All the lepton masses from G, pi, e

[arivero]

Plus c and h, of course.

The idea is to collect here in only a thread all the approximations voiced out during the summer. Surely this is to be quarantinised in TheoryDev, but it is interesting enough to be kept open as a thread (if closed, please be free to use my weblog (http://www.infoaragon.net/servicios/blogs/conjeturas/index.php?idarticulo=200410041))

First we get Planck Mass, M_P, from G, c and h as usual.

Then we solve for \alpha, the fine structure constant, in HdV second equation (http://physicsforums.com/showthread.php?t=43862) (*),


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


The term in parenthesis, well, is sort of a first order correction.

Then we use Nottale's remark (http://wwwusr.obspm.fr/~nottale/ukmachar.htm) to get the mass of the electron


\ln (M_P/m_e) = \alpha^{-1} \sin^2 \theta_W


where the square sine of Weinberg angle \theta_W is to be rather misteriously taken at the GUT value, 3/8 (http://www.arxiv.org/abs/hep-ph/0410023). I have not checked if the need of Schwinger correction above counterweights this need of a value running up to GUT scale.

Now we use HdV first equation set (http://www.chip-architect.com/news/2004_07_27_The_Electron.html) to get in sequence the mass of the muon, via the rather strange


\ln {m_\mu \over m_e}= 2\pi - 3 {1\over \pi}


and the mass of tau via the simpler


\ln {m_\tau \over m_\mu}= \pi - {1\over \pi}


Alternatively HdV set of equations can be presented from a quotient m_e m_\tau^n / m_\mu^{n+1}, but the original presentation hints of a hyperbolic sine, or perhaps a q-group scent.

We could try to follow towards the full mass matrix, including neutrinos, via some empirical approximations collected by Mikanata and Smirnov.

----
(*) Update: HdV has uploaded to his webpage an indication (http://www.chip-architect.com/news/2004_10_04_The_Electro_Magnetic_coupling_constant. html) of the origin of his formula, as a 3rd term truncation of a peculiar series.

[marcus]

I do not imagine that this will be quarantined, these numerical relations are not your personal theory that you are developing and promoting. Rather, you are reporting on the possible follies (or wisdom) of others.

Even if this may be the work of dope-crazed hippies and Wiccan moon-worshipers, we should not be blind to the knowledge that it exists. How else shall each of us know what numerology is? At the very least, it is a very good vaccination against the temptations of numerical coincidence.

[arivero]

Also it is a high barrier for guessers. I do not know anyone deriving such high precision with so small formulae (specially HdV ones have a small error). So any posting of numerology should be requested to have at least this effectivity, or to shut up. In this sense yes, it is a very effective vacunation.

BTW, while I am using here "numerology" to adhere to the broader view of this word, to me it is numerology only the use or arbitrary dimensional constants (inches/year for a famous example), while using arbitrary adimensional ones is more an issue of compacteness of infomation. Here, at least, one can use HdV relations as a mean to remember almost exactly the values of some fundamental constants; for somebody it can be easier to remember the formulae than the decimal number expressions.

[Hans de Vries]

Mathematics used for the Fine Structure constant formula.

Yes it should be called Mathematics rather than "Numerology"
This is especially true for the way how the formula for the
fine structure constant was generated.

It's possible to derive a function's complete Taylor expansion
from a single point and its infinitesimal small environment.
A similar "Taylor like" method was used here to derive an
entire function from a single (but highly accurate) number:

The tricky point is the start. We need to find the start of the
function presuming that a function exists and it can be expressed
as a converging series.

A successful ansatz turned out to be to express the amplitude
\alpha^{1/2} as a "Gaussian like" value:

\alpha^{1/2}\approx e^{-{\pi^{2}/4}}

From here on we can develop a corrective series A by taking
successive differences so that:

\alpha^{1/2}\equiv Ae^{-{\pi^{2}/4}}

This generates the following series A:

A = 1+{\alpha \over (2\pi)^0 }(1+{\alpha \over (2\pi)^1 }(1+{\alpha \over (2\pi)^2 }( 1 + ........

or separated:

A = 1+{\alpha \over (2\pi)^0 }+{\alpha^2 \over (2\pi)^1 }+{\alpha^3 \over (2\pi)^3 } + ........

The series converges straightforward to reproduce the value
of the fine structure constant exact in all its digits:


after term 0: 0.0071918833558268
after term 1: 0.0072972279174862
after term 2: 0.0072973525456204
after term 3: 0.0072973525686533
f.s. constant 0.007297352568.(+/-24)


I hope that everybody realizes how incredible small the
possibility is that something like this happens by chance.
Say the fine structure constant is some arbitrary number
and then we get the whole series out it with an overall
precision better than one in ten billion !!

Basically the only thing I have "put in" the formula was the
ansatz e^{-{\pi^{2}/4}}. The rest is generated by repeatedly
subtracting the two numbers.


Regards, Hans.

Fine Structure constant article online (http://www.chip-architect.com/news/2004_10_04_The_Electro_Magnetic_coupling_constant. html)

PS. The formula given in Alejandro's opening post represents
the series truncated after term 2.

[Nereid]

Now we use HdV first equation set (http://www.chip-architect.com/news/2004_07_27_The_Electron.html) to get in sequence the mass of the muon, via the rather strange


\ln {m_\mu \over m_e}= 2\pi - 3 {1\over \pi}


and the mass of tau via the simpler


\ln {m_\tau \over m_\mu}= \pi - {1\over \pi}


Alternatively HdV set of equations can be presented from a quotient m_e m_\tau^n / m_\mu^{n+1}, but the original presentation hints of a hyperbolic sine, or perhaps a q-group scent.Your challenge, should you choose to accept it, is to estimate the probability of finding two numbers (any two of the three lepton mass ratios) using only some 'simple' functions, pi, e (and any other 'cool' transcendentals), a few small integers (preferably <5, though a bunch of contiguous primes would also count as 'cool'), to within 0.1% (more points if 0.01%). You may also write a simple program - to generate the 'formulae' - or even an 'evolutionary' meta-program (which will all but guarrantee to find at least one formula meeting the input criteria).We could try to follow towards the full mass matrix, including neutrinos, via some empirical approximations collected by Mikanata and Smirnov.Better yet, since the neutrino masses are known to have only upper limits, why not make some firm predictions (based on goat entrails, er, sorry, magic formulae)? While we're at it, how about the Higgs? the LSSP? neutralinos, axions, wimpzillas, ... the whole zoo?

[humanino]

wimpzillas :surprised
I lost mine and have been unable to replace them :tongue2:

I agree with Neired : the probability that you find a relevant relation with "numerical experiments" is far from obviously non-vanishing.

[arivero]

wimpzillas :surprised
I lost mine and have been unable to replace them :tongue2:

I agree with Neired : the probability that you find a relevant relation with "numerical experiments" is far from obviously non-vanishing.

Yes, because we expect there is at least one such relation. And some of us hope it to be simple (well, a quantum group sounds "simple" to me :biggrin:) and geometrical. Still, it should be good if Mathematica (TM) skilled people were able to make some symbolic experiments to find out such formulae and their probability.

[arivero]

Better yet, since the neutrino masses are known to have only upper limits, why not make some firm predictions (based on goat entrails, er, sorry, magic formulae)? While we're at it, how about the Higgs? the LSSP? neutralinos, axions, wimpzillas, ... the whole zoo?

Stop, stop (I mean the verb, not the SUSY particle)... the relations I was referring to, for neutrino -thus leptonic- mixing angles, are still considered good taste in the real world. Of course, your mileage can vary.

[marcus]

Hans, let us focus on your approximate equation about alpha

I expect you know that Oxford U. Press is publishing a book called Universe or Multiverse containing an article by Smolin that presents a testable scientific theory that the dimensionless fundamental constants have gone through an interation process called CNS (cosmic natural selection).

CNS would have favored convergence to constants optimized for starformation and black hole production. This is not the "Anthropic Principle" in one of its non-empirical forms---it makes testable predictions, it is falsifiable, it is not predicated on life or "consciousness", it is a simple physical theory which may in fact be wrong but seems worth testing.

I would like to understand your equation for alpha because it seems to me to be the sort of thing which might arise from an optimality condition in some iterative cosmology like CNS.

It is, as you say, mathematics. And therefore I have quoted arivero introduction


Then we solve for \alpha, the fine structure constant, in HdV second equation (http://physicsforums.com/showthread.php?t=43862) (*),


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


The term in parenthesis, well, is sort of a first order correction.

Now you have a longer equation representing successive approximations.
I would like to see this written out more explicitly.

And by the way, the value I have at hand for alpha is
0.007 297 352 533(27)

and the value which you get for alpha is
0.007 297 352 568 653...

In case you want to look at Smolin's article in that Universe or Multiverse book, the preprint is
http://arxiv.org/abs/hep-th/0407213

I do not think that your equation is necessarily in conflict with an iterative evolutionary model (although such equations in isolation, like a mysterious man dressed only in a raincoat, may scandalize and outrage propriety)

Here is the abstract on hep-th/0407213

Scientific alternatives to the anthropic principle
Lee Smolin
Comments: Contribution to "Universe or Multiverse", ed. by Bernard Carr et. al., to be published by Cambridge University Press.

"It is explained in detail why the Anthropic Principle (AP) cannot yield any falsifiable predictions, and therefore cannot be a part of science. Cases which have been claimed as successful predictions from the AP are shown to be not that. Either they are uncontroversial applications of selection principles in one universe (as in Dicke's argument), or the predictions made do not actually logically depend on any assumption about life or intelligence, but instead depend only on arguments from observed facts (as in the case of arguments by Hoyle and Weinberg). The Principle of Mediocrity is also examined and shown to be unreliable, as arguments for factually true conclusions can easily be modified to lead to false conclusions by reasonable changes in the specification of the ensemble in which we are assumed to be typical.
We show however that it is still possible to make falsifiable predictions from theories of multiverses, if the ensemble predicted has certain properties specified here. An example of such a falsifiable multiverse theory is cosmological natural selection. It is reviewed here and it is argued that the theory remains unfalsified. But it is very vulnerable to falsification by current observations, which shows that it is a scientific theory.
The consequences for recent discussions of the AP in the context of string theory are discussed."

[marcus]

\alpha^{1/2}\equiv Ae^{-{\pi^{2}/4}}

This generates the following series A:

A = 1+{\alpha \over (2\pi)^0 }(1+{\alpha \over (2\pi)^1 }(1+{\alpha \over (2\pi)^2 }( 1 + ........

or separated:

A = 1+{\alpha \over (2\pi)^0 }+{\alpha^2 \over (2\pi)^1 }+{\alpha^3 \over (2\pi)^3 } + ........



If I apply a step of Middleschool algebra to your equation, what I get is:

e^{\pi^{2}/4}\alpha^{1/2} = 1+{\alpha \over (2\pi)^0 }+{\alpha^2 \over (2\pi)^1 }+{\alpha^3 \over (2\pi)^3 } + .......}


e^{\pi^{2}/4} = \alpha^{-1/2} +{\alpha^{1/2} \over (2\pi)^0 }+{\alpha^{3/2} \over (2\pi)^1 }+{\alpha^{5/2} \over (2\pi)^3 } + .......}

apologies for tinkering like this, just want to see how it looks when kneaded a bit

so let's imagine that instead of the number 137.0359991 (or whatever it is believed to be now) we want its square root 11.7062376... and let us define
a function

F(X) = X +{X^{-1} \over (2\pi)^0 }+{X^{-3} \over (2\pi)^1 }+{X^{-5} \over (2\pi)^3 } + {X^{-7} \over (2\pi)^6 }....

So then if we solve the equation

F(X) = e^{\pi^{2}/4}

then according to you, HdV, the solution X will turn out to
be this famous universal number 11.70623762...
which so far has only been determined by experimental measurement.

heh heh

shocking

:smile:

[Hans de Vries]

And by the way, the value I have at hand for alpha is
0.007 297 352 533(27)

and the value which you get for alpha is
0.007 297 352 568 653...


I did use the NIST value of 0.007 297 352 568 (24)

Nist web page for alpha (http://physics.nist.gov/cgi-bin/cuu/Value?alph|search_for=fine)

Regards, Hans

[arivero]

Mathematics used for the Fine Structure constant formula.

Yes it should be called Mathematics rather...

Hmm the kind of mathematics of Ramanujan (http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Ramanujan.html), perhaps

[Nereid]

Stop, stop (I mean the verb, not the SUSY particle)... the relations I was referring to, for neutrino -thus leptonic- mixing angles, are still considered good taste in the real world. Of course, your mileage can vary.Apologies :redface:

Neutrino (and other lepton) mixing angles are, as you say, not at all like numerology, and coming at ratios from a good theoretical direction is of course a worthwhile exercise, in very good taste.

Wrt to your other post, I did some playing around a while ago, and found that it's extraordinarily easy to get some 'cool' numbers, to within 1%. For example, you can always change a really big number to something between 0 and 10 (or maybe 20 or so), simply by taking a log (natural or otherwise). \pi^2 is pretty close to 10, and various x-n, where x is a cool integer or trancendental, make a nice toolkit for fine tuning (+ or -, integer powers, etc). It's only slightly more difficult to create things that look like they're the first couple or three terms of a series. Which is why I think it's important to demand a prediction before any proposal is treated seriously. Or perhaps there's too much diamond dust (http://www.physicsforums.com/showpost.php?p=323591&postcount=36) in my drink.

[marcus]

I did use the NIST value of 0.007 297 352 568 (24)

Nist web page for alpha (http://physics.nist.gov/cgi-bin/cuu/Value?alph|search_for=fine)

Regards, Hans

I see that your value is more up-to-date, and more accurate than what I was using. Thanks.

[arivero]

Wrt to your other post, I did some playing around a while ago, and found that it's extraordinarily easy to get some 'cool' numbers, to within 1%
I partly agree, but with a couple of caveats. First, that even if it is easy, it has some merit, as I have never seen extraordinary precision in most of the cranks webpages. Some of them even alter the value of the constants to fit into their theories! Some 'cool' relationships can be always found, of course. Larsson, and indirectly Lubos, brought (http://groups.google.com/groups?hl=es&lr=&ie=UTF-8&selm=24a23f36.0408072208.cf75b1d%40posting.google. com)a couple of famous ones:

1/alpha = pi + pi^2 + 4*pi^3 = 137.0363038
m_proton / m_electron = 6*pi^5 = 1836.118109

TL attributes the second one to Lubos, but I have read it attributed to Feynman himself, just as an example of the kind we are discussing, random relationships. It could be interesting to know the origin of the first one to alpha, as it is a variant of KdV idea.

(Incidentaly, the third part of Wilczek "Plannk mountain" article touchs a bit the relationship between Planck scale and proton scale)

Secondly, I find in the first set of de Vries' relations a bit more of symmetry that usual. The form (q^1 - q^-1) is very geometrical, and it is very appealing that it happens between muon and tau, somehow marking the end of the number of generations. The fact of being able to find the same format for the quotient between first and second generation is also amazing as a coincidence. It lets you a lot of freedom to manipulate the relationship, for instance you can write a pair of equations
{m_e m_\tau^2 \over m_\mu^3}=e^\pi, \;
{m_e m_\tau^3 \over m_\mu^4}=e^{\frac1\pi}

The appeal of this format is, again, that it does not give trivial room to "predict" more generations.


PS: The Mikanata Smirnov paper I was referring before is hep-ph/0405088. They refer to a conjectured relation between Cabibbo angle and the quotient tau/muon

[marcus]

Alejandro, and anyone else interested, have any of us checked the full HdV equation to see if the approximation is as good as he says?

With all respect to HdV, sometimes people make numerical error and it is good to have an independent verification.

In fact i tried to check it with my calculator and perhaps I did something wrong but I did not get such a good approximation as his post led me to expect So my calculation may be at fault. I would appreciate if someone else try it.

Here is my calculation

The inverse alpha number is 137.0359991 and I will assume that we want a formula for its square root 11.7062376...

Let us define this function

F(X) = X +{X^{-1} \over (2\pi)^0 }+{X^{-3} \over (2\pi)^1 }+{X^{-5} \over (2\pi)^3 } + {X^{-7} \over (2\pi)^6 } +....

HdV post leads me to expect that if we solve for X in the equation

F(X) = e^{\pi^{2}/4}

then the solution X will turn out to be 11.70623762...

Conversely, if I plug in this value for X then I should get (to a good approximation) the exp(pi^2/4) number, which is 11.79176...

So I expect

F(11.70623762...) = 11.79176...

But I didn't get very close to this. BTW I was avoiding roundoff error by using 137.0359991 to build even powers of 11.70623762, so there is not the most obvious explanation. Can anyone help?

[marcus]

Oh oh,
I redid the calculation, using inverse alpha = 137.03599911, which I get from NIST website.
And using as the square root the approximation 11.70623762

and what do you think?
when I plug these into the function F(X) defined here, then I get 11.79176139.

But this is also equal, to the limit of the accuracy of my hand calculator, the exp(pi^2/4) number, which is also 11.79176139, or so my calculator says.

such coincidences are rare, maybe I am still doing something wrong, or the calculator battery is low. or I am dreaming this. Please someone else check the calculation

...
...
Here is my calculation

The inverse alpha number is 137.03599911 and I will assume that we want a formula for its square root 11.70623762...

Let us define this function

F(X) = X +{X^{-1} \over (2\pi)^0 }+{X^{-3} \over (2\pi)^1 }+{X^{-5} \over (2\pi)^3 } + {X^{-7} \over (2\pi)^6 } +....

... if I plug in this value for X then I should get (to a good approximation) the exp(pi^2/4) number, which is 11.79176139...

So I expect

F(11.70623762...) = 11.79176139...

... BTW I was avoiding roundoff error by using 137.03599911 to build even powers of 11.70623762....

Here is the form of F(X) I used in the calculation

F(X) = X(1 +{X^{-2} \over (2\pi)^0 }+{X^{-4} \over (2\pi)^1 }+{X^{-6} \over (2\pi)^3 } )

this way all the powers of X are even and one can use 137.03599911, reducing the amount of arithmetic, then at the end there is one multiplication by 11.70623762

[Hans de Vries]

It's OK :smile:

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]

It's OK :smile:

X = 11.7062376154

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

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]

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?

I did checked it before I posted :smile:

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

Regards, Hans

[marcus]

so when did you first see this?
...

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]

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.

Schön :smile:

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 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

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 :shy: ) :smile:

Regards, Hans

[marcus]

Fun, isn't it?

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]

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...

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]

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.


Yep, I know from the ArXiV :frown: , 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:

3^2 + 4^2 = 5^2

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:
.
.
3^2 \ ln(m_e) \ \ \ + \ \ \ (4^2+n)\ ln(m_\tau) \ \ \ = \ \ \ (5^2+n)\ ln(m_\mu)
.
.

We then look at the value of n. We do find n \approx 1.00086
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, ln(m_\tau) 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
\{ m_e, m_\mu, m_\tau \} \rightarrow \{ xm^y_e, \ xm^y_\mu, \ xm^y_\tau \} 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.


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

I can see your conflict; your first set of formulae and this third one have different assignations for the quotient of logarithms,
{\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}
so surely at least one if them is a random coincidence.
(EDITED: Or, perhaps, there is some happy circunstance (http://www.clowder.net/hop/pi2ovr12.html) where an expression happens during some series expansion of the another.
(EDITED again: in fact, using the expansion \pi^2/6=\zeta (2)=\sum_n 1/n^2, its first term is 10/18, ie 9+1 \over 17+1 so both expresions are no so far away: the second formula can be seen, at least, as the first term of the expansion of {\pi^2-1-1/2 \over 2\pi^2-3-1/2}. 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:

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


Now if and only if the term \pi-1/\pi was exact then all three
formulae would be within experimental range. Now it isn't but the
\pi-1/\pi 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:


\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


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

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

\tanh \xi=v/c ____\exp \xi=Doppler Ratio

The Doppler Ratio now becomes \pi interestingly enough (for
blueshift) and 1/\pi (for redshift) corresponding with a speed
v/c (of rotation?) The term \pi-1/\pi 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 |\exp( )|^2 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:

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

with:

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

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, \nu, e, u, d and the tetrad of relativistic coordinates t, \rho, \theta, \phi. 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 (http://arxiv.org/abs/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]

TL attributes the second one to Lubos, but I have read it attributed to Feynman himself, just as an example of the kind we are discussing, random relationships. It could be interesting to know the origin of the first one to alpha, as it is a variant of HdV idea.


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 (http://groups.google.com/groups?hl=es&lr=&ie=UTF-8&selm=43opuv%24c5n%40sparcserver.lrz-muenchen.de) in the net give it as folklorical knowledge, sometimes related to a couple of footnote-quoted papers (http://quantumfuture.net/quantum_future/Born_reciprocity.htm) 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 (http://arxiv.org/abs/hep-th/9302030), 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",
\alpha=(9/8 \pi^4)(\pi^5/2^4 5!)^{1/4}=1/137.03608
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]

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.

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
:smile:

[arivero]

I found finally the Mp/Me=6 pi^5 estimate. It appears in the Physical Review,

Friedrich Lenz "The Ratio of Proton and Electron Masses" Phys. Rev. 82, 554 (1951) (http://prola.aps.org/abstract/PR/v82/i4/p554_2)
so neither Motl nor Feynman.

Bayley and Ferguson (http://crd.lbl.gov/~dhbailey/dhbpapers/newrel.pdf) adscribe it to I.J. Good (http://www.stat.org.vt.edu/holtzman/IJGood/CV_IJG.pdf), "On the masses of Proton, Neutron and Hyperons", Journal of the Royal Naval Science Service, v 12, p 82-83 (1957), an article which is not included in the standard biblography of this author. But it is true that Good worked on this topic (http://www.statslab.cam.ac.uk/~grg/books/hammfest/9-jg.ps) and published some comments about it. The bibliography gives
Some numerology concerning the elementary particles or things, JRNSS 15 (1960) 213
The Scientists Speculates: An Anthology of Partly-Baked Ideas (London, 1962)
The proton and neutron masses and a conjecture for the gravitational constant. Phys Lett A 33, 6 (1970) p 383-384

As it happens, Gustavo R. Gonzalez-Martin (http://arxiv.org/abs/physics/0405094), who wrote me earlier this year (and I ignored him), gives all the correct references. I believe he uses a volume quotient very in the spirit of Wyler, so it is not strange he has done the right research of earlier work.

Note finally a funny coincidence: both De Vries and F. Lenz got his names as a famous heritage, of older scientists. But note that while HdV seems to exist in the network today, F. Lenz did not published anything in the Phys Rev except for that three-lines letter (hmm, it seems there is a contemporary F. Lenz publishing in the Z. Naturforsch.)

[Hans de Vries]

.
.
This one is the most “physical” yet. It’s also the simplest
and most accurate expression involving lepton mass ratios.
(fully within experimental range and exact to 0.0000073%)
.
.
.
\frac{d_{crad}}{d_{spin}} \ \ = \ \ \frac{m_e}{m_\mu} \ (1 \ + \ \ a \ \frac{m_e}{m_\tau} \ ) , \ \ \ \ \ \ \ \ \ \ \ \ \ where \ \ a = \frac{1}{\frac
{1}{2}(1+\frac{1}{2})} = \frac{4}{3}
.
.
It relates the ratio of two “classical distances” with the
lepton mass ratio’s. Since distances scale (inversely)
proportional to mass we may expect the distance ratio
to be directly proportional to the lepton masses as well.
So far so good.


The distance ratio is dependent only on the fine structure
constant and consequently can be calculated with a very
high precision:

1/206.68905011 (69) _ dist. ratio from fine structure const.
1/206.689035 (19) ___ dist. ratio calculated with mass ratios

1/206.7682838 (54) __ electron/muon mass ratio
1/3477.48 (57)_______ electron/tau mass ratio

The electron/muon ratio already equals the distance
ratio to within 0.038% A simple correction term using
the electron/tau ratio then brings it to 0.0000073%



The distance ratio is the same for all spin ½ particles.
The classical distances are:


dcrad:___ Classical (Electron) Radius
=====================================
A lepton and anti-lepton spaced at this distance,
(which is inversely proportional to their mass),
have a (negative) potential energy which is equal
to their rest mass energy.

dspin:___Classical (Electron) Spin ½ Orbital
=====================================
A lepton and anti-lepton spaced at this distance,
(which is inversely proportional to their mass),
and orbiting at the frequency corresponding to their
rest mass, have an angular momentum of a spin ½
particle: \sqrt{(\frac{1}{2}(1+\frac{1}{2}))} \ \hbar


It turns out that the velocity at which the leptons
would have to orbit here is equal for all spin ½ particles.
It’s a dimension less constant when written as v/c and
the solution of the following equation:

\frac{v^2/c^2}{\sqrt{1-v^2/c^2}} \ \ = \ \ \sqrt{\frac{1}{2}(1+\frac{1}{2})}

Where the right hand term corresponds to the angular
momentum. When solved it gives:

\frac{v}{c} \ \ = \ \ v_{spin\frac{1}{2}} \ \ = \ \ 0.754141435281767 \ \ = \ \ \sqrt{\frac{1}{8}\sqrt{57} - \frac{3}{8}}

The relation between the distance ratio and the fine
structure constant is:

\frac{d_{crad}}{d_{spin}} \ \ = \ \ \frac{1}{2}\alpha / v_{spin\frac{1}{2}}

So that we can write exact to within 0.0000073%

\alpha \ \ = \ \ 2 \ v_{spin\frac{1}{2}} \ \ \frac{m_e}{m_\mu} \ (1 \ + \ \frac{1}{\frac{1}{2}(1+\frac{1}{2})} \ \frac{m_e}{m_\tau} \ ) \ \


The following CoData 2002 values were used so you
can test it yourself. (Don’t forget to apply a factor of
\frac{1}{\sqrt{1-v^2/c^2}} to the mass when calculating the angular
momentum)


fine structure const 0.07297352568 ____ 0.00000000024
Planck constant ____ 6.626 0693e-34 ___ 0.0000011e-34 J s
speed of light _____ 299792458 ________ (exact) m/s
electron mass ______ 9.1093826e-31 ____ 0.0000016e-31 kg
speed of light _____ 299792458 ________ (exact) m/s
electric constant __ 8.854187817e-12 __ (exact) F/m
electric charge ____ 1.60217653e-19 ___ 0.00000014e-19 C
clas_electron_rad __ 2.817940285e-15 __ 0.000000028e-15 m



These "classical distances" are loaded with theoretical problems.
The classical electron radius has been used to try to cut of
the electric field close to the electron in the hope to avoid
the infinity problem of the 1/r^2 field: Doesn’t work.

The spin angular momentum is much to high to fit into any
guess of the electron size. The angular orbit here is much
larger then electron radius! The electric force is more then
40 times higher then the centrifugal force to allow such an
orbit, et-cetera.

However, as a side remark, I've been looking at not so very
different scenarios were it seems that it may be possible to
bring the spin back into classical EM field theory without the
use of any new physics.


Regards, Hans

[arivero]

However, as a side remark, I've been looking at not so very
different scenarios were it seems that it may be possible to
bring the spin back into classical EM field theory without the
use of any new physics.

Well, for sure spin is a non-classical parameter. It is measured as integer or half-integer multiples of the Planck Constant. So If you have a classical theory with spin, you have a classical theory predicting Planck Constant. And my, that's really new physics.

I am giving a glance to Gonzalez-Martin estimates, as well as Wyler's. At different decades, they postulated the use of quotient of volumes as a mean to determine mass ratios. It does not seem as accurate as this thread work.

[arivero]

1) Hans, the thing that worries me about your last try is the dissappearance of the logarithmic expressions. It seems to be thus radically different.

2) An history of numerological approaches to the fine structure constant is contained in
H. Kragh, "Magic Number: A Partial History of Fine-Structure Constant", Arch Hist Exact Sci 57 (2003) p 395-431
According copyright, private emailing of the paper, not massive, is allowed. So if someone wants to have a read of it, please tell me at my standard email address, arivero@... (where ...=unizar.es)

[Hans de Vries]

1) Hans, the thing that worries me about your last try is the dissappearance of the logarithmic expressions. It seems to be thus radically different.

We have another situation now. We relate the mass ratio's
with two "classical physical quantities" In fact these quantities
are about the two most elementary you can think of.

The 1st relates the rest mass to the electric field energy.

The 2nd relates the rest mass to the spin 1/2 ang. momentum.

It's remarkable that the ratio between these two equals the
electron/muon mass ratio to within 0.038%

It's even more remarkable that a simple correction term using
the other (electron/tau) mass ratio produces the right result
to within 0.0000073%! And all within experimental accuracy:


___________\frac{d_{crad}}{d_{spin}} \ \ = \ \ \frac{m_e}{m_\mu} \ (1 \ + \ \ \frac{4}{3} \ \frac{m_e}{m_\tau} \ )


The only coefficient (4/3) is equal to the inverse square of
the spin 1/2 angular momentum value:

___________________\frac{1}{\frac{1}{2}(1+\frac{1} {2})} = \frac{4}{3}

Further, it's remarkable (and good news) that the ratio is
directly proportional to the mass ratio's. Both quantities are
distances and distances are supposed to behave that way.

For instance the classical electron radius is 206.7682838 times
larger than the classical muon radius, exactly the same ratio as
their masses have (but reversed).

These classical radii have another long forgotten property:
If something would rotate around this radius with a frequency
corresponding to the rest mass frequency then the speed
comes out to be c times alpha. This is true for the electron,
muon and tau.

The classical radii can be found in the Codata list. The other
value however has no public status. One has to solve a few
equations to get it. I called it a "classical value" because
probably nobody with a background in QFT or just QM in general
would ever bother to calculate it!

It simply looks which orbit a particle, with a certain rest mass
and a rotation frequency corresponding with that rest mass,
needs to have in order to give it the spin angular momentum
of a spin 1/2 particle.


Regards, Hans

[arivero]

Renormalization group running of the coupling constant seems to inspire some of the formulae for alpha. While the real coupling is sensitive to the particle content and it should break down at the GUT scale, these theories prefer to work with some naive exponential coupling from the Planck scale, where they impose a simple value of alpha at these scale, say 2pi or 1 or infinity. So Nottale formula,

M_P/m_e = e^{\frac38 \alpha^{-1}}

has alpha going down from infinity to 1/137.41.

By aumenting the complexity of the formula, the "prediction" can be more precise. Recently an anonymous nicknamed "Quantoken" has suggested
(M_P/m_e)^2=2 \pi \alpha^{-1} e^{\frac23 \alpha^{-1}}
which gives alpha a value near 2 pi at the planck scale, and a "prediction" of 1/137.07

I don't what to do of these formulae. My current opinion, as I said above, is that they work because they are kind of approximations to the running constant formulae. I had some curiosity because the factors 1/3 and 3/8 in the exponents differ a 1/24, so I wondered if another formula with such factor could be proposed. But to compare both formulae is not useful here, because the exponentials are nearly parallel.


EDITED: Probably Nigel (http://members.lycos.co.uk/nigelbryancook/)'s electrogravity theory (http://nigelcook0.tripod.com/) falls also in the family here described. The key is to notice that any reference to Newton constant is, at the same time, a reference to the Planck mass scale.

[Hans de Vries]

Magnetic Anomalies and Mass Ratios


I found this numerical relation intriguing as well:
Taken into account that there are only a handful
of mass-ratios to play with.


1 \ \ + \ \ \frac{m_\mu}{m_Z} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \ \ 1.001158692 (27)

1 \ \ + \ \ \frac{m_\mu}{m_Z} \ \ + \ \ \frac{m_e}{m_W} \ \ = \ \ 1.001165046 (30)


We recognize the magnetic moments of the leptons:


1.001159652187 ____ Electron’s Magnetic Moment
1.001158692(27) ___ 1 + mμ/mZ

1.0011659160 ______ Muon’s Magnetic Moment
1.001165046(30) ___ 1 + mμ/mZ + me/mW


The small term me/mW doesn’t do too bad either in
bridging the difference between the electron's and
muon’s (normalized) magnetic moments:


0.000006263813 ____ Difference of Magnetic Moments
0.000006353(3) ____ me/mW

The weak contributions are very small in the normal
anomaly calculations. For the electron they are
simply neglected. For the muon they are smaller than
the hadronic vacuum polarization terms.

At this scale one would expect to see nothing but QED.


Regards, Hans


PS: The CODATA mass values used:

W mass = 80.425___(38) GeV
Z mass = 91.1876__(21) GeV
e mass = 0.51099892(4) MeV
μ mass = 105.658369(9) MeV

[arivero]

This seems to be a case for "add 1 effect", as I.J. Good, above quoted, explains. The approximation becomes more impressive by counting from 1.00XXX... that from 0.00XXX...

Still, it could be pretty standard physics. The corrections to the magnetic moment are no very far away from formulae of this kind, with some powers of the quotient of masses etc. I'll check my QED.


EDITED: a recent review of muon magnetic moment calculations:
http://xxx.unizar.es/abs/hep-ph/0411168

[arivero]

The weak contributions are very small in the normal
anomaly calculations. For the electron they are
simply neglected. For the muon they are smaller than
the hadronic vacuum polarization terms.
At this scale one would expect to see nothing but QED.


Yep you are right here, we only expect some corrections depending on alpha. Perhaps it is coincidence, perhaps a fine tunning mechanism for a whole summation of diagrams.

By the way, while checking this I have got a clue of where Nottale could have got his 3/8 coefficient from. While he claims to relate it to angular prediction from GUT, a older apparition of 3/8 appears associated to the infrared-catastrophic term of the traditional calculation for the anomalous moment. For instance Sakurai' formula 4.462 contains
{\alpha \over 3 \pi}{q^2 \over m_e^2} (\ln {m_e\over \lambda_{IR}} -\frac38)
So perhaps some argument about the control of the infrared catastrophe up to planck mass scale could be retorted somehow to justify Nottale's formula.

Obviusly this 3/8 term was very apparent in all the old books when doing this calculation. I had seen it also in formulae 10.3 of Aitchison, who in turn refers to Bjorken Drell sect 8.6 and pg 172--6 for the infrared problem.

[Hans de Vries]

These classical distance ratio's, R_f for fermions and R_b
for bosons are realy full of surprices!! The whole list for
sofar:

1)

The ratio R_b/R_f of the two is equal to the m_W/m_Z ratio!
(to within 0.063% or sigma 1.2)


\cos \theta_W \ \ = \ \ \frac{m_W}{m_Z} \ \ = \ \ \frac{R_b}{R_f} \ \ = \ \ \frac{2 \beta_b/\alpha}{2 \beta_f/\alpha} \ \ = \ \ \frac{\beta_b}{\beta_f}

where \beta_f, \beta_b are the classical spin 1/2 and spin 1 velocites.
This would mean that the electro-weak mixing angle would be a
numerical constant! with a physical geometric origin rather than
an arbitrary symmetry breaking parameter.


2)

Rf is equal to the muon-electron mass ratio to within 0.038%

R_f^{-1} \ \ =\ \ \frac{m_e}{m_{\mu}}

which improves to 0.0000073% if we add a correction term
involving the third lepton like this:

R_f^{-1} \ \ = \ \ \frac{m_e}{m_\mu} \ (1 \ + \ \ \frac{4}{3} \ \frac{m_e}{m_\tau} \ )

1/206.6890501 first expression
1/206.7682987 second expression
1/207.7682838 mass ratio

3)

We can do something similar for the tau-muon mass ratio:

\beta_b R_f^{-1/2}}\ \ =\ \ \frac{m_{\mu}}{m_{\tau}}

which is accurate to within 0.090%. This becomes 0.000040%
when we add the following correction term with the other lepton:

\beta_b R_f^{-1/2}}\ \ =\ \ \frac{m_{\mu}}{m_{\tau}} \ (1 \ + \ \ \frac{3}{16} \ \frac{m_e}{m_\mu} \ )

1/16.80305 first expression
1/16.81829 second expression
1/16.8183 mass ratio

----
----

We did see that the clasical velocity can be defined as:

“The velocity of a mass with spin s rotating on an orbit
with a frequency corresponding to its rest mass and an
angular momentum \sqrt{ s(s+1}\ \hbar"

One gets a general solution for the 'classical velocity' of spin s:


\beta_s \ \ \ \ \ \ = \ \ \ \ \sqrt{\sqrt{\ \ s(s+1) \ \ + \ \ ( \frac{1}{2}
\ s(s+1) \ )^2 } \ \ - \ \ \frac{1}{2} \ s(s+1) }

Which solutions are dimensionless constants, independent
of the mass of the particle. The values of the common spins
are given below:


spin 0.0: __ 0.00000000000000000000000000000000
spin 0.5: __ 0.75414143528176709788873548859945
spin 1.0: __ 0.85559967716735219296923576621118
spin 1.5: __ 0.90580479773844104117525862119228
spin 2.0: __ 0.93433577808377694874713811004304
spin inf: __ 1.00000000000000000000000000000000




Weinberg’s Electro-Weak mixing angle becomes a dimension-
less constant as well and is given in the \sin^2 \theta_W form as:


\sin^2 \theta_W
\ \ \ \ = \ \ \ \ 1 \ - \ \frac{\beta^2_f }{\beta^2_b}
\ \ \ \ = \ \ \ \ 0.22310132230086634541466926662604

\sin^2 \theta_W
\ \ \ \ = \ \ \ \ 1 \ - \ \frac
{ \sqrt{\ \ \frac{1}{2}(\frac{1}{2}+1) \ \ + \ \ ( \frac{1}{2} \ \
\frac{1}{2}(\frac{1}{2}+1) \ )^2 } \ \ - \ \ \frac{1}{2} \
\ \frac{1}{2}(\frac{1}{2}+1) }
{\sqrt{\ \ 1(1+1) \ \ + \ \ ( \frac{1}{2} \ \
1(1+1) \ )^2 } \ \ - \ \ \frac{1}{2} \
\ 1(1+1) }

The usual electro-weak parameters g1 and g2 would become:

g_1^2 \ \ = \ \ e^2 \frac{\beta_b^2}{\beta_f^2} \ \ \ \ \ \ \ \ \ g_2^2 \ \ = \ \ \frac{e^2}{1-\frac{\beta_f^2}{\beta_b^2}}

where e^2 = \alpha


Regards, Hans

[Hans de Vries]

1)

The ratio R_b/R_f of the two is equal to the m_W/m_Z ratio!
(to within 0.063% or sigma 1.2)


\cos \theta_W \ \ = \ \ \frac{m_W}{m_Z} \ \ = \ \ \frac{R_b}{R_f} \ \ = \ \ \frac{2 \beta_b/\alpha}{2 \beta_f/\alpha} \ \ = \ \ \frac{\beta_b}{\beta_f}

where \beta_f, \beta_b are the classical spin 1/2 and spin 1 velocites.
This would mean that the electro-weak mixing angle would be a
numerical constant! with a physical geometric origin rather than
an arbitrary symmetry breaking parameter.


Now there is an interesting story behind this:

I was looking at this peculiar coincidence:

0.00115869 = muon / Z mass ratio
0.00115965 = electron magnetic anomaly
0.00000635 = electron / W mass ratio
0.00000626 = difference between muon and electron magnetic anomaly

Now whatever, what we can say is that the magnetic
anomaly is totally dominated by photon (spin 1) interactions
coming from the first order [itex]\alpha/2\pi[/tex] term while the difference
of the muon and electron anomaly is almost entirely vacuum
polarization interaction (spin 1/2), the result of virtual electrons
and muons. This causes an asymmetry that results in a different
anomaly for the muon and the electron which otherwise would
be the same.

when I saw this I immediately tried if Rf/Rb had any relation
with mW/mZ. To my big surprise it turned out that they were
equal!

Regards, Hans

[arivero]

0.00115869 = muon / Z mass ratio
0.00115965 = electron magnetic anomaly
0.00000635 = electron / W mass ratio
0.00000626 = difference between muon and electron magnetic anomaly

Now whatever, what we can say is that the magnetic
anomaly is totally dominated by photon (spin 1) interactions
coming from the first order [itex]\alpha/2\pi[/tex] term while the difference
of the muon and electron anomaly is almost entirely vacuum
polarization interaction (spin 1/2), the result of virtual electrons
and muons.

I see. In the first case we are having neutral particles, and the Z ratio gets a role. In the second case we have a contribution from charged particles, and the W ratio gets a role. And Z is a neutral particle, and W is a charged particle. So at least we have a logical link.

[arivero]

Some remarks from the historic literature: According H. Kragh (above quoted article), Sommerfeld definition of alpha is not very far from the quotient you are using for the rest of the electroweak thing. To be precise, the fine structure constant is defined as the quotient between wave K angular momentum, h/2pi and "limiting momentum", e^2/c. Or equivalently (H. Kragh, "the fine structure constant before quantum mechanics"), between the orbital velocity in wave K and the maximum velocity c.

(Let me recall here that the eigenvalues of electron velocity are +-c only).

Related to this we have the so-called "Sommerfeld puzzle". In the early time, before Dirac, he applied his quantization procedure to an orbiting electron plus relativistic mass correction, and well, he got the right formula for the fine structure of hidrogen. Without any reference to spin! When Dirac equation was known, time after, its only real news were to justify an arbitrary selection rule for level intensity.

Hans, I can not read Sommerfeld german, but I guess you can, so I'd suggest you to try to find his relativistic calculation. It is around 1915-1916, a couple articles in Annalen der Physik and Akad. der Wiss Muenchen Sitzungsberichte.

[zefram_c]

Related to this we have the so-called "Sommerfeld puzzle". In the early time, before Dirac, he applied his quantization procedure to an orbiting electron plus relativistic mass correction, and well, he got the right formula for the fine structure of hidrogen. Without any reference to spin! When Dirac equation was known, time after, its only real news were to justify an arbitrary selection rule for level intensity He may have got the right correction, but he did not get the total number of states right - he only got half because he didn't have spin. The Sommerfeld calculation was one of many false steps in the early days of QM, unlike the Dirac equation which actually made for progress.

[Nereid]

This is all fascinating stuff PFers!

Are you any closer to making some specific, concrete predictions? Testable ones would be nice.

[arivero]

He may have got the right correction, but he did not get the total number of states right - he only got half because he didn't have spin. The Sommerfeld calculation was one of many false steps in the early days of QM, unlike the Dirac equation which actually made for progress.
Actually, if my lecture notes are right, he got not half but double, and then he was forced to impose a selection rule to have it right. The selection rule being a change in one h unit of angular momentum, it is now clear, as you say, that it was related to a spin flip.

But again, please count that I can not read German so I am looking at third hand -or more. lecture notes.

[arivero]

Are you any closer to making some specific, concrete predictions? Testable ones would be nice.
Nereid, I don't know. I plan to redo the perturbative calculation holding also the cubic v/c term -usually wiped out, but cointaining the infrared divergence- to see if some clue comes from there. I can not tell of Hans's plans -our main contact is via this thread-. Also I am not sure what kind of predictions should we be looking for; Weinberg angle is "predicted" by Hans, but this quantity very commonly predicted in theoretical works.

Another scent to follow comes from L. De Broglie Phys. Rev. 76, 862–863 (1949) (http://prola.aps.org/abstract/PR/v76/i6/p862_1). There, inspired by Feynman "Relativistic Cut-off" results, he suggests that electron charge self-energy can be regulated by asking for additional massive spin 1 bosons interacting with the electron. He argues that the total sum of charges for these bosons must counterweight the electron charge, but I do not quite follow his argument yet.

[arivero]

Point is, that there is an small nuisance in the calculation of the magnetic anomaly: it is calculated in the non-relativistic approximation v<<c of quantum fields, but the result contains the fine structure constant, which is a genuine relativistic quantity. So perhaps -only perhaps- HdV is seeing some conjure of the rest of the electroweak structure to grant a soft landing in the non-relativistic world. On other hand, there are other phenomena at the NR limit that could cover this role instead of the HdV equations. For an instance, the change in divergence rate of lepton self-energies.

[Hans de Vries]

Actually, if my lecture notes are right, he got not half but double, and then he was forced to impose a selection rule to have it right. The selection rule being a change in one h unit of angular momentum, it is now clear, as you say, that it was related to a spin flip.

But again, please count that I can not read German so I am looking at third hand -or more. lecture notes.

Hi, Alejandro

I found this subject very well covered in the book: "The story of spin"
from Sin-Itiro Tomonaga (the QED one). Chapter one covers the pre-
Dirac models of Sommerfield, Landé and Pauli with all the splitting
levels. Landé basically improved Sommerfields "Fine Structure Formula"
to make it suitable for atoms with a larger Z. It's covered in chapter 2.

Landé 's "Erzats Model" made it to circa 1926. Tomonaga describes
were it eventually failed in chapter 2. Pauli get's very close to the
right solution but he doesn't accept the concept of spin until Thomas
handles spin relativistically and gets the factor 2 that bothered Pauli
so much.

Chapter 3 then handles Pauli's Spin theory and then finally how Dirac
derived his equation from the Klein Gordon one. How Dirac could make
the orbital angular momentum a conserved quantity by adding the
electron spin and how Dirac subsequently found the factor 2 in the
magnetic moment of the electron when he applied an external field.

Regards, Hans

[Nereid]

This is all fascinating stuff PFers!

Are you any closer to making some specific, concrete predictions? Testable ones would be nice.I'd like to amend this ... since all measurements are inexact, and since there's ample history in physics of key constants being hard to pin down with accuracy (my favourite is G (http://www.npl.washington.edu/eotwash/gconst.html)), a good way to test the numerology work here is to state unambiguously what predictions of well-observed ratios (etc) are, and point out a) what the next few digits will be found to be, and b) which best measurements today are actually 'wrong'!

A succinct, bold summary anyone?

[arivero]

I'd like to amend this ... since all measurements are inexact, and since there's ample history in physics of key constants being hard to pin down with accuracy (my favourite is G (http://www.npl.washington.edu/eotwash/gconst.html)), a good way to test the numerology work here is to state unambiguously what predictions of well-observed ratios (etc) are, and point out a) what the next few digits will be found to be, and b) which best measurements today are actually 'wrong'!

A succinct, bold summary anyone?

First let me to tell that I have my doubts about the possibility of testing numerology. A senior bayesian, quoted somewhere above, has some ideas about measuring complexity and then how likely a numerological proposal is. But my point of view is that at most numerology can signal a subyacent theory. For a famous example, the fact of the three running constants being near equal at very high scale is traditionally interpreted a a signal of an unknown GUT theory.

Said this, let me summarize the thread up to now. There are are least two or three different approaches condensed here; I have tryed to bring here all the interesting coincidences noted in the literature, and if I have failed to name any, I should be glad to head about it (here or privately). We have seen the idea of studying the logarithms of mass quotients, which seem very approachable with simple geometrical quantities. Then we have played a bit with a diversity of formulae, and then be have jumped to a very different area of coincidences with quotients of magnitudes in the electroweak group.

About precision of each coincidence, I think HdV has done a very careful effort to remark it at each step. A beyond 0.1% has been more or less systematically requested in order to consider interesting a coincidence, as well as simplicity of the formula.

[arivero]

\beta_b R_f^{-1/2}}\ \ =\ \ \frac{m_{\mu}}{m_{\tau}} \ (1 \ + \ \ \frac{3}{16} \ \frac{m_e}{m_\mu} \ )

I understood that this coefficient 3/16 is got from trial an error, and it is unrelated to angular momentum (while the 4/3 is). Am I in a mistake?

Alejandro

[Hans de Vries]

\beta_b R_f^{-1/2}}\ \ =\ \ \frac{m_{\mu}}{m_{\tau}} \ (1 \ + \ \ \frac{3}{16} \ \frac{m_e}{m_\mu} \ )

I understood that this coefficient 3/16 is got from trial an error, and it is unrelated to angular momentum (while the 4/3 is). Am I in a mistake?

Alejandro

It's indeed trial and error. I did find a note a I made about:

16/3 = \frac{(1(1+1))^2}{(\frac{1}{2}(\frac{1}{2}+1)) } \ \ \ because \ \ \ \beta_b R_f^{-1/2} \ \ \ contains \ \ \ \beta_b \beta_f^{-1/2}

but that might be too far fetched.

Regards, Hans

[Hans de Vries]

--
--
I came across this one in "Introduction to the Standard Model" from
Cottingham and Greenwood, where it is used to explain the large
meson and baryon masses compared to the quark masses.

So a quark confined in a ~1 fm core results in momentum of p = \hbar/1 fm
~ 200 MeV/c or equivalent to a mass of E = pc ~ 200 MeV, which is
much larger than the individual quark masses.

This is very different from say the electromagnetic mass model which
leads to the "classical radius of the electron" for instance. The use of
Heisenberg's uncertainty relation is of course a perfect way to obtain a
cut-off for any 1/r^2 field in order to avoid infinities.

Just like the 1/r^2 field energy models we get the relation that the
radius is inversely proportional to the mass. If we now define a
"Heisenberg radius" for the electron or any other arbitrary particle
in the above way then we see the following:


r_h = 137.03599911 \ r_0 \ \ \ \ or \ \ \ \ 3.861592678. 10^{-13} m


where r_0 is the classical (electron) radius. It relates to it with
the value of alpha. We could thus assign a "coupling constant"
of 1.0000000 to the "Heisenberg field".

Such an Heisenberg radius fits much better to the other classically
derived radii. We have these two classical velocities for spin 1/2 and
spin 1 particles defined by:

“The velocity of a mass with spin s rotating on an orbit
with a frequency corresponding to its rest mass and an
angular momentum \sqrt{ s(s+1}\ \hbar"


And one can now express our radii for fermions and bosons as:


r_f = \beta_f \ r_h
r_b = \beta_b \ r_h

with the classical velocities:

\beta_f = 0.75414143528176709788873548859945
\beta_b = 0.85559967716735219296923576621118


So these fit much better here than with the so much smaller
classical electron radius. It's also independent of the charge.
The "Heisenberg radius" itself corresponds to an orbital speed
of c when you require that the frequency equals the rest mass
frequency.

If we do the same for the classical calculation of the magnetic
moment then we get.


r_m = 1.0011596521859 \ r_h


So it's the small anomaly that tips it over the edge (the speed
would have to pass c by the same amount if not explained
otherwise)

Regards, Hans

[arivero]

--
--
I came across this one in "Introduction to the Standard Model" from
Cottingham and Greenwood, where it is used to explain the large
meson and baryon masses compared to the quark masses.

So a quark confined in a ~1 fm core results in momentum of p = \hbar/1 fm
~ 200 MeV/c or equivalent to a mass of E = pc ~ 200 MeV, which is
much larger than the individual quark masses.
Yep, I remember this argument. I believe it evolved from the other use of the distance-momentum undeterminacy, to relate the range of a virtual particle and its mass. Thus, considering that nuclear force is a short interaction about 1 fm, it was suggested to look for a massive particle around 200 MeV, which we know now to be the pion (and its family).

The argument for quarks seems more touchy: glueballs, asymptotic freedom, partons, mass gap...

The use of
Heisenberg's uncertainty relation is of course a perfect way to obtain a
cut-off for any 1/r^2 field in order to avoid infinities.

And jointly with the relativistic cut-off (which Bethe uses for the Lamb shift), they explain why QFT is less divergent than classical field theory.



If we do the same for the classical calculation of the magnetic
moment then we get.

r_m = 1.0011596521859 \ r_h


Hmm, this needs a little more elaboration; the other distances are related to forces, but what about this one?

[Hans de Vries]

The argument for quarks seems more touchy: glueballs, asymptotic freedom, partons, mass gap...


The idea of kinetic glueball energy as the main mass generator
for hadrons is not unlike this kind of Heisenberg-like mass generation
via the momentum for particles confined to a small space.

Heisenberg being behind the fermionic degeneracy force preventing
White Dwarfs (electrons) and Neutron stars (neutrons) from collapsing
further:

http://scienceworld.wolfram.com/physics/ElectronDegeneracyPressure.html

With Pauli's exclusion principle mentioned just as often for the same.....

http://hyperphysics.phy-astr.gsu.edu/hbase/astro/whdwar.html#c3

So it's a fermionic thing then with glueballs being bosons...




Hadronic mass... It's basically what attracts us to earth.

In the case of kinetic glueball energy we are then attracted to
earth mainly by massless particles not coupled to Higgs field...

For glueballs one would expect some form of field energy, as the
result of the self interaction, and something similar to potential
energy...

Well, questions enough and LHC still 3 years away!


Hmm, this needs a little more elaboration; the other distances are related to forces, but what about this one?
Just using the classical equation \mu = \frac{1}{2}evr for the magnetic moment
to obtain a radius if we require the rest mass frequency.

Feliz Navidad!

Hans.

[arivero]

hep-ph/0502200 does a regression fit of ln(m/M) for quark masses. The three points of each generation are in line, after they choose the right "\bar M. S. masses" (hmm).

Intriguingly, a fourth generation b-quark predicted with this scheme should have exactly the same mass that the Z boson.

[Hans de Vries]

.
.
.

It's interesting to organize the mass-ratio coincidences given in this thread as
generations of a coupling constant. Instead of e=\sqrt{\alpha} we will use the slightly
adapted value of \varrho = \sqrt{\alpha/2\beta_f} where \beta_f is our "classical fermion velocity" and
\alpha/2\beta_f is our "classical distance ratio"



\begin{array}{llclccll}
\mbox{Coupling Generation 4: } & m_e/m_Z & = & \beta_f/\pi & \mbox{times} & \varrho^4 & \mbox{ (Z boson) } & 1.0027295 \\
\mbox{Coupling Generation 4: } & m_e/m_W & = & \beta_b/\pi & \mbox{times} & \varrho^4 & \mbox{ (W boson} & 1.0033602 \\
\mbox{Coupling Generation 3: } & m_e/m_{\tau} & = & \beta_b & \mbox{times} & \varrho^3 & \mbox{ (tau lepton) } & 1.0012880 \\
\mbox{Coupling Generation 2: } & m_e/m_{\mu} & = & 1 & \mbox{times} & \varrho^2 & \mbox{ (mu lepton) } & 1.0003833 \\
\mbox{Coupling Generation 0: } & m_e/m_e & = & 1 & \mbox{times} & \varrho^0 & \mbox{ (electron) } & 1.0000000 \\
\end{array}


So we get very simple expressions with reasonable precision (last column)
for all known ElectroWeak masses based on the fine structure constant and
our two classical velocities \beta_f and \beta_b for spin 1/2 and spin 1 particles which
define the ElectroWeak mixing angle.



The classical velocities are defined by:

“The velocity of a mass with spin s rotating on an orbit
with a frequency corresponding to its rest mass and an
angular momentum \sqrt{ s(s+1}\ \hbar"

These are exact dimensionless constant ratio's with c given by:

\beta_f = 0.75414143528176709788873548859945 for fermions
\beta_b = 0.85559967716735219296923576621118 for bosons

From the general formula for arbitrary spin s:

\beta_s \ \ \ \ \ \ = \ \ \ \ \sqrt{\sqrt{\ \ s(s+1) \ \ + \ \ ( \frac{1}{2}
\ s(s+1) \ )^2 } \ \ - \ \ \frac{1}{2} \ s(s+1) }

We see that the ratio of the two \beta_f/\beta_b is equal to the m_W/m_Z ratio (to within
0.063% or sigma 1.2) and thus defines (as another numerical coincidence) the
electro weak mixing angle.


\cos \theta_W \ \ = \ \ \frac{m_W}{m_Z} \ \ = \ \ \frac{\beta_f}{\beta_b} \ \ = \ \ 0.8814185598789792074327801204579



All the mass ratio's are now defined here by the coupling constant alpha
and the two components that define the electroweak mixing angle.


Regards, Hans

[arivero]

Magnetic Anomalies and Mass Ratios
I found this numerical relation intriguing as well:
Taken into account that there are only a handful
of mass-ratios to play with.
1 \ \ + \ \ \frac{m_\mu}{m_Z} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \ \ 1.001158692 (27)
1 \ \ + \ \ \frac{m_\mu}{m_Z} \ \ + \ \ \frac{m_e}{m_W} \ \ = \ \ 1.001165046 (30)


Let me add that a possible next term is 1/2 (mu/W)^2


\ \ \frac{m_\mu}{m_Z} \ \ + \ \ \frac{m_e}{m_W} + \ \ \frac12 ({m_\mu \over m_W})^2 \ \ = \ \ .00116590899

to be compared with current experimental value .0011659208(6)

[arivero]

.00116590899 to be compared with current experimental value .0011659208(6)
... or to the theoretical QED-only value .0011658471 or to the theoretical electroweak value .0011658487

This is sort of a problem when aiming to fit more terms: should the expansion go towards the experimental value (which is thought to include hadronic contributions) or towards the pure electroweak value (then revealing some hidden property of the electroweak theory)?

If we opt towards the second possibility, we have the advantage of being not constrained by experimental data anymore. But the theoretical value, I read, was calculated using a Z0=91.1875 and W=80.373, then giving worse agreement. On the other hand, higgs (and quarks?) is involved in the theoretical calculation, correcting the mass of W.

Incidentally, state-of-art theoretical calculation, Phys Rev D 67, 073006 (2003), is very readable.

[arivero]

generations of a coupling constant. Instead of e=\sqrt{\alpha} we will use the slightly
adapted value of \varrho = \sqrt{\alpha/2\beta_f} where \beta_f is our "classical fermion velocity" and
\alpha/2\beta_f is our "classical distance ratio"


I supposse this adjustment is done so that the m_\mu/m_Z correction of the anomalous magnetic moment becomes exactly the first order perturbative correction \alpha/2\pi. Another possibility is to define \varrho (or an \alpha') so that the correction to the anomalus magnetic moment of electron becomes the whole experimentally known correction.

[Hans de Vries]

I supposse this adjustment is done so that the m_\mu/m_Z correction of the anomalous magnetic moment becomes exactly the first order perturbative correction \alpha/2\pi. Another possibility is to define \varrho (or an \alpha') so that the correction to the anomalus magnetic moment of electron becomes the whole experimentally known correction.

If you take the magnetic moment anomaly of the electron times 2pi as an adjusted "coupling value" instead
of alpha then \varrho becomes 1/√207.0023039 and:

\begin{array}{llclccllll}
\mbox{Coupling Generation 4: } & m_e/m_Z & = & \beta_f/\pi & \mbox{times} & \varrho^4 & \mbox{ (Z boson) } & 91187.6 & 91215.2 & 1.0003031 \\
\mbox{Coupling Generation 4: } & m_e/m_W & = & \beta_b/\pi & \mbox{times} & \varrho^4 & \mbox{ (W boson} & 80425 & 80398.8 & 1.0003257 \\
\mbox{Coupling Generation 3: } & m_e/m_{\tau} & = & \beta_b & \mbox{times} & \varrho^3 & \mbox{ (tau lepton) } & 1776.99 & 1778.73 & 1.0009848 \\
\mbox{Coupling Generation 2: } & m_e/m_{\mu} & = & 1 & \mbox{times} & \varrho^2 & \mbox{ (mu lepton) } & 105.658369 & 105.777953 & 1.0011318 \\
\mbox{Coupling Generation 0: } & m_e/m_e & = & 1 & \mbox{times} & \varrho^0 & \mbox{ (electron) } & 0.51099892 & 0.51099892 & 1.0000000 \\
\end{array}


Which gives a significant improvement for the electroweak bosons. (Last 3 columns: measured in MeV,
calculated in MeV, precision)


Regards, Hans



PS: mathematical symbols directly as characters instead of latex:

√ α β γ ψ Ψ μ τ φ π Σ Π ∫ ∂ ∞ ≈ ≠ ≤ ≥ ≡ ½ ⅓ ¼ ⅔ ¾ ⅛ ⅜ ⅝ ⅞

The're nicer in Times New Roman though:

√ α β γ ψ Ψ μ τ φ π Σ Π ∫ ∂ ∞ ≈ ≠ ≤ ≥ ≡ ½ ⅓ ¼ ⅔ ¾ ⅛ ⅜ ⅝ ⅞

[marcus]

PS: mathematical symbols directly as characters instead of latex:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ½ ? ¼ ? ¾ ? ? ? ?


Hans I dont know how to make these characters.
If you have a moment could you say how to type a few of them?

I have tried [ alp*ha ] and & alp*ha ;

removing * and spaces but dont get anything that can survive the next edit.

[arivero]

Speaking of precision, our result for the muon anomalous moment quotients to 0.99998987, 1.0000530 or 1.0000517 depending if we compare to experimental measure, QED standard prediction or electroweak standard prediction. I think our estimate relates to some leptonic-only calculation in the electroweak setup. I wonder if there is some way to cancel Adler' kind of triangles across generations.

And I am using the "honest" version of the quotient, referring only to the additive correction. Putting the 1s inside, as if it were a multiplicative correction (to the nonanomalous g=2), the quotient is 0,9999999882 and so on.

I am really furious about this being an unpublishable unscientific claim. When I think Nobel, I am not thinking prize anymore... but dynamite.

[arivero]

I have tried [ alp*ha ] and & alp*ha ;
removing * and spaces but dont get anything that can survive the next edit.

Haha! Marcus, I am also asking myself what happens with alpha. More exactly, why our mass-based formulae have lost any reference to the coupling constant alpha, while Schwinger first estimate was just alpha/2pi.

To me, some interplay of the SU(2)xU(1) triangle anomaly, generations, couplings and constants, can be suspected. And it could even be consistent with Hans's considerations on angular momentum and velocities.

But even if coincidental or unrelated (or perhaps related!), it could be time to name the recent numerology (well, halfbaked theory) of Jay R. Yablon, who has suggested that Higgs electroweak vacuum, <v>=246 GeV, should be invited to the game. For instance mass of tau= alpha times <v>. Or some other things involving powers of alpha and the sin of weinberg angle. Hmm.

[Hans de Vries]

Hans I dont know how to make these characters.
If you have a moment could you say how to type a few of them?

I have tried [ alp*ha ] and & alp*ha ;

removing * and spaces but dont get anything that can survive the next edit.

Hi, Marcus.

It seems you've got only the characters in the basic Ascii set. The first 256
characters of the Western. ISO 8859-1 characterset. You might need to set
the character encoding of your Browser to this character set under the
"View" menu.

I got the other via Word (Insert, Symbol) You should be able to copy and
paste them from here.


α β γ δ ε ζ η θ ι κ λ μ ν ο π ρ ς σ τ υ φ
Γ Δ Θ Λ Ξ Π Σ Φ Ω ∏ ∑ ∩ ∆ ∂ ∫ ∂ ∞
√ ≈ ≠ ≤ ≥ ≡ ½ ⅓ ¼ ⅔ ¾ ⅛ ⅜ ⅝ ⅞


Regards, Hans

[arivero]

http://prola.aps.org/abstract/PRD/v17/i7/p1854_1
m_e / m_\mu=N \alpha / \pi \sin^2 \theta_W where N is a pure number of order 1 which depends on the specific model

[Hans de Vries]

http://prola.aps.org/abstract/PRD/v17/i7/p1854_1
m_e / m_\mu=N \alpha / \pi \sin^2 \theta_W where N is a pure number of order 1 which depends on the specific model

"Calculating the electron mass in terms of measured quantities"
December 1977 Barr, S. M.; and Zee, A.

is this the A.Zee as the A.Zee in "QFT in a Nutshell"?

I don't have access to APS unfortunately.

Regards, Hans.

[arivero]

is this the A.Zee as the A.Zee in "QFT in a Nutshell"?

I guess so. But from a glance to preprints and bibliography, it seems that the model builder is Barr.

In any case, it is just one of the older (pre-GUT, practically) articles on extended symmetries, and its value here is only to point out that a good bunch group theoretical models contain predictions for mass quotients.

[arivero]

Let me add that a possible next term is 1/2 (mu/W)^2
\ \ \frac{m_\mu}{m_Z} \ \ + \ \ \frac{m_e}{m_W} + \ \ \frac12 ({m_\mu \over m_W})^2 \ \ = \ \ .00116590899
to be compared with current experimental value .0011659208(6)

This travel to madness goes from surprise to surprise. Considering that in this sum we had first order term for the anomalous moment of electron, then a first order term for the difference between an. m. of electron and muon, and then a second order term for the an. m. of electron; thus, it seems, it is natural to look for another second order term. I have been trying terms with m_e^2 with no success. But, to my astonishment, it is possible to use terms on m_e m_\tau!!!!

(examples: mm/mz+me/mw+.5*(mm*mm/mw/mw)-.5*(me*mt/mz/mw) = .00116584708, the pure QED value. Or mm/mz+me/mw+(1/2)*(mm^2/mw^2)+(1/6)*(me*mt/mz^2)= .0011659272,
nearer to the experimental value than the uncorrected term)

A minor problem with all the second order terms is that it is not clear when to use W and when to use Z; the election at so small corrections is almost an aesthetic issue.

Also other adjustments are possible if we consider separately the difference between moments and the electron moment.

[Hans de Vries]

A minor problem with all the second order terms is that it is not clear when to use W and when to use Z; the election at so small corrections is almost an aesthetic issue.

I see, It's the mZ in the first term which ultimately determines the precision.

Regards, Hans

[arivero]

I see, It's the mZ in the first term which ultimately determines the precision.
Yep Hans. Even if your angular momenta devices could have a role somewhere. Your hat is always surprising.

Let me -marcus style :smile:- restate the equations as a pair. First, our experimental input is
mw=80525 (+-38)
mz=91187.6 (+-2.1)
me=0.51099892 (+-0.00000004)
mm=105.658369 (+-0.000009)
mt=1776.99(+0.29-0.26)
ae=0.001159652187 (+-0.000000000004) (ie +-4 10^-12)
am=0.0011659208 (6)

The pdg 2004 has 0.0011659203 (+-0.0000000007), surely the previous run of the g-2 experiment.

Lets take separately ae, and the difference d=am-ae=.0000062686; so at first order our comparisions are
0,00115869=mm/mz
0,001159652187=ae
with a quotient 0.99917
and
0,000006345=me/mw
0,0000062686=d
quotient 1.01219

Note I am shortening decimal precision to follow approximately the experimntal precision of W and Z measuremente. Wel, now lets enter our second order corrections:

(1/2) (mm/mw)^2 =.000000861 is to be added to the first comparation, s that now we have

.00115955=mm/mz+(1/2) (mm/mw)^2
.001159652187=ae

and now put for instance (1/2) (me*mt/mw^2)=.00000007001 to be substracted, so that we compare

.000006275=me/mw - (1/2) (me*mt/mw^2)
.0000062686=d=am-ae

The quotients are respectively .999911 and 1.00102. It is possible to reverse the pair and to use the experimental anomalous magnetic moments to calculate mass of W and mass of Z.

The total sum .001165825 fails the experimental point 0.0011659208 (6) by a quotient .999918, ie less than 0.01%. This total formula, although, is already a bit ugly:
a_\mu=
{m_\mu \over m_Z}+ {m_e \over m_W}+
\frac12 {m_\mu^2-m_e m_\tau \over m_W^2}=.001165825

[arivero]

On the other hand, the theoretical electroweak value (some quark loops?, but no hadronic correction) is .0011658487 and the pure QED value is 0011658471. The corresponding quotients are .999979 and .999981

For the first order approximation, the quotient was in any case (including against the experimental) about .9993, which was already less than 0.1%

[arivero]

What about the anomalous magnetic momentum of the tau lepton? Well, it is not a measured quantity, but at least the QED value has been calculated by Samuel, Li and Mendel to be 0.0011732 (a previous calculation by Narison, giving 0.0011696, was retired by its author, in favour of the Samuel et al. result). The electroweak contribution, scaled from the muon calculation, should increase the value up to 0.0011737. This info comes from Phys Rev Letters v67 p 668 and a further erratum at v 69 p 995.

To look for a "first order" correction, we should find a <0.1% match for the QED differences a_\tau-a_\mu=.0000073 or a_\tau-a_e=.0000135, or to the corresponding EW values, .0000005 higher.

I will not try to land on these values using Hans' generation mechanism. Instead, let me see if we can do something from phenomenological values. A first inspection shows us that any quotient using the mass of muon, or using the mass of tau, will ask for a denominator mass fairly above 10 TeV. So we are left with the electron mass.

Now, to fit the electron mass to values .0000073 .0000078 .0000135 .0000140, we need respective masses of 69.9, 65.5, 37.8, 36.4 GeV. Looking at the catalogue of forgotten experimental deviations, we had a mass around 40-45 GeV back in 1984 (Nature 12 July 1984, p. 310) and, hey, a mass around 68-70 GeV in 1999 from the L3 collaboration: hep-ex/9909044, hep-ex/0009010, hep-ex/0105057. Nowadays, the statistics of this L3 excess has been reevaluated so the final deviation is under three sigmas; but it could be said that it joins the 115 GeV event in waiting further clarification at LHC.

The putative assignment for the 68 GeV particle was a charged scalar. If we have also faith in the 115 GeV range for the Higgs, we can give -speculate- a prediction:

a_\tau=
{m_\mu \over m_Z}+ {m_e \over m_{W^+}}+{m_e \over m_{H_{L3}^+}}
+{m_\mu^2-m_e m_\tau \over m^2_H_0}

[arivero]

hep-ph/9810512 is already an up-to-date source for the tau data, and besides it has an interesting remarck for the muon: "In models where the muon mass is generated by quantum loops... under very general asumptions, the induced \delta a_\mu is given by \delta a_\mu= C m_\mu^2 / \Lambda^2 where Lambda is the scale of 'New Physics' responsible for generating m_\mu". (C is a constant of order unity)

So our square terms are not so rare, after all.

According Brodsky et al. hep-ph/0406325, we are meeting a challenge from Feynman in the 12th Solvay conference, where he asked: "Is there any method of computing the anomalous moment of the electron which, on first approximation, gives a fair approximation to the \alpha term and a crude one to \alpha^2; and when improved, increases the accuracy of the \alpha^2 term, yielding a rough estimate to \alpha^3 and beyond"

[Kea]

A = 1+{\alpha \over (2\pi)^0 }+{\alpha^2 \over (2\pi)^1 }+{\alpha^3 \over (2\pi)^3 } + ........

The series converges straightforward to reproduce the value
of the fine structure constant exact in all its digits:

after term 0: 0.0071918833558268
after term 1: 0.0072972279174862
after term 2: 0.0072973525456204
after term 3: 0.0072973525686533
f.s. constant 0.007297352568.(+/-24)


This actually agrees after term 2 already, no? Why not develop Alejandro's truncated formula then?

Cheers
Kea
:smile:

[arivero]

This actually agrees after term 2 already, no? Why not develop Alejandro's truncated formula then?
(Just to fix attributions, and lacking Bourbaki here, the trunc. formula is also from Hans). I supposse the idea was to enter inside the 1sigma level, +-24, as a proof of concept. 568-24 is 544, so the previous iteration is only in the border. It has sense if you think that nowadays a lot of people translates sigmas to "Confidence Levels".

[arivero]

I am also asking myself what happens with alpha. More exactly, why our mass-based formulae have lost any reference to the coupling constant alpha, while Schwinger first estimate was just alpha/2pi.

But even if coincidental or unrelated (or perhaps related!), it could be time to name the recent numerology (well, halfbaked theory) of Jay R. Yablon, ...


Lets keep calm. The whole webpage of Yablon (http://home.nycap.rr.com/jry/Personal Web Page.htm) is about a method to get the masses of the three leptons as a perturbative expansion in alpha. Really a methodology of this kind could explain our results: For some reason, the alpha expansion generating lepton mass comes to coincide with the expansion of the vertex diagram, and we can absorb Schwinger-like terms (or whole series of them!) into lepton masses.

From the seventies, there is a whole market of theories having perturbative lepton masses, usually via the so called "horizontal symmetries". Here we are looking for a theory based on expansion of the vertex, using the symmetry breaking scale of the electroweak group, and having no new coupling constants. Pity there is not a catalogue of "beyond SM theories".

[Hans de Vries]

A bit about Jay's numerical series.

Jay, in his enthusiasm, is bringing this out in the open at the moment he
discovers something. Leaving the reflections for later. If he plays a little bit
more with this then he will find out how careful one must be with numerical
coincidences.

At one hand it makes sense in a QFT world to develop a series based on a
coupling constant. On the other hand it's always possible to transform a decimal
number to any other number system. For instance a number system based on
1/e = 11.72... instead of 10.

If one needs 5 terms to approximate the 0.511.. MeV mass of the electron
to 0.509 MeV then there's no more "numerical coincidence" value left in it.
Jay will find out that there are many other combinations that will lead to
equal or better results.


Still the approach makes sense and the most intriguing hint may be the anomalies.
To bring them up one more time:

0.00115869 = muon / Z mass ratio
0.00115965 = electron magnetic anomaly
0.00000635 = electron / W mass ratio
0.00000626 = difference between muon and electron magnetic anomaly


Regards, Hans.

[Hans de Vries]

This actually agrees after term 2 already, no? Why not develop Alejandro's truncated formula then?

Cheers
Kea
:smile:

Hi Kea,

What I looked for was a series with a single generating principle. In a sense
such a series is simpler than a formula with three terms.

Regards, Hans

[Hans de Vries]

Hi, Alejandro

first this:

Mass = inversely proportional to Square Root of the Giro Magnetic Ratio

Simply from the classical magnetic moment: \mu \ = \ I A \ = \ e f_0 \pi r^2 where:
I is the current, A the area enclosed by the orbit, e the charge and for f0 we
take the rest-frequency of the particle. Masses are inversely proportional
to size so the r2 factor leads to the inverse square root dependency. We
get fractions which are about half the size:

ge = 1.001159652187 --> 1/√ge = 0.9994206777
gμ = 1.0011659208__ --> 1/√gμ = 0.9994175489


Now Look at this:

\begin{array}{lcl}
1 - \frac{1}{2} \frac{m_{\mu}}{m_W} & = & 0.999420654 \\
1/\sqrt{g_e} & = & 0.999420678 \\
\mbox{the absolute error} & = & 0.000000024 \\
\end{array}


\begin{array}{lcl}
1-\frac{1}{2} \frac{m_{\mu}}{m_W}-\frac{1}{2} \frac{m_e}{m_Z} & = & 0.999417477 \\
1/\sqrt{g_{\mu}} & = & 0.999417549 \\
\mbox{the absolute error} & = & 0.000000072 \\
\end{array}


The absolute error becomes 40 times smaller for the electron case!


Regards, Hans

[Jay R. Yablon]

If one needs 5 terms to approximate the 0.511.. MeV mass of the electron
to 0.509 MeV then there's no more "numerical coincidence" value left in it.
Jay will find out that there are many other combinations that will lead to
equal or better results.

Hans, this is Jay. I am not fixed for sure on the 5 terms for the electron, but what is very nice about this result is that the electron mass is then characterized in leading order by only four-loop terms and that all five of the four loop terms go into the electron mass. So, the fact that every one of these terms has a common Feynman diagram interpretation as a four-loop term seems to be based on some physics beyond coincidence and beyond picking and choosing terms

Still the approach makes sense and the most intriguing hint may be the anomalies.
To bring them up one more time:

0.00115869 = muon / Z mass ratio
0.00115965 = electron magnetic anomaly
0.00000635 = electron / W mass ratio
0.00000626 = difference between muon and electron magnetic anomaly


Our friend arivero has now got me looking at these anomalies using the terms I have developed. I will let you know what I find.

Jay

[arivero]

A small corrected improvement. While mm/mZ is 99.76% of the Schwinger correction, the following approximation
{m_\mu\over m_Z}={\alpha \over 2 \pi} - 0.5 ({\alpha\over\pi})^2
is 99.9982% accurate

Still I hope the 0.5 comes from addition of multiple Feynman diagrams, then being approximate itself. I am afraid than an exact -1/2 coefficient in second order QED comes not from a finite term, but associated to the infrared divergence.

[arivero]

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

Now whatever, what we can say is that the magnetic
anomaly is totally dominated by photon (spin 1) interactions
coming from the first order \alpha/2\pi term while the difference
of the muon and electron anomaly is almost entirely vacuum
polarization interaction (spin 1/2), the result of virtual electrons
and muons.

Hmm but at second order of alpha, there is a vacuum polarisation term due to pairs electron positron, and it is not negligible. It is 0.015687 \alpha^2/\pi^2, ie .0000000846.

Er, wait... lets supposse this term is not in your ratio. Add it!
0.0011586922+.0000000846=.0011587768

Ok it does not seem very much of an improvement. But if we add also my term (1/2) (mm/mW)^2 we have
0.0011586922+.0000000846+.0000008608=.0011596376
to be compared with experimental 0.0011596521. No bad.

I would conclude that our series (well, two terms) on masses does not approach to the experimental magnetic moment, but to the two loop QED (or full electroweak, does not matter) vertex correction, excluding the vacuum polarisation loop. On first examination, it seems that this loop is recovered when we introduce the product of electron and tau masses, but I have not examined the expansion for this anomalous moment.

Just for the record, the two loop correction for electron is 0.5 a/pi - 0.3284794 (a/pi)^2. In our case, disregarding the vacuum polarisation amounts to a second term coefficient -.3441636 We have
a_e^{\mbox{2qed-v.p.}}=.00115955280
{m_\mu \over m_Z}+ \frac12 {m_\mu^2 \over m_W^2}=.001159553
agreement about, well, dammn, it is already inside the experimental error for Z0... if you want, respective to central values it is of Z and W, it is 99.99997%.

If one feels bad about having the 2 m_W^2 in the denominator, you can always use the square of an unknown neutral mass about 114 GeV... the LHC start will wait some time yet.

[Hans de Vries]

Hmm but at second order of alpha, there is a vacuum polarisation term due to pairs electron positron, and it is not negligible. It is 0.015687 \alpha^2/\pi^2, ie .0000000846.


That's just the value you want it to have! Do you have you any more
data like that. Like the explicit value of the same term in the Muon?
Really interesting would also be the relation with the muon/electron
mass with these terms in the formula's (which are known analytically.
I suppose that we can ignore the same second order terms for virtual
muon-anti muon pairs for the time being.


0.000 006 263 813 : Difference between electron and muon anomaly
0.000 000 084 639 : First vacuum polarization term of the electron
----------------------------------------------------------------------
0.000 006 348 452 : Sum of the vacuum polarization terms (the above)

0.000 006 353 732 : (+/-3 000) mass ratio of electron and the W boson.
0.000 000 002 998 : uncertainty from the Z mass


Regards, Hans.

[arivero]

That was the point. We have hit experimental precision in both expressions.

Sunday night I mailed you a separate email, the bibliography tells where to find such terms. You would not like me to write the closed expression for the "vacuum polarised" terms of the muon, it is about four lines in the Phys Rev. Moreover, the next order terms are also relevant.

Alejandro

[arivero]

0.000 006 263 813 : Difference between electron and muon anomaly
0.000 000 084 639 : First vacuum polarization term of the electron
----------------------------------------------------------------------
0.000 006 348 452 : Sum of the vacuum polarization terms (the above)

0.000 006 353 732 : (+/-3 000) mass ratio of electron and the W boson.
0.000 000 002 998 : uncertainty from the Z mass


Hmm, I had tried the same with the data truncated at two loops for electron and three for the muon and I got a somehow weaker result. It seems one needs to add the three loop data for the electron, but it is a mess because then we have three new diagrams to exclude.

[Hans de Vries]

The other interesting observation is that the missing term
becomes equal in both cases to within experimental value.


0.00115877693 : mu/mZ + VP2
0.00115965218 : electron anomaly
0.00000087525 : missing

0.00116504602 : mu/mZ + me/mw (+ VP2 - VP2)
0.0011659208_ : muon anomaly
0.00000087478 : missing

0.00000000047 : missing1 - missing2
0.00000002668 : uncertainty due to Z (cancels if missings are subtracted)
0.00000000299 : uncertainty due to W mass


So there may be a single missing term.


Regards, Hans

PS: VP2 = 0.00000008464 : First vacuum polarization term of the electron
which is a second order term.

PPS: I'm not entirely sure if the term 0.00116504602 : mu/mZ + me/mw (+ VP2 - VP2)
should indeed not include VP2.

me/mW - VP2 = difference between muon and electron anomaly.
mu/mZ + VP2 = all self energy terms + first vacuum polarization term of the electron anomaly

[arivero]

Making it in a reverse way: as the missing term is already in the electron anomalous moment, assume it is sort of square of the first term. So it is mu^2/X^2. Solve for X:
X=sqrt(mu^2/.000000875015)=112.95 GeV

The quantity is interesting in two ways. We suspect of a neutral scalar H0 at 115 GeV, and it could have this role. But also mw*sqrt(2) is 113.87 GeV, so we can use the W particle, which was my first attempt for the missing term. Lacking of more theory, both are equally suitable: values up to 114.5 GeV are covered by the Z indeterminacy. The first has the advantage of not using an arbitrary 1/2 coefficient and it is neutral as the Z, but it has not been discovered (yet?), the second is an already discovered particle but we have used it for the "vacuum polarised graphs", and it is surprising to have it here too, even if squared.

[arivero]

For the sake of completeness, references. The industry of calculation of the anomalous moment seems to be based in Cornell, around a veteran named T. Kinoshi-ta. Other group does exist in North America around A. Czarnecki

http://arxiv.org/abs/hep-ph/9810512, from Czarnecki and Marciano, is the main entry point for the calculation up to order alpha^4. It is regretly a short preprint and it does not separate loop by loop, so one is referred to more detailed bibliography, which is not in the arxiv :frown: :frown:

The five diagrams for order alpha^2 appear well separated in Levine and Wright Phys. Rev. D 8, 3171-3179 (1973) http://prola.aps.org/abstract/PRD/v8/i9/p3171_1. I got from here the specific value we were using above.

Also some sums for 40 diagrams of order alpha^3 are presented there. Note that of these, 12 diagrams are vacuum polarisation loops, amounting perhaps to a contribution of 0.37 (alpha/pi)^3

The alpha^2 "polarisation loop", depending of the mass quotient of the external and internal lepton, is studied both analytic and numerically by Li Mendel and Samuel, Phys. Rev. D 47, 1723-1725 (1993) http://prola.aps.org/abstract/PRD/v47/i4/p1723_1

Also Samuel Li and Mendel provide a calculation of tau lepton, Phys. Rev. Lett. 67, 668-670 (1991) http://prola.aps.org/abstract/PRL/v67/i6/p668_1 up to order alpha^3. For this lepton, the contributions for quarks are already noticeable at this scale.

[Jay R. Yablon]

Also Samuel Li and Mendel provide a calculation of tau lepton, Phys. Rev. Lett. 67, 668-670 (1991) http://prola.aps.org/abstract/PRL/v67/i6/p668_1 up to order alpha^3. For this lepton, the contributions for quarks are already noticeable at this scale.

Alejandro"

Do you and Hans have any close fits relating the tau magnetic moment with any lepton-to-electroweak-boson ratio?

Jay.

[arivero]

Alejandro"
Do you and Hans have any close fits relating the tau magnetic moment with any lepton-to-electroweak-boson ratio?


Hmm the answer past yesterday was yes, the answer today is more towards no. On one side the quantity m_e m_\tau/m_W^2 in of the right order of magnitude to do further corrections in our calculations, but we do not need it anymore, giving the uncertainty in the mass of Z. On other hand, and more concretely answering your question, the difference between anomalous moment of mu and tau can only be covered with a new quotient m_e/m_{X^+}, and the mass of the new X+ particle should be around 68-70 GeV. At these energy range, the LEP2 presented an slight "statistical" deviation, but no particle :frown:

[arivero]

mw=80525 (+-38)

80425

So I was getting a discrepance between calculations at home and calculations at work.

[Jay R. Yablon]

Dear Alejandro and Hans:

I just posted to my web site http://home.nycap.rr.com/jry/FermionMass.htm, a Gordon-like decomposition of the Fermion mass into a term due to electromagnetic charge, and a term due to magnetic moment. This is still an early working draft.

I hope this can help you in your efforts by providing a covariant field theory context for your efforts to characterize the magnetic moments. I know that your efforts have helped me recognize that consideration of magnetic moments is likely to be a critical aspect of what I am attempting to do.

Best,

Jay.

[arivero]

I just posted to my web site http://home.nycap.rr.com/jry/FermionMass.htm, a Gordon-like decomposition of the Fermion mass into a term due to electromagnetic charge, and a term due to magnetic moment. This is still an early working draft.

Probably Hans is also exploring this way, but I am not so optimistic about a direct connection; perhaps a semiclassical effect, could be. But even that is strange to manage. To me, it seems more as if the symmetry breaking mechanism of the electroweak group (and its vacuum value) were needing of the lepton masses is some misterious way.

From our quadratic formulae we can get rather intriguing equations. For instance this one:

{m_\tau\over m_Z} + {m_\mu\over m_W}=
{m_\tau\over m_\mu} a_\mu^{s.e.} + {m_\mu\over m_e} a_\mu^{v.p.}


Where a_\mu^{s.e.},a_\mu^{v.p.} are the self-energy and vacuum polarisation parts of the muon anomalous magnetic moment; note that the v.p. part depends internally of lepton mass quotients, while the s.e. is mass independent, in QED (in the full electroweak theory new dependences appear).

[arivero]

Some of the development of the thread uploaded at http://arxiv.org/abs/hep-ph/0503104

[Hans de Vries]

Hmm,

I was doing something else and ran just incidently into this one:

\sqrt{ \ 2 \ \frac{m_V}{m_Z} \ \frac{m_{\tau}}{m_e}} \ = \ 137.038 (12)

mV is the vacuum expectation value of 246.22046 GeV (according to Jay)
The biggest uncertainty is from the tau mass.


Regards, Hans


\ \ \alpha \ \ \ = 1/137.03599911
mV = 246220.46
mZ = 91187.6 (+-2.1)
mτ = 1776.99(+0.29-0.26)
me = 0.51099892 (+-0.00000004)

[arivero]

vacuum expectation value of 246.22046 GeV (according to Jay)
The biggest uncertainty is from the tau mass.
Also this vacuum should have a high uncertainness. I wonder where did Jay got so many digits from.

[Hans de Vries]

Also this vacuum should have a high uncertainness. I wonder where did Jay got so many digits from.

It's basically "one over the square of" : The Fermi Coupling constant 1.16637 (1) times sqrt(2)

So it should be 246.2206 (11)

Regards, Hans.

[arivero]

That's just the value you want it to have! Do you have you any more data like that.
Funny, I have found a paper which, while aiming toward other goals, also uses a variant of QED discarding vacuum polarisation terms in order to get more explicit formulas. It happens in section 3 of
Phys. Rev. 95, 1300-1312 (1954) (http://prola.aps.org/abstract/PR/v95/i5/p1300_1)
which is titled "3. EXAMPLE: QUANTUM ELECTRODYNAMICS WITHOUT PHOTON SELF ENERGY PARTS".

The authors are a M.Gell-Mann and a F.E.Low, from Illinois.

[Hans de Vries]

Funny, I have found a paper which, while aiming toward other goals, also uses a variant of QED discarding vacuum polarisation terms in order to get more explicit formulas. It happens in section 3 of
Phys. Rev. 95, 1300-1312 (1954) (http://prola.aps.org/abstract/PR/v95/i5/p1300_1)
which is titled "3. EXAMPLE: QUANTUM ELECTRODYNAMICS WITHOUT PHOTON SELF ENERGY PARTS".

The authors are a M.Gell-Mann and a F.E.Low, from Illinois.

There are related follow-ups:

Quantum Electrodynamics at Small Distances,
Baker, Johnson, 1969.
http://prola.aps.org/abstract/PR/v183/i5/p1292_1


Quantum Electrodynamics Without Photon Self-Energy Parts
S. L. Adler and W. A. Bardeen 1971
http://prola.aps.org/abstract/PRD/v4/i10/p3045_1


Regards, Hans

[arivero]

Baker and Johnson have actually a whole forest of papers.

Following this review of classic bibliography, I also come across to the formula

{M_0^2 \over M_V^2}= {3 \over 2 \pi} \alpha

which is world-famous, but I was unaware. Regretly it is about a single scalar charged particle, not a fermion, and the quotient against the vector boson gets this square dependence. The formula was found in
Radiative Corrections as the Origin of Spontaneous Symmetry Breaking by Sidney Coleman and Erick Weinberg, Phys. Rev. D 7, 1888-1910 (1973) (http://prola.aps.org/abstract/PRD/v7/i6/p1888_1) They even have a generalisation to SU(3)xU(1).

Incidentaly, one of these authors was contacted about our preprint 0503104 (http://arxiv.org/abs/hep-ph/0503104), here is his statement: Given the current state of knowledge in the field, speculations concerning approximate numerical coincidences such as the ones you discuss do not constitute the degree of substantial new physics that is required for publication

[CarlB]

Hans, I am impressed. It's too many digits for the accuracy.

If the standard model is an effective field theory from a deeper level, then the fine structure constant should be calculated from a series in that deeper (unified) level.

One supposes that such a unified field theory would be extremely strongly coupled (otherwise it'd be visible), and that our usual perturbation methods would fail, and therefore that calculations would be impossible. However, this is a way out of this.

Our usual experience with bound states is that when two particles are bound together, we expect the bound state to have a higher mass than either of the particles contributing to it. Of course the total mass is a little less than the sum of the masses, E=mc^2 and all that. But if the particles are extremely strongly bound, then the mass of the bound state could be negligible compared to the mass of either free particle.

For example, the mass of a free up quark is unknown, but all indications are that it would require a lot of energy to make one, so its mass should be extremely large. Present experimental limits say it should be much larger than the mass of a proton.

Now doing quantum mechanics in such a nonperturbational region might seem impossible, but this is not necessarily the case. In fact, infinite potential wells make for simple quantum mechanics problems. Perturbation theory may not be needed or appropriate.

Looking at QFT from the position eigenstate representation point of view, the creation operators for elementary particles have to work in infinitesimal regions of space. Suppose we want to do physics in that tiny region. The natural thing we'll do is to use a Gaussian centered at the position.

One way of representing the potential energy between two objects bound by extreme energies is to suppose that they each stress space-time (in the general relativistic stress-energy manner), but in ways that are complementary. Thus the sum makes for less stress to space-time than either of the separate particles.

In that case, if we represent the stress of each particle with a Gaussian, we end up deriving a potential energy that, for very low energies, works out as proportional to the square of distance. This is the classic harmonic oscillator problem, and the solution in QM is well known without any need for perturbation theory.

Now your series for the fine structure constant used a Gaussian form. Coincidence? I doubt it.

My suspicion is that this is a clue. My guess is that there is a unified field theory with equal coupling constants for everything, and that the strong force is strong because it has fewer coupling constants multiplied together in it. That would have to do with the factor in the exponential. This all has to do with my bizarre belief that even the leptons are composite particles.

Carl Brannen

As an aside, I once decided to see if the sum of inverses of cubes \zeta(3) = \sum 1/n^3 could be summed similarly to how the sum of inverses of squares or fourth powers could be summed. I wrote a C++ program that computed the sum out to 5000 decimal places (which requires a lot of elementary mathematics as the series converges very very slowly), and then did various manipulations on it to search for a pattern.

The most useful thing to know, when trying to determine if a high precision number is rational, is the series obtained by taking the fractional part of an approximation and inverting it. It's been over a decade, but I seem to recall that the name for this is the "partial fraction expansion".

[arivero]

Attached (!) I have drawn the whole elementary particle spectrum at logarithmic scale. Honoring Yablon, I have put a 1/137 line also between the tau and the electroweak vacuum.

There are four clearly distinguished zones, usually called the electromagnetic breaking, the chiral breaking, the hadronic scale (or SU(3) gap) and the electroweak breaking scale. SO I have encircled them with green rectangles.

(EDITED: If you are using the Microsoft Explorer, you will need to expand the jpg to full screen or almost)

[Jay R. Yablon]

Hi Alejandro:

Thanks for the recognition.

While I have stayed off the board for awhile I have not been inactive. I am working on a paper with a well-known nuclear theorist in Europe. I won't get into details yet, but I think you all will find it interesting once we are ready to "go public."

Best,

Jay.

[arivero]

Still more references, if only for a future observer/reader of the thread...

Stephen L. Adler (of the anomaly fame) pursued during the seventies an eigenvalue condition in order to pinpoint the value of the fine structure constant:

Short-Distance Behavior of Quantum Electrodynamics and an Eigenvalue Condition for alpha, Phys. Rev. D 5, 3021-3047 (1972) (http://prola.aps.org/abstract/PRD/v5/i12/p3021_1) (Spires (http://www.slac.stanford.edu/spires/find/hep/www?j=PHRVA,D5,3021))

PS: specially for new readers, please remember we are trying an at-a-glance view of this thread (and other results) in the wiki bakery http://www.physcomments.org/wiki/index.php?title=Bakery:HdV

[arivero]

We missed this one. It is a published formula, its exactitude is one of the better ones in the thread, and a very short review can be checked online as hep-ph/9603369 (http://arxiv.org/abs/hep-ph/9603369)
m_e+m_\mu+m_\tau=\frac 23 (\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau})

Or, if you prefer: The vector having by components the square roots of leptonic masses helds and angle of 45 degrees with the vector (1,1,1). If you use experimental input, the angle is 45.0003 plus/minus 0.0012 degrees, according Esposito and Santorelli.

Jay can be specially interested on this property, because it derives almost trivially from asking trace preservation both of the mass matrix M and its square root (the 45 degrees being the maximum possible aperture without negative eigenvalues in square root of M).

EDITED: http://ccdb3fs.kek.jp/cgi-bin/img_index?198912199 is the first paper from Koide. (even before the improved values of tau?) From other references, it seems that the research was framed in the general context of "democratic family simmetry" ( a degenerated mass matrix filled with ones, so that when rotating to eigenvectors only the third generation is naturally massive).

[CarlB]

Here's another lepton mass formula, one that gives all the mass ratios, and involves the Cabibbo angle:

Consider the matrix:

M = \left( \begin{array}{ccc}
\sqrt{2} & e^{i\delta} & e^{-i\delta} \\
e^{-i\delta} & \sqrt{2} & e^{i\delta} \\
e^{i\delta} & e^{-i\delta} & \sqrt{2} \\ \end{array} \right)

where \delta = .222 = 12.72 degrees, is the Cabibbo angle. Let

r = e^{2 i \pi/3}, s = 1/r.

Then the eigenvectors and eigenvalues of M are approximately:

(1,r,s), \sqrt{m_e / 157},
(1,s,r), \sqrt{m_\mu / 157},
(1,1,1),\sqrt{m_\tau / 157},

with the lepton masses in MeV.

Note that the sum of the eigenvalues of the matrix M are 3 \sqrt{2}, and the square of this is 18. And M^2 has diagonal entries of 4, so a trace of 12. Since 12/18 = 2/3, the relationship between the lepton masses and their square roots already discussed on this thread is automatically provided by the choice of the diagonal value as \sqrt{2}

The reason for using the Cabibbo angle is that the Cabibbo angle gives the difference between the quarks as mass eigenstates and the quarks as weak force eigenstates. If you believe, as I do, that the quarks and electrons are made from the same subparticles (which I call binons and which correspond to the idempotents of a Clifford algebra), then it is natural that the Cabibbo angle enters into the mass matrix for the leptons. Note that the Cabibbo angle is small enough that its sine is close to the angle.

Carl

[Hans de Vries]

Here's another lepton mass formula, one that gives all the mass ratios, and involves the Cabibbo angle:

Consider the matrix:

M = \left( \begin{array}{ccc}
\sqrt{2} & e^{i\delta} & e^{-i\delta} \\
e^{-i\delta} & \sqrt{2} & e^{i\delta} \\
e^{i\delta} & e^{-i\delta} & \sqrt{2} \\ \end{array} \right)

where \delta = .222 = 12.72 degrees, is the Cabibbo angle. Let

r = e^{2 i \pi/3}, s = 1/r.

Then the eigenvectors and eigenvalues of M are approximately:

(1,r,s), \sqrt{m_e / 157},
(1,s,r), \sqrt{m_\mu / 157},
(1,1,1),\sqrt{m_\tau / 157},

with the lepton masses in MeV.

Carl

Carl, this is really very interesting.

If one takes 0.222222047168 (465) for the Cabibbo angle then you
get the following lepton mass ratios:

\frac{m_\mu}{m_e}\ =\ 206.7682838 (54)

\frac{m_\tau}{m_e}\ =\ 3477.441653 (83)

\frac{m_\tau}{m_\mu}\ =\ 16.818061210 (38)

Which is well within the experimental range:

\frac{m_\mu}{m_e}\ =\ 206.7682838 (54)

\frac{m_\tau}{m_e}\ =\ 3477.48 (57)

\frac{m_\tau}{m_\mu}\ =\ 16.8183 (27)

The first one is no surprise because we solved the Cabibbo angle
to get this result, but the second surely is! This means that your
formula is predictive to 5+ digits!

Furthermore, the values given also are exact for Koide's formula
which is due to the \sqrt 2's on the diagonal of your matrix as you said:

m_e+m_\mu+m_\tau=\frac{2}{3} (\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau})^2

1+206.7682838+3477.441653=\frac{2}{3} (\sqrt{1}+\sqrt{206.7682838}+\sqrt{3477.441653})^2

Regards, Hans

P.S. The scale factor would be 156.9281952 (123)

[arivero]

The first one is no surprise because we solved the Cabibbo angle
to get this result, but the second surely is! This means that your
formula is predictive to 5+ digits!

Hmm??? Ah, you mean that you have used Carl's matrix to find a value for the Cabibbo angle, do you? It is actually very surprising to be able to find this angle from leptons; it is the mixing parameter ... of quarks!

Incidentally, the first model that originated Koide's formula also had a prediction for this angle, it was

\tan \theta_c={\sqrt 3 (x_\mu - x_e) \over 2 x_\tau - x_\mu -x_e}

with x_l\equiv \sqrt{m_l}- Ref http://prola.aps.org/abstract/PRL/v47/i18/p1241_1 Note that the context was similar to Carl's argument, ie a composite model for quarks and leptons.

Surely this result can be combined into Carl's matrix to get Koide's democratic mass matrix (http://prola.aps.org/abstract/PRD/v28/i1/p252_1,http://prola.aps.org/abstract/PRD/v39/i5/p1391_1)
which is more of less similar to Carl's but having \delta=0

[arivero]

M = \left( \begin{array}{ccc}
\sqrt{2} & e^{i\delta} & e^{-i\delta} \\
e^{-i\delta} & \sqrt{2} & e^{i\delta} \\
e^{i\delta} & e^{-i\delta} & \sqrt{2} \\ \end{array} \right)

As a matter of notation, I'd call such matrix "A^\frac 12" or something similar, given that its eigenvalues relate to square roots of masses.

[ohwilleke]

Keep up the good work! This is really facinating reading.

[Hans de Vries]

Hmm??? Ah, you mean that you have used Carl's matrix to find a value for the Cabibbo angle, do you? It is actually very surprising to be able to find this angle from leptons; it is the mixing parameter ... of quarks!


I actually liked the close proximity to 2/9 of \delta\ = 0.2222220.. without
the necessity of the latter actually being the angle used to systemize
the quark eigenstate mixing observed in weak hadronic decay.

After all the goal is to reduce the number of arbitrary parameters :^)

Regards, Hans

[CarlB]

Hans, thanks for supplying more accuracy. The fit to the lepton masses can't be any better or worse than the 2/3 formula fit. From there, my guess is that the Cabibbo angle fit is at least partly based on chance. That is, if you look at the CKM matrix, the other numbers aren't showing up. Maybe that's due to suppression from the high quark masses, and the Cabibbo angle being close is due to the SU(3) associated with the up, down and strange quarks having roughly similar masses.

By the way, I wonder what happens to the fit if you use \theta_C = 2/9. One wonders if there is a relationship between the masses of the up, down and strange, and the deviation of the formula from 2/9. If there were, then maybe the same relationship would clear up the rest of the CKM matrix.

Arivero, thanks for the references. I'm not associated with a university, so if it doesn't show up on a google search I have to drive over to the University of Washington to look it up. I'll go by there this afternoon, I can hardly wait. And you're right that the formula isn't really any more predictive than the 2/3 formula, except for the Cabibbo angle coincidence.

If one assumes that the leptons and quarks, along with their various families and colors, are made from various combinations of just two subparticles each of which come in three equivalent colors, which I've been calling the |e_r>, |e_g>, |e_b>, |\nu_r>, |\nu_g>, |\nu_b>, then the mass matrix shown is a result of a branching ratio for interactions of the form:

|e_r> -> |e_r> 50%
|e_r> -> |e_g> 25%
|e_r> -> |e_b> 25% (and cyclic in r, g, b)

with the Cabibbo angle providing the phases for the last two interactions. That is, a particle of a given color has a 50% chance of staying that same color, and 25% chances of switching to one of the other two colors. This has a direct interpretation in terms of angles, if you break it up into left and right handed parts.

I should mention what all this has to do with Higgs-free lepton masses.

Consider the Feynman diagrams (in the momentum representation) where each vertex has only two propagators, a massless electron propagator coming in, and a massless electron propagator coming out, and a vertex value of m_e. When you add up this set of diagrams, the result is just the usual propagator for the electron with mass. Feynman's comment on this, (a footnote in his book, "QED: The strange theory of matter and light"), is that "nobody knows what this means". Well the reason that no one knows what it means is because these vertices can't be derived from a Lorentz symmetric Lagrangian.

But what the above comment does show is that it is possible to remove the Higgs from the standard model (along with all those parameters that go with it), if you are willing to assume Feynman diagrams that don't come from energy conservation principles.

You can take the same idea further by breaking the electron into left and right handed parts, and then assuming that the Feynman diagrams always swap a left handed electron travelling in one direction with a right handed electron travelling in the opposite direction. This preserves spin and is similar to the old "zitterbewegung" model of the electron.

If you take the same idea still further, and assume the electron, muon and tau are linear combinations of subparticles, as described above, then the lepton mass formula is natural to associate with the Cabibbo angle. By the way, the standard model includes a Higgs boson to take away the momentum from the left handed electron reversing direction, but its otherwise the same thing.

Mass is weird because it is only the mass interaction that allows left and right handed particles to interact. That's why the Higgs bosons are supposed to be spin-0, to allow the coupling of left and right handed fermions.

The zitterbewegung model was based on noticing that the only eigenvalues of the electron under the operator that measures electron velocity are +c and -c. So the assumption was that the electron moved always at speed c. Of course having the electron suddenly reverse direction in order for the zitterbewegung to work is a violation of conservation of momentum, but the fact is that you do get the usual electron propagator out of all this.

By the way, to do these calculations, it helps to choose a representation of the gamma matrices that diagaonalizes particle/antiparticle and spin-z. That is, the four entries on the spinors will correspond to left-handed electron, right handed electron, left handed positron and right handed positron. When you do this, the mass matrix will no longer be diagonal.

There are a lot of other things that come out of this, and most of them are quite noxious to physicists. For example, in order to have the left handed electron be composed of three subparticles, the subparticles have to carry fractional spin. Like fractional charge, the idea is that fractional spin is hidden from observation by the color force. Another example is that one tends to conclude that the speed of light is only an "effective" speed, and that the binons have to travel faster. Also, the bare binon interaction violates isospin but is also a very strong force.

Convincing the physics world to simultaneously accept so many hard things to swallow is essentially impossible.

Carl

[Hans de Vries]

By the way, I wonder what happens to the fit if you use \theta_C = 2/9.

I did try this yesterday, :^)

M^\frac{1}{2} = \left( \begin{array}{ccc}
\sqrt{2} & e^{i\delta} & e^{-i\delta} \\
e^{-i\delta} & \sqrt{2} & e^{i\delta} \\
e^{i\delta} & e^{-i\delta} & \sqrt{2} \\ \end{array} \right)

With \delta \ =\ \frac{2}{9}

one gets the following eigenvalues:

3.3650337331519900946141875218449
0.82054356652318236464766296162669
0.057063387444112687143215689158409

Squaring the ratios gives the following mass ratios:

3477.4728371045985323130012254729 = m_\tau /m_e
206.77031597272938861255033931149 = m_\mu /m_e
16.818046733377517056533911325581 = m_\tau /m_\mu

Which are exact to circa one part per million.


Effectively two parameters are predicted, The first comes from
Koide's formula which was brought to our attention by Alejandro
and which you reworked into your matrix. The second comes from
the parameter \delta and which one may hope to be either a simple
mathematical constant (2/9) or another SM parameter. (Cabibbo)


Regards Hans

[CarlB]

Hans, I just realized that the branching ratios I gave, i.e. |e_r> -> |e_g> 50%, 25%, 25%, were probably incorrect. The reason is that when you compute a probability in QM you do it by taking
P = |<a|b>|^2,
and that means that the mass matrix has to be included four times, not twice. That means that the branching ratios actually are:

|e_r> -> |e_r> 66.67%
|e_r> -> |e_g> 16.67%
|e_r> -> |e_b> 16.67%

I realized that the numbers had to be wrong when I was working out the branching ratio from another point of view. This is somewhat speculative, so bear with me, please. Remember that the unusual thing about SU(3) color is that it appears to be a perfect symmetry...

Suppose that nature has a particle that travels at some fixed speed near the speed of light and exerts an extreme stress on space-time. The stress being very high says that the energy contained in the particle is very high. Nature wishes to reduces this stress by cancelling it.

Suppose that there is a hidden dimension, and the stress has sinusoidal dependence on that hidden dimension. That is, when you average the stress over the hidden dimension, you get zero, but when you integrate the square of the stress over the hidden dimension you get a number that corresponds to a very high energy per unit volume.

One way that nature could arrange to minimize the total energy of the particle is by ganging it up with another particle of the same sort, but arranging for the phase of the other particle, in the hidden dimension, to cancel the first. Thus you would compute the potential energy of the combined particles by first summing their stresses, and then integrating over all space.

Because the cancellation would depend on how far apart the particles were, this would result in a force. It turns out that the force that results from this sort of thing is, to lowest order, compatible with the usual assumptions about the color force. That is, the force is proportional to the distance separated.

The reason the calculation works out this way is quite generic. That is, any force based on minimization of a potential, with the potential having a nice rounded bottom, will be approximately harmonic. So this coincidence really doesn't mean much in and of itself.

If it weren't for the Pauli exclusion principle, nature could cancel the first with another particle travelling in the same direction. So nature instead reduces the stress by combining several particles travelling in somewhat different directions.

It turns out that if you analyze this problem from the point of view of Clifford algebra (that is, you assume that the stress take the form of a Clifford algebra), there is no way to get a low energy bound state out of two particles. Instead, you have to go to three. Details are beyond the scope of this post. So lets assume that nature combines three particles, with their phases (as determined by an angular offset in the hidden dimension) different by 120 degrees. The fact that 360 degrees divides equally into three multiples of 120 degrees gives the explanation for why SU(3) is a perfect symmetry, but the details are beyond the scope of this post.

Let's assume that the "center of mass" of the three particles travels in the +z direction. Let the red particle be offset in the +x direction, with the green and blue offset appropriately around the z-axis. The three particles travel on a cone centered around the z-axis.

Let the opening angle of the cone be \theta_b, where b stands for "binding angle". We expect that b will be as small as nature can get away with, but that it will be balanced by Fermi pressure. That is, if \theta_b is too small, the probability of the three particles being near each other goes down, and this raises the total energy.

Then the unit velocity vectors for the three particles are:
V_R = (s_b , 0,c_b)
V_G = (-s_b/2, s_b\sqrt{3}/2,c_b)
V_B = (-s_b/2,-s_b\sqrt{3}/2,c_b),
where s_b, c_b = sin, cos(\theta_b),

If the speeds of the individual particle are c', then the speed of the bound particle is c'\cos(\theta_b). If we assume that the bound particle is a handed electron, then this says that c' is faster than the speed of light by a factor of \sec(\theta_b).

In other words, the subparticles would have to be tachyons that travel at some fixed speed faster than the speed of light. There is some experimental evidence for the existence of this sort of thing. It consists of observations of gamma ray bursts. EGRET observed a double gamma ray burst with a delay of about an hour between bursts. It's called 940217 in the literature and there are plenty of theoretical attempts (failures) to explain it.

If the gamma ray burst were caused by a collection of tachyonic particles all travelling in the same direction, then a single burst of tachyons would generate time separated bursts of gamma rays as the tachyons travelled through regular matter if the regular matter was distributed into two lumps.

I made the argument that binons might be an explanation for high energy cosmic rays at the PHENO2005 meeting. There are about a half dozen good reasons for expecting exactly the odd sort of behavior seen in the Centauro events from a binon. The reasoning is beyond the scope of this post. I will soon get around to putting up a copy of the argument on the PHENO2005 website.

Anyway, the above description of a tachyon bound state would also apply to the left handed electron. Assume that the red for the left handed electron is also oriented in the +x direction, but the bound particle is travelling in the -z direction.

Then the unit vectors for the left-handed electron are:
Then the unit velocity vectors for the three particles are:
V_R' = (s_b , 0,-c_b)
V_G' = (-s_b/2, s_b\sqrt{3}/2,-c_b)
V_B' = (-s_b/2,-s_b\sqrt{3}/2,-c_b).

This means we can now compute angles between <R'|R>, and <G'|R>:

<R'|R> = s_b^2 - c_b^2,
<G'|R> = -s_b^2/2 - c_b^2,
<B'|R> = -s_b^2/2 - c_b^2,

The above give the cosines of the angles between the unit vectors. It's well known that probabilities in QM for things with an angle theta between them follow a law proportional to 1+cos(theta). After normalizing to unit probability, it turns out that the branching ratios do not depend on \theta_b. (This is also what you'd expect from relativistic length contraction in the z direction.) Instead, you get exactly the branching ratios that just happen to fit the fermion mass matrix:

P_{R'R} = 2/3,
P_{G'R} = 1/6,
P_{B'R} = 1/6,

Carl

[arivero]

M = \left( \begin{array}{ccc}
\sqrt{2} & e^{i\delta} & e^{-i\delta} \\
e^{-i\delta} & \sqrt{2} & e^{i\delta} \\
e^{i\delta} & e^{-i\delta} & \sqrt{2} \\ \end{array} \right)
where \delta = .222 = 12.72 degrees, is the Cabibbo angle. Let

The use of circulant or retrocirculant mixing matrices in order to implement a permutation symmetry between generations was abogated by Adler in the late nineties. He missed this formula of course because he wasn't interested on square roots, and besides he used the antidiagonal version of the matrix. Koide's arrived to Carls's version a bit after Adler, in hep-ph/0005137 (http://arxiv.org/abs/hep-ph/0005137), and hoping to relate it to trimaximal mixing. Intriguingly, his parametrisation in this paper does not get Cabbibo angle.

A footnote in Weinberg's The Problem of Mass points to old attempts to derive Cabibbo's angle, not only from m_d/m_s but also from pre-quark formalism, m^2_\pi/2m^2_K. Fascinating.

Carl could be interested on the preon models that motivated the formula time ago in the eighties. Someones are available in the KEK preprint server. http://ccdb3fs.kek.jp/cgi-bin/img_index?8208021


Let me note here a personal communication from Dr. Koide.

The most difficult point in my mass formula is that
the charged lepton mass term is \Delta I =1/2 if we
consider mass generation by Higgs scalars, while if
we consider that mass term is given by a bilinear form,
it means that the term is \Delta I =1.
We need something beyond Standard Model.

[CarlB]

Dear Dr. Rivero, Thanks for the Koide references. I picked up photocopies of the ones you listed yesterday.

Could you please explain your interpretation of Koide's comment on \Delta I =1/2? I have no idea what \Delta I is.

The problem with making the fundamental fermions out of a combination of a fermion and a boson is that it requires a force that isn't described, and it would imply the existence of more particles that haven't been observed. But to make them out of fermions requires either that the spins cancel, which also has problems with too many particles, or, more naturally, that the spin angular momentum of the subparticles be h-bar/6. But that violates the usual understanding of the relation between spin and statistics. On the other hand, the hiding of fractional spin by a subcolor force does remind one of the hiding of fractional charge by the color force.

There is no way that I would have come to these very radical and speculative conclusions if I'd started out by looking for a method of unifying the elementary particles. Instead, I started out by trying to understand Lorentz symmetry. So if what I've written seems insane, please understand that you're not getting the story as it unfolded to me. Instead, I'm including here only conclusions, and only those conclusions that contribute to a connected understanding.

My difficulty is that no one has enough patience to go through the reasoning I went through to get where I am now.

Suppose you discovered the unified field theory, but the theory required a modification of every assumption of both quantum mechanics and relativity. Maybe you can convince a few people to listen to an alternative interpretation of the foundations of one of those two theories, but long before they have heard your alternatives for both theories they will have concluded you are insane.

And to show equivalence to the standard model requires not only rewriting the foundations for both theories, but then to make simple but non trivial calculations in the new theory. There is not a chance in hell that you will convince anyone to sit through this.

So what you do instead is you use your understanding to find relationships between the parameters of the standard model. After I get papers written up to correspond to talks I gave at the PHENO2005 and APSNW meetings (on centauro cosmic rays and binding calculations for binons, respectively), I will start working on the fine structure coupling constant. When I'm done, I hope to have a complete derivation of the standard model from first principles. It will be impossible to get anyone to read it, but it should allow me to derive relationships between standard model parameters, and who knows, maybe then. Or perhaps new observations of the Centauro events will match my predictions.

Carl

[arivero]

Could you please explain your interpretation of Koide's comment on \Delta I =1/2? I have no idea what \Delta I is.

I guess it is about Isospin.

which also has problems with too many particles,

Personally I dislike composite models because of this; it is not easy to see how to interpret them in a realistic way without generating excessive particles at the same time.

Instead, I started out by trying to understand Lorentz symmetry

A very honourable quest.

Suppose you discovered the unified field theory, but the theory required a modification of every assumption of both quantum mechanics and relativity.
I would try to make sure that both QM and Relativity were got in the adequate limit, as well as QFT. On the contrary I would need to reproduce all the known results, a lifelong work.


long before they have heard your alternatives for both theories they will have concluded you are insane.

This is a different problem, and this thread is long enough already without philosophical discussions. As you can see, we have focused the thread on mass predictions, including analysis of the exactitude and full description of the input. I'd not like to miss focus. Perhaps in other thread.

So what you do instead is you use your understanding to find relationships between the parameters of the standard model.

Just a question here: are you claiming that your matrix above has been derived from your theory? Iff so, please give here a link to a corresponding webpage.

[Kea]

My difficulty is that no one has enough patience to go through the reasoning I went through to get where I am now.

Hi Carl

That's not for you to say. Maybe someone will have the patience. Could you give us a link, or explain some more of the fundamentals that you have in mind?

All the best
Kea
:smile:

[CarlB]

Dear Kea, Dr. Rivero is right, a long discussion really doesn't belong on this thread. At this time, I'm busily texing up a report for the PHENO2005 meeting. I should be done in a few days, and I'll link it in then.

Basically, the idea is to assume that one must generalize the Dirac propagator to form a Dirac equation that simultaneously contains multiple particles. One does this with a Clifford algebra. There are very few choices that one can make, and the choices pretty much amount to assumptions about the nature of the manifold that the Clifford algebra is defined on. (In other words, this is a "geometric algebra" variety of Clifford algebra as promoted by David Hestenes.)

Having a propagator that contains multiple particles, one derives the symmetries that the propagator must have, and sees whether these symmetries correspond to those observed in the natural world. Fitting the two together take a lot of work.

Carl

[CarlB]

> I guess it is about Isospin.

I understand now, but I doubt I understand to the extent that Koide meant. From my point of view, the problem with mass is that it is the only thing in QM that couples opposite handedness.

> I would try to make sure that both QM and
> Relativity were got in the adequate limit, as
> well as QFT.

Fortunately, I don't have to worry about relativity, as several other physicists have already worked out the details. I'm not a general relativity expert, but since there are 5 physicists publishing papers on the theory, I expect that at least one of them got it right. The GR theory goes by various names that differ according to the researcher, but a common name that seems to be coming to the fore is "Euclidean relativity". A place to start reading is Hestenes's review article, Almeida's "4DO" version, or Montanus's famous work:
http://modelingnts.la.asu.edu/pdf/GTG.w.GC.FP.pdf
http://arxiv.org/abs/physics/0406026
Hans Montanus, (1997) "Arguments Against the General Theory of Relativity and For a Flat Alternative," Physics Essays, Vol. 10, No. 4, pp 666-679.

It probably doesn't help with the physics community that Ron Hatch (engineer who developed GPS which depends on relativity calculations) supports the underlying modification of relativity (sometimes called "Lorentzian relativity"):
http://www.egtphysics.net/Gravity/Gravity.htm

> On the contrary I would need to reproduce all the
> known results, a lifelong work.

Not really a lifelong work. All you need to be able to reproduce are the underlying assumptions and, to the extent that you can, the parameters. From those, all the rest of the standard model follows. But it certainly isn't the sort of thing that a worker cranks out (or that a crank works out) in 3 months.

> Just a question here: are you claiming that
> your matrix above has been derived from
> your theory?

Yes. I gave a very brief talk on this at the University of Victoria at the APSNW2005 meeting two weeks ago. After I finish typing up the PHENO2005 paper from their meeting of May 1st, I'll type up the paper associated with this.

The mass matrix problem, after one assumes that the leptons are composites of (R,G,B), boils down to the question of how one calculates |<R|G>| as compared to |<R|R>|. If you make the assumption that R, G and B correspond to 120 degree rotations around a direction of propagation of the (handed) particle, then it is required that the mass matrix have the \sqrt{2} down the diagonal (if the off diagonal elements are to have magnitude 1). The 120 degree assumption is not so radical as it might appear at first as it certainly explains why SU(3)_c is a perfect symmetry.

The freedom to apply real world geometry to internal states of quantum particles comes from the use of the alternative version of relativity mentioned above. Within the assumptions of standard relativity, perfect Lorentz symmetry obtains, and this leads to the "no-go" theorems of Coleman-Mandula &c.

Carl

[Kea]

There are very few choices that one can make, and the choices pretty much amount to assumptions about the nature of the manifold that the Clifford algebra is defined on. (In other words, this is a "geometric algebra" variety of Clifford algebra as promoted by David Hestenes.)


Ahh. How interesting. My colleagues here are very interested in Hestenes' approach. I look forward to seeing your report.

Cheers
Kea :smile:

[arivero]

Dear Kea, Dr. Rivero is right, a long discussion really doesn't belong on this thread.

Perhaps I have exagerated the point. Of course any direct justification of any of the formulae we are cataloging is on-topic. It is just that as I see it, it is better to wait for Carl to write down his note (next month? next couple of months?) and then to announce it here. It is no news, to theoretically specullate about preons, because the original paper was already from a preon theory.

On my side, I am trying to thing how a m^\frac 12 factor can appear in a modern theory. Classical non-relativistic theory needs such factors in order to go from Energy to Velocity, but in a relativistic theory we have the "c" highway.

On other hand the scattering matrix (whose poles are the masses) has two traditional expresions, as funcion of energy or as function of momentum, but we are in the same situation than above: when going relativisitic, there are not m^1/2 factors around.

[CarlB]

I probably have another 7 days of texing before I release the first of those papers. It won't have quite enough to get the lepton masses, but that should follow a few weeks later.

By the way, I should note an interesting fact that I don't think has been mentioned here. If you set the angle \delta to zero in that M^{1/2} matrix, you won't get a result that satisfies the square root mass formula unless you conclude that the square root of the electron mass is negative.

In that sense, it's a good thing that \delta is as large as it is, otherwise we'd have missed the square root mass formula completely.

On the other hand, one could also take the point of view that the fundamental mass matrix is M, in which case the existence of a convenient square root of it is just a coincidence.

Carl

[CarlB]

I don't know what to make of this.

The problem with the binons that I believe underly the fundamental particles is that they have their spins aligned with their velocity vectors. That makes it necessary to have fractional spin or something similar. In the search for something similar, I found an unusual coincidence having to do with the structure of the eigenvectors for the fermion square root mass matrix, M^{1/2}.

The eigenvectors, with their eigenvalues, were:
(1,1,1),\sqrt{m_\tau / 156.9281952 (123)}
(1,s,r), \sqrt{m_\mu / 156.9281952 (123)}
(1,r,s), \sqrt{m_e / 156.9281952 (123)}

where r = e^{2 i \pi/3}, s = 1/r..

The problem is that I have to have three subparticles have a spin that somehow unites to produce \hbar/2. The natural way of doing this would be to use the usual Clebsch-Gordon coefficients. What I'd like to point out here is that the Clebsch-Gordon coefficients have a pattern curiously similar to the pattern of the above eigenvectors.

First, consider three spin-1/2 particles:
\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{3}{2} + \frac{1}{2} + \frac{1}{2}'
It's natural to associate one of the \frac{1}{2} with the combined spin-1/2, but it turns out that if you look at the states that have S_z = 1/2, they have exactly the coefficients as the above three eigenvectors. That is:

Let
| R \rangle = |\frac{1}{2}\; \frac{1}{2}\rangle|\frac{1}{2}\; \frac{1}{2}\rangle|\frac{1}{2}\; \frac{-1}{2}\rangle

| G \rangle = |\frac{1}{2}\; \frac{1}{2}\rangle|\frac{1}{2}\; \frac{-1}{2}\rangle|\frac{1}{2}\; \frac{1}{2}\rangle

| B \rangle = |\frac{1}{2}\; \frac{-1}{2}\rangle|\frac{1}{2}\; \frac{1}{2}\rangle|\frac{1}{2}\; \frac{1}{2}\rangle

Then

|\frac{3}{2} \;\frac{1}{2} \rangle = |R\rangle + |G\rangle + |B\rangle

|\frac{1}{2} \;\frac{1}{2} \rangle = |R\rangle + r|G\rangle + s|B\rangle

|\frac{1}{2} \;\frac{1}{2}' \rangle = |R\rangle + s|G\rangle + r|B\rangle

In other words, the electron and muon families are alone, but there should be a spin-3/2 version of the tau. That is, according to this logic, there should be a spin-3/2 set of quarks and leptons.

Anyone seen them? I would think that the experimental results having to do with the number of lepton families based on counting neutrinos would apply, but maybe spin-3/2 neutrinos don't buy it. Or maybe it's a candidate for dark matter.

Carl

P.S. Part of the problem of quantum mechanics is the huge number of accidental symmetries. It's like navigating in a house of mirrors.

[arivero]

P.S. Part of the problem of quantum mechanics is the huge number of accidental symmetries. It's like navigating in a house of mirrors.

I can not but agree. Take SU(3)_color. It was introduced in order to solve the problem of (anti)symmetrizating protons and pions. But at that time there was only three quark flavours, thus another, approximate, SU(3). So the people involved on colour needed to remark, in their preprints, that SU(3)_c was independent of any SU 3, 4, 5, or 6 of flavour.

About m^1/2: I am surprised nobody mentioned harmonic oscillator frequency.

[marcus]

Congratulations on your new arxiv posting!

http://arxiv.org/abs/hep-ph/0505220
The strange formula of Dr. Koide
Alejandro Rivero (Universidad de Zaragoza), Andre Gsponer
10 pages, 1 figure
"We present a short historical and bibliographical review of the lepton mass formula of Yoshio Koide, as well as some speculations on its extensions to quark and neutrino masses, and its possible relations to more recent theoretical developments."

[Hans de Vries]

About m^1/2: I am surprised nobody mentioned harmonic oscillator frequency.

The formula works fine with the square roots of the quark masses as
well. Within the experimental values given by codata it seems.


M^\frac{1}{2} = \left( \begin{array}{ccc}
D & e^{i\delta} & e^{-i\delta} \\
e^{-i\delta} & D & e^{i\delta} \\
e^{i\delta} & e^{-i\delta} & D \\ \end{array} \right)

Leptons:
D\ = \sqrt{2}
\delta\ \ = 0.2222220471

BSD quarks:
D\ = \sim 1.2987
\delta\ \ = \sim 0.1089

TCU quarks:
D\ = \sim 1.1320
\delta\ \ = \sim 0.0706

The angles are different for the various cases. Altough we do
define 2 parameters to get 2 mass ratio's per set of 3 masses,
this doesn't mean we can find solutions for any arbitrary set of
masses. This is not the case.



Regards, Hans

P.S. The following values fit with the \overline{MS} running mass of the
bottum quark (~4.25GeV) rather than the 1S value (~4.75GeV)
used above.

BSD quarks:
D\ = \sim 1.3150
\delta\ \ = \sim 0.1149

[arivero]

My last headache comes from a Yuri Danoyan, who wrote to tell me of some hadronic relationships. Basically it can be noticed that some products of mesons map approximately into barion masses. Particularly a pair of products are near the mass of the nucleon (proton or neutron):

m_B m_\pi \approx m_D m_K \approx m_p^2

So Yuri proceeds by taking the quotient of every (pseudoscalar, no excited) meson against proton mass, and then plotting the arctan of this quotient. The pattern that emerges is a clustering reflecting the above approximation, but the unexpected phenomena is that all the intervals are about the same, some 18 degrees, symmetric around the 45 degrees or arctan(1), exhausting the quadrant.

If we believe Gell-Mann view of chiral perturbation theory, [square of] the first product should be or order K m_b (m_d+m_u)/2, while the [square of the] second one should be K m_c m_s, so the first equality amounts to m_b (m_d+m_u)/2 \approx m_c m_s. Somehow poor result, because, after all, chiral p.t. does not apply to the bottom mesons (and very badly to charmed ones). Worse, chiral p. t. does not predict, afaik, a value for the nucleon mass, so even a way to analise the second equality is barren. Another problem is that the pures\bar smeson is not a mass eigenstate, and it should be extracted from the eta-eta' mixing so see how it fits in the patterns.

So I am afraid I will not be able to progress in this line, but anyway I thought it was worthwhile to mention it here.

[arivero]

Recently I noticed hep-ph/0504256, from W Krolikowski, and I have read a bit across its bibliography. In the early nineties, he proposed a model of "algebraic partons" (yep, more preons (http://www.physicsforums.com/showthread.php?t=76937)) based on a Clifford-guided generalisation of Dirac-Kaehler equation. It was published in some minor journals, but also reported in Phys. Rev. D 45, 3222-3227 (1992) (http://prola.aps.org/abstract/PRD/v45/i9/p3222_1) (and having a sequel about composite higgs (http://prola.aps.org/abstract/PRD/v46/i11/p5188_1) that I havent read yet).

Well, the point is that in his Phys Rev article, after some summations and normalisation of that extended Dirac formalism, and including a guesswork of a very Barut-like formula, he arrived to a prediction for the mass of the tau:
m_\tau={6\over 125}(351m_\mu+136m_e) =1783.47 MeV

The numbers are ugly, but K. did a good work of justifying them, at least good enough for the Phys Rev D referee... And well, you can remember that at these times the reported value of tau mass was 1784 MeV (+2.7, -3.6) and how this value was problematic for Koide's formula. Here on the contrary the value was right.

Ok for the old times, but now the measured value is 1777. So what of Krolikowski?. Well, he reviewed his guesswork for the massformula and he found a sign to change somewhere (EDITED: he only needed to remove a somehow arbitrary -1 imposed subtly in the Phys Rev article to keep the three masses at positive values).

m_\tau={6\over 125}(351m_\mu-136m_e) =1776.80 MeV

Amusing.

Unfortunately it is not exactly compatible with Koide's 1776.97 prediction, so we can not straighforwardly to intersect both equations. As Koide in his first papers, also Krolikowski has forseen some places to tweak its relationship, see eg hep-ph/0108157.

[arivero]

where degrees, is the Cabibbo angle.

Convincing the physics world to simultaneously accept so many hard things to swallow is essentially impossible.

Well, no need to proceed simultaneously.

The Minakata Smirnov relation,
\theta_{\mbox{sun}} + \theta_{\mbox{cabibbo}} = {\pi \over 4}

(hep-ph/0405088; I noted it some months later at sci.phys.research, and at PF (http://www.physicsforums.com/showthread.php?t=38407) the day after my birthday) has got some attention during the last year. A review by Minakata appeared yesterday as hep-ph/0505262. It has been renamed to "quark-lepton complementarity". The point is that, if true, it is evidence of a common origin of leptons and quarks. So it makes a good selling argument for Cabibbo in charged leptons. If it is Cabibbo and not the 2/9 of Hans!

The formula had been mentioned before in the thread (message #15) but not explicitly. I though the observation was vox-populi, but acording Minakata it was first voiced by Smirnov in a conference in December 2003, hep-ph/0402264. In july 2004 I was somehow depressed and I went on holidays to Benasque, simultaneusly to a neutrino workshop (http://benasque.ecm.ub.es/benasque/2004neutrinos/2004neutrinos-talks.htm), so perhaps I overheard it around there.

[CarlB]

Dr. Rivero, On the W Krolikowski papers,

Thanks for the links. The mass formula is stunning in its length, and surely the Koide is more attractive. The funny thing is that W Krolikowski is touching on a lot of subjects that are similar to mine, that is Clifford algebra.

And the links for the neutrino mixing angle coincidence with the Cabibo angle is very helpful.

My paper is coming along. I've recently found a fascinating verification in the Centauro high energy cosmic ray events and I can barely wait to get the paper finished. You will have provided many of the references that I will have to include, if you fail to request otherwise, I will include a note of appreciation in the paper.

I mentioned that it is difficult to get the physics community to accept too many impossible ideas at the same time. When you see the write-up, you will see that I was, if anything, understating the problem. But the Centauro data are very convincing. I am so ashamed to be still sitting on this, I'll go home now and write some more.

Carl

[arivero]

The formula works fine with the square roots of the quark masses as
well. Within the experimental values given by codata it seems.

BSD quarks:
D\ = \sim 1.2987
\delta\ \ = \sim 0.1089

TCU quarks:
D\ = \sim 1.1320
\delta\ \ = \sim 0.0706

The angles are different for the various cases.
Regards, Hans

P.S. The following values fit with the \overline{MS} running mass of the
bottum quark (~4.25GeV) rather than the 1S value (~4.75GeV)
used above.

BSD quarks:
D\ = \sim 1.3150
\delta\ \ = \sim 0.1149

Indeed some researchers have already noticed the BSD quarks seem close to be compatible with Foot's version of Koide's, ie that the vector (\sqrt{m_b},\sqrt{m_s},\sqrt{m_d}) is at 45 degrees of (1,1,1).

For the other flavour of unix, :confused: er, not, I mean, for the TCU set, it should be possible to fit a 45 degrees rule but against the vector (1,1,0). This should be
\sqrt{m_t}+\sqrt{m_u} = \sqrt{ m_t+m_c+m_u}
which simplifies to
{2 m_u \over m_c} = {m_c \over m_t}

In fact decades ago this proportion, whithout the factor two, was suggested by an expert on textures (H. Fritzsch?) as a clue for a mass of the top higher than the then current expectations. The argument was drawn over a logarithmic plot, causing Feynman to protest that "In a log plot, even Sofia Loren adjusts to a straight line"

[arivero]

Malcolm H.Mac Gregor (http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+A+MACGREGOR%2C+M), Electron generation of leptons and hadrons with reciprocal alpha-quantized lifetimes and masses, hep-ph/0506033 (http://arxiv.org/abs/hep-ph/0506033), Int J of Mod Phys A V. 20, No. 4 (2005) 719-798 & 2893-2894, is mostly about hadrons, but it could be [part of] an answer to the Danoyan headache I mentioned before.

Nice plots.

Hmm, and who is this McGregor? First time I heard of him. Lets research SPIRES: As it seems (hep-ph/0309197), he is a senior, now retired, of the LLNL. Early in his worklife, he become interested in a mass systematics for hadrons (Phys. Rev. D 13 (http://prola.aps.org/abstract/PRD/v13/i3/p574_1), 574) which did not got its way up to the mainstream (I wonder if because of excessive predictions, just as Barut's leptons, lets say), and time after he played with the idea of spinless quarks as well as the concept of "large electron" with pointlike charge.

It seems that he become impatient about the low impact of his model; one of the last publications before going "preprint only" was titled "Can 35 Pionic Mass Intervals Among Related Resonances Be Accidental?" (Nuovo Cim.A58:159,1980). In some sense, he shares adventure with another insider, Paolo Palazzi (http://arxiv.org/abs/physics/0301074)

[arivero]

This one does not score in the same quality that the rest of the thread, but well... Get \pi^0 and Z^0 and consider both their mass m and their total decay width \Gamma. We have for the 2004 PDG central values,

{m_\pi^3 \over \Gamma_\pi} {\Gamma_Z \over m_Z^3} = 1.04...

The listed error for the decay width of pion is relatively high, so the result is compatible with the unit. Separately we had, considering errors
\sqrt{m_\pi^3 \over \Gamma_\pi}= 541...580 GeV
\sqrt{m_Z^3 \over \Gamma_Z}= 551.0 ... 551.5 GeV

Another way to see the same thing is to "calculate" the lifetime of neutral pion:

\Gamma_{\pi^0}= \Big({M_{\pi^0} \over M_{Z^0} }\Big)^3 \; \Gamma_{Z^0}


We get 8.13E-17 s (8.1 eV), compare with a theoretical effective calculation such as hep-ph/0206007, and against experimental 8.4E-17 +-3*0.2E-17 s (7.8 eV) at PDG 2004.

[arivero]

Let me put numbers, from PDG 2004, into the formula


\Gamma_{\pi^0}= \Big({M_{\pi^0} \over M_{A} }\Big)^3 \; \Gamma_{A}


where mass of pi0 is 0,134976 GeV and we expect to find a value for its gamma around experimental
Gp(exp) =0,000 000 007 8 (-0 5, +0 6) GeV
or theoretical
Gp(the) =0,000 000 008 10 (+-0 08) GeV

Well, I found two valid neutral (but no scalar) particles.

First, for the aforementioned Z0, we have M= 91,1876 , G=2,4952 (all the values in GeV) and it results in a prediction
Gp(Z0) =0,000 000 008 1 (+-0.000 000 000 0)

And second, for J/Psi !! We have M=3,096916 G=0,000091 and
Gp(J/P) =0,000 000 007 5 (+-.000 000 000 3)

No bad. It should be nice to have a theory backing it.

Ah, the other neutral boson, the upsilon \Upsilon, does not fit, but it is possible to make use of MacGregor's alpha pattern and tweak a bit with the data; the best fit is with the least bounded state, upsilon(3), that after multiplication times alpha gives 0.000 000 007 9.


EDITED, VERY TECHNICAL: One argument is that as vectors can not decay to two photons except virtually, they compose the whole decay amplitude. As for Upsilon, perhaps it is decaying using the square anomaly instead of the triangular one.

[Kea]

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 :cool:

[arivero]

Kea, you can prettify your formula, imho, by writting it as the square of a hyperbolic cosine... and adding a reference to that Temperley-Lieb factor; I am afraid it is not a object of my mathematical bagage. In any case it does not sound as bad, against the measured 137.0359, if it were the first term of a series.

Let me note that the updated Wolfram-Weisstein webpage has a pair of recent approximations, formula 6 and 7 (http://scienceworld.wolfram.com/physics/FineStructureConstant.html). They refer to work from Bailey (http://www.cecm.sfu.ca/organics/papers/bailey/) and Plouffe (http://www.lacim.uqam.ca/~plouffe/).

About alpha from a physics point of view, I do not remember if we quoted already here Adler's program, the unsuccessful idea of looking for an eigenvalue equation for it: Phys. Rev. D 5, 3021-3047 (href=http://prola.aps.org/abstract/PRD/v5/i12/p3021_1). Also, Kinochita has some old listings (KEK (http://ccdb3fs.kek.jp/cgi-bin/img_index?9009168)) of experimental values of alpha which are interesting to see (they are quoted in Peskin/Schroeder about pages 90-100, depending on edition)

Unrelated to this... I have started a note on the pion relationship, you can see the draft here (http://dftuz.unizar.es/~rivero/research/pion.pdf)

[arivero]

My last formula in hep-ph/0507144 involved a summatory in all the fermions the Z0 can decay to,
\sum_f C_f (|V_f|^2 + |A_f|^2)
where C is a coefficicient 1 for leptons, 3 for quarks, and V and A are the vector and axial couplings.

After releasing it, I read again the calculation of the Z0 decay width in "QCD and Collider Physics" (Ellis-Stirling-Weber) and I noticed that, when summed for a complete generation of fermions, this coefficient is very near of the number e.

[arivero]

Let me describe the calculation. The A coupling is always \pm 1/2, but the V coupling depends of charge and Weinberg angle. Table 8.3 in the above quoted book lists the values V_u \sim +0.191, V_d\sim -0.345, V_\nu=0.5, V_e=-0.036. And as I said, C is 3 for quarks (it receives corrections if the colour coupling constant is enabled, but the book does not worry about this, so neither me) and 1 for leptons.

So the sum is 2.717814
versus e^1= 2.71828... it makes a 0.017% discrepancy.

Now, the book calculation was for (running) sin^2 of Weinberg angle 0.232 and the intermediate rounding was against us (we get 2.7181 if we use this angle without the intermediate rounding). Lets forget about the possible corrections from C, and ask for which value of Weinberg angle is this equality exact. We need then the formulae,
A_f=T^3_f
V_f=T^3_f-2Q_f \sin^2 \theta_W
and we get sin^2 \theta_W = 0.231948

Of course, we are speaking of Z0 couplings for Z0 decay, so the value of angle is to be taken to this scale, in distintion to Hans approximation early in this thread, which was unrenormalized. Check table 10.5 of the pdg review (http://pdg.lbl.gov/2005/reviews/stanmodelrpp.pdf).

Hmm if we believe both Hans estimate plus the exactitude of this relationship, the running of Weinberg angle could be interpreted as a prediction of the mass scale of Z0. Then in turn the mass of W is predicted (via Weinberg angle again!) and then the scale of electroweak vacuum follows.

On other hand, it should be remarked that the minimum of this expression, 2.5, is got for Sin^2 W equal to 3/8, a value known from GUT theories, and that has the additional virtue of cancelling the top [,up, charm] V_f couplings (but not A_f ones, of course).

[arivero]

I was worried because in this thread we have got two very high precision inventions for \sin^2 \theta_W and they are very different in spirit; the one by Hans about middle way in the thread, the other here in the message above.

But now I think about, this later one is about Weinberg angle as parametrisation of the coupling constants while the first one (Hans will correct me if I am wrong) was in the scent of the mass quotient between Z and W. One number can be calculated directly from the another if SU(2)xU(1) is broken via the minimal Higgs mechanism, but they are different concepts and it is possible to have different unrelated ways to estimate them. Uff.

Now I think about, I haven't put here explicitly the estimate from previous message yet; it is
\sin^2\theta_W=\frac 38 \Big(1-\sqrt\frac 23 \sqrt{(\sum_{n=0}^\infty {1\over n!}) - \frac 52}\Big)

Or, if you prefer
4(1-2 \sin^2\theta_W + \frac 83 \sin^4\theta_W) =\sum_{n=0}^\infty {1\over n!}
Or

4 ( \cos^4\theta_W + \frac 53 \sin^4 \theta_W) =\sum_{n=0}^\infty {1\over n!}

or :biggrin:
\ln ( 4 (\cos^4\theta_W + \frac 53 \sin^4 \theta_W)) -1 =0

[arivero]

I was thinking to start a new thread, but well anyway... this last number, 0.231948... when checked against the aforementioned table 10.5 (http://pdg.lbl.gov/2005/reviews/stanmodelrpp.pdf) selects only a particular measurement: 0.23185(18), the one from the Forward/Backward asymmetry!!!

Why the exclamation marks? because this measurement is a well known problem of standard model fits; depending of the data it stands between 2.9 and 3.5 sigmas of the expected value, no bad enough to claim new physics, but no good enough to consider it in the same bag than the rest. And while the NuTeV anomaly comes from only an experimental group, this one comes from different teams, so it is usually assigned to "unknown systematic error".

Hans' number from posting #44 (http://www.physicsforums.com/showpost.php?p=382642&postcount=44), 0.2231013... , when used for the second column of table 10.5 selects the first three measurements, ie the accepted standard model value.

Perhaps it is worthwhile to remember that the relationship between Weinberg angle (ie the mixing of couplings in Weinberg model) and the masses of Z and W depends on the structure of the Higgs sector; the usual M_W/M_Z is a result for the minimal Higgs.

[Jay R. Yablon]

Hans, Please view the attachment (if this works right), I have summarized some possible relationships between your Fine Stucture Constant Work and my Earlier formula for the Tau Electron mass.

Jay.

[Jay R. Yablon]

Hello to all:

I wanted to let you know about my new paper just posted at
http://arxiv.org/abs/hep-ph/0508257, titled Magnetic Monopoles and Duality
Symmetry Breaking in Maxwell's Electrodynamics.

This paper summarizes the main direction of my research over these past
eight months.

The abstract is as follows:

It is shown how to break the symmetry of a Lagrangian with duality symmetry
between electric and magnetic monopoles, so that at low energy, electric
monopole interactions continue to be observed but magnetic monopole
interactions become very highly suppressed to the point of effectively
vanishing. The "zero-charge" problem of source-free electrodynamics is
solved by requiring invariance under continuous, local, duality
transformations, while local duality symmetry combined with local U(1)_EM
gauge symmetry leads naturally and surprisingly to an SU(2)_D duality gauge
group.

As regards this thread, I note the extremely accurate formula (no "mere coincidence" in my
view) that Hans has posted for the fine structure constant and the
appearance of 2pi and (pi/2)^2 as a key numeric drivers in this formula. In section 6, I call your attention particularly
to equation (6.7) which brings pi into the fine structure formula based on
phenomenological origins, and to the derivation leading to this including
the Dirac Quantization Condition expressed as (5.22) where both the 2pi and
the a^.5 (the electric charge) are apparent ingredients. Regarding the
(e^pi/2)^2 factor, pi/2 for the "complexion" angle I am using in this work
is what rotates between the electric and magnetic monopoles. And, with some
rejuggling of (5.22), one can clearly get pi^2 factors to show up there.
The exponentials do not yet have a clear phenomenological origin, but, since
running couplings a connect via a log relationship to probe energy u, that
is, ln u ~ a, so that u ~ e^a, one can conceivably get this exponential onto
phenomenological footing as well once a connection is made to probe
energies. The connection to probe energies, I will make the topic of a
follow up paper.

In other words, the complexion angle, which is central to this
paper, is a direct function of the fine structure constant a, but is an angle
of rotation which naturally introduces factors like 2pi and pi/2. I have a
strong suspicion that this might provide a basis for going beyond
"mathematical fitting" of the numbers (I am being careful not to use the
word "numerology" because of its negative connotations) to explain how Hans' Fine Structure Constant
mathematics can possibly be given a phenomenological origin.


I would be interested in any feedback, public or private, that you may wish
to provide.

Sincerely,

Jay R. Yablon
_____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com

[Kea]

Hi Jay

Very interesting! I see on page 18 you begin to discuss monopole masses. I assume you know that one of the LHC experiments will be searching for such things in the right ballpark....

http://moedal.web.cern.ch/moedal/

:smile:

[Jay R. Yablon]

Hi Kea:

Yes, thank you, a few people have pointed the LHC plans out to me.

I do want to clarify for the readers of this post that the ~ 2.35 TeV is the predicted mass of the vector boson which mediates magnetic monopole interactions (assuming the Fermi vacuum expectation value applies here), and is not the mass of the magnetic monopoles themselves, which thus far are not predicted.

Jay.

[arivero]

Funny. Terry Prachett's "Guards! Guards!", a old book of the Discworld series, contains a formula including Planck Length, alpha, and the quotient between proton and electron mass. It seems thus an old maladie.

I will try to locate the formula and plot it here, but the Spanish translation has typographical problems, so if someone can check the UK one please do.

[Hans de Vries]

A short latex style paper on the alpha result discussed earlier on this thread:

A simple radiative series leading to the exact value of the fine structure constant (http://www.chip-architect.com/physics/fine_structure_constant.pdf)


Regards, Hans.

[arivero]

Hans, my problem with your result, and the secret because I did not concentrate on it, is that I am not able to see how standard the method of successive differences is. If could be good if you had time to put some examples using other ansatzes, ie what number will we get if instead of (1) we take \sqrt \alpha \approx e^{-\pi}? or if we take \sqrt \alpha \approx \pi^2? Or \sqrt \alpha \approx \alpha^2? Neither I understand the systematics to get all these \pis in denominators along the process of expansion. Do they come from the exponent in the exponential somehow?

[Hans de Vries]

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

[Jay R. Yablon]

Hello to all:

I am pleased to announce that my newest paper, "Magnetic Monopoles, Chiral
Symmetries, and the NuTeV Anomaly," has now been published at
http://arxiv.org/abs/hep-ph/0509223.

This paper is a follow up to my earlier publication at
http://arxiv.org/abs/hep-ph/0508257, and takes a closer look at the magnetic
monopoles themselves as fermionic particles. I have reported interim
progress along the way on the sci.+ boards; now you can see the full
picture.

This paper calculates widths and cross sections associated with the
predicted magnetic charge, and determines that there is a very slight
cross-section enhancement at sqrt(s) = M_z ~ 91 GeV due to magnetic
monopoles.

If one were to do experiments and NOT understand the magnetic monopole
origin of this small cross section enhancement, one might instead conclude
that the weak mixing angle had decreased for e/ebar scattering, in relation
to neutino/neutrino-bar scattering, by a small amount. How small? This
paper predicts a reduction of approximately .003, which is right near the
magnitude of the NuTeV anomaly and goes in the right direction as well.

Fundamentally, the NuTeV anomaly is thus seen to be the first experimental
evidence of the existence of the magnetic monopole charges, which have been
a mystery ever since Maxwell's era.

Also, some fundamental connections are drawn between the magnetic / electric
symmetries, and chiral symmetries.

If you want the quick tour, look at equations (9.12) to (9.15) which contain
the final numeric results. Then look at (8.16) through (8.20) which shows
these same results represented in term of the cross section enhancements
from which they were derived.

If you are doing NuTeV experiments, and even if not, look at (7.34) to
(7.44), which show the full and differential cross sections in the most
general form. This should help you with the NuTeV anomaly even if you don't
believe as I do that the magnetic monopole charge at least contributes to
this anomaly. Because these equations tell you how a vector boson (call it
the Z^u' if you wish) with mass > M_z would enhance the cross section
generally, whether the origin of that vector boson is from magnetic
monopoles or somewhere else. So, these give you a theoretical framework to
fit the data under a variety of assumptions that you may wish to make.

If you assume two or more massive bosons with mass > M_z, then there will be
further cross section terms for each new vector boson, as well as further
cross terms between pairs of vector bosons, the form of which can readily be
understood and deduced from (7.34) to (7.44). My own suspicion is that
there is also an electroweak-based Z^u' in the 1.3 TeV range in addition to
the M^u which mediates the magnetic monopole interaction here. This will
require extending the entire electroweak theory to consider weak and weak
hypercharge magnetic monopoles, and may well be the subject of my next
paper.

Once the cross section enhancement is known under whatever scenario one may
assume, the apparent impact on sin^2 theta_w can be deduced following the
steps shown in section 9. So, there is some good grist here for the NuTeV
folks. And for anyone who is interested in understanding magnetic monopoles
and chiral symmetries.

I also suggest a look at the conclusion.

From there, look at whatever you want.

Happy reading.

Jay.
_____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com

[CarlB]

Jay, that is a cool paper. The mechanism you're suggesting for giving mass to the fermions seems to me to be similar to the old Zitterbewegung theory. My understanding is that this is in violation of Lorentz symmetry.

Another way of putting this: Your paper writes: "It may well be that here, the \psi_e = \psi_R and \psi_m = \psi_L are swallowing up one another to go from being two twocomponent fermions which are each massless, to being a single four-component Fermion which is massive."

This is a Higgsless mechanism for mass, a mechanism that I also believe in. The problem is Lorentz violation. To get an electron of spin +1/2 in the z direction, one must have a right handed electron moving +z and a left-handed electron moving -z. To move from one to the other is a violation of conservation of momentum. Any comments?

Carl

[Jay R. Yablon]

Hi Carl:

Thanks for your reply. I just saw your post or I would have replied sooner.

I guess I'd be interested in seeing mathematically how the Lorentz violation comes about in Zitterbewegung theory and how what I am doing matches up or not. It is a common practice to construct a four-component Fermion \psi from the \psi_R and \psi_L using \psi = \psi_R + \psi_L, where Lorentz transformations essentially shift the relative magnitudes of the various components of the Dirac spinor, and in a sense, in addition of course to the (I believe, novel) connection with electric and magnetic charges, that is all we are suggesting here.

Beyond that, the comment about fermion mass is really just an aside from the main thrust of development. I'm curious about your overall view of this apporach to magnetic monopoles and whether you think we may be on to something for the NuTeV anomaly?

Thanks.

Jay.


Jay, that is a cool paper. The mechanism you're suggesting for giving mass to the fermions seems to me to be similar to the old Zitterbewegung theory. My understanding is that this is in violation of Lorentz symmetry.

Another way of putting this: Your paper writes: "It may well be that here, the \psi_e = \psi_R and \psi_m = \psi_L are swallowing up one another to go from being two twocomponent fermions which are each massless, to being a single four-component Fermion which is massive."

This is a Higgsless mechanism for mass, a mechanism that I also believe in. The problem is Lorentz violation. To get an electron of spin +1/2 in the z direction, one must have a right handed electron moving +z and a left-handed electron moving -z. To move from one to the other is a violation of conservation of momentum. Any comments?

Carl

[CarlB]

Dear Jay

I guess I'd be interested in seeing mathematically how the Lorentz violation comes about in Zitterbewegung theory and how what I am doing matches up or not.

I'm not sure if what you're doing matches up, but here's my understanding of the problem for my own version of fermion masses:

The problem with Lorentz violation when one attributes mass to left handed electrons turning into right handed electrons and vice versa appears when you write down the interactions in Feynman diagrams.

It's not called the "Zitterbewegung" problem in the literature. Since this method of giving mass to the electron violates Lorentz symmetry, it's not in the literature. The objection is what will happen when you talk to physicists about this sort of thing at a conference.

I believe that this method was known to Feynman, and must have been common knowledge back in the late 1940s. The reason for this is that Feynman implies that a version of this exists for adding mass to scalar particles (I seem to recall that they were massless Klein-Gordon). He wrote something like "Nobody knows what this means" with regard to the derivation. But the reference to this is not in the physics literature. Instead, it's a footnote for his excellent (high school math level) introduction to QED:

QED: The Strange Theory of Light and Matter
http://www.amazon.com/gp/product/customer-reviews/0691083886

As I said earlier, I believe that this is the correct way to add mass to massless particles (rather than Higgs). I partially wrote up a paper that included the Zitterbewegung creation of mass in QFT using (illegal) Feynman diagrams a year ago but it turns out that I've never released it. My hesitation to do so is entirely because of the problem with Lorentz invariance. I've talked about it over a napkin at a physics conference, to appreciative laughter (The idea is very simple in that it gives mass to the electron without any need for Higgs etc., but it is wildly heretical. The effect, to a physicist, is sort of like when you divide by zero in a proof you show to a mathematician.), but I've not written it up completely. Let me try and write it up here, but without the graphics.

The Zitterbewegung problem is that when you look for velocity eigenstates of the Dirac equation, one finds solutions only for speed c. For example:

"Even more surprising result [2] is obtained if we consider velocity \vec{\dot{x}} in the Dirac formulation. Instantaneous group velocity of the electron has only values ±c in spite of the non-zero rest mass of electron. In addition, velocity of a free moving electron is not a constant of motion."
http://www.hait.ac.il/jse/vol0103/sep040706.pdf

Particles that travel at speed c are massless, so how does the electron get mass? Normally Zitterbewegung stuff is done at the QM level. It is when you take it to the QFT level and use it to explicitly create mass that you get into trouble with Lorentz invariance.

If the electron is to be made from massless subparticles, I suggest we take a bow to weak theory and assume that the electron is a composite made up of two particles that alternate, a left-handed electron and a right-handed electron. Let us work in the momentum representation. The propagators for a massless left/right-handed e_L, e_R electron is:

i/p_L \; , i/p_R.

Notice that I'm leaving off the \gamma stuff. You can add it back in, but the results is the same. That is, when you resum a simple set of Feynman diagrams you will turn a pair of massless spin-1/2 particles into a single massive spin-1/2 particle.

[Long post continued]

[CarlB]

I guess I'd be interested in seeing mathematically how the Lorentz violation comes about in Zitterbewegung theory and how what I am doing matches up or not.

[Long post continued]

Now consider the Feynman diagram that annihilates an e_L and creates an e_R with momentum p_R = p_L. This is a trivial type of Feynman diagram in that it has only one input and only one output. Add another diagram, one that does the reverse transformation. Together, these form a Zitterbewegung model for the electron.

Since we are considering the electron as a composite particle made up of a part e_L and a part e_R, we will have to have a two-component vector to model the electron. The two components of this vector will

correspond to the left and right handed parts of the electron. Since the two halves of the electron are scalar particles, we will require a vector of two complex numbers to model the electron. This is convenient because it is just the number of degrees of freedom we need.

To derive the propagator for the electron from the propagators for the e_L and e_R, we need to take into account the Zitterbewegung effect. That is, we need to take into account the interaction that converts a left-handed electron into a right-handed electron and vice-versa.

As is usual in QFT, we will need to resum the whole series of Feynman diagrams (that is, the series generated by the above two) in order to find the propagator for the electron. Fortunately, these two Feynman diagrams are particularly simple.

Let me write down the Feynman diagrams that we must resum, in order of their complexity:

e_L \to e_L
e_R \to e_R
e_L \to e_R
e_R \to e_L
e_R \to e_L \to e_R
e_L \to e_R \to e_L
e_R \to e_L \to e_R \to e_L
e_L \to e_R \to e_L \to e_R
...

To calculate these we need the usual Feynman rules. Note that since there are no loops, there is no need to do any integration. I need to specify a value to associate with the nodes, that is a probability, let us choose $im$.

Now I can resum the propagators. Note that the above series will resum not to a single propagator, but instead to 4. They will be the e_L -> e_L, the e_L -> e_R, the e_R -> e_L and the e_R -> e_R. Let's do

the e_L->e_L propagator first:

e_L \to e_L
+ e_L \to e_R \to e_L
+ e_L \to e_R \to e_L \to e_R \to e_L
+ ...

To sum these, remember the propagators defined above and that p = p_L = p_R.

L \to L =(i/p_L) + (i/p_L) im (i/p_R) im (i/p_L) + ...
L \to L =(i/p) + i (m/p)^2p + i (m/p)^4 p + ...
L \to L =(i/p)(1+(m/p)^2 + (m/p)^4 + ...
L \to L =\frac{i/p}{1-(m/p)^2}
L \to L =\frac{ip^2/p}{p^2-m^2}
L \to L =\frac{ip}{p^2-m^2}

I've left more stages in the algebra than necessary in order to hint where you will have to work harder to put the gamma matrices back in.

The p_R to p_R propagator will be similar:
L \to R =\frac{ip}{p^2-m^2}

The p_L to p_R propagator is as follows. The Feynman diagrams are:

e_L \to e_R
+ e_L \to e_R \to e_L \to e_R
+ e_L \to e_R \to e_L \to e_R \to e_L \to e_R
+ ...

These work out to be:

L \to R =(i/p_L) im (i/p_R) + (i/p_L) im (i/p_R) im (i/p_L) im (i/p_R) + ...)
L \to R = \frac{im}{p^2}(1 + (m/p)^2 + (m/p)^4 + ...)
L \to R = \frac{im}{p^2(1-(m/p)^2)}
L \to R = \frac{im}{p^2-m^2}

Similarly, R->L works out as:
R \to L = \frac{im}{p^2-m^2}

With all four propagators computed, we can put them together into a single equation using matrices. We assume our (spinors) consist of left-handed massless particles on the top part of a 2-vector, and right-handed massless particles on the bottom part. Then the desired resummation of the propagators is:

\left(\begin{array}{cc} L \to L \;&\; L \to R \\ R \to L \;&\; R \to R \end{array} \right)
=\left(\begin{array}{cc} \frac{ip}{p^2-m^2} \;&\; \frac{im}{p^2-m^2} \\ \frac{im}{p^2-m^2} \;&\; \frac{ip}{p^2-m^2} \end{array} \right)
=\frac{1}{p^2-m^2}\left(\begin{array}{cc} ip \; & \; im \\ im \; & ip \end{array}\right)

Uniting the left and right handed electron fields back to the same particle, one ends up with a propagator of:

\frac{i(p+m)}{p^2-m^2}

I tend to lose factors of i so you might check this. Also, note that the positron works out the same way and increases the size of the matrix from 2x2 to 4x4 as in the usual Dirac matrices.

Now in the above, there isn't any Lorentz violation. But if you consider an electron at rest with spin +1/2 in the z direction, for it to be composed of e_L and e_R portions, one must have that the velocities of these two subparticles be in opposite directions. This is incompatible with their momenta being identical. You can fix this by going back and making p_R = - p_L, but then you've lost conservation of momentum at the vertices of your Feynman diagrams.

Carl

[arivero]

This thread is now three citation steps away from a paper of Wilczek. It is cited by hep-ph/0505220 (Rivero-Gsponer), which is cited by hep-ph/0508301 (Koide), which is cited by hep-ph/0509295 (an article of Brian Patt, David Tucker-Smith and Frank Wilczek).

[Kea]

Perhaps someone has already commented on this, but it might be interesting to note the following:

The Koide mass formula

m_{e} + m_{\mu} + m_{\tau} = \frac{2}{3} (\sqrt{m_{e}} + \sqrt{m_{\mu}} + \sqrt{m_{\tau}})^{2}

can be rewritten in terms of the squareroots as

s_{e}^{2} + s_{\mu}^{2} + s_{\tau}^{2} = 4(s_{e}s_{\mu} +
s_{e}s_{\tau} + s_{\tau}s_{\mu})

Via a simple change of coordinates

s_{e} + s_{\tau} = x_{1} \hspace{11mm} - s_{e} - \frac{s_{\mu}}{2} = x_{2} \hspace{11mm} \frac{s_{\mu}}{2} + s_{\tau} = x_{0}

the relation becomes

x_{1}^{2} = 4 x_{0} x_{2}

which is an instance (the \rho = 0 'ground' case) of a coadjoint orbit for the obvious SL(2,\mathbf{C}) action on \mathbf{C}^{3}. In other words, the mass triplet is being described by a representation of (the double cover of) the Lorentz group, and the constraint on the vector is simply a quantization condition (a la Kirillov).

:smile:

[arivero]

Via a simple change of coordinates

s_{e} + s_{\tau} = x_{1} \hspace{11mm} - s_{e} - \frac{s_{\mu}}{2} = x_{2} \hspace{11mm} \frac{s_{\mu}}{2} + s_{\tau} = x_{0}


Simple, but not uninteresting. One can rewrite as

2 s_\tau= x_0+x_1+x_2
2 s_e= -x_0+x_1-x_2
s_\mu= x_0 -x_1 -x_2

Or, for the masses
4 m_\tau= (x_0+x_1+x_2)^2=(x_0^2+x_1^2+x_2^2)+2(x_0x_1+x_1x_ 2+x_2x_0)
4 m_e= (-x_0+x_1-x_2)^2=(x_0^2+x_1^2+x_2^2)+2(-x_0x_1-x_1x_2+x_2x_0)
m_\mu= (x_0 -x_1 -x_2)^2=(x_0^2+x_1^2+x_2^2)+2(-x_0x_1+x_1x_2-x_2x_0)

The sums are,

m_\tau+m_e+m_\mu=\frac 32 (x_0^2+x_1^2+x_2^2) - 2 x_0x_1+2x_1x_2-x_2x_0


(s_\tau+s_e+s_\mu)^2=(x_0-x_2)^2=(x_0^2+x_2^2)-2x_0x_2

Hmm... I am tired. There is a mistake somwhere.

[Kea]

In other words

\frac{m_{\tau}}{m_{\mu}} = \frac{1}{4} \frac{(1 + y_{1} + y_{2})^{2}}{(1 - y_{1} - y_{2})^{2}}

\frac{m_{e}}{m_{\mu}} = \frac{1}{4} \frac{(- 1 + y_{1} - y_{2})^{2}}{(1 - y_{1} - y_{2})^{2}}

where (y_{1},y_{2}) = (0.91,0.12)

:smile:

[CarlB]

Hmm... I am tired. There is a mistake somwhere.

Re:
s_{e} + s_{\tau} = x_{1} \hspace{11mm} - s_{e} - \frac{s_{\mu}}{2} = x_{2} \hspace{11mm} \frac{s_{\mu}}{2} + s_{\tau} = x_{0}

This makes the left hand side and right hand side of:

x_{1}^{2} = 4 x_{0} x_{2}

have differing signs.

Also, what is the "obvious SL(2,C) action on C^3".

Oh, is it that you take the usual representation of SL(2,C) in 3x3 matrices and do matrix multiplication. As in

\left(\begin{array}{ccc}0 & 1 & 0 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{array}\right)

\left(\begin{array}{ccc}0 & 0 & 0 \\ 2 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right)

\left(\begin{array}{ccc}2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -2 \end{array}\right)



Carl

[arivero]

Hmm, no. In the LHS

x_1^2=s_e^2+s_\tau^2+2 s_e s_\tau


And in the RHS
4 x_0 x_2=-(s_\mu^2+ 2s_\mu s_e + 2 s_\mu s_\tau + 4 s_e s_\tau)


Thus keeping the sign as it is, moving the first term to the LHS and the last of the LHS to the RHS we get

s_e^2+s_\tau^2+s_\mu^2=-2 s_\mu s_e -2 s_\mu s_\tau -6 s_e s_\tau

Or changing the sign, we get

s_e^2+s_\tau^2-s_\mu^2=2 s_\mu s_e +2 s_\mu s_\tau +2 s_e s_\tau


I can not see how Kea's is equivalent to Koide's.

[Kea]

Oops - I'd better recheck sums! :smile:

[CarlB]

An argument for 1/\alpha^3 = 2573380, based on requiring that the ratios of two actions be an exact integer is here:

http://www.fervor.demon.co.uk/noteto.htm

From:
http://www.fervor.demon.co.uk/

Meanwhile, I'm finding a suspicious relationship between Fibonacci series and the equation for the phase angle \delta in my version of Koide's lepton mass relationship. After a bunch of theory, you get:

\tan(2 \delta)/\sqrt{3} = \kappa

then an important cosine is

\cos(\theta_B) = \frac{1-\kappa}{1+3\kappa} = 29/73

29 = F_3^2 + F_5^2
73 = F_4^2 + F_6^2

where F_N is the Nth Fibonacci number, 1, 1, 2, 3, 5, 8 ...

I don't think that this is of any physical significance. It gives a value for \delta = .22226 that is too high. But I'm including it here for entertainment only.

Carl

[CarlB]

A Very Simple Empirical Neutrino Mass Formula
Wojciech Krolikowski
http://www.arxiv.org/abs/hep-ph/0510355

This references Koide's lepton mass formula, but the neutrinos don't follow the 1.5 factor, at least according to my calculator, even when you include possible negative square roots.

Note that this is another application of that interesting number, 29.

Also, my idea for using a Lorentz violation in a hidden dimension to arrange for the breaking of electroweak symmetry is no longer unique as a group in Europe is submitted a similar idea:
http://arxiv.org/abs/hep-ph/0510373

The difference is that my paper (so far it's at http://www.brannenworks.com/PPANIC05.pdf but it needs corrections) is written from the assumption that the wave equation is the fundamental physical object while their's is written with the usual Lagrangian / Hamiltonian formalism.

Carl

[arivero]

I happenned to give a look today to hep-ex/0511027 (http://arxiv.org/abs/hep-ex/0511027), from the CERN (http://lepewwg.web.cern.ch/LEPEWWG/). There in page 126 the new measure estimate to W mass is given as
m_W=80.392\pm0.039. It is still to be seen how this new fit enters in the next edition of the particle data group evaluation, but any influence will affect possitively the estimates in messages #41 to #88 of this thread (and leaves out the need for corrections such as the one of message #63). Lets wait.

Note also hep-ex/0509008v2 pg 224, where the prediction of m_W from the rest of parameters of electroweak model is said to be calculated as 80.363\pm0.032

[arivero]

Funny, tonight I have found the first approximation of the equations starting this thread,

\ln {m_\mu \over m_e}= 2\pi - 3 {1\over \pi}


\ln {m_\tau \over m_\mu}= \pi - {1\over \pi}

in
Andreas Blumhofer, Marcus Hutter Nucl.Phys. B484 (1997) 80-96
http://arxiv.org/abs/hep-ph/9605393

[CarlB]

Arivero,

I guess I should mention that I'm soon to release a paper that describes an alternative method of calculating probabilities than the spinor method.

Basically, the idea is to go back to the density matrix and write probabilities using geometric principles directly from the density matrix, ignoring the fact that it was derived from a spinor. I can show that when one does this, if one chooses a restricted type of representation for a Clifford algebra, then one obtains the same probabilities that are seen in the usual spinor representation. But it turns out that ALL the representations that one sees in the literature are of these restricted type.

The usual spinor probabilities, when translated into density matrix form, become traces. The way I'm calculating probabilites is with the geometric squared magnitude instead.

As an example, consider some 2x2 matrix over the complexes. Such a matrix can always be written as a sum over complex multiples of unity and the Pauli spin matrices. That is:

M = \alpha_1\hat{1} + \alpha_x\sigma_x + \alpha_y\sigma_y + \alpha_z\sigma_z

where \alpha_\chi are complex constants. Then define

|M|^2_G = \sum_\chi |\alpha_\chi|^2.

If you happen to have two pure density matrices, for example \rho and \rho', then the probability of transition between the two states described by the two matrices is

P_{\rho,\rho'} = \tr(\rho\rho')

It turns out that the above probability is proportional to |\rho\rho'|_G^2. Furthermore, the same thing applies to all the usual representations of the Dirac algebra. The constant of proportionality is just tr(\hat{1}) or 2 for the Pauli algebra and 4 for the Dirac algebra.

However, there do exist representations where the geometric probability is not equivalent to the spinor probability (or the trace), and these apply to the masses of the fermions.

As an example of the "general" representations where the geometric probability is not equivalent to the standard ones, one must understand more about representations than I can put into this short note. I'll put it into a successor.

The reason one ends up with square roots in the mass matrix is because the fundamental probability relation, when defined in terms of spinors, is 4th order. That is, P = <A|B><B|A> has four contributions from spinors. The natural fields are the density matrices, and in that representation, probabilities are proportional to squares (or products) of the objects, as we would expect. The usual requirement that a wave function be normalized, <A|A>=1 would better be written as <A|A><A|A> = 1.

Carl

[CarlB]

A geometric characterization of the representation of a Clifford algebra in matrices using primitive idempotents (ideals):

We can fully characterize a representation of a Clifford algebra in complex matrices NxN by knowing the Clifford algebra equivalents of N+1 simple matrices that are all primitive idempotents. There are N "diagonal primitive idempotents". The first N are just the diagonal matrices that have a single one somewhere along the diagonal and are everywhere else zero. We will label these by \iota_j where "j" gives the position on the diagonal, 0 to N-1, where the one is. But this is not quite enough to specify a representation. To complete the specification, we can add the "democratic primitive idempotent". This is the matrix that has 1/N in every location, which we will call \iota_D.

To get any specific position on the matrix, for example the (j,k) spot, we can simply multiply:

M_{jk} = \iota_j \; \iota_D \; \iota_k

Thus it is clear that if you know these N+1 elements of the Clifford algebra, then you know the representation.

Now the odd thing about ALL the usual representations of Clifford algebras used in physics is that the degrees of freedom contained in \iota_D are completely independent of the degrees of freedom contained in the set of \iota_j. For example, in the Pauli algebra, the diagonal primitive idempotents have z as a degree of freedom, while the counter diagonal elements have x or y. These are completely independent, in terms of degrees of freedom. The same thing is true for the various representations of the Dirac matrices that appear in the literature.

Now it turns out that it is possible to make representations where there are degrees of freedom shared between the diagonal and the democratic primitive idempotents. And it is these representations where the geometric probability comes out different from the usual trace.

To explain more about this, I would have to go into more detail on how one obtains the primitive idempotents (or primitive ideals) of a Clifford algebra. But it's very beautiful and I can't wait to release it. Right now, I'm working on designing a biodiesel plant design instead.

Perhaps I should remind the reader that the "democratic mixing matrices" also appear naturally in the Koide mass formula.

Carl

[arivero]

Right now, I'm working on designing a biodiesel plant design instead.


You mean you are being paid? Interesting, even if offtopic. Lot of deforestation will follow (as it is said of soja beans and of palm oil).


Perhaps I should remind the reader that the "democratic mixing matrices" also appear naturally in the Koide mass formula.


Yeah, you should :approve: . It has been a long time ago since this thread was alive, and without this reference it seems a bit off.

[CarlB]

Now it turns out that it is possible to make representations where there are degrees of freedom shared between the diagonal and the democratic primitive idempotents. And it is these representations where the geometric probability comes out different from the usual trace.

I should exhibit a representation of SU(2) which does not satisfy the requirement that the geometrically defined squared magnitude is equivalent to the usual spinor squared magnitude (and therefore to the traces). So here it is:

\sigma_x = \left(\begin{array}{cc}0.0&0.2\\5.0&0.0\end{array}\right)

\sigma_y = \left(\begin{array}{cc}0.0&-0.2i\\5.0i&0.0\end{array}\right)

\sigma_z = \left(\begin{array}{cc}1.0&0.0\\0.0&-1.0\end{array}\right)

This corresponds to a democratic primitive idempotent of

\iota_D = (1.0 + 2.6\sigma_x + 2.4\sigma_x\sigma_z)/2.0

and the diagonal primitive idempotents are the usual:

\iota_\pm = (1.0 \pm \sigma_z)/2.0

From the above, you can see that the degrees of freedom of both the diagonal primitive idempotents and the democratic primitive idempotent include the sigma_z. That is, (sigma_x)(sigma_x sigma_z) = sigma_z so that the democratic primitive idempotent encroaches on the turf of the diagonal primitive idempotents.

To see that the above violates the relationship between squared magnitude and spinor squared magnitude one may consider the element |x><x| where |x> is the spin-1/2 eigenvector for spin in the x direction. Of course <x|x> = 1, but ||x><x||x><x|| in the geometric language is very large.

Oh, and I get paid for the biodiesel design if the design gets accepted by the customer. As far as deforestation, this shouldn't contribute much. It wil use US oil crops and US surplus oils (i.e. yellow grease, etc.).

In the tropics, its pretty clear that the oil palm is the way to go. Those things are prolific. But I really don't think science has found the optimum oil producing plant to grow in the US type climate.

Carl

[arivero]

I should exhibit a representation of SU(2) which does not satisfy the requirement that the geometrically defined squared magnitude is equivalent to the usual spinor squared magnitude (and therefore to the traces). So here it is:

\sigma_x = \left(\begin{array}{cc}0.0&0.2\\5.0&0.0\end{array}\right)

\sigma_y = \left(\begin{array}{cc}0.0&-0.2i\\5.0i&0.0\end{array}\right)

\sigma_z = \left(\begin{array}{cc}1.0&0.0\\0.0&-1.0\end{array}\right)



It is more a sort of deformation of SU(2) than a representation of [the infinitesimal generators of] SU(2), isn't it? People expects selfadjointness to get unitary generators, thus unitary group, which is the thing that the U stands for, after all. Just terminology issue, but it can be a communication problem.

[CarlB]

I'm sure I've been sloppy with my terminology. I tend to be a very calculation oriented person. Let me try again. SU(2) is the group, su(2) is the algebra.

SU(2) refers to the fact that one can represent the elements of the group by special unitary 2x2 matrices. But that is not the only representation of the group, nor are the Pauli matrices the only representation of the algebra su(2).

The set of deformed matrices are a representation of su(2) in that they satisfy the usual commutation relation:

\sigma_j\sigma_k-\sigma_k\sigma_j = 2i \epsilon_{jkm}\sigma_m.

If we wanted to describe the SU(2) that this defines, then we just take exponentials of the above. But since the above is identical to the usual su(2), we'll just get a deformation of the usual SU(2). Uh, what's the correct word for "deformation"? In this case it's something like isomorphism or homeomorphism or...

By the way, the better (i.e. geometric) way of writing the above relation is to replace i with \sigma_x \sigma_y \sigma_z.

Carl

[arivero]

But since the above is identical to the usual su(2), we'll just get a deformation of the usual SU(2). Uh, what's the correct word for "deformation"? In this case it's something like isomorphism or homeomorphism or...

Hmm I think it is not going to be an isomorphism; supposse that in the original representation of SU(2) I build unitary elements A_i=\exp ( a_i^\mu \sigma_\mu) from the sigmas and triplets of real numbers; this is the standard business. Now you should have an invertible map from this real triplet in in the standard representation to the real triplet components in your representation, ie a map (a^\mu) \leftrightarrow (b^\mu). And then in order to check that you representation it really an iso-, and not a deformation, you should be sure of the details, for instance that a composition of any three elements giving the identity, A_1 A_2 A_3 = I in the original representation should still give the identity in your mapped representation. I doubt it, because your sigmas are not idempotent anymore. So let it to be called "deformation".

[arivero]

By the way, the better (i.e. geometric) way of writing the above relation is to replace i with \sigma_x \sigma_y \sigma_z.

Yeah, it is a very depth trick because somehow you want the product of the three sigma to have a meaning as volume element (or dual to it), the product of two as surface elements (or dual to it) and so on. But all these topics should be theme for a separate, well indexed, thread somewhere. Clifford things.

[CarlB]

I doubt it, because your sigmas are not idempotent anymore.

I'm not sure where you're going here. \sigma_x is never idempotent. They square to unity, in both cases. And the idempotents you get from them, for example,

(1+\sigma_x)/2.0

is idempotent in both representations. In the usual, one obtains:

\left(\begin{array}{cc}0.5&0.5\\0.5&0.5\end{array}\right)

In the deformed representation, one obtains:

\left(\begin{array}{cc}0.5&2.5\\0.1&0.5\end{array}\right)

One can easily verify that both these representations of (1+\sigma_x)/2.0 are, in fact, idempotent.

One can define a Clifford algebra as a vector space. On doing this, one obtains that addition is automatically identical between representations, and in this case the multiplication is identical too. So I'm pretty sure that they're both equally valid representations of su(2), and after exponentiation, SU(2).

Anyway, all this gets back to quarks and leptons in that I've discovered a beautiful representation of both the quarks and leptons that requires this as a basis for the theory.

What is going on here, in the making of the representations non unitary, is that we are preserving the usual quantum mechanics for the individual particles (i.e. the ideal generated by (1+\sigma_x) is isomorphic to the usual spin-1/2 no matter the representation), but are changing the relationship between different particles.

This shows up in the lepton mass formula because the different particles are different types (in terms of color). So you need a little more freedom than is available when you assume that the spinors (different particles) are separated from each other in the usual method of unitary representations. But they're still good su(2) reps.

It's kind of a long and involved thing and I'm splitting it up into two papers both of which are long and involved. The one having to do with representations also gets into the meaning of gauges and why Hestenes' factorization of the Dirac equation has an orientation built into it. For more on that subject, see the excellent short comment by Baylis:
http://www.arxiv.org/abs/quant-ph/0202060

Yet another way of describing this is that I am rejecting the splitting of the Banach space (or density matrix) into a Hilbert space (or spinors) because there are always multiple ways (i.e. a gauge) to do this. The Banach space is more physical because it does not depend on gauge. Along this line, but restricted to a single particle and the U(1) complex phase gauge, those who are fans of Bohmian mechanics need to read Hiley's paper which generalizes Bohmian mechanics to density matrices:
http://www.arxiv.org/abs/quant-ph/0005026

In terms of Schwinger's elegant measurement algebra, this amounts to writing the theory strictly in terms of simple measurements M(a), and rejecting the use of general measurements M(a,b). My paper shows that when one uses M(a,b), one automatically implies an orientation gauge similar to the one that Hestenes uses. And this gets back into the assumption that the vacuum is a physical state.

Carl

[CarlB]

Maybe this is the isomorphism that is needed to explain this better:

\left(\begin{array}{cc}a&b\\c&d\end{array}\right)
\left(\begin{array}{cc}A&B\\C&D\end{array}\right) =
\left(\begin{array}{cc}aA+bC&aB+bD\\cA+dC&cB+dD\end{array}\right).

then the isomorphism consists of multiplying the off diagonal elements by r and 1/r. Multiplication is preserved:

\left(\begin{array}{cc}a&br\\c/r&d\end{array}\right)
\left(\begin{array}{cc}A&Br\\C/r&D\end{array}\right) =
\left(\begin{array}{cc}aA+bC&(aB+bD)r\\(cA+dC)/r&cB+dD\end{array}\right).

and addition obviously is preserved too. Also note that the transformation preserves the trace as the diagonal elements are unchanged, and it is the trace that gives probabilities in the standard density matrix approach. But what is not preserved is the natural squared magnitude of matrices. That is, the squared magnitude of the untransformed matrix is |a|^2 + |b|^2 + |c|^2 + |d|^2, but the squared magnitude of the transformed matrix gets some r activity. The geometric squared magnitude is also not preserved. In particular, the coefficients for x and y are changed in magnitude by the r.

So this gives a way of rewriting QM where the traces (and therefore the usual way of calculating QM probabilities) is preserved, but the geometric method is not. This allows one to explore cases that are outside the standard model, but with a method that includes the standard model in that the geometric probability is identical to the trace in every standard representation of the standard model. It is only when we use these unstandard representations, (such as the su(2)/SU(2) modified by r) that the predictions can be made to differ, and only then in the geometric probabilities. The trace is unchanged.

I don't think that getting into the details of the "r" transformation advances understanding much. The same trick works in the Dirac matrices, but the number of degrees of freedom available to the representation is far far larger and you can't fit into a simple description like the above "r".

The better way of seeing everything is to just look at where the (matrix) diagonal primitive idempotents and the (matrix) democratic primitive idempotents map to (in the Clifford algebra), the rest follows automatically from that. If you want to know how many representations of the Dirac algebra exist that share the same diagonal representation (i.e. the same commuting operators) as the Weyl represenstation, for example, I think you should look at it geometrically. If you restrict yourself to representations where the spinor probabilities match the geometric probabilities, and you ignore the infinite number of cases you get by rotating x into y, you end up with 96 Weyl reps that share identical diagonal representations. The representations that give different geometric probabilites that I have found are all rotations, but are rotations that use cosh and sinh instead of cos and sin. That is, they rotate different signature elements of the canonical basis elements of the Clifford algebra.

For understanding all this it helps to understand how primitive idempotents appear in Clifford algebras. That is kind of complicated, but if you've heard of Radon-Hurwitz numbers you're probably already there. The paper I'm writing up on it is very very long and has lots and lots of examples and exercises so it should be easy, if boring, to understand. Give me a week to finish it off.

Carl

[arivero]

[QUOTE=CarlB]I'm not sure where you're going here. \sigma_x is never idempotent. They square to unity[/tex]
Sorry, my fault. If was referring to the fact that they should square to the unity, and you are right they do (doe to the decimal notation I didn't notice it rightly). Thus OK it is the same algebra.

[CarlB]

Another paper on the Koide mass formula:

On the Koide-like Relations for the Running Masses of Charged Leptons, Neutrinos and Quarks
Zhi-zhong Xing, He Zhang
Current experimental data indicate that the Koide relation for the pole masses of charged leptons, which can be parametrized as Q^{pole}_l = 2/3, is valid up to the accuracy of O(10^{-5}). We show that the running masses of charged leptons fail in satisfying the Koide relation (i.e., Q_l(\mu) \neq 2/3), but the discrepancy between Q_l (\mu) and Q^{pole}_l is only about 0.2% at \mu=M_Z. The Koide-like relations for the running masses of neutrinos (1/3 < Q_\nu(M_Z) < 0.6), up-type quarks (Q_{U}(M_Z) \sim 0.89) and down-type quarks (Q_{D}(M_Z) \sim 0.74) are also examined from M_Z up to the typical seesaw scale M_R \sim 10^{14} GeV, and they are found to be nearly stable against radiative corrections. The approximate stability of Q_{U}(\mu) and Q_{D}(\mu) is mainly attributed to the strong mass hierarchy of quarks, while that of Q_l(\mu) and Q_\nu(\mu) is essentially for the reason that the lepton mass ratios are rather insensitive to radiative corrections.
http://www.arxiv.org/abs/hep-ph/0602134

Carl

[arivero]

Another paper on the Koide mass formula:

http://www.arxiv.org/abs/hep-ph/0602134

Carl

Hmm it seems that at least we have contributed to create awareness on Koide's. In this case the paper does not quote any webpage thread, but it still quotes a couple papers that redirect towards the web discussions.

As for the paper, it is more cautious that hep-ph/0601031, I hope it will be published somewhere.

Also, we missed hep-ph/0510289 because it fails to mention Koide in the abstract.

[CarlB]

I suppose I should give the calculation that shows that, in the case of SU(2), the geometric squared magnitude gives the same number as the more usual trace.

Begin with two spinors:

|\alpha\rangle = \left(\begin{array}{c}1\\ \alpha\end{array}\right)
|\beta\rangle = \left(\begin{array}{c}1\\ \beta\end{array}\right)

These aren't normalized, to normalize divide by, for example, \sqrt{1+|\alpha|^2}. Then the probability associated with transititions from one of these forms to the other is given by:

P_{\alpha\beta} = \frac{1+\alpha^*\beta+\beta^*\alpha+|\alpha\beta|^ 2}
{(1+|\alpha|^2)(1+|\beta|^2)}

The density matrices associated with these spinors are (leaving off the normalization):

\rho_\alpha = \left(\begin{array}{cc}1&\alpha^*\\ \alpha&|\alpha|^2\end{array}\right)

and similarly for beta. The products of the two density matrices is:

M_{\alpha\beta} = \rho_\alpha\rho_\beta = \left(\begin{array}{cc}1+\alpha^*\beta&\beta^*+\alpha^*|\beta|^2\\ \alpha+\beta|\alpha|^2& \beta^*\alpha+|\alpha\beta|^2\end{array}\right)

again leaving off the normalization. Note that the trace gives the correct probability. I now write

M_{\alpha\beta} = c_1\hat{1} + c_x\sigma_x + c_y\sigma_y + c_z\sigma_z

where c_* are complex coefficients (you can redo this with real coefficients if you wish by going to the real basis that includes bivectors like sigma_x sigma_y and the pseudoscalar sigma_x sigma_y sigma_z). The coefficients are:

c_1 = (1 + \alpha^*\beta\ + \beta^*\alpha + |\alpha\beta|^2)/2
c_x = (\alpha+\beta|\alpha|^2+\beta^*+\alpha^*|\beta|^2)/2
c_y = (\alpha+\beta|\alpha|^2-\beta^*-\alpha^*|\beta|^2)/2i
c_z = (1 + \alpha^*\beta\ - \beta^*\alpha - |\alpha\beta|^2)/2

The geometric squared magnitude is |M_{\alpha\beta}|^2_G = |c_1|^2 + |c_x|^2 + |c_y|^2 + |c_z|^2. This can be computed easily by multiplying by the complex conjugate and adding. Parts of the 1 and the z calculations cancel each other, as do parts of the x and y calculations. What is left is equal, after a small amount of algebra, to:

|M_{\alpha\beta}|^2_G = 2(1 + \alpha^*\beta+\beta^*\alpha+|\alpha\beta|^2)(1+|\a lpha|^2)(1+|\beta|^2)

this is the same as the usual probability, after one corrects the normalization (i.e. the two things multiplying on the right), and divides by 2 to account for the fact that tr(1) = 2.

Thus we have that in the case of one spin-1/2 spinor transitioning into another, we can replace the spinor computation of the probability with a purely geometric calculation based on the geometric squared magnitude.

Now the trace is also geometrically defined in that it is the scalar part of the matrix. But the above shows that the geometric squared magnitude also gives the same result as the trace.

It should be noted that all the above calculations follow over into the various standard representations of the Dirac algebra, in particular the Weyl representation and the usual one.

We can distinguish between the trace and the geometric squared magnitude by going to representations of a Clifford algebra that are different from the usual ones. In particular, it is in the masses of the leptons that one can find evidence for the necessity of the squared magnitude, rather than the trace, as a measure of probability.

Carl

[arivero]

from Kea in other thread (http://www.physicsforums.com/showpost.php?p=929574&postcount=14 )

In this spirit:

Note that the reduced link of the electron (figure 17) is just the Hopf link, when the ends are connected up. Now taking Jones polynomials J at 5th roots of unity for universal quantum computation, one finds that

J_{\textrm{Hopf}} = d = 1.618 \cdots

the golden ratio. Now let a deformation parameter be

q = e^{\frac{2 \pi}{2 + d}}

namely the SU(2)_{q} conformal field theory expression. It is well known that the spin half rep quantum dimension is given by [ 2 ]_{q} = q + q^{-1}. Putting two of these electron graph invariants together one estimates

\alpha = 4 (q + q^{-1})^{2} = 137.08

This is of course an extremely coarse estimate, and hence not particularly accurate. :smile:

[CarlB]

I found how to apply the Koide formula to the neutrino masses.

Several researchers have noted that the Koide formula is excluded from being applied to the masses of the neutrinos. For example, see section (III) in:
http://arxiv.org/abs/hep-ph/0601031

The experimental data that supposedly excludes the Koide formula is:

m_2^2 = m_1^2 + 8.0 \times 10^{-5}\; eV^2,\;\;\; (1)

m_3^2 = m_2^2 + 250.0 \times 10^{-5}\; eV^2,\;\;\; (2)

where I've left off the error bars as they are fairly small. Also note, I'm assuming the "normal" ordering, where m_1 < m_2 << m_3, which removes an absolute value from the LHS of (2).

Some time ago on this thread I published a formula for the Koide masses based on the eigenvalues of matrices of the type:

\mu\;\left( \begin{array}{ccc}1&\eta e^{+i\delta}&\eta e^{-i\delta}\\\eta e^{-i\delta}&1&\eta e^{+i\delta}\\\eta e^{+i\delta}&\eta e^{-i\delta}&1\end{array}\right)\;\;\; (3)

The above is derived as a form for cross family operators in my latest paper in equation (217):
http://brannenworks.com/GEOPROB.pdf

For the charged leptons, one uses \eta = \sqrt{0.5}. The three eigenvalues for a matrix of type (3) are given by:

\sqrt{m_n} = \mu(1 + 2\eta\cos(\delta + 2 n \pi/3)),\;\;\;\;\;\;n = 1,2,3\;\;\; (4)

If one is given a set of three eigenvalues, one can determine the value of \eta by dividing the following formulas:

\left(\sum \sqrt{m_n}\right)^2 = (3 \mu)^2 = 9\mu^2,\;\;\;\; (5)

\sum\left(\sqrt{m_n}\right)^2 = 3\mu^2(1 + 4\eta^2<\cos^2>) = \mu^2(1+2\eta^2),\;\;\;\; (6)

where "< >" means average value (i.e. average over the three roots), and the average value of the cos^2 is 1/2. (Alternatively, you can apply trigonometry.) The above formulas are obtained by taking traces of the square root mass matrix and its square, and they allow the computation of eta^2 from sets of possible eigenvalues.


If one tries various values for m_1^2 in the neutrino squared mass difference formula, one soon finds that the Koide mass formula is not possible, assuming one assumes positive square roots for the square roots of the masses. The Koide formula has eta^2 = 1/2, and for various choices of m1, the values of eta^2 are:

\begin{array}{cc}
m_1^2&\eta^2\\
0.0&0.3753\\
1.0&0.1628\\
8.0&0.0823\\
20.0&0.0491\end{array},\;\;\;\; (7)

Thus, eta is too low when m1=0, and increasing m1 simply makes eta even smaller. However, if we instead assume that sqrt(m1) < 0, then the Koide formula can be achieved with the following approximate neutrino masses:

m_1 = 0.00039\;eV,
m_2 = 0.00895\;eV,
m_3 = 0.05079\;eV,

and the Koide mass formula is met by the relation:

\frac{(-\sqrt{m_1} + \sqrt{m_2} + \sqrt{m_3})^2}{m_1+m_2+m_3} = \frac{3}{2}

Now so far this really hasn't shown anything. There was a free variable in the definition of the neutrino masses and I used it to satisfy the Koide relation by assuming a negative square root. However, it gets interesting when we compute the delta values for the charged leptons and compare it with the delta value for the neutral leptons. Turns out that the two angles differ, to within experimental error, by 15 degrees. That is,

\delta_0 = \delta_+ + \pi/12.

Is this a coincidence? I doubt it. Much more on this later. By the way, if you follow the logic of my paper, you will understand why I suspect that the square roots of the masses of the charged leptons should all be taken to be negative. This transforms the delta to a value that is consistent with a hidden dimension interpretation of the modification of the spinor probability formula P = (1+\cos(\theta))/2.

This sort of transformation takes delta to 60-delta. Another obvious transformation, (that leaves the eigenvalues unchanged) is to replace delta with -delta. Together, these give four representations possible for the charged leptons and also four for the neutral leptons. I expect that one (okay, two, since you can always complement the imaginary unit) of these 16 possibilities will be picked out by the mixing matrix for neutrinos, that is, the MNS matrix. This will complete my classification of the leptons, and it should be up here tomorrow.

Carl

[arivero]

Carlb perhaps you have an opportunity (take it with care) to pester the bosses. It seems that ours awareness-campaing on Koide's (I say ours, but also Li & Ma, and Koide himself or course) has culminated with three pages (46-48) in the last neutrino review of Mohapatra and Smirnov (http://arxiv.org/abs/hep-ph/0603118)

[Hans de Vries]

Charge Renormalization and the value of Alpha.

I came across a nice passage in Feynman's book "The Theory of
Fundamental processes" which has everything to do with our alpha
result. It shows Feynman's ideas (around 1961). In chapter 29 he
speculates that the theoretical value of alpha e_{th}^2 could be a nice
simple result from some future theory. The experimental value e_{th}^2
is then a corrected value due to vacuum polarization.

He gives as an example e_{th}^2 = 1/141 which is getting renormalized
to e_{exp}^2 = 1/137.036. (Just as example, no calculation) So the renor-
malized value is slightly higher as the theoretical one. (In Zee's
treatment it's the other way around)

This seems pretty much in the same ballpark as what we have:

e_{th} = e^{-\pi^2/4}, \quad e_{th}^2 = 1/139.045636

With our empirical corrective series:

e_{exp}\ =\ \left(1 + \alpha + \frac{\alpha^2}{2\pi}\right) e_{th}

We get a value e_{exp}^2 = 1/137.03599952 which coincides
with the two most accurate measurements:

1/137.03599911(46) From the Quantum Hall effect.
1/137.03599976(50) Magnetic Moment Anomaly
1/137.03599941(15) The two Combined

The series discussed by Feynman is the "string of loops": photon to
loop to photon to loop ... He has a formula:

e_{exp} = e_{th}/(1-Y)^{1/2}

Where Y approaches a constant for low (photon) q's and where the
contribution of the electron positron loops is given by X=q^2Y.
There's not so much more in the book on the (divergent) series.

Has there been more work from Feynman along these lines?


Regards, Hans



P.S: Our short series can be rewritten in order to express the series in
e=e_{th} instead of alpha. (although in a much more complicated fashion):


e_{exp}\ = \ e\ +\ e^3\ +\ \left( 2+\frac{1}{2\pi} \right)e^5\ +\
\left( 5+\frac{6}{2\pi} \right)e^7\
+\ \left( 14+\frac{28}{2\pi}+\frac{4}{4\pi^2} \right)e^9\ +\ \left(
48+\frac{104}{2\pi}+\frac{37}{4\pi^2} \right)e^{11} \ +\ ......

[arivero]

Charge Renormalization and the value of Alpha.

We get a value e_{exp}^2 = 1/137.03599952 which coincides
with the two most accurate measurements:

1/137.03599911(46) From the Quantum Hall effect.
1/137.03599976(50) Magnetic Moment Anomaly
1/137.03599941(15) The two Combined


Figure 6 of eprint http://arxiv.org/abs/nucl-ex/0404013 and figure 5 of http://arxiv.org/abs/hep-ex/0512026 are historic plots of the measurements of alpha during the last twenty years. It seems the Magnetic Moment result is being favoured.

[arivero]

Charge Renormalization and the value of Alpha.

I came across a nice passage in Feynman's book "The Theory of
Fundamental processes"

...

The series discussed by Feynman is the "string of loops": photon to
loop to photon to loop ... He has a formula:

e_{exp} = e_{th}/(1-Y)^{1/2}

Where Y approaches a constant for low (photon) q's and where the
contribution of the electron positron loops is given by X=q^2Y.
There's not so much more in the book on the (divergent) series.

Has there been more work from Feynman along these lines?


There are some approximations based on resummations of a series, I can recall for instance the "ladder approximation". On other hand, the point about renormalised series is that usually are self-generated by insertion of a finite number of counterterms that generates insertions in all orders of the expansion.

Historically, it could be that Feynman was following here, or hinting to, the "eigenvalue approach" of Adler.

[CarlB]

[QUOTE=CarlB]Another unusual number, this one having to do with the ratio of the charged leptons to the neutral lepton masses.

As mentioned above, the Koide mass formula can be rewritten in a form which removes a degree of freedom from the masses. That is,

\sqrt{m_n} = \mu_\chi(1 + \sqrt{2}\cos(\delta_\chi + 2n\pi/3)),

where \chi is either 0 for the neutral leptons (or neutrinos) or 1 for the charged leptons (i.e. electron, muon and tau). Having done this, the previous note found that the neutrino squared mass formulas can be met, and the usual charged lepton masses can be obtained by putting \delta_0 = 0.222222047 and \delta_1 = \delta_0 + \pi/12.

This note is to mention an unusual coincidence with the ratio \mu_1/mu_0:

\mu_1/\mu_0 = 3^{11}.

Now the masses are proportional to \mu^2, so the difference in scale between the charged and neutral leptons comes out to be 3^{22}. Using the best numbers for the squared neutrino mass differences, this relation is obtained to an accuracy of 0.1%, far greater than the purported accuracy of the squared neutrino mass measurements.

Assuming that the relation is exact gives a prediction for the squared neutrino masses a bit lower than the center of the current estimates. Instead of m_2^2-m_1^2 = 8.0 \times 10^{-5} eV^2, we have, if I recall correctly, 7.95 \times 10^{-5} eV^2. I'll see if I can get back to a computer fast enough to edit in the actual numbers, unless someone computes them out.

By the way, it is also possible to put the difference between 2/9 and \delta_1 in the form of a small constant times a power of three, but I'll leave that to the reader to find as its not as simple. If you happen to find one, please post it and we will see if we happened to find the same one.

As to why the charged leptons should have a mass with such a relationship, I can only suggest that in the mysterious reaction that allows a right handed lepton to transform itself into a left handed lepton and vice versa, there might be a pathway that is 11 steps longer for the neutrinos, with each step having a probability of 1/3. For example, perhaps the neutrinos require 12 steps while the charged leptons require only one.

The reason I was looking for powers of three is because the mysterious factor 2/9 in the delta function suggested that threes were important. Also, of course, there are three colors, three generations, and three elements in the Holy Trinity.

Carl

[arivero]

\mu_1/\mu_0 = 3^{11}.

Hmm without a model backing it, this is probably one of the most numerological formulae we have catalogued.

[arivero]

Recently I have got scent of the work of G. Rosen. Check eg SPIRES, because he does not arxiv:
http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=find+a+rosen%2C+g&FORMAT=WWW&SEQUENCE=
He did some attempts at combinatorial values for Sommerfeld constant, and in 1971 he was one of the people peering at the bandwagon of using alpha to exponentiate back from planck scale to electron scale, directly withut any renormalisation group trayectory. This was the paper Phys. Rev. D 4, 275–277 (1971) (http://prola.aps.org/abstract/PRD/v4/i2/p275_1)

Recently he is advocating that in all the four triplets of fermions one can find a pair related by m/M=e^{2 \sqrt 2} and that in each case the extant fermion can be adjusted from a third term depending on Barion number and Charge.

[CarlB]

Hmm without a model backing it, this is probably one of the most numerological formulae we have catalogued.

I agree completely. The hope is that some of these numerical coincidences will give a clue to the underlying theory, and under that assumption, I couldn't help but mention the coincidence. And of course the Koide formula doesn't have a model backing it yet.

My feeling is that the physics of the relationship between the generations is at the same stage as when the early physicists were puzzling out the relationships between the wavelengths of light emitted by excited states of the hydrogen atom.

Carl

[arivero]

The centrifugal force goes as
F={L \over r} {v \over r}
and you fix the second factor using a quantum condition so that

F={L \over r} m {c^2 \over \hbar}

Now, comparing with k/r^2 we have
k= L m r {c^2 \over \hbar} = L v
so the velocity (in units c) times the required angular momentum (in units \hbar) produces the coupling constant (in units c\hbar) of the Coulombian force needed to meet your requisites.


Er... I am telling an stupidity, or a triviality. The fact is that always k=L v, unrelated to your quantum condition.

The point I wanted to do is that your definition fixes L and v, so there are no more freedom to say that the coupling constant of the force is such or such, it is compulsorily L times v. Funny you have a method to generate coupling constants, but instead 1/137 you hit Weinberg angle.

Also there is a transition hidden somewhere. After all, your condition sets L as a function of spin s and then v as a function of L (via \gamma (\frac vc)^2 = L). But if the coupling increases enough (above unity in bohr-schroedinger, but not sure in your model) there are spiral orbits in the allowed range, and naive relativistic quantum mechanics fails. So at highers spins we fall into a strong coupling regime probably forbidden in r.q.m.


We did see that the clasical velocity can be defined as:

“The velocity of a mass with spin s rotating on an orbit
with a frequency corresponding to its rest mass and an
angular momentum \sqrt{ s(s+1}\ \hbar"

One gets a general solution for the 'classical velocity' of spin s:


\beta_s \ \ \ \ \ \ = \ \ \ \ \sqrt{\sqrt{\ \ s(s+1) \ \ + \ \ ( \frac{1}{2}
\ s(s+1) \ )^2 } \ \ - \ \ \frac{1}{2} \ s(s+1) }

Which solutions are dimensionless constants, independent
of the mass of the particle. The values of the common spins
are given below:


spin 0.0: __ 0.00000000000000000000000000000000
spin 0.5: __ 0.75414143528176709788873548859945
spin 1.0: __ 0.85559967716735219296923576621118
spin 1.5: __ 0.90580479773844104117525862119228
spin 2.0: __ 0.93433577808377694874713811004304
spin inf: __ 1.00000000000000000000000000000000




Weinberg’s Electro-Weak mixing angle becomes a dimension-
less constant as well and is given in the \sin^2 \theta_W form as:


\sin^2 \theta_W
\ \ \ \ = \ \ \ \ 1 \ - \ \frac{\beta^2_f }{\beta^2_b}
\ \ \ \ = \ \ \ \ 0.22310132230086634541466926662604

\sin^2 \theta_W
\ \ \ \ = \ \ \ \ 1 \ - \ \frac
{ \sqrt{\ \ \frac{1}{2}(\frac{1}{2}+1) \ \ + \ \ ( \frac{1}{2} \ \
\frac{1}{2}(\frac{1}{2}+1) \ )^2 } \ \ - \ \ \frac{1}{2} \
\ \frac{1}{2}(\frac{1}{2}+1) }
{\sqrt{\ \ 1(1+1) \ \ + \ \ ( \frac{1}{2} \ \
1(1+1) \ )^2 } \ \ - \ \ \frac{1}{2} \
\ 1(1+1) }

[Hans de Vries]

\beta_s \ \ \ \ \ \ = \ \ \ \ \sqrt{\sqrt{\ \ s(s+1) \ \ + \ \ ( \frac{1}{2}
\ s(s+1) \ )^2 } \ \ - \ \ \frac{1}{2} \ s(s+1) }

Which solutions are dimensionless constants, independent
of the mass of the particle. The values of the common spins
are given below:


spin 0.0: __ 0.00000000000000000000000000000000
spin 0.5: __ 0.75414143528176709788873548859945
spin 1.0: __ 0.85559967716735219296923576621118
spin 1.5: __ 0.90580479773844104117525862119228
spin 2.0: __ 0.93433577808377694874713811004304
spin inf: __ 1.00000000000000000000000000000000





These dimensionless numbers related to angular momentum may also be
derived from Quantum Mechanics. In fact this does solve one of the
ad-hoc presumptions that I had to make about a “missing” relativistic
increase in energy at very small radii where v must become relativistic.

The post which summarizes various numerical coincidences with these
numbers can be found here:

http://www.physicsforums.com/showpost.php?p=382642&postcount=44


Let me archieve the math of the QM derivation here below.


================================================== ===


There are 3 steps:

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.



- STEP 1 –


We want to re-introduce speed into QM as a function of (orbital)
angular momentum and radius.


Let the angular momentum be:


\mathbf{p} \times \mathbf{r}\ =\ \sqrt{l(l+1)} \hbar


Entering the speed v via SR the momentum as a function of
The radius becomes:


p\ =\ \frac{m_0 v}{\sqrt{1-v^2/c^2}}\ =\ \sqrt{l(l+1)} \frac{\hbar}{r}


We can write with r_c = \hbar/(mc) as the Compton radius:


\frac{v/c}{\sqrt{1-v^2/c^2}}\ =\ \sqrt{l(l+1)} \frac{\hbar}{m_0 c
r}\ =\ \sqrt{l(l+1)}\ \frac{r_c}{r}


Which we can rework to get gamma as a function of the radius:


\gamma \ =\ \frac{1}{\sqrt{1-v^2/c^2}}\ =\ \sqrt{1+l(l+1) \frac{r_c^2}{r^2}}


We can now also assign a physical speed as a function of
the radius:


v/c \ =\ \sqrt{\frac{l(l+1)}{\frac{r^2}{r_c^2}+l(l+1))}}


The apparent increase in mass needed to produce the angular
momentum at this speed:


\gamma m_0\ =\ \frac{m_0}{\sqrt{1-v^2/c^2}}\ =\ m_0\sqrt{1+ l(l+1)
\frac{r_0^2}{r^2}}


Problem: We don’t see this radius dependent increase in energy in QM
solutions with (orbital) angular momentum. The energy is independent
of the radius.


- STEP 2 –


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



E^2= p_r^2 c^2 + p_\theta^2 c^2 + p_\phi^2 c^2 + m_0^2 c^4\


We now want a negative square radial momentum term which
compensates for the relativistic mass increase:



E^2 - E_0^2\ \ =\ \ E^2 - m_0^2 c^4\


\mbox{So we need:}\ \ \ p_r^2 c^2 \ =\ - (p_\theta^2 c^2 + p_\phi^2 c^2)\



p_r^2 c^2 \ =\ m_0^2c^4 - E^2
\ =\ (1-\gamma^2)E_0^2



From the expression for gamma above:


(1-\gamma^2)\ =\ 1 - \left(\ \sqrt{1+l(l+1) \frac{r_c^2}{r^2}}\ \right)^2 = -\frac{l(l+1)}{r^2}\ r^2_c


The last term is exactly the term which enters the radial equation
from the theta equation when solving the Spherical Laplacian in the
Schroedinger or Dirac equations:


-\left( \frac{l(l+1)}{r^2}r^2_c \right)\ E_0^2\ =\ -\frac{l(l+1)}{r^2} \hbar^2 c^2


This term you can find in any textbook which handles the derivation
of the Hydrogen solutions.



- STEP 3 -


We want to know the radius r/r_c where the physical speed v/c
is such that the orbital frequency becomes equal to the de Broglie
frequency.


At one hand we now have a physical orbital frequency:



f\ =\ \frac{v}{2\pi r}\ =\ \frac{c}{2\pi r }\ \sqrt{\frac{l(l+1) }
{\frac{r^2}{r_c^2}\ +\ l(l+1)}}


At the other hand we have the deBroglie frequency:


f\ =\ \frac{mc^2}{h}\ =\ \frac{mc^2}{2\pi\hbar}\ =\
\frac{c}{2\pi}\frac{1}{r_c}


Substituting the latter in the first:


\frac{r}{r_c}\ =\ \sqrt{\frac{l(l+1) } {\frac{r^2}{r_c^2}\ +\
l(l+1)}}


Giving the quadratic equation:


\left(\frac{r^2}{r_c^2}\right)^2\ +\
l(l+1)\left(\frac{r^2}{r_c^2}\right)\ -\ l(l+1)\ =\ 0


Which is solved by:


\left(\frac{r^2}{r_c^2}\right)\ =\ -\frac{1}{2} l(l+1) \pm \
\sqrt{\left(\frac{1}{2} l(l+1) \right)^2 + l(l+1)}


So, finally we get our dimensionless numbers related to
quantum mechanical angular momentum:


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





Regards, Hans

[arivero]

I was thinking, should we add to the list the suggested relationship between mass of top and fermi constant?

[arivero]

I was thinking, should we add to the list the suggested relationship between mass of top and fermi constant?

Lets go. From the pdg (http://pdg.lbl.gov/2005/reviews/stanmodelrpp.pdf) we can get the important data: Fermi constant is measured in muon decay, being G_F=1.16637(1) 10^-5 GeV^-2 and it relates to electroweak model in the

{G_F \over \sqrt 2} = \frac 18 {g^2 \over M^2_W}

Where the inverse of the quantity \frac 12 {g \over M_W} is usually referred as the "electroweak vacuum", and from the above equation it has a value about 246.22 GeV.

On the other hand the top, as any other fermion, gets mass via a Yukawian term y_i\bar\psi_i h \psi_i that under breaking generates a mass term
\bar \psi_i {y_i \over \sqrt 2} { 2 M_W \over g} \psi \equiv m_i \bar \psi_i \psi_i

(Fermions couple to the Higgs boson via this same Yukawa-originated term
\bar \psi_i {y_i \over \sqrt 2} H \psi=\bar \psi_i {g m_i \over 2 M_W} H \psi
and it is interesting to note that we have already met m_\mu/ m_W before in this thread.)

(Above, h is the higgs field in unbroken coordinates, while H is the extant Higgs boson of the Minimal Standard Model.)


Well, the point is that m_i^2 G_F = y_i^2 {1 \over 2 \sqrt 2 }
so that
y_i^2= 2 \sqrt 2 G_F m_i^2
and in particular with m_t=172.5 \pm 2.3 (http://arxiv.org/abs/hep-ex/0603039) we have (assuming G_F exact)

2 \sqrt 2 G_F m_t^2 = .9816 \pm 0.026

Or


y_t = 0.991 \pm 0.013


There is some theoretical justification, from infrared fixed points of the renormalisation group, to have a value of order unity, but nothing about exactly unity (In quantized Bohr-Sommerfeld relativity, a unit coupling in a coulombian interaction corresponds to the situation where the trajectory of the lowest state becomes unstable and spirals into the origin of the potential. In fact such situation could be used to define Planck's constant as coincinding with the limit angular momentum). A mass equal to the value from fermi constant, ie equal to 174.1042 GeV (\pm 0.00075), should be sort of theoretical paradise or nightmare, so it is important to refine the measure.

Perhaps it should be clarified that the sqrt(2) factors come only from two sources: the traditional normalisation of Fermi interaction gives the sqrt(2) that stands even in the final formula. The other sqrt(2) is the traditional normalisation of the vacuum and it goes both to the top yukawa coupling and, squared, to the first equation so that it cancels out at the end. The only consequence of this factor is the value of the "electroweak vacuum". Some books change normalisation so that this value is also equal to 174 GeV instead 246 but most books keep at 246 so surely there is some good argument to stay so.

[arivero]

Lets go. From the pdg (http://pdg.lbl.gov/2005/reviews/stanmodelrpp.pdf) we can get the important data: Fermi constant is measured in muon decay, being G_F=1.16637(1) 10^-5 GeV^-2 and it relates to electroweak model in the

{G_F \over \sqrt 2} = \frac 18 {g^2 \over M^2_W}

Where the inverse of the quantity \frac 12 {g \over M_W} is usually referred as the "electroweak vacuum", and from the above equation it has a value about 246.22 GeV.

On the other hand the top, as any other fermion, gets mass via a Yukawian term y_i\bar\psi_i h \psi_i that under breaking generates a mass term
\bar \psi_i {y_i \over \sqrt 2} { 2 M_W \over g} \psi \equiv m_i \bar \psi_i \psi_i

(Fermions couple to the Higgs boson via this same Yukawa-originated term
\bar \psi_i {y_i \over \sqrt 2} H \psi=\bar \psi_i {g m_i \over 2 M_W} H \psi
and it is interesting to note that we have already met m_\mu/ m_W before in this thread.)

(Above, h is the higgs field in unbroken coordinates, while H is the extant Higgs boson of the Minimal Standard Model.)


Well, the point is that m_i^2 G_F = y_i^2 {1 \over 2 \sqrt 2 }
so that
y_i^2= 2 \sqrt 2 G_F m_i^2
and in particular with m_t=172.5 \pm 2.3 (http://arxiv.org/abs/hep-ex/0603039) we have (assuming G_F exact)

2 \sqrt 2 G_F m_t^2 = .9816 \pm 0.026

Or


y_t = 0.991 \pm 0.013


There is some theoretical justification, from infrared fixed points of the renormalisation group, to have a value of order unity, but nothing about exactly unity (In quantized Bohr-Sommerfeld relativity, a unit coupling in a coulombian interaction corresponds to the situation where the trajectory of the lowest state becomes unstable and spirals into the origin of the potential. In fact such situation could be used to define Planck's constant as coincinding with the limit angular momentum). A mass equal to the value from fermi constant, ie equal to 174.1042 GeV (\pm 0.00075), should be sort of theoretical paradise or nightmare, so it is important to refine the measure.

Perhaps it should be clarified that the sqrt(2) factors come only from two sources: the traditional normalisation of Fermi interaction gives the sqrt(2) that stands even in the final formula. The other sqrt(2) is the traditional normalisation of the vacuum and it goes both to the top yukawa coupling and, squared, to the first equation so that it cancels out at the end. The only consequence of this factor is the value of the "electroweak vacuum". Some books change normalisation so that this value is also equal to 174 GeV instead 246 but most books keep at 246 so surely there is some good argument to stay so.

[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 . :confused:

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} =
\; 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 lenght 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
\ \ \ \ = \ \ \ \ 1 \ - \ \frac{\beta^2_f }{\beta^2_b}
\ \ \ \ = \ \ \ \ 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 (http://www.physicsforums.com/showpost.php?p=382642&postcount=44). 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 From c numbers to q numbers (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)


:bugeye:
{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)}
:eek:



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, in german (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, in german (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 :wink: 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 uncertanty principle for J).

[arivero]

So, finally we get our dimensionless numbers related to
quantum mechanical angular momentum:

... =\ \sqrt{-\frac{1}{2} l(l+1)\ \ +\ \
\sqrt{\left(\frac{1}{2} l(l+1) \right)^2\ +\ l(l+1)} }



A fast trivial observation is that operator-wise this is still

\beta^2= \left<\bar \Psi \left| {-\frac{1}{2} J^2\ \ +\ \
\sqrt{J^2 + \left(\frac{1}{2} J^2 \right)^2\ } } \right| \Psi \right>

and thus


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


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

[arivero]

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


(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

\left| \Psi_{J=1} \right> = Q \left| \Psi_{J=1/2} \right>


We could even define the operatorial object


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


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. :smile:

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]

While looking for the concept of hyperzitterbewegung (http://www.springerlink.com/(xxt5cuergasy5u2cyeweirbm)/app/home/contribution.asp?referrer=parent&backto=issue,6,13;journal,898,2386;linkingpublicat ionresults,1:100479,1), 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]

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



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)


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


or



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


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...


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

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]

I should mention this too :rofl:

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 has IP# 137.036.004.003


Cobleskill is approx "160 miles northwest of New York City and midway between Albany and Oneonta"

[arivero]

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


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

\beta^2_{-1}= -2.73205080756887729352

Fix the above mentioned quartic coupling to be \lambda=1. Then we can produce the "vacuum mass"

v=\sqrt{-m^2 \over \lambda} = M_{Z0}\sqrt{ -\beta^2_{-1} \over \lambda \beta^2_1} = M_{Z0} 1.9318516525781365


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 o:)

[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

M_W^2 Sin^2 \theta = {\pi \alpha \over \sqrt 2 G_F } = {2 v^2 \pi \alpha }


Thus

({\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

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


which for \lambda=1 gives (ooooh!)

\alpha=.00739161112923688931...

\alpha=1/135.28... :redface:

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
\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... :redface:



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... :redface:

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
\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', you can eliminate the O(\alpha^2).

This gives a value

\delta_1 = 2/9 - \frac{4\pi\alpha'}{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"

v=\sqrt{-m^2 \over \lambda} = M_{Z0}\sqrt{ -\beta^2_{-1} \over \lambda \beta^2_1} = M_{Z0} 1.9318516525781365


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:


i\frac{\beta_-}{\beta_+}\ = \frac{1}{\sqrt{1-\beta^2_+}}


and thus:


\tanh{(\cosh^{-1}{(1.9318516525781365)})}\ =\ \beta_1


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 :cool:

:cool: 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 :biggrin: but one is almost tempted to predict that the extant term \beta_{-1/2} is to be used to get the Higgs boson mass. :cool:


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 :redface: 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 (http://dorigo.wordpress.com/2006/04/24/how-well-will-cdf-do-with-the-top-quark-mass/)

[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 =
\begin{vmatrix}{ 0 -x & \sqrt b \cr \sqrt b & -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 writen as asking for the eigenvalues of the object


\begin{pmatrix} 0 & 1 \cr 1 & 0 \end{pmatrix} \sqrt{-C_1 C_2} + \begin{pmatrix}{0 & 0 \cr 0 & -1}\end{pmatrix} (-C_2)}


Or, more generally, of


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


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 :frown: :confused:. 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). :biggrin: 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

A \equiv \sigma_\perp \otimes \sqrt{-C_1 C_2} +
{ {\bf I} - \sigma_z \over 2 } \otimes C_2}


and now an auxiliar operator

B \equiv \sigma_\perp \otimes \sqrt{-C_1 C_2} +
{ \sigma_z } \otimes C_2}


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... (http://www.physicsforums.com/showpost.php?p=969833&postcount=214)
the sin of Weinberg angle at mass shell =0.2231013223... (http://www.physicsforums.com/showpost.php?p=382642&postcount=44)

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). :biggrin: .

arivero, sorry I'm not a musician, :frown: William B. Conner was. :biggrin: 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, :frown: William B. Conner was. :biggrin: 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 :devil: :mad: 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. :cool:
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] :rolleyes:

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. :cool:


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] :rolleyes:


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... (http://www.physicsforums.com/showpost.php?p=382642&postcount=44)


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


{\alpha \over 2} {m \over M_W} \approx {m_e \over m_\mu}



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


and then derived from both,

\left({\alpha \over 2}\right)^3 {m m_Z^2 \over M_W^2} \approx ({m_e \over m_\tau})^2


But we also have Yablon observation,

m_\tau \approx \alpha \sqrt 2 \ 172.18 GeV


And then of course we have Koide's

\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})


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

http://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....? :surprised 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:


\ln_2{\left(\frac{\mbox{value}}{\mbox{error}}\righ t)}


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

Add for each number used:


\ln_2{\left(\mbox{number}\right)}


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:


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


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]

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



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:


M_s^4 - C_2M_s^2 + C_1C_2 = 0


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


M_s^4 + C_2M_s^2 + C_1C_2 = 0


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:


M_s^4 \pm C_2M_s^2 - C_1C_2 = 0


We get complex masses but:


|M_s|^2 \propto S


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:


M_s^4 - C_2M_s^2 + C_1C_2 = 0


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 :smile: ).

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. :smile:

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 :smile: ).

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)}{\le ft(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)}{\le ft(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}}=\fra c{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}}=\lef t(\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 Zwiebach (http://web.mit.edu/physics/facultyandstaff/faculty/barton_zwiebach.htm)'s book (available from time to time in P2P networks). and also about middleway of his MIT 8.251 (http://web.mit.edu/8.251/www/index.html)"String Theory for Undergraduates" course (now also offered as OpenCourseWare (http://ocw.mit.edu/OcwWeb/Physics/8-251Spring-2005/CourseHome/index.htm)). Similar courses are offered elsewhere, for instance I googled into others from ucsb (http://www.physics.ucsb.edu/~phys230B/lectures_homework_old.html) and caltech (http://www.theory.caltech.edu/~mschulz/Ph135c/).

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 :tongue2:

[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)}{\le ft(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) (http://arxiv.org/abs/math.NT/0401406) 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:


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


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


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


The Energy Density is given by:


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


We integrate over r.


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



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


To get the result:


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


and with,


e^x \ =\ 1 + x + \frac{1}{2}x^2 + ....


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


{\cal E} \ =\ \frac{q^2}{8\pi\epsilon r_o}\


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


Regards, Hans

[Hans de Vries]

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 :smile:

[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:

[itex]\alpha_{exp}[/tex] = 1 /137.035 999 528 369 196 352 446 647 041



If we now calculate g/2 with the parameters of Kinoshιto then we find:

1.001 159 652 180 85____(+/-76) Measured by Gabrielse
1.001 159 652 180 68____QED version
1.001 159 652 180 71____ElectroWeak version
1.001 159 652 182 38____Full SM version including hadronic contributions

The last three ones are calculated from our result. So this seems to exclude
the version which includes the hadronic contributions.


Regards, Hans

[physmike]

... 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]

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

[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 realised 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]

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....

http://www.amazon.com/gp/product/customer-reviews/1579550088/103-1417335-7282252?redirect=true&_encoding=UTF8


Regards, Hans.

[Ganzfeld]

Hans

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

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

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

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

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

[arivero]

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

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

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

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

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

[arivero]

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

[physmike]

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


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

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


Some general references to show partially related physics:

On the Sensations of Tone , Hermann Helmholtz

Fundamentals of Musical Acoustics, Arthur H. Benade

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

Music, Physics and Engineering, Harry F. Olson

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

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

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

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


Physics and pyramids ... some history,

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

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

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

[arivero]

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

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

[physmike]

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

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

All the best, physmike

[CarlB]

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

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

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

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

Carl

[Ganzfeld]

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

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

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

[arivero]

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


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

[arivero]

2) An history of numerological approaches to the fine structure constant is contained in
H. Kragh, "Magic Number: A Partial History of Fine-Structure Constant", Arch Hist Exact Sci 57 (2003) p 395-431
A related paper by the same author is "The fine structure constant before quantum mechanics". There, a subsection "Dimensionless speculations" tells about early numerology and conjectures.

[Kea]

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

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

where e is electron charge.

[Hans de Vries]

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

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

where e is electron charge.

Hi, Kea.

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

1/137.035 999 528 369


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

1/137.035999710 (96)

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

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

1/137.035999110(460)

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

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

e^{-\pi^2/4}

and then use a Charge Renormalization Factor given by:

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

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

The last term brings the result from:
1/137.038339943 to:
1/137.035999528
So it improves the result by a factor 12800 ...


Pfff, I've spend so much time to explain the truncated series from
geometrical physics... I've countless numbers of scratch pages with
potentials, propagators, Green functions, Fourier transforms.

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


Regards, Hans.

[Hans de Vries]

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

0.00115965218085 (76).

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

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

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

g/g' = 1.0022828 = 1.001140762

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


Regards, Hans.

[Hans de Vries]

Mass relations between the vector bosons and the leptons.


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

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

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

Now, we already found the following on this thread:


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


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

The following we also found earlier on this thread:


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


This gives another lepton/vector_boson mass ratio. The numbers fit very
well. Remarkable is that the properties of the muon and Z explain the
electron/W mass ratio, while in the first case it was the other way around:
The properties of the electron and W explained the muon/Z mass ratio....


And then the tau. where does this leave the tau-lepton? If this is also a
precession/spin ratio, then it would need a significantly stronger coupling
because its mass (=frequency) is 16.8183 times higher as that of the muon.

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

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

This leads us to a new numerical coincident:


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


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

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


Regards, Hans.


PS: See also http://arxiv.org/abs/hep-ph/0503104
(and http://arxiv.org/abs/hep-ph/0604035 for \alpha_s(m_Z))

[Severian]

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



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



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

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

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

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

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

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

[Hans de Vries]

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

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


Hi, Severian.

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

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

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

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


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

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

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

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

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


Regards, Hans

[Severian]

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


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


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


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


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


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

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

[arivero]

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

...


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



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

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

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

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

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

[arivero]

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

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


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


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

[arivero]

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

[Hans de Vries]

Actually we have already an uninvited aparision of alpha_s .

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


Regards, Hans

[arivero]

Just adding this one to the thread for reference

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

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

[arivero]

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

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

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

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

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

Now the meson sum is

134.9766+139.57018+139.57018+547.51+497.648+497.64 8+493.677+493.677
=2944.277

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

938.27203+939.56536+1115.683+1192.642+1189.37+1197 .449+1321.31+1314.83=9209.12139=
9.815 M_p, while pi square is about 9.87. So off by 0.55% this time but no mixing to be blamed here.

[Demystifier]

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

[arivero]

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

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

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

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

[Demystifier]

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

[arivero]

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

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

I do not know what to do of his "social experience". I remember, as undergradute student, how excited I was about the articles of Cvitanovic on Chaos theory, and how a time later I was not so fond of them. But he got to bail out into another physics area, at least. On the other hand, 20 years later, Kino****a (could someone edit it our of the politically correct spelling rules of PF, please!) is the leader of the perturbative calculation effort (http://es.arxiv.org/abs/hep-ph/0507249), and it is because of him, perhaps, that the (g-2) is still an important test of the standard model.

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

[arivero]

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

[CarlB]

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

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

[arivero]

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

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


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

[Hans de Vries]

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

Good to see you posting!

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



We had two numerical coincidences for the mW/mZ mass ratio on this thread:

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

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


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

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

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


Regards, Hans

[arivero]

Good to see you posting!

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

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

[arivero]

Via Woit's blog, a pointer to a talk from Giorgi:
http://www2.yukawa.kyoto-u.ac.jp/~yt100sym/files/yt100sym_georgi.pdf

There, in slides 88 ff., the history of Bj's relation for m_e\over m_\mu is discussed.

[CarlB]

Nice talk, and very good slides. They give the talking points along with the images.

This past weekened I was looking for (one of my many alleged) copy of Georgi's book. I didn't find it.

The objective was to tie down the 3x3 circulant matrices as an example of an SU(3) symmetry. I managed to make some progress.

With the usual representation of SU(3), the canonical particles are the eigenvectors of the diagonalized operators. Turning these into density operators (like I always do), the states are:
\left(\begin{array}{ccc}1&&\\&0&\\&&0\end{array}\right)
\left(\begin{array}{ccc}0&&\\&1&\\&&0\end{array}\right)
\left(\begin{array}{ccc}0&&\\&0&\\&&1\end{array}\right)

These are primitive idempotents. Call the associated spinors |1>, |2>, and |3>. Their eigenvalues are (+1,+1), (+1,-1), and (-2,0). We want to map these into the circulant primitive idempotents:

\frac{1}{3}\left(\begin{array}{ccc}1&w^{+n}&w^{-n}\\w^{-n}&1&w^{+n}\\w^{+n}&w^{-n}&1\end{array}\right)
where w = \exp(2i\pi/3) and n=1, 2, and 3. Call the three associated spinors |R>, |G>, |B>.

The mapping is then given by S = |R><1| + |G><2| + |B><3|, and its inverse and one takes

a -> S a S^{-1}

where a is any of the 8 generators of SU(3).

On doing this, one finds that, sure enough, the diagonalized (commuting) SU(3) generators become commuting circulant matrices. And since the trace is conserved by the S mapping, the diagonal terms are zero. What one ends up with for the commuting circulant generators of SU(3) are:

\left(\begin{array}{ccc}0&1&1\\1&0&1\\1&1&0\end{array}\right)
\left(\begin{array}{ccc}0&+i&-i\\-i&0&+i\\+i&-i&0\end{array}\right)

where I have left off some unimportant multiplications by constants. Note that the above are Hermitian and circulant. The other generators of SU(3) end up non circulant, just as in the usual representation of SU(3) they are non diagonal.

Carl

[CarlB]

Alejandro, I've been looking at how one would apply Koide's mass formula to the mesons and baryons. There are some really cool things you can do with resonances, but the mass measurements of resonances are not that accurate so the statistics are not terribly convincing. However, the masses of the neutron and proton are very carefully measured so I will talk about them now.

In rewriting Koide's formula into eigenvector form, you may recall that there was an overall scale factor. It was equal to the average of the square roots of the charged leptons:

\begin{array}{ccc}
\textrm{Particle}&\textrm{Mass (eV)}&\sqrt{\textrm{eV}}\\
\textrm{Electron}& 510998.91& 714.84187762049867\\
\textrm{muon}&105658369.2&10279.02569312870236\\
\textrm{tauon}&1776990000&42154.35920518778328\\
\end{array}

The sum of square roots is 53148.22677593698431 and so the average square root charged lepton mass is: 17716.07559197899477.

In my version of preons, the "average root" is the contribution of the valence preons to the amplitude of a mass interaction in the charged leptons. The mass comes about by squaring the amplitude. The rest of the amplitude comes from the sea preons, and can be negative or positive. The sea preon amplitude is sqrt(1/2) times the valence preon amplitude, but there are twice as many sea preons, hence the sqrt(2) in the Koide formula.

The idea is that the electron is light is not because it is made of parts that are light (compared to the muon and tau), but instead because the sea and valence contributions cancel almost completely. And the neutrinos are unnaturally light because their mass is coupled through 11 stages of sterile neutrinos. The sterile neutrinos aren't seen because they don't interact weakly (or strongly), but their light weight is not an indication that the particles they are made of have a weight different from the ones making up the charged leptons. Instead, the charged and neutral leptons are just orthogonal primitive idempotents (somehow).

Mass is additive, so when we compute masses based on objects interacting with forces no greater than the strong force, we have to add together masses, not square roots of masses. Accordingly, to get a generic preon mass, we square the above average square root to get:
313859334.4 eV = 313.859334 MeV.

There are three quarks in a baryon. Surprisingly, tripling the above mass gives a number remarkably close to the neutron and proton masses: 941.5780031 MeV.

From a year ago, I suspect that the neutrinos are lighter because of 11 sterile species of neutrinos, and this causes a 3^11 ratio in the average square root mass of the charged versus neutral leptons. I found that power of 3 by writing out the ratio in base 3. In base 3, the ratio 177082.0 works out to be 22222220121.0 (base 3) a number which is very close to 100000000000.0 (base 3), hence the 3^11 ratio.

The reason for looking for this is because one suspects that the natural probability in a system of three preons is 1/3, and that given several different ways that something can happen, one expects that one will pick up powers of this probability the same way that one picks up powers of alpha in QED. That is, tree level diagrams will contribute a certain negative power of 3, and the next level diagrams will be some power of 3 smaller, depending on the number of probabilities that have to be picked up.

Applying this sort of reasoning to the neutron and proton masses, I find the following odd coincidences:

(Neutron - Proton)/Proton
(939565360 - 938272029)/938272029 = 0.001378418
= .0000010000101122101 (base 3)

(Neutron - Proton)/Neutron
(939565360 - 938272029)/939565360 = 0.001376520
= .0000010000021121200 (base 3)

(Neutron - Proton)/3xPreon
(939565360 - 938272029)/941578003 = 0.001373578
= .0000010000002221000 (base 3)

To compare these numbers, let us write them together. I've grouped digits into sets of six to show the structure better. Note that the actual number of significant digits is a couple less than shown, but I haven't worked out the details over variation in experimental measurement of the input masses:


\begin{array}{ccccc}
.000001 &000010 &112210 &1 &\textrm{/Proton}\\
.000001 &000002 &112120 &0 &\textrm{/Neutron}\\
.000001 &000000 &222100 &0 &\textrm{/3xPreon}\end{array}


Putting the probability p = 1/3, all the above have a leading term of p^6, and the next leading terms are of order p^{12}. Of the three, note that the division by 3xPreon is the simplest, in the third group it has the sequence 222100 which suggests the base 3 number 222,222 ~ 1,000,000. That the preon mass division gives the simplest form for the mass is what we would expect if the Preon is to be the simplest and most basic quantum of mass.

Cleaning up the above, we can guess that the first three terms are:

3(p^6 + p^{12} - N p^{18})

That is, the differences between proton and neutron mass work out to be something involving three diagrams with 6 vertices at first order, three diagrams with 12 vertices at second order, and possibly some multiple of three diagrams each with 18 vertices at third order. The reason for their being multiples of three diagrams at each order is because of RGB symmetry.

Other than this, I've found more amazing things with the masses of other mesons and baryons, specifically repetitions of the mysterious 0.22222204717 number, but the above mass formulas are easily more important

The AMU data for all but the tau mass are more accurate than the eV data shown above. And you can presumably compute the tau mass from the Koide formula. That means that you can redo the above computations with AMU data and get more accurate numbers.

What this gives is two levels of mass splitting in the baryons. The first a sort of fine-preon splitting with a mass of Preon/3^5 = 1.291602 MeV, and a next level of mass splitting with a value of Preon/3^11 = 1771 eV = .001771 MeV. Of course the first level of mass is just the preon mass itself.

The objective is to classify as many of the several thousand mesons, baryons, and resonances according to these sorts of mass splittings, and then look for patterns. In addition to the mass splittings, there is also a lot of information about phase angles that will eventually be useful, but I think that getting a good classification of the mesons and baryons by this sort of mass splitting will be a useful thing. One then looks for patterns in the relation between the quantum numbers of the states and the splitting counts.

Summing up the mass formula, this one is that the baryons masses are approximately

m_{N,P} = (1 + O(3^{-6}))(\sqrt{m_e}+\sqrt{m_\mu} + \sqrt{m_\tau})^2/3

and their splitting is approximately

m_N - m_P = (3^{-6} + O(3^{-12})(\sqrt{m_e}+\sqrt{m_\mu} + \sqrt{m_\tau})^2/3

[arivero]

The analysis starts more crackpotty than usual...
17716.07559197899477, good heavens! It doesnt mean anything; all these digits are under the experimental error. Fortunately you do not use them in the the real discussion, but I was very tempted to stop reading here and I am sure some people has.

The main point in the first part is that you suggest the constituent mass of a u or d quark is nearby

({\sqrt m_e + \sqrt m_\mu + \sqrt m_\tau \over 3})^2

and so the proton and neutron masses, having three of such quarks, are near


({\sqrt m_e + \sqrt m_\mu + \sqrt m_\tau})^2 \over 3


Ok, it could be. Or it could simply to reflect the already misterious fact of having the mass of the tau in the GeV range.


The second part of your posting is your quest for powers of 3. There, the last of the relationships,
939565360 - 938272029)/941578003 = 0.001373578
gets the impact of the starting mistake... 941578003 is a fake, the four or five lesser digits are under the experimental error and then without meaning. Amusingly, if you put the experimental error in, then you need to reject the two or three last digits in 0.001373578 and your fit is more impressive.

The moral: control the numbers against the errors.

Ah, note that using Koide, we have


{({\sqrt m_e + \sqrt m_\mu + \sqrt m_\tau})^2 \over 3 }
=
{({ m_e + m_\mu + m_\tau}) \over 2 }

So a related coincidence is that the masses of muon and tau average near of the proton mass. Or m_\mu + m_\tau \approx 12 \pi^5 m_e ...

[CarlB]

The analysis starts more crackpotty than usual.

Well, this is hot off the press. There are thousands of mesons, baryons and resonances so I've got many many hours of effort left on this. One of the things I'm putting together is a Java calculator to make the calculations automatic. What you see above are the first calculations from that calculator.

The second part of your posting is your quest for powers of 3. There, the last of the relationships,
939565360 - 938272029)/941578003 = 0.001373578
gets the impact of the starting mistake... 941578003 is a fake, the four or five lesser digits are under the experimental error and then without meaning.

I could have screwed this up. The objective is to use the Koide relation to eliminate the need to use the tau mass data. In the original calculation, I used the AMU figures from my MASSES2 paper (equation 17), using this technique. For this, the scale factor is:

\mu_1 = 0.5804642012(71)

which is a little less than 8 digits of accuracy for the 941578003 number. Note that this is an AMU accuracy, which is 15x as accurate as the eV number. The AMU data at the PDG is more accurate than the eV because the measurements are made in AMU and then converted to eV. So the error is limited by the accuracy of the conversion. However, this conversion error should cancel as all the above calculations were done with eV data that were converted from AMU measurements (presumably the PDG uses the same conversion ratio for their best guess). (Provided I avoided the tau mass number.)

In fact, the first time I made these calculations was by hand. I used the AMU data, of course. I was so shocked at the result that I went back and wrote up some Java code to assist in the calculation, and as a check I redid them with eV data instead of AMU. The result was substantially the same.

Let me redo it with the AMU data the right way and show it here as an edit in the next hour or two:

[edit]

Proton and neutron masses in AMU:

m_P = 1.00727646688(13)
m_N = 1.00866491560(55)

m_N-m_p = 0.00138844872(68) (Error is 5 x 10^-7 )

From the MASSES2 paper calculation:

m_L = [0.5804642012(71)]^2 = 0.3369386889(83) (Error is 2.5 x 10^-8)

Therefore, the ratio has an overall error of 5 x 10^-7 and we have:

(m_N-m_P)/m_L = 0.0041207755(20)

or

0.0041207735 < (m_N-m_P)/m_L < 0.0041207775

If you're going to convert numbers to base 3 by hand, I suggest first converting them to base 9, and then taking each digit and converting it to base 3. To check my work, you should get the following:

\begin{array}{cccc}
0.0041207735 &=& 0.0030028471 &(base 9)\\
0.0041207775 &=& 0.0030028486 &(base 9)
\end{array}

\begin{array}{cccc}
0.0041207735 &=& 0.00001000000222112101 &(base 3)\\
0.0041207775 &=& 0.00001000000222112220 &(base 3)
\end{array}

Now I'm claiming that 3^6 = 729 is the correct equivalent to the fine structure constant here. So it makes sense to convert the above into digit groups of six tribits:

\begin{array}{ccccccc}
0.0041207735 &=& 0.000010 &000002 &221121 &01 &(base 3)\\
0.0041207775 &=& 0.000010 &000002 &221122 &20 &(base 3)
\end{array}

To write this as 3 times a sum in base 729, we have:

0.0041207745(1) = 3 (3^{-6} +3^{-12} - 12.5 \times 3^{-18} ).

In the above, the "12.5" was chosen to get the number in the range above. I.e., in the earlier calculation, this is the O(3^{-18}) figure. It is only 1.7 percent of the next higher term, so it seems likely that once I understand the sequence, I can make a calculation that will give this term.

You may possibly recall that my 8-digit predictions for the neutrino masses were based on the assumption that a factor of 3^{12} was involved. I realize that this looked pretty cracked at the time. I didn't make much effort to publish that paper because I knew that it would look pretty insane. What can I say, there are things one learns by doing that cannot be easily explained to others. And I didn't want to waste my time on a "gee whiz, look at this unexplainable coincidence" paper just to see it in print.

There are a lot of things in my Clifford algebra calculations that I've never explained to you. As far as I can tell from my DNS logs, very few people have even downloaded all my papers on Clifford algebra. If people were reading it, they'd be pointing out typos or asking for clarification. No one is doing this, but even if they did, there is a lot of other results that are not available on the net. Perhaps my intuition is good, perhaps it is not, that is for time to tell. But before you reject this as another coincidence involving BIG powers of 3, you should consider the possibility that I am still sitting on a lot of information I haven't explained to you.

Carl

[arivero]

Proton and neutron masses in AMU:

m_P = 1.00727646688(13)
m_N = 1.00866491560(55)

m_N-m_p = 0.00138844872(68) (Error is 5 x 10^-7 )

From the MASSES2 paper calculation:

m_L = [0.5804642012(71)]^2 = 0.3369386889(83) (Error is 2.5 x 10^-8)

Ok here is my objection: m_P and m_N have experimental errors, but m_L is a calculation. The error in m_L if you take it as experimental is dominated by the error in the mass of the tau, of order 10^-4, and you are taking it to be 10^-8 because you are using the calculated prediction from Koide instead of the experimental value.

In the first post you listed the experimental mass of the tau, no the Koide prediction, and you caused me to switch to fast reading mode (*) . But perhaps you really need to use the prediction and to take only electron and mu as inputs. It is OK to do it; it is only that in the first post you were not doing it, or not telling about. The second post is a lot better.

(*) and In fact I missed the point of that you were already explaining this detail in the second posting :frown:

[arivero]

But even if the third triplet of your decimals (er, not 10-cimals, but 3-cimals ... tricimals?)happens to be just noise, the main fit is interesting. You connect the mass of the leptons to the constituent mass of quarks, or to the whole mass of the proton if you wish.


{({\sqrt m_e + \sqrt m_\mu + \sqrt m_\tau})^2 \over 3 }
=
{({ m_e + m_\mu + m_\tau}) \over 2 }
=m_{p,n}\approx 0.5 m_D

EDITED: furthermore, the (electromagnetic??) mass difference between a neutral pion (quarks with atractive electric charges) and a charged pion (repulsive) is about 4.6 MeV, then it is pausible that your difference m_p - m_L comes from the electromagnetic binding, and than a "pure QCD" barion had a mass exactly m_L. I set a question mark because some mass differences can be told to come from the difference of quark masses (still assuming no preonic quarks here)

[CarlB]

First, a few more interesting coincidences. When the fine structure constant is used, the first order change to energy is proportional to alpha^2. This turns out to be very close to 27^{-3}. I would prefer to write this as 27 \times 729^{-2} where 729 is the 12th power of 3.

The gravitational coupling constant and the weak coupling constant both have a factor of M^2. If one divides the gravitational coupling constant by 4pi, as is done in the first chapter of the recent text book "High Energy Physics" by Perkins:
http://www.amazon.com/Introduction-Energy-Physics-Donald-Perkins/dp/0521621968
then the gravitational constant =5.33 \times 10^{-40}. The Fermi force coupling constant is already so divided, it is 1.16637(1)\times 10^{-5}
see http://pdg.lbl.gov/2006/reviews/consrpp.pdf

These two coupling constants have the pure number ratio of 2.188\times 10^{+34} = 729^{11.99}, very close to an exact power of 729.

So the weak and gravitational coupling constants are related by a power of three (to first order in 1/27). To relate these to the fine structure constant, we must pick an energy or mass. Of course it is possible to choose a mass that makes all three coupling constants be related by powers of three. What would that mass be? Well, since there is a division by M^2, one has different scales that one could choose for this. But a natural scale is the mass of the proton / neutron. And this turns out to work, at least to first order in 1/27.

I know that what I've written is going to be interpreted as just more unimportant coincidences from the forklift driver, but to explain the theory behind this requires more time than my readers have available. If you want to get started understanding my solution of the problem of how one combines quantum states to produce bound states you will just have to begin reading my book on the subject of the density operator formalism of quantum mechanics, http:\\brannenworks.com\dmaa.pdf

The above book is not up to date. I will know when someone is reading it because they will generate many dozens of questions, requests for clarification, and complaints about errors, I will then start writing more. I see none of these, so I know that I have about a 3 month head start on the rest of the planet - as it now, I'm the only person who understands how to apply primitive idempotents to elementary particle theory.

The new theory uses a sort of Feynman diagrams in a sort of perturbation theory. Instead of having incoming fermions treated as point objects, they are treated as bundles of six preons. (This is derived from very simple principles in the above book.) There is only one coupling constant, it is 1/3. To properly count diagrams, one must understand that all the particles must be dressed. That is, the bare propagators of the usual theory are dressed composite propagators in this theory, so one must dress the incoming and outgoing fermion propagator bundles to make them identical in form to a free fermion propagator bundle.

There are 27 tree level diagrams that contribute to \alpha^2. In these diagrams, there are three places where an arbitary color phase can be chosen. The choices of color phase are red, green, and blue, and this gives the 27 = 3x3x3 diagrams. These tree level diagrams each have 12 vertices, and each contributes an overall probability of 1/3. The result is that alpha^2 is given to first order by 27\times 3^{-12}.

Carl

[arivero]

The gravitational coupling constant and the weak coupling constant both have a factor of M^2.

This is one of my motivations to be against natural units :!!) : At the end of the day, Fermi coupling is short range (sort of dirac delta, or a 3-dim 1/r^3 potential) while gravitational coupling is long range. But in natural units they seem the same.

[Kea]

They will find my sites of townships -- not the cities that I set there.
They will rediscover rivers -- not my rivers heard at night.
By my own old marks and bearings they will show me how to get there,
By the lonely cairns I builded they will guide my feet aright.

Kipling

[CarlB]

They will find my sites of townships -- not the cities that I set there.
They will rediscover rivers -- not my rivers heard at night.
By my own old marks and bearings they will show me how to get there,
By the lonely cairns I builded they will guide my feet aright.

Kipling

Well I'm a little more optimistic than that. I think that they will eventually get quite stuck and will be forced to more carefully examine the compass. But until then, there is plenty to do in examining the promised land (that the compass pointed to). The numerical coincidences are fun to find, but dealing with powers of three is a pain unless you happen to have a base-N calculator like this one:
http://www.measurementalgebra.com/CarlCalc.html

I think it's rather clunky and will be improving it to a more efficient model soon. The source Java code is here:
http://www.measurementalgebra.com/Calc_Top.java

First improvement: The "accuracy" of some of the inverse trig and hyperbolic functions has beem "improved".

Carl

[mrigmaiden]

03 14 07

Hello Carl:
I found this on a thread on Kea's blog. Nice. I share Alejandro's thoughts on accuracies in your approximations. I don't, however, knock the idea of looking for meaning in powers of three. To be honest, there is a beauty in that approach. Ultimately, just check your significant figures such that your truncation errors can be reduced to nada!!!

I havta say that natural units always drive me crazy because I must remember from whose perspective they are natural. hehehehehe I wasn't able to read the whole thread, so I was simply wondering about the approximation for the fine structure constant you listed above.

When you recommend changing from base ten to base nine to base three, I think each transformation you are losing accuracy of numerical approximation. This introduces systemmatic error into your calculations. Why not use the Summation convention generated by geometric series no matter which base you are in? Then you are limited by only truncation error in power of logarithm which is rational that you are setting to an infinite sum of terms.

Recall that post I did on a non standard approximation for 1/3 in binary with alternating coefficients. The alternating coefficients may not fit into your framework, but you can find infinite number of geometric series to represent your numbers in base three! Interestingly enough, I think that likening the fine structure constant to the twelfth power of three is pretty darned clever! You are seeing the importance of such correspondence. HOWEVER, that approximation introduces errors as well because (from what I gather) the fine structure constant is actually irrational. Recall that you may produce a series expansion of an irrational number in any base. You will see some beauty fall out but here is the process that I might take and then go out as many terms as possible to keep as accurate as data allows:


1. Solve equation: 3^x=fsc (fine structure constant)
2. Solution will be infinite geometric series in log(fsc)/log(3). See below:

3^x=fsc
Log(3^x=fsc)
xLog(3)=Log(fsc)
x=Log(fsc)/Log(3)

3. Now this is with the understanding that you previously series expanded the term LOG(fsc) and you can do that a number of ways!

4. Nice approach, now it is time for cranking down truncations!!!

[arivero]

mrigmaiden, nice blog you have.

Yes indeed, natural units are a sort of misnamer. I was happy with calling them "Planck units", from the paper of Planck.

About the approximations, in CarlB and in all the thread, it makes sense to have two layers, "exact", and "exact at order alpha" (or alpha^2 or alpha^n), and then of course to consider the "order alpha" correction and see if it is exact on its own.

Since the start of the thread, we have been insisting on simple numbers percent accuracy, and from this point of view 939 vs 941 GeV qualifies! The powers of 3 are not so simple at first try, but I do not deny them. They have a flavour near to Krolinowski's formulae.

[CarlB]

Mahndisa,

Your comments on accuracy are important and need to be taken account of. Modern computers use IEEE (http://www.psc.edu/general/software/packages/ieee/ieee.html) standard double precision 64-bit floating point for calculations. IEEE is optimal in that any calculation is wrong only by at most one bit. My Java program uses the "double" format. This gives 16 decimal digits (51 binary bits) of accuracy. Each time one does a multiplication or division, one can end up wrong by 1 part in 10^16, or 1 part in 3^33. Subtraction and addition can be worse, if the numbers cancel to high accuracy, but ignoring this, their accuracy is similar.

The calculation for the conversion to base 3 requires a multiplication and a subtraction for each digit produced. The calculator I provided gives base 3 numbers with 33 digits. To do this calculation requires 66 arithmetic operations and the maximum error will be 2110 base 3, which means that the bottom four 3-bits are iffy. In other words, the calculator is only good to 29 digits of 3-digit numbers (13 digits of 10-digit numbers).. Fortunately, this is far more accuracy than is available in scientific measurements of masses.

Base 9 uses half as many cycles so its error is half as big. Conversion from base 9 to base 3 has no error at all as a single base 9 digit converts into two base 3 digits directly with no arithmetic operations. So if you used base 9 as an intermediary to base 3, you could get 30 3-digits of accuracy, worst case.

The thing you did with the ln(fsc)/ln(3) is actually how I do calculations, and is how the calculator finds base 3 versions of numbers. The natural log calculation has an error of only one bit so doing this introduces very small errors compared to the conversion to base 3.

To test the conversion, you might try converting to base 3 various fractions that you happen to know the repeating 3-base expansion for. For instance, 0.5, in base 3, is 0.1(1) And 1/7 is 0.102120(102120) in base 3. The calculator happens to give these numbers exact. If you can find one that gives a significant error, do tell. I've considered running the numbers out to higher accuracy. I just don't look forward to writing the trig and hyperbolic functions.

To test the accuracy, you can compute various numbers that you happen to know exactly. For example ((1/729 + 1)^5)( 3^78) gives

1.00001200010100010100001200000022 x 3^(78)

The binomial theorem for (x+1)^5 has terms 1, 5, 10, 10, 5, and 1. In base 3 these are 1, 12, 101, 101, 12, and 1. Thus the exact result is:

1.00001200010100010100001200000100 x 3^(78)

And the above calculation has an error of only one bit in the smallest displayed digit. This sort of thing shows that the inherent accuracy of the calculator is more than sufficient for the first 3 powers of 729 for numbers in the range of what is involved here, even with scientific notation. (This is better than my back of the envelope worst case calculation, but that's what back of the envelope worst calculations are supposed to tell you.)

And anyway, if I had problems that led to random bits, then why the heck am I getting sweet results for powers of 3? That would be a result in probability theory or computational theory that would be no less shocking than the physics result. No, the powers of 3 are present in the spectrum of pure numbers used in particle theory and these are unexplained in the standard model. I have found many more results than are discussed above. I've provided the calculator so that you can find them too. If you don't publish anything, I will keep feeding them to you guys here on this thread.

To find powers of 3 in this stuff induces fairly severe cognitve dissonance in any reasonable physicist. It is very natural to conclude that there is arithmetic or precision errors. Furthermore, this is as it should be; unexpected results need to be examined very carefully and critically.

Suppose that God came to you and provided you with a peek at the unified field theory. Suppose that it required nearly everything that is known and believed about physics to be overturned. Suppose that you were an engineer who hadn't been in physics for years. What would you do?

You can try writing a book on the theory. The resulting book (like any other textbook giving substantial new results in physics) requires many weeks of difficult study to understand. People educated in the previous way of doing things will accept your new way of looking at things with the same warm acceptance that the older physicists accepted quantum theory back 100 years ago (i.e. not at all).

You will find that people capable of understanding your book are too busy pushing their own extensions of the standard physics to bother with it, and people with time available to read are not equipped with a sufficient understanding of the standard model to appreciate the relationships. Furthermore, the more you know about physics, the more you know how the assumptions of physics are self reinforcing, and the less inclined you are to continue reading anything that rejects those things that you already "know". A collection of self reinforcing beliefs is not a proof of truth, instead it only implies that if something is wrong with it as a whole, (in the present case, QM is not compatible with relativity), then it is shot through with self reinforcing errors.

So what do you do next? What you do is you start applying your understanding to standard physics. A major source of dissatisfaction with the status quo is the large number of arbitrary constants. These arbitrary constants mostly appear in the quantum theory. This is the weakest spot in the foundations of physics, the place where the cracks are widest. While the foundations are broken from one end to the other, for sociological reasons, you can only sneak in under the "crackpot radar" at this weak point.

So you obtain formulas that relate things that should have no relation. If the relationships are simple enough, your formulas will attract attention, and maybe, eventually, someone will take the trouble to understand what you are saying. Before they do this, they will assume every other possible thing about your formulas. (1) You made a mistake. (2) They are accidents. The next explanation will be: (3) They are correct but can be explained by minor modifications to the standard model.

Communicating with physicists is extremely difficult. One must write formulas that are so blindingly simple that they do not allow cognitive dissonance to cause the reader to jump to explanations that are compatible with the many assumptions built into a physicist. The simplest explanation for a paper giving new results in physics is that it just wrong.

I don't think that the formulas I've written here are sufficient to cause anyone to undergo the pain needed to understand density operator theory. However, this is just the nose of the lead camel in a very long pack train.

Of the mass numbers in the Particle Data Group, the mesons are a mess and will require some heavy lifting to figure out. The structure of the baryons is fairly obvious. They come in 3s and use modified versions of the Koide relationship in its eigenvector form. I'll write them up next, but don't hold your breath, I'm fairly busy in my day job right now and am a little surprised that I haven't already been fired.

Masses are just the tip of the iceberg of PDG information. There are also widths, phase angles, lifetimes and branching ratios. These are shot through with unexplained coincidences.

Carl

[kneemo]

You can try writing a book on the theory. The resulting book (like any other textbook giving substantial new results in physics) requires many weeks of difficult study to understand. People educated in the previous way of doing things will accept your new way of looking at things with the same warm acceptance that the older physicists accepted quantum theory back 100 years ago (i.e. not at all).

You will find that people capable of understanding your book are too busy pushing their own extensions of the standard physics to bother with it, and people with time available to read are not equipped with a sufficient understanding of the standard model to appreciate the relationships.

Hi Carl

One of the problems is that the strictly operator approach opens up a theoretical can of worms. A question that arises is: what happens to gauge symmetry in the strictly operator approach? You address this question in 8.4 of your book, but I haven't yet found any explicit calculations of the actions of SU(3), SU(2) and SU(1) on the PIs. If you have any of these in print, I'd like to check them out, thanks!

[mrigmaiden]

03 14 07


Carl thanks for the response, but I think you missed the point I was trying to make. Yes I know how a calculator works, but the crucial point I was trying to make was that you get truncation error when you solve log equation for arbitrary base and that error gets carried around with your calculations systemmatically. My question about using infinite geometric series to represent numbers in any base is a valid one and I am not sure if you addressed that. My question also had to do with you approximating the fine structure constant, An IRRATIONAL number in base three. How exactly are you doing that? The way I would do it would be to use an expanion of Log(fsh) and tak that to large accuracy first. Next, I would express fsh as an infinite series in Log(fsh), which can be done for arbitrary irrational numbers.

As to commnication with physicists, I am not here to do anything but learn. Sakurai used density matrices in Chapter 4 maybe? So density operators aren' new, but I dont know enough about your theory to say much else.

The only thing I was really curious about was how you were expressing the fine structure constant, because I believe your approximation for it in base three is a bit too crude.

[mrigmaiden]

03 14 07

Oh lastly Carl, I never specify which base in Log I am using because that depends upon easiest way to parse problem. I want a pure interpretation of a number and a generating series base three. At this point we are speaking around one another because our desire is to express a number in an arbitrary base. Natural log is only good for when you have e popping about. Otherwise use Log base arbitrary! heheheh So when solving 3^x=2 you could write:

3^x=2
exp(log(3^x))=2
then go through the bs rigomorole with LN, OR you could also say:

3^x=2
Log[3^x=2]; Taking Log base three of both sides to get that x= log2/log3 But this is log base3.

You can do a Taylor series approximation for log base three quite nicely so ...

[CarlB]

One of the problems is that the strictly operator approach opens up a theoretical can of worms. A question that arises is: what happens to gauge symmetry in the strictly operator approach?

A gauge symmetry means that one has a bunch of distinct ways of representing the exact same physical situation. This in itself is a theoretical can of worms. There is only one universe, why should we assume that it may be faithfully represented by more than one mathematical object? The only reason we require gauge symmetries to define boson interactions is because it has worked so far, but as physics has seen so many times before, this is not proof that it will suffice for all future versions of physics.

The gauge principle is one of the self-reinforcing beliefs about reality that has locked physics into not advancing for so many years. Like I said above, when you have a large number of self-reinforcing beliefs, it does not imply that they are all true. Instead, it implies that if you fiddle with one of them, you will also have to fiddle with the others.

When you convert a spinor into a density matrix (operator), you eliminate the arbitrary complex phase. This is an elimination of a gauge freedom, and yet the density operator has the same physically relevant information as the original wave function. I think that this is a clue. What I'm doing with the SU(3) and SU(2) gauge symmetries is analogous to this. They no longer exist per se.

SU(2) symmetries are very easy in Clifford algebra. Any two primitive idempotents that anticommute and square to unity define an SU(2), if I recall correctly. For the standard model, you need to get SU(2) reps four at a time in the particular form of a doublet and two singlets. These appear naturally in Clifford algebras when you combine two primitive idempotents as is discussed at length in DMAA.

In short, given four primitive idempotents with quantum numbers (-1,-1), (-1,+1), (+1,-1), (+1,+1) the nonzero sums compatible with the Pauli exclusion principle are (-2,0),(+2,0) the doublet, and (0,-2) and (0,+2), the two singlets. To put it into Dirac gamma matrices, the four primitive idempotents could be (1\pm \gamma_0\gamma_3)(1\pm i\gamma_1\gamma_2)/4 with the signs giving the quantum numbers. On the other hand, SU(3) is built into the assumptions of circulant symmetry. I'd type up more on this, and was sort of getting inclined to, but the PDG is too much fun to play with.

The use of the gauge symmetries in QM is to allow a derivation of the forces of physics. But after one "derives" those forces from this "principle", one need not retain the gauge symmetry itself. Instead, one could suppose that some particular gauge was the correct one and make all ones calculations based on that choice.

In other words, the assumption of gauge symmetry is not required to make calculations in QM, but instead to derive the laws. Gauge symmetry is an attribute of the equations, but it does not need to be an attribute of all possible equations unless you assume that the equations you know are all that can be. Since any new theory of QM need only be equivalent to the old theory in terms of comparing the results of calculations that can be verified by current experiments, there is no need to preserve a gauge symmetry per se.

Things like "no preferred reference frame" are not experimental facts, they are only theoretical facts. An experimental fact is a description of an experiment combined with the observed result of that experiment. For example, electrons interfere with each other when sent through a double slit experiment. All realistic theories must agree on this. They do not need to agree on how the calculation is done. For example, one theory might assume that the electron is composite, another that it is elementary.

The question of the compositeness of the electron is a theoretical fact, it's an assumption of the theory, not something that can be proven experimentally. The theory that produces a calculation is just wrapping around a calculational result, even the calculation itself, is just wrapping around that numerical result. The accuracy of the calculation cannot prove the uniqueness of the wrapping.

Regarding the use of density operators (for the continuous degrees of freedom instead of the discrete degrees I play with), you might find this article interesting, which also discusses how one need not have a Hilbert space to do QM:
http://www.arxiv.org/abs/quant-ph/0005026]Brown & Bohm- Schrodinger revisited: an algebraic approach

Carl

[CarlB]

Carl thanks for the response, but I think you missed the point I was trying to make. Yes I know how a calculator works, but the crucial point I was trying to make was that you get truncation error when you solve log equation for arbitrary base and that error gets carried around with your calculations systemmatically.

Mahndisa,

I'm just a very practical working man, an engineer by trade mostly. Mathematics is very attractive and there are mathematical beauties hiding behind everything, but if I spent the time to explore them I would not have time to do the physics. One thing I think is interesting is the lengths of repeating decimals. It should be clear that the repeating decimal for p/q cannot repeat with more than q-1 bits in an arbitrary base. Do you suppose there could be something else here, perhaps something that has to do with Fermat's Little Theorem (http://en.wikipedia.org/wiki/Fermat's_little_theorem)?

But life is short. To remain free of the grasp of pure mathematics, one must make the decision to not waste ones time on the beauty of pure mathematics every second of the day.

Glad to see you're feeling good enough to be posting more.

[mrigmaiden]

03 15 07


<b>"To remain free of the grasp of pure mathematics, one must make the decision to not waste ones time on the beauty of pure mathematics every second of the day."</b>


Yes Carl, this is where we have a difference in approach. I don't think that studying the beauty of pure mathematics is ever a waste and I don't see it separate from physics either. Yes Fermat's Little Theorem is part of what I was alluding to. However, still not quite. You don't have to be obsessed with the beauty of math to come up with additive discrete representations of numbers in arbitrary base using geometric series. And although there exist practical limitations to computing power, there are go-arounds for the efficient.

Since you are using density matrix formalism etc, how could you distinguish that formalism from revelling in mathematics? I see no distinction.

What I will say is that more physicists and engineers might wish to study optimization science, and intense error analysis. It can only help.

One of the issues that I faced when I took a Graduate Laboratory Seminar was that I knew a lot of the theory quite well and could build circuits etc, but my methods for error propagation were not as complete. I studied this and now try to avoid the dominant sources of systemmatic error one might come across in such computations.

As a matter of practicality, a geometric series representation used to represent a number makes the most sense and by using double or float (as you aptly mentioned) you can go out quite far!

[CarlB]

Continuing the story, we return to the case of the baryons.

The Koide formula gives the masses of the electron, muon, and tau by the formula:
\sqrt{m_n} = \mu_v + \mu_s \cos(2n\pi/3 + \delta)

where the constants are defined as:

\begin{array}{rcl}
\mu_v &=& 17.716 \sqrt{MeV},\\
\mu_s &=& \mu_v \sqrt{2},\\
\delta &=& 0.22222204717
\end{array}

The above formula has one degree of freedom removed with the square root of 2. This is the formula that Koide discovered in the early 1980s. The angle \delta, is surprisingly close to 2/9, and this post is devoted to the application of this angle to the baryon resonances.

The authors model of the leptons supposes that they are composite with three elementary objects in each (simplifying here a bit), and that these elementary objects are held together with a force similar to the color force. That is, the claim is that the electron, muon and tau are color singlets and the generation structure arises from a similar effect.

The author extended the above formula to the neutral leptons, the neutrinos, by jumping to the conclusion that the above numbers have something to do with quantum numbers. The justification for this is beyond the scope of this post, but the formula published a year ago was:

\begin{array}{rcl}
\sqrt{m_{\nu n}} &=& 3^{-11} (\mu_v + \mu_s\cos(2n\pi/3 + \delta + \pi/12))
with the same constants given above.

A short form reason for the pi/12 is that it appears when you convert a 3x3 array of complex multiples of primitive idempotents of the Pauli algebra into a 3x3 array of complex numbers that preserves matrix addition and multiplication. In short, the equation that relates the 0.5 with the pi/12 is:
\begin{array}{rcl}
P_x &=& 0.5(1 + \sigma_x),\\
P_y &=& 0.5(1 + \sigma_y),\\
P_z &=& 0.5(1 + \sigma_z),\\
(0.5\exp(-i\pi/12))^4 P_xP_yP_xP_x &=& (0.5 \exp(-i\pi/12)) P_x,
\end{array}
That is, one can eliminate the nastiness of the product of these three projection operators, and turn them into just another complex multiplication if you multiply each by some numbers that have to do with the sqrt(2) in the Koide formula, and the difference between the charged and lepton delta angles. (Note, I haven't checked the above with care. If you play around with it, you can fix any errors.)

There are hundreds of baryon resonances / excitations and understanding their masses is an ongoing project. We will use the word "resonance" to mean a set of baryons that all have the same quantum numbers. We will use the word "excitations" to distinguish between baryons that have the same quantum numbers.

Other than lepton number, the leptons all have the same quantum numbers. So our analogy between the leptons and the baryons, is between generations of leptons, and excitations of baryons. This makes a certain amount of sense in that leptons do not have excitations other than the generation structure. One could also imagine looking at the generation structure of the baryons. For such an analogy, one would want to compare stuff like (ddd,sss,bbb) and (uuu,ttt,ccc). Unfortunately, these more charmed states do not have very good data.

When a baryon has only two or fewer excitations, we suppose that the others are either yet to be detected, or are hidden as we will discuss later. For this program, a worse situation is when an excitation comes in a multiplicity greater than 3. For the baryons, this happens only one time, with the N_{1/2+}. The fourth state carries only one reliability star. In the PDG data, this means that "evidence of existence is poor". Accordingly, we will ignore this state.

With the leptons, we saw that the angle 0.22222204717 had something to do with the difference between the charged and neutral leptons. We suppose it has to do with the Weinberg angle. The charged leptons and the neutral ones differed by pi/12 = 15 degrees. We therefore speculate that the excitations of the baryon resonances will carry this same relation, that is, that they will have angular dependency of the form:
\cos(2n\pi/3 + \delta + m\pi/12).
where m depends on the resonance.

Adding 8 to m is the same as subtracting 1 from n, so we need only consider 8 different values of m, for instance, from 0 to 7. Since the cosine is an even function, we cannot distinguish between positive and negative angles. This causes a reflection in the data. Consequently the algorithm for finding the angle from the mass data will return angles from 0 to 60 degrees rather than 0 to 120 degrees. As a result, the cases for m > 3 are folded over those same 60 degrees and we will bin the calculated values into the following 8 bins

\begin{array}{rcr}
\delta + 7\pi/12&==& 2.27\\
\delta + 0\pi/12&==&12.73\\
\delta + 6\pi/12&==&17.27\\
\delta + 1\pi/12&==&27.73\\
\delta + 5\pi/12&==&32.27\\
\delta + 2\pi/12&==&42.73\\
\delta + 4\pi/12&==&47.27\\
\delta + 3\pi/12&==&57.73\end{array}

For example, in the first line, \delta + 7\pi/12 gives 117.73 degrees. Adding 1 to n is the same as subtracting 120 degrees from the angle, so this is the same as -2.73 degrees and is indistinguishable from +2.73 degrees because the cosine is an even function.

Note that the first and last bins are very close to 0 and 60 degrees. An angle of 0 degrees corresponds to a degenerate case with two excitations at the lower mass value while the angle 60 degrees puts the degeneracy at the upper mass value. These degeneracies would correspond to excitations of baryons that only appear with two masses.

The above set of bins have gaps of length 4.54 and 10.46 degrees. The rms average for a random value in a bin of length D is:
\frac{2}{D} \int_{x=0}^{D/2} x^2 dx = D^2/12
The 4.54 degree gap will be hit 4.54/15 of the time, while the other will be hit 10.46/15 of the time. The average rms is therefore:
((4.54^3 + 10.46^3)/12*15)^{1/2} = 2.622

I will present the PDG data in the next post.

[CarlB]

Here's the results of the calculations:
\begin{array}{lccccc|l}
Bin/Set & mu_v & mu_s & \delta&Error & L_{IJ} &Notes \\ \hline
m=7 & & & 2.27 & & & Hidden \\ \hline
m=0 & & &12.73 & & &\\
e,\mu,\tau & 17.716 &25.05 &12.73 & & & ****, ****, ****\\
N_{1/2-} & 41.89 & 3.92 &12.97 &+0.24 & S_{11} & ****, ****, *\\
\Lambda_{3/2-} & 42.77 & 5.58 &12.67 &-0.06 & D_{03} & ****, ****, *\\\hline
m=6 & & &17.27 & & & \\
\Sigma_{3/2-} & 41.52 & 2.45 &16.08 &-1.19 & D_{13} & ****, ***, **\\
N_{3/2-} & 41.95 & 3.87 &19.22 &+1.95 & D_{13} & ****, ***, **\\ \hline
m=1 & & &27.73 & & & \\
\Sigma_{1/2-} & 42.33 & 2.60 &23.03 &-4.70 & S_{11} & ***, **, *\\ \hline
\end{array}

\begin{array}{lccccc|l}
Bin/Set & mu_v & mu_s & \delta&Error & L_{IJ} &Notes \\ \hline
m=5 & & &32.27 & & & \\
\Delta_{1/2-} & 43.51 & 3.36 &31.43 &-0.84 & S_{31} & ****, **, *\\
\Delta_{3/2+} & 39.80 & 5.16 &35.70 &+3.43 & P_{33} & ****, ***, ***\\ \hline
m=2 & & &42.73 & & & \\
\Sigma_{3/2+} & 41.90 & 4.95 &41.57 &-1.16 & P_{13} & ****, **, *\\
N_{1/2+} & 36.69 & 6.34 &42.65 &-0.08 & P_{11} & ****, ****, ***\\
\Sigma_{1/2+} & 39.55 & 5.23 &43.22 &+0.49 & P_{11} & ****, ***, **\\
\Lambda_{1/2-} & 40.21 & 2.81 &43.51 &+0.78 & S_{01} & ****, ****, ***\\ \hline
m=4 & & &47.27 & & & \\
\Lambda_{1/2+} & 38.74 & 5.46 &47.46 &+0.19 & P_{01} & ****, ***, ***\\ \hline
m=3 & & &57.73 & & & Hidden\\ \hline
\end{array}

The units of \mu_v^2, \mu_s^2 are MeV. The next column is the calculated delta value, then the error. The final column are notes. The asterisks are from, the PDG and describe how certain the three states are. I've split the table into two because the PF LaTex editor just couldn't quite handle it as one.

The rms error is 1.88 degrees, somewhat below the expected 2.623. The worst fit is for the \Sigma_{1/2-}, which coincidentally also happens to carry the worst asterisk rating from the PDG. The second worst fit is the \Delta_{3/2+}. While this set is well supported by experiment, it includes the \Delta(1600) whose mass is only loosely constrained. About this particle, the PDG writes: "The various analyses are not in good agreement." Together, these two bad fits contribute 80% of the error among the 13 sets of 3 masses each.

The S_{1/2-} excitations, in addition to carrying a high error, are also in a class by themselves. They would seem to fit better in the \delta + 0\pi/12 class, with the N_{1/2-} which is also S_{11}.

What you're seeing here is ALL the data from the baryons. I suspect that the data will look better when the particle mass ranges are taken into account. My mass calculator does not yet have the software to take errors in the data into account. Before this can be published, it needs to have the errors in the excitations taken into account. Hopefully the bad fits will correspond to loose mass ranges.

[CarlB]

Whoops. I left off the \Sigma_{1/2+}. Oh, no I didn't!

Looking at the data, it seems that the really well supported classes are the even ones. Also, there are two low-lying excitations with two masses where one of the two masses is said, in the PDG, to be possibly doubled. These would be the N_{3/2-} N(1700), N(2080)^2 = D_{13} and the N_{3/2+}N(1720)^2,N(1900) = P_{13}.

I've finished the user interface for an online applet that will compute Koide parameters with error bars, but I've not yet put the math code into it. I've got that code in another program so it's just a matter of putting it in there and ironing out any bugs. But I really need to get back to the day job.

Carl

[CarlB]

Okay, I've got a tool that allows you to compute error bars for these Koide type parameterizations of three masses.

In addition, the mesons are famous for being messy with duplicate masses hard to distinguish. One ends up with six masses instead of three. The hope is that one can split those six mesons into two groups of three that are decent. Accordingly, the tool holds six masses at the same time and automatically steps through the various permutations:
http://www.measurementalgebra.com/KoideCalc.html

Source code is available at the above.

Using new tools is difficult. I've initialized the above program with the data for the electron, muon and tau. So all you have to do to calculate your first error bars is hit the "KOIDE" button. Also, I've set it up to give the angles in degrees rather than radians.

[arivero]

In addition, the mesons are famous for being messy with duplicate masses hard to distinguish.

Indeed that is the idea, isn't it? We have six mesons with charge +1, due to combinations of three families of antidown quarks and two families of up quarks. But they have spin 0; then we have the same number of degrees of freedom that in the case of leptons of charge +1.

Have you spotted now some interesting pattern in the mesons, Carl? I tried some pages ago, up in the thread, and I was not very happy.

[CarlB]

Dr Rivero,

Right now I'm busily working on the theoretical side rather than the phenomenological side. I wrote up the software so that one could divide up a set of six mesons into two groups each with decent Koide numbers. But I don't think that alone would be very convincing. To do it right, I think you have to split six states into two groups of three in such a way that the other properties of the states make sense.

There are four S=C=0 mesons for which only three excitations exist, the pi, the omega, the f(1) and the rho(3). Their mass data are:

\begin{array}{cccclll}
\pi & \pi &\pi(1300) &\pi(1800) &= 134.9766, &1300(100), &1812(14)\\
\omega & \omega(782) &\omega(1420) &\omega(1650) &= 782.65, &1425(25), &1670(30)\\
f(1) & 1285 &1420 &1510 &= 1281.8(0.6), &1426.3(0.9), &1518(5)\\
\rho(3) & \rho(1690) &\rho(1990) &\rho(2250) &= 1688.8(2.1), &1982(14), &2250(?100?)
\end{array}

The rho is ugly in that two of its states are single star, and they don't give limits on the mass of the last one. I've arbitrarily put +-100MeV. Typing the data into the Koide calculator, the resulting angles are (in degrees, min, typ, max):

\begin{array}{c|ccc}
\pi & 45.5 & 48.5 & 51.3\\
\omega & 43.8 & 46.6 & 49.5\\
f(1) & 36.9 & 38.0 & 39.1\\
\rho(3)& 26.1 & 32.7 & 41.8
\end{array}

All but the f(1) are consistent with delta angles, that is, 47.27 and 32.27 degrees. The f(1), which unfortunately has the most accurate mass measurements, is between 32.27 and 42.73. The errors are +1.23, -.67, -4.27, -0.43. RMS error is 2.27, less than what chance would suggest, but not by a lot.

When you read the description of the f(1) resonances, at least they are pretty weird. Their lightest entry shows fairly heavy, maybe they are something more complicated. The PDG commentary on the states is interesting. See page 3:
http://pdg.lbl.gov/2006/listings/m027.pdf

[verdigris]

I think that it would be more useful if you had some physical theory to tune these numbers and equations into.Because without one what use will the numbers and equations be anyway!

[arivero]

verdigris, I can think of some uses: Even without a theory, it is an attack to the ideology of GUT unification at high scales, because in such setup the masses run via renormalisation group and you should not be able to find any simple relationship at low energy. And we find some.

The problem was that the best model builders got in love with this business of Very High Energy unification, and most of the standard material is focused from such approaches. If you neglect this point of view, Carl (and others?) effort is not very out from the hopes of group theory based unification. His adopted Fermion Cube holds spinors in the same way that it was done in the late seventies (or early eighties). The unification for one family need to be extended in a way to produce three generations and no more, and this is the usual problem in model building.

(Note for instance the idea of using spinors, or fermionic creation operators, to build the representation. Using six of them you get 32 fermionic degrees of freedom, with the right charges of one generation, but if you add another two then you get not 96=32*3 but 128=32*4. Thus you need a very exotic symmetry breaking scheme for the extra two generators, or alternatively a different way to produce the generation-wise symmetry).

[Hans de Vries]

General Mass Formula:

We possibly have neglected our own mass formula which
we had already at the start of this thread. New is a possible
link to real physics which I will talk more about in coming posts.

M(a,b)\ =\ a\pi-b/\pi\ =\ 2\sqrt{ab}\ \sinh\left( \ln\left(\pi \mbox{\Large $\sqrt{\frac{a}{b}}$}\ \right)\right)

The inputs are always two small integer numbers and the
output is the log ratio of two masses. Almost all the
elementary combinations are close to actual ratios of
the (base states) of existing particles:


.
M(0,2) : log(mτ/mp) ---> 0.636619 : 0.638635 = 0.316%
M(1,1) : log(mτ/mμ) ---> 2.823282 : 2.822461 = 0.029%
M(1,2) : log(mτ/mπ) ---> 2.504972 : 2.544107 = 1.562%
M(1,3) : log(mp/mμ) ---> 2.186662 : 2.183828 = 0.129%
M(1,4) : log(mp/mπ) ---> 1.868353 : 1.905472 = 1.986%
M(2,2) : log(mπ/me) ---> 5.646565 : 5.609955 = 0.652%
M(2,3) : log(mμ/me) ---> 5.328255 : 5.331598 = 0.062%
M(3,4) : log(mτ/me) ---> 8.151538 : 8.154063 = 0.030%
M(4,2) : log(mw/me) ---> 11.92975 : 11.96646 = 0.307%
M(3,6) : log(mp/me) ---> 7.514918 : 7.515427 = 0.0068%

me = electron
mμ = muon lepton
mτ = tau lepton
mπ = pion (+/-)
mp = proton
mw = W-boson

The possible relation with real physics would be this:

M(a,b)\ =\ \sum_{k\ =-\infty}^\infty\ \mbox{\huge
J}_k(2\sqrt{ab})\ \left(\pi \mbox{\Large $\sqrt{\frac{a}{b}}$}\
\right)^k

e^{ieA \sin(\omega t)}\ =\ \sum_{k\ =-\infty}^\infty\ \mbox{\huge
J}_k(eA)\ \ e^{ik\omega t}

The later expression describes the non-pertubative interaction
factor with a sinusoidal field.


Regards, Hans

[CarlB]

I think that it would be more useful if you had some physical theory to tune these numbers and equations into.

There is a physical theory behind this. It's based on Schwinger's measurement algebra. The equations are rather distant from what you can get with the usual physics and so is the theory. I've written most of a book dedicated to explaining the principles:
http://brannenworks.com/dmaa.pdf

I know that no one has read the above book with any degree of care because if they did, they'd have pestered me with questions, complaints, pointed out typos, and suggested improvements. People are too busy to spend a lot of time reading an amateur's textbook.

What I do get are comments from people who don't understand my theory, haven't carefully read my many documents, do not understand much about Clifford algebras, but who want to use the equations for promoting their own physics theories.

I don't blame them for this. It is very similar to what I did with Koide's original equation. After a couple hours of playing with it I saw that it had an easy derivation in terms of the principle idempotents of Clifford algebras and ran with it. The new interpretation put his equation into eigenvalue form and I posted it here. But I've ignored completely Koide's work in explaining the equation.

There are something like 30,000 physicists on the planet. I think that the papers of only about 1000 of them are carefully read and studied. The other 29,000 write stuff that is ignored.

[Hans de Vries]

A mass relation for all six principal charge 1 particles:



\begin{array} {|ccc|c|c|c|c|c|c|c|}
\hline
& & & & & & & & & \\
& & \ \ &\ \ \frac{0}{\pi}\ \ &\ -\frac{1}{\pi}\ &\ -\frac{2}{\pi}\ &\ -\frac{3}{\pi}\ &\ -\frac{4}{\pi}\ &\ -\frac{5}{\pi}\ &\ -\frac{6}{\pi}\ \\
& & & & & & & & & \\
\hline
&0\ \pi & &2VeV&\cdot&\cdot&\cdot& & &\cdot\\
\hline
&1\ \pi & &\cdot&\cdot&\cdot&\cdot& W &\cdot&\cdot\\
\hline
&2\ \pi & & p &\cdot&\tau &\cdot&\cdot&\cdot&\cdot\\
\hline
&3\ \pi & &\cdot&\cdot&\cdot& \mu & \pi^\pm &\cdot&\cdot\\
\hline
&4\ \pi & &\cdot&\cdot&\cdot& &\cdot&\cdot&\cdot\\
\hline
&5\ \pi & & & &\cdot&\cdot&\cdot&\cdot& e \\
\hline
\end{array}



We can put all six principal charge 1 particles in a simple 2D grid.
All grid positions with "." are forbidden via a simple rule that says:

"No two pair of particles may the same mass ratio."


The log mass ratio calculation is: Y\pi -X/\pi, where X and Y are the
2D grid's axis. The origin of the grid is 2VeV. (Vacuum expectation Value)


Some examples:

1) Electron-Proton mass ratio: log(1836.1526726) (natural log)
7.515427 = experimental
7.514918 = calculated = 3\pi -6/\pi
accuracy: 0.0000677


2) Electron mass ratio with 2VeV (Vacuum exp.Value): log(963699)
13.77853 = experimental
13.79810 = calculated = 5\pi -6/\pi
accuracy: 0.00142


3) Electron-Muon mass ratio: log(206.7682838)
5.331598 = experimental
5.328255 = calculated = 2\pi -3/\pi
accuracy: 0.000627


4) Proton-Pion mass ratio: log(6.72258237)
1.905472 = experimental
1.868353 = calculated = \pi -4/\pi
accuracy: 0.01986


5) Electron W-boson mass ratio: log(157387)
11.96646 = experimental
11.92975 = calculated = 4\pi -2/\pi
accuracy: 0.00307


Try it yourself!

Regards, Hans.


\begin{array} {|clc|c|rc|}
\hline
& & & & & \\
& electron & & e & 0.51099892(40)& MeV \\
& muon\ lepton & & \mu & 105.658369(9) & MeV \\
& tau\ lepton & &\tau & 1776.99(29) & MeV \\
& pion\ \pm & & \pi & 139.57018(35) & MeV \\
& proton & & p & 938.27203(8) & MeV \\
& W boson & & W & 80398(25) & MeV \\
& & & & & \\
\hline
\end{array}

[CarlB]

Try it yourself!

I've violated our tradition of ignoring ideas that don't immediately strike us as useful for our own silly ideas and tried it myself.

Your accuracy numbers are exaggerated. You're using the natural logs of the masses. And your formula is linear in those logs. So there's no reason to divide by the logarithm.

You are not fitting to a linear formula like y = ax + b where you would be testing for, for example, b = 0. In such a case it would make sense to divide the error by y because you would want a relative error.

Instead, you are fitting a sort of Diophantine equation and the errors are compared to integers. Another way of putting this is that the way you are calculating errors, the larger the ratio, the more accurate your ratios will be. But the steps between different choices of m-n/\pi are constant, so the expected error does not depend on the ratio.

Here, let me make an error calculation the way you are doing it. The year is 2007, which turns out to have approximately a worst case error for fitting to stuff with a small value for n/\pi. We find that:

639*\pi - 1/\pi = 2007.16

The way you are calculating errors, this would have an accuracy of

0.16/2007 = 0.0000794

Do you really want to claim this for a fit to 2007? By the way, a better approximation for 2007 is:

639*\pi - 1.5/\pi = 2007.00024

Let me put it this way, if you were claiming that the ratios were integers, then the error would obviously be the difference between a ratio and the nearest integer. The worst you could do would be 1/2, and this is the number you should divide your errors by, not for example, 2007 or whatever the nearest integer.

When one describes the points on the real line of the form n\pi - m/\pi for small values of n and m, one finds that they are mostly separated by 1/\pi = 0.3183. When one assigns random numbers to the nearest one of these, the worst one can do is half the interval, or 1/(2\pi) = 0.1591 so this is the number you need to divide the differences by, not the log of the mass ratio.

With these changes, the mass ratios you've listed have errors as follows:

0.3%
12.3%
2.1%
23.3%
23.1%

I think that this is impressive enough as it is. What I would like to see is the results of a computer program that can find these sorts of fits, and see various randomized data thrown at it.

Now the other thing I wanted to point out is that the mass formula you have here is not at all incompatible with the Koide formula or the formulas that I've described. Furthermore, the Koide formula ends up putting an angle of 0.22222204717(48) radians into an exponential in the mass matrix for the charged leptons. This suggests that taking natural logs of masses is likely to be a useful thing to do.

Carl

[Hans de Vries]

Your accuracy numbers are exaggerated. You're using the natural logs of the masses. And your formula is linear in those logs. So there's no reason to divide by the logarithm.

Carl,

These accuracies correctly describe the number of prediction bits
according to information theory. Don't forget that a dynamic range is
needed next to the precision. In terms of floating point numbers:
You need an "exponent" as well as a "mantissa".

Let us simply do the calculation: Take example 1:


1) Electron-Proton mass ratio: log(1836.1526726) (natural log)
7.515427 = experimental
7.514918 = calculated = 3\pi-6/\pi
accuracy: 0.0000677

The number of bits predicted is -log2(0.0000677) = 13.850 ~ 14 bits.

If I would express the number exp(7.514918) = 1835.2181 which has an
accuracy of 0.000509, as a floating point number, then I would need:

-log2(0.000509) = 10.939 ~ 11 bits for the mantissa,

but I also need another 4 bits for the exponent which would have the
value 10 for 2^10 = 1024 to express the range between 1024 and 2048
in which the value 1835.2181 falls.

To be entirely exact: For the exponent I need -log2(10) = 3.321 bits.

Now 10.939 + 3.321 bits ~ 14 bits or the same number of bits as in the
case of the logarithm!

The total number of predicted bits in the 5 relations is 50. There are
six independent relations which provide a total of 61 bits. Good for
18.4 correct decimal digits.

Now we are also inputing information here which we need to subtract.
These are the grid positions. The information we provide for the six
relations = 6 x( log2(5) + log2(6)) = 29.441 bits.

Subtract these from the 61 bits and we are left with 31.6 bits which is,
by far, the best result I've presented on this entire thread until now...

Now here is something else very interesting. The exclusion rule: Note
that almost all grid positions are forbidden by it and a random placement
would very likely break the rule. Thus, the 29 bits number would be
much lower if the rule is correct.


Regards, Hans

[whatta]

this thread needs only one number, 42. this will probably be the page number when it gets locked.

[Hans de Vries]

this thread needs only one number, 42.

Dear Whatta,

This thread was initiated to allow, (but also to contain) posts with
the purpose of the archival of numerical coincidences. These
post are allowed (here on this thread) within the following restrictions:

1) The reported numerical coincidences should be independent of
the units used (meters, kg..). Only coincidences with dimensionless
numbers are allowed.

2) The numerical coincidences must be independent of the number
system used, like, decimal numbers, binary, hexadecimal.

3) The reported numerical coincident should have a sufficient
"predictability". That is, It should produce significantly more result
bits as the number of bits used as input.


Kind Regards, Hans de Vries

[CarlB]

These accuracies correctly describe the number of prediction bits according to information theory.

I would like to think that information theory is appropriate for the calculation of a fit, but I suspect that even a well accepted calculation like the g-2 value for the electron will fail an information theory test. That is, the amount of bits required to describe the thousands of Feynman diagrams, or the rules to generate those Feynman diagrams along with coupling constants plus etc., etc., etc., will be greater than the amount required to simply code g-2 to experimental error.

But if you insist on using information theory, you need to count the number of bits needed to describe your formula, along with the number of bits used to describe the exponent and the number of bits needed to describe the mantissa. This will blow up the accuracy. If the formulas really were that good they wouldn't be ignored so much. You haven't faulted my calculation for the approximation of e^{2007} in the form e^{n\pi - m/\pi} because it was correct.

The calculation I provided has the useful feature that for masses distributed uniformly over relatively small ratio spaces, for example, if the mass ratio is from e^{12.5} < R < e^{13.0}, it will give approximately the correct probability for getting a hit, in that the error will be approximately uniformly distributed from 0 to 100%. This feels to me like the right way of looking at it.

There are some other things I forgot to mention. The vertical column would make more sense to me if it was turned upside down relative to the pi values. The way it's set up, increasing values of the n cause decreases in masses and that is counterintuitive. So the electron would have the 0\pi rather than 5\pi.

Second, I'm not sure what "principle" means with respect to charge 1 particles. Why not include the \Delta^+? Did you try to fit other charge +1 particles and they didn't fit or what? I guess there are most of a thousand meson / baryon resonances and excitations with charge +1.

Carl

[Hans de Vries]

But if you insist on using information theory, you need to count the number of bits needed to describe your formula.

I see I was too conservative on the input bits side.

There are only 30 grid positions which are the same for each result.
6 of the 30 are hits. I should have discounted the input space only once
and not 6 times. This brings the prediction back to 61-log2(30)~56 bits

The expression itself, discounting the small integers, is about 10 bit, which
is three times a basic operation like +-x/ (two bit each) plus the use of
a single elementary constant (pi) which we presume to be in small group
together with the small integers. (3-4 bit)

This leaves us with ~45 bits prediction which is three to four times more
as the alpha result.


There are some other things I forgot to mention. The vertical column would make more sense to me if it was turned upside down relative to the pi values. The way it's set up, increasing values of the n cause decreases in masses and that is counterintuitive. So the electron would have the 0\pi rather than 5\pi.


It was like this until I decided to put the Vacuum expectation Value at the
origin of the grid.


Second, I'm not sure what "principle" means with respect to charge 1 particles. Why not include the \Delta^+? Did you try to fit other charge +1 particles and they didn't fit or what? I guess there are most of a thousand meson / baryon resonances and excitations with charge +1.


Many, many indeed, and there are only 30 grid positions....

It's more that the particles select them self by fitting. The resonances
and excitations don't fit but the base states do. Hadrons with masses
dominated by a single (fractional charge) quark probably won't fit either
but I have yet to try.

Heck, this formula is one of the subjects of the very first post in this
thread and I never even bothered to try a hadron mass because of
me being convinced that these can only have incredibly complicated
QCD determined mass values.

But then, looking at the infamous 'proton spin crisis', something can be
incredibly complex inside, and then, all the small spin contributions from
the quarks, the gluons, the sea-quarks and all the various angular
momenta add up to a very simple number determined by geometry only.


Regards, Hans

[CarlB]

Hans, I've been playing around with the numbers and now I see that I was too hasty to judge your work as numerology. (I didn't say so, but that was what I was thinking.) In fact, now that I understand the method better, I think I can contribute to this exciting branch of phenomenology. Before, I just had trouble seeing what the heck an exponential could be doing in the mass spectrum.

First, I guess I should mention some theory that drove me in this particular direction. The fact is that there is a lot of periodicity in the elementary particle masses having to do with 3s. 3 is sort of midway between pi and e. The next important number smaller than e is 5/2. Also, 5/2 is the average of the two smallest prime numbers, which suggests that p-adic field theory could be important here, just like the string theorists say. And I like to think of QFT as a probability related theory so equations such as

5/2 = 1 + \sum_n3^{-n}

naturally led me to explore the use of 5/2 in the elementary particles. Enough for the theory (maybe it still needs some work); so here's my formula:

m_{N,M} = (5/2)^{3(N - M/(4\pi^2))}.

This formula works very well for the charged leptons, with the electron, muon and tau taking N=0,2,3, and M= 0, 2, and 1, respectively. I need not point out how suggestive these small constants are! The experimental and calculated exponents are:

\begin{array}{rcc|l|l|l|}
&N&M& 3(N+M/(4\pi^2)) & \log_{5/2}(m/m) & accuracy \\ \hline
\mu/e &2& 2& = 5.848018& 5.818676 & 0.00501\\
\tau/e &3& 1& = 8.924009& 8.898993 & 0.00280\\
\tau/\mu&1&-1& = 3.075991& 3.0803163& 0.00141\\ \hline
\end{array}

The above makes a grid with 4x3 = 12 boxes, three of which are filled. This is suggestive, especially when you look at those tight accuracy figures and the very convincing division by pi. But I don't think that this is nearly enough to write a paper on.

The real test is to see if we can put the other charged particles into the same formula. Admittedly, one might suppose that the color force, being a sort of charge, would contribute something to the mass of a quark or baryon, who reallly knows? In phenomenology, it makes sense to boldy go where no sane man has gone before and simply see what particles naturally fit together.

\begin{array}{|ccc|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
&&&&&&&&&&&&&&&\\
&&M=&0&1&2&3&4&5&6&7&8&9&10&11&12\\
\hline
&N=0&&&&e&&&&&&&&&&\\
\hline
&N=1&&3d&&&&&&&&&&3u&&\\
\hline
&N=2&&\pi^+&&&&\mu&&&&&&&&\\
\hline
&N=3&&&&&\tau&\Omega&&&\Xi&\Delta&\Sigma&\Lambda&&p\\
\hline
\end{array}

In the above, I'm not so sure about the up and down quark. The masses aren't known very accurately. I've made the assumption that the masses that one needs to use are at the low end of the PDG figures. And since the quarks also have the color force (as well as E&M), I've multiplied their masses by 3.

Only one meson fell into the fit, but it is the most fundamental one. I guess this is evidence that the mesons really are a mess. But I could fit all of the basic (i.e. no charm, bottom or top) low lying baryons. Furthermore, no two particles ended up in the same slot, which is cool.

Some of the fits that I checked are remarkably good. For example, the Lambda/e accuracy is 0.00013 and the 3u/Omega is 0.00041. That last ratio really is stunning accuracy given that the mass of the up is listed as 1.5 to 4.0 MeV in the PDG! I'm not sure about how this happened, I'm just crunching numbers on my calculator.

Anyway, the formula fits all the principle baryons, all the charged leptons, the lightest quarks, and the lightest meson.

There are only 30 grid positions which are the same for each result. 6 of the 30 are hits. I should have discounted the input space only once and not 6 times. This brings the prediction back to 61-log2(30)~56 bits

I'm kind of a dummy and I don't really see how your calculation here makes any sense. But since I've got 12 hits in 52 grid positions, it does look like this should be okay. In addition, all the fermions are bunched together on one side of the diagram so it kind of makes me wonder if it wouldn't be useful to define a "baryon bit" and use three quantum numbers to define the mass exponent.

The expression itself, discounting the small integers, is about 10 bit, which is three times a basic operation like +-x/ (two bit each) plus the use of a single elementary constant (pi) which we presume to be in small group together with the small integers. (3-4 bit) This leaves us with ~45 bits prediction which is three to four times more as the alpha result.

Since I've got 12 masses listed I bet that I should be really positive. Since I was employed for 15 years designing digital logic, and one of my specialties was coding theory, it really bothers me that I don't understand your bit calculations. Maybe I should hit the books -- I learned theory back in the stone age when we mostly used our fingers. Heck, I still remember punch card codes.

I find it hard to believe that you could send someone your expression in just 10 bits. Of course when mapping expressions to bit sequences, it makes sense to code the important expressions in smaller number of bits.

My HP calculator is fairly efficient general purpose scientific calculator. It has over 32 keys so pressing anything that can be coded in one key stroke costs 6 bits. Looking up a data value takes two keystrokes or 12 bits. Pressing the <enter> key costs 6 bits.

It was like this until I decided to put the Vacuum expectation Value at the origin of the grid.

Yes, the factor of 2 was brilliant! I'd have never had the idea to triple the quark masses if you hadn't led the way.

enjoy,
Carl

[whatta]

was that post sarcastic

[Kea]

...which suggests that p-adic field theory could be important here...

Yes, Carl. A nice paper on this point is

On the universality of string theory
K. Schlesinger
http://arxiv.org/abs/hep-th/0008217

which introduces the concept of a tower of quantizations. All of String theory fits into the prime 3 on the tower, from the point of view of the true M Theory. In other words, the unified theory can address all scales on an equal footing. The classical landscape is Mathematics Itself.

How do you like that, whatta?

[kneemo]

On the universality of string theory
K. Schlesinger
http://arxiv.org/abs/hep-th/0008217

which introduces the concept of a tower of quantizations. All of String theory fits into the prime 3 on the tower, from the point of view of the true M Theory. In other words, the unified theory can address all scales on an equal footing. The classical landscape is Mathematics Itself.


T. Pengpan and P. Ramond showed in hep-th/9808190 (http://arxiv.org/hep-th/9808190) that the 11D supergravity triplet of SO(9) representations sits at the base of an infinite tower of irreps of SO(9), describing an infinite family of massless states of higher spin. They muse that such higher-spin states describe degrees of freedom of M-theory.

[CarlB]

was that post sarcastic

Somehow I forgot to give the calculations for the absolute errors and the masses of the predicted particles. Of course these aren't as pretty as the selected ratios, but are probably more of an indication of how good the fit is:

Calc = 0.5743319321193890 MeV * {(5/2)^{Exp}}

\begin{array}{cccc}
Exponent&particle&Mass&Calc/Mass\\
3(0 + 2/(4\pi^2))&e&0.51099892&0.9778\\
3(1 + 10/(4\pi^2))&3u&4.5&0.9940\\
3(1 + 0/(4\pi^2))&3d&9.0&0.9971\\
3(2 + 4/(4\pi^2))&\mu&105.658369&1.0044\\
3(2 + 0/(4\pi^2))&\pi^+&139.57018 &1.0046\\
3(3 + 12/(4\pi^2))&p&938.27203&1.0126\\
3(3 + 10/(4\pi^2))&\Lambda&1115.683&0.9788\\
3(3 + 9/(4\pi^2))&\Sigma&1189.37&0.9843\\
3(3 + 8/(4\pi^2))&\Delta&1232.0&1.0188\\
3(3 + 7/(4\pi^2))&\Xi&1321.31&1.01845\\
3(3 + 4/(4\pi^2))&\Omega&1672.45&0.9915\\
3(3 + 3/(4\pi^2))&\tau&1776.99&1.0005
\end{array}

Getting back to the charged leptons, their masses (and ratios) are exact numbers and one expects that to store them requires an infinite number of binary bits. Our experimental measurements, on the other hand, are inexact, so one can certainly expect that data they contain can be easily compressed.

The charged lepton mass numbers run over a very wide ratio and so the natural way to compress them is by a power series. Koide's formula uses a square root, which is also a compression method and so at first I suspected it as well.

But if Koide were looking for a compression formula rather than physics, he would have done well to begin with the 7th root of masses rather than the square root. Then the masses of the charged leptons are fairly close to 1^7, 2^7, and 3^7, and one could write a generation formula in the form

m_n = (n+f(n) )^7

for n=1,2,3. In fact, I'm quite certain that I could find such a formula, and then bend it around to pick up the baryon masses which form such a convenient linear series. But I think my point here is made.

In the face of how easy it is to find compression algorithms for sparse data, where the Koide formla is more convincing is that it is consistent with exactly three generations and no more. The short form for the Koide formula is:

\sqrt{m_n} = 1 + \sqrt{2}\cos(2n\pi/3 + 2/9 + \epsilon)

where \epsilon = 0.22222204717(48) - 2/9 and I've left off an overall scaling factor. Since 2(n+3m)\pi/3 = 2n\pi/3 + 2m\pi, the formula gives exactly three masses so there are only three generations implied. These are the electron, muon, and tau for n the generation number, 1, 2, 3. The \sqrt{2} is what Koide found in 1981, the \epsilon is what I found a year ago.

So the Koide formula is exact to experimental error, and it's not really in a form that is obviously convenient for simply hiding a compression algorithm.

[Mentz114]

"It is shown here that the rest energies and magnetic moments of the basic elementary particles are given directly by the corresponding Planck sublevels."

http://uk.arxiv.org/pdf/physics/0611100

Enjoy.

[CarlB]

This was released by Hans on sci.physics.foundations:

A non-perturbative derivation of the exact value of the SU(2) coupling value g from the standard Electroweak Lagrangian itself.
Hans de Vries, March 30, 2007
http://chip-architect.com/physics/Electroweak_coupling_g.pdf

I wasted two days playing with the charged particle formulas and ended up quite disgusted and angry. Having learned my lesson, I'm leaving this one for others to analyze.

In short, it relates a smaller number of constants but to a higher accuracy and with simpler formulas, which is about what one would expect.

[Hans de Vries]

Hi, Carl

I must say you did impress me with your feverish activity the in last few days.
From experience I know that these extreme bursts of mental activity have a
risk of ending up in the type of exhaustion you're describing here. :blushing:
Relax, Rome wasn't build in a day, they say. I was still considering a
response before starting this subject.

This was released by Hans on sci.physics.foundations:A non-perturbative derivation of the exact value of the SU(2) coupling value g from the standard Electroweak Lagrangian itself.

Hans de Vries, March 30, 2007
http://chip-architect.com/physics/Electroweak_coupling_g.pdf
In short, it relates a smaller number of constants but to a higher accuracy and with simpler formulas, which is about what one would expect.

Before jumping onto this one I should maybe first point out the "somewhat
vague" mathematical relation between the new paper and the numerical
coincidents in these mass ratios:



\mbox{\huge $ e^{\left( m\pi-\frac{n}{\pi} \right) } $}\ \ =\
\mbox{mass ratio numerical coincidents}


Where m and n should be small integer values. This can be written as a
power expansion like this:


\mbox{\huge $ e^{\left(m\pi-\frac{n}{\pi}\right)} $}\ \ =\ \sum_{k\
=-\infty}^\infty\ \mbox{\huge J}_k(2\sqrt{nm})\ \left( \pi
\mbox{ $\sqrt{\frac{n}{m}}$}\ \right)^k




This now relates to the core of the new paper:


\mbox{\huge $ e^{iQ \sin(\omega t)}$}\ \ =\ \sum_{k\
=-\infty}^\infty\ \mbox{\huge J}_k(Q)\ \mbox{\huge $ e^{ik\omega t} $}


The later is the phase a charged particle acquires in a sinusoidal
electromagnetic (electroweak) field. This is a superposition where
the Bessel coefficients can be interpreted as amplitudes:

J0(x) = amplitude to absorb 0 quanta
J1(x) = amplitude to absorb 1 quanta
J2(x) = amplitude to absorb 2 quanta
J3(x) = amplitude to absorb 3 quanta
...............

These Bessel coefficients have the unique property that they
are Unitary for both amplitudes as well as probabilities
for any value of Q:


\sum_{k\ =-\infty}^\infty \mbox{\huge J}_k(Q)\ = 1, \quad \ \
\sum_{k\ =-\infty}^\infty \left|\mbox{\huge J}_k(Q)\right|^2\ = 1



Regards, Hans

PS: Try the link below. There's a lot on these Bessel coefficients
in regard with frequency modulation on the internet:

http://images.google.nl/images?hl=en&q=frequency%20modulation%20bessel&btnG=Google+Search&ie=UTF-8&oe=UTF-8&um=1&sa=N&tab=wi

[CarlB]

I guess I should probably update three things.

First of all, I've been using the constant 17.716 sqrt(MeV). Squaring to get to the units everyone else uses, this is 313.85 MeV.

Nambu uses 35 MeV in his empirical mass formulas. The relation to my 313.85 MeV is that 35 x 9 = 315 MeV. Of course 9 is a power of 3 and powers of 3 are important in my theoretical stuff, the majority of which is not published. Some links for the Nambu theory are:

http://www.google.com/search?hl=en&q=nambu+mass+formula

Also

http://www.arxiv.org/abs/hep-ph/0311031

refers to it as Y. Nambu, Prog. in Theor. Phys., 7, 595 (1952). I've not yet read much on the theory. I'd like to thank Dr. Koide for noting that my mass formulas reminded him of the Nambu stuff.

The Nambu formula has probably been discussed around here but I haven't found it. I'll drop by the local university and read the articles on it sometime in the next week or so. We should discuss the Nambu formulas here or maybe on another thread. Dr. Koide also mentioned the Matsumoto formula, which I've not yet looked up.

Since the mass I'm using comes from the electron and muon masses, I can calculate the "Nambu mass" to much higher accuracy. The number starts out as 34.87 MeV.

Second, on the analogy between the force that composes the electron, muon and tau, and the excitations of the elementary particles: At first I was thinking that the analogy should be strongest when the three quarks making up the baryon were identical, as in the Delta++. But the spin of a Delta++ has to be 3/2 which is different from that of the electron.

In the theory I'm playing with, the 3 preons inside an electron are assumed to be in an S state and can transform from one to another by a sort of gluon. To get that kind of wave function, one should instead look at the baryons that are made up of three different quarks.

Among the low lying baryons, there are two that are composed of one each of u, d, and s. These are the Lambda and Sigma. The charged lepton Koide formula is:

\sqrt{m_n} = 17.716 \sqrt(MeV) (1 + \sqrt{2}\cos(2n\pi/3 + 0.22222204717(48) ))

and the neutrinos by a similar formula (multiplied by 3^{-11}), but with the angle \pi/12 added to the angle inside the cosine. These are the m=0 and m=1 mass formulas listed above, though the neutrinos are not included above.

To get the analogy between the charged leptons and the baryons as close as possible, one naturally looks for a set of three "uds" baryons that have the same angle as the charged lepton mass formula. Such a triple does exist, it is the \Lambda_{3/2-} D03. The triple consists of the \Lambda(1520), \Lambda(1690), \Lambda(2325). Putting these into the Koide formula gives the form:

\sqrt{m_n} = 42.769 + 5.5856 \cos(2n\pi/3 + 0.22186)\;\;\;\sqrt{MeV}

The angle is close to the 0.22222204717, though it cannot be distinguished from 2/9. The other two constants are related to the 17.716 constant approximately as
\mu_v = 42.769 = (1+\sqrt{2}) \;\;\;17.716
\mu_c = 5.5856 = \sqrt{2}\;\;\; 17.716\times 2/9

Making the assumption that these are exact allows one to "predict" the associated resonances as:

\Lambda(1520) = 1520.408
\Lambda(1690) = 1690.673
\Lambda(2325) = 2323.355

These numbers are well within the PDG estimates. This is the only uds excitation that falls in the m=0 class. The other Lambda and Sigma excitations have some interesting numbers as well, but are not as nicely suggestive.

The suggestion is that \mu_v comes from the internal energy of the particles. Looking at a quark as a system, its internal (square root) energy is the 1+\sqrt{2} number in the charged lepton formulas when you ignore the cosine.

The idea here is that if you ignored the color effects and the energy of the stuff that glues them together, all quarks would weigh the same amount. The "1" is the length of the mass vector that the preons differ in, while the sqrt(2) is the length of the mass vector that they share. This sqrt(2) gets modified by the cosine according to how well they cancel their fields. (And the generations arise from glue effects.)

The \mu_s comes from the color force. The color force between quarks is only 2/9 of the force between the preons. One can provide various unconvincing arguments for why this should be 2/9. Suffice it to say that 2/9 shows up fairly frequently in these formulas.

The third thing I need to mention is that I made an error in a calculation for the delta angles from the baryon excitations. I was making calculations by calculator, this was before I coded it up into Java. There were two excitations that gave particularly bad errors in their delta calculation. The primary change is that these errors decreased considerably and the fit is much better than advertised.

The \Sigma_{1/2-} delta error was -4.7 degrees in the m=1 class, now it is 20.44 and is in the m=6 class with an error of +3.17. The \Delta_{3/2+} error was 3.43. Now the best angle is 34.10 and the error is 1.83 degrees. There are still wide error bands on the calculated angles, but the RMS error is close to halved as these two outliers contributed 80% of the old RMS error.

Eventually I'll write this up in a LaTex article and check the numbers carefully. Right now, I'm amusing myself by alternately pushing from the theoretical and phenomenological sides. Also I should mention that I found and fixed an unrelated minor Java programming error in the Koide calculator.

When I finally get around to writing up the LaTex article, I will try to figure out how Alejandro and Andre wrote the "Gim" symbol in this paper:

http://www.arxiv.org/abs/hep-ph/0505220

and redefine it as a vector, so that mass = |Gim|^2.

There are obvious reasons for expecting powers of e in physics. Powers of 3 are more rare. One way of getting a power of 3 is by exponentiating ln(3). Lubos Motl's blog recently brought the subject of how ln(3) shows up in black hole calculations here:
http://motls.blogspot.com/2007/04/straightforward-quasinormal-calculation.html

[Hans de Vries]

The latest Standard Model prediction for the tau magnetic moment is:

1. 00117721 (5)

The theoretical value is six orders of magnitude more accurate
as the experimental one due of course to the short lifetime.


The tau lepton anomalous magnetic moment
S. Eidelman, M. Giacomini, F.V. Ignatov, M. Passera
http://arxiv.org/abs/hep-ph/0702026


Theory of the tau lepton anomalous magnetic moment
S. Eidelman, M. Passera
http://arxiv.org/abs/hep-ph/0701260


Regards, Hans

[Hans de Vries]

Another numerical coincident of the vertex correction (magnetic anomaly)
in a quite elementary mass ratio. This time the square of the pion mass delta:

\left|\ \frac{\pi^\pm}{\pi^0} - 1\ \right|^2\ =\ 0.00115821 (26)

So we have as numerical coincidences:

0.001159652________ Electron Magnetic Analomy
0.001159567________ Mass independent Magnetic Analomy
0.001158692_(27)___ Muon / Z boson mass ratio.
0.00115821__(26)___ Pion mass delta square.

The latter two relations are as good as sigma 1.8. Not as good but similar
is this one concerning the proton neutron mass delta:

\left|\ \frac{m_p}{m_n} - 1\ \right|\ \ =\ 0.0013765212 (6)


0.00131419__(41)___ Muon / W boson mass ratio
0.0013765212_(6)___ Proton-neutron mass delta


Regards, Hans

http://arxiv.org/PS_cache/hep-ph/pdf/0503/0503104v1.pdf

[Hans de Vries]

A pretty amazing coincident isn't it? The relation:

\mbox{\Huge $\frac{m_{\pi^\pm}}{m_{\pi^0}}\ =\ 1+ \left(\frac{m_\mu}{m_Z}\right)^\frac{1}{2}\ =\ 1.0340344(55)$}

following from the previous post is as exact as:

1 : 1.0000067 (42)

Where more than half the error is experimental uncertainty.


And secondly. The delta isn't just any value. It's square, which is:

\mbox{\Huge $\frac{m_\mu}{m_Z}\ =\ 0.001158692(27) $}

is to a very high degree equal to the magnetic anomaly. Most notably
the mass independent value (without the vacuum polarization terms),
which is 0.001159567


I've used the latter two of following pion mass data:

m_{\pi^\pm} = 139.57018 \pm 0.00035\ MeV
m_{\pi^0} = 134.9766 \pm 0.0006\ MeV
m_{\pi^\pm}-m_{\pi^0} = 4.5936 \pm 0.0005\ MeV

Where the difference is experimentally known better as the two
absolute values. So the error I use is the 0.0005 MeV. The
combined error of the charged and neutral pions is larger as
the error in the numerical coincident!


Regards, Hans.

[Jay R. Yablon]

Dear friends over at Physics Forums:

I have for more than two years been researching the possibility that
baryons may in fact be non-Abelian magnetic sources.

The result of this research are now formally and rigorously presented in
a paper at:

http://home.nycap.rr.com/jry/Papers/Baryon%20Paper.pdf

Among other things, I believe this paper fundamentally solves the
problem of quark and gluon confinement within baryons, and origin of
mesons as the mediators of nuclear interactions. It may also resolve
the question of fermion generation replication.

I think you guys may enjoy playing with the mass formula which I first develop in (3.7). It bears a resemblance to Koide formula.

I would very much appreciate your constructive comments.

Best to all.

Jay
_____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com
Web site: http://home.nycap.rr.com/jry/FermionMass.htm
sci.physics.foundations co-moderator

[Hans de Vries]

Dear friends over at Physics Forums:

I have for more than two years been researching the possibility that
baryons may in fact be non-Abelian magnetic sources.

The result of this research are now formally and rigorously presented in
a paper at:

http://home.nycap.rr.com/jry/Papers/Baryon%20Paper.pdf


Thank you for presenting your work over here Jay. It's becoming more
interesting with each new paper.

I think we do share the viewpoint that much of the interesting and
important physics is the result of the special terms, like the spin term in
the Dirac equation for the fermions, and the extra non-Abelian term for
the Bosons. My feeling is that the universe would be just a dull homo-
geneous "soup" without these two terms.


Regards, Hans.

[Hans de Vries]

I never did expect numerical coincidences with hadron masses but
the pion shows a second one, again magnetic anomaly related.
Half the mass ratio of the (charged) two quark pion and the electron
is roughly equal to alpha. Now if we divide this value by 2pi then we
find a coincidence with the Muon's magnetic anomaly:

\mbox{\Huge $\frac{1}{\pi} \frac{m_e}{\ \ m_{\pi\pm}}\ =\ 0.001165407 (3) $}

We already had an Electron magnetic anomaly coincidence:

\mbox{\Huge $\left(\ \frac{m_{\pi^\pm}}{m_{\pi^0}} - 1\ \right)^2\ =\ 0.00115821 (26) $}

This adds to the list of Magnetic Anomaly related numerical coincidences:


0.001165920________ Muon Magnetic Analomy
0.001165407__(3)___ Electron / Pion mass ratio /pi.
0.001165892________ Electrom / Pion mass ratio +Vpi-Vmu

0.001159652________ Electron Magnetic Analomy
0.001159567________ Mass independent Magnetic Analomy
0.001158692__(27)__ Muon / Z boson mass ratio.
0.00115821___(26)__ Pion mass delta square.

0.0000063532_______ Muon Mag.Anomaly mass dependence
0.0000063558_(20)__ Electron / W mass ratio (2007)


The third value (+Vpi-Vmu) is closer. What we have done here
is to add the one-loop QED vacuum polarization difference for
a particle of muon mass and a particle of pion mass. A lepton
with the mass of a pion should have a higher magnetic anomaly
as the muon by this amount plus another 4-8% or so for higher
order VP loops which I can't calculate.

The "Mass independent Magnetic Anomaly" is the same for
all leptons. It is calculated here by subtracting the one loop VP
contribution from the electrons magnetic anomaly.

The third numerical coincidence was found earlier and relates
the electron / W mass ratio with the mass dependent part
of the Muon's magnetic anomaly. It was calculated by taking
the difference between the Muon and Electrons magnetic
anomalies and adding the Electrons largest (one-loop) VP term.


Regards, Hans


One loop vacuum polarization:

m1/m2 = 273.132044975751, one-loop vpol = 0.000006389950 (pion +/-)
m1/m2 = 206.768283800000, one-loop vpol = 0.000005904060 (muon)
m1/m2 = 1.00000000000000, one-loop vpol = 0.000000084641 (electron)

[Jay R. Yablon]

Thank you for presenting your work over here Jay. It's becoming more
interesting with each new paper.

I think we do share the viewpoint that much of the interesting and
important physics is the result of the special terms, like the spin term in
the Dirac equation for the fermions, and the extra non-Abelian term for
the Bosons. My feeling is that the universe would be just a dull homo-
geneous "soup" without these two terms.

Regards, Hans.

Hi Hans, thank you for your encouragement. :smile:

As you know I mentioned over on SPF, the most important results in the baryon paper have to do with confinment. I have extracted and consolidated these results and posted them at the link below.

http://home.nycap.rr.com/jry/Papers/Confinement%20Paper.pdf

This paper may provide an exact analytical solution to the problem of QCD confinement and the existence of short-range meson mediators of the nuclear interactions.

As you noted over on SPF, this paper evolves from the non-vanshing three form P = dF which would be the Yang-Mills (non-abelian) "magnetic current."

If I were to put the paper in one sentence it would be:

Baryons are non-Abelian (Yang-Mills) magnetic monopoles.

And if permitted a second sentence:

These Yang Mills monopoles exhibit all the properties of a baryon, including three fermions which via exclusion we connect with quarks, confinement of these quarks and their mediating gluons such that there is never any net flux of gluons or individual quarks across any closed two-dimensional surface of integration through the baryon current density, and emission of mesons which have short range such that these mesons, which account for interactions between baryons, are the only entities for which there is a net flux through the integration surface.

I appreciate any comments which the Physics Forums readers may have.

Jay.
_____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com
Web site: http://home.nycap.rr.com/jry/FermionMass.htm
sci.physics.foundations co-moderator

[arivero]

0.001165407__(3)___ Electron / Pion mass ratio /pi.


Well, that is rare, as every you find :smile: ! I would expect the quotient between charged and neutral pion to be related somehow to the fine structure constant, and then this natural connection would propagate to the rest of findings in the thread. But again, too much exactitude.

In a related thems, did I tell that I was interested on the history of the calculation of g-2? It seems that a guy, who helped Kino****a to set up the software, is now in the dark side of accidental numeric coincidences.

I have been for the last four months working in a BOINC supercomputing platform, Zivis. I wonder how much time could take to repeat K. et al calculations; I guess that programming time should be higher than computing time.

[Jay R. Yablon]

Dear Physics Forum friends:

I have posted an updated draft of my confinement paper to:

http://home.nycap.rr.com/jry/Papers/Confinement%20Paper.pdf

Last evening, I uncovered and added to the paper, a line of calculation
which explicitly and formally gives a non-zero rest mass to the meson
mediators of QCD without resort to the Higgs Mechanism, naturally
eliminates the propagator poles and therefore allows one to have "real"
rather than "virtual" particles without ad hoc tricks, formally renders
QCD a short-range interaction, and solves the Yang-Mills mass gap
problem. This is in addition to the confinement solution which has
already been posted for several days now.

Interested in feedback.

Jay.
_____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com
Web site: http://home.nycap.rr.com/jry/FermionMass.htm
sci.physics.foundations co-moderator

[arivero]

I believe we have not included G. Rosen in the references for the topic of this thread.

His http://prola.aps.org/abstract/PRD/v4/i2/p275_1 , for instance, gives one of these formulae with exponential relationship between Planck and electron masses. Other papers look for masses of electron and quark, value of the fine structure constant, etc. See Spires (http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=find+a+rosen%2C+g&FORMAT=WWW&SEQUENCE=)

[arivero]

Then we use Nottale's remark (http://wwwusr.obspm.fr/~nottale/ukmachar.htm) to get the mass of the electron


\ln (M_P/m_e) = \alpha^{-1} \sin^2 \theta_W



Note that this remark is mostly an elaborate rewritting of the argument about the self energy of the electron. For instance in a modern text such as Polchiski's string textbook, it appears as

\delta m \approx \alpha m \ln {1 \over m l}

without any reference except "known since the 1930s", and then it is explained that l should to be expected to ultimate cutoff, thus the lenght of Planck, and that \delta m \over m should be expected to be of order O(1).

Thus the formula can be rewritten as

\alpha^{-1} {\delta m_e \over m_e} \approx \ln {m_P \over m_e}


And Nottale's statement translates, in standard knowledge, as telling that the electromagnetic contribution to the mass of the electron is about 3/8.

While most of the "rediscoverers" of this relationship are outsiders, "out of the loop" people, in the case of Nottale it is sort of astonishing that he forgets to tell that he is recovering a "known" relationship and not a new one (he does not have, at all, new arguments for the 3/8 adjustment neither).

Another remark nobody does is that the argument can be reverted to predict the GUT-Planck-String scale from the low energy scale. Thus one could to try to predict muon or electron masses from the electroweak scale, then use the cutoff argument to predict the GUT scale, and the to use the seesaw to predict the neutrino scale.

EDITED: Nottale presentation can be found in http://luth2.obspm.fr/~luthier/nottale/ukmachar.htm Also, it is argued in http://luth2.obspm.fr/~luthier/nottale/arDNB.pdf that better results are met if one uses in a very peculiar way the fine structure constant running value instead of the infrared one. Indeed using alpha at m_e, one really "predicts" (except the justification of the 3/8) the value of Newton Constant. In any case, I am ashamed 1) about our inability to note that it was just the renormalisation of electron mass as usual, and 2) that Nottale hides or ignores this fact, after being in contact with QED theory during decades.

[arivero]

For numerical "coincidences", hep-ex/0412028 remembers that "the infamous Florida 2000 presidential election recount with the official result of 2,913,321 Republican vs. 2,913,144 Democratic votes, with the ratio equal to 1.000061".

[Hans de Vries]

Once in a while one runs into a nice numerical coincident like this one
relating alpha to a Yukawa potential:

\mbox{\Huge $\frac{\alpha}{2\pi}\ =\ \frac{e^{-2\pi r}}{2\pi r}$}

Where r is the quotient of two radii:

r\ =\ \frac{r_c}{r_l}

in which rc is the Compton radius of the electron, muon or tau lepton and
rl is the corresponding cut-off radius for which the electrostatic energy
is equal to the magnetostatic energy of the classical electron, muon or
tau lepton. It's the magnetic moment which is responsible for the magneto-
static energy. One gets the following relations:

\alpha =1/137.05268 -\ \quad \mbox{for the electron}
\alpha =1/137.04743 -\ \quad \mbox{for the muon}
\alpha =1/137.03796 -\ \quad \mbox{for the tau lepton}
\alpha =1/137.03599971\ \quad \mbox{experimental}

The differences stem from the differences of the magnetic moment analomies.


Regards, Hans





==================================================

PS: For the EM fields the following expressions were used:


\textsf{E}_x\ = \frac{q}{4\pi\epsilon_o}\ \frac{x}{r^3}, \quad
\textsf{E}_y\ = \frac{q}{4\pi\epsilon_o}\ \frac{y}{r^3}, \quad
\textsf{E}_z\ = \frac{q}{4\pi\epsilon_o}\ \frac{z}{r^3}



\textsf{B}_x\ = \frac{\mu_o\mu_e}{2\pi}\left(\frac{3z}{r^5}x
\right), \quad \textsf{B}_y\ =
\frac{\mu_o\mu_e}{2\pi}\left(\frac{3z}{r^5}y \right), \quad
\textsf{B}_z\ =
\frac{\mu_o\mu_e}{2\pi}\left(\frac{3z}{r^5}z-\frac{1}{r^3} \right)


And for the total energy:



\mbox{Energy:} \qquad E\ =\ \frac{1}{2}\left(\epsilon_o\textsf{E}^2
+ \frac{1}{\mu_o}\textsf{B}^2\right)



E\ =\ \int \left\{ \frac{q^2}{32\pi^2\epsilon_o r^4}\ +\
\frac{\mu_o\mu_e^2}{8\pi^2}\left(\frac{3z^2}{r^8}+ \frac{1}{r^6}
\right) \right\} dx^3\ =\ \frac{q^2}{8\pi\epsilon_o r_o}\ + \
\frac{\mu_o\mu_e^2}{4\pi}\ \frac{3}{r_o^3}


For an electron:


\frac{q^2}{8\pi\epsilon_o} = 1.1535385\
10^{-28},\quad \frac{3\mu_o\mu_e^2}{4\pi}\ =\ 2.5862051\ 10^{-53}


and ro is the cut-off radius.

[arivero]

This last comment reminders me about Born calculation of the Lamb shift,
in the train cominb back from Shelter Island. He had no serious argument to fix the cut-off in the renormalized equation, so he simply choose the compton lenght of the electron because it is the cutoff of particle creation.

[Walter L. Wagne]

See:

"Evidence for Detection of a Moving Magnetic Monopole", Price et al., Physical Review Letters, August 25, 1975, Volume 35, Number 8.

This was the last of a series of balloon flights, launched in 1973, but not analyzed by myself until 1975, due to higher priority cosmic rays analysis then ongoing.

The suggestion that the anomalous track could have been caused by a doubly fractionating normal nucleus is untenable. One would have expected to have seen billions of similar tracks, not quite as closely matched to the expected track of a magnetic monopole, first. No such similar events were ever detected.

For further information, contact the administrator who can email me, as I do not regularly post at this forum. Or check www.sciforums.com where I do regularly post, and PM me.

Whether the Large Hadron Collider [LHC] will create a magnetic monopole is highly debatable. It might also create miniature black holes, or strangelets.

[taarik]

Please see my paper about mass quantization @ arxiv. hep-ph/0702140

[arivero]

Please see my paper about mass quantization @ arxiv. hep-ph/0702140

Indeed your paper, your reference list
http://www.slac.stanford.edu/spires/find/hep/wwwrefs?key=7074930
and the list of citations of McGregor
http://www.slac.stanford.edu/spires/find/hep/www?c=NUCIA,A58,159
are interesting for the topics of this thread. A problem of quantisation of M instead of quantisation of M^2 is that it has some scent of classical group theory, thus one needs to see how many of the relationships are already explained in the quark model and check the extant cases.

[taarik]

Thiis paper has been accepted for publication in Modern Physics Letters A.
The important thing is that while the charged pions and muon are related in sense via the decay of former into latter, the neutral pion and muon are not related in any sense. Yet there mass difference serves as a basic mass unit for both leptons and hadrons.

[arivero]

Thiis paper has been accepted for publication in Modern Physics Letters A.
The important thing is that while the charged pions and muon are related in sense via the decay of former into latter, the neutral pion and muon are not related in any sense. Yet there mass difference serves as a basic mass unit for both leptons and hadrons.
very good news.

the question, to me, is not why muon and pion have different mass, but why have they got almost the same mass. A conjecture is SUSY.

[taarik]

Here is another surprise
the t lepton mass can be obtained by taking 57 jumps of 29.318 MeV from the muon mass i.e t mass= 57x 29.318. Now this 57 number also helps us to include the electron mass as 57 times electron mass= 29.127 very close to 29.318. This leads us to thing that like in Nambu's and many other cases the basic unit appears from the electron mass.
Now this also means the pion -muon= 57x electron , tau - muon =57x 29.318 =57x57x electron. which in turn leads to tau- pion =56x 29.318 =56 x 57 x electron.
Hence the lightest hadron i.e pion and lightest unstable lepton i.e muon , two leptons muon and tau , lightest hadron and heaviest lepton i.e tau are all related through electron mass.

[arivero]

Now this also means the pion -muon= 57x electron , tau - muon =57x 29.318 =57x57x electron. which in turn leads to tau- pion =56x 29.318 =56 x 57 x electron..

I prefer to write then

{ m_\pi - m_\mu \over m_e}= \sqrt {m_\tau - m_\mu \over m_e}

It should be nice to have a mathematical (group theoretical) argument for 57.

EDIT: It is a bit puzzling that if we fix the mass of tau, mu and electron to the experimental values, the above formula "predicts" 134.88 MeV, to be compared with the mass of the neutral pion (134.976 MeV). Naively one could expect the result to be more related with the mass of the charged pion, which is 4.6 MeV above.

EDIT2. Perhaps Krolikowski has some argument for 58/2. Also, Ramanna (eg pg 16 of nucl-th/9706063)

[arivero]

I prefer to write then...
Hmm, funnier:

( m_{\pi_0} - m_\mu )= \sqrt {m_e} \sqrt {m_\tau - m_\mu}


LHS and RHS still agree within a 0.3 %. No bad.

The above comments still apply. On other hand, if I recall correctly, the question about why the mass of the charged pion is higher, and not lower, than the neutral one was a touchy issue decades ago, and it required very high level theoretists to explain it.

[taarik]

Refering to first version, it ia again amazing that the mass of the neutral pion is deteremined very precisely in terms of the three leptons. Now neutral pion has no relation with the three leptons. it does not decay into any of these particles. On the other hand the charged pion decays into electron and muon. Hence charged pion mass should have been related to lepton masses.

[arivero]

Refering to first version, it ia again amazing that the mass of the neutral pion is deteremined very precisely in terms of the three leptons. Now neutral pion has no relation with the three leptons. it does not decay into any of these particles. On the other hand the charged pion decays into electron and muon. Hence charged pion mass should have been related to lepton masses.

I agree, it is misterious. Furthermore, forgetting the issue of integer multiples and the squaring of masses, the formula is very reminiscent of charged pion decay, you know, these \prop m_\mu^2 ( m_\pi_+^2 - m_\mu^2) from textbooks.

To put more intrigue, the mass difference between eta and the average of pion and muon (say, diff=427.2 MeV) also fits roughly in the obvious permuted formulae:

\sqrt {m_\mu} \sqrt {m_\tau - m_e} \approx \sqrt {m_\tau} \sqrt {m_\mu - m_e}
\approx \sqrt {m_\tau \pm m_e} \sqrt {m_\mu \pm m_e}
\approx \sqrt {m_\mu} \sqrt {m_\tau} = 433.27 MeV


EDITED: a purpose of the above formulas is to consider the limit m_\mu \approx m_\tau where the former formula cancels and the two first ones in the above become the same. Also, the same cancellation and similarity happens in the other limit m_e \to 0. Simultaneous limit conflicts with Koide's.

[arivero]

My thinking during the last two years was:
initially there is a symmetry where neutrinos have the same mass than neutral mesons and charged leptons have the same mass than charged mesons. Note the count of degrees of freedom. Of course one could also expect the dirac mass of neutrino and charged lepton to coincide.
Then seesaw moves the mass of neutrinos out of reach
and mixing, including CKM, and/or other unknown mechanism alter the mass eigenvalues of the mesons.
The mechanism could be related to a mismatch between isospin in mesons and leptons. Namely, third generation mesons do not exist except bB.

[arivero]

To put more intrigue, the mass difference between eta and the average of pion and muon (say, diff=427.2 MeV) also fits roughly in the obvious permuted formulae:

\sqrt {m_\mu} \sqrt {m_\tau - m_e} \approx \sqrt {m_\tau} \sqrt {m_\mu - m_e}
\approx \sqrt {m_\tau \pm m_e} \sqrt {m_\mu \pm m_e}
\approx \sqrt {m_\mu} \sqrt {m_\tau} = 433.27 MeV


Hmm, Gell-Mann Okubo value for unmixed \eta_8 is
569.32 GeV, so \eta_8 - \pi^0 = 434.34 MeV

[arivero]

Hmm, Gell-Mann Okubo value for unmixed \eta_8 is
569.32 GeV, so \eta_8 - \pi^0 = 434.34 MeV

For amateurs, it could be worthwhile to explain what the Gell-Mann Okubo is, or refer to a textbook. I like Donoghue Golowich Holstein, "Dynamics of the Standard Model". The formula appears in chapter VII, expression (1.6b). You get the formula from the following set of equations:
m^2_\pi=B_0 (m_u + m_d)
m^2_{K^0}=B_0 (m_s + m_d)
m^2_{K^\pm}=B_0 (m_s + m_u)
m^2_{\eta_8}=\frac 13 B_0 (4 m_s + m_u + m_d )
and so on.

Asuming isospin, up and down have the same mass, and thus you can get a combination of neutral kaon, pion and eta8.

If works well with the neutral particles; it is not only that it does not account for isospin; the idea does not account for EM interactions neither. Old timers extract an extra EM relation via "Dashen's theorem", but I think to remember there was some work of Witten or some other genious about this kind of corrections.

EDITED: Indeed we could use the above expressions to reformulate our equations in terms of the mass m_s and \hat m \equiv m_u = m_d, with SU(3) flavour breaking to global SU(2) isospin x U(1) as it happened in the papers of 1960s on global symmetries.

m^2_\pi = (m_\mu + \sqrt { m_e (m_\tau-m_\mu)})^2 = B_0 \hat m
m^2_{\eta_8} = (m_\pi + \sqrt { m_\mu (m_\tau-m_e)})^2 = \frac 23 B_0 (2 m_s + \hat m)
Here you can see also one of the themes which were debatable in the sixties: the use of mass square instead of plain mass. For instance, it is because of it that our resulting equations
do not allow to cancel B_0 out.

[arivero]

Here is another surprise
the t lepton mass can be obtained by taking 57 jumps of 29.318 MeV from the muon mass i.e t mass= 57x 29.318. Now this 57 number...

http://adsabs.harvard.edu/abs/1962RSPSA.268...57D

Dirac gets 53 times the electron mass for the muon, in a paper that has been lately recalled by the guys working on strings and branes.

[arivero]

A pretty amazing coincident isn't it? The relation:

\mbox{\Huge $\frac{m_{\pi^\pm}}{m_{\pi^0}}\ =\ 1+ \left(\frac{m_\mu}{m_Z}\right)^\frac{1}{2}\ =\ 1.0340344(55)$}

following from the previous post is as exact as:

1 : 1.0000067 (42)


About this one, it can have interesting implications: if the mass of the muon goes to zero, assume so it happens with the masses of up and down, then (global) isospin symmetry is restored. On the contrary, if Z goes to cero (and W) but the quark masses are different, the restored symmetry is only gauge G-W-S isospin and the charged pion decays into the neutral one. What is amazing in this argument is that if Z goes to infinity the two pions can not tunnel one into another, but from the point of view of the electroweak scale the masses of quarks are negligible, thus global isospin is restored again.

The traditional current algebra formula for the pion mass(^2) difference puts it in terms of the fine structure constant and the pion decay constant, e^2 / F_\pi^2 times some other factors.

Entering the octet, we are touching deep problems of the elders. There is a short work of witten in 1983 (http://www.slac.stanford.edu/spires/find/hep/www?j=PRLTA,51,2351) about how the mass of the charged pion must always be higher than the neutral pion, even if only to avoid tachions in the limit of zero pion mass. Also, the mixing between [itex]\eta_8[\itex] and [itex]\eta_0[\itex] to give [itex]\eta[\itex] and [itex]\eta'[\itex] was the U(1) headache, addressed by t'Hoft, Veneziano (http://www.slac.stanford.edu/spires/find/hep/www?j=NUPHA,B159,213) and Witten (http://www.slac.stanford.edu/spires/find/hep/www?j=NUPHA,B156,269) independently, and according Okubo still unclear. I have found even some recent work in the context of strings: Armoni 2004 (http://www.slac.stanford.edu/spires/find/hep/wwwrefs?key=5864437)

[arivero]

http://arxiv.org/abs/0710.2429 (g-2)_mu status and prospects

[Count Iblis]

You could try to use the LLL algorithm (http://mathworld.wolfram.com/LLLAlgorithm.html) to find formulae.

[arivero]

You could try to use the LLL algorithm (http://mathworld.wolfram.com/LLLAlgorithm.html) to find formulae.
It is an interesting idea. Actually, the team of D.H. Bailey (http://crd.lbl.gov/~dhbailey/dhbpapers/) have tried in the past to input the standard model masses etc in some of their algorithms; but I had no idea about their new work using LLL.

[Count Iblis]

Actually, the LLL algorithm is a bit older than Bailey's PSLQ algorithm. The LLL is a little less efficient, but that's only a problem if you have thousands of significant digits of some number and try to find a formula in terms of known constants for it.

For your work, the LLL may be better, because with the LLL you can look for simultaneous relations, the PSLQ can't do that (at least not in general, you can work with complex numbers, quaternions,... but you'll reach a limit beyond which you can't go). If you have 5 numbers that are known to ten digits then you have 50 digits of information. Ten digits may not be enough to detect a relation, but 50 may be enough provided, of course, that the five numbers are given by formulae of the same form that are specified by the same constants.

See the Appendix of this article (http://arxiv.org/abs/0708.1763) for details.

[arivero]

The significance of numerical coincidences in nature
Authors: Brandon Carter

http://arxiv.org/abs/0710.3543

[jal]

I found that your link gave a good and understandable explanation.
However,
p. 9
...They can be categorised as three coupling constants,
and three mass ratios, and their empirically determined numerical
values are approximately:12
gS .=4, e.=1/12, mN.=1/2× 10−10
The values of the coupling constants are rather more familiar in their
squared forms: thus we have the gravitational fine structure constant
… the ordinary (electromagnetic) fine structure constant, e2 .= 1/137, and …
----------
I need explanation with e.=1/12 and , e2 .= 1/137
I always thought that 12X12=144 not 137
jal

[arivero]

Well, I guess he takes the nearest simple fraction to sqrt(137), and he chooses 1/12. So it really compares 12 against 11.7, arguably not so bad.

A peculiarity of this article is that a couple ways ago I have been told that the relationship between mass of electron and pion (!) had circulated as a conjecture in the sixties. I though it came from McGregor, but it is formula (6) in pg 11 of this paper.

[jal]

"So it really compares 12 against 11.7, arguably not so bad."
If you find anything else that can help I'll put it in "How to build a universe"
jal

[CarlB]

I've had some interesting results in rewriting the Koide equation as a sort of "field energy" equation. The idea is to treat the square in the mass as coming from the energy of a field.

Field energies are quadratic, for instance, E&M field gives mass as m = (E^2 + B^2)/c^2, where I've left off some units. So begin with electromagnetism as a toy example.

Then the thing to notice is that E and B end up quantized at different amplitudes. Magnetic monopoles are much heavier than electrons, so assume that when you quantize E and B, the contribution of B dominates, giving you m = B^2/c^2.

From there, you assume that the angle I've called "delta" is 2/9exactly, and that the reason this doesn't exactly fit the electron, muon and tau masses is cause "E" does contribute slightly.

That converts the Koide formula from being a two parameter fit, with mu and delta, (the mass scale and the angle), to being a two parameter fit with a B scale and an E scale. To write the masses we have

m_n = |B|^2(1 + \sqrt{2}\cos(2/9+2n\pi/3))^2 + |E|^2(\sqrt{2}\sin(2/9+2n\pi/3))^2

where B and E are constants. The contribution to B is split into two parts, 1 + \sqrt{2}\cos(2/9+2n\pi/3), so we write B_v = B, B_s = \sqrt{2}\cos(2/9+2n\pi/3), and E_s=\sqrt{2}\sin(2/9+2n\pi/3). That is, the "v" field is a valence field that is shared between the electron, muon, and tau, and the "s" field is a sea field that distinguishes the three generations.

Then the mass equation is m = (B_v + B_s)^2 + E_s^2.

What's more interesting is that if you write down the vectors (B_v,B_s,E_s) for the electron, muon, and tau, you get the tribimaximal mixing matrix (after scaling the B stuff and E stuff so that each vector has length 1).

Another way of saying this is that the vectors (B_v,B_s,E_s) are orthogonal. Making them orthonormal defines the tribimaximal neutrino mixing matrix. (Except that when you see it in the literature, it is usually has two columns reversed so you should put the three contributions in the order (B_s,B_v,E_s) instead.)

Using the best PDG numbers for the electron and muon masses to predict the tau mass, the equations for the charged lepton masses are (ignore the precision, I haven't had time to compute the ranges and fix everything up yet):

\begin{array}{rcl}
m_n &=& 313.8561002547\;\textrm{MeV}\;(1 + \sqrt{2}\cos(2/9 + 2n\pi/3))^2\\
&&+4.6929703\;\textrm{eV}\; (\sqrt{2}\sin(2/9+2n\pi/3))^2
\end{array}

And the three vectors (which ignore the phase angle 2/9 because it is presumably cancelled in the neutrinos) are:
\begin{array}{ccc}
(1,& \sqrt{2},& 0)\\
(1,& -\sqrt{2}/2,& +\sqrt{3/2})\\
(1,& -\sqrt{2}/2,& -\sqrt{3/2})
\end{array}

In the above, note that the angle 2/9 has been removed as it is presumably cancelled in the neutrinos, which also use a similar angle. And the scaling to B and E has been removed because in computing phases, one needs to normalize by particle number rather than energy.

After dividing by the lengths of the vectors, sqrt(3), and turning the three vectors into a matrix, one has:
\left(\begin{array}{ccc}
\sqrt{1/3},& \sqrt{2/3},& 0\\
\sqrt{1/3},& -\sqrt{1/6},& +\sqrt{1/2}\\
\sqrt{1/3},& -\sqrt{1/6},& -\sqrt{1/2}
\end{array}\right)


Carl

Koide paper giving Tribimaximal mixing matrix, see eqn (3.2):
http://arxiv.org/abs/hep-ph/0605074

[Hans de Vries]

After dividing by the lengths of the vectors, sqrt(3), and turning the three vectors into a matrix, one has:
\left(\begin{array}{ccc}
\sqrt{1/3},& \sqrt{2/3},& 0\\
\sqrt{1/3},& -\sqrt{1/6},& +\sqrt{1/2}\\
\sqrt{1/3},& -\sqrt{1/6},& -\sqrt{1/2}
\end{array}\right)

Carl

Koide paper giving Tribimaximal mixing matrix, see eqn (3.2):
http://arxiv.org/abs/hep-ph/0605074

Interesting, I noticed that one can write the above tribimaximal
matrix for neutrino mixing as the x,y,z coordinates of a tetrahedron
with sides of \sqrt{1/2} with its top down and the origin in h/2,
thus:

\left(\begin{array}{ccc}
z_1,& y_1,& x_1\\
z_2,& y_2,& x_2\\
z_3,& y_3,& x_3
\end{array}\right)\ =\
\left(\begin{array}{ccc}
\sqrt{1/3},& \sqrt{2/3},& 0\\
\sqrt{1/3},& -\sqrt{1/6},& +\sqrt{1/2}\\
\sqrt{1/3},& -\sqrt{1/6},& -\sqrt{1/2}
\end{array}\right)

The angle of 2/9 radians is then a simple rotation around the z-axis
to get your form of Koide's lepton mass formula:

\sqrt{m_n}\ =\ \sqrt{1/3} + \sqrt{2/3}\cos(2/9 + 2n\pi/3)


I see that this "A4-symmetry" was already found here by Ma:
http://arxiv.org/PS_cache/hep-ph/pdf/0606/0606024v1.pdf

and that there is another group X24 which is larger which could
incorporate quarks here:
http://aps.arxiv.org/PS_cache/hep-ph/pdf/0701/0701034v3.pdf

On the other hand, Garrett Lisi uses a 3d quark matrix here:
http://arxiv.org/PS_cache/arxiv/pdf/0711/0711.0770v1.pdf
http://www.physicsforums.com/showthread.php?t=196498

which is the same except for some coordinate switching:

\left(\begin{array}{ccc}
-\sqrt{1/3},& -\sqrt{1/3}, & -\sqrt{1/3} \\
-\sqrt{1/2},& +\sqrt{1/2}, & 0 \\
-\sqrt{1/6},&-\sqrt{1/6},& \sqrt{2/3}
\end{array}\right)

see (2.4) in the paper and also page 18

Regards, Hans

[CarlB]

Hans,

I ran into the Garrett matrix at Bee's blog:
http://backreaction.blogspot.com/2007/11/theoretically-simple-exception-of.html

Of all the places where it comes up, the physically most convincing one to me is the neutrino mixing angle. If I had known in advance when I was writing the above post that I would get that matrix, I might have started from that end.

I need to go back and look at the baryon masses and see if this sort of formula fits them better.

[Hans de Vries]

Hans,

I ran into the Garrett matrix at Bee's blog:
http://backreaction.blogspot.com/2007/11/theoretically-simple-exception-of.html

Of all the places where it comes up, the physically most convincing one to me is the neutrino mixing angle. If I had known in advance when I was writing the above post that I would get that matrix, I might have started from that end.

I need to go back and look at the baryon masses and see if this sort of formula fits them better.


One can see the matrix elements as vertices of the unit cube as well,
with one vertex at 0.0 and the three nearest given by:

\left(\begin{array}{ccc}
z_1,& y_1,& x_1\\
z_2,& y_2,& x_2\\
z_3,& y_3,& x_3
\end{array}\right)\ =\
\left(\begin{array}{ccc}
\sqrt{1/3},& +\sqrt{2/3},& 0\\
\sqrt{1/3},& -\sqrt{1/6},& +\sqrt{1/2}\\
\sqrt{1/3},& -\sqrt{1/6},& -\sqrt{1/2}
\end{array}\right)

Connected to the Lepton mass ratios by a rotation around the z-axis:

\left(\begin{array}{c}
\sqrt{m_\tau} \\
\sqrt{m_\mu} \\
\sqrt{m_e}
\end{array}\right)\ =\ C
\left(\begin{array}{ccc}
\sqrt{1/3},& +\sqrt{2/3},& 0\\
\sqrt{1/3},& -\sqrt{1/6},& +\sqrt{1/2}\\
\sqrt{1/3},& -\sqrt{1/6},& -\sqrt{1/2}
\end{array}\right)\left(\begin{array}{c}
1 \\
\cos(2/9) \\
\sin(2/9)
\end{array}\right)

with "2/9" replaced with 0.222222047168 (465) one gets
the precise lepton mass ratios, as we know since your post here:

http://physicsforums.com/showthread.php?t=46055&page=8



\begin{array}{llll}
\mbox{equation:} &
\mbox{\huge $\frac{m_\mu}{m_e}$}\ =\ 206.7682838 (54) &
\mbox{\huge $\frac{m_\tau}{m_e}$}\ =\ 3477.441653 (83) &
\mbox{\huge $\frac{m_\tau}{m_\mu}$}\ =\ 16.818061210 (38) \\
\mbox{experim:} &
\mbox{\huge $\frac{m_\mu}{m_e}$}\ =\ 206.7682838 (54) &
\mbox{\huge $\frac{m_\tau}{m_e}$}\ =\ 3477.48 (57) &
\mbox{\huge $\frac{m_\tau}{m_\mu}$}\ =\ 16.8183 (27)
\end{array}


Well within experimental precision

Regards, Hans

[Hans de Vries]

The value 0.222222047168 (465) does very nice indeed
as the Cabibbo angle as originally guessed.



\mbox{Cabibbo-Kobayashi-Maskawa:}\ \ \left(
\begin{array}{lll} 0.9753 & 0.221 & 0.003 \\ 0.221 & 0.9747 & 0.040 \\ 0.009 & 0.039 & 0.9991
\end{array}
\right)



\left(
\begin{array}{lll} \cos(2/9) & \sin(2/9) & 0 \\ \sin(2/9) & \cos(2/9) & 0 \\ 0 & 0 & 1
\end{array}
\right)\ \ \ =\ \ \
\left(
\begin{array}{lll} 0.9754 & 0.2204 & 0.000\ \\ 0.2204 & 0.9754 & 0.000 \\ 0.00 & 0.00 & 1.000
\end{array}
\right)



Regards, Hans

http://en.wikipedia.org/wiki/CKM_matrix

[Hans de Vries]

We can put this all in a picture (see below) like this:

Place the charged leptons in a 3d coordinate space using the
tribimaximal neutrino mixing matrix: The coordinates determine
the percentage of neutrino mass eigen-states each neutrino
flavor has, with the matrix mirrored, swapping e and \tau

\left(\begin{array}{ccc}
\tau_z & \tau_y& \tau_x\\
\mu_z & \mu_y& \mu_x\\
e_z& e_y& e_x
\end{array}\right)\ =\
\left(\begin{array}{ccc}
\sqrt{1/3},& +\sqrt{2/3},& 0\\
\sqrt{1/3},& -\sqrt{1/6},& +\sqrt{1/2}\\
\sqrt{1/3},& -\sqrt{1/6},& -\sqrt{1/2}
\end{array}\right)

http://en.wikipedia.org/wiki/Tribimaximal_mixing


Now:

1) The projections P on the vector (sin θ, cos θ, 1) lead to the
exact charged lepton masses if we use 0.22222204717 (47) for θ,
(the Cabibbo angle for Quark mixing ?)


\begin{array}{llll}
\mbox{equation:} &
\mbox{\huge $\frac{P^2_\mu}{P^2_e}$}\ =\ 206.7682838 (54) &
\mbox{\huge $\frac{P^2_\tau}{P^2_e}$}\ =\ 3477.441653 (83) &
\mbox{\huge $\frac{P^2_\tau}{P^2_\mu}$}\ =\ 16.818061210 (38) \\ \\
\mbox{experim:} &
\mbox{\huge $\frac{m_\mu}{m_e}$}\ =\ 206.7682838 (54) &
\mbox{\huge $\frac{m_\tau}{m_e}$}\ =\ 3477.48 (57) &
\mbox{\huge $\frac{m_\tau}{m_\mu}$}\ =\ 16.8183 (27)
\end{array}




2) The projections P obey the Koide relation (exact for any θ):

\left(\ P_e+P_\mu+P_\tau\ \right)^2\ =\ \frac{3}{2}\left(\ P^2_e+P^2_\mu+P^2_\tau\ \right)


3) The coordinates could even be real coordinates using the
Pauli-Weisskopf interpretation of the wave function as a continous
charge-spin density distribution:

The angles with the z-axis of the charged leptons are equal to the
precession angle of spin 1/2 particles and the precessing speed would
be equal to phase frequency of the charged leptons if the torque is 2.
Also, the angle of (sin θ, cos θ, 1) with the z-axis is same as the
precessing angle of a spin 1 vector boson.


Regards, Hans

[CarlB]

I loved the thumbnail, how did you do it?

I'm still mulling over the concept of torque here. I spent the weekend making a java applet that graphs the discrete Fourier transform of the baryons. I got the graphical user interface (GUI) to run, but didn't see the patterns I expected, just noise. That could be defects in the program, defects in my understanding of how to use it, etc.

The reason for looking at discrete Fourier transforms with respect to masses was given by Marni Sheppeard here:
http://arcadianfunctor.files.wordpress.com/2007/11/fouriermass07.pdf

[Hans de Vries]

I loved the thumbnail, how did you do it?



It's done with Povray, The projections are just shadows, they come
basically for free. One could also draw the 3D object multiple times with
each time one of the coordinates fixed. You would get "2.5 D" projections
with the balls and cylinders all aligned in the same 2D plane. Might give
a nice effect as well.

I'm still mulling over the concept of torque here. I spent the weekend making a java applet that graphs the discrete Fourier transform of the baryons. I got the graphical user interface (GUI) to run, but didn't see the patterns I expected, just noise. That could be defects in the program, defects in my understanding of how to use it, etc.


If the input is a number of mass spikes then the "noise" may well
be the correct result.


Regards, Hans.

[Soshnikov_Serg]

Me = sqr((h/(2*Pi))*C/(4*Pi*G)) * Exp( - 16 * Pi)

Me = 9.08086 * 10 ^ -31 kg

it`s not for real electron, bat for "naked" electron
Soshnikov_Serg

[arivero]

Me = sqr((h/(2*Pi))*C/(4*Pi*G)) * Exp( - 16 * Pi)

Indeed even Polchinski book explains about this kind; one expects that Me*sqrt(G)*Exp(1/alpha or something so) to be of order unity, if G is the ultimate cutoff for electroamgnetism.

[Hans de Vries]

The charged leptons are placed in a 3d coordinate space using the
tribimaximal neutrino mixing matrix: The coordinates determine the
percentage of neutrino mass eigen-states each neutrino flavor has.

\left(\begin{array}{ccc}
e_z& e_y& e_x \\
\mu_z & \mu_y& \mu_x\\
\tau_z & \tau_y& \tau_x
\end{array}\right)\ =\
\left(\begin{array}{ccc}
\sqrt{1/3},& +\sqrt{2/3},& 0\\
\sqrt{1/3},& -\sqrt{1/6},& +\sqrt{1/2}\\
\sqrt{1/3},& -\sqrt{1/6},& -\sqrt{1/2}
\end{array}\right)

http://en.wikipedia.org/wiki/Tribimaximal_mixing


Now:

1) The projections P on the vector (cos θ, sin θ, 1) lead
to the exact charged lepton masses if we use for the angle:

θ = 7/6 pi + 0.22222204717 (47)


\begin{array}{llll}
\mbox{equation:} &
\mbox{\huge $\frac{P^2_\mu}{P^2_e}$}\ =\ 206.7682838 (54) &
\mbox{\huge $\frac{P^2_\tau}{P^2_e}$}\ =\ 3477.441653 (83) &
\mbox{\huge $\frac{P^2_\tau}{P^2_\mu}$}\ =\ 16.818061210 (38) \\ \\
\mbox{experim:} &
\mbox{\huge $\frac{m_\mu}{m_e}$}\ =\ 206.7682838 (54) &
\mbox{\huge $\frac{m_\tau}{m_e}$}\ =\ 3477.48 (57) &
\mbox{\huge $\frac{m_\tau}{m_\mu}$}\ =\ 16.8183 (27)
\end{array}




2) The projections P obey the Koide relation always
independent of the angle θ, while one of the projections
can become negative. ( to please Carl :^) )

\left(\ P_e+P_\mu+P_\tau\ \right)^2\ =\ \frac{3}{2}\left(\ P^2_e+P^2_\mu+P^2_\tau\ \right)

Angle θ with pointer inside the cube: all projections positive.
Angle θ with pointer outside the cube: one projection negative.

The latter is a requirement for Koide's relation to describe the
neutrino mass-eigen-states. This setup does so naturally.
There are the following regions for θ:

075 - 105 degrees: inside
105 - 195 degrees: outside
195 - 225 degrees: inside
225 - 315 degrees: outside
315 - 345 degrees: inside
345 - 075 degrees: outside


3) What fixes the angle θ? For the charged leptons we have the
following numerical coincident involving the projection of tau:

The maximum value of a projection is 0.9855985... is (1+\sqrt{2})/\sqrt{6}

P_{max}^2\ =\ 0.97140452079...

P_\tau \quad \ =\ 0.97140158810...

It has to be said that the masses are still very sensitive to the very small
error (The muon mass uncertainty determines the uncertainty because of
Koide's relation)



4) The coordinates could even be interpreted as real coordinates
with the Pauli-Weisskopf interpretation of the wave function as a
continuous charge-spin density distribution:

The angles with the z-axis of the charged leptons are equal to the
precession angle of spin 1/2 particles and the precessing speed would
be equal to phase frequency of the charged leptons if the torque is 2.
Also, the angle of (cos θ, sin θ, 1) with the z-axis is same as the
precessing angle of a spin 1 boson. In other words: A spin coupling
like adaption of the Yukawa coupling.


Hypothetical Neutrino mass-states eV from here: (page 48)
http://arxiv.org/abs/hep-ph/0603118
are used in the drawing below.


Regards, Hans

[CarlB]

On the subject of applying Koide's mass formula to the baryon resonances, I've added a <a href="http://carlbrannen.wordpress.com/2007/12/16/regge-trajectories-and-koides-formula/">blog post</a> on how this fits in with Regge trajectories.

The short version is that Regge trajectories look like M = sqrt(L), where M and L are mass and angular momentum. This comes from an assumption of flux tubes that have energy (and therefore mass) proportional to their length R, but angular momentum proportional to R^2. These flux tubes have different energies per unit length. The formula applies to baryons with the same quantum numbers except for angular momentum.

Koide's mass formula looks like M = E^2 where E is a field strength, and it applies to baryons with identical quantum numbers. That is, it applies to groups of three resonances that share identical angular momentum.

If you combine the two mass formulas, you end up concluding that the flux tubes that give the Regge trajectories have diameters that are proportional to the Koide field strength.

[arivero]

a power law n^5, reported on spp

http://groups.google.es/group/sci.physics.particle/browse_thread/thread/af72a41b7539e298?hl=es

[CarlB]

Soon I am going to release a paper with some new coincidences based on square roots of masses. Here's an example.

Let \lambda_{en} be the (positive) square root of the mass of the nth electron, that is the square roots of the masses of the electron, muon, and tau. The Koide equation can be written as

\lambda_{en} = 25.0544\sqrt{\textrm{MeV}}(\sqrt{1/2} + \cos(2/9 + 2n\pi/3)\;)

The lightest meson is the pion. It comes in three varieties (with the same quantum numbers), the pion, pi(1300), and pi(1800). The square roots of their masses are given by an equation similar to the above:

\lambda_{\pi n} = 25.0544\sqrt{\textrm{MeV}}(6/5 -3/4 \cos(2/9 + 2n\pi/3)\;)

Accuracy is very high.

To express the relationship as a linear one in terms of square roots of masses, we have:

4\lambda_{\pi_1} + 3\lambda_{e_3} =
4\lambda_{\pi_2} + 3\lambda_{e_2} =
4\lambda_{\pi_3} + 3\lambda_{e_1} = \lambda_{\pi_1} + \lambda_{e_3}
=\sqrt{138}+\sqrt{1777}\;\sqrt{\textrm{MeV}}.

Linear relationships on mass are suggestive that the elementary particles are collections of objects that don't interact enough to completely change their character. An example is the masses of the atomic nuclei. The number of nucleons is approximately proportional to the mass of the nuclei (and atom). And nuclei mass can be thought of as mostly having to do with neutrons and protons and only a little to do with the force that keeps them together.

Linear relationships on square root of mass are suggestive that the mass (or energy) comes from an object which is linear but is proportional to the square root of energy. For example, the energy in a magnetic field is:
\int\;|\vec{B}|^2\;d^3r
The magnetic field is a linear object. That is, if two objects each have magnetic fields and they are superimposed, then by linear superposition, the total magnetic field is the sum of the magnetic fields. The energy in the field is proportional to the square of the field and so if we wish to do a linear operation on the object creating the energy / mass, we need to first take the square root of the energy / mass.

The first thing a physicist does to a linear data stream is to take the Fourier transform of it. For a discrete set of 3 masses this would be the discrete Fourier transform. Marni Sheppeard wrote a short paper showing that the Koide formulas are related to discrete Fourier transforms:

http://arcadianfunctor.files.wordpress.com/2007/11/fouriermass07.pdf

I'll eventually get back with more.

[arivero]

Carl, if you are going to review the pion thing, you could perhaps to include also Hans #349 ff and Taarik post #366.

[CarlB]

Arivero,

I'm looking at things that relate the generation structure of the fermions with the excitation structure of the mesons and baryons through square roots. The use of square roots is supposed here to get the situation to one where there is something linear going on.

Hans #349 is possibly appropriate because it has a square root in it, but it's written as ratios. Maybe I could rewrite it in square root form. Hmmm, let's see. Hans has, writing things in square root mass terms:
\lambda^2_{\pi +}/\lambda^2_{\pi 0} = 1 + \lambda_\mu/\lambda_Z
Multiply by the square root of the mass of the Z:
\lambda_Z (\lambda^2_+/\lambda^2_0) = \lambda_Z + \lambda_\mu
The right hand side is nice because it is linear in square roots but the left hand side is not. So rewrite the ratio of the pion masses..

\lambda_Z(1 + (\lambda^2_{+}-\lambda^2_0) / \lambda^2_0) = \lambda_Z + \lambda_\mu

\lambda_Z(\lambda^2_{+}-\lambda^2_0) = \lambda_\mu \;\lambda^2_0

Write \lambda_+ = \lambda_0 + \lambda_Q where Q is the contribution to the mass field that comes from charge and is small compared to \lambda_0. Keeping first order in Q we have:

\lambda_Z\; (2 \lambda_Q) = \lambda_\mu \;\lambda_0

which is not quite linear in square root mass.

[CarlB]

I should write down my derivation of the pi meson mass formula.

The pi+ is made from a up quark and an anti-down quark. They have to have opposite colors, but other than that, if you know the quantum numbers of one, you know the quantum numbers of the other.

Arbitrarily, consider the QM problem of an up quark moving in the field of an anti down quark. Make this a qubit kind of problem by ignoring position and momentum information. Then the only thing the up quark can do is change its color. The anti down quark will also change color, but this is defined by conservation laws and so we don't need to worry about it; if the up quark is blue, then the anti down quark is anti blue.

Over the long term, just looking at the up quark, there are nine transitions going on, from {R, G, B} to {R, G, B}. This amounts to a scattering matrix. We have nine amplitudes, write them as a matrix:
\left(\begin{array}{ccc}RR&RG&RB\\GR&GG&GB\\BR&BG&BB\end{array}\right)

SU(3) is an unbroken symmetry so we can assume RR = GG = BB, as well as RG = GB = BR and GR = BG = RB. This requires the matrix to be circulant. Let I = RR, J = RG, K = GB, so the matrix has to be of the form:
\left(\begin{array}{ccc}I&J&K\\K&I&J\\J&K&I\end{array}\right)

The RG and GR amplitudes are the time reversals of each other. It follows that they must be complex conjugates, that is, K = J*. That makes the above circulant matrix Hermitian.

Suppose that J and K are nonzero, but I is zero. Is this possible?

The action of J on an arbitrary state is to increment its color (that is, if you think of the three scattering matrix terms that all must be equal to J, their action can apply to any state and increments the color). The action of K is to decrement its color. Since both these processes are possible, it is also possible to have one followed by the other. Such a process would leave the colors unchanged. Therefore I cannot be zero.

This is an argument similar to the one Feynman made when he contemplated what happens when you insert an intermediate state between the initial and final states; the path integral formulation says that you have to sum over the intermediate states.

Suppose my initial state is G and my final state is B. Following Feynman, we insert an intermediate state, which can be R, G, or B. Then we sum over intermediate states. We find that:

GB = GR RB + GG GB + GB BG

We can insert intermediate states between all the other 8 amplitudes. We end up with a full set of nine equations:

RR = RR RR + RG GR + RB BR,
RG = RR RG + RG GG + RB BG,
RB = RR RB + RG GB + RB BB,
GR = GR RR + GG GR + GB BR,
GG = GR RG + GG GG + GB BG,
GB = GR RB + GG GB + GB BB,
BR = BR RR + BG GR + BB BR,
BG = BR RG + BG GG + BB BG,
BB = BR RB + BG GB + BB BB.

We can conveniently rewrite the above equations in matrix form:
\left(\begin{array}{ccc}RR&RG&RB\\GR&GG&GB\\BR&BG&BB\end{array}\right)
= \left(\begin{array}{ccc}RR&RG&RB\\GR&GG&GB\\BR&BG&BB\end{array}\right)^2

or

\left(\begin{array}{ccc}I&J&K\\K&I&J\\JK&I\end{array}\right)
= \left(\begin{array}{ccc}I&J&K\\K&I&J\\J&K&I\end{array}\right)^2

The above has three solutions:
\begin{array}{rcl}
I&=&1/3,\\
J&=&\exp(+2n\pi/3)/3,\\
J&=&\exp(-2n\pi/3)/3\end{array}
for n = 1,2,3.

This is good because there are exactly three pi mesons. Let's see if we can get Koide's formula for them.

We have nine processes describing the meson, that is, 3 copies of I, 3 copies of J, and 3 copies of K. The J and K processes are related by time reversal, but we really don't know how these relate to the I process. To convert these nine processes into a description of the meson, let's suppose that two constants convert the above amplitudes into field amplitudes, say "v" for the I processes and "s" for the J and K processes. The J and K processes are complex conjugates and are complex numbers. We also don't know how to convert these complex numbers into field strengths. Let's suppose that to convert them into a field strength requires a complex phase (maybe due to non commutativity like Berry phase), so they need to be multiplied by exp(i \delta) for the J and the complex conjugate for the K.

Then the total field strength for the three particles is given by

v + 2 s \cos(\delta + 2n\pi/3)

With delta = 2/9, v = 6/5 and s = -3/4, this is the Koide formula for the lowest energy mesons, the pi mesons. With delta = 2/9, v = sqrt{1/2}, and s = +1, this is the Koide formula for the leptons. More to come later.

[CarlB]

After the pion, the next most basic meson is the rho. There are 5 rho resonances, the rho(770), rho(1450), rho(1700), rho(1900), and rho(2150). The last two are "omitted from the summary section" by the Particle Data Group. The lower three can be put into Koide form as follows:

\lambda_{\rho n} = 25.0544\sqrt{\textrm{MeV}}(10/7 -1/3 \cos(2/9 + 2n\pi/3)\;)

This is fairly similar to the formula for the pion:
\lambda_{\pi n} = 25.0544\sqrt{\textrm{MeV}}(6/5 -3/4 \cos(2/9 + 2n\pi/3)\;)

And of course the Koide formula uses the same angle 2/9.

I don't feel very comfortable about the rational values of the v = 10/7, 6/5, and s= 1/3 and 3/4 numbers. I feel better about the fact that v+s/2 tends to be constant when you are looking at two different resonance series. That is, 10/7 + (1/3)/2 is about equal to 6/5 + (3/4)/2 to 1%. There's a good example of this 2 to 1 ratio in some of the longer resonance series where you have more terms in the series.

The reason I like this 2 to 1 relationship between v and s is because in the derivation of the previous post, it comes up with v + 2 s \cos(\delta + 2n\pi/3), so changing v and s by 2 to 1 means that the vector length is split between the constant part (the valence part) "v", and the variable or sea part "s".

I'm still thinking on Hans' equation. The kind of thing I'm looking at is geometric as in:
http://www.sparknotes.com/math/geometry2/theorems/section5.rhtml

[Kea]

After the pion, the next most basic meson is the rho...

Carl, of course it is nice to have a derivation of so many particle masses, but I am guessing that people will still complain that v and s constitute 2 parameters for 2 outcomes in each case (forgetting for the moment the nice 'coincidence' of choice of scale etc), so could you please elaborate a little on the field idea, and the choice of these variables.

[CarlB]

Well Kea, there are three masses so three degrees of freedom. The "v" and "s" only provide two degrees of freedom. The coincidence is in the 2/9. I'm working on a nice graphic that will show how the coincidence works, for inclusion in the paper, which I guess I ought to give the first cut on:
http://www.brannenworks.com/qbs.pdf

The idea is to plot the angle delta as a function of the power one chooses to take. That is to explore mass formulas that look like:

m^{k} = v + s\cos(\delta + 2n\pi/3)

where the three parameters are v, s, and delta, and to plot delta as a function of k. I believe that a nice coincidence will show up when k=1/2, that is the curves for the various mesons the approrpriate baryons, and the leptons will all converge to 2/9 at that value of k.

Carl

[arivero]

I think that one must consider some logical grouping for the mesons. For instance, when we recorded the sqrt(m1) sqrt(m2-m3) permutations, we related them to the level spliting of the flavour octet.

[CarlB]

Here's a mass formula for the pions that I hope the reader will find amusing. Begin with the mass formula for the charged leptons (since I want to use "n" for radial excitations of mesons, I will use "g" for the quantum number that gives the generation):

\sqrt{m_{e g}}/25.0544 = \sqrt{0.5} + \cos(2/9 + 2g\pi/3)

where 25.0544 \sqrt{\textrm{eV}} is a mass scaling constant that makes the leptons nice and we will use for the mesons as well.

The basic idea is to think of the mesons as having radial excitations "n" with quantum numbers like the spherical harmonics of the hydrogen atom, but also having color excitations following Koide's formula. This means that we get the hydrogen wave functions, but tripled.

Guess that the three lowest mass pions = (\pi, \pi(1300), \pi(1800)) with masses (139.57, 1300, 1812) are the n=1, l=0 states with three Koide generations distinguishing them. Let's use "g" for the generation number. Then the formula for these three pion masses turns out to be:

\sqrt{m_{\pi 10g}}/25.0544 = 1.196797 -0.743543\cos(2/9 + 2g\pi/3).

Make the same guess about the three lowest mass J=1 pions = (\pi_1(1400), \pi_1(1600), \pi_1(2015)). Their masses are (1376, 1653, 2013). Since J=1, these have n=2, l=1. Note the last of these, along with the last two in the next group of three, is hard to find in the PDG. It's listed on the page titled "further states":
http://pdg.lbl.gov/2007/listings/m300.pdf
The Koide formula for these j=1 pion masses is:

\sqrt{m_{\pi 21g}}/25.0544 = 1.6313 + 0.1792986\cos(2/9 + 2g\pi/3 + \pi/12).

The extra angle pi/12 means that these have a formula like the neutrinos.

And the three lowest J=2 pions are the (\pi_2(1670), \pi_2(1880), \pi_2(2005)) with masses (1672.4, 1880, 1990). These presumably have n=3, l=2. They also have a Koide formula:

\sqrt{m_{\pi 32g}}/25.0544 = 1.71477 + 0.0864566\cos(2/9 + 2g\pi/3 + 2\pi/12).

The above 3 equations give 9 masses and they work pretty well.

For the hydrogen excitations, the total energy (or mass) of the hydrogen atom is approximately:
m_{H n} = 10^9 - 13.6 /n^2 eV.
The second term, the binding energy, is very small compared to the total energy of the hydrogen atom (about 1GeV) which is mostly due to the rest mass of the proton. Consequently, if I rewrite this as a formula for the square root of the mass of the hydrogen atom, the square root energy levels will still follow a 1/n^2 law:
\sqrt{m_{H n}} = 10^{9/2} - (10^{-9/2}\times 13.6/2) /n^2 \sqrt{eV}.

So based on the energies of the hydrogen atom, one might look for 1/n^2 dependency in the three Koide mass formulas given above, in order to unify the three equations. The three "v" terms need to be: (1.19680, 1.6313, 1.71477). These have n=(1,2,3). They can be fairly well approximated with the formula v_n = 16/9-\sqrt{1/3}/n^2. The three "s" terms need to be (-0.743543, +0.1792986, +0.0864566 ). The sign change can be accounted for by making all three signs negative, but taking an extra phase of pi for the n=1 and n=2 case. This can be accomplished by adding a phase of nj\pi/2=n(n-1)\pi/2 to the phase angle. Then the numbers are fairly closely approximated by s_n = -0.75/n^2. The resulting formula for nine pion masses is fairly compact:

\sqrt{m_{\pi nlg}}/25.0544 = 16/9 - \sqrt{3}/n^2 -(3/4)\cos(2/9 + 2n\pi/3 + jn\pi/2)/n^2

The above simplifications give an approximations of the v's as follows:
1.19680 -> 1.2004275
1.6313 -> 1.63344
1.71477 -> 1.71362
And the s's are approximated by:
-0.743543 -> -3/4
+0.1792986 -> 0.1875,
+0.0864566 -> 0.08333
which is fairly close.

The nine computed masses are as follows:
137.99 1271.2 1834.1
1365.5 1659.6 2032.4
1676.1 1883.0 1977.3

The nine actual masses are:
136.5(2.5) 1300(100) 1812(14)
1376(17) 1652(21) 2014(?)
1672.4(3.2) 1876-2003(?) 1974-2005(?)

Note: The error in the pi mass is due to the mass difference between the pi+ and pi0. And the masses labeled with (?) do not have mass spreads in the PDG. If more than one measurement is given, the range of measurements are given.

[CarlB]

Ooops. Correction on that mass formula. Managed to make three errors in it. (And what happened to the edit function within 24 hours?)

\sqrt{m_{\pi nlg}}/25.0544 = 16/9 - \sqrt{3}/n^2 -(3/4)\cos(2/9 + 2g\pi/3 + l\pi/12 + ln\pi/2)/n^2

In the above, for the pion states it applies to, l is assumed to be equal to n-1. The full set of quantum numbers is nlmg rather than nlg, but there is no energy dependency on m.

And I guess I should note that 16/9 = 2 - 2/9. So the valence part of the formula can be interpreted as the contribution from the seas of the two quarks (whose valence parts have cancelled because one is a quark and the other is an anti-quark), and the 2/9 may be related to the 2/9 in the cosine.

I've got a lot more of these. One of the cool ones to analyze is the J/Psi. There are 6 states with masses fairly close together. Is there a way of dividing them up into two groups of three, where each group of three fits the above kind of formula? Let's see if I can get a solution edited into this post before it times me out.

[CarlB]

Remember Foot's gemoetric interpretation of Koide's formula in terms of the vector (1,1,1)? It turns out that the same may be said of the tribimaximal mixing matrix. This should be of interest to those interested in Koide's formula:
http://carlbrannen.wordpress.com/2008/06/29/the-mns-matrix-as-magic-square/

Maybe we will finally get a complete Koide type formulation for the elementary particles as a similar property applies to the CKM matrix as well. (That is, Kea is says that the CKM matrix can be written as a sum of a 1-circulant and 2-circulant matrix, and this is related to the Foot geometry, as shown above.)

Also, I should mention that I've got good results applying the Koide formula to the b-bbar and c-cbar mesons. It could be an interesting summer.

[arivero]

CITATION: C. Amsler et al. (Particle Data Group), Phys. Lett. B667, 1 (2008) has some changes in W mass and width, as well as others.


The strange news: [/B]

with the move of W mass from 80.425 down to 80.398 ± 0.025 GeV the fit of me/mW to the muon anomalous difference, in post #41, should be worse. But the pdg now lists .62686, an increased difference. So both moves sort of cancel.

the positive review:

Alpha formula, in post #4, survives. But term 4 should be included in next revisions!
after term 3: 0.0072973525686533
f.s. constant 0.0072973525680.(+/-240) old
f.s. constant 0.0072973525692 .(+/-27) http://arxiv.org/abs/0712.2607

post #28, Pythagorean triples, would go better or similar, with actual tau mass.

#44, Weinberg angle, enters the 1-sigma! In 2004 it was to within 0.063% or sigma 1.2). Now with the 2008 pdg, it predicts W mass within 0.029% and sigma 0.94

The prediction is more spectacular if you see post #66, column "calculated in MeV"

[arivero]

Some works are borderline between standard "texture" research and our thread.
In http://arxiv.org/pdf/hep-ph/9703217v1, they suggest
[tex]
\sqrt {\m_t\over\m_c} = {m_\tau\over m_\mu}
[\tex]
In http://arxiv.org/pdf/hep-ph/0106286v2, a lot of such sqrt textures are invoked as popularly known.
In http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?198812215 and http://dx.doi.org/10.1016/0370-2693(87)91621-2 Christof Wetterich leads other attempts.

[arivero]

Alpha formula, in post #4, survives. But term 4 should be included in next revisions!
after term 3: 0.0072973525686533
f.s. constant 0.0072973525680.(+/-240) old
f.s. constant 0.0072973525692 .(+/-27) http://arxiv.org/abs/0712.2607


Correction, the full calculation is in http://www.physics-quest.org/, so we can compare
0.00729735256865385342269 theoretical
0.0072973525692 .(+/-27) measured http://arxiv.org/abs/0712.2607
quotient:

1.000000000075 \pm 0.000000000369

[Hans de Vries]

Correction, the full calculation is in http://www.physics-quest.org/, so we can compare
0.00729735256865385342269 theoretical
0.0072973525692 .(+/-27) measured http://arxiv.org/abs/0712.2607
quotient:

1.000000000075 \pm 0.000000000369

Wow!

That's a nice surprice! New measurements plus a revised value of the eighth-order
QED contribution to the anomalous magnetic moment of the electron from Kinoshita
brings our series in line with experiment to a fraction of the (improved) error.

\alpha^{-1}~=~~ 137.035 999 710 (96) Old Kinoshita/Gabrielse (July 2006)
\alpha^{-1}~=~~ 137.035 999 084 (51) New Kinoshita/Gabrielse (Feb 2008)
\alpha^{-1}~=~~ 137.035 999 095 829 Theoretical value from alpha series.

New measurement from Gabrielse's group: http://arxiv.org/abs/arXiv:0801.1134v1
Revision of the eight-order QED term: http://arxiv.org/abs/0712.2607
The alpha series: http://physics-quest.org/fine_structure_constant.pdf

This happened a while ago already. somehow we missed it. Thank you for posting!



Regards, Hans.

[arivero]

To complete the record:

\alpha^{-1}~=~~ 137.035 999 108 (450) Original conject. (http://physicsforums.com/showthread.php?t=43862) from Codata 2004.
\alpha^{-1}~=~~ 137.035 999 710 (96) Old Kinoshita/Gabrielse (July 2006)
\alpha^{-1}~=~~ 137.035 999 084 (51) New Kinoshita/Gabrielse (Feb 2008)

\alpha^{-1}~=~~ 137.035 999 095 829 Theoretical value from alpha series.

Wow!
This happened a while ago already. somehow we missed it. Thank you for posting!

I somehow fused it with the already reported, somewhere in the middle of the thread, 2006 update.

As for the calculation, the first order formula

\alpha^{-1/2}+ \alpha^{1/2}=e^{\pi^2 \over 4}


is crying a word: duality. Problems are:
1) It is not so clear how the succesive corrections are applied.
2) the formula for dyon energy (particle with electric+monopole charge) is

M^2 \approx e^2 + e^{-2} = \alpha^{1}+ \alpha^{-1}

so this formula is a kind of rare square root of the usual duality formula.

3)Still we havent got a clue for the precise choosing of e(pi^2/4) except that it works.

[Hans de Vries]

To complete the record:

\alpha^{-1}~=~~ 137.035 999 110 (450) Original conject. from Codata 2004.
\alpha^{-1}~=~~ 137.035 999 710 (96) Old Kinoshita/Gabrielse (July 2006)
\alpha^{-1}~=~~ 137.035 999 084 (51) New Kinoshita/Gabrielse (Feb 2008)

\alpha^{-1}~=~~ 137.035 999 095 829 Theoretical value from alpha series.


Both the direct measurement from the Quantum Hall effect (CODATA 2004) and
the indirect one from the Harvard g/2 measurements are now in agreement.


As for the calculation, the first order formula

\alpha^{-1/2}+ \alpha^{1/2}=e^{\pi^2 \over 4}


is crying a word: duality. Problems are:
1) It is not so clear how the succesive corrections are applied.
2) the formula for dyon energy (particle with electric+monopole charge) is

M^2 \approx e^2 + e^{-2} = \alpha^{1}+ \alpha^{-1}

so this formula is a kind of rare square root of the usual duality formula.

3)Still we havent got a clue for the precise choosing of e(pi^2/4) except that it works.

To explain it is the hard thing..... Do you have a link for the this dyon
mass formula?


Regards, Hans

[arivero]

To explain it is the hard thing..... Do you have a link for the this dyon
mass formula?
In fact, the point of not having recognised it immediately is proof of the amateurish character of the internet forums :redface:

Remember the lore: if a theory goes with a coupling constant a, the dual theory goes usually with a coupling constant 1/a.

In the case of T-duality the coupling have dimensions. Or said otherwise, Tduality is possible because the theory has a dimensional scale. But for ElectroMagnetic duality, the coupling is adimensional. The (square of the?) magnetic energy of a monopole goes as hbar/alpha.

Actually, a lot of the initial formulae you suggested in 2004 were of the form x+1/x for some constant x; it is hard of believe that you were not inspired, at least inconsciently, by the usual popular remarks on duality. Again, I failed to notice it, so perhaps both did.

A dyon is a particle having both elemental and monopole charges. So its mass square has two contributions.

I saw the formula for the mass in a remark about Montonen-Olive duality (http://www.slac.stanford.edu/spires/find/hep/www?irn=251658). It seems that this duality is the oldest generalization of EM to (super)gauge theories, and in the long run it has been argued that it can be derived from T duality, eg Vafa http://arxiv.org/abs/hep-th/9707131

The dyon mass formula appears in the last page of Montonen-Olive preprint, and they quote towards to older articles.

To put some stuff in handwritten from Witten itself, see slide 25 of http://math.berkeley.edu/index.php?module=documents&JAS_DocumentManager_op=downloadFile&JAS_Document_id=2101 in http://math.berkeley.edu/index.php?module=documents&JAS_DocumentManager_op=viewDocument&JAS_Document_id=116
and then slide 28

[arivero]

Perhaps I have cited very heavy artillery. I apologize; I have never put a lot of interest on these topological objects; it is a long history.

So another source is Zee's book on QFT, and it redirects to Electromagnetic duality for children (http://www.maths.ed.ac.uk/~jmf/Teaching/EDC.html), from Figueroa-o'Farrill.

In section 1.3.1 we are told that the mass of a monopole always meets the Bogomol’nyi bound. In section 1.3.2 the bound is "saturated" in the Prasad–Sommerfield limit. So we have the initials BPS. The saturated bound is the formula for the mass of a monopole with both electric and magnetic charge I told above, depending unfortunately of \alpha^1 instead of \sqrt{\alpha} \; (=e).



Alert: Section 1.4.2 explains another earlier calculation of Witten, and it builds the "Noether" charge of a dyon as funtion of charges q g plus a theta-vacuum, in the shape

N={q \over e} + (e g) {\theta \over 8 \pi^2}

and then argues that N must be an integer. But note that q and g are not fully adimensional; they are defined as, say q=(n e) and (g=4 pi m/e), so the factors of e there in the formula are intended only to cancel the internal ones. The argument in 1.4.2 amounts to prove that when m=1, n is also quantized.

[arivero]

Could it be better to write the first order approximation squared, ie


e^{\pi^2/2} -2 \approx \alpha + \alpha^{-1}


In such case, how could the corrective terms be applied?

[CarlB]

Or is it useful to begin with a first order equation:
\sqrt{\alpha} = i^{\ln(i)} = i^{i\pi/2}
= (i^i)^{\pi/2} = (e^{-\pi/2})^{\pi/2} = e^{-\pi^2/4}

Or did I make an algebra mistake here?

[Kea]

Assuming that the first order relation is indicative of duality, this would be Langlands, or electric magnetic duality, most simply understood via relations coming from the modular group, or its braid group cover. This suggests interpreting the Gaussian term as a trace factor (usually called d in knotty algebra maths), or a normalisation arising from geometry of non integral dimension, not unlike Deligne's Gaussian for the discrete Fourier transform, except that the factor of \pi presumably results from an infinite sequence of terms.

[arivero]

Or is it useful to begin with a first order equation:
\sqrt{\alpha} = i^{\ln(i)} = i^{i\pi/2}
= (i^i)^{\pi/2} = (e^{-\pi/2})^{\pi/2} = e^{-\pi^2/4}

Or did I make an algebra mistake here?
:rofl::rofl: No, I think it is fine. Of course i=e^{i\pi /2} is a nice trick to get this power of pi. Actually it shows that the seed chosen by Hans is not a random number.

Most important perhaps, you can -as Hans hinted in 2004, calling this a "gaussian"- also write in a formal way


{i \over \sqrt \pi} \int_{-\infty}^\infty e^{ (x-0) (x\pm \pi)} dx = e^{- {\pi^2 / 4}}
or if you prefer
{1 \over \sqrt \pi} \int_{-\infty}^\infty e^{ - (x-0) (x\pm i \pi)} dx


it is because of this second detail that I have wasted, poorly, the weekend looking into soliton, instanton, dyons etc...ons. Not only the x-(1/x) part of the equality is typical of this kind of theories; also the integral of gaussians is the part to get exact values of energy, barrier penetration, mass etc. So the "1-st order" (Hans call the one you wrote "0-th order") formula has common points with non-perturbative QM/QFT/Qthing in both sides of the expression, which is intriguing.

[Count Iblis]

You only got ten digits, that's not enough to guess the correct formula.

[Hans de Vries]

I'm sure the magnetic moment plays a major role, which reminds me of another electro-
magnetic duality: There are types of two magnetic dipoles with indistinguishable fields,
the vector dipole and the axial dipole. The first is a combination of two opposite magnetic
monopoles and the second is a point like circular electric current.

So, associating the field with a continuous charge/spin distribution, which of the two
types is involved in case of the magnetic moment? Well, the net force in a magnetic
field is quite different:

Force on a Vector dipole in a B field:


\begin{array}{l l |l l l| l | l | l}
& & \partial_x \textsf{B}_x & \partial_y \textsf{B}_x & \partial_z \textsf{B}_x & & \mu_x & \\
\vec{F}_{magn} & =\ \ & \partial_x \textsf{B}_y & \partial_y \textsf{B}_y & \partial_z \textsf{B}_y & \cdot & \mu_y & \quad =\ \partial_j B_i\ \mu_i \\
& & \partial_x \textsf{B}_z & \partial_y \textsf{B}_z & \partial_z \textsf{B}_z & & \mu_z
\end{array}


Force on an Axial dipole in a B field:


\begin{array}{l l |l l l| l | l | l}
& & \partial_x \textsf{B}_x & \partial_x \textsf{B}_y & \partial_x
\textsf{B}_z & & \mu_x & \\
\vec{F}_{magn} & =\ \ & \partial_y \textsf{B}_x & \partial_y \textsf{B}_y & \partial_y \textsf{B}_z & \cdot & \mu_y & \quad =\ \partial_i B_j\ \mu_i \\
& & \partial_z \textsf{B}_x & \partial_z \textsf{B}_y & \partial_z \textsf{B}_z & & \mu_z

\end{array}


Note the exchange of the indices. The second case is what you would expect:

\vec{F} ~~=~~\mbox{grad}(\vec{B}\cdot\vec{\mu})


So, it should be easy to distinguish between the two cases you would say,
and this rules out the pair of magnetic monopoles. There is an interesting twist
however: The net force between either two vector dipoles or two axial dipoles
turns out to be same:


\vec{F}\ =\ \frac{3\mu_o\mu_e^2}{4\pi r^4}\left[ \left\{\left(
\hat{\mu}_1 \cdot \hat{\mu}_2 \right) - 5\left( \hat{r} \cdot
\hat{\mu}_1 \right)\left( \hat{r} \cdot \hat{\mu}_2 \right)\right\}
\mbox{\Large $\hat{r}$} + \left( \hat{r} \cdot \hat{\mu}_2
\right)\mbox{\large $\hat{\mu}_1$} + \left( \hat{r} \cdot
\hat{\mu}_1 \right)\mbox{\large $\hat{\mu}_2$}\ \right]


At least in the case that both are at rest, which reminds me that I should find some
time to work out the general case too really proof the exclusion of the first case.


Regards, Hans

[CarlB]

Okay, if the algebra works out, (it is usual for me to get 2s and factors of pi wrong the first time), then I should give credit where credit is due. The formula for i^i was discussed at Built On Facts:
http://scienceblogs.com/builtonfacts/2008/08/sunday_function_4.php

So what are the attributes of the complex mapping
f(u) = u^{\ln(u)}
and why would you be interested in this function at the point i?

What are the poles of ln(u)?

I bet Kea can tie this into the Riemann zeta function.

[arivero]

Just to fix the idea, the point about the exp is that it is gaussian but a peculiar one. It is a wavepacket with the momenta distribution centered in pi instead of the origin. To be precise, the normalized solution of schroedinger equation, at t=0, for a gaussian packet of mean wavenumber k_0 and distribution width a=1.

\Psi(x)= ({2\over \pi})^{1/4} e^{i k_0 x} e^{-x^2}

so that at least formally

\int_{-\infty}^{\infty} \Psi(x) dx = ({2 \pi})^{1/4} e^{-k_0^2\over 4}


And NOW I see that I do not understand why I am trying to fit the minus sign; Hans "first order" formula uses i^(-ln i). CarlB reversed it only to use the 0th-order.

So it is the reverse situation: we actually have a wavepacket of imaginary average wavenumber k_0=i \pi. I'd say that this kind of beast fits with the definition of instanton.

[arivero]

An extra rewrite: \sqrt \alpha= e /\sqrt {2 \pi} so

e + {2 \pi \over e} = \sqrt{2 \pi} \rm e^{\pi^2/4}

(typography problem: should we use \rm for the exponential or for the electron charge?)

While it seems lot uglier, it agrees with the conventions for Dirac monopoles, whose charge g is such that
e g = 2 \pi n
for n any integer.

[arivero]

So, associating the field with a continuous charge/spin distribution, which of the two
types is involved in case of the magnetic moment? Well, the net force in a magnetic
field is quite different:


Witten's http://ccdb4fs.kek.jp/cgi-bin/img_index?197909065
quotes some old (1936-1949) works on "heuristic derivations" of the relationship between electric and magnetic charge when both kinds are present in the same point.

Thoughts of today, if the exponential is actually a gaussian wavepacket:

- there is a imaginary momentum, and this is Euclidan wick rotation.
- such momentum is a multiple of pi, this is periodicity or circular configuration: U(1)??
- self duality is usually the hallmark of a BPS state.

the problem is how has a field theoretical beast, the electromagnetic instanton, descended to the realm of naive quantum mechanics. Of course we "known" the answer: it can only descend for a particular value of alpha. But we do not know what the question is.

[CarlB]

the problem is how has a field theoretical beast, the electromagnetic instanton, descended to the realm of naive quantum mechanics. Of course we "known" the answer: it can only descend for a particular value of alpha. But we do not know what the question is.

Hmm, interesting that you would put it that way. All the Koide formulas seem to be reduction of a QFT problem to QM methods.

And the paper I'm working on with the hadron mass formulas amounts to the same thing.

Naive QM wise, we can think of the meson as one quark moving in the field of the other, that is, we go to center of mass coordinates in both space and color. As a first approximation, assume the quark is in a 1S state and we will ignore spatial degrees of freedom. What's left is color degrees of freedom, R, G, and B.

Let H be the Hamiltonian for the meson. Since there's no spatial dependency, H is only a 3x3 matrix, with color indices:
H = \left(\begin{array}{ccc}
H_{RR}&H_{RG}&H_{RB}\\
H_{GR}&H_{GG}&H_{GB}\\
H_{BR}&H_{BG}&H_{BB}\end{array}\right)
By color symmetry,
H_{RR} = H_{GG} = H_{BB} = v,
H_{RG} = H_{GB} = H_{BR} = se^{i\delta},
H_{RB} = H_{GR} = H_{BG} = s'e^{i\delta'}.

To get real eigenvalues H must be Hermitian so v is real, s=s' is real, and \delta' = -\delta. The three eigenvectors are (1,1,1), (1,w,w*), (1,w*,w), where w is the cubed root of unity, and the three eigenvalues are:
s+2v\cos(\delta + 2n\pi/3),
for n = 0,1,2.

The above is almost in the form of Koide's formula. The difference is that Koide's formula is for sqrt(H) instead of H. To get that last step, note that, without color, the non relativistic Hamiltonian is in the form:
H = \nabla^2 + V
where V is a potential, and a slightly more complicated equation for relativistic energy. To get this into clean form, we do the same thing Dirac did to get the gamma matrices, that is, we take the square root. The difference is that in our case, we are taking the square root of a 3x3 color matrix instead of the square root of a d'Alembertian (or whatever the 4-d gradient is called).

I'm working on the statistics of the fits for 20 hadron triplets to this formula and should get something done in the next few days. I've discussed various fits here and elsewhere, but this wil be the first time that everything is brought into one paper, along with the statistical fits. This involves a fair amount of computer programming.

[arivero]

Hmm, interesting that you would put it that way. All the Koide formulas seem to be reduction of a QFT problem to QM methods.

And the paper I'm working on with the hadron mass formulas amounts to the same thing.


Yes, and Hans' version of the Weinberg angle was also a QM object. In general, this thread is defying the lore of the origin of constants from HEP (from planck scale GUT groups). Of course, if we don't go high (on energy, I mean) we do not need to create/annihilate particles, and QM should work.

What worries me today is that the we are not looking in a far dark corner; your angles on neutrinos were a "minor" mainstream topic in recent years. The reinterpretation of Hans alpha as coming from self dual objects is not a minor stream, it is a major fluent of the Amazon river. Damn, it is just a "wrong sign version" of expression 1.2 of hep-th/9407087.

[Kea]

What worries me today is that the we are not looking in a far dark corner.....

Why is that a concern? Are you worried that the string theorists will swallow this whole? If they had really understood these kind of observations, we would have discovered it by now.

[arivero]

Why is that a concern? Are you worried that the string theorists will swallow this whole? If they had really understood these kind of observations, we would have discovered it by now.

More or less, this is the point. It is hard to think that they can overlook a critical value of alpha in a theory whose main paper has almost two thousand citations. They should have discovered it by now. The gate is not very hidden; probably it amounts to consider f(e+g) instead of f(e+ig) as they usually do. And there are some hints that they have tried, notably Poliakov and a lot of stuff on condensates.

[arivero]

I'm sure the magnetic moment plays a major role, which reminds me of another electro-
magnetic duality: There are types of two magnetic dipoles with indistinguishable fields,
the vector dipole and the axial dipole. The first is a combination of two opposite magnetic
monopoles and the second is a point like circular electric current.


And I guess you can also compare circular monopole currents against electric charges.

[arivero]

Or

e^{\pi^2/4} - e^{-\pi^2/4} \approx \sqrt{ \alpha + {1 \over \alpha}}

instead of

e^{\pi^2/4} \approx (\sqrt \alpha + {1 \over \sqrt\alpha})


Actually it is not so good as starting point
11.706956417... / 11.7065492967 (22) = 1.000034777
11.791761389... / 11.7916621597 (22) = 1.000008415
and a sinh is less atractive, to me, than a clean gaussian. And I doubt it could receive the same corrective series than the original guess. It is just that it is closer to popular shapes.

[Hans de Vries]

At least the new 2008 value for alpha from Gabrielse/Kinoshita begins to
solidify the n=3 term which now gives some convidence in the whole series.

\alpha\ =\ 1/137.035999084 (51)

The alpha "radiative" series:

\mbox{\Huge i}^{~ln\, i }\sum_{n=0}\frac{\mbox{\Huge $\alpha$}^{n-\frac{1}{2}}}{(2\pi)^{b_n}} ~~ = ~~ \mbox{\Huge 1}

Where b_n is the binominal series 0,0,1,3,6,10.. with successive increments
of 0,1,2,3,4....

The result of the series after each extra term is:

n=0: ..... 0.992 747 158 626 634
n=1: ..... 0.999 991 584 655 288
n=2: ..... 0.999 999 998 402 186
n=3: ..... 0.999 999 999 957 418
n=4: ..... 0.999 999 999 957 464



Regards, Hans

[CarlB]

Re: string theory, writing down simple equations like this puts a crimp in the anthropic theories.

[jal]

I apologize if I'm am interupting the flow with an irelevant question.
(I can barely understand what you are doing.)
From what you are doing,

arivero,
Just to fix the idea, the point about the exp is that it is gaussian but a peculiar one. It is a wavepacket with the momenta distribution centered in pi instead of the origin. To be precise, the normalized solution of schroedinger equation, at t=0, for a gaussian packet of mean wavenumber k_0 and distribution width a=1.
Can the numbers of wavepacket that are on the shell be calculated?
jal

[CarlB]

Getting things to work at low energies instead of the traditional perturbation theory.

I wonder if recent papers by Marco Frasca are related:
http://marcofrasca.wordpress.com/2008/09/03/a-stunning-summer/

[arivero]

Re: string theory, writing down simple equations like this puts a crimp in the anthropic theories.

It puts a crimp in GUT almost. The thread is telling that no yukawa couplings are to be predicted from GUT. Moreover, Weinberg angle and fine structure constant comes from Hans, so the "GUT coincidence" can only predict the SU(3) coupling. Furthermore, there is probably a high-low consistence rule: one climbs up from the electroweak scale, get the SU(2) and U(1) couplings to meet, then climbs down the SU(3) coupling until it becomes large and all the QCD nonperturbative stuff (as Marco's) applies, and then surprise, after all this walk we are in a energy scale no far from the original one.

(Of course, it is not fair in a thread on coincidences to discard the One of GUT coupling constants).


Can the numbers of wavepacket that are on the shell be calculated?
jal

Yep, when the packet halfwidth is taken as 1, the wavenumber is k0= i pi, or perhaps i pi/4 depending on what normalization do we aim to. It stinks to periodic potential or periodic configuration space, as a first conjecture.

[CarlB]

Yep, when the packet halfwidth is taken as 1, the wavenumber is k0= i pi, or perhaps i pi/4 depending on what normalization do we aim to. It stinks to periodic potential or periodic configuration space, as a first conjecture.

Or perhaps one hidden dimension, cyclic, which is what my Clifford algebra density matrix analysis of the situation requires. That is, using primitive idempotents (projection operators or pure density matrices) you can count the hidden dimensions of spacetime by looking at the weak hypercharge and weak isospin quantum numbers:
http://brannenworks.com/a_fer.pdf

With this sort of idea, the arbitrary complex phase universal to all quantum mechanics becomes a position coordinate in the hidden dimension.

[arivero]

In the spirit of keeping this thread for numbers and leaving (most of) the standard theory in other hands, I have hitchhiked to the thread

http://www.physicsforums.com/showthread.php?t=240247

to discuss on EM duality and like. Here, let me just to note that a fix of notation, from Dirac 1948:

e_0^2={1 \over 137} \hbar \ c
g_0^2=4 {137 \over 1} \hbar \ c

for n=1 in {1 \over 2} n \hbar \ c

The point is that there was other papers where it is argued that n must be even. Let me call g_2 to this forceful even constant. In this case
g_2^2= {137 \over 1} \hbar \ c

As for the 4 pi factor, it seems, looking at Dirac's paper, that it was because some other publications use h instead of \hbar, and this use has propagated until today.

I need to locate the paper where it is argued that n=2 is the minimum case.

[arivero]

In the spirit of keeping this thread for numbers and leaving (most of) the standard theory in other hands, I have hitchhiked to the thread

http://www.physicsforums.com/showthread.php?t=240247

to discuss on EM duality and like. Here, let me just to note that a fix of notation, from Dirac 1948:

e_0^2={1 \over 137} \hbar c = \alpha_e \hbar \ c
g_0^2=4 {137 \over 1} \hbar c= 2^2 {1 \over \alpha_e} \hbar \ c

for n=1 in
eg= {1 \over 2} n \hbar c

The point is that there was other papers where it is argued that n must be even. Let me call g_2 to this forceful even constant. In this case
g_2^2= {137 \over 1} \hbar \ c

As for the 4 pi factor, it seems, looking at Dirac's paper, that it was because some other publications use h instead of \hbar, and this use has propagated until today.

I need to locate the paper where it is argued that n=2 is the minimum case.

[Hans de Vries]

In the spirit of keeping this thread for numbers and leaving (most of) the standard theory in other hands, I have hitchhiked to the thread

http://www.physicsforums.com/showthread.php?t=240247

to discuss on EM duality and like. Here

One can of course consider the continuous spin-density distribution of an electron
field as a distribution of parallel Dirac strings. (Which are basically solenoids)
In one way or another this could lead to charge quantization in the Dirac sense.

Jackson, in section 6.12 mentiones this issue of n versus n/2. There are semi-classical
derivations from Saha (1936) and Wilson (1949) of the Dirac condition which lead to n.


Regards, Hans

[arivero]

One can of course consider the continuous spin-density distribution of an electron
field as a distribution of parallel Dirac strings. (Which are basically solenoids)
In one way or another this could lead to charge quantization in the Dirac sense.

It seems that the interaction between "Dirac-Schwinger-Zwinger-Winger" quantization and topological solutions of electromagnetism is a well known candidate to fix the fine structure constant. Ketov 9611209v3 starts his lecures on Seiberg-Witten underlining that "the sole existence of duality symmetry allows one to exactly determine the critical temperature which must occur as the self-dual point where K=K* or sinh(2J/k T)=1". And, more in our electromagnetic constext, Julia and Zee 1975 tell that


If for some reason ... can only take on discrete values ... and if the argument of Schwinger and Zwanziger is relevant for the present case, the one would apparently be faced with a misterious condition saying that the theory will only make sense for some definite value of e^2

[Hans de Vries]

It seems that the interaction between "Dirac-Schwinger-Zwinger-Winger" quantization and topological solutions of electromagnetism is a well known candidate to fix the fine structure constant. Ketov 9611209v3 starts his lecures on Seiberg-Witten underlining that "the sole existence of duality symmetry allows one to exactly determine the critical temperature which must occur as the self-dual point where K=K* or sinh(2J/k T)=1". And, more in our electromagnetic constext, Julia and Zee 1975 tell that

Stimulating to follow this line are Tonomura's video's of real life toplological solutions.

http://www.hqrd.hitachi.co.jp/global/movie.cfm
Tonomura's nicely illustrated booklet (http://www.amazon.com/Quantum-World-Unveiled-Electron-Waves/dp/9810225105/ref=sr_1_2?ie=UTF8&s=books&qid=1220711444&sr=1-2)

Specially movie no. 5 with shows the annihilation of "particle/ anti-particles" (vertices/anti-
vertices) There's a close relation of these vertices with the Dirac's charge quantization.

Concerning introductions in the major QFT textbooks, there's Ryder and more recently
Zee for those interested in the general topic.


Regards, Hans

[arivero]

It just dawned on me: the square root of 4 pi is not a normalisation factor nor a volume factor; it is the charge of the electron when the fine structure constant is unity.

So Hans formula is


e_{\alpha=1/137...} + g_{\alpha=1/137...} = e_{\alpha=1} \ {\bf e}^{\pi^2 \over 4}


where
e_{\alpha=1/137...}= \sqrt{4 \pi \alpha} is the electric charge of the electron
g_{\alpha=1/137...}= 4 \pi / e_{\alpha=1/137...} is the DSZ charge of the monopole.
e_{\alpha=1}= \sqrt{4 \pi} the charge of the electron in the fixed point of duality (alpha=1)

Note that if we restore units h,c, they appear congruently in the three terms, because the 4pi of DSZ inversion is actually 4pi hbar c, or 2hc

As for the scaling/rotation factor, the two best ansatzes I have got are still
{\bf e}^{\pi^2 \over 4} = (\sinh {\pi^2 \over 4} + \cosh {\pi^2 \over 4} )
and
\sqrt {4 \pi} \ {\bf e}^{\pi^2 \over 4} = \int {\bf e}^{\pm \frac 12 \pi x } {\bf e}^{-{x^2 \over 4}} dx = k \int_{-\infty}^{\infty} {\bf e}^{\pm \frac 12 k \pi x} {\bf e}^{-{k^2 x^2 \over 4}} dx for any k.

the idea of the second one is that the integral is the square modulus of an Euclidean wavepacket (thus of imaginary wavenumber i k pi). Wick rotating back to Minkowski, the wavenumber gets real, its exponential regains the usual imaginary factor, and then its "Minkoswki" square modulus is
k \int_{-\infty}^{\infty} {\bf e}^{-{k^2 x^2 \over 4}} dx = \sqrt {4 \pi}

[arivero]

Er did I say "natural units". Of course in natural units 1=hbar = h/ 2 pi. So a new conjecture for the formula is



e_{\alpha=1/137...} + g_{\alpha=1/137...} = e_{\alpha=1} \ {\bf e}^{h^2 \over 16}



It has a satisfactory point, all the objects are now quantum objects. But it has a couple problems: it is not clear how units are rebuilt in the exponent, and the 16 is a pity; I had hoped it to dissapear too :-(.

In any case, the point is that the pi is not coming from the compactification of a KK circle: it is coming directly from Heisenberg principle when applied to the wavepacket. The wavepacket is actually a regularisation family for a dirac delta; one could even hope that in the cero radius limit the exponential factor could be the same for every regularisation; it is perhaps too much to ask for.

[CarlB]

For some reason, the semiclassical approximation to QCD shown in this paper seemed to m to be germane to the alpha calculation:


Yang-Mills Propagators and QCD
Marco Frasca
We present a strong coupling expansion that permits to develop analysis of quantum field theory in the infrared limit. Application to a quartic massless scalar field gives a massive spectrum and the propagator in this regime. We extend the approach to a pure Yang-Mills theory obtaining analogous results. The gluon propagator is compared satisfactorily with lattice results and similarly for the spectrum. Comparison with experimental low energy spectrum of QCD supports the view that  resonance is indeed a glueball. The gluon propagator we obtained is finally used to formulate a low energy Lagrangian for QCD that reduces to a Nambu-Jona-Lasinio model with all the parameters fixed by those of the full theory.
http://arxiv.org/abs/0807.4299

[Hans de Vries]

at least the new 2008 value for alpha from gabrielse/kinoshita begins to
solidify the n=3 term which now gives some convidence in the whole series.

\alpha\ =\ 1/137.035999084 (51)

the alpha "radiative" series:

\begin{aligned}&. \\ &.~~~~\sum_{n=0}^\infty~\frac{\mbox{\huge $\alpha$}^{n-\frac{1}{2}}}{(2\pi)^{b_n}} ~~ = ~~ \mbox{\huge $ e^{\frac{~\pi^2}{4}}$}~~~~. \\ &. \end{aligned}

where bn is the binominal series 0,0,1,3,6,10.. With successive increments
of 0,1,2,3,4....


apparently somebody solved it for 1294 digits :smile:

http://science6.2ch.net/test/read.cgi/sci/1091534329/l50


1/α=
137.0359990958 2970048964 7400982482 4649832472 5408221072 8280453419 8236353775 3291508251 5164063679 7696140100
0562638972 3955307219 1131054898 8411966378 9253597119 3431450591 9378868411 2253831430 9814642191 1084940907
8132259876 7066274120 1689553474 9369387943 1203244561 3909631074 6327549332 6111965801 5656960227 1639347243
0757007082 5600347556 4972841179 8177526459 2238581338 1031279567 7161638183 9778886042 0778973428 2353736066
0555206708 9800179254 4185014141 4229664797 4602586290 4504613346 3047999465 7523213673 2504266689 6696242124
0527823171 5115527412 2534727726 0326504468 2081772605 6146751666 7986300114 2671896058 3800062750 9682551272
0001496187 4633319304 9496438649 4087857886 0022706370 1852275955 5736144810 3351280327 2115723542 9277212344
6351827920 3207784574 9582376454 9389618767 5666772174 2641243315 1604981948 7898184100 7657369800 7939844563
0773213335 5609920096 1118610374 0331148902 0684994710 0221083527 2203073949 1900453584 3300826440 5980415900
4846789648 2011040441 7960811223 2721740912 7687806462 3818793279 0079147864 9620502089 5698441703 6636685100
2052169799 1019507098 3686916619 8730033698 5715452876 0094017710 1259020172 1556975359 8282754919 5804007348
2935098701 8839649369 8755374879 7396746681 2007580127 6048718855 2240202268 0518075853 2654287399 3572658304
7624575104 9403524357 1121045001 8139733681 5436362649 7016835615 5476863905 7693063483 1562871355 1･･･

Regards, Hans

[granpa]

from post #342:

where the Koide formla is more convincing is that it is consistent with exactly three generations and no more. The short form for the Koide formula is:

\sqrt{m_n} = 1 + \sqrt{2}\cos(2n\pi/3 + 2/9 + \epsilon)

where \epsilon = 0.22222204717(48) - 2/9 and I've left off an overall scaling factor. Since 2(n+3m)\pi/3 = 2n\pi/3 + 2m\pi, the formula gives exactly three masses so there are only three generations implied. These are the electron, muon, and tau for n the generation number, 1, 2, 3. The \sqrt{2} is what Koide found in 1981, the \epsilon is what I found a year ago.
.

it allows 4 generations if you count m=0

[Vanadium 50]

Wow...that post you are responding to is two years old.

There is no massless 4th generation of leptons.

[granpa]

just in case it wasnt clear here is the koide formula

http://upload.wikimedia.org/math/2/f/a/2faa10c9790027937a525b23039fc688.png

zero can obviously be added to both numerator and denominator without changing anything.

http://en.wikipedia.org/wiki/Koide_formula

I dont know if that changes anything or not. the math in the rest of this thread is far beyond me.

[arivero]

Wow...that post you are responding to is two years old.

There is no massless 4th generation of leptons.

Still, from time to time I wonder if there is a massless 3rd generation of leptons. Meaning that if the electron mass is put equal to zero, the masses of muon and tau still are near of Koide's conditions. On the other hand, in the limit where all the e,u,d generation is massless, the pion should be, and it strikes me: in this limit, it should have the same mass that the electron. In the actual, broken situation, it has almost the same mass that the muon. Besides, if it had exactly the same mass that the muon, the charged pion would be an stable particle.

[CarlB]

from post #342:
it allows 4 generations if you count m=0

If you'll look carefully at the formula, you will find that putting 4 into the equation gives you the same masses as if you'd put 1. This is because trig functions repeat every 2 pi.

I'm preparing a submission for Phys Math Central on the subject of Koide's mass formulas for the hadrons. I'm supposed to have something ready by January 10th. The latest cut is here:
http://www.brannenworks.com/koidehadrons.pdf

Carl

[granpa]

I was using this formula:

http://upload.wikimedia.org/math/2/f/a/2faa10c9790027937a525b23039fc688.png

a mass of zero can be added to numerator and denominator without changing anything.

[arivero]

I
I'm preparing a submission for Phys Math Central on the subject of Koide's mass formulas for the hadrons. I'm supposed to have something ready by January 10th. The latest cut is here:
http://www.brannenworks.com/koidehadrons.pdf

Carl

Hi Carl,

I would use some more group notation. Instead of your 6x6=36, I'd vote by 6 x bar6 = sum of irreps.

Second, besides to the "extraordinary" case you put, I'd include more of the "Ordinary" case, meaning the original argument of Koide about his formula for composite particles, along the lines of the original PhysRev.

[granpa]

there are obviously an infinite number of formulas relating the masses of the electron muon and tauon. the beauty of the koide formula is its simplicity and elegance. especially the fact that Q=2/3 which is exactly halfway between the upper and lower limits of 1 and 1/3.

as wikipedia puts it:
Not only is this result odd in that three apparently random numbers should give a simple fraction, but also that Q is exactly halfway between the two extremes of 1/3 and 1.

the formula involving cos given above seems to me to lack this elegance. it should be obvious that anyone can always find a formula involving cos that will pass through any 3 points quite easily.

[CarlB]

Hi Carl,

I would use some more group notation. Instead of your 6x6=36, I'd vote by 6 x bar6 = sum of irreps.

Hmmm. The only place I've got 36 in the thing right now is the calculation for the square of a set of MUBs. In terms of color, this would be
(3+\bar{3})\times (3+\bar{3})
i.e. (R+G+B+/R+/G+/B)^2,
and I guess the symmetry would be, (hems and haws whilst trying to recall Georgi), uh, let's see, maybe
(3+\bar{3})\times (3+\bar{3})
= 8+\bar{8}+6+\bar{6}+3+\bar{3}+1+\bar{1}?

Yes, as usual, your comment is brilliant and I will incorporate it.

In a certain sense, what I'm doing is taking the
3\times \bar{3} = 8+1
definition of gluons and claiming that I can compute in qubits with it using 3x3 matrices. [If you can read this, then PhysicsForums let me edit out a silly comment about 6+3-bar.]

Because both can be faithfully represented by 3x3 complex matrix addition and multiplication. From the symmetry point of view, it's a matter of selecting a certain ratio of the coupling constants.

Second, besides to the "extraordinary" case you put, I'd include more of the "Ordinary" case, meaning the original argument of Koide about his formula for composite particles, along the lines of the original PhysRev.

You mean the 1982 article I suppose. I guess I'll have to hike over to the university and read the article. Somehow I don't think I've ever downloaded it. One of the things I'm trying to do is to avoid talking about the preon model. There are two reasons for this. First, it doesn't apply to hadrons, and second, it gets into hot water with Coleman Mandula or (worse) spin statistics.

I think I should be ready to release a first cut in a few days. It still doesn't have "results" and the abstract is obsolete. But I've redone the results section to be make an argument that hangs together better.

[CarlB]

Okay, I've got this thing past the first cut. It is time to ask for advice and for people to point out obvious problems etc.

http://www.brannenworks.com/koidehadrons.pdf

[CarlB]

The most recent arXiv included a hep-th article that I thought was pretty cute.

Symmetries of Nonrelativistic Phase Space and the Structure of Quark-Lepton Generation

Piotr Zenczykowski.

According to the Hamiltonian formalism, nonrelativistic phase space may be considered as an arena of physics, with momentum and position treated as independent variables. Invariance of x^2+p^2 constitutes then a natural generalization of ordinary rotational invariance. We consider Dirac-like linearization of this form, with position and momentum satisfying standard commutation relations. This leads to the identification of a quantum-level structure from which some phase space properties might emerge. Genuine rotations and reflections in phase space are tied to the existence of new quantum numbers, unrelated to ordinary 3D space. Their properties allow their identification with the internal quantum numbers characterising the structure of a single quark-lepton generation in the Standard Model. In particular, the algebraic structure of the Harari-Shupe preon model of fundamental particles is reproduced exactly and without invoking any subparticles. Analysis of the Clifford algebra of nonrelativistic phase space singles out an element which might be associated with the concept of lepton mass. This element is transformed into a corresponding element for a single coloured quark, leading to a generalization of the concept of mass and a different starting point for the discussion of quark unobservability.
http://arxiv.org/abs/0901.2896

[arivero]

Let Me be the mass of the electron, Mp be the mass of the proton, and Mn be the mass of the neutron. Then observe that

Mn/Me - Mp/Me is approximately equal to ln(4*pi) = 2.5310...


Of course, it is really the quotient between the electromagnetic isospin breaking, (Mn-Mp), and the mass of the electron.

ellipsoid of volume

4pi(4pi-1/pi)(4pi-2/pi) = 1836.15...

Which is approximately equal to Mp/Me (a very old result)


Hmm not sure how old is this "ellipsoid". A really old approximation to Mp/Me as 6 pi^5 has been discussed in the middle of this thread.


I hesitate to put in measured values since Mp/Me has changed several times. At one time the ellipsoid volume was within one standard deviation of the accepted CODATA value, but the accepted value changed. I simply rest my case with the mathematical formulae.


A pseudoconvention in this thread is to put both the exact quotient between left and right sides of the equality AND the one-sigma error of this quotient. In this way it is clear how much we can expect the fit to change.



A tradition in this thread is to put both the exact quotient between left and right sides of the equality AND the one-sigma error of this quotient. In this way it is clear how much we can expect the fit to change.

In any case, according current pdg data:

mn − mp = 1.2933317 ± 0.0000005 MeV
Me= 0.510998910 ± 0.000000013 MeV

thus
(mn-mp)/me=2.530987199 ± .000001 MeV
and
(mn-mp)/me :: ln(4*pi) = 0.999 985 36 ± 0.000 000 4

As Hans says, the important point is that the agreement is of 99.998 5 %. The one-sigma adjustment is not important because the explerimental precision is a lot better. But this thread stresses that reaching the empirical bar of 99.99% in a simple formula in not easy.

[CarlB]

Walter Smilga has rewritten his justification of Wyler's approximation of the fine structure constant in a nice clean article:

Probing the mathematical nature of the photon field
Walter Smilga

The mathematical content of the interaction term of quantum electrodynamics is examined under the following assumption: It is presumed that the apparent degrees-of-freedom of the photon field reflect the kinematical degrees-of-freedom of the two-particle state space of massive fermions, rather than independent degrees-of-freedom of the photon field. This assumption is verified by reproducing the numerical value of the fine-structure constant.
http://arxiv.org/abs/0901.4917

Of course I see everything through the lens of what I'm doing. So I see the above as evidence that in understanding the color force, one should do something similar with gluons. My recent paper analyzes the mesons in terms of what happens to the quarks, pretty much ignoring the gluons. And it also finds that things turn out simpler than expected.

[legna777]

http://foros.astroseti.org/viewtopic.php?t=4529&postdays=0&postorder=asc&start=180


Entonces Legna, (perdona mi ignorancia) ¿qué significa la cifra resultante de 522?



saludos

Existen, como sabes, 5 grupos excepcionales ( a veces los llamo especiales ) de Lie y son:

\[
E8,E7,E6,F4,G2\]


\[
dim(E8)=248\]


\[
dim(E7)=133\]


\[
dim(E6)=78\]


\[
dim(F4)=52\]


\[
dim(G2)=14\]


En el grupo excepcional \[
E8\] están contenidos los otros 4 grupos excepcionales de Lie.
Este grupo \[
E8\] tiene 240 raíces no nulas, número que es el producto
de los 5 primeros términos de la sucesión de Fibonacci.

\[
240=1\times2\times3\times5\times8\]

\[
dim(E8)+dim(E7)+dim(E6)+dim(F4)+dim(G2)=\Omega\]



\[
\Omega+(8-5-3-2-1)=522\]

Saludos

La tabla de caracteres del grupo \[
E8\] se compone de una matriz de \[
453060\times453060\]

http://gaussianos.com/category/noticias/page/3/

\[
\frac{\pi^{5}}{120}\] es el factor de volumen de una hiperesfera de 10 dimensiones

http://es.wikipedia.org/wiki/3-esfera

Momento magnético anómalo del electrón

Límite del error experimental máximo:






\[
1+\frac{\Omega+(\frac{\pi^{5}}{120})^{-1}}{453060}=1+\frac{\alpha}{2\pi}+\frac{2\alpha^{2 }}{3}(\frac{\alpha}{2\pi})-\frac{4}{3}(\frac{\alpha}{2\pi})^{2}\]

http://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment

Saludos

[legna777]

http://foros.astroseti.org/viewtopic.php?t=4529&postdays=0&postorder=asc&start=210

[legna777]

Hola

\[
(\frac{m_{e}}{m_{e}}+\frac{m_{\mu}}{m_{e}}+\frac{m _{\tau}}{m_{e}})\approxeq(2^{7}+\frac{1}{\sqrt{2^{ 7}}})^{-1}(\frac{m_{W-}}{m_{e}}+\frac{m_{W-}}{m_{e}}+\frac{m_{W-}}{m_{e}})\]

\[
(\frac{m_{e}}{m_{e}}+\frac{m_{\mu}}{m_{e}}+\frac{m _{\tau}}{m_{e}})\approxeq(\alpha^{-1}(M_{Z})-\sin(\frac{2\pi}{6}))^{-1}(\frac{m_{W-}}{m_{e}}+\frac{m_{W-}}{m_{e}}+\frac{m_{W-}}{m_{e}})\]


\[
m_{e}=\] masa del electrón

\[
m_{\mu}=\] masa del muón

\[
m_{\tau}=\] masa del tauón

\[
m_{W-}=\] masa del bosón W con carga negativa -1= 80,398 Gev



\[
\alpha^{-1}(M_{Z})=128,95\] o valor de la "constante" estructura fina a la escala de unificación electrodebil o masa del bosón Z

muon-electron mass ratio

http://physics.nist.gov/cgi-bin/cuu/Value?mmusme|search_for=atomnuc!

tau-electron mass ratio

http://physics.nist.gov/cgi-bin/cuu/Value?mtausme|search_for=atomnuc!

Saludos

\[
[\frac{M_{Z}}{m_{e}}-(\frac{m_{e}}{m_{e}}+\frac{m_{\mu}}{m_{e}}+\frac{m _{\tau}}{m_{e}})]=174764=\sum_{p=2}^{137}p^{2}\]

Donde los corchetes en
\[
[\frac{M_{Z}}{m_{e}}-(\frac{m_{e}}{m_{e}}+\frac{m_{\mu}}{m_{e}}+\frac{m _{\tau}}{m_{e}})]

significa la función parte entera de lo que contiene

Y
\[
\sum_{p=2}^{137}p^{2}\] es el sumatorio del cuadrado de todos los números primos menores o iguales a 137

\[
M_{Z}=91,1876\; Gev\]




\[
[(\frac{M_{W}}{m_{e}}){\cos\theta_{W}}]=\sum_{p=2}^{127}p^{2}=138834\]

Recuérdese que para la inversa de la "constante" de acoplamiento electromagnética en la escala de unificación elctrodébil ( masa bosón Z ), 128,95; el número primo inmediatamente inferior a su parte entera [128,95] es el primo de Mersenne 127

\[
127=2^{7}-1\]

Saludos

[legna777]

Hola

\[
N_{C}(E8)+(1+\cos\theta_{W})(1^{3}+2^{3}+3^{3}+5^{ 3}+8^{3})=(137\cdot\ln137)^{2}\]

\[
1\times2\times3\times5\times8=240\] Cantidad de raíces no nulas del grupo E8. Los 5 primeros números de la sucesión de Fibonacci.



\[
N_{C}(E8)=453060=\] Número de la matriz generadora de los caracteres del grupo E8

\[
(1^{3}+2^{3}+3^{3}+5^{3}+8^{3})=[\sqrt{N_{C}(E8)}]=[\sqrt{453060}]=673\]

[legna777]

Hola

\frac{1}{\ln(\frac{\alpha^{-1}-21\varphi}{2})}=\frac{\pi^{4}}{384}

Donde Pi^4/384 es la densidad de enpaquetamiento de hiperesferas en dimensión 8. Grupo E8.

http://mathworld.wolfram.com/HyperspherePacking.html


Alpha= constante estructura fina= (137,035999084...)^-1
Phi= 1+SQR(5) /2 = número aureo= 1,618033989...

[legna777]

Hola

\[
(\frac{m_{PK}}{m_{e}})^{2}\cdot\varphi\cdot\cos(2\ pi/10)\approxeq\alpha^{-21}\]

\[
m_{PK}=\] masa de Planck= 2.176 44 x 10E-8 kg

http://physics.nist.gov/cgi-bin/cuu/Value?plkm|search_for=universal_in!




\[
m_{e}=\] masa del electrón= 9.109 382 15 x 10E-31 kg

http://physics.nist.gov/cgi-bin/cuu/Value?me|search_for=atomnuc!

\[
\alpha^{-1}=137,035999084\]

\[
\varphi=\frac{1+\sqrt{5}}{2}\]= límite del cociente entre

2 términos consecutivos de la sucesión de Fibonacci.

\[
21=1\times3\times7\]= septimo término de la sucesión de Fibonacci

1, 2, 3, 5, 8, 13, 21,....

\[
[\alpha^{-1}]=137\]

\[
[\alpha^{-1}]=137=2^{2}-1+2^{3}-1+2^{7}-1\]

Saludos

[legna777]

Hola

\[
\ln((\frac{\ln\ln\varphi}{\sum_{p}^{137}1/P}+137)/21)\approxeq\sum_{p}^{137}1/p\]


Donde: \[
\sum_{p}^{137}1/p=1,872603466...\]
es el sumatorio del inverso de los números primos menores o iguales a 137


\[
\varphi=\frac{1+\sqrt{5}}{2}=1,618033989...\]

[legna777]

Hola


\[
(\frac{M_{Z}}{m_{e}})(\frac{\alpha}{2\pi})=\frac{m _{\mu}}{m_{e}}a_{\tau}a_{e}\]


\[
m_{Z}=91,1876\]
$\: Gev$= 1.625566473 E-25 Kg


\[
m_{e}=9.10938215E-31\: Kg\]


\[
\alpha=1/137,035999084\]

\[
a_{\tau}\]= momento magnético anómalo del tau=\[
\frac{g-2}{g}\]=
\[
1.00117721\]

The tau lepton anomalous magnetic moment

http://arxiv.org/abs/hep-ph/0702026




\[
a_{e}=\frac{g_{e}-2}{g_{e}}=1.00115965218\]= momento magnético anómalo de electrón.

http://physics.nist.gov/cgi-bin/cuu/Value?ae|search_for=atomnuc!



\[
m_{\mu}=1.8835313E-28\: Kg\]=masa del muón

[legna777]

Hola

http://img522.imageshack.us/img522/8949/27nx3mz9.jpg

Espacio de 10 dimensiones ( 9 espaciales y una temporal )
6 dimensiones enrolladas-compactadas.

http://img224.imageshack.us/img224/9769/29pm8.jpg

http://foros.astroseti.org/viewtopic.php?t=4529&postdays=0&postorder=asc&start=165

[legna777]

Hola

http://img385.imageshack.us/img385/4326/30ya8.jpg

Masa de la Tierra= 5,9736E 24 Kg

[legna777]

Hola

exp(137 + 2Phi) X Tiempo de Planck= Edad del Universo=

13700 millones años= 13700 x 1E6 x 365 x 24 x 60 x 60= 4,320432E 17 segundos

Phi= (1 + SQR(5) )/2= numero aureo= 1,618033989....

exp( 137 + 2Phi ) = 8.012176286E 60

Tiempo de Planck= 5,3912 E -44 segundos

http://es.wikipedia.org/wiki/Tiempo_de_Planck

Good is the architect of Universe

[legna777]

http://foros.astroseti.org/viewtopic.php?t=4529&postdays=0&postorder=asc&start=165

[legna777]

Esto se pone interesante con la colaboración de todos :lol:

Cabibbo–Kobayashi–Maskawa matrix

In the Standard Model of particle physics, the Cabibbo–Kobayashi–Maskawa matrix (CKM matrix, quark mixing matrix, sometimes also called KM matrix) is a unitary matrix which contains information on the strength of flavour-changing weak decays. Technically, it specifies the mismatch of quantum states of quarks when they propagate freely and when they take part in the weak interactions. It is important in the understanding of CP violations. A precise mathematical definition of this matrix is given in the article on the formulation of the standard model. This matrix was introduced for three generations of quarks by Makoto Kobayashi and Toshihide Maskawa, adding one generation to the matrix previously introduced by Nicola Cabibbo. This matrix is also an extension of the GIM mechanism, which only includes 2 of the 3 current families of quarks

Perdón, señores, se me olvido el link de la cita

http://en.wikipedia.org/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix

\[
\theta_{C}=\] Ángulo de Cabibbo

\[
\theta_{W}=\] ángulo mezcla electrodebil

\[
\theta_{W}=\frac{4\varphi\theta_{C}}{3}\]

\[
\varphi=\frac{1+\sqrt{5}}{2}\]

\[
\alpha^{-1}=137,035999084\]


\[
\frac{20\pi\alpha}{3^{2}}\approxeq\sin^{2}\theta_{ C}=(0,2257)^{2}\]

\[
1+\sin(3\theta_{C})+2\cdot3^{2}\approxeq\frac{1}{\ sin^{2}\theta_{C}}\]

Ángulo áureo= \[
\frac{2\pi}{\varphi}\]

\[
\frac{\tan(2\pi/\varphi)}{2\cdot3^{2}}\approxeq\mid V_{cd}\mid^{2}=(0,2256)^{2}\]





\[
\left(\frac{1}{\mid V_{ud}\mid\cdot\mid V_{us}\mid\cdot\mid V_{ub}\mid}+\frac{1}{\mid V_{cd}\mid\cdot\mid V_{cs}\mid\cdot\mid V_{cb}\mid}+\frac{1}{\mid V_{td}\mid\cdot\mid V_{ts}\mid\cdot\mid V_{tb}\mid}\right)^{-1}=\pi^{4}+\frac{2}{5^{2}}\]


Saludos:smile:

[legna777]

\left(\sum^{137}_{p=2}p^{2}\right)\frac{\sqrt{8}}{ \pi}=\frac{M_{W}}{m_{e}}

Sum over all prime numbers <= 137

M_{W}=80,398 Gev

[legna777]

Gravitational radiation from binary systems

http://en.wikipedia.org/wiki/Gravitational_waves




P=\frac{32\cdot G_{N}^{4}\cdot M_{\odot}^{2}\cdot M_{\oplus}^{2}\cdot(M_{\odot}+M_{\oplus})}{5c^{5}d }\cdot\frac{1}{(1-e^{2})^{\frac{7}{2}}}\cdot(1+\frac{73e^{2}}{24}+\f rac{37e^{4}}{96})$


P=\frac{32\cdot G_{N}^{4}\cdot M_{\odot}^{2}\cdot M_{\oplus}^{2}\cdot(M_{\odot}+M_{\oplus})\cdot t_{year}}{5c^{5}d}\cdot\frac{1}{(1-e^{2})^{\frac{7}{2}}}(1+\frac{73e^{2}}{24}+\frac{3 7e^{4}}{96})=M_{pk}\cdot c^{2}\cdot\sqrt{10}\cdot\left[\frac{1}{(1-e^{2})^{\frac{7}{2}}}\cdot(1+\frac{73e^{2}}{24}+\f rac{37e^{4}}{96})\right]^{2}$

M_{\oplus}=5,9722\cdot10^{24}Kg$\medskip{}

M_{\odot}=1,98842\cdot10^{30}Kg$ \medskip{}

Where :
A.U, d=1,49597870700\cdot10^{11}m$\medskip{}

t_{year}=315581449,76\, s$\medskip{}


M_{pk}= Planck'mass = 2,17644 E-8 Kg


e=1,6710218\cdot10^{-2}$\medskip{} ( Earth's Orbit eccentricity )

Particle Physics Booklet (rpp-2008-booklet ), pág 6 :http://pdg.lbl.gov/2008/download/rpp-2008-booklet.pdf

[legna777]

Efecto casimir: energía del vacio entre 2 placas cuadradas conductoras de area L y a una distancia d


\[
E_{vacio\; Casimir}=-\frac{\pi^{2}\hbar cL^{2}}{720d^{3}}\]

Fuerza de Casimir:


\[
F_{vacio\; Casimir}=\frac{\pi^{2}\hbar cL^{2}}{240d^{4}}\]

Presión de Casimir:


\[
P_{vacio\; Casimir}=\frac{\pi^{2}\hbar c}{240d^{4}}\]

http://en.wikipedia.org/wiki/Casimir_effect

Investigation into Compactified Dimensions: Casimir Energies and Phenomenological Aspects

http://arxiv.org/abs/0901.3640v1

Página 31

================================================== ========



\[
E_{vacio\; Casimir}=-\frac{\hbar cL^{2}}{[\frac{240}{\alpha^{-1}\pi^{3}}+\ln(\frac{M_{PK}}{m_{e}})]\sqrt{2}d^{3}}=-\frac{\pi^{2}\hbar cL^{2}}{720d^{3}}\]

Donde: \[
\alpha^{-1}=137,035999084\]

\[
M_{PK}=2,17644E-8Kg\]

\[
m_{e}=9,10938215E-31Kg\]

http://foros.astroseti.org/viewtopic.php?p=77266#77266

[arivero]

I noticed a classical argument from Nigel
http://nige.wordpress.com/carl-brannen-and-the-koide-formula-how-to-derive-the-koide-formula/
about Koide-ish square roots, which could be of some value: Simply consider the total kinetic energy of the mass center of a system of three particles and compare it with the sum of the energies of the components.


E_{CM}=\frac 12 {{ (\sqrt{2E_1m_1}+ \sqrt{2E_2m_2}+\sqrt{2E_3m_3})^2} \over m_1+m_2+m_3}


and compare it with the total Energy E_T=E_1+E_2+E_3[/tex].

In the simplest "dirac delta" partition where [itex]E_1=E_2=E_3
we have that


{ E_{CM} \over E_T} = \frac 13 {{ (\sqrt{m_1}+ \sqrt{m_2}+\sqrt{m_3})^2} \over m_1+m_2+m_3}


and then Koide "3/2" is the condition that the CM energy is one half of the total, free, kinetic energy. Funny.


Disclaimer: I am not sure if I am being careful or standard when calling E_{CM} to this energy.
Edit: A guess is that some related but quantum conditions could be found by playing with the available phase space for scattering, using golden rule and all that.

[arivero]

and then Koide "3/2" is the condition that the CM energy is one half of the total, free, kinetic energy. Funny.
.

Of course, the total energy in an arbitray reference frame is the energy of the CM in this frame plus the ("rotational?") kinetic energy in the CM reference frame. No jokes about Lubos website, please :rofl:

So Koide is the relationship for masses in a frame where
- the kinetic energy of all the particles is the same, and
- The "traslational" energy of the CM in this frame is equal to the "rotational" energy of the particles in the CM frame.
- all the particles travel in the same line and orientation. (this is, to take the positive square roots and one dimensional sum)

[CarlB]

Hi Alejandro,

This is interesting. I'm going to have to play around with it as I'm not very intuitive about relativistic mechanics.

Marni and I just finished up the FFP10 conference in Perth, Australia. It went very well. I had a couple grad students ask for copies of my latest paper, which puts the Koide relation in the context of spin path integrals:
http://www.brannenworks.com/Gravity/EmergSpin.pdf

One of the talks was about quasars. It seem that there's evidence that they have intrinsic redshifts. They're born with very high redshifts, and then the redshifts decrease, in quantum jumps. The jumps correspond to changes in z of about 0.229, pretty close to the 2/9 in the Koide formula.

So I typed up a few extra PPT slides and wrote up an outline of a theory of gravity where gravity becomes repulsive inside of a black hole so that instead of having a single event horizon, they end up with a series of shells where their gravity is neutralized. The basic idea is to describe gravity as a vector field (see The River Model of Black Holes), and describe the vector length as a Wick rotated boost of something proportional to the gravitational flux. I'll post something at my blog when I get the time.

[arivero]

This is interesting. I'm going to have to play around with it as I'm not very intuitive about relativistic mechanics.

Actually, it is classical. It is not easy to read across Nigel's blog, it is not very concrete and it deviates from the path or misses the target very easily. But it is really straightforward, what he does in the deep, even without knowing it, is to ask for classical quantities where the angle with the vector \vec \sqrt m_i is relevant. And he finds one.

Let me rewrite it in one line:

{E_{CM} \over E_{CM} + E_{Rot}}={E_{CM} \over E_{Tot}}=
{ \frac 12 (\sum m_i) ({\sum m_i \sqrt {2 E_i/m_i}\over \sum m_i})^2 \over \sum E_i}
= { ({\sum \sqrt m_i \sqrt {E_i} })^2 \over \sum m_i \sum E_i}
= \Big({ \vec m^{1/2} . \vec E^{1/2} \over |\vec m^{1/2}| |\vec E^{1/2}| }\Big)^2


This is a general formula for freshman physics, and then if we put all the three energies equal it simplifies to
{E_{CM} \over E_{Tot}}=
{ E_{1/3} ({\sum \sqrt m_i })^2 \over 3 E_{1/3} \sum m_i } = \frac 13 {({\sum \sqrt m_i })^2 \over \sum m_i }


And thus Koide's happens to be the mass restriction that allows necessary conditions to find a frame where simultaneusly E_{CM}=E_{Rot} and E_1=E_2=E_3. Note that E_{Rot} in classical kinematics is frame-invariant.

With a little more of work, using the approximation v << c, it should be possible to do the same arguments in the relativistic kinematics of 3 body decay.


Also, It could be seen not as an explanation of the origin of the relationship but as a consequence with kinematic repercusions.


Marni and I just finished up the FFP10 conference in Perth, Australia. It went very well. I had a couple grad students ask for copies of my latest paper, which puts the Koide relation in the context of spin path integrals:
http://www.brannenworks.com/Gravity/EmergSpin.pdf


Good to hear of it.

[arivero]

Of course, if all the three masses are equal, then it is the kinetic energy triple the one which holds Koide's relationship. Amusing.

The only thing I am worried here is about taking the positive sign of the square root for all the three velocities; this is an extra requirement to the reference frame, to see all the three particles in the same direction. And also that they are all the three in the same line. On other hand, the "different sign" case is the one used by Carl in the neutrinos.

[enotstrebor]

Plus c and h, of course.

The idea is to collect here in only a thread all the approximations voiced out during the summer. .

For a single 4\pi relationship for the electron generations you might want to see Apeiron 16(4) 475-484 (2009).

The \mu (n=1)to e (n=0) mass ratio m_{\mathrm{e_1}}/m_{\mathrm{e_0}} is \sqrt{2}(4\pi\varrho_1)^{(3-1)} where \varrho_1=.96220481
while the \tau (n=2) to \mu (n=1) mass ratio m_{\mathrm{e_2}}/m_{\mathrm{e_1}} is \sqrt{2}(4\pi\varrho_2)^{(3-2)} where \varrho_2=.94635968.

Thus the first and second (n=1,2) generation mass ratio (m_{\mathrm{e}(n)}/m_{\mathrm{e}(n-1)}) form is \sqrt{2}(4\pi\varrho_n)^{3-n}.

As the W, proton and electron masses are a function of (4\pi\varrho)^{3} , i.e

m_x = M_{\mathrm{sp}} ~(2S ~(4\pi\varrho)^3/\varsigma )^{(S ~C ~M)}

whereM_{\mathrm{sp}}=\sqrt{m_p \cdot m_e}, \varrho=0.9599737853, S is the spin quantum number (1/2,1), C is the charge quantum number (+1,-1) and M is the matter quantum number (matter= +1 antimatter= -1),

You can get the masses of the W, proton, electron and its generations using the single formula for particle x (x=W,p,e,mu,tau)

m_x = M_{\mathrm{sp}(n)} ~(2S~(4\pi\varrho)^3/\varsigma)^{(S ~C ~M)} ,

where M_{\mathrm{sp}(n)} = M_{\mathrm{sp}} ~S^{-n/2}(4\pi\varrho_n)^{(6S n - S n(n+1)))} and \varrho_n = 1 - log(1 + 64.7564~n/S)/(112S) are used, and generation n is \{0,1,2\}.

[legna777]

m_{e}=\frac{V_{H}}{\sqrt{2}\cdot\Bigl(e^{12}+\frac {m_{Z}}{m_{e}}\Bigr)}



$V_{H}=\; vacuum\; Higgs\; \; "mass"=\frac{246,2205691\cdot E^{9}\cdot1,602176487\cdot E^{-19}\: Gev}{c^{2}}$\medskip{}




$m_{Z}=mass\; Z\; boson=91,1876\; Gev$



$e^{\Bigl(12+\frac{\cos(2\pi/5)}{10}\Bigr)=\frac{m_{Z}+m_{W}}{2m_{e}}\;\; m_{e}=\: electr\acute{o}n\; mass\; m_{W}=bos\acute{o}n\: W=80,398\: Gev}$\medskip{}




\[
\frac{V_{H}}{\sqrt{2}}+\frac{V_{H}}{\pi^{4}}=\sum_ {q=1}^{6}m_{q}-\sum_{l=1}^{6}m_{l}\]
\medskip{}



\[
\sum_{q=1}^{6}m_{q}=sum\; over\; all\; six\; quarks\;\;\]
\medskip{}



\[
\sum_{l=1}^{6}m_{l}=m_{e}+m_{\mu}+m_{\tau}+\nu_{e} +\nu_{\mu}+\nu_{\tau}=sum\; leptons\; masses\]





\[
\frac{V_{H}}{\sqrt{2}}-\frac{V_{H}}{\pi^{4}}=m_{Z}+m_{w}\]
\medskip{}



\[
bos\acute{o}n\; Higgs\; mass=\frac{\sqrt{2}\cdot V_{H}}{e}=m_{H}\backsimeq128\; Gev\]
\medskip{}



$e^{\bigl(12+(\sin^{2}eff^{lep}(\phi_{W}))^{-1}/10\bigr)}=\frac{m_{H}}{m_{e}}$




$\sin eff^{lep}(\phi_{W})=\sqrt{0,2315...}=\ln(\varphi)$ \medskip{}




$\varphi=\frac{1+\sqrt{5}}{2}$



\[
\frac{\sin\theta_{C}}{\sqrt{4\pi\alpha}}\Bigl(\fra c{V_{H}}{\pi^{4}}\Bigr)=\sum_{l=1}^{6}m_{l}=m_{e}+ m_{\mu}+m_{\tau}+\nu_{e}+\nu_{\mu}+\nu_{\tau}\]
\medskip{}



$\theta_{C}=Cabibbo\; angle=13,04^{\circ}$\medskip{}



$\alpha=fine\; structure\; constant=(137,035999084...)^{-1}$







"En este punto aun tenemos 4 bosones gauge (Wi\textgreek{m}(x) y B\textgreek{m}(x)) y 4 escalares $\xi\overrightarrow{(x)}$ y h(x)), todos ellos sin masa, lo que equivale a 12 grados de libertad (Conviene notar que un bosón vectorial de masa nula posee dos grados de libertad, mientras que un bosón vectorial masivo adquiere un nuevo grado de libertad debido a la posibilidad de tener polarización longitudinal: 12 = 4 × 2 + 4[escalares sin masa]). P. W. Higgs fue el primero en darse cuenta de que el teorema de Goldstone no es aplicable a teorías gauge, o al menos puede ser soslayado mediante una conveniente selección de la representación. Así, basta con escoger una transformación:"


http://es.wikipedia.org/wiki/Mecanismo_de_Higgs


http://arxiv.org/PS_cache/hep-ph/pdf/0001/0001283v1.pdf





[b]A Finely-Predicted Higgs Boson Mass from A Finely-Tuned Weak Scale

Lawrence J. Hall, Yasunori Nomura
(Submitted on 13 Oct 2009 (v1), last revised 19 Oct 2009 (this version, v2))

Abstract: If supersymmetry is broken directly to the Standard Model at energies not very far from the unified scale, the Higgs boson mass lies in the range 128-141 GeV. The end points of this range are tightly determined. Theories with the Higgs boson dominantly in a single supermultiplet predict a mass at the upper edge, (141 \pm 2) GeV, with the uncertainty dominated by the experimental errors on the top quark mass and the QCD coupling. This edge prediction is remarkably insensitive to the supersymmetry breaking scale and to supersymmetric threshold corrections so that, in a wide class of theories, the theoretical uncertainties are at the level of \pm 0.4 GeV. A reduction in the uncertainties from the top quark mass and QCD coupling to the level of \pm 0.3 GeV may be possible at future colliders, increasing the accuracy of the confrontation with theory from 1.4% to 0.4%. Verification of this prediction would provide strong evidence for supersymmetry, broken at a very high scale of ~ 10^{14 \pm 2} GeV, and also for a Higgs boson that is elementary up to this high scale, implying fine-tuning of the Higgs mass parameter by ~ 20-28 orders of magnitude. Currently, the only known explanation for such fine-tuning is the multiverse.

http://arxiv.org/abs/0910.2235

[legna777]

\[
\theta_{C}=Cabibbo\; angle\]
\medskip{}




\[
\tan\theta_{c}=\sin^{2}eff^{lep}(\phi_{W})=0,2315. ..=\ln^{2}(\varphi)\]








http://pdg.lbl.gov/2009/reviews/rpp2009-rev-vud-vus.pdf

[legna777]

"Standard" parameters


A "standard" parameterization of the CKM matrix uses three Euler angles ($\theta_{12}\,,\,\theta_{13}\,,\,\theta_{23}$)and one CP-violating phase $(\delta_{13})$
Couplings between quark generation i and j vanish if $\theta_{ij}=0$ .Cosines and sines of the angles are denoted $c_{ij}$ and $c_{ij}$ $s_{ij}$ respectively. $\theta_{12}$ is de Cabibbo angle.

The currently best known values for the standard parameters are:


$\theta_{12}=13.04\pm0.05^{\circ}$
$\theta_{13}=0.201\pm0.011^{\circ}$
$\theta_{23}=2.38\pm0.06^{\circ}$
$\delta_{13}=1.20\pm0.08^{\circ}$





\[
\begin{bmatrix}c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta_{13}}\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{13}} & c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{13}} & s_{23}c_{13}\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{13}} & -c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta_{13}} & c_{23}c_{13}\end{bmatrix}\]



http://en.wikipedia.org/wiki/Cabibbo–Kobayashi–Maskawa_matrix (http://en.wikipedia.org/wiki/Cabibbo%2DKobayashi%2DMaskawa_matrix)


\[
\frac{\alpha}{\sin(\phi_{W})}=\frac{\theta_{13}}{\ theta_{12}}\]




\[
\theta_{12}+\theta_{13}+\theta_{23}-\delta/2=\frac{2\pi}{24}\; rad.\]




\[
2\theta_{12}+\theta_{13}+\theta_{23}\backsimeq\phi _{W}\]

[legna777]

Alexis Monnerot-Dumaine: The Fibonacci Fractal

http://alexis.monnerot-dumaine.neuf.fr/articles/fibonacci fractal.pdf


\[
\frac{\sin eff^{lep}(\phi_{W})}{\frac{m_{W}}{m_{Z}}}=\frac{D_ {H}}{3}=\frac{\ln\varphi}{\ln(1+\sqrt{2})}\]


Hausdor Dimension=3=Dh


FRACTAL GEOMETRY IN QUANTUM MECHANICS, FIELD THEORY AND
SPIN SYSTEMS

H. Kröger

Physics Reports 322 (2000) 81-181

[Link Deleted]

It´s free, not my work. Great work

In geometry, an icosahedron (Greek: εικοσάεδρον, from eikosi twenty + hedron seat; pronounced /ˌaɪkɵsəˈhiːdrən/ or /aɪˌkɒsəˈhiːdrən/; plural: -drons, -dra /-drə/) is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of five Platonic solids.

It has five trianglular faces meeting at each vertex. It can be represented by its vertex figure as 3.3.3.3.3 or 35, and also by Schläfli symbol {3,5}. It is the dual of the dodecahedron, which is represented by {5,3}, having three pentagonal faces around each vertex.

http://en.wikipedia.org/wiki/Icosahedron



\[
-\frac{\sin\theta_{C}}{\sqrt{4\pi\alpha}}\backsimeq \cos(138,186965^{\circ})=\cos(\widehat{\Omega})_{d }icoSHD=\]

cosine icosaedral dihedral angle

Symmetry :dodecahedral manifold


\[
\frac{2\pi}{12+\varphi^{-1/2}}=\widehat{\phi_{W}}\; rad\]




\[
\frac{2\cdot m_{Z}}{m_{e}}+\alpha_{s}^{-1}(m_{Z})\cdot\varphi^{2}=\exp(12+\varphi^{-1/2})\]
\medskip{}



\[
\exp(\varphi^{5}+1)+\alpha_{s}^{-1}(m_{Z})\cdot\varphi^{2}=\frac{m_{Z}}{m_{e}}\]
\medskip{}



\[
-\sin\Bigl(\frac{2\pi}{\varphi^{2}}\Bigr)-\cos\Bigl(\frac{2\pi}{\varphi^{2}}\Bigr)=\frac{\al pha(q=0)}{\alpha_{s}(m_{Z})}=\frac{(137,035999084) ^{-1}}{0,1178}\]

[Vanadium 50]

First, the paper link you posted to is not "free". It is "stolen". I have since deleted it.

Second, Kroeger doesn't mention icosahedra - or polyhedra at all - in his paper.

Third, if this thread has taught us anything, it's that one can always toss a simple equation together to get some "interesting" value. By itself, this means nothing. This is why physicists call this "numerology".

Fourth, the ratio of two fundamental constants at two different scales cannot possibly be physical - as they are at two different scales.

Fifth, even if one takes everything at face value, the agreement is not impressive. Plugging in alpha_s at the Z, one gets a value for alpha at q=0 of 1/(137.4 ± 2.3). While that covers the correct value, the agreement is not so impressive when the error bars are added.

Finally, I'd like to remind everyone in this thread about the PF rules for overly speculative posts.

[arivero]

Yes, I am sorry that while this thread was active people was more aware of how touchy the issue is. We could close it perhaps.

[legna777]

First, the paper link you posted to is not "free". It is "stolen". I have since deleted it.

Second, Kroeger doesn't mention icosahedra - or polyhedra at all - in his paper.

Third, if this thread has taught us anything, it's that one can always toss a simple equation together to get some "interesting" value. By itself, this means nothing. This is why physicists call this "numerology".

Fourth, the ratio of two fundamental constants at two different scales cannot possibly be physical - as they are at two different scales.

Fifth, even if one takes everything at face value, the agreement is not impressive. Plugging in alpha_s at the Z, one gets a value for alpha at q=0 of 1/(137.4 ± 2.3). While that covers the correct value, the agreement is not so impressive when the error bars are added.

Finally, I'd like to remind everyone in this thread about the PF rules for overly speculative posts.


First, the paper is not "stolen": is public

Second: I do not say that the paper mentions nothing around icosahedra. I´ts is my opinion

Third: The fine structure constant QED at transfer momentun 0 ( or mass =0 )is:

(137,035999084... )-1

The
effective coupling equals the fine-structure constant a at
the Thomson limit(q2=
* 0)and is expected to increase
logarithmically as jq2
j increases at large jq2

PHYSICAL REVIEW LETTERS Volume 81 number 12 21 September 1998

"Measurement of the Running of Effective QED Coupling at Large Momentum Transfer
in the Space like Region"

The link is: PUBLIC, not stolen

http://dspace.lib.niigata-u.ac.jp:8080/dspace/bitstream/10191/1710/1/p2428_1.pdf





Four: Alpha_s is the coupling strong

Fifth: the ratio of two fundamental dimensionless constants at two different scales can possibly be physical

[Vanadium 50]

First, the paper is not "stolen": is public

No, it's not. It's in Physics Reports, which means that Elsevier owns the copyright. The fact that you have to download it from a file-sharing site rather than a scientific site should have been a big hint.

Second: I do not say that the paper mentions nothing around icosahedra. I´ts is my opinion

I'd like to remind everyone in this thread about the PF rules for overly speculative posts. .

[legna777]

No, it's not. It's in Physics Reports, which means that Elsevier owns the copyright. The fact that you have to download it from a file-sharing site rather than a scientific site should have been a big hint.



I'd like to remind everyone in this thread about the PF rules for overly speculative posts. .

Your are in true:The fact that you have to download it from a file-sharing site rather than a scientific site should have been a big hint.

correctely deleted the link, I am sorry around this fact

[CarlB]

With journals, you used to get a pile of paper "reprints" to mail around to people. Now some journals instead give you an electronic copy of the paper. You're allowed to put this on your own website but it still isn't quite a public copy in that no one else is allowed to do this. I've got a paper to be published this month in IJMPD and here's what they say:

You may post the postprint on your personal website or your institution's repository, provided it is accompanied by the following acknowledgement:
Electronic version of an article published as [Journal, Volume, Issue, Year, Pages] [Article DOI] © [copyright World Scientific Publishing Company] [Journal URL]
http://www.worldscinet.com/authors/authorrights.shtml

So when you see a copy of a scientific paper on the web it's not necessarily stolen. If it's on a file sharing website that's a bad sign, but there are also papers that have been released by the journals.

[Hans de Vries]

I'd say that this last remark reveals all. Either legna is mocking of the rest of the thread (which is reasonable, but perhaps and argument more to close the thread after all these years) or a part of legna is mocking of other part of self. On other hand, it is true that the relationships are, if not under 1%, at least within 3%. But it is clear that they fail on the side of simplicity, when compared to the rest of the thread.

To make people realize that one can come up with hundreds of thousands
of formulas with an equal or lower error you could introduce the following
posting requirements:

1) Every numerical coincidence posted should include the error percentage.
2) The posts should contain the "prediction power" of the "formulas"

The "prediction power" can for instance be defined by the simple formula:

\mbox{PP} ~~=~~ \frac{10^{-\frac43 S}}{\mbox{error}}

where S is the number of symbols like 1,2,3,+,-,*/,sin,log...

The value of 1/PP is then the amount of similar complex formulas which give
an equal or better prediction, most likely to be a very big number....


Regards, Hans

[jtbell]

This thread is closed pending a moderation decision.

Update: it has been decided to leave this thread closed.