M Theory Lesson 276
Thanks to Phil for commenting about magic matrices and MUB operators. The interaction of the matrix $R_3$ with the Fourier operator $F_3$ is expressed in relations such as 
where all entries have the same norm, and under normalisation are just phases. Note that $2 + \overline{\omega} = 1 - \omega$. The odd phase differs from the other phase by a special angle. The moduli of these matrices are all permutation matrices, which are also trivially magic with row sum $1$. The special angle, in radians, is given by $\theta = 0.2928428$, which corresponds to a sin squared of $1/12$. Actually, the phase difference is $\pi - 2 \theta$. By cubing $1 - \omega$, we see that the basic phase here is just $\pi/6$, a $12$th root. Unsurprisingly, the value of $\textrm{sin} \theta$ also turns up in the Fourier transform of the neutrino tribimaximal mixing matrix in circulant form.
where all entries have the same norm, and under normalisation are just phases. Note that $2 + \overline{\omega} = 1 - \omega$. The odd phase differs from the other phase by a special angle. The moduli of these matrices are all permutation matrices, which are also trivially magic with row sum $1$. The special angle, in radians, is given by $\theta = 0.2928428$, which corresponds to a sin squared of $1/12$. Actually, the phase difference is $\pi - 2 \theta$. By cubing $1 - \omega$, we see that the basic phase here is just $\pi/6$, a $12$th root. Unsurprisingly, the value of $\textrm{sin} \theta$ also turns up in the Fourier transform of the neutrino tribimaximal mixing matrix in circulant form.
posted by Kea | 8:07 PM
6 Comments:
I believe I now have a full (non-constructive) solution to your matrix decomposition problem which you can find in the comments on Carl's blog. I'll try to put it into a more coherent form, so that you or Carl can check it and use it as you wish.
Hey Kea, how are you? hope this isnt too off topic but i wasnt sure where else to put it lol i was just wondering what your opinion was on the work of Joy Christian (of Oxford and the Perimeter Institute, particualrly his latest paper on the subject of entanglement,'Disproofs of Bell, GHZ, and Hardy Type Theorems and the Illusion of Entanglement' ( http://arxiv.org/abs/0904.4259 ) Your thoughts and opinion would be greatly appreciated. Take care, Chris
I disagree with these conclusions. Category theory is much more elegant.
would you wish to elborate? when i first saw the paper i thought 'crackpot' perhaps to my own discredit. But on further inspection, albeit brief, im not so sure, though far be it for an amateur such as myself to judge the veracity of the claims made there-in. Oh, and please excuse my ignorance on these matters, im just a interested outsider trying to get an insiders perspective. Im fascinated by these sorts of thngs and am reliant on those better positioned to explain the more technical and abstruse aspects of the subject : ) Chris
Latest PI lecture from Bob Coecke is available at the Perimeter Institute. I think what is needed here is a set of non trivial examples of what is going on.
I think the central concept is that of elimination of summation in favor of cross products and product of composition. I'll write up a blog post on it.
One of the problems with summation as used in QM is that it doesn't work in the density operator formalism I prefer. I've heard all the arguments in favor of it, and I can write down examples where it is supposed to be critically important but actually is not needed at all.
For example, the concept of "quantum interference" where there are two paths that interfere with each other. It's hard to get people to deny that summation is a part of reality when they see those nice interference patterns.
But interference is something that happens to wave functions when you add them together. This is not possible in pure density operator formalism, but it's a fact that density operators generate all the same results as the usual state vector form.
To get that sort of example to be finite, you make the number of interfering legs finite. The other typical examples are spin-1/2 particles.
Meanwhile, I'm still spinning along on CKM and MNS matrices. I've got the Pauli algebra MUBs built into 3x3 matrices of projection operators and have verified the annihilation and idempotency equations.
Gee Phil, sorry we didn't acknowledge your excellent work sooner! I have been rather busy, shoe shopping and filling out my new social calendar.
Carl, I have been thinking about qubits and projective geometry, and no doubt so has kneemo.
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