### Neutrinos Again III

Carl Brannen rightly pointed out that the matrices appearing in the tribimaximal mixing papers are in fact basically the same as those that characterise MUBs and the quantum Fourier transform for the prime 3. In fact, let where $(231)$ (oops, it should be $(312)$) denotes as usual the cyclic permutation in $S_{3}$ (sometimes drawn as ribbon diagrams) and $M$ is Carl's notation. Both $(231)$ and $3M$ cube to the identity. The democratic matrix is given by $D = \frac{1}{3}[1 + (231) + (231)^{2}]$, which can be thought of as a vector $(\frac{1}{3} , \frac{1}{3} , \frac{1}{3})$. Observe that the operator $S$ from the $A_{4}$ representation obeys the rules

$D \cdot 3S = 3D$

$2D - 3S = 3I$

where $I$ is the identity. I'm beginning to wonder if those poor experimenters are ever going to detect a $\theta_{13} > 0$. Note also that the (norm square) $2 \times 2$ form of the neutrino mass matrix, which was used by Harrison et al, is expressed as which utilises the 2-circulant $3 \times 3$ matrix that happens to square to the identity. In other words, this is a $3 \times 3$ representation of the Pauli spin Fourier polynomial. Thus tribimaximal mixing is expressed as a composition of mass Fourier and spin Fourier components.

$D \cdot 3S = 3D$

$2D - 3S = 3I$

where $I$ is the identity. I'm beginning to wonder if those poor experimenters are ever going to detect a $\theta_{13} > 0$. Note also that the (norm square) $2 \times 2$ form of the neutrino mass matrix, which was used by Harrison et al, is expressed as which utilises the 2-circulant $3 \times 3$ matrix that happens to square to the identity. In other words, this is a $3 \times 3$ representation of the Pauli spin Fourier polynomial. Thus tribimaximal mixing is expressed as a composition of mass Fourier and spin Fourier components.

## 5 Comments:

Kea, either I confused myself horribly last night or I found a formula that spits out quite exact meson mass numbers for nine pion mesons. It's very very simple and has just what you would expect, a combination of radial excitations of the hydrogen atom and the Koide formula

Well, the only way to see if you're not just confused is to show us so we can all take a look. I'm pretty confused most of the time myself, but I can always plug the PDG data into a calculator to check formulas. Heh, if you've sorted the mesons out, maybe you can move on to polishing off some other particles ...

Kea, the calculation is at physics forums. It gives the masses for the nine lowest pions to the accuracy of the splitting between pi+ and pi0.

Meanwhile, I'm having great difficulty explaining basic economics to Sabine. Perhaps you could look at the comments and see what it is that I'm having trouble explaining. Honestly, my opinion of her sinks deeper with each additional statement she makes on this.

Or am I being unfair? Debating ethanol with a guy who is a VP of engineering at an ethanol plant may not be the best way to look brilliant.

Ah, thanks for the links. Those coincidences are beginning to pile up. And no, I think it's silly for people to argue with you about something you clearly know much more about than they do, whether it be biofuels or physics.

Carl mentions a corn ethanol biofuel discussion with Bee.

From scanning the discussion there, one thing that I did not notice being discussed was the issue of total capacity.

Consider the USA.

What is the (obviously finite) maximum number of acres in the USA that could be used to grow corn?

How many bushels per acre could be produced?

So,

what is the total capacity (in barrells of fuel) can the USA maximally produce each year as ethanol from corn?

How does that compare with the USA consumption of fuel (in barrells of fuel per year) ?

Tony Smith

PS - I suppose other competing USA fuel sources might include coal (converted to gasoline) and/or tar sands (converted to gasoline),

but

burning them would increase atmospheric CO2.

Don't growing plants consume atmospheric CO2? If so, would corn ethanol be CO2 neutral in that the CO2 produced in burning ethanol would be the same CO2 consumed by the growing corn?

PPS - Could sugar cane grown in the USA South (think of the Everglades etc) also be useful for ethanol, and if so how much capacity would that add?

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