All the lepton masses from G, pi, e

[arivero]

y_i^2= 2 \sqrt 2 G_F m_i^2

Hmm how do we put units back in this formula? If G_F is measured in (\hbar c)^3 \over GeV^2 and mass is measured in GeV \over c^2 then the square of yukawa coupling has units, er, \hbar^3 \over c.

Hmm it is already strange here
{G_F \over \sqrt 2} = \frac 18 {g^2 \over M^2_W}
Because in the LHS we have again (hc)^3 / Gev^2

The point of this question is, if we agree that g and y_i are adimensional coupling constants, then we need to add the h and c to restore dimensionality. Furthermore if we postulate that the yukawa coupling of Top is unity, the formula is

1= {c \over \hbar^3} 2 \sqrt 2 G_F m_{top}^2

This is, if we had a theory implying the exact value of y_t=1, this theory give us a formula for the constant \hbar^3 / c. I had expected a formula for \hbar.

Well, at least we can tell that the Area of Planck is

A_P= l_P^2 = {G_N \hbar \over c^3} = {1 \over (\hbar c) ^2} 2 \sqrt 2 G_N G_F m_{top}^2

or, using (\hbar c)=m^2_P G_N

G_N= {2 \sqrt 2 G_F m_{top}^2 \over m_P^4 l_P^2} =<br /> \; 2 \sqrt 2 \; ({ m_{top} \over m_P })^2 { 1 \over m_P^2 \; l_P^2 } G_F

But there should be more fundamental ways of defining the conversion constant (hc) because after all it is the critical value of coulombian dirac equation. This is because classically a coulombian relativistic coupling hc has minimum angular momentum h (when the orbit radius goes to cero and the orbit speed goes to c). In fact, in the formula above, the denominator m^2_X \; l^2_X has the only goal of producing a conversion factor and it works at any scale, not compulsorily the Planck scale. It is more physical perhaps to write

m_P^2 G_N= { 2 \sqrt 2 \over m^2_X \; l^2_X} m_{top}^2 G_F
The funny thing of this formula is that it does not use explicitly h nor c, the necessary combination is hidden in the condition of having m_X and l_X defined at the same scale. As for the powers of m and l, they come from dimensional considerations: Fermi force goes as 1/r^4, gravitation as 1/r^2 so we need a length square. Fermi force does not depend of mass, gravitations depende on product of two masses, so we need a mass square. The only ad-hocness is, as said, to ask the needed mass and the needed length to be defined at the same scale (ie the length must be compton length of the corresponding mass), even if arbitrary; in this way mass and length cancel to produce h/c factors.

 

[arivero]

Equation group (7) in page 8 of hep-ph/0604035 lists

0.22306 \pm 0.00033

for the vector mass-based weinberg angle on-shell, so the kinematical result

\sin^2 \theta_W<br /> \ \ \ \ = \ \ \ \ 1 \ - \ \frac{\beta^2_f }{\beta^2_b}<br /> \ \ \ \ = \ \ \ \ 0.22310132230086634541466926662604

is still in and for sure already for three digits. It was "(to within 0.063% or sigma 1.2)" in 2004 at the start of the thread. Now it is 0.01% and sigma 0.13 if we buy the global fit of this paper (which is from Erler, so it will most probably appear in the 2006 PDG).

 

[arivero]

These dimensionless numbers related to angular momentum may also be
derived from Quantum Mechanics.
...
1- We’ll derive a radius dependent speed (in QM) which we still need.
2- Show how QM can solve the missing relativistic mass increase.
3- Get the formula which gives us the dimensionless numbers.

Well, at the moment all we have is a derivation from De Broglie Quantum Mechanics, not new, not old, quantum mechanics. In fact some afirmations are very DeBroglian, as when you speak of a dependence or independence between radius and energy; in new quantum mechanics all the observables are got via averaging against the wavefunction squared, so all the three coordinates are integrated out; no observable has dependency in coordinates.

Step 1 and 2 are even valid in classical mechanics simply using L^2 instead of l(l+1)

Ah, a minor problem with the use of l(l+1), of course, is that while total angular momentum can be half-integer, orbital angular momentum as produced by quantising L=RxP according Schroedinger rules produces an operator with only integer numbers. This is the typical lore of SU(2) being the double covering of SO(3).

We want to show how QM can compensate for this increase with a
negative p^2_r radial momentum term. In general we have:

It seems a good idea, even if I am unsure about the process in step 2. For circular orbits, we have only p_\phi different of zero, have we?

But yep, either we use some requisite relating the kinetik rotational (centrifugue term, if you wish) energy with the rest mass energy, or we look for some confining potential imposing the radius ad-hoc.

 

[arivero]

Well, at the moment all we have is a derivation from De Broglie Quantum Mechanics, not new, not old, quantum mechanics. In fact some afirmations are very DeBroglian,

In fact it was ringing me some bell so I have revised my archives, and well, I have found your formula in De Broglie himself!

Ondes et Quanta, Note de M. Louis de Broglie, presentee par M. Jean Perrin. Seance du 10 Septembre 1923, Academie des Sciencies, pages 507 to 510. There is a footnote "au sujet de la presente Note, voir M. Brillouin, Comptes Rendus, t. 168, 1919, p 1318" . The paper is one using a small non null mass for the photon and declaring a tachionic wave over it, so it is not very popular in mainstream I guess. But after finishing with the photon, he focuses in a orbiting particle, and then in page 509 it gives

{m_0 \beta^2 c^2 \over \sqrt {1-\beta^2} }T_r= n h

With T_r the period of revolution of the particle in the orbit. This is the justifiable part of Hans Argument. Over it we need two extra assumptions

1) fix the period from the rest mass,
T_r = { h \over m_0 c^2}

2) Move n to be a value alike to the ones from Schroedinger wave mechanics, instead of the one from Einstein stability condition ("old" Bohr-Sommerfeld QM}.
n \to \sqrt {j (j+1)}

One could expect at least 2) to be addressed by De Broglie in some later work but I haven't found any. I keep searching.

The j(j+1) value for L^2 appears for sure in Born version of QM, and in the article I am using (Nov. 1925, published 1926) it remarks "this result is formally reminiscent of the values of M^2 which enter the Lande g formula". Or something so, in German. The result is also general enough to do not rule out semintegers, as Schroedinger result does.

It seems (I am reading in a history report of J Brandmuller) that the substitution rule was proposed by Lande in 1923 after some previous turmoil about delaying publications for coordination with Back. The original substitution is J^2 ---> (J^2-1/4). Then Pauli in 1904 proposes a change of variables and the well known j(j-1) emerges there, empirically, with a narrow two years advantage against theoretist publications. The paper of Pauli should be Z Phys 20 p 371, and perhaps also 31 p 765.

Zur Frage der Zuordnung der Komplexstrukturterme in starken und in schwachen äußeren Feldern

The paper of pauli is restricted to subscribers of the full electronic series, I can not reach it from my campus. A commentary appears in Olivier Darrigol's http://content.cdlib.org:8088/xtf/view?docId=ft4t1nb2gv&chunk.id=0&doc.view=print, from formula 170 to 177 and ff. It seems that Pauli suggested that a derivative had been substituted by a difference (remember j has units of \hbar so it is plausible)

<br /> {1 \over j^2} \to - {d \over dj } ({1 \over j}) \to {1 \over j-1} - {1 \over j } \to {1\over j(j-1)}<br />
​

and this hint was used by Heisenberg to fit with his own development. Schroedinger does not need it as it comes automagically from functional analysis. And De Broglie? Does he miss the point?

Some papers put l(l+1) in Sommerfeld formulae, so perhaps some edition of his Atombau corrected for it (with proof?)

A recent reference recalling the Lande g formula is hep-ph/0209068, page 7. It also discusses the semiinteger vs integer issue, but does not quote sources. Barut wrote a small biography of Lande, http://www.physik.uni-frankfurt.de/paf/paf38.html .

 

[Hans de Vries]

Well, at the moment all we have is a derivation from De Broglie Quantum Mechanics, not new, not old, quantum mechanics. In fact some afirmations are very DeBroglian, as when you speak of a dependence or independence between radius and energy; in new quantum mechanics all the observables are got via averaging against the wavefunction squared, so all the three coordinates are integrated out; no observable has dependency in coordinates.

Well the path-integral did bring back the idea of the moving point particle.
de Broglie was the first to use the plane wave solutions of the Klein Gordon
equation without knowing it. On the other hand, I just as well may have
some "Bohr"-ian coincidence here.

It seems a good idea, even if I am unsure about the process in step 2. For circular orbits, we have only p_\phi different of zero, have we?

Actually, p_\phi attributes l^2/r^2\ \hbar and p_\theta attributes l/r^2\ \hbar

So looking to the local phase behavior, the motion in a pure z-spin state
is still at an angle with the x,y plane: \psi \ =\ \pm\arctan{(1/\sqrt{l})}
Plus or minus relates to a 50:50 superposition of the +/- x-spin and y-spin
states. This still may be related to "circular orbits" The circle at a tiled
angle crosses the x-y plane twice, at plus and minus psi.

All tiled circles combined over 360 degrees of phi then gives the
"precession picture", although it's more like happening all simultaneously.
All the circular orbits are 100% either left handed or right handed around
the z-axis and 50%-50% around the x-axis and y-axis like Stern-Gerlach
would like it to be.

I actually don't believe in moving point particles around circular orbits
but there's the correspondence between the local phase behavior and
the phase behavior of a plane wave representing a moving particle.

But yep, either we use some requisite relating the kinetik rotational (centrifugue term, if you wish) energy with the rest mass energy, or we look for some confining potential imposing the radius ad-hoc.

There are lots of interesting candidates for stable localized Klein-Gordon/
Dirac solutions for free particles which do not extend to infinity.
They are merely modulated by the plane wave solutions but finite in size.
(They are not super positions of on the shell plane wave solutions, these
aren't stable at all)

Some of them counter a repulsive potential just by geometry (like the
hydrogen solutions counter an attractive potential) Some cancel
radius dependent relativistic mass increase (like what I was talking
about). Yet again others can modify the rest-mass by geometry alone
and mimic exactly the wave behavior of particles with another mass.I might give some specific examples later together with the math.


Regard, Hans

 

[Hans de Vries]

The j(j+1) value for L^2 appears for sure in Born version of QM, and in the article I am using (Nov. 1925, published 1926) it remarks "this result is formally reminiscent of the values of M^2 which enter the Lande g formula". Or something so, in German. The result is also general enough to do not rule out semintegers, as Schroedinger result does.

I have this as a reprint in van der Waerdens book: Sources of Quantum Mechanics.
Born together with Heisenberg and Jordan: On Quantum Mechanics II

A recent reference recalling the Lande g formula is hep-ph/0209068, page 7. It also discusses the semiinteger vs integer issue, but does not quote sources. Barut wrote a small biography of Lande, http://www.physik.uni-frankfurt.de/paf/paf38.html .

Lots of nice links Alejandro, A good source is also "The Story of Spin"
from Tomonaga.


Regards, Hans

 

[arivero]

There are lots of interesting candidates

To me this should be an atractive point of the argument, it is very general old relativistic QM, so a lot of modellers can do use of it, potentially

Lots of nice links Alejandro, A good source is also "The Story of Spin"
from Tomonaga..

Thanks. The e-book of Olivier Darrigol is also a good source for spin histories. I have asked him about why De Broglie did not try the j(j+1) view, he suspects that "To him it would have been only one among many other symptoms of the failures of classical models". It is amusing because once you get the idea of using this trick the path to speeds or radiouses is clear.

(note that one of the consequences of j(j+1) is that the angular momentum is always greater than the third component of angular momentum; Born argues this is the way nature implements uncertainty principle for J).

 

[arivero]

So, finally we get our dimensionless numbers related to
quantum mechanical angular momentum:
<br /> ... =\ \sqrt{-\frac{1}{2} l(l+1)\ \ +\ \ <br /> \sqrt{\left(\frac{1}{2} l(l+1) \right)^2\ +\ l(l+1)} }<br />

A fast trivial observation is that operator-wise this is still
<br /> \beta^2= \left&lt;\bar \Psi \left| {-\frac{1}{2} J^2\ \ +\ \ <br /> \sqrt{J^2 + \left(\frac{1}{2} J^2 \right)^2\ } } \right| \Psi \right&gt;

and thus

<br /> \sin^2 \theta = 1 - { <br /> \left&lt;\bar \Psi_{J=1/2} \left| {- J^2\ \ +\ \ <br /> \sqrt{4 J^2 + \left( J^2 \right)^2\ } } \right| \Psi_{J=1/2} \right&gt;<br /> \over<br /> \left&lt;\bar \Psi_{J=1} \left| {- J^2\ \ +\ \ <br /> \sqrt{4 J^2 + \left( J^2 \right)^2\ } } \right| \Psi_{J=1} \right&gt;<br /> }<br />

(note that if we do not use natural units, we must write {1\over \hbar^2} J^2 for the algebraic operator).

 

[arivero]

<br /> \sin^2 \theta = 1 - { <br /> \left&lt;\bar \Psi_{J=1/2} \left| {- J^2\ \ +\ \ <br /> \sqrt{4 J^2 + \left( J^2 \right)^2\ } } \right| \Psi_{J=1/2} \right&gt;<br /> \over<br /> \left&lt;\bar \Psi_{J=1} \left| {- J^2\ \ +\ \ <br /> \sqrt{4 J^2 + \left( J^2 \right)^2\ } } \right| \Psi_{J=1} \right&gt;<br /> }<br />

(note that if we do not use natural units, we must write {1\over \hbar^2} J^2 for the algebraic operator).

Hmm of course if we had an operator Q such that
<br /> \left| \Psi_{J=1} \right&gt; = Q \left| \Psi_{J=1/2} \right&gt; <br />

We could even define the operatorial object

<br /> \left(Q^+ f({1\over \hbar^2} L^2) Q \right)^{-1} <br /> f({1\over \hbar^2} L^2)<br />

Which will produce the desired quantity "M_W/M_Z" when evaluated on a solution of Dirac equation.

And of course Q is a supersymmetry generator.

The problem here is that when L is not the total angular momentum but the orbital momentum it does not commute with J (and what about Q?). But on other hand our algebraic account is independent of the starting point, it "only" serves to define f()

 

[arivero]

hyperzitterbewegung

While looking for the concept of hyperzitterbewegung, I happened to find an article very in the spirit of this thread,

E. Schönfeld
Electron and Fine-Structure Constant
Metrologia 27, p 117-125 (1990)
http://www.iop.org/EJ/article/0026-1394/27/3/002/metv27i3p117.pdf

It is cited by James G. Gilson in his preprint http://www.maths.qmul.ac.uk/~jgg/gilj.pdf , also overly speculative work I think we have already met along the thread.

 

[arivero]

<br /> \frac{r}{r_c}\ =\ \sqrt{-\frac{1}{2} l(l+1)\ \ +\ \ <br /> \sqrt{\left(\frac{1}{2} l(l+1) \right)^2\ +\ l(l+1)} }<br />

I have only noticed it now; of course the funny thing about r_c is that it depends of the mass. So if we ask rto keep the same value when we jump from l=1/2 to l=1, then we are asking the mass of the orbiting particle to jump from \propto M_{W} to \propto M_Z.

We can put also (with L^2 adimensional here)

<br /> r {M_l c \over \hbar}\ =\ \sqrt{-\frac{1}{2} L^2\ \ +\ \ <br /> \sqrt{\left(\frac{1}{2} L^2 \right)^2\ +\ L^2} }<br />

or<br /> M_l \ =\ {1 \over \sqrt 2}{\hbar\over c r} \sqrt{- L^2\ \ +\ \ <br /> \sqrt{\left( L^2 \right)^2\ + 4 \ L^2} } <br />

or, with M \equiv {\hbar\over c r} (hmm, we could even to hide here the sqrt(2), could we?), and natural units to become grouptheoretical...

<br /> M_s^2 = \frac 12 ( - M^2 S^2 + \sqrt{ (M^2S^2)^2 + 4 M^2 (M^2 S^2) })<br />
and now we should go to check the Casimir invariants of unitary representations of the Lorentz group and see if this combination has some meaning for physmathematicians. Note that in the semiclassical limit \hbar \approx 0 we have a tautological M_s^2 \approx M^2 so the formula is not very bad at all.

 

[arivero]

Tilburg

I should mention this too

host www.uvt.nl is IP # 137.056.000.224

I can not find a host having 137.036.000.001 This range is assigned to SUNYat Cobleskill, but www.suny.edu web uses a different range, 141... Ah, here:

www.cobleskill.edu has IP# 137.036.032.031
telcobilling.cobleskill.edu[/URL] has IP# 137.036.004.003Cobleskill is approx "160 miles northwest of New York City and midway between Albany and Oneonta"

 

[arivero]

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

What about the negative square speeds in the original proposal? Here correspond to negative mass square and there is a situation where negative mass square has sense for a force field, namely when it is an scalar field with a cuartic self interaction: then we get the "mexican hat" symmetry breaking scheme.

Let me use the notation \beta^2_{-s} for the negative square speed got as solution of the same equation than \beta^2_s. The only interesting one I have noticed is
<br /> \beta^2_{-1}= -2.73205080756887729352<br />
Fix the above mentioned quartic coupling to be \lambda=1. Then we can produce the "vacuum mass"
<br /> v=\sqrt{-m^2 \over \lambda} = M_{Z0}\sqrt{ -\beta^2_{-1} \over \lambda \beta^2_1} = M_{Z0} 1.9318516525781365 <br />

and for M_{Z0} = 91.1874 \pm 0.0021 GeV we get

v = 176.1605 \pm 0.0041 \; GeV

a value slightly higher that the measured 174.1042 GeV (\pm 0.00075) coming from Fermi constant. We could of course "adjust" the quartic coupling to be \lambda=(176.1605/174.1042)^2=1.0238 but it seems bit of cheating; we can better blame radiative corrections

 

[arivero]

Now a tour of force: if we have the vacuum and we have sin weinberg, we have all the GSW model. In particular we have
<br /> M_W^2 Sin^2 \theta = {\pi \alpha \over \sqrt 2 G_F } = {2 v^2 \pi \alpha }<br />

Thus
<br /> ({\beta_{1/2}^2 \over \beta_1^2})(1-{\beta_{1/2}^2 \over \beta_1^2})= 2 \pi \alpha { -\beta^2_{-1} \over \lambda \beta^2_1}

and
<br /> \alpha={\lambda \over 2 \pi} ({\beta_{1/2}^2 \over -\beta^2_{-1} }) (1-{\beta_{1/2}^2 \over \beta_1^2})<br />

which for \lambda=1 gives (ooooh!)

\alpha=.00739161112923688931...

\alpha=1/135.28...

Again blame radiative corrections, or put \alpha to predict \lambda, or both things. Or set two different vacua to get an additional parameter or claim GSW, SM, nor MSSM are the real things... a lot play here. The most puzzling thing looking only at our setup is the lack of a role for <br /> \beta_{-1/2}, and the meaning of spins (the vacuum is scalar, the W is vector, and we have respective spins 1 and 1/2 instead).

 

[Hans de Vries]

which for \lambda=1 gives (ooooh!)

\alpha=.00739161112923688931...

\alpha=1/135.28...

It seems to get quite interesting Alejandro. Let's see, I suppose the
coupling lambda is like the one in:

{\cal L}\ =\ \frac{1}{2}\left( (\partial \vec{\psi})^2 \ +\ m^2\vec{\psi}^2 \right) - \frac{\lambda}{4}\vec{\psi}^4

Where the sign of the mass term is inverted ?Regards, Hans

Ok I see, Using +\sqrt{m^2/\lambda} for symmetry breaking leads then
to one component becoming the massless (Goldstone) boson.

 

[CarlB]

which for \lambda=1 gives (ooooh!)

\alpha=.00739161112923688931...

\alpha=1/135.28...

Again blame radiative corrections, or put \alpha to predict \lambda, or both things. Or set two different vacua to get an additional parameter or claim GSW, SM, nor MSSM are the real things... a lot play here. The most puzzling thing looking only at our setup is the lack of a role for <br /> \beta_{-1/2}, and the meaning of spins (the vacuum is scalar, the W is vector, and we have respective spins 1 and 1/2 instead).

Interesting. I've seen that number before. Look at the correction for \delta_1 in my forbidden paper on the neutrino masses:

Eqn (14)
http://brannenworks.com/MASSES.pdf

If you replace my \alpha + O(\alpha^2) with your \alpha&#039;, you can eliminate the O(\alpha^2).

This gives a value

\delta_1 = 2/9 - \frac{4\pi\alpha&#039;}{3^{12}}

= 0.22222204743

well within the error bars set by experiment: .22222204717(48) given as equation (12) in my reference.

Carl

 

[Hans de Vries]

Fix the above mentioned quartic coupling to be \lambda=1. Then we can produce the "vacuum mass"
<br /> v=\sqrt{-m^2 \over \lambda} = M_{Z0}\sqrt{ -\beta^2_{-1} \over \lambda \beta^2_1} = M_{Z0} 1.9318516525781365 <br />

and for M_{Z0} = 91.1874 \pm 0.0021 GeV we get

v = 176.1605 \pm 0.0041 \; GeV

Just to note that we have generally for any spin:

<br /> i\frac{\beta_-}{\beta_+}\ = \frac{1}{\sqrt{1-\beta^2_+}}<br />

and thus:

<br /> \tanh{(\cosh^{-1}{(1.9318516525781365)})}\ =\ \beta_1 <br />

Regards, Hans

 

[physmike]

The following may not be in the same league as the main contributions to this thread, but I think it's cool. It doesn't work to many significant figures. Now, never mind why I'm doing this, but if we let \phi = 1.618 \cdots be the golden ratio and we try to relate \alpha to the Temperley-Lieb d factor, we come up with \alpha = 137.08 from

\frac{\sqrt{\alpha}}{2} = e^{\frac{2 \pi}{2 + \phi}} + e^{\frac{- 2 \pi}{2 + \phi}}

Cheers
Kea

Kea, Have you seen this?

Fine-structure Constant, Anomalous Magnetic Moment, Relativity Factor and the Golden Ratio that Divides the Bohr Radius
Authors: R. Heyrovska (1), S. Narayan (2)


" ... Here the mysterious fine-structure constant, alpha = (Compton wavelength/de Broglie wavelength) = 1/137.036 = 2.627/360 is interpreted based on the finding that it is close to 2.618/360 = 1/137.508, where the Compton wavelength for hydrogen is a distance equivalent to an arc length on the circumference (given by the de Broglie wavelength) of a circle with the Bohr radius and 2.618 is the square of the Golden ratio, which was recently shown to divide the Bohr radius into two Golden sections at the point of electrical neutrality. From the data for the electron (e) and proton (p) g-factors, it is found that


(137.508 - 137.036)= 0.472 = [g(p) - g(e)]/[g(p) + g(e)]
(= 2/cube of the Golden ratio),

(360/φ2) - α-1 = 2/φ3 = (gp - ge)/(gp + ge),


and that (2.627 - 2.618)/360 = (small part of the Compton wavelength corresponding to the intrinsic radii of e and p/de Broglie wavelength) = 0.009/360 = (1- gamma)/gamma, the factor for the advance of perihilion in Sommerfeld's theory of the hydrogen atom, where gamma is the relativity factor.

ge/gp = (φ3 – 2)/(φ3 + 2)

...Figure 1. The Golden ratio, point of electrical neutrality (Pel) and magnetic center (Pµ) of the hydrogen atom."

http://arxiv.org/abs/physics/0509207


alphaly, physmike :shy:

 

[arivero]

Ok I see, Using +\sqrt{m^2/\lambda} for symmetry breaking leads then
to one component becoming the massless (Goldstone) boson.

Exactly. You use different signs in mass and quartic coupling to get the "mexican hat". It is almost fine, except that in the standard model this symmetry breaking measures 174 gev, not 176 gev. But it is funny because this quantity is in the same context that the others, and in fact we only collected all the know ones: W, Z, and the breaking. It could be a bit risky to say that the photon is got from spin 0 but one is almost tempted to predict that the extant term \beta_{-1/2} is to be used to get the Higgs boson mass. EDITED: on second though, perhaps I should try to understand the models with two Higgs fields to break
Ideally it should be possible to implement all the new insights in a single lagrangian; it was because of this that I started to look to the negative solutions too. It is fascinating also that this 176 GeV has been hidden in front of our eyes a whole year (and a pity that top quark is now down to 172; it was at 178 last year).

 

[physmike]

Exact match of the fine-structure constant,
...with fibonacci and golden ratio.


α = 7.297 352 568 x 10-3 [1]


(α/3)^-12 = 2.33061803... x 10^31

(α/3)^-12 = 233 x 10^29 + 61803... x 10^23


This result was inspired by the harmonic system of William B. Conner [2].
A harmonic scale of 12 "musical intervals" from 144, ..., 233, ...270.
Fibonacci number, 233, is in the 13th place of the series that begins with
1,1,2,3,..., and 89 + 144 = 233. 233 is the tone SE in the harmonic system.
144 is the fundamental tone DO, light harmonic, and a decagon
angle. 144^1/2 = 12 & 27^1/3 = 3 . 270 is the tone of "action" TI,
and 27 is the "time" harmonic. Inverse golden ratio, φ^-1 = 0.61803...


According to Conner, 233 represents, among other things here;
the minimal compression density of the formative forces
in the quadrispiral cycle of interlocking compressive/expansive
vortices.

And the inverse golden ratio reflects the spiral geometry.


[1] http://physics.nist.gov/cgi-bin/cuu/Value?alph|search_for=abbr_in!

[2] Conner, William B. Harmonic Mathematics: A Phi-Ratioed Universe as
Seen through Tone-Number Harmonics. Chula Vista, CA: Tesla Book Company, 1982

 

[CarlB]

The function

f = 1 + \sqrt{2}\cos(\delta + 2n\pi/3)

gives the masses of the leptons according to the Koide formula. But it turns out that there was another crank at the APS meeting that is using the same formula, but in a slightly different context. I don't know how he got it, but it had to do with assuming that the electron was on a helical path.

This is somewhat similar to some of the stuff you guys are doing, in that it is applying classical mechanics to the electron, and somewhat similar to my insane ideas (my lecture was not well attended, by the way), so here it is:

A spatial model of a free electron (or a positron) is formed by a proposed helically circulating point-like charged superluminal quantum. The model includes the Dirac equation's electron spin \textstyle{1 \over 2}\hbar and magnetic moment e\hbar /2m as well as three Dirac equation measures of the electron's \textit{Zitterbewegung} (jittery motion): a speed of light velocity c, a frequency of 2mc^2/h=2.5\times 10^{20} hz, and a radius of \textstyle{1 \over 2}\hbar /mc=1.9\times 10^{-13}m. The electron's superluminal quantum has a closed double-looped helical trajectory whose circular axis' double-looped length is one Compton wavelength h/mc. The superluminal quantum's maximum speed in the electron model's rest frame is 2.797c. In the electron model's rest frame, the equations for the superluminal quantum's position are:

\begin{array}{l} x(t)=R_0 (1+\sqrt 2 \cos (\omega _0 t))\cos (2\omega _0 t) \\ y(t)=R_0 (1+\sqrt 2 \cos (\omega _0 t))\sin (2\omega _0 t) \\ z(t)=R_0 \sqrt 2 \sin (\omega _0 t) \\ \end{array} \]

where R_0 =\textstyle{1 \over 2}\hbar /mc and \omega _0 =mc^2/\hbar. A photon is modeled by an uncharged superluminal quantum moving at 1.414c along an open 45-degree helical trajectory with radius R=\lambda /2\pi.
http://meetings.aps.org/Meeting/APR06/Event/47453

His website:
http://www.superluminalquantum.org

Now one odd thing is that the way I came up with the Koide extension was by looking very carefully at Clifford algebra and I also ended up with superluminal components for the electron.

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

 

[Hans de Vries]

EDITED: It is fascinating also that this 176 GeV has been hidden in front of our eyes a whole year (and a pity that top quark is now down to 172; it was at 178 last year).

If you look at Weinberg's Vol.II 21.3.37 and 21.3.38 you'll see that he uses
the value for alpha at mZ ( 1/127.904(19) ) This then gives you a lamda of
1.057 and the use of this value makes the 176 GeV go back to 171.3 GeV.

Regards, Hans

 

[arivero]

If you look at Weinberg's Vol.II 21.3.37 and 21.3.38 you'll see that he uses
the value for alpha at mZ ( 1/127.904(19) ) This then gives you a lamda of
1.057 and the use of this value makes the 176 GeV go back to 171.3 GeV.

Yes, I saw that our alpha at least goes in the right side (increases respect to the standard non-renormalised value) and I though on checking evaluations at mZ, but I am too lazy and I put it in the to-do stack; also because we do not now how the things move in our context, where some quantities are clearly non-renormalised and some other (coupling constants, surely) are.

BTW, the people at CDF are working hard on nailing an other \lambda, the one of the top quark, as Dorigo tells

 

[arivero]

by looking very carefully at Clifford algebra and I also ended up ...

In any case, it was an interesting coincidence.

Well, while it could be physics, also math has a word in such coincidences. Surely some of the formulae that started the thread can be got by approximation techniques from the ones with irrational numbers, sort of fractional approximations of pi and such.

As for Clifford-like objects, a lot of coincidence can come from the fact that the solutions to a second degree equation can be put as eigenvalues of a 2x2 matrix, and the sigma matrices are a basis for 2x2 matrices. One could wonder if it generalises to higher degree equations and gamma matrices, and if it can be related to Galois theory, but this could be a thread for math research.

For our recent cases, it is enough to notice that
x^2 + a x - b= (-x) (-x-a) - b = <br /> \begin{vmatrix}{ 0 -x &amp; \sqrt b \cr \sqrt b &amp; -a -x}\end{vmatrix}

We used a=M^2S^2=-C_2 and b=M^4 S^2=-C_1 C_2 in operatorial notation, where C1 and C2 are the Casimir Invariants of Poincare group, with eigenvalues m^2 and -m^2 s (s+1) respectively. Then our equation can be written as asking for the eigenvalues of the object

<br /> \begin{pmatrix} 0 &amp; 1 \cr 1 &amp; 0 \end{pmatrix} \sqrt{-C_1 C_2} + \begin{pmatrix}{0 &amp; 0 \cr 0 &amp; -1}\end{pmatrix} (-C_2)}<br />

Or, more generally, of

<br /> \sigma_\perp \otimes \sqrt{-C_1 C_2} + <br /> { {\bf I} - \sigma_z \over 2 } \otimes C_2}<br />

with \sigma_\perp being any combination \sigma_x \sin t + \sigma_y \cos t

It is interesting to consider some different forms for the second term. We can use two parameters r, \theta (aggh, to many angles here) to put it as r {\bf I} \sin \theta - r \sigma_z \cos \theta; then for r to infinity we get a sin^2 weinberg of 5/8, the opposite of the GUT value . And for \sin \theta=0, r=1 we can retort the argument to surface a 114 GeV value in the place where we were expecting it.

 

[arivero]

A harmonic scale of 12 "musical intervals"

Sorry I disgress in the offtopic, but I would like remark that the proof that 12 musical intervals do not build up an octave is one of the important results from Euclide (a detail that Pythagoreans and Platonists prefer discretly to forget). In the scale of popular sci-myths, it compares to the one telling that the Lunar Cicle is 28 days (it is more than 29 if you care to look at the sky instead the equations).

 

[arivero]

Let me resume the situation numerically too. We have the original object
<br /> A \equiv \sigma_\perp \otimes \sqrt{-C_1 C_2} + <br /> { {\bf I} - \sigma_z \over 2 } \otimes C_2}<br />

and now an auxiliar operator
<br /> B \equiv \sigma_\perp \otimes \sqrt{-C_1 C_2} + <br /> { \sigma_z } \otimes C_2}<br />

The eigenfunctions adscribe to representations (m,s) of the Poincare Group and we are interested on the eigenvalues. We use that C1 and C2 are the casimir invariants, with respective eigenvalues m^2 and -m^2 s (s+1)

From M_Z we fix m to be 106.5732 +- 0.0024 GeV. You can not see it in this formulation, but it is the orbit radius of Hans's original formulation.

Then we get

Eigenvalues of A for s=1/2
+ (80.3717 +- 0.0019 GeV) ^2
- ( 122.384 +- 0.003 GeV) ^2
Eigenvalues of A for s=1
+ (91.184 +- 0.0021 GeV)^2 %MZ is the experimental input
- (176.154 +- 0.004 GeV)^2
Eigenvalues of B for s=1/2
+- ( 114.07 +- 0.003 GeV)^2
Eigenvalues of B for s=1
+- ( 166.796 +- 0.005 GeV)^2

Note that the Eigenvalues of B are same module, different sign; sort of degenerate. Of these six values, three are well known and another one (114 GeV) is the best estimate of Higgs mass.

We have used an experimental input and predicted two new ones (and the 114). Alternatively we can use the parametrisation of the standard model to cancel the experimental input and then we predict two adimensional quantities:
the fine structure constant =1/135.28...
the sin of Weinberg angle at mass shell =0.2231013223...

The later of the two quantities is a best prediction than the former, reflecting the fact that 176 GeV misses the 174 GeV of the electroweak vacuum, while M_W is targeted accurately. But it is good enough being as they are, unrenormalised, sort of tree level, estimates.

 

[physmike]

Sorry I disgress in the offtopic, but I would like remark that the proof that 12 musical intervals do not build up an octave is one of the important results from Euclide (a detail that Pythagoreans and Platonists prefer discretly to forget). .

arivero, sorry I'm not a musician, William B. Conner was. And his 12 musical intervals, as whole tone-numbers, do form an octave. Guess you were referring to Pythagorean ratios, as a bit off-topic, and I'm not familiar with Euclid's proof on this; found a reference [1].

However, while we're a little off-topic; Euclid did know about the beginning of the Fibonacci series,
in Book VII, Proposition 28 of Euclid, as explained by Ben Iverson [2].

On a path through history bringing us back toward the topic,
the Great Pyramid has a height of 233 sacred cubits by Conner's measurements,
and a slant height of 618 in a measurement equivalent to our foot [3].

And a related descriptive example from nature, Mario Livio reports the largest of sunflowers can have a 233/144 spiral ratio, clockwise and counterclockwise spiral patterns [4].




[1] Tonalsoft:Encyclopedia of Microtonal Music Theory

"The Pythagorean comma is the difference between 12 just perfect fifths up and 7 octaves up:

(3/2)^12 = 312/212 = 531441/4096
(2/1)^7 = 128

531441/4096 x 1/128 = 531441/524288 Pythagorean comma

The ratio 531441/ 524288, in JustMusic prime-factor notation designated as 312, with an interval size of approximately 0.23 Semitones [= ~23.46001038 cents].

The pythagorean comma was first described c. 300 BC by pseudo-Euclid in Divisions of the Canon."

("Equal temperament does away with the Pythagorean comma, ..." [5]).

http://tonalsoft.com/enc/p/pythagorean-comma.aspx


[2] Iverson, Ben. Pythagoras and the Quantum World Volume II (Revised). Tigard, OR: ITAM, 1995


[3] Turbeville, Joseph. "The Great Pyramid Architect Had A Secret", A Glimmer of Light From the Eye of a Giant: Tabular Evidence of a Monument in Harmony with the Universe, A mathematical combination of the Fibonacci series with a process of number reduction by distillation led to the development of several numerical tables that provide evidence of a direct numerical connection to the Great Pyramid and the cosmological order of the universe. http://www.eyeofagiant.com


[4] Livio, Mario. The Golden Ratio, The Story of Phi, the World's Most Astonishing Number. Broadway Books: New York, 2002


[5] Garland, Trudi Hammel and Kahn, Charity Vaughan. Math and Music: Harmonious Connections. Dale Seymour Publications: Palo Alto, CA, 1995

 

[arivero]

arivero, sorry I'm not a musician, William B. Conner was. And his 12 musical intervals, as whole tone-numbers, do form an octave. Guess you were referring to Pythagorean ratios, as a bit off-topic, and I'm not familiar with Euclid's proof on this; found a reference [1].

Yep, that is. And it is the pest of musicians since then. A question of harmonics.

On a path through history bringing us back toward the topic,
the Great Pyramid has a height of 233 sacred cubits by Conner's measurements,
and a slant height of 618 in a measurement equivalent to our foot [3].

And a related descriptive example from nature, Mario Livio reports the largest of sunflowers can have a 233/144 spiral ratio, clockwise and counterclockwise spiral patterns [4].

NO This is no the topic. First of all, no masses involved. Second, we all know of geometrical ratios. Third, geometrical ratios are GEOMETRICAL, they can be built without knowledge of algebra or numbers, so you can expect it in geometrical constructions (eg buildings) without needing to resort to strange or arbitrary measurement units. A drawing of a circle is not the same that knowledge of pi.

Note that in the topic of this thread a lot of relationships are adimensional coupling constants or adimensional mass quotients. A joke on the use of units to derive results is Bethe et al short note deriving the fine structure constant (adimensional quantity) from Celsius temperature scale (an ad-hoc arbitrary unit).

There is a math incognita about pyramids but unrelated to Egyptians: the fact that the formula for the pyramid can not be got from a finite process of slicing and pasting known volumes; it needs of differential calculus or other similar infinite involved process. Archimedes says that this volume was first calculated by Democritus, but indefinitely iterated proofs have not survived in greek texts; there are some proofs on the same spirit in later chinese texts, eg Lin Hui. The final proof of impossiblity was given in 1900-1903 as answer to one of the problems of Hilbert.

Enough for the off-topic. I hope you were just trying to do a joke (look for instance some messages before, the host IP joke). If you are serious, you are, er, dissonant.

 

[arivero]

- ( 122.384 +- 0.003 GeV) ^2
+- ( 166.796 +- 0.005 GeV)^2

Reviewing Tony Smith website I become aware again of the work of Dalitz and Goldstain in the early 1990, about some events at 122 GeV that were rejected as background.

As for the 166. Some methodologies of analysis happen to get this set of values due to systematic error.

It is interesting to note that the difference between operator A and B amounts to replace a Pauli matrix by a projector matrix. Sort of selecting one chirality?

 

[physmike]

Electron Mass: from Tone Number and Golden Ratio

Trying to get to the topic.
Beginning with the same basic algebra in post #220.

me = 9.109 3826 x 10-31 kg [1]

(me/3)^-12 = 1.627750...


Tone Number 162 & (60)^1/2 ~ 7.75


162, Sun Tone RE in base octave, resonant to the Golden Ratio φ = 1.618...
60 is the second suboctave of Tone Number 240, LA, harmonic of "tachyon motion" [3]


Curious numerical correlations? While neither in the Bethe or Eddington camp, we'll quote John Barrow: "Our purpose in revealing some of its examples (alpha, numerical gymnastics) is not without serious object." (p. 75) [2]

Quoting Conner on the origin of Tone-Numbers:

"The Pythagorean Table, as it is known today, is a reconstruction of earlier versions. This is the work of one Albert von Thimus, a lawyer/scholar who lived in the 1800's...
My adaptation is based on the Table as it appears in Siegmund Levarie's and Ernst Levy's
"The Pythagorean Table" (Main Currents, March-April 1974). ...does not in any way involve a restructuring of the basic ratios. However, for string-length on the vertical arm I have substituted frequency (pitch), and I have used the specific number value of 144 as the fundamental." (p.75) [4]



[1] http://physics.nist.gov/cgi-bin/cuu/Value?me|search_for=abbr_in!

[2] Barrow, John D. The Constants of Nature. New York, NY: Pantheon Books, 2002

[3] Conner, William B. Harmonic Mathematics: A Phi-Ratioed Universe as Seen through Tone-Number Harmonics. Chula Vista, CA: Tesla Book Company, 1982

[4] Conner,William B. PsychoMathematics: The Key to the Universe (revision of Math's Metasonics).
Chula Vista, CA: Tesla Book Company, 1983

 

[Hans de Vries]

Trying to get to the topic.

Not really, You've provided a classical example of how not to search for
numerical systematics in physics:

Beginning with the same basic algebra in post #220.

me = 9.109 3826 x 10-31 kg [1]

First of all: m_e is a dimensional number, its value depends on the
totally arbitrary definition of the kilogram. Its numerical value as such
does not mean anything.

(me/3)^-12 = 1.627750...

Tone Number 162 & (60)^1/2 ~ 7.75

Then you use more numbers and symbols than you predict, There
are zillions of ways to do this so you're not predicting anything.

Next you change the value 1.62 in 162, Where does the factor 100
come from? In any other number system for instance on base 16 or
base 12 this "resemblance" doesn't exist.

Curious numerical correlations? While neither in the Bethe or Eddington camp, we'll quote John Barrow: "Our purpose in revealing some of its examples (alpha, numerical gymnastics) is not without serious object." (p. 75) [2]

Serious it should be.

Quoting Conner on the origin of Tone-Numbers:

"The Pythagorean Table, as it is known today, is a reconstruction of earlier versions. This is the work of one Albert von Thimus, a lawyer/scholar who lived in the 1800's...
My adaptation is based on the Table as it appears in Siegmund Levarie's and Ernst Levy's
"The Pythagorean Table" (Main Currents, March-April 1974). ...does not in any way involve a restructuring of the basic ratios. However, for string-length on the vertical arm I have substituted frequency (pitch), and I have used the specific number value of 144 as the fundamental." (p.75) [4]

This sounds all very poetic and maybe it will even produce some nice
music, but as Arivero noted: It's dissonant on a Physics webside.

[3] Conner, William B. Harmonic Mathematics: A Phi-Ratioed Universe as Seen through Tone-Number Harmonics. Chula Vista, CA: Tesla Book Company, 1982

[4] Conner,William B. PsychoMathematics: The Key to the Universe (revision of Math's Metasonics).
Chula Vista, CA: Tesla Book Company, 1983

The Mentors of Physicsforums would call the above "crackpot references"
which are not allowed here. They will "hunt" you down and complain
that you're not observing the PF Guidelines to which you've agreed.

Sorry,

Regards, Hans.

 

[arivero]

From M_Z we fix m to be 106.5732 +- 0.0024 GeV. You can not see it in this formulation, but it is the orbit radius of Hans's original formulation.

Then we get

Eigenvalues of A for s=1/2
+ (80.3717 +- 0.0019 GeV) ^2
- ( 122.384 +- 0.003 GeV) ^2
Eigenvalues of A for s=1
+ (91.184 +- 0.0021 GeV)^2 %MZ is the experimental input
- (176.154 +- 0.004 GeV)^2

the sin of Weinberg angle at mass shell =0.2231013223...

Hans kindly remembered me that the above linked post contained also some relationships for charged leptons. In fact even at first order these relations are good; in our new notation (with this m=106.5732 GeV deduced from measured MZ) we could put them as

<br /> {\alpha \over 2} {m \over M_W} \approx {m_e \over m_\mu}<br />

<br /> {\alpha \over 2} {m_Z^2 \over m M_W} \approx ({m_\mu \over m_\tau})^2<br />

and then derived from both,
<br /> \left({\alpha \over 2}\right)^3 {m m_Z^2 \over M_W^2} \approx ({m_e \over m_\tau})^2<br />

But we also have Yablon observation,
<br /> m_\tau \approx \alpha \sqrt 2 \ 172.18 GeV<br />

And then of course we have Koide's
<br /> \frac 23 \left(\sqrt{1}+\sqrt{m_\mu\over m_e}+\sqrt{m_\tau \over m_e}\right)^2 = ({1}+{m_\mu\over m_e}+{m_\tau \over m_e})<br />

And even Krolinowski (I should check it, I think he produced some sqrt(19) somewhere) could have a hit.

Too many!

 

[arivero]

(and I wonder if it could be interesting to look systematically for small integers of pi multiples or so as I did at the start of the thread, now that we have a couple new masses to think about. For instance
{ m_\mu m_\tau \over (122.384 GeV) \ m_e} \approx 3

 

[physmike]

:-)

...You've provided a classical example of how not to search for
numerical systematics in physics:

Have you thought about the relation of your exponential
alpha expression with a logarithmic spiral, and the golden ratio?
We'd like to see more of your work as promised in post #155.
Please make it a "knot".

Alejandro,

I'll add a second section to the paper demonstrating this successive
difference method. Indeed, currently only the end-result is presented
without any explanation.

Regards, Hans

https://www.physicsforums.com/showpost.php?p=742908&postcount=155 :

First of all: m_e is a dimensional number, its value depends on the
totally arbitrary definition of the kilogram. Its numerical value as such
does not mean anything.:

Yes, that's the consensus. Have you ever wondered about the
"invisible hand"?

Then you use more numbers and symbols than you predict, There
are zillions of ways to do this so you're not predicting anything.

Next you change the value 1.62 in 162, Where does the factor 100
come from? In any other number system for instance on base 16 or
base 12 this "resemblance" doesn't exist.:

Are you serious? ... Perhaps we oversimplified the presentation
to highlight a "coincidence".
...floating decimal point arithmetic, and

har·mon·ic

1 a. Any of a series of musical tones whose frequencies
are integral multiples of the frequency of a fundamental tone.

...

3 Physics. A wave whose frequency is a whole-number multiple
of that of another.


http://www.answers.com/topic/harmonic?method=22 :

This sounds all very poetic and maybe it will even produce some nice
music, :

Yes, the Table is poetic, in the best sense of the word.
It has interesting connections with Onar Aam's poetic
octonionic structures of music, and Tony Smith's Witting
polytope nursery rhymes:

"The Witting polytope can be constructed from the
4-dim 24-cell by a Golden Ratio expansion
of the 24-cell to a 24+96 = 120 vertex 4-dim polytope, the 600-cell.

It has octonionic structure and lives in 8-dim space.

Since the Witting polytope has 240 = 20x12 vertices,
it has 12-structure."

Though we question his Wyleresque attempts to formulate the
masses, and his recently acquired status. A status similar to the
references you were concerned about.

 

[CarlB]

I've been thinking about formulas and information theory. Perhaps a good rule for choosing which formulas have merit and which do not would be by comparing how much information is required to describe the formula.

For example, 355/113 gives 3.1415929 ... which is fairly close to pi = 3.1415926, the error is one part in 1.0 E+7, or about 24 bits of accuracy. Now we know that 355/113 is simply an entry in the continued fraction expansion of pi, and so is not some magic formula with an accuracy that we can't expect to find with the continued fraction expansion of any "typical" real number.

355 has 9 binary bits and 113 has 7 more, so at first glance it appears that one has taken about 8 bits of information out of pi. But if one take the point of view that this particular equation must be picked out of all possible equations, then one must also include coding for the termination of the binary number and the division.

Now Shannon's theory of encoding says that you should use short sequences of bits for the "characters" that are more frequently used. Surely binary numbers and division are fairly frequent used items, so they should be encoded efficiently. The encoding of the binary approximation of pi also has some overhead, that required to define where the decimal place goes. Surely these should also be efficient things.

Looking at the numbers, it seems like these sorts of things take about 3 or 4 bits each.

Looked at this way, a formula, in order to truly represent information more efficiently than a simple decimal approximation, had better be quite terse.

One way of approximating the information required to encode a number occurs to me. Count the number of keystrokes required to obtain the number on a calculator. For the two approximations of pi given above, the number of keystrokes is each about 8 so neither is significantly more efficient than the other. For stuff like the Koide formula, one requires a supplementary key that says "solve" I suppose.

Carl

 

[arivero]

Yes, that's the consensus. Have you ever wondered about the
"invisible hand"?

Physmike, an invisible hand for kilograms...? Er, you are pushing conspiracy theory to new heights!

You are telling that someone at the age of the French Revolution or so, knowing secretly the value of the mass of the electron, manipulated the definition of Kilogram in order to get the numbers of the Golden Ratio. But as the definition of Kilogram comes from a cubic decimeter of agua, it follows is that this Secret Fellow actually manipulated the definition of the meter, ie He took the final decision about where to mark the signals in the platinum-iridium bar at Paris so that the people two hundred years after him were able to detect the existence of this secret advanced scientific society.

And you have found Them! Congratulations.

(but still, Conspiracy is off topic for this thread, I am sorry. I hope moderators will be notifyed of your postings at some time)

 

[arivero]

I've been thinking about formulas and information theory.

I.J. Good did a try on it decades ago; his conclusion (or his method) was disapointing, the only formula he got to select was the infamous 6 pi ^5 for proton/electron ratio.

 

[Hans de Vries]

I've been thinking about formulas and information theory.
Carl

It's a good exercise. Very few relations pass the "Shannon Test"

The predictive value (in bits) can be defined as:

<br /> \ln_2{\left(\frac{\mbox{value}}{\mbox{error}}\right)}<br />

To determine how many bits your formula uses to make this prediction
you can do something like this:

Add for each number used:

<br /> \ln_2{\left(\mbox{number}\right)}<br />

Add two bits or so for each use of the most basic constants like pi, e ...

Add \ln_2{(6)} for each of the six elementary operations:

<br /> +, -, x, /, {a}^n, \sqrt[n]{a} <br />

Add an extra bit for the three latter ones since they are non-commutative.
(swapping the operands changes the result)

Add 1 bit for each pair of brackets since it divides an expression in two.

Add for every basic function exp, ln, sine, cosine, tangent a value
of ln2(10) (there are about 10 elementary functions)etcetera.

It turns out that very few formulas actually predict more bits than
they use. A number very hard to quantify is how much a formula
resembles a "physical" formula. The above quantification doesn't
discriminate between very odd looking formulas and more realistic
ones so it might be a tad too negative.Regards, Hans.

 

[physmike]

Physmike, an invisible hand for kilograms...? Er, you are pushing conspiracy theory to new heights!
...
(but still, Conspiracy is off topic for this thread, I am sorry. I hope moderators will be notifyed of your postings at some time)

Incorrect allusion, ...you were off topic, ... Regards, er, not Hans

 

[arivero]

Eigenvalues of A for s=1/2
+ (80.3717 +- 0.0019 GeV) ^2
- ( 122.384 +- 0.003 GeV) ^2
Eigenvalues of A for s=1
+ (91.184 +- 0.0021 GeV)^2 %MZ is the experimental input
- (176.154 +- 0.004 GeV)^2

Thus
Tr A_{s=1/2} = \frac 38 Tr A_{s=1}

Eigenvalues of B for s=1/2
+- ( 114.07 +- 0.003 GeV)^2
Eigenvalues of B for s=1
+- ( 166.796 +- 0.005 GeV)^2

Tr A_{1/2} = Tr A_{s=1}=0

This is worth remarking because a lot of breaking systems have mass formulae extracted from trace or supertrace of the M^2 operator.

 

[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