thank you for keeping this thread updated. I will visit the physicsforum thread this weekend.

Cheers,
T.

There is also an aproved “independent research” thread in physicsforums, to discuss the topic of this thread. No posts at this moment, but some people could prefer to use the BBoard system insted of Blog comments
http://www.physicsforums.com/showthread.php?t=172821

I really think that string theory could help with the most troublesome problem of this approach: to interpret the pairs cc, cu and uu, which are not obviously forbidden, forming three (and times color) degres of freedom with no obvious arrangement as Dirac fermions, and pretty exotic electric charge. They could be arranged in three chiral (and charged!) fermions, but then they are massless and can not be uplifted to the mass of the top quark. The hope is that the mechanism barring these particles out of existence is forced to be a chirality argument and then it introduces chirality in sBootstrap.

a related meditation is pure numerology: that the main numbers of string theory revolve around 8 and 24, with some secondary numbers being 26, 10, 10/2=5, and 2^5=32. The sBootstrap approach seems to visit similar places.

funny how you are being drawn to some of ST’s ideas by your own theory. I think it is a very interesting convergence. Keep working at it!

Cheers,
T.

Now, decay rate has units of mass (or energy, if you wish). A theory without dimensional constants can only produce a mass dependence m^1. If the theory contains some fundamental scales La Lb Lc etc, then you can combine them to get different depencences. Ten dimensional string theory has a lot of these: the string scale, to start with, plus the compactification scale in each compact direction.

If you only consider the simplest combinations, only with the string scale A, of dimension [M^-2], then you can get a mass dependence in two ways: either A/M, or M^3/A. We need the later, but amusingly most of the papers on “string decay” like to consider the former, simply because it is the most paradoxical (the lifetime increases with M!).

I shall not comment on the many interesting contributions to the discussion above, but I wish to thank all who are contributing to it. It is “science in the making” and I am glad this creative process is taking place here.

Cheers,
T.

I have only a limited understanding of “… “Regge trajectories”: a dependence between energy and angular momentum …’, from papers such as the following.

D. A. Kulikov, R. S. Tutik. Renormalization of expansions for Regge trajectories of the Schrödinger equation
http://arxiv.org/ftp/quant-ph/papers/0609/0609066.pdf

other Tutik papers
http://arxiv.org/find/all/1/all:+tutik/0/1/0/all/0/1

This comment may be an over generalization, focusing on my perceived importance trajectories.
I speculate that the importance of trajectories as a “dynamic” dimension or degree of freedom has been overlooked in physics.
The classical 3 spatial dimensions, whether complex, real or “imaginary”, are essentially static on some type of coordinate system, whether commutative or non-commutative, polar or Euclidian.
Trajectory sets may include the sum of all paths with some paths more optimal subsets than others and may be non-commutative even in a commutative coordinate system.
Consider the travel of spacecraft from Earth to Mars and back.
A trajectory from planetary bodies in relative motion with coordinates centered on E0: E0 to M1 on M2 return to E3 where E is Earth, M is Mars and 0,1,2,3 are the relative coordinate positions at times 0, 1, 2, 3.
E0 ……. E1 ……………….. E2 ……………. E3
From to M1 … when …………… when
when …………. on M1 to M2 … from to M3
M0……. M1 ………………. M2 …………… M3
The simultaneous trajectory from M0 to E1 on E2 back to M3 is not commutative.

On an Earth roadmaps the relative positions of Start to Finish do not change but many trajectory routes are available; not all are optimal.

A reason I discuss this is that I suspect that physics is not using all the mathematical tools available to engineering. These disciplines are historically related.
Just as CP Steinmetz used phasor equation based upon Grassmann Algebra in electrical engineering about 25-30 years before ERJA Schrodinger used Hamiltonian equations based upon Clifford Algebra in quantum mechanics; today engineers are using dynamic game theory and variants in robotics to correlate electrical calculations with mechanical movements. This is somewhat like naval gunnery using the electronics of fire control radar to operate the mechanical ballistics [trajectory] of gunfire.

The SIAM classic Basar and Olsder, Dynamic Noncooperative Game Theory (Classics in Applied Mathematics), is very complicated mathematics leading to pursuit evasion games.
Paul J Nahin has easy to read engineering based books on extrema and pursuit evasion games:
1- Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills,
2 – When Least Is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible,
3 – Chases and Escapes: The Mathematics of Pursuit and Evasion.

There are more flexible, dioid, idempotent “tropical” variants of this type of mathematics.
I am most familiar with Max Plus Algebra. I have not found a blog on this subject. There is a web homepage. There are many ArXiv articles with Stéphane Gaubert of INRIA among the more prominent authors.
Geert Jan Olsder with others has written Max Plus Algebra as an introductory text.

There are other engineering books that I need to read to better understand applied as opposed to theoretical mathematical solutions.
Much of this material may indirectly be related to John Conway and his work on game theory. I do hope to one day complete ‘On Quaternions and Octonions’.

However, I was more inspired by the K-matrix papers of Asakawa, Sugimoto and Terashima, where a D-brane is realized as a spectral triple and classified by K-homology. In this construction, a (noncommutative) Dp-brane is constructed by D-
instantons or D0-branes whose positions are represented by eigen values of the scalar field Hermitian operators (hep-th/0505184,hep-th/0305006).

Taking such Dp-brane constructions over to Connes’ model, the involutive algebra A=M(3,C)+H+C may actually be producing a noncommutative brane of KO-dimension 6 (mod 8). We already saw how the C*-algebra M(3,C) itself gives rise to a three-point NCG space, Connes’ algebra would be an extension of this case. From a naive perspective, using the SU(2) H relation, I’m thinking Connes’ is using a six-point space for his finite geometry. In string theory, the points of this space would likely be identified as D-instantons or D0-branes.

The full geometry of Connes is the product geometry MxF, where M is a 4-dimensional smooth compact Riemannian manifold of KO-dimension 4 and F is the finite space of KO-dimension 6 (mod 8), giving MxF KO-dimension 2 (mod 8).

From pondering Connes’ model and its possible relation to Matrix theory, I have also questioned if M could be constructed by D-instantons or D0-branes. This is tantamount to dispensing with commutative C*-algebras all together and using only their noncommutative counterparts.

Cheers,
T.

About using U(6), or SU(6), I doubt. If you check again the section “predictions” on the post, or in sect 3 of the preprint, you will see that it depends very heavily of not linking all the quarks to strings, and in fact the matching of bosons with fermions predicts that you can only link five kind of quarks. But one should want to have this prediction as an outcome, so perhaps leave U(N), or the number of branes, free and then to ask for exact matching again, to see if the answer is N=5. Risking a bit of numerology, it could be easy to get this answer as one half of the number of dimensions of space time; already Marcus and Sagnotti (and Weinberg after them) hinted of a link between D/2 and the number of classical Chan-Paton factors.

I agree that in Carl’s lepton model, U(3) is more akin to a flavor group. The three orthonormal primitive idempotents of the spectral decomposition serve as a projective basis for CP^2. U(3) merely provides a change of basis for CP^2, leaving the corresponding real coefficients (eigenvalues of the Hermitian matrix/square roots of lepton masses) invariant.

I’m not sure how the above generalizes for quarks, but as you stated, one may have to introduce more D-branes to accommodate all the quark flavors. Perhaps there is a U(6) symmetry in the mass matrix approach to quarks.

Kea, yes, the discrete Fourier transform/circulant connection is promising.