Neutrinos Again VIII
The vector sum of the last post, representing neutrino mixing, traces a two step path on a cube in the group algebra space for $S_{3}$. Observe that a similar picture for the CKM matrix must be higher dimensional, since the normalisation of the diagonal (to 1) can only result in an edge of amplitude $4 \times 10^{-3}$ if many edges sum to 1. Although this magic version of the neutrino mixing matrix uses different phases from the simple Fourier product, the relation to a quantum Fourier transform is still evident. The face diagonal may be decomposed into two steps of length $\frac{1}{\sqrt{3}}$, in the directions $(123) = \mathbf{1}$ and $(31)$. This tribimaximal path is then of the form $(XY)Z$, using the categorical convention of performing $Z$ first. Recall that the six paths of this type label the vertices of an important hexagon axiom.
posted by Kea | 6:38 PM
4 Comments:
Your sticking to your beliefs is admirable. Like Einstein working in the patent office, you prefer not to be seduced by "money science." Sacrificing life to the grant paper chase leads to studying fashionable subjects. like strings or "dark energy."
This is interesting, and I wonder if this does have something to do with magic, sort of a statement about the axes.
As far as I know, the question "is every 3x3 unitary matrix equivalent to a 3x3 magic unitary matrix?" is still not answered, though I believe the question is "yes". Does this post give a hint on how to solve this?
And on the theory that sometimes progress is made by looking at other things for a while, I've been at it again with Java programming, this time to Monte-Carlo statistics for Koide mass fits.
Preliminary numbers for the significance (i.e. probability of a better fit arising from random chance) of the Upsilon and Psi fits (where one chooses 3 of 6 masses to be electron-like and the other 3 to be neutrino-like) are 10% and 2%, respectively.
One could add other things that would make the %s smaller, such as requiring that the neutrino fit be the smaller, or that both fits should use a negative "S" parameter, but these, I think, are significant enough.
By the way, I'm now doing statistics on these things from the point of view of looking at the ratio of (b-a)/(c-b) where the masses are a < b < c. When one reverses the sign of S, the ratio R becomes 1 - R. And the electron and neutrino assume different theoretical values, 0.231 and 0.466, so it's simply a matter of computing the ratio from 3 masses, and seeing what they're closest to.
Then to get the %s, I generated 10 million collections of 6 random masses each, took all the ways of splitting them into two groups of 3, and computed the errors against the electron and neutrino ratios. And then used all that data to compute the distribution of errors.
Anyway, all this is stuff that is needed in order to write a real article on the subject. So work progresses.
Of course the reason for using simple ratios instead of the more elegant technique of splitting into S, V and delta is that this will be easier to understand. (I'm reducing the problem from high school trig back to middle school algebra, LOL.)
Great to hear you've been working on the stats, Carl. I'm afraid I've been busy waitressing and playing in the snow with friends.
Can you tell me sketching program (PC based I assume) you are using, for your diagrams?
I found out 2 things at the "Art of Grant Writing" course, at SIGGRAPH 2008:
1) they don't fund "ideas", but concrete projects
they leans towards the conservative side, so "way out" ideas tend not to get approved. Look at Einstein, his GR theory was so far out (no one really understood it). But, many yrs later they said "I think Einstein has something here, let's give him a Nobel for..say, his work on Photoelectric Effect ("concrete" thing)
2) effective use of Graphics
color, 3D animation, movies, etc.
I would highly recommend, that your next submission of your FQXI proposal, utilize some state-of-the-art Scientific Visualization techniques. I was at the "Fast-forward Technical Paper Preview" (where each technical paper presenter, does a 5 minute blitz presentation). Many of them, used a short movie to illustrate their paper. You should try exploring this new avenue (your friend Louise is dabbling in movies), it falls right into your characteristic of "succinctness". Heck, people all over the World (incl proposal reviewers), can simply download a Youtube movie (or iTunes video-podcast episode).
Take a look at some TED talks, G. Lisi presented at the last one in Feb (?). Again, it's along the lines of a blitz presentation.
BTW, my concept of an Interdisciplinary R&D Inst is along your concept of a Category Theory Inst. A few weeks ago, I happened upon a Plummers furniture store & saw some neat configurations.
R&D Inst, furniture configurations
I like the conservative/classical style. What is your idea of furniture configuration design for your Category Theory Inst (you mention that hut in an isolated scenic spot in NZ)? Or, do you prefer historical/rustic theme:
Balikun/China rustic furniture
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