. It's not confusing (at least not to me and others in this thread besides you) about whether it claims to be a complete first principles derivation of $\alpha$ with no experimental input. It's not, and it doesn't claim to be.

 

[Meir Achuz]

The size of what? Did you possibly intend this post for a different thread?

The FSC=1/137.

 

[Meir Achuz]

?

 

[PeterDonis]

?

?

 

[Meir Achuz]

The FSC=1/137.

I was trying to ask what was skeptical about

The FSC=1/137.

 

[PeterDonis]

@Meir Achuz I don't have any idea what your actual question is. Nobody is "skeptical" about the value of the fine structure constant.

 

[vanhees71]

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).

 

[Meir Achuz]

@Meir Achuz I don't have any idea what your actual question is. Nobody is "skeptical" about the value of the fine structure constant.

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