All the lepton masses from G, pi, e

[Hans de Vries]

It was noted that using the \beta_1= 0.855599677 radius (relative to the Compton
radius) as the cut-off radius for the muon we get as the self energy of the
remaining Electric field the value of the electron mass to within 0.038%.

However one would expect that an abrupt cut-off at a certain radius can only
give an approximation since since we do not expect something like that to happen
in Nature. Therefor we apply a more realistic gradual cut off on the 1/r potential
by subtracting a Yukawa potential:

<br /> V \ =\ \frac{q}{4\pi\epsilon r}\ <br /> \left(\frac{1}{r}-\frac{e^{-r/r_o}}{r}\right)<br />

We will see that this leads to exactly the same result as the abrupt cut-off.
The Electric field follows from differentiation:

<br /> E \ =\ \frac{dV}{dr}\ =\ \frac{q}{4\pi\epsilon}\ \left( <br /> -\frac{1}{r^2}+\frac{1}{r^2}e^{-r/r_o} +\frac{1}{r_or}e^{-r/r_o} <br /> \right)\ <br />

The Energy Density is given by:

<br /> {\cal E}_{(r)} \ =\ \epsilon E^2\ =\ \frac{q^2}{(4\pi)^2\epsilon}\ <br /> \left( -\frac{1}{r^2}+\frac{1}{r^2}e^{-r/r_o} <br /> +\frac{1}{r_or}e^{-r/r_o} \right)^2\ <br />

We integrate over r.

<br /> {\cal E} \ =\ \frac{q^2}{(4\pi)^2\epsilon}\ \int_0^\infty 4\pi r^2 <br /> \left( -\frac{1}{r^2}+\frac{1}{r^2}e^{-r/r_o} <br /> +\frac{1}{r_or}e^{-r/r_o} \right)^2 dr<br />

<br /> {\cal E} \ =\ \frac{q^2}{4\pi\epsilon}\ \int_0^\infty \left( <br /> -\frac{1}{r}+\frac{1}{r}e^{-r/r_o} +\frac{1}{r_o}e^{-r/r_o} <br /> \right)^2 dr<br />

To get the result:

<br /> {\cal E} \ =\ -\frac{q^2}{4\pi\epsilon}\ \left| \frac{1}{2r_o}\ <br /> e^{-2r/r_o}\ +\ \frac{1}{r}\left(1-e^{-r/r_o} \right)^2\ <br /> \right|_0^\infty<br />

and with,

<br /> e^x \ =\ 1 + x + \frac{1}{2}x^2 + ... <br />

Using only the linear term for the limit r goes to 0 we obtain the result:

<br /> {\cal E} \ =\ \frac{q^2}{8\pi\epsilon r_o}\ <br />

Which is equal to that of the abrupt cut-off.Regards, Hans

 

[Hans de Vries]

Alpha series formula holds!

This thread started with a series to calculate the value of alpha from a series
with an arbitrary number of digits:

\alpha = 0.00729735256865385342269

From Luboš Motl we now hear of a new measurement of g/2 with a
sixfold improvement of precission.

http://motls.blogspot.com/
http://motls.blogspot.com/2006/05/new-values-of-g-and-fine-structure.html#comments

Gerry Gabrielse, an experimenter from Harvard University, and his collaborators are going to announce new, more accurate values of the fundamental constants. Using their single-electron quantum cyclotron, they can see that the new magnetic moment of the electron is

* g/2 = 1.001 159 652 185 85 (76).

As you can see, there are 13 significant figures or so - the value is six times more accurate than ever before. Using the cyclotron result for "g" above plus QED theorists from other universities, they can also deduce the value of the fine-structure constant...

We can take our value given above and calculate g/2 with the use of the
latest results of the famous work of Kinos hita and Nio. hep-ph/0512330 v2
(of 7 Mar 2006)

http://arxiv.org/abs/hep-ph/0512330

The result one gets is:g/2 = 1.001 159 652 186 038 ____ (Calculated value)
g/2 = 1.001 159 652 185 85 (76). (Measured by Gabrielse)


So it's still within the 13 digits of experimental precision, notwithstanding
a sixfold increment in measurement precision, and within a sigma of 0.25
(This is the full series, not the truncated one)Regards, Hans

 

[physmike]

It was noted that using the \beta_1= 0.855599677 radius (relative to the Compton
radius) as the cut-off radius for the muon we get as the self energy of the
remaining Electric field the value of the electron mass to within 0.038%.

Interesting minor sidenote, 100 - 62 = 38

62 ~ harmonic to inverse golden ratio, φ^-1

 

[physmike]

This thread started with a series to calculate the value of alpha from a series
with an arbitrary number of digits:

\alpha = 0.00729735256865385342269

From Luboš Motl we now hear of a new measurement of g/2 with a
sixfold improvement of precission.

...

So it's still within the 13 digits of experimental precision, notwithstanding
a sixfold increment in measurement precision, and within a sigma of 0.25%
(This is the full series, not the truncated one)


Regards, Hans

Thanks for posting this new information Hans,

good to compare with your "predictive" version.

 

[arivero]

http://arxiv.org/abs/hep-ph/0512330

Amusingly I was using input from 0507249 where (page 7) the A(4) and A(6) muon corrections were still listed as "insignificant at present"; it is just fortunate I also mentioned 0512330

 

[Hans de Vries]

Oopss,

Luboš is a very fast typer, but sometimes a bit to fast, The preprint of
Gabrielse and Kinoshιto has the correct new alpha and g/2 values:

http://hussle.harvard.edu/~gabrielse/gabrielse/papers/2006/NewFineStructureConstant.pdf

We find the correct value for g/2: 1.001 159 652 180 85 (76).
This does not bode well for our series version so we're left with this one:

Charge Renormalization Factor: ____ \Gamma\ =\ 1+\alpha+\frac{\alpha^2}{2\pi}

This is a correction factor between the theoretical value and the measured
expirimental value to correct the value of the charge for vacuum polarization.
This relates the theoretical and experimental alpha as follows:


\alpha_{th} = \Gamma^2\ \alpha_{exp}

We use the square since alpha is proportional to the square of the of
the electric charge value . For a theoretical alpha value of:

e^{-\pi^2/2}

This produces a renormalized charge giving an experimenal alpha of:

\alpha_{exp}[/tex] = 1 /137.035 999 528 369 196 352 446 647 041<br /> If we now calculate g/2 with the parameters of Kinoshιto then we find:<br /> <br /> 1.001 159 652 180 85____(+/-76) Measured by Gabrielse<br /> 1.001 159 652 180 68____QED version <br /> 1.001 159 652 180 71____ElectroWeak version <br /> 1.001 159 652 182 38____Full SM version including hadronic contributions<br /> <br /> The last three ones are calculated from our result. So this seems to exclude <br /> the version which includes the hadronic contributions.Regards, Hans

 

[physmike]

Some "numerical analysis":

... numerical analysis of part of the calculation for the
new value of the fine structure constant.


"The most extensive calculation made so far in QED is for the magnetic moment of the electron. Ignoring parts that depend on particle masses the result (derived in successive orders from 1, 1, 7, 72, 891 diagrams) is

2 ( 1 + α/(2 Pi) + (3/4 Zeta[3] - Pi^2/2 Log[2] + Pi^2/12 +
197/144) (α/Pi)^2 + (83/72 Pi^2 Zeta[3] - 215/24 Zeta[5] -
239/2160 Pi^4 + 139/18 Zeta[3] + 25/18 (24 PolyLog[4, 1/2] +
Log[2]^4 - Pi^2Log[2]^2) - 298/9 Pi^2 Log[2] + 17101/810 Pi^2 +
28259/5184) (α/Pi)^3 - 1.4 (α/Pi)^4+ …)," :-)

From Stephen Wolfram: A New Kind of Science


Maybe this is mere coincidence, most of these numbers are harmonics
in Conner's system. Notice the prevalence of 144, his fundamental tone.

144, 144/2, 144^1/2, 2(144^1/2), 15 x 144 = 2160 ...

and 5184 = Re 5 = 162 x 32, 162 is the phi tone-number.

The 891 feynman diagrams are interesting. 891 = 27 x 33, 27 + 33 = 60

The Quadrispiral has 4 spirals of 15 loops each, for a total of 60 loops,
3 x 4 x 5 = 60 , each with a particular harmonic value.
27 is the time harmonic, 33 is the inward time harmonic.

The first loop of the inward time energy spiral, the minimal compression of density, has something of an inverse relation with the first loop of the energy spiral, minimal loss of density for the energy spiral, first loop harmonic value of 144^2.

In our formulation of alpha we used the time energy spiral,
144 x phi ~ 233, harmonic value of the first loop representing minimal compression of density.
The inward time energy spiral was not included
because it was not really understood what the "inward time energy"
interaction was, according to Conner's monograph of "laboratory notes";
and our result matched the Quantum Hall Effect experiment. While Gabrielse Research Group at Harvard have a value closer to the previous magnetic moment anomaly result. If this is really 10 times more accurate, then our alpha calculation is to be corrected by the means briefly indicated here.

Refs. post #230, #227, #220

 

[physmike]

From the April APS Meeting:

Derivation of lepton masses from the chaotic regime of the linear sigma model
Ervin Goldfain

"Our work suggests that the lepton mass spectrum may be recovered from the chaotic dynamics of the simplest prototype for classical gauge boson-fermion interaction, the linear sigma model."

Derivation of the fine structure constant using a fractional dynamics approach
Ervin Goldfain

"It is shown that the fine structure constant can be recovered from the fractional evolution equation of the density matrix under standard normalization conditions."

flux.aps.org/meetings/YR03/APR03/baps/abs/S540.html#SC14.008

 

[arivero]

"Our work suggests that the lepton mass spectrum may be recovered from the chaotic dynamics of the simplest prototype for classical gauge boson-fermion interaction, the linear sigma model."

A curious case of crossing. I expect chaotic dynamics to be used someday to explay Bode's law.

 

[physmike]

A curious case of crossing. I expect chaotic dynamics to be used someday to explay Bode's law.

Just wondering...

"An alternative theoretical approach to describe planetary systems through a Schrodinger-type diffusion equation"
Authors: Neto, M. de Oliveira; Maia, L. A.; Carneiro, S.

References...
M. S. El Naschie, E. Rossler, and I. Prigogine, Quantum Mechanics, Diffusion and Chaotic Fractals (Pergamon Press, Oxford), 1995.
G/A [] B. G. Sidharth, Chaos, Solitons and Fractals, 15 (2003) 25.

Leads to some Golden Ratio speculation.

www.citebase.org/cgi-bin/citations?id=oai:arXiv.org:astro-ph/0205379[/URL]

 

[Ganzfeld]

The 6 pi^5 ratio for the proton-electron mass ratio was also independently published in an item by me in Nature ( July 7th 1983 I think the date was )...which the editor rather scathingly titled ' The Temptations Of Numerology '. The same stuff was also published in 1984 in 'Speculations In Science And Technology'

In that article, I also included a number of other fascinating 'coincidences'

For example...the combined mass of the particles of the meson octet is 3.14006 times the proton mass. Pretty close to pi. Amazingly, the combined mass of the particles in the baryon octet is around 9.8 times the proton mass. Close to pi squared.

Perhaps the most striking 'coincidence' is...alpha is very close to 4 pi^3 + pi ^ 2 + pi. This value is 137.03630. I think this result may have actually been published earlier by Armand Wyler. But it ties in with a truly REMARKABLE series of coincidences that were the core of my item in Nature :

Where I expressed the proton mass as pi ^ 6 , the neutral pion mass turns out to be almost exactly pi ^ 4 + pi ^ 3 + pi ^ 2. A very close fit. And the muon mass turns out to be quite close to pi ^ 4 + pi ^ 2 + 1. Amazingly...these two and the alpha formula all fit together into a single mathematical operation !

The following is the mathematical ‘operation’ table for positive times negative integers. For the range 0 to 2 for positive numbers and 0 to –2 for negative numbers. ( I hope all this lines up...it did when I typed it )


A B C
* 0 -1 -2

X 0 0 0 0
Y 1 0 -1 -2
Z 2 0 -2 -4


Now take the values in the above table and use them as powers to which pi is raised ( i.e pi ^ -1 is pi to the power of minus 1)

A B C
* 0 -1 -2

X 0 1 1 1
Y 1 1 Pi ^-1 Pi ^ -2
Z 2 1 Pi ^-2 Pi ^ -4


The fine structure constant 137.03604 happens to be very close to (4 * pi ^ 3 + pi ^ 2 + pi , or 137.03630
This happens to be pi ^ 3 times the sum of the values in colums A and B ( or rows X and Y ) in the table.

If we take the proton mass to be pi ^ 2, then the sum of the values in column B ( or row Y ) gives you the neutral pion mass to extremely close accuracy, and the sum of the values in column C ( or row Z ) gives the
Muon mass very closely. The overall accuracy here is remarkable !

Consider this……that a simple operation on a multiplication operator table gives a simple matrix that provides a remarkably accurate relationship between fine structure constant, muon mass, neutral pion mass,
and proton mass. ( The proton mass does not have to be pi ^ 2 but can clearly be scaled up or down as any power of pi along with the matrix itself being scaled up or down by the same power….so all the relationships can be defined as between integral powers of pi .)

Fine structure constant = 137.03600 Above table gives 137.036303

Ratio of proton to neutral pion mass = 6.95230 Above table gives 6.95223189

Ratio of proton to muon mass = 8.87678 Above table gives 8.87883898

If this is a coincidence, then it is a remarkable one ! No other piece of 'numerology' ( except perhaps Bode's law ) provides such a close relationship between so many physical constants.

My original Nature article did not include the fact that the whole thing can be linked to the operation table for positive times negative integers...which I only realized was the case some years later.

 

[Ganzfeld]

Nope...it didn't line up...but it shouldn't be too hard to get the gist. It's a simple mathematical operation table.

 

[arivero]

The 6 pi^5 ratio for the proton-electron mass ratio was also independently published in an item by me in Nature ( July 7th 1983 I think the date was )...which the editor rather scathingly titled ' The Temptations Of Numerology '.

Amazing. So you are the other popular source of lore for this result. Lubos tells somewhere in his blog how he found it as a youngster playing with the handheld calculator. But you both are predated by the Physical Review article.

By the way, the subheading of the Nature article is also frightening:
"SUMMARY: Too much innocent energy is being spent on the search for numerical coincidences with physical quantities. Would that this Pythagorean energy were spent more profitably."

I had missed this article because of our local mode of subscription to Nature. Thanks for pointing it out.

If this is a coincidence, then it is a remarkable one ! No other piece of 'numerology' ( except perhaps Bode's law ) provides such a close relationship between so many physical constants.

Well, I do not know if you have got time to go along the whole thread (it is too long) but we are also proud of our links between (g-2), (g_mu-g_e), m_e, m_mu, m_Z and m_W

As far as I know, the widely referenced numerological results of Wyler are just decomposition on factors.

 

[arivero]

The Nature thread run at least along five articles, according databases:

Nature 313, 524-524 (14 Feb 1985) Correspondence
NATURE 308 (5962): 776-776 1984
Nature 306, 530-530 (08 Dec 1983) Correspondence
Nature 305, 672-672 (20 Oct 1983) Scientific Correspondence
Nature 304, 11-11 (07 Jul 1983) News and Views

 

[Ganzfeld]

Well...the editor of Nature ( John Maddox, at that time ) originally described my results as 'spectacular' when I first sent them in July 1981. He then sat on them for 2 whole years before finally publishing in July 1983.

I was at a bit of a loss to understand what his mention that I was a 'factory worker' ( I am currently a systems analyst ) had to do with it all. I think he originally thought I was a scientist...and sat on my stuff for 2 years when he realized ( probably with a sense of horror ) that the stuff had been worked out by a mere 'factory worker' rather than Nature's usual subscribers. LOL...there goes my Nobel Prize prospects !

In any case, a slightly more detailed article appeared ( without sceptical comment ) in Speculations In And Technology the following year. I forget which edition ( it's all crammed away in a draw somewhere ).

The 'matrix' that I provide above IS spectacular ( all the more so when it all lines up properly and you can see how simple it is ) I'd agree with the scathing comments on most 'numerology', but here we have a very simple table based on very simple mathematical operations. It contains just 9 values...yet from those 9 we can derive an overall very accurate relationship between alpha, muon mass, neutral pion mass, and proton mass. And I think I am correct in saying that the pion actually carries the strong nuclear force, which is alpha ^ -1 times stronger than the electromagnetic force...so there OUGHT to be a relationship between alpha and pion.

Of course, if one goes looking for numerical coincidences one is bound to find them. It's a bit like Nostradamus prophecies...accurate after the event. But even now I look at that 'matrix' and I have to say I find myself thinking that it's highly improbable that one would get a relationship between 4 physical constants...to that degree of accuracy...and in such a simple little table...purely by chance.

What's more...it's fascinating that the table expands on the original alpha ^ -1 = 4 pi^3 + pi^2 + pi observation, which I learned had been published before me though I was unaware at the time. It's not often that a piece of numerology actually independantly expands on a prior observation like this and includes more constants.

Unfortunately I don't have anything like the nuclear physics knowledge to say if it all means anything. But if anyone else wants to have a go...feel free.

 

[Ganzfeld]

Interestingly...the original Nature article does not contain the operation table as presented above. It contains the 9 values in a different form. And it was not till several years later that it dawned on me that the whole thing could be derived simply from the operation table for positive times negative integers...used as powers to which pi is raised.

I think the original article would have appeared far more impressive had I been able to point out that relationship at the time. I've considered writing a new article to point this out...but never really got round to it. And in any case...if it ever were found to be of significance I could always claim that the relationship was implicit in my original article anyway.

 

[physmike]

The Fine Structure Constant

Ok Ganzfeld! Here's some of that "Pythagorean energy".

108^2 + 144^2 = 180^2

108^2 x 144^2 = 2.41864704 x 10^8

Now, if we consider a nearly Euclidean space where pi is reduced by,

pi - 2.41864704 x 10^-6 = p then:

4p^3 + p^2 + p = 137.035999707... ~ alpha^-1

& this compares to 137.035999710 Gabrielse Research Group

"New Determination of the Fine Structure Constant. from the Electron g-Value and QED"
hussle.harvard.edu/~gabrielse/gabrielse/ papers/2006/NewFineStructureConstant.pdf

The Pythagorean triangle above is a remarkable convergence of decagon angular measure, semicircle, 144 light harmonic, fibonacci number, and fundamental tone-number. 108 as a harmonic of the sixth root of phi. And 180, one of the 4 basic tone-numbers generating the Quadrispiral mentioned in post #220 and #257.

Thanks for explaining your pi matrix, and hope to see more detail.
And thanks to arivero for the Nature quote, which initiated this calculation.

 

[Hans de Vries]

108^2 x 144^2 = 2.41864704 x 10^8
Now, if we consider a nearly Euclidean space where pi is reduced by,

pi - 2.41864704 x 10^-6 = p then:
4p^3 + p^2 + p = 137.035999707... ~ alpha^-1

Dear Physmike:

Please note that you introduce an unexplained factor 10¹⁴ here which is:

38D7EA4C68000 in hexadecimal notation or
34327724461500000 in octal notation or
11100011010111111010100100110001101000000000000000 in binary notation

The numbers you use only look alike in your calculator display because
it pure accidentally uses the decimal notation. See chapter 4 in your
Stephan Wolfram book.

Physmike, The hardest thing after finding a valid numerical coincidence
is to put it in the context of real physics. Please don't throw in all these
things like pyramids, tonal systems, and other stuff. It's much better to
refrain from all this wired stuff here if there isn't at least some kind of link
to accepted mainstream physics since this is after all a physics website.

You don't need a Physicist's education to find numerical coincidences,
However if you want to interpret the coincidence then you do so.
You'll be more respected if you present a numerical coincidence without
all the numerological stuff.

Just to show you. Look how Stephan Wolfram (famous inventor of
Mathematica) gets lots of cynical reviews from physicist on his book
"A new kind of physics" and keep in mind that his book is ten times
better as the other ones you're quoting from...

https://www.amazon.com/gp/product/1579550088/?tag=pfamazon01-20Regards, Hans.

 

[Ganzfeld]

Hans

I don't disagree that if one starts with the maths, one has to end up with some physics to support it. But surely half the basis of particle physics is in itself mathematical...all the different SU models are derived from group theory and so on. The wonder is that the physical world just happens to follow abstract mathematical concepts.

I dislike the term 'numerology'. It smacks of Uri Geller proclaiming that the number 11 turns up all over the place. Well of course it does ! But that's a far cry from finding a single simple symmetry, as I did, that accurately relates the values of 4 physical constants...and I really don't think it belongs in the same boat as Uri Geller.

When I published the article in Nature, it was not with a view to announcing ' here's some breakthough in physics '. The essence is that of presenting something which appears to be way beyond what one would expect purely by chance. The ONLY thing that makes such numerical relationships noteworthy ( something Geller has yet to grasp ) is that element of probability. Is this something that one would expect to arise purely by bog standard chance alone ?

If the answer is 'yes'...then it's just another in the long line of odd coincidences that can safely be forgotten.

If the answer is 'no'...then we may not have gone from maths to physics just yet but we most certainly have crossed the threshold to where the results cry out for an explanation and more serious attention.

 

[arivero]

I'd add another property: they are not numeric but algebraic coincidences the ones we are finding in this thread. I mean algebraic to imply two consequences:

-they do not depend on the number base (ten, binary, octal, etc)
-they do not depend on a measuring unit (Meters, GeV, kilograms, etc)

Numerology in real world fails to meet these constraints. And then, some occassional visitors of this thread forget these properties. On one hand I would like to ignore them, on the other it seems appropiate to defend the thread.

Numerology in physics has become a more restricted meaning in the sense that it meets the two above requeriments and sometimes it becomes very akin to be "topolology" or some rare branch of math. Still it is modernly rejected since are the measured low energy quantities are believed to evolve under renormalisation group. But some rare coincidences, as Koide's (see the thread), or the one of the mass ot the top against fermi constant, seem to point that RG evolution is not the whole history and that it worths to keep an eye to low energy relationships.

Another thing that most numerologist ignore is the experimental error; I am proud that in this thread we have kept some care with the sigmas.

 

[arivero]

Let me add, Ganzfeld, that I must apologize by not noticing your equations during the two years we have run this thread. In my discharge let me explain that I am 200km away from the nearest deposit of Nature, and our electronic access does not include access to the historic repository.

 

[physmike]

Physmike, The hardest thing after finding a valid numerical coincidence
is to put it in the context of real physics. Please don't throw in all these
things like pyramids, tonal systems, and other stuff. It's much better to
refrain from all this wired stuff here if there isn't at least some kind of link
to accepted mainstream physics since this is after all a physics website.

Hans, Conner's work is partially based on "accepted mainstream physics", specifically including the work of Kepler, Helmoltz, Einstein, Schrodinger, and Bohm. He also uses admittedly "off the beaten track" sources of Pythagoras, Tesla, Russell, Cathie, and others. His books could use some good editing work too. I reference him because I think he is "on to something" in bringing together these different sources toward an explanation of the physics involved here. Though his phi-nesting spirals, fibonacci mirrors, and a few other ideas are original, unique, and brilliant. And I agree, it is hard to put a "valid numerical coincidence" in the "context of real physics"; the new concepts introduced here, relate to the understanding of this.
Judgments should respect the possiblity of physical reality, and regard the width of our educational conditioning.

In constructing his Quadrispiral, he used a transposition technique from electrical engineering which gives the result that, in particular, some harmonic values are increased by powers of ten, most notably, 10^4.
The harmonics of the Pythagorean Table produce a factor of 10 as well.
And finally, the harmonic system itself, addresses this too, as noted before.


Some general references to show partially related physics:

On the Sensations of Tone , Hermann Helmholtz

Fundamentals of Musical Acoustics, Arthur H. Benade

Theoretical Acoustics, Philip M. Morse, K. Uno Ingard

Music, Physics and Engineering, Harry F. Olson

On the mathematical structure of Tonal Harmony, Segre, Gavriel
"Some little step forward is made in the analysis of the mathematical structure of Tonal Harmony, a task begun by Galilei, Euler and the Lagrange" http://arxiv.org/abs/math/0402204

The Geometry of Musical Chords
Dmitri Tymoczko, Princeton University
"Musical chords have a non-Euclidean geometry..."
http://music.princeton.edu/~dmitri/voiceleading.pdf

G. Mazzola. The Topos of Music. Geometric Logic of Concepts...

The Geometry of White’s Dimensional-Shift Operator
Douglass A. White, Observer Physics www.dpedtech.com/Geo.pdf


Physics and pyramids ... some history,

Mystery of 'chirping' pyramid decoded, Philip Ball
http://www.nature.com/physics/highlights/7020-2.html

http://www.physicstoday.org/pt/vol-57/iss-9/p29b.shtml[/URL]

l1. L. W. Alvarez et al., Science 167, 832 (1970); see also L. Alvarez, Adventures in Exp. Phys. 1, 157 (1972).
2. Tesla Foundation, Unfolding Pyramids' Secrets Using Modern Physics, film, narrated by L. W. Alvarez and B. C. Maglich, directed by Victoria Vesna (1988).

 

[arivero]

I expect at least you will notice you are mad for common standards, physmike. It seems we can not help you here (I *really* wished to be able to help you), you can not help us, and our interaction does not help physics as a whole pursuit.

Worst, your random launching of unvalorated data-links murks any geometrical or algebraic truth which could be extracted from the proportion law you are interested in (the infamous a/b = b / (a+b) )

 

[physmike]

Sorry arivero, Perhaps you made an unconsidered connection with selected, non-random, references posted for general review by Hans (concerned about standards of physics, and should not have been needed), with a proportion that I did not talk about?

Thanks for really wanting to help, ...you did. (Guess it does not show)

All the best, physmike

 

[CarlB]

-they do not depend on a measuring unit (Meters, GeV, kilograms, etc)

Amusingly enough, my version of the Koide coincidence, got accused of depending on units by a professional physicist:

The paper on Lepton masses is interesting, but there is not really any physics, just a coincidence of numbers. Even that may not be as remarkable as it seems, since you predict 3 charged Lepton masses using 4 variables: \mu, \nu , \delta and the units of mass.
http://groups.yahoo.com/group/QM_from_GR/message/1056

I consider this another piece of evidence in favor of the thesis that nowadays, physicists will faithfully attempt to read out of the ordinary papers only to the extent that they trust the author. Instead of reading the paper, they will instead scan it for the first "error" they can find, and then move on to more interesting things.

Carl

 

[Ganzfeld]

Insofar as the essence of any piece of numerology is along the lines of 'surely this is beyond chance ?', perhaps the most useful thing would be to be able to determine just exactly what the probability is for any result. To be able to say ' the odds against chance are 1 in a million ' would at least be impressive.

Of course, it's not exactly clear just exactly how one would do so. It would require some sort of 'a priori' statement of just exactly what sort of results one was looking for...and that's not something that exists after the event. But I do think some sort of 'contrivance factor' could be incorporated, whereby all results with a similar degree of simplicity or complexity get weighed accordingly.

Surely the best response to any numerology, rather than saying there's no physics so it can't be physics, is to show that any result is just exactly what one would expect by chance.

 

[arivero]

I consider this another piece of evidence in favor of the thesis that nowadays, physicists will faithfully attempt to read out of the ordinary papers only to the extent that they trust the author. Instead of reading the paper, they will instead scan it for the first "error" they can find, and then move on to more interesting things.

Yes, and this is the goal of the introductory section of the paper: to get the trusting of the reader/referee. Old papers were a lot shorter, and I think it was because they did not rely on this need of introduction.

 

[arivero]

2) An history of numerological approaches to the fine structure constant is contained in
H. Kragh, "Magic Number: A Partial History of Fine-Structure Constant", Arch Hist Exact Sci 57 (2003) p 395-431

A related paper by the same author is "The fine structure constant before quantum mechanics". There, a subsection "Dimensionless speculations" tells about early numerology and conjectures.

 

[Kea]

Alpha

Now might be a good time to resurrect the de Vries truncated formula

e^{2} \textrm{exp}(\frac{\pi^{2}}{4}) = e + e^{3} + \frac{e^{5}}{2 \pi}

where e is electron charge.

 

[Hans de Vries]

Now might be a good time to resurrect the de Vries truncated formula

e^{2} \textrm{exp}(\frac{\pi^{2}}{4}) = e + e^{3} + \frac{e^{5}}{2 \pi}

where e is electron charge.

Hi, Kea.

If anything it would be the truncated version to pursue now. It leads to
a value of:

1/137.035 999 528 369 Interesting now would be a new direct measurement of alpha maybe from
a photon recoil experiment. The newly indirect measurement derived from
the new record setting precision electron's magnetic anomaly experiment
is:

1/137.035999710 (96)

http://hussle.harvard.edu/~gabrielse/gabrielse/papers/2006/NewFineStructureConstant.pdf

So we are off with about 1.9 sigma here. While the paper mentions many
direct measurements of alpha, it remains unclear to me why they ignore
the NIST/CODATA value which is currently the best direct measurement:

1/137.035999110(460)

http://physics.nist.gov/cgi-bin/cuu/Value?alphinv|search_for=fine+structure

But OK, We get our value for alpha by starting of with an nice analytical
'bare' vale for e (= sqrt(alpha)) of:

e^{-\pi^2/4}

and then use a Charge Renormalization Factor given by:

\Gamma\ =\ 1+\alpha+\frac{\alpha^2}{2\pi}

To get the renormalized alpha of 1/137.035 999 528 369

The last term brings the result from:
1/137.038339943 to:
1/137.035999528
So it improves the result by a factor 12800 ...Pfff, I've spend so much time to explain the truncated series from
geometrical physics... I've countless numbers of scratch pages with
potentials, propagators, Green functions, Fourier transforms.

Well it's at least useful for something I guess.Regards, Hans.

 

[Hans de Vries]

The electrons magnetic anomaly, being the result of a (very) complex
series, is:

0.00115965218085 (76).

Now could there be a direct analytical calculation of this series?
An interesting starting point seems to be:

\frac{e^{-e-e^{-1}}}{(2\pi)^2}\ =\ 0.0011570109\ \approx \ \frac{\alpha}{2\pi} + ...

That's in the right range and it becomes more interesting if we look at the
error in the ratio between the two:

g/g' = 1.0022828 = 1.00114076²

So in the error we again see our target value, which may be a sign that
our expression could be a simplified version of a more complex analytical
formula.Regards, Hans.

 

[Hans de Vries]

Mass relations between the vector bosons and the leptons.If we interpret the frequency m\ c^2/h of the leptons as a precession,
then there must also be another frequency: The frequency at which it spins.

(Simply like: Spinning top frequency versus the frequency at which it
precesses. The harder we try to tilt the top, the faster it precesses.)

The first which comes to mind is the magnetic anomaly. This is the ratio
between the orbit frequency of a (lepton) in a magnetic field and the
frequency at which its spin direction precesses.

Now, we already found the following on this thread:


0.00115965__ = electron magnetic anomaly
0.00115869__ = muon / Z mass ratio


It gives a lepton/vector_boson mass ratio. Now, can a charge-less
particle precess in a EM field? What about light by light scattering via
charged virtual particles (vacuum polarization).

The following we also found earlier on this thread:


0.0000063522 = muon g vacuum polarization terms.
0.0000063537 = electron / W mass ratio.


This gives another lepton/vector_boson mass ratio. The numbers fit very
well. Remarkable is that the properties of the muon and Z explain the
electron/W mass ratio, while in the first case it was the other way around:
The properties of the electron and W explained the muon/Z mass ratio...And then the tau. where does this leave the tau-lepton? If this is also a
precession/spin ratio, then it would need a significantly stronger coupling
because its mass (=frequency) is 16.8183 times higher as that of the muon.

Well, we naively use the (photon) diagrams of the magnetic anomaly
in first order to try to see how large this coupling should be. (even though
the only particles which have similar propagators are the gluons)

\mbox{Anomaly}\ \ \ =\ \ \frac{\alpha_?}{2\pi} + ...

This leads us to a new numerical coincident:


0.1224 ________ = required coupling constant.
0.1216 (0.0017) = the coupling constant \alpha_s(m_Z)


So, the coupling constant required to give the tau / Z mass ratio,
assuming massles propagators, leads us to the strong coupling
constant at mZ energies...

Now, "who ordered" the s of strong here? Well, at least the use of
massless (gluon) propagator diagrams fits ... :^)Regards, Hans.PS: See also http://arxiv.org/abs/hep-ph/0503104
(and http://arxiv.org/abs/hep-ph/0604035 for \alpha_s(m_Z))

 

[Severian]

I have sort of missed this entire thread, and I am not going to read through 11 pages, so I apologise if this point has already been made.

<br /> \alpha^{-1/2}+ (1+{\alpha \over 2 \pi }) \alpha^{1/2}=e^{\pi^2 \over 4}<br />

If this equation is fundamental (and I agree that it is rather a huge coincidence if it is not) then the obvious question is: Why should there be a relation for the asymptotic low energy value of alpha? Why not alpha(Mz) or alpha(MGUT)? The latter would seem to make more sense to me, but clearly it won't work with the formula.

In other words, usually we think of the most fundamental physics existing at high energies, but this is a low energy equation.

Incidentally, I have seen a similar but slightly different form:

\alpha = \Gamma^2 e^{-\pi^2/2}

with
\Gamma = 1+\frac{\alpha}{(2\pi)^0} \left(1+\frac{\alpha}{(2\pi)^1} \left(1+\frac{\alpha}{(2\pi)^2} \left(1+ ... \right. \right. \right.

This is not quite as nice, but also gives the right low energy alpha. Both of these equations can't be fundamental(?), so there has to be at least one coincidence.

 

[Hans de Vries]

If this equation is fundamental (and I agree that it is rather a huge coincidence if it is not) then the obvious question is: Why should there be a relation for the asymptotic low energy value of alpha? Why not alpha(Mz) or alpha(MGUT)? The latter would seem to make more sense to me, but clearly it won't work with the formula.

In other words, usually we think of the most fundamental physics existing at high energies, but this is a low energy equation.

Hi, Severian.

I would expect such a formula to be much more complicated and
depending on all what's out there in the vacuum, including the things
which are still hiding out there.

Looking at, say, the calculations of the magnetic anomaly of the muon,
then it's always the low energy limit of alpha which is used, and the
vacuum polarization comes in from explicit terms defined in mass relations
like A2(mμ/me), A4(mτ/me). After that you get all the hadronic terms
and electroweak terms.

So a running alpha (which includes all these vacuum polarization terms)
is dependent on a complex function of all kinds of SM parameters like
the lepton mass ratios for the QED only part and getting much worse
for the hadronic contributions.

The point is that, on this thread we're looking for simple numerical
coincidences which just might have a physical origin. So, the shorter
the expression is, the better. Complex things generally lead to complex
expressions and we try to avoid them.

Incidentally, I have seen a similar but slightly different form:

\alpha = \Gamma^2 e^{-\pi^2/2}

with
\Gamma = 1+\frac{\alpha}{(2\pi)^0} \left(1+\frac{\alpha}{(2\pi)^1} \left(1+\frac{\alpha}{(2\pi)^2} \left(1+ ... \right. \right. \right.

This is not quite as nice, but also gives the right low energy alpha. Both of these equations can't be fundamental(?), so there has to be at least one coincidence.

Both formula's are from here The second one was an attempt to
extend the first one into a series. The first expression is just the first 3
terms of the series.Regards, Hans

 

[Severian]

Looking at, say, the calculations of the magnetic anomaly of the muon,
then it's always the low energy limit of alpha which is used, and the
vacuum polarization comes in from explicit terms defined in mass relations
like A2(mμ/me), A4(mτ/me). After that you get all the hadronic terms
and electroweak terms.

But there is a good reason that the alpha used in the magnetic moment of the muon (or the electron for that matter) is 1/137. It is a low energy observable. So it is really not the same thing as an equation that is supposed to derive alpha itself.

So a running alpha (which includes all these vacuum polarization terms)
is dependent on a complex function of all kinds of SM parameters like
the lepton mass ratios for the QED only part and getting much worse
for the hadronic contributions.

My theoretical prejudice would be that the number at high energies would have a simple form, while that at low energies was simply obtained by running to the low scale. The low energy value would then be the one which contains all the messy corrections. So I would be happier if you could you could reproduce an alpha at the GUT scale (or perhaps the Planck scale) which would provide the low energy value after running.

The point is that, on this thread we're looking for simple numerical
coincidences which just might have a physical origin. So, the shorter
the expression is, the better. Complex things generally lead to complex
expressions and we try to avoid them.

Fair enough. The coincidence I find most intriguing is the match between the Higgs vev and the top mass. Or why, to current experimental accuracy, is the top Yukawa exactly 1?

I see now that the two equations I quoted are the same (with one truncated). Silly me! Though I saw it on a different site (I will have a look for the link).

 

[arivero]

My theoretical prejudice would be that the number at high energies would have a simple form, while that at low energies was simply obtained by running to the low scale.

...


Fair enough. The coincidence I find most intriguing is the match between the Higgs vev and the top mass. Or why, to current experimental accuracy, is the top Yukawa exactly 1?

Yes, in a way this thread defies the current prjudice by suggesting that it is possible to find relationships in the asymptotic low energy limit. This could be explained in some ways:

- For mass relationships, it could be that the high energy GUT masses are zero, thus the mass is generated radiatively at low energy. This has been tryed a lot of time in the seventies to get the electron mass out of the muon one, but abandoned after (or because of) the third generation.

- For coupling constants, some eigenvalue in the renormalisation group flow could be reached when running to low energy, sort of universality. Adler did some speculative tries on this sense.

- For relationships between coupling constants, it could be related to the symmetry breaking mechanism. Meaning, that the symmetry breaking mechanism triggers at a energy scale with the coupling constant meeting some algebraic relationships. I have no idea of a mechanism of such kind.

Of course, we could also consider that our prejudice about high energy GUT is just that, a prejudice. For instance, Koide's relationship, related to mass quotients across the three generations, seems hard to be married with a radiative generation principle.

 

[arivero]

So, the coupling constant required to give the tau / Z mass ratio,
assuming massles propagators, leads us to the strong coupling
constant at mZ energies...

Do you mean alpha/2pi = mtau / mZ approx? Hmm the only thing I am afraid is about the experimental value of the strong coupling, a fuzzy bussiness.

Now, "who ordered" the s of strong here? Well, at least the use of
massless (gluon) propagator diagrams fits ... :^)

Actually we have already an uninvited aparision of alpha_s in the relationship between Z0 decay and Pi0 decay (somewhere in the middle ot the long long thread). It is hidden because we speak of the "pion decay constant", but this pion decay constant is actually a sum of radiative corrections coming from the strong force.

 

[arivero]

I see. In other words you are telling that the relationship between tau and muon mass is as the one between electromagnetic and strong coupling constants. Gsponer did me a related observation time ago, that if we drove electron mass to zero, Koide relationship should imply a quotient between tau and muon, and this quotient was about the same magnitude that the nuclear strong force (the pion-mediated force between nucleons). Now, electron mass to zero with a fixed muon mass should be equivalent to electromagnetic coupling going to zero. Hmm.

 

[Hans de Vries]

Actually we have already an uninvited aparision of alpha_s .

One might blame it on the infamous ninth gluon... :^) , the white/anti white one,
sometimes speculated to be the photon. ( which would leave it with the
wrong coupling constant)Regards, Hans

 

[arivero]

Just adding this one to the thread for reference

http://es.arxiv.org/abs/hep-ph/0609131
G. Lopez Castro, J. Pestieau
The unit of electric charge and the mass hierarchy of heavy particles

Some more:
Jean Pestieau http://es.arxiv.org/abs/hep-ph/0105301
G. Lopez Castro, J. Pestieau http://arxiv.org/abs/hep-ph/9804272
Jean Pestieau* and Probir Roy http://prola.aps.org/abstract/PRL/v23/i6/p349_1

 

[arivero]

For example...the combined mass of the particles of the meson octet is 3.14006 times the proton mass. Pretty close to pi. Amazingly, the combined mass of the particles in the baryon octet is around 9.8 times the proton mass. Close to pi squared.

Let me to evaluate these quantities because, as we know, these multiplets at at the heart of SU(3) flavour mass formulae.

The meson octect is \pi^0 \pi^+ \pi^- K^0 \bar K^0 K^+ K^- \eta_8. The latter mixes with the singlet to produce \eta and \eta' which are the actually measured masses, and we can use \eta if we think that the mix is very small. But we could also consider the full non irreducible nonet or the extended 16-plet with the charmed particles.

The barion octect is p n \Lambda \Sigma^0 \Sigma^+ \Sigma^- \Xi^- \Xi^0. No mixing issue here, but Sigma^0 is a very fascinating piece of physics on itself, particularly its decay mechanism. As for adding charm, we could, going then to a 20-plet.

figures are drawn in http://pdg.lbl.gov/2006/reviews/quarkmodrpp.pdf

Now the meson sum is

134.9766+139.57018+139.57018+547.51+497.648+497.648+493.677+493.677
=2944.277

2944.27696/938.27203=3.1379, which is 0.12% off from pi, thus qualifies for the thread. It fails pi for about 3.5 MeV, so the error bars can not be blamed, perhaps the mixing can.

938.27203+939.56536+1115.683+1192.642+1189.37+1197.449+1321.31+1314.83=9209.12139=
9.815 M_p, while pi square is about 9.87. So off by 0.55% this time but no mixing to be blamed here.

 

[Demystifier]

Numerology

I do not know if somebody already said that, but I suspect that it is possible to write a computer program that will find a "numerical coincidence" (up to a reasonably specified accuracy) between any two (or more) specified numbers. Essentially, such a program tries various algebraic relations and combinations of small integer numbers and constants such as "pi" and "e", until it finds a "good" one.

 

[arivero]

I do not know if somebody already said that, but I suspect that it is possible to write a computer program that will find a "numerical coincidence" (up to a reasonably specified accuracy) between any two (or more) specified numbers. Essentially, such a program tries various algebraic relations and combinations of small integer numbers and constants such as "pi" and "e", until it finds a "good" one.

Indeed this program has been written and it is quoted somewhere in the middle of the thread; even the output is available in a web page, sorted by decimal ordering. It was done by a couple of computer scientists and the motivation is to try to define some kind of complexity of an algebraic expression. Another researcher, I.J.Good, tryed to use the same approach to single out "low entropy" expressions. Such GIGO (measure garbage In to determine the Garbage out) methods usually trip on the electron/proton quotient

In some sense a problem with these programs is that they concentrate in algebra instead of geometry (nor to speak of dynamics). So 6 \pi^5 is reported as "less complex" than, for instance, e^\pi - {1 \over e^{\pi}}

Let me add that the study if the rings generated by the rationals plus some finite set of irrational numbers are a very prolific field of study in algebra. Still, its truncation to some n-digits decimal expansion is not very studied as far as I know; a friend of me, J. Clemente, tried time ago to work out the algebraic setting of IEEE "real" numbers and we did not found too much bibliography on it.

 

[Demystifier]

I am really happy to know that such a program has been made.
Thanks arivero.

 

[arivero]

Via an old comment of Baez in Woits blog a year ago, I note the http://www.nbi.dk/~predrag/papers/finitness.html about the perturbative expansion of QED. It its weak form, it claims that the growth of diagrams is not combinatorial. In its strong form, it seems to claim that the coefficients are of order unity, a point that Kino****a considers refuted after the 8th-order calculation.

Lovers of gossip and theoretists of science would like to check also the remarks
http://www.nbi.dk/~predrag/papers/g-2.html
http://www.nbi.dk/~predrag/papers/DFS_pris.ps.gz

I do not know what to do of his "social experience". I remember, as undergradute student, how excited I was about the articles of Cvitanovic on Chaos theory, and how a time later I was not so fond of them. But he got to bail out into another physics area, at least. On the other hand, 20 years later, Kino****a (could someone edit it our of the politically correct spelling rules of PF, please!) is the leader of the perturbative calculation effort, and it is because of him, perhaps, that the (g-2) is still an important test of the standard model.

And yes, the 8 order term seems nowadays very much as -sqrt(3). The 6th order term is now 1.181241..., it was 1.195(26) already in the 1974 paper; the 0.922(24) refers to a particular subset, see http://www.nbi.dk/~predrag/papers/PRD10-74-III.pdf

 

[arivero]

On sociology, it is sad to think that a "pet theory" had privated QED of one of its more dedicated calculators. Now consider if the observations of Hans about g-2 could attract some interest, or more probably to excite old memories of conflict.

 

[CarlB]

Did we include this one already?
http://federation.g3z.com/Physics/#MassCharge

I can't give you the name of the author as I don't know it. Mark Hopkins maybe?

 

[arivero]

Did we include this one already?
http://federation.g3z.com/Physics/#MassCharge

I can't give you the name of the author as I don't know it. Mark Hopkins maybe?

It refers to Yablon, and some of the mass formulae have been discussed in usenet news.

 

[Hans de Vries]

It refers to Yablon, and some of the mass formulae have been discussed in usenet news.

Good to see you posting!

We should compare the recently improved W mass with the results of the
thread. The new world average for the W mass is now 80.398 (25) GeV.
http://www.interactions.org/cms/?pid=1024834
The value for the Z mass is still 91.1876 (21) GeV as far as I know.
We had two numerical coincidences for the mW/mZ mass ratio on this thread:

0.881418559878 ___ from the spin half / spin one ratio
0.881373587019 ___ the value arcsinh(1)

Using the more precise value of Z we can get values for the W mass:

80.374  ( 2)    Derived W mass from spin half / spin one ratio
80.370  ( 2)    Derived W mass from arcsinh(1) 
80.376  (19)    Experimental W mass: from sW on page 8 of hep-ph/0604035 
80.398  (25)    Experimental W mass: New world average
80.425  (38)    Experimental W mass: Old world average

This is certainly an improvement for both the values as well as the sigmas
which are around 1 for both now. (mid value difference: 0.030% and 0.035%)

The last value was discussed here:
https://www.physicsforums.com/showpost.php?p=958122&postcount=202Regards, Hans

 

[arivero]

Good to see you posting!

Hi! Instead of a sabbatical I got increased workload, so I read the blogs and forums but I do not calculate

In fact I had not checked the new values. So, before, the 1 sigma low point was 80.387 and now it is 80.377 so the results are better than in 2004. No surprise, as the word average is to be calculated with similar patterns than hep-ph/0604035. This means that the coincidence is here to remain, unexplained or not. Any future deviations could be covered with radiative corrections, if it comes from a fundamental theory.