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Guest post: Alejandro Rivero, “sBootstrap” *October 16, 2007*

*Posted by dorigo in Blogroll, mathematics, physics, science.*

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*Alejandro Rivero** is a theoretical physics PhD from Zaragoza University who, to his own regret, was too good with computers and too stubborn about research paths. As there was a lot more of work for computer scientists than postdocts for non commutative geometry with the orthodox Connes approach, he went into the former. But he still has more confidence on NCG than in strings. Currently he works in the **BIFI**, where he is involved in the project of a national infrastructure for volunteer computing.*

**CHEWISH PHYSICS**

Coming back to the RadLab from his exile, Chew developed in the early sixties a new approach to strong interactions: the bootstrap. He observed that there was nothing in a Feynman diagram allowing to know which particles were composite and which ones were fundamental ones. Raising the flag of “nuclear democracy”, and feeling himself backed by observations of Feynman and Heisenberg, he launched a S-matrix-based program aiming to find an unique self-consistent theory of the nuclear strong force. “Nuclear democracy” was to mean that a particle can be interpreted as composite or elementary depending of its role in a particular diagram, so there were no elementary particles at all, or all the particles were equally elementary. The pursuit of further pieces of matter, subcomponents and preons and prepreons and so on, was to come to an end by asking some general properties to the scattering matrix, the particles coming ready from its pole structure.

During the first years the approach enjoyed some attention from labs everywhere, because even if you did not agree with Chew’s goals, the S-matrix was one of the very few tools available to approach to the study of strong interactions. But for the late sixties it started to be clear that the answer for strong force was more of the same: the hadrons were equal between them… because they were all composite, made of quarks. QCD became the new game, and S-matrix theory was abandoned. At the same time, one of the arguments of Chew theory, s-t duality, evolved by itself coming to beget “dual models” and then “string theory”, and then no a democracy but an unique ruler, the Planck-scale superstring.

Now, open bosonic strings, the ones you can make with two quarks, or a quark and an antiquark, lie in “Regge trajectories”: a dependence between energy and angular momentum, in a way such that all the strings of a same trajectory have a same composition: they are just excitations of its fundamental, spin 0 state. In this way, the number of strings depends of the number of different flavours of quarks you have

available to combine. And these strings are bosons. Assume each spin 0 state is not degenerate (choose the lowest energy one). Let’s postulate that for each spin 0 string there is a fermionic degree of freedom with the same electric and colour charge, but elementary. If it is so, we have saved not the “nuclear democracy” nor the bootstrap, but part of its philosophy: each elementary particle is not composite, but it is supersymmetric to a composite.

**COULD THE sBOOTSTRAP WORK ? PREDICTIONS**

A first test is to check for matching in degrees of freedom. For **n** generations of quarks and leptons, we have **n **charged leptons, and then **2 n** degrees of freedom of charge +1 and** 2n** d.o.f. of charge -1. On the other side, we can combine quarks U and D and their antiquarks to form **n^2** strings of charge +1 and the same of charge -1. Then **2n=n^2** implies **n=2**. This is already bad, because we know there are three generations. But in the neutral side it is worse: the **n** neutrinos give us **4 n** degrees of charge 0, while in the composite side we have either **2 n^2** neutral combinations of quarks and antiquarks that should reduce to **2 n^2 -1** after taking care of the U(1) singlet (as in SU(3) you get an octet instead of a nonet).

Even if we are willing to accept the result** n=2** and the extra U(1), we still need to confront quarks. And here the bartering definitely breaks. On one side, the UD sector has **n^2** combinations to form charge +1/3, for which there are **2n** degrees of freedom in the elementary side. Well, **n^2=2n** still implies **n=2**. But on the other side, when combining D quarks to get charge -2/3 you get **n(n+1)/2** of these, while you need **2n**. So from the DD sector we obtain **n=3**. The idea fails in the quark sector.

Thus the sBootstrap conjecture predicts 2 generations and can only work assuming an extra U(1) in the combinations for neutral strings and neglecting the possibility of partners for down quarks. Kind of failure for our idea.

Really? Let’s review… we have done an extra assumption that actually does not appear in Nature: we have assumed that all the quarks bind into strings. This is not true: the top meson has a mass greater than the W, and then it is disintegrated under electroweak force before being allowed to link into a QCD bind.

Back to the blackboard: let be **n** the number of generations, **s** the number of U quarks and **r** the number of D quarks. Obviously **n** equal or greater than **r** and **s**, and we ask both to be greater than zero. The rest of the history is section 3 of my e-print arxiv:0710.1526. Now the equations are

**2 n = rs**,

**4n = r^2 + s^2 -1**

for the lepton side, and

**2n = rs**,

**2n = r (r+1)/2**

for the quark side, and all of them solve uniquely to **n=3, s=2, r=3**. The sBootstrap conjecture predicts three generations AND predicts that while all the D quarks are light, only two U quarks are light.

**PROSPECTIVE**

There are some hints that the framework of superstring theory could be fit to keep developing on this conjecture. If we look to the colour side, the standard model with massive neutrinos has 24 fermionic degrees of freedom for each colour, including the neutral possibility. Generation-wise, it makes 8 degrees of freedom by generation; then we could have some hope of fitting them into the massless states of some sector of a superstring.

Furthermore, it is worthwhile to think of the quark-antiquark composites as oriented open strings, while the quark-quark and antiquark-antiquark are different sectors of unoriented open strings. To close an unoriented string you need to zip it against another unoriented string from the opposite sector, and then the resulting closed string carries a charge of the kind colour+anticolour, similar to a gluon. On the other side, an oriented string can close upon itself, giving a closed string uncoloured but perhaps still with electric charge… pretty much as the electroweak bosons, and it could explain the strange similarity in decay rates between the Z, W particles and the most stable mesons, of which Dorigo was kind enough to speak time ago in this blog.

A different question is, what will the LHC find, if all the superpartners have already being found? “*And the prize for the experimental finding of supersymmetry goes to… hmm, Cecil Frank Powell again???*“. Giving that we do not expect a QFT to have elementary fermions beyond spin 1/2, it can happen we will not find any superpartner more. And yet, we are telling that the mechanism giving mass to the stringy-bosons is the same that the mechanism giving mass to the elementary-fermions. So there should be definitely something to decrypt in the electroweak scale.

**THE BALL BACK TO HEP ENERGIES**

The lightness of the five quarks, and the heavy character of the top, is predicated with reference to the electroweak scale. Moreover, the superpartners are the particles bound by the QCD strong force. If we can work out some model based on superstrings, they will not be superstrings at the Planck scale but superstrings down under the TeV. It is hard to guess if a modern string theorist will be happy or sad about the sBootstrap idea.

A string-like binding fits well with other expectations of the amateur spectrologists in physicsforums.com, all of them from HEP energy data: the quotient between the mass of Z and W has been noticed, by Hans de Vries, to be very much as the one of the binding two relativistic particles with total angular momentum 1 and 1/2. Some breaking in the neutral vs charged pion was also related to the quotient between the mass of a lepton and those of a gauge boson, and some of the octet breaking or mixing was very near of relationships involving only lepton masses. Furthermore, Koide’s formula works better when mixing composite particles, and then it could be a formula to be met in the initial three supermultiplets, before further symmetry breaking and mixing in the strong sector. This fact has been another of the guiding principles towards our suggestion of a composite elementary symmetry. Of course all the amateur findings, without a backing theory, could be birthday coincidences: *how many people must meet in a room to have a 50% probability of finding two persons with a common birthday?* But note that some formulae are grouped in very nicely symmetric families.

In any case, the plot associated to the Z width mystery provides an approximation of the scale range: from the mass of the pion -and muon- to the point where the scaling of charged pion meets the scaling of neutral pion. The latter goes with the cube of the mass, the former with the quintic power. The lines meet at 2.6 TeV, and of course the effect of the breaking of electroweak symmetry starts to blur already before, as we climb beyond the 0.1 TeV of the Z mass.

## Comments

Sorry comments are closed for this entry

First, allow me to thank Tommaso by allowing my guess post. I hope ths idea I am presenting will be inspiring to other physicists. Hey, I even offer myself to travel to present it in seminars; I could even allow me to suffer again another transoceanic jump (I only went USA once, to Austin in the past century).

To put a bit more of discussion on the table, and to wink preon-theoretists (hey Carl!), this week I have been speculating about how the sBootstrap should be in a preonic theory. The subcomponents of an elementary fermion should be a couple of spin zero squarks joined with a spin 1/2 string; and if we think of this join as a force exchange then it is impossible (a spin zero can not emit a spin 1/2)… except, perhaps, in the case of tadpoles: the two spin zero particles being in the same position of the space. So at least a preon sBoot would have an autonomous explanation of the point-like character of the particles. I am not sure if there are in the market preonic theories with five squark-like and antisquark spin 0 objects.

Thanks for a great post, arivero! I’d love to invite you to NZ, but my waitressing job doesn’t pay me highly enough for funding such ventures. Actually, the bootstrap appears in my thesis, where the Veneziano amplitudes are seen to arise from operad combinatorics – as you say, very stringy but not stringy.

By the way, Heisenberg pointed out long ago that particles in QFT could not be considered ‘fundamental’ since they contained others potentially in their interactions, and this feature should be accounted for in their description.

Alejandro, having a spin-1/2 particle connect up two spin-0 particles seems natural in a classical way to me. Spin-1/2 is a vector (direction oriented) sort of particle, spin-0 is a scalar. What better way to connect two points then with a direction vector.

And a tadpole seems pretty natural for a process that is supposed to happen to make a point like particle.

Tommaso, what an excellent series of guest posts!

Hi Carl, if you liked it, you are welcome to be the next contributor, as long as you make it simple and you look for a “negotiation basis” with orthodoxy – and you promise to answer comments about it yourself 😉

Cheers,

T.

Hi Kea, I had read Chew to say he was building on suggestions of Feynman and Heisenberg, for Feynman he quoted some “personal communication” and for H. I thought he was referring to S matrix stuff but your reference is more adequate to the bootstrap philosophy. Interesting.

About operads, it hear time ago on the topic because the advisor of Alessandra Frabetti, a mathematician called Jean-Luis Loday, was very enthusiastic about them in the middle nineties. It seems they have some hope of using this kind of structure in “non commutative” physics.

Carl, but do you think a preonist could be happy about building all the structure out of squarks?

As I say, this suggestion was an added speculation after the post and it seems a lot weaker than the rest of the discussion (Hey, I proved n_gen=3! How many theories, orthodox or non, get to it?). The spin seems very troublesome; it is exchanged either from one particle or the other, so the idea of using it to fixing a direction does not seem valid, except for the fact that if the composite particle has definite spin, so it should happen with the sum of constituents.

By the way: who got the awful idea of calling “sfermions” to bosonic particles?

I should alert too that I was not thinking on one-legged tadpoles but the two-legged version, which is a kind of bubble attached to the line of other particle. So in a second thought it is not so clear how it would imply point like particles. The two bosons could “exchange” a bubble without needing to be in the same position. Or it could be possible to propose a Feynman diagram with six legs, two to attach the bubble, and the other two pairs for each scalar worldline.

In the large N-limit of pure YM (no susy), where one finds a string theory on a 5D geometry, Juan Maldacena has stated that adding quarks corresponds to adding D-branes extended along all five dimensions. Moreover, the open strings ending on these D-branes are the mesons. In this picture, a D0 brane in the bulk corresponds to a baryon.

…the advisor of Alessandra Frabetti, a mathematician called Jean-Luis LodayAh, yes. I believe Frabetti and Loday are two of the leading Mathematical Physicists today with their emphasis on operads and trialgebras.

I have lost the track of Frabetti, she did some work with Brouder on renormalization trees but I have not got news since then.

Kneemo, about the 5D way, AdS/QFT etc, a big problem is that I can not take large N, my proof needs SU(3) in order to build an antiquark from the partner of a diquark. Still, the lepton/meson part could work, then implying at least three generations but not fixing the exact number.

Alejandro, yes that was what I thought you meant by “tadpole”. It will appear point-like provided that the mass of the exchange particle is high enough. I see what you mean about spin. If the spin is inherent to the exchange particle, then where does it reside in between exchanges? That is a problem.

Tommaso, if I write a guest post, it will be labeled as something like “[n] remarkable physics papers that are largely ignored” and will be devoted to stuff that I think is cool, but few people pay any attention to.

Alejandro, I see your point. The other way to approach the problem is by looking at N coincident D-branes whose positions are given by scalar field matrices phi^m in the adjoint rep of the unbroken gauge group U(N). In your case, N=3, so such scalar fields are Hermitian elements of the C*-algebra M(3,C). By diagonalizing these Hermitian matrices (Phi^m=U^m(D)U^m*), we recover three real eigenvalues, giving the classical positions of the three D-branes. The off diagonal elements of the unitary matrices U^m_ij describe fluctuations in spacetime, arising from the short open strings connecting D-branes i and j, while the off diagonal elements Phi^m_ij act as “Higgs fields” for the symmetry breaking mechanism.

Through the lense of NCG, our C*-algebra would be A=M(3,C), with corresponding noncommutative space X=spec(A) consisting of only three points (a zero-dimensional manifold). We can complete the spectral triple by including the Hilbert space H=C^3 and Dirac-like operator D.

The three-point space identified in NCG doesn’t seem very interesting at first, but upon closer inspection, the spectral decomposition of our Hermitian scalar field matrices reveals that the D-branes are actually mapped to three points in the projective space CP^2, in the form of three primitive idempotents. This is where we can make the connection to Carl’s lepton mass matrix work (see The Lepton Masses).

A minor difference with Carl’s work and the coincident D-brane picture is that upon separation of the D-branes, the U(3) symmetry is broken, giving a set of massive fields with mass equal to that of the stretched strings. In Carl’s lepton paper, the eigenvalues (i.e., classic D-brane positions) correspond rather to the the square roots of the masses of the charged leptons. Another minor difference is that Carl uses circulant Hermitian matrices, rather than just general Hermitian element of A=M(3,C). Despite these minor differences, there seems to be strong evidence that Carl’s lepton model has a dual D-brane interpretation.

And as you know, kneemo, the circulants are related to the eigenvalues via the discrete Fourier transform, which introduces a 6 point phase space satisfying the Weyl rules.

Kneemo, I had not though approaching the problem via coincident branes, but I will think about it. In the description towards Carl’s model, N=3 seems more as a flavour group than a colour group, is it? If it is flavour, most probably I need an stack of five coincident D-branes, to label u,d,s,c,b terminations. Well, at least it avoids these pesky N->infty limits, buy I’d favour some mechanism more old-fashioned, with diquarks as unoriented strings, mesons as oriented strings, etc.

In any case, independently of the way, what we have here is that if someone gets to build a superstring model having all the spin zero mesons and diquarks of QCD then the partners of these modes will amount exactly to the same particle content than the standard model. This is a minor claim than the prediction of the number of generations, and in fact I did this claim already two years ago, but I did not thought on asking for a string based technique to pair bosons and fermions (note my “biographical snippet” at the start: I do not like string theory, the results are forcing me into it).

I am a bit surprised about how fluently you build up an spectral triple to be associated with an stack of Dbranes. Is it a mechanism of your own design, or is it some standard lore I have missed due to my phobia?

Alejandro, I attended a talk in New York last year where he discussed gauge fields and NCG (see hep-th/0408012. He identified the spectral triple for the NxN case and presented a Born-Infeld lagrangian for the corresponding non-abelian gauge theory. In the D-brane picture such a Born-Infeld lagrangian describes a model of non-linear electrodynamics on a fluctuating p-brane. Kerner didn’t mention any D-brane intepretations, but this is more likely just a matter of language not physics.

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.

My apologies, “he” as in Richard Kerner.

Thanks for the pointer, I will check recent work from Kerner. I think that now that we know how to build NCG with a KO dimension different from the spectral dimension, one should review the issue of string theory on NCG spaces. It could be a surprise if just KO dimension 2 mod 8 meets the needs of superstrings!

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.

Carl, please go ahead. I will be happy to publish it here – I see a lot of interest arising from the latest few guest posts here, so why not continuing with that ? Also, it is much more relaxing to have others write in my place 😉

Cheers,

T.

Alejandro, for more recent work on NCG, KK-theory and branes, see the papers by Richard Szabo (e.g. arXiv:0709.2128v2 [hep-th], arXiv:0708.2648v1 [hep-th]).

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.

Hi Alejandro,

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

Click to access 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’.

Tommaso, I’ll type up a post tentatively titled “4 magnificent papers by authors who think I’m a complete idiot”.

Hi all,

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.

Sounds like a great title, Carl. I’m confident the body will match my expectations.

Cheers,

T.

Thank you for the words of support Tommaso.

Indeed thank very much. Althought I hope the casual readers will notice this is “amateur science in the making”; profesional science is done by a sequence of setting and answering problems inspired in the problems already solved or hinted by your advisor and your peers.

About using string theory for the simplest mesons, the problem is how to calculate the decay rates, which should be proportional to m^3 or to m^5. This is because the strong decay channels are closed.

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!).

Alejandro, for a clear discussion on quarks and string-brane configurations, see 15.6 and 15.7 of Barton Zwiebach’s string theory book. There he constructs an intersecting brane model on T^6 (6-torus) which closely resembles the standard model, but also mentions the more subtle issues that arise such as electroweak symmetry breaking and the non-uniqueness of the brane configuration.

Thanks, I have checked it now and it is actually clearer than some preprints of the UAM I had got to read (Zwiebach says that his example clones or follows some others from the Spanish group of strings, which is mostly around the Universidad Autonoma; regretly I have not relationship with them, but I was considering to ask some students I knew).

[…] Tommaso Dorigo advises to approach orthodoxy on a negotiations basis. (See comments to this post.) […]

At the kind requeriment of the owner of this blog, Woit answers about supersymmetry putting the finger in the ttraditional problem: how to break it. Our idea here still lacks of a mechanism, but it should be very different from the usual ones, specially if we want the leptons to meet Koide; it should imply that most of the breaking is mixing between QCD objects, leaving leptons untouched.

On the other hand, about compositeness and dynamics, we have already a part of the dynamics: QCD. The problem is the compositeness or not of the gauge bosons. Here we start to approach string theory, imagining a W boson as the result of joining the extremes of a charged meson to form a closed string. Because of this I explicitly asked Dorigo to stress, in the introduction above, that I do not like string theory. I have always been against it, and now the perversion of being forced into its use gives me mixed feelings, the more positive being that if it works here it probably disproves or discourages its use at Planck scale.

Hi Alejandro,

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.

Hi,

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.

Hi Alejandro,

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

Cheers,

T.

Hi T.

Perhaps the comment in #32 is an important advance. I was thinking today: if we admit that the only possible arrangement of these degrees of freedom is as “coloured” Majorana fermions, then it seems that they truncate themselves out, because QCD is vector like. The usual folklore about the need for the neutrino to be neutral.

On other hand, allowing for this chiral beasties and its truncation would force similar forgiveness in the equations of the main argument, clouding it 😦

To resume, I need more time to understand what is going on with chirality, and I haven got any, at least until the summer.