All the lepton masses from G, pi, e

Hans de Vries

It's OK

X = sqrt(137.03599911) = 11.7062376154766

11.7062376154766 = X
00.0854245431237 = X^-1 /(2pi)^0
00.0000992128957 = X^-3 /(2pi)^1
00.0000000183389 = X^-5 /(2pi)^3

=================================
11.7917613898350 = exp(pi^2/4)

_3.1415926536222 = pi (measured...)
_3.1415926535897 = pi (exact)


Regards, Hans

Tip: The Windows calculator is exact to 32 digits.

marcus

so when did you first see this?

also it is maybe not so important but what do you mean by the "measured" value of pi?

marcus

Rilke has a funny sonnet about mohammed being in a cave and
a visitor comes and shows him something. it could be like the experience of seeing this. do you read German? I will see if I can find the poem
Yes, here it is:

Mohammeds Berufung


Da aber als in sein Versteck der Hohe,
sofort Erkennbare: der Engel, trat,
aufrecht, der lautere und lichterlohe:
da tat er allen Anspruch ab und bat
bleiben zu dürfen der von seinen Reisen
innen verwirrte Kaufmann, der er war;
er hatte nie gelesen - und nun gar
ein solches Wort, zu viel für einen Weisen.


Der Engel aber, herrisch, wies und wies
ihm, was geschrieben stand auf seinem Blatte,
und gab nicht nach und wollte wieder: Lies.
Da las er: so, dass sich der Engel bog.
Und war schon einer, der gelesen hatte
und konnte und gehorchte und vollzog.

Hans de Vries

I did checked it before I posted

"measured" just because it's derived from a measured value, (Only
to distinguish it from the exact value )

Regards, Hans

marcus

I mean, when did you first discover this numerical relation?
Have you known for weeks, for months? Or did you just
notice this a day or two before posting it?

I think it must have been a strange experience
so I am curious about what it was like

Hans de Vries

Schön

Fun, isn't it? At least it might give an entry from another
direction to reveal something of the underlaying
physics (or geometry)

Regards, Hans

Hans de Vries

I found it a week or two before posting. tried to see if it
could be further simplified but couldn't resist to post it
anymore. (Even though I'm the kind of person who will
never feel 100% sure about anything )

Regards, Hans

marcus

I do not know what to say.

if it is a coincidence then
it is the most decimal places coincidence I have
seen in my lifetime

I wish some of the others would say something

marcus

well I am glad you posted it here
where we can see and discuss some

I suppose the next thing is two things:

1.write email to some physicists
(I would think immediately to write Lee Smolin because
it could have some connection with his CNS scheme for
iteratively generating the fundamental dimensionless constants)

2. prepare an article to post on http://arxiv.org

Alejandro knows about arXiv, has some experience.
Once it is on arXiv then it will always come up when people do
keyword searches. Then it may be of use to someone who
finds it by accident and is building a theory to explain alpha.

I think anyone here would be glad to help. It is quite interesting
that a coincidence (let us call it that) should go out 10 decimal places

arivero

Yep, I know from the ArXiV , and I can see it is not easy to fit there, as the goal of the preprint distributions is to keep researchers up-to-date about advancements in *their* area. It is clearly not a hep-th/ , as it does not have a theory under, and probably not a math-ph/ . It could be focused as sort of "state-of-art memotecnics" or something in a line able to fit as physics/ (think about the American Journal of Physics kind of articles).

Hans de Vries

I found an interesting one:

The idea is to see if Nature's limitation to three gene-
rations might have some relation with the identity:

[tex]3^2 + 4^2 = 5^2[/tex]

For the three consecutive integers 3,4 and 5, leading
to only 3 states rather than a whole series. We write
for the three lepton masses:
.
.
[tex]3^2 \ ln(m_e) \ \ \ + \ \ \ (4^2+n)\ ln(m_\tau) \ \ \ = \ \ \ (5^2+n)\ ln(m_\mu) [/tex]
.
.

We then look at the value of n. We do find [itex] n \approx 1.00086[/itex]
using the following Codata values for the mass ratio’s:

__3477.48 ____ (57)__ tau/electron
___206.7682838 (54)__ muon/electron
____16.8183 ___(27)__ tau/muon

The least well known value, [tex]ln(m_\tau)[/tex] has to be
exact to within 0.005% to get a result so close to
1.0000. This is less than the current experimental
uncertainty which means that the exact value of
n=1.0000 is still within the experimental uncertainty.

Regards, Hans
.
.
.
PS. Please be aware that the chance for coincidences
might be bigger than you think!

PPS: The formula is invariant under the transformation
[tex]\{ m_e, m_\mu, m_\tau \} \rightarrow \{ xm^y_e, \ xm^y_\mu, \ xm^y_\tau \}[/tex] where x and y
are arbitrary numbers.

arivero

Actually there are two "pythagorean sequences", the other one is (-1,0,1), having as square the 1,0,1 sequence. Intriguingly, in your combination there is a twist, because the -1 is not going along with the 3, but with the 4.

I can see your conflict; your first set of formulae and this third one have different assignations for the quotient of logarithms,
[tex]{\ln m_\mu/m_\tau \over \ln m_e/m_\mu}={\pi^2-1 \over 2\pi^2-3}; \,
{\ln m_\mu/m_\tau \over \ln m_e/m_\mu}={3^2\over 4^2+1}[/tex]
so surely at least one if them is a random coincidence.
(EDITED: Or, perhaps, there is some happy circunstance where an expression happens during some series expansion of the another.
(EDITED again: in fact, using the expansion [tex]\pi^2/6=\zeta (2)=\sum_n 1/n^2[/tex], its first term is [tex]10/18[/tex], ie [tex]9+1 \over 17+1[/tex] so both expresions are no so far away: the second formula can be seen, at least, as the first term of the expansion of [tex]{\pi^2-1-1/2 \over 2\pi^2-3-1/2}[/tex]. Good enough if one takes into account that both HdV formulae are approximate guesses of an hypothetical exact formula.
Said this, both relations have an intense geometrical flavour. We are exploring two avenues of discretization of mechanics which could, in the long run, provide some justification: either to parametrise the ambiguity in taking two derivatives of the position to get Newton law, or to parametrise the ambiguity in the two sequencial derivatives of Lie Bracket.

arivero

Perhaps it is physics after all; I have just revised (ah, google!) a point that I should have noticed instantaneously: that the Pythagorean condition appears naturally in the definition of Cayley spinors. A guy called Andrzej Trautman has worked out the diophantine case SL(2,Z). If interested in this line, check http://math.ucr.edu/home/baez/week196.html and references therein.

Hans de Vries

Alejandro,

I'm trying to go back to more physics but first this. The two
lepton ratio formula's may also be combined into the following:

[tex]\ln {m_\tau \over m_\mu} = {\pi-{1 \over \pi}}[/tex]_______ [tex]\ln {m_\mu \over m_e} = {4^2+1 \over 3^2} (\pi-{1 \over \pi})[/tex]_______[tex]\ln {m_\tau \over m_e} = {5^2+1 \over 3^2} (\pi-{1 \over \pi})[/tex]


Now if and only if the term [itex]\pi-1/\pi[/itex] was exact then all three
formulae would be within experimental range. Now it isn't but the
[itex]\pi-1/\pi[/itex] term is the only thing that I could connect to some real
physics up to now. It's inspired on the way how you rewrote:


[tex]\ln {m_\tau \over m_\mu} = {\pi-{1 \over \pi}}\ \ \ \ \ \ \equiv \ \ \ \ \ \ \ln {m_\tau \over m_\mu} = 2\sinh(\ln \pi)\ \ \ \ \ \ \equiv \ \ \ \ \ \ {m_\tau \over m_\mu} = |\exp( \sinh(\ln\pi) )|^2[/tex]


the sinh() gives us something in the space-time domain or in
momentum space if we consider it to be a boost like in:

[tex]\sinh \xi=\frac{ v/c}{\sqrt{1-v^2/c^2}}[/tex]____[tex]\cosh \xi=\frac{1}{\sqrt{1-v^2/c^2}}[/tex]
with:

[tex]\tanh \xi=v/c[/tex] ____[tex]\exp \xi=Doppler Ratio[/tex]

The Doppler Ratio now becomes [itex]\pi[/itex] interestingly enough (for
blueshift) and [itex]1/\pi[/itex] (for redshift) corresponding with a speed
v/c (of rotation?) The term [itex]\pi-1/\pi[/itex] could for instance
correspond with the imbalance in momentum change when
absorbing a photon from the back and a photon from the front.

The term [itex]|\exp( )|^2[/itex] may possibly be associated with going from
phase space (defined in x and ct) to probability space. if the
masses were to be defined as "mass density probabilities"
(mass * wave function) then the imaginary part of the
argument would define phase while the real part would
lead to the mass ratio at each point of the wave function.

Another, although numerological, reason to become interested
in this approach is that the most exact equations I got up to
now are found in the "boost domain", that is:

[tex]\ln {m_\mu \over m_e} \ = \ 2\sinh(a) * 1.000000093 , [/tex]____[tex]\ln {m_\tau \over m_\mu} \ = \ 2\sinh(b) * 1.0000047[/tex]

with:

[tex]a = 1 +\sqrt{1/2} \ \ \ and \ \ \ ab^2 = \sqrt 5[/tex]

I don't know if a and b are really a pair but at least there is a
simple relationship.

That's the best I can do for sofar in the hope to get some
physical meaning out of it.

Regards. Hans

arivero

Yep, I intended a connection with relativity when casting sinh here, but I am still far to understand how it goes. There is some dinamical analogy between the tetrad of elementary particles, [tex]\nu, e, u, d[/tex] and the tetrad of relativistic coordinates [tex]t, \rho, \theta, \phi[/tex]. On one side quarks are unobservable, so they feel very much an angular coordinate -which lacks of metric scale-. On another neutrinos are sort of special, so they feel as the trivial time coordinate. But we are very far from getting real meaning of this analogy; my own research program is in the backburner fire already for more than five years (see hep-th/9804169 and "cited by" there) with no remarkable milestones.

Hans, are you using some symbolic algebra program to search for your equations? I ask because even if the results are not publishable as a physics article, it could then to be a very valid article for computational journals.

arivero

Let me to update on this. While Lubos Motl seems to have proposed the 6 pi^5 equation in an independent way, some previous postings in the net give it as folklorical knowledge, sometimes related to a couple of footnote-quoted papers by Armand Wiler:
Wyler,A., 'On the Conformal Groups in the Theory of Relativity and their Unitary Representations', Arch. Rat. Mech. and Anal.,31:35-50, 1968
Wyler,A., L'espace symetrique du groupe des equations de Maxwell' C. R. Acad. Sc. Paris,269:743 745
Wyler,A., 'Les groupes des potentiels de Coulomb et de ¥ukava', C..R. Acad. Sc. Paris,271:186 188
According F. D. T. Smith, this work of Wyler was noted in the Physics Today Aug and Sept 1971 (vol 24, pg 17-19 according M. Ibison).

Regretly old comptes rendues are not in the internet so I can not verify these papers. The Physics Today comment attracted some discussion in the Physical Review Letters,
http://prola.aps.org/abstract/PRL/v27/i22/p1545_1
http://prola.aps.org/abstract/PRL/v28/i7/p462_1
http://prola.aps.org/abstract/PRD/v15/i12/p3727_1
And well, there they refer to a "Wyler equation for alpha",
[tex]\alpha=(9/8 \pi^4)(\pi^5/2^4 5!)^{1/4}=1/137.03608[/tex]
not to an equation for proton mass. So we are still in doubt about the priority of the proton/electron quotient.

PS: please do not believe the Journal-Ref of Tony Smith, he is always exchanging trickeries with the arxiv (and please do not comment about it in this thread!!!). But Tony is always a good starting point to remember exotic, sometimes forgotten, research.

Kea

By the way, Trautman (I guess it's the same guy) considered
applying categories to gravity a long time ago. He presented
an article on it at Dirac's 70th birthday party.

The HdV work is cool. I'm tired of arguing with a string
phenomenologist that I know, who believes that it's in
principle impossible to calculate (from a fundamental theory)
the mass ratios.

Cheers
Kea