All the lepton masses from G, pi, e

[CarlB]

Hmmmm. Interesting...

The fluid measure of 1000 pounds of biodiesel is 137 gallons.
http://www.soypower.net/calculator.asp#Conversions

Can this be a coincidence? Or is the fine structure constant involved in agricultural fuels?

Carl

 

[physmike]

Hmmmm. Interesting...

The fluid measure of 1000 pounds of biodiesel is 137 gallons.
http://www.soypower.net/calculator.asp#Conversions

Can this be a coincidence? Or is the fine structure constant involved in agricultural fuels?

Carl

Amazing work Carl, could that be related to the 137 atoms

in the chlorophyll molecule? Has 12 branches too? :-)

 

[Hans de Vries]

<br /> M_s^2 = \frac 12 ( - M^2 S^2 + \sqrt{ (M^2S^2)^2 + 4 M^2 (M^2 S^2) })<br />

So with your use of the two Casimir Invariants of the Poincaré group
C_1=m^2,\ \ C_2=-m^2s(s+1), you could write:

<br /> M_s^4 - C_2M_s^2 + C_1C_2 = 0<br />

From which we want to obtain the masses. Changing the sign of the
middle term:

<br /> M_s^4 + C_2M_s^2 + C_1C_2 = 0<br />

Does give the same solutions but multiplied by a factor i. This would be
equivalent to your "pauli-sigma" formula where the "chirality-selecting" like
term becomes (1+\sigma_z)/2. For both formulas we get the Weinberg angle:

s_W\ =\ 1 - \frac{M_{1/2}^2}{M_1^2}\ =\ 0.223101322.. \qquad \mbox{experimental = 0.22306 (33)}

Interesting is also when we change the sign of the 2nd term:

<br /> M_s^4 \pm C_2M_s^2 - C_1C_2 = 0<br />

We get complex masses but:

<br /> |M_s|^2 \propto S<br />

Showing a desirable Regge Trajectory behavior (which can also be
found in the other solution, as you pointed out. edit: this new one
now actually follows the straight line in (M^2,S) coordinates )Regards, Hans

 

[arivero]

So with your use of the two Casimir Invariants of the Poincaré group
C_1=m^2,\ \ C_2=-m^2s(s+1), you could write:

<br /> M_s^4 - C_2M_s^2 + C_1C_2 = 0<br />

Yep, in fact my idea was that we can use this equation to jump from the pure argumentation section in the paper to the "model" section, ie to jump to explain your original formulation.

...
Showing a desirable Regge Trajectory behavior (which can also be
found in the other solution, as you pointed out. edit: this new one
now actually follows the straight line in (M^2,S) coordinates )

Actually I have been a bit sloppy about Regge things; I looked at some books past Friday but all the stuff of complex angular momentum is so old that we did not touch it during graduate; of course it is of some importance today because strings follow straight Regge trajectory. It is very sad that we could actually be on the track of some stringy thing (my bet: a technisuperstring, ie a supersymmetric string from the topcolor/technicolor interaction. No papers of such beast do exist up to now ).

I explored the idea of further jumping from your formulation to a relativistic spinning rod and then to a spinning string, but I am not satisfyied with the exploration I have done up to now in the spinning rod.

 

[Hans de Vries]

Yep, in fact my idea was that we can use this equation to jump from the pure argumentation section in the paper to the "model" section, ie to jump to explain your original formulation.

Exactly, That was what I was looking for.

Actually I have been a bit sloppy about Regge things; I looked at some books past Friday but all the stuff of complex angular momentum is so old that we did not touch it during graduate; of course it is of some importance today because strings follow straight Regge trajectory. It is very sad that we could actually be on the track of some stringy thing (my bet: a technisuperstring, ie a supersymmetric string from the topcolor/technicolor interaction. No papers of such beast do exist up to now ).

I'm still looking for measurement data on the actual resonances that follow
this pattern. There should be a lot actually.

I explored the idea of further jumping from your formulation to a relativistic spinning rod and then to a spinning string, but I am not satisfyied with the exploration I have done up to now in the spinning rod.

One point is that there would be one radius which is special, but why?

-Effective cutoff radius? The me/mu ratio could point to something like this.
-A 'vacuum' which vibrates in x an y with 90 degrees phase difference
has rotation but has only one specific radius.
-A string has a single radius as well. Probably one of the reasons for
your explorations.


Regards, Hans

 

[arivero]

One point is that there would be one radius which is special, but why?

For the relativistic spinning rod, one can select the radius to be inverse to the angular speed, so that r \omega=c marks the extreme of the rod orbiting at lightspeed. I think this is true of some QCD strings also, I do not know of fundamental strings.

Then for a rod of mass M one can ask for what radius at the same rotational speed will a point particle of the same mass reproduces the same angular momentum (this angular mometum is actually fixed from the above condition r w = c if assume for instance that the mass is equally distributed along the rod). And then one goes across your equations again, but with a fixed angular momentum, this is mostly unsatisfactory.

 

[kmarinas86]

\frac{m_{\mu}}{m_{e}}=\left(\frac{\alpha^{-2}}{2\pi}\right)^{2/3}\frac{\left(1+2\pi\frac{\alpha^2}{2}\right)}{\left(1+\frac{\alpha}{2}\right)}=206.76828 (206.76827)
\frac{m_{\tau}}{m_{\mu}}=\left(\frac{\alpha^{-1}}{2}\right)^{2/3}\frac{\left(1+\frac{\alpha}{2}\right)}{\left(1-4\pi\alpha^2\right)}=16.817 (16.817)
\frac{m_{\tau}}{m_{e}}=\left(\frac{\alpha^{-3}}{4\pi}\right)^{2/3}\frac{\left(1+2\pi\frac{\alpha^2}{2}\right)}{\left(1-4\pi\alpha^2\right)}=3477.2 (3477.3)
\frac{m_{N}}{m_{e}}=\frac{12\pi^2}{1-\alpha}\sqrt{\frac{\sqrt{3}}{\alpha}}=1838.06 (1838.68)
\frac{m_{ssc}}{m_{ddu}}=\frac{m_{ssc}}{m_{N}}=\frac{1}{2\pi}\left(\frac{1-\alpha}{3\alpha^2}\right)^{1/3}=2.926
\frac{m_{bbt}}{m_{ssc}}=\left(\frac{2\pi^2}{\alpha^2}\right)^{1/3}=71.8
\frac{m_{bbt}}{m_{ddu}}=\frac{m_{bbt}}{m_{N}}=\left(\frac{1-\alpha}{12\pi\alpha^4}\right)^{1/3}=210

 

[arivero]

Then for a rod of mass M one can ask for what radius at the same rotational speed will a point particle of the same mass reproduces the same angular momentum (this angular mometum is actually fixed from the above condition r w = c if assume for instance that the mass is equally distributed along the rod). And then one goes across your equations again, but with a fixed angular momentum, this is mostly unsatisfactory.

To center the matter, this is more or less the kind of discussions on chapters 7 and 8 of http://web.mit.edu/physics/facultyandstaff/faculty/barton_zwiebach.htm.

The rod or open string problem appears frequently, I see it at
-problem 5 of March 16 2005 Test.
-problem 2 of March 18 2004 Test (and solutions)
-Short questions 1,2 of May 6 2004 Quiz
-Problem set 5 of Ph135c at Caltech

So if some student is reading this thread he can evend find useful to go across our discussions

 

[arivero]

\frac{m_{\mu}}{m_{e}}=\left(\frac{\alpha^{-2}}{2\pi}\right)^{2/3}\frac{\left(1+2\pi\frac{\alpha^2}{2}\right)}{\left(1+\frac{\alpha}{2}\right)}=206.76828 (206.76827)
...

Hmm if you expand these formulae towards a shape

=(first order correction) (1+second order correction)

or something so, I think they will not give nothing already listed. But did you use computer algebra or manual inspection? If the former, it could be good to give a pointer to your algorithm.

In a not completely unrelated matter, via J Baez blog I am made aware of math.NT/0401406 A Faster Product for Pi and a New Integral for ln(Pi/2) where some funny integer series are exposed.

 

[rgauthier]

One possible relation is that his theory has something to do with what would happen if you convert my theory over to Bohmian mechanics. Bohmian mechanics adds a particle trajectory to the usual wave function.

Now my theory involves three particles that are moving under the influence of a potential that is fairly easily to calculate. But the potential is given by a Clifford algebra and is kind of complicated. It is at least conceivable that one would find that the solution gives helices for the preon (binon) trajectories. That would give a classical interpretation for angular momentum that would match the quantum interpretation.

In any case, it was an interesting coincidence.

Carl[/QUOTE]

Hello Carl and others,
It was an interesting coincidence about the similarity of our formulas. I do think that there is a wave equation associated with the superluminal quantum trajectory in my electron model. That way there can be an internal self-interference which can give rise to the de Broglie wavelength when the electron model is moving with some velocity v. It is the case mathematically that when two waves both having the Compton wavelength h/mc interfere coming from opposite directions (this would be the case in my electron model which is a circulating photon-like object of wavelength h/mc) , the interference pattern when relativistically Doppler shifted gives rise to exactly the relativistic de Broglie wavelength L=h/gammamc. That is, the superluminal electron model, having an associated wave equation) would generate the de Broglie wavelength as it moved with velocity v.
With best wishes,
Richard Gauthier
http://www.superluminalquantum.org

 

[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.