Display MoreThe problem is that you have never refuted this statement by referring to the actual
data in the Durr et al 2016 arxiv paper that you cited
as having 6 digit significance for the n-p mass difference
Yes, but I rowed back on that shortly after: and you ignored that and kept on repeating this weird 6 digit mantra!
I am however quite interested in the general topic of who makes better predictions: Mills or QED/QCD. The problem here is QCD which is thoroughly difficult to get highly accurate results from for calculation reasons.
However, we have QED - the "world's most accurately tested ever theory". I'm fascinated by, for example, the weird 2pi values that enter into Mills claimed calculation for the anomalous magnetic moment of the electron.
Stefan - have you looked at this value's derivation (it is a cubic in alpha). I'd like to go through it in detail to understand how the anomalous 2pis get there (the alpha^2 part of the alpha cubed term not divided by 2pi).
My reference for Mills is: http://zhydrogen.com/wp-content/uploads/2013/04/test6.pdf
1 + alpha/2pi is simply stolen from the first order (in alpha) QED expansion coefficient, which is known analytically to be exactly this.
Mills' semiclassical derivation based on the Poynting Power Theorem agrees with QED to this order, which should not surprise us.
Let us work this out. Current experimental value for ae (the above value is supposed to be ae+1):
ae = 0.001 159 652 181 643(764) (from 2011 Wikipedia)
also
(from Control of a Single-Electron Quantum Cyclotron: Measuring the Electron Magnetic Moment" 2011 is given in Wikipedia and consistent.
Also alpha is known as roughly:
α−1 = 137.035999049(90) (from 2010,2011 ref 3,4 in https://arxiv.org/pdf/1705.05800.pdf)
also cf with the 2014 CODATA value: 137.035999139(31) that is consistent and only a tiny bit more accurate.
For convenience we calculate alpha/2pi = 0.00116 140 973 3
Using just the first order QED-only term (ignoring higher order QED, hadronic and electro-weak components):
alpha/2pi = 0.0011641409733
That is 6 sig figures on g/2, but since the first figure is 1 this is really only 5 sig figures. If you take the value of ae (g/2-1) which is the anomalous part of course it is only 3 sig figures accurate.
So our starting point is this first order analytical QED approximation. How much extra accuracy does Mills next two terms give us?
Subtracting first order terms from real value we get:
-0.0000044888 - this is the number that Mills has to use numerology to hit!
This differs by 1 in the 12th decimal place.
Mills reference from 2006
He quotes alpha-1 = 137.03604(11) from
R.C. Weast, CRC Handbook of Chemistry and
Physics, 68th edition (CRC Press, Boca Raton,
FL, 1987–88), pp. F-186–F-187. - this is consistent with the 2010 value but 3 sig figures less accurate
ae = .001 159 652 188(4) from
R.S. Van Dyck Jr., P. Schwinberg, and H.
Dehmelt, Phys. Rev. Lett. 59, 26 (1987). This is slightly inconsistent with the current value, being two SD too high.
Mills, calculates
ae= 0.001 159 652 120
which he compares with (his 2006 experimental value)
ae= 0.001 159 652 188(4)
Excellent agreement
Mills (in 2006) notes that values for the fine structure constant are variable. Indeed his alpha-1 value has error 137.03604(11) or 10^-7.
Propagating this error to (ae -1) we see that the fractional (ae - 1) error is the same as the fractional alpha error which is the same as the fractional alpha-1 error.
that gives a Mills calculated ae error (from his 2006 alpha data) of:
0.001 159 652 120(100)
The calculated value is coincidentally 50X better than would be expected if his formula was precisely correct given his stated error in alpha!
Mills spends some time discussing different values for alpha: but he is cheating! He talks about the remarkable agreement between his value and the correct value, when he cannot have a value of alpha that justifies this level of accuracy! So his 11 significant figures accuracy for ae+1 is the same as 8 significant figures accuracy for alpha.
Let us see what happens if we use more recent values. The key value is that of alpha - which is less precise than ae by 2 sig figs.
Using the current (CODATA 2014) value for alpha of
137. 035 999 139(31)
We have (alpha/2pi) = 0.00116 140 973 241(25)
We get a Mills value for ae of:
+0.001 161 409 732 41(25) (1st order - same as QED 1st order)
-0.000 001 798 496 75 (2nd order)
+0.000 000 041 231 02 (3rd order)
+0.001 159 652 466 68(25) (total)
versus CODATA value for ae of
+0.001 159 652 180 91(26)
Using recent values Mills is wrong by a factor of 500X the error bound
Mills' calculation is precise. If his theory is correct. So something must give.
Mills claims that the CODATA values for alpha may be wrong (by a factor of 1000X larger than the error bound?) because they involve QED. We may have to investigate this is anyone here (Stefan, RB?) feels that is a plausible claim.
Alternatively Mills must now invoke otherwise unspecified errors to explain his lack of correspondance with theory.
How does QED do? A 2017 improved QED 10 loop calculation is https://arxiv.org/pdf/1712.06060.pdf
ae = +0.001 159 652 182 03 (72)
But evaluating these calculations precisely is complex: the values given all depend on alpha - with error to first order the same (fractionla) as alpha. Neverthless this is 1000X better than Mills, unless you conclude that the CODATA value of alpha is wrong by 1000X the stated error.
Stefan: there are quite possibly mistakes in this, though I've put some effort into it, let me know if you find any!
THH
THH, Thanks for interesting thoughts!
The derivation of the g factor is something I looked into and as far as I can tell Mills uses an ok approach in the beginning that one can follow and apply it through the integrals of the fields. I can only see a possible fudge of a factor of 2 in those fields. But as you say, the 2 pi are a weird factor and I think that most of us will fail to understand those. What I can tell is that he views the fields in different frames of references, one in the lab, one at c speed which are really strange and I have not seen any references of this view elsewhere. But if you look into it, it has a structure that is reused many times to yield correct results all over the place. You can do this fudge one time or not and you get a few bits of extra presition of this fudge which is way too small to explain the correctness of the formula. Anyhow I have an idea of what these frames of references are. In the c reference you could consider the solution as a standing wave that basically has a radial component (moves in and out radially) that has a period of r. But when you spin off a photon and move it it will in the lab frame (not in the c-frame) circulate the orbit and hence the period is 2pi r. That is at least my hand waving to try explain this strange fact (there doesn't seam to be room enough to fudge so it looks to have some kind of unknown validity)
I agree that QED is exact in it's results, but here I miss the correct statistical approach to fix more digits. It looks like the theory always follows the new experimental accuracy. Also as far as I understand the expansions need to decide which terms to add and not to add at least that is the critique that Mills claim e.g. fudging.
Anyhow I think you did a little mistake in the claim that Mills once was over exact in his value. You had alpha-1 = 137.03604(11) that's +/- 2 (2 sd) on the eights figure and it is quite likely
to get the eight's figure correctly by chance. which is what he basically has.
The accuracy is pretty high in Mills derivation and I am open for both QED and Mills as valid approaches to derive the result and this indicates that perhaps QED is based on a more exact formulation in certain setups but uses a way more complicated model compared to Mills theory. Perhaps W will one day see how his further work can shed light over what this correspondence really are because he has ideas of how to enrich Mills theory to improve accuracy. I certainly view QED as a non fudge formula.