All the lepton masses from G, pi, e

CarlB

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:
[tex]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)[/tex]
By color symmetry,
[tex]H_{RR} = H_{GG} = H_{BB} = v,[/tex]
[tex]H_{RG} = H_{GB} = H_{BR} = se^{i\delta},[/tex]
[tex]H_{RB} = H_{GR} = H_{BG} = s'e^{i\delta'}.[/tex]

To get real eigenvalues H must be Hermitian so v is real, s=s' is real, and [tex]\delta' = -\delta[/tex]. 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:
[tex]s+2v\cos(\delta + 2n\pi/3),[/tex]
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:
[tex]H = \nabla^2 + V[/tex]
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

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

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

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

And I guess you can also compare circular monopole currents against electric charges.

arivero

Or
[tex]
e^{\pi^2/4} - e^{-\pi^2/4} \approx \sqrt{ \alpha + {1 \over \alpha}}
[/tex]
instead of
[tex]
e^{\pi^2/4} \approx (\sqrt \alpha + {1 \over \sqrt\alpha})
[/tex]

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.

[tex]\alpha\ =\ 1/137.035999084 (51)[/tex]

The alpha "radiative" series:

[tex]\mbox{\Huge i}^{~ln\, i }\sum_{n=0}\frac{\mbox{\Huge $\alpha$}^{n-\frac{1}{2}}}{(2\pi)^{b_n}} ~~ = ~~ \mbox{\Huge 1}[/tex]

Where [itex]b_n[/itex] 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,

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/200...unning-summer/

arivero

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).

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

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:

[tex]e_0^2={1 \over 137} \hbar \ c[/tex]
[tex]g_0^2=4 {137 \over 1} \hbar \ c[/tex]

for n=1 in [tex]{1 \over 2} n \hbar \ c[/tex]

The point is that there was other papers where it is argued that n must be even. Let me call [tex]g_2[/tex] to this forceful even constant. In this case
[tex]g_2^2= {137 \over 1} \hbar \ c[/tex]

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:

[tex]e_0^2={1 \over 137} \hbar c = \alpha_e \hbar \ c [/tex]
[tex]g_0^2=4 {137 \over 1} \hbar c= 2^2 {1 \over \alpha_e} \hbar \ c [/tex]

for n=1 in
[tex] eg= {1 \over 2} n \hbar c[/tex]

The point is that there was other papers where it is argued that n must be even. Let me call [tex]g_2[/tex] to this forceful even constant. In this case
[tex]g_2^2= {137 \over 1} \hbar \ c[/tex]

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

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

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

Hans de Vries

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

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