for the parameters $r > 0$ and $\delta$, and with $\omega_{i}$ the three cubed roots of unity. So for the leptons, $r = 1/2$ and a good choice for $\delta$ is $2/9$. Now using the trigonometric identities 
we see that the average mass number of a Koide triplet is given by 
which is independent of $\delta$. For example, the spectrum given by $r \in 0.5(0, 1, 2, 3, \cdots)$ is the integer spectrum $\mu \in 1, 2, 3, \cdots$ The general Koide rule is now given by 
Positivity of the $\lambda_{i}$ says that $0 < r < 1$, so that the (charged) lepton value is central to the range. In 1994, Foot observed that the quantity $\mu^{-1}$ may itself be thought of as the cosine squared of the angle between two vectors, namely the vectors $(1,1,1)$ and $(\lambda_{1}, \lambda_{2}, \lambda_{3})$. At $r = 1/2$, this angle will be $\phi = \pi /4$. In general, the angle $\phi$ results in$r = \frac{1}{2} \textrm{tan}^{2} \phi$
showing how the parameter $r$ measures the angular deviation from the lepton value.
Good luck with this research. I hope you can get a complete theory of mass sketched out soon, thinking about the big theoretical picture, not just looking at modelling details. Even if the abstract modelling you are doing is right, you still have to work out a complete theory of mass replacing the Higgs field or the charge of quantum gravity, before you can really claim to have an alternative to string theory. It's a good idea to avoid trying to formulate a complete theory until you have enough detailed modelling of empirical data available to base a complete theory on, but there is a danger that it is possible to end with mathematically impressive and predictive, yet physically unhelpful, epicycles.
ReplyDelete- A. R. Rogant
Mr Arrogant, if you yourself have a fully working theory of quantum gravity (including physical computations using ideas that go well well beyond field concepts) then one does wonder why you waste so much time writing such dull comments on the blogs of lesser mortals. If, on the other hand, you do not have a fully working theory of quantum gravity, then one wonders from what remarkable dearth of experience you have managed to ascend to such a lofty tower.
ReplyDeleteI'm interested in what you are doing with respect to neutrino masses and mixing parameters like the CKM matrix.
ReplyDeleteI've had a functional quantum gravity theory published in 1996 (not on the arXiv or in an appropriate journal, after CQG sent it for review by string theorists who had no interest in predictive rival ideas), that predicted the cosmological acceleration later discovered observationally (before it was confirmed, it was rejected as speculative; after it was confirmed, it was rejected as ad hoc), and which also calculates all the coupling parameters in the Standard Model (plus gravity). I'm minimizing the changes necessary to the Standard Model, in order to predict gravity and masses.
BTW, if all else goes wrong, you would make a brilliant mainstream string theorist (responding to new ideas with paranoid suspicion and insults).
Um, you haven't shown us any interesting new papers. You are not even using your name. Perhaps you haven't heard, but insulting people anonymously is considered cowardly.
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