View Poll Results: Multiple poll: Check all you agree. Logarithms of lepton mass quotients should be pursued. 24 27.91% Alpha calculation from serial expansion should be pursued 22 25.58% We should look for more empirical relationships 26 30.23% Pythagorean triples approach should be pursued. 21 24.42% Quotients from distance radiuses should be investigated 16 18.60% The estimate of magnetic anomalous moment should be investigated. 26 30.23% The estimate of Weinberg angle should be investigated. 21 24.42% Jay R. Yabon theory should be investigate. 16 18.60% I support the efforts in this thread. 47 54.65% I think the effort in this thread is not worthwhile. 30 34.88% Multiple Choice Poll. Voters: 86. You may not vote on this poll

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## All the lepton masses from G, pi, e

 Quote by arivero 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.

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 Quote by 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.
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.

 Quote by arivero 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.

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 Quote by 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.
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.

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 Quote by 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.
And I guess you can also compare circular monopole currents against electric charges.
 Blog Entries: 6 Recognitions: Gold Member 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.
 Recognitions: Science Advisor 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
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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
 Recognitions: Homework Help Science Advisor 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/

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 Quote by CarlB 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).

 Quote by jal 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.

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 Quote by arivero 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.
 Blog Entries: 6 Recognitions: Gold Member 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.
 Blog Entries: 6 Recognitions: Gold Member 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.

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 Quote by 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
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

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 Quote by 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.
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

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 Quote by 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
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

 Tags koide formula, lepton masses