Koide Formulas and circulant matrices:

Introductory paper, Koide formulas for charged leptons and neutrinos using circulant (or "1-circulant") matrices. (If you got to this website from the link in the paper "Neutrino Mass and New Physics" by R. N. Mohapatra and A. Y. Smirnov, Annual Reviews of Nuclear and Particle Science, 56 (2006) 569-628, then this is the unpublished paper you're looking for.

B-bbar and c-cbar meson spectrum, using 1-circulant matrices. There are six Upsilon and six J/Psi meson resonances. This link shows that their masses follow two similar Koide formulas.

Marni Dee Sheppeard (Kea) observed that the CKM matrix can be written as the sum of a 1-circulant and a 2-circulant in this blog post, leading to recent work in putting the mixing matrices into circulant form.

MNS matrix remarkably simple when written as a 1-circulant + 2-circulant. In these forms, the mixing matrices are "doubly magic", that is, their rows and columns all sum to 1 before converting to squared magnitude as well as after.

Weak hypercharge and weak isospin quantum numbers written using 1-circulant and 2-circulant matrix idempotents. The 1-circulant 3x3 matrices correspond to the even permutations on three elements, while the 2-circulants correspond to the odd permutations. From solving the idempotency relation, one obtains the weak hypercharge and weak isospin quantum numbers.

My popular but yet incomplete book on the standard model is available for download free here:
Operator Guide, 144+ pages, pdf, LaTex /memoir format
It's for two sided printing on 8.5x11 paper. This is a book giving a new foundation for quantum mechanics, and of course it has appropriate cover art, here:
Cover JPG, 655KB
If you absolutely must have a preprint copy hard bound, I've set it up so that you can buy it for under $30 at LuLu, which will print and bind a copy for you.

Some Java applets
Sudoku solver
Crystal Drawing
Pyrite photos
Simulation of General Relativity orbits
Koide paramaterization calculator
Scientific calculator with non base-10 display
Clifford algebra calculator


Now you might wonder how it comes to be that an amateur finds a formula for the neutrino masses. The answer is that I've been working on a variant of the usual field theory for the last three years. Instead of the state vector formalism, it's based on the density matrix formalism.

It turns out that the density matrix formalism is quite elegant and has natural geometric interpretations when it is written in Clifford algebra (which are a generalization of the Dirac gamma matrices). This is similar to what David Hestenes does with the Geometric algebra, but instead of being applied to the state vector formulation of quantum mechanics, it's instead applied to the density operator formalism.

One of the interesting features of the density operator formalism is that it is linear in a different way than the state vector formalism. In the state vector formalism, you can take linear superpositions between two kets and get a new ket. A density operator still lives in a linear space so you can add two density operators together. But if you translate the result of doing such an addition back into the ket formalism, you will find that what is linear in density operator formalism is nonlinear in ket formalism. This means that you can do certain nonlinear interactions very easily in the density operator formalism, but only with some difficulty in the spinor formalism.

Converting the standard model over to density operator form turns out to be quite a task. The biggest problem is that if you want this formalism to be distinct from the state vector formalism, you are not allowed to split a quantum state, which is represented by a density operator, into a bra and ket. This prevents you from having a vacuum. This stops the Higgs mechanism from giving mass to the fermions because there is no vacuum expectation values to play around with.

Fortunately, one can eliminate the Higgs by assuming a preon model of the elementary particles. See the article by J.-J. Dugne, S. Fredriksson, J. Hansson, and E. Predazzi hep-ph/9709227 for example. Unfortunately, there are very many possible preon models.

However, in the context of density operator theory, there is pretty much only one way to put preons into the standard model. And it was while messing around with those preons that I found how to write the Koide relation as an eigenvector equation, and then found how to extend it to the neutrinos.

Going back to the density operator formalism is a fairly extensive rewrite of quantum mechanics. The assumptions are very very simple, but the calculations are involved and it uses some mathematical methods that are not common knowledge in the physics community. So rather than write articles that no one can understand, I've decided to write a textbook introduction to density operator formalism that will go through the rather long and complicated derivation of the standard model.

At this writing, the first chapter is done, and the second chapter just needs a section on the physics of the bilinears of the Dirac algebra, plus marginal notes, and entries in the index:
Density Operators and the Standard Model

And now back to the website in progress. This is an older version of the lepton mass paper that was briefly on Arxiv until the moderators at Cornell decided it didn't belong there:
Koide Mass Formula for Neutrinos, 3 pages, pdf, ver 1.03 04/07/2006

The above paper was delivered at the April APS 2006 meeting in Dallas Texas. A link to the talk is here:
9:54 AM–10:06 AM, Sunday, April 23, 2006 Abstract: H12.00008 : Schwinger's Measurement Algebra, Preons and the Lepton Masses
The slides and notes for the 10 minute speech that goes with it is here: Very short APS06 slides and speech.

While the talk will cover some theory, the arXiv paper given above is phenomenological. I found the formula while messing around with a modification for the foundations of quantum mechanics. A rather hard to read and out of date write up exists here:
Geometric Probabilities, 32 highly defective pages, pdf, ver 0.00 03/06/2006

The above theory paper is defective in many ways. I regret the alternate probability interpretation it includes -- it is useful only if one wants the theory to work in mixed signature Clifford algebras. Some of the later sections having to do with the fundamental fermion are incomplete. I haven't even started the conclusion. The abstract and synopsis are in shambles. The second appendix, on astrophysics, is the result of only an hour of typing. Undoubtedly the whole paper is shot through with typographical errors and arithmetic mistakes which will make reading it difficult.

I went to the Particle And Nuclei International Conference in Santa Fe, New Mexico in late October (2005) and will presented a poster entitled "Particle Symmetry Breaking in Density Matrix Formalism with Geometric Algebra". I've uploaded the poster pages here:
Santa Fe Poster, 14 pages, pdf, ver 1.00 10/15/2005

I am also releasing an early version of a rather long paper that supports the above. This is very incomplete, has not been thoroughly edited and is in the process of being written. But it has some applications of geometric algebra to the problem of analyzing the Stern-Gerlach experiment that may be of interest: Incomplete Unfinished Error-Ridden long PANIC paper (pdf from LaTex)

And I've uploaded some pictures I took in New Mexico here: Pictures from Santa Fe PANIC 2005 meeting

A June 2005 introduction to binons, with the astrophysical evidence for their existence:
A Hidden Dimension, Clifford Algebra, and Centauro Events, 27 pages, pdf, ver 1.00 06/14/2005


Some older papers, in the order in which I think that they should be read. Some of these are referenced by the above, but the above paper is complete in itself:
The Proper Time Geometry, 5 pages, pdf, ver 1.0 10/19/2004
Phase Velocity of de Broglie Waves, 2 pages, pdf, ver 1.0 11/20/2004
The Geometry of Fermions, 11 pages, pdf, ver 1.01, 10/21/2004
The Geometric Speed of Light, 5 pages, pdf, ver 1.0 11/07/2004
Nonlinear Waves on the Geometric Algebra, 4 pages, pdf, ver 1.1 12/02/2004

I don't like heights, but here I am taking a self picture at 80 feet above the ground. I'm marking a grain alcohol plant so that it can be taken down and reassembled in Iowa. That's the Seattle equivalent of sunshine in the background.
[Carl Self Picture]

Markings are done with a waterproof pen. You can see some marked "TM", which means "Tower, Mid height". It's a beautiful machine. After it's taken apart, it will be shipped by rail. Then I appear to have been volunteered to go to Iowa to reassemble it. I hope it's not in the middle of winter. (As it turns out, the plant will be reassembled without me.)

Medium sized industrial equipment like this has lot more small parts than it looks. Reassembling them basically means putting together a steel jigsaw puzzle, with power wrenches, welding torches, and a 120 foot crane. My specialty is in getting the electronics and control systems running again. Sometimes this can be tricky, but this plant comes with about a 6" stack of blueprints so I don't expect any major problems.

You take lots of photographs beforehand, and it helps if you label small things according to their approximate position in the machine. But mostly factories are reassembled about the same way that jigsaw puzzles are reassembled. (See, these bird droppings and rust stains? These two pipes have to fit together.) Unlike a jigsaw puzzle, you can buy or build replacement parts for stuff that didn't make the trip.
[Mid Tower picture]

If you're interested in the philosophy behind the mathematics of the above papers, you can read some of my earlier html formatted write-ups:

A note on the weak gravity limit of the interaction between entropy, or wave function collapse, and gravity, where it is shown that the MOND relation is not unexpected:
Weak Gravity, MOND, and State Vector Reduction

A new topology for space time, with a calculation showing an explanation for the MOND gravitational anomaly:
Ether, Relativity, Gauges and Quantum Mechanics

The above paper is the second in the series. The first is here:
Searching for a fundamental reality in physics

If I'm selling any stuff on eBay, it's here.

To send me email, use carl@brannenworks.com: (Carl Brannen)


Upon Julia's Clothes


Whenas in silks my Julia goes,

Then, then, methinks, how sweetly flows

That liquefaction of her clothes.


Next, when I cast mine eyes, and see

That brave vibration each way free,

Oh, how that glittering taketh me!


Robert Herrick, 1591-1674