### A connection with hadronic string model

In the previous posting I described the realization that so called super-canonical degrees of freedom (super Kac-Moody algebra associated with symplectic (canonical) transformations of M

^{4}

_{+/-}× CP

_{2}(light-cone boundary in a loose terminology) is responsible for the non-perturbative aspects of hadron physics. One can say that the notion of hadronic space-time sheet characterized by Mersenne prime M

_{107}and responsible for the non-perturbative aspects of hadron physics finds a precise quantitative theoretical articulation in terms of super-canonical symmetry. Note that besides bosonic generators also the super counterparts of the bosonic generators carrying quantum numbers of right handed neutrino are present and could give rise to super-counterparts of hadrons. It might not be easy to distinguish them from ordinary hadrons.

** 1. Quantitative support for the role of super-canonical algebra**

Quantitative calculations for hadron masses (still under progress) support this picture and one can predict correctly the previously unidentified large contribution to the masses spin 1/2 baryons in terms of a bound state of g=1 (genus) super-canonical gluons with color binding conformal weight of 2 units reducing the net conformal weight of 2-gluon state from 18 to 16. An alternative picture is that super-canonical gluons suffer same topological mixing as U type quarks so that the conformal weights are (5,6,58). In this case ground state could contain two super-canonical gluons of first generation and one of second generation (5+5+6=16).

I thought first that in the case of mesons this contribution might not be present. There could be however single super-scanonical meson present inside pion and rho meson with conformal weight 5 (!) and it would prevent color magnetic binding conformal weight to make pion a tachyon. The special role of π-ρ system would be due to the fact that pion mass is below QCD Λ. If no mixing occurs, g=0 gluons would define the analog of gluonic component of parton sea and bringing in additional color interaction besides the one mediated by ordinary gluons and having very strong color coupling strength α_{s}=α_{K}=1/4. This contribution is compensated by the color magnetic spin-spin splitting and color Coulombic energy in the case of pseudoscalars in accordance with the idea that pseudoscalars are Golstone bosons apart from the contribution of quarks to the mass of the meson.

Quite generally, one can say that super-canonical sector adds to the theory the non-perturbative aspects of hadron physics which become important at low energies. This contribution is something which QCD cannot yield in any circumstances since color group has geometric meaning in TGD being represented as color rotations of CP_{2}.

**2. Hadronic strings and super-canonical algebra**

Hadronic string model provides a phenomenological description of the non-perturbative aspects of hadron physics, and TGD was born both as a resolution of energy problem of general relativity and as a generalization of the hadronic string model. Hence one can ask whether something resembling hadronic string model might emerge from the super-canonical sector. TGD allows string like objects but the fundamental string tension is gigantic, roughly a factor 10^{-8} of that defined by Planck constant. An extremely rich spectrum of vacuum extremals is predicted and the expectation motivated by the p-adic length scale hypothesis is that vacuum extremals deformed to non-vacuum extremals give rise to a hierarchy of string like objects with string tension T propto 1/L_{p}^{2}, L_{p} the p-adic length scale. p-Adic length scale hypothesis states that primes p≈2^{k} are preferred. Also a hierarchy of QCD like physics is predicted.

The challenge has been the identification of quantum counterpart of this picture and p-adic physics leads naturally to it.

- The fundamental mass formula of the string model relates mass squared and angular momentum of the stringy state. It has the form
M

^{2}=M_{0}^{2}J ,M

_{0}^{2}≈ .9 GeV^{2}.A more general formula is M

^{2}=kn. - This kind of formula results from the additivity of the conformal weight (and thus mass squared) for systems characterized by same value of p-adic prime if one constructs a many particle state from g=1 super-canonical bosons with a thermal mass squared M
^{2}=M_{0}^{2}n, M_{0}^{2}=n_{0}m_{107}^{2}. The angular momentum of the building blocks has some spectrum fixed by Virasoro conditions. If the basic building block has angular momentum J_{0}and mass squared M_{0}^{2}, one obtains M^{2}= M_{0}^{2}J, k=M_{0}^{2}, J= nJ_{0}. The values of n are even in old fashioned string model for a Regge trajectory with a fixed parity. J_{0}=2 implies the same result so that basic unit might be called "strong graviton". Of course, also J=0 states with the same mass are expected to be there and are actually required by the explanation of the spin puzzle of proton. - g=1 super-canonical gluons has mass squared
M

_{0}^{2}= 9m_{107}^{2}.The bound states of super-canonical bosons with net mass squared M

_{0}^{2}= 16m_{107}^{2}are responsible for the ground state mass of baryons in the model predicting baryon masses with few per cent accuracy. The value of M

_{0}^{2}is .88 GeV^{2}to be compared with its nominal value .9 GeV^{2}so that also hadronic string tension is predicted correctly!

_{q}=1/2) state and (J=2,J

_{q}=3/2) state in which super-canonical bound state has spin 2.

To sum up, combining these results with earlier ones one can say that besides elementary particle masses all basic parameters of hadronic physics are predicted correctly from p-adic length scale hypothesis plus simple number theoretical considerations involving only integer arithmetics. This is quite an impressive result. To my humble opinion, it would be high time for the string people and other colleagues to realize that they have already lost the boat badly and the situation worsens if they refuse to meet the reality described so elegantly by TGD. There is enormous amount of work to be carried out and the early bird gets the worm;-).

For the revised p-adic mass calculations hadron masses see the revised chapter p-Adic mass calculations: hadron masses.

## 2 Comments:

Very interesting! Thanks. Yes, the anthropic principle would appear to have been wiped off the map already, along with the string physics that suggests it. However, 'M theory' as a categorical axiomatics for this framework shows promise of organising the calculations nicely.

Dear Kea,

I like M-theory in the monadic and motivic sense. The introduction of spontaneous compactifications was a mistake. The tragedy of string theory was that it established itself as a hegemony and paralyzed its own development: otherwise people had probably soon realized that 3-D generalization of strings must exist and found it sooner or later.

I am convinced that category theoretical concepts give excellent hopes of understanding also TGD at S-matrix level. Going to the core of the category theoretical philosophy: the fact that you have instead of single S-matrix infinite collection of S-matrices (or M-matrices or

Matrixes, somehow I like this word even I have not seen the movie;-),) forming a groupoid like structure and having also the analog tensor product satisfying a lot of axioms (do not forget theoretical constraints and super-conformal symmetries) could help to understand the quantum dynamics in a situation where path integrals are not of much help.Of course, there is a functional integral involving Chern-Simons phase factor plus exponent of Kahler function making the integral also mathematically well defined. I expect this to be exactly calculable at quantum criticality in the sense that radiative corrections vanish since perturbation series would not allow any control about whether or not results are algebraic numbers.

Of course, category theoretical formulation is a challenge for young mathematicians! I am a physicist with primitive technical skills in math but with animal instict for what is physically important;-) so that now and then I am almost free of feelings of shame and inferiority;-).

Returning to the topic of the posting: the correct prediction of hadronic string tension from p-adic thermodynamics and p-adic length scale hypothesis plus a fit of baryon masses within accuracy better than one per cent is something extremely non-trivial since the quark masses are exponentially sensitive to integer parameters involved and color magnetic spin-spin splittings and Coulombic binding conformal weights are also multiples of small integers.

One must also feed in information about the masses of spin 3/2 baryons and spin 1 mesons so that a full prediction is not in question. One should some day be able to predict the possible values of p-adic primes characterizing the quarks for a hadron with given electroweak quantum numbers and deduce the integers characterizing spin-spin splittings.

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