Does the spin of hadron correlate with its super-canonical boson content?

The revision of hadronic mass calculations is still producing pleasant surprises. The explicit comparison of the super-canonical conformal weights associated with spin 0 and spin 1 states on one hand and spin 1/2 and spin 3/2 states on the other hand (see this) demonstrates that the difference between these states could be understood in terms of super-canonical particle contents of the states by introducing only single additional negative conformal weight s_c describing color Coulombic binding . s_c is constant for baryons(s_c=-4) and in the case of mesons non-vanishing only for pions (s_c=-5) and kaons (s_c=-12). This leads to an excellent prediction for the masses also in the meson sector since pseudoscalar mesons heavier than kaon are not Golstone boson like states in this model. Deviations of predicted and actual masses are typically below per cent and second order contributions can explain the discrepancy. There is also consistency with string bounds from top quark mass.

The correlation of the spin of quark-system with the particle content of the super-canonical sector increases dramatically the predictive power of the model if the allowed conformal weights of super-canonical bosons are assumed to be identical with U type quarks and thus given by (5,6,58) for the three generations. One can even consider the possibility that also exotic hadrons with different super-canonical particle content exist: this means a natural generalization of the notion of Regge trajectories. The next task would be to predict the correlation of hadron spin with super-canonical particle content in the case of long-lived hadrons.

For the revised p-adic mass calculations hadron masses see the revised chapter p-Adic mass calculations: hadron masses of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".

Revised p-adic calculations of hadron masses

The progress in the understanding Kähler coupling strength led to considerable increase in the understanding of hadronic masses. I list those points which are of special importance elements for the revised model.

1. Higgs contribution to fermion masses is negligible

There are good reasons to believe that Higgs expectation for the fermionic space-time sheets is vanishing although fermions couple to Higgs. Thus p-adic thermodynamics would explain fermion masses completely. This together with the fact that the prediction of the model for the top quark mass is consistent with the most recent limits on it, fixes the CP₂ mass scale with a high accuracy to the maximal one obtained if second order contribution to electron's p-adic mass squared vanishes. This is very strong constraint on the model.

2. The p-adic length scale of quark is dynamical

The assumption about the presence of scaled up variants of light quarks in light hadrons is not new. It leads to a surprisingly successful model for pseudo scalar meson masses using only quark masses and the assumption mass squared is additive for quarks with same p-adic length scale and mass for quarks labelled by different primes p. This conforms with the idea that pseudo scalar mesons are Goldstone bosons in the sense that color Coulombic and magnetic contributions to the mass cancel each other. Also the mass differences between hadrons containing different numbers of strange and heavy quarks can be understood if s, b and c quarks appear as several scaled up versions.

This hypothesis yields surprisingly good fit for meson masses but for some mesons the predicted mass is slightly too high. The reduction of CP₂ mass scale to cure the situation is not possible since top quark mass would become too low. In case of diagonal mesons for which quarks correspond to same p-adic prime, quark contribution to mass squared can be reduced by ordinary color interactions and in the case of non-diagonal mesons one can require that quark contribution is not larger than meson mass.

3. Super-canonical bosons at hadronic space-time sheet can explain the constant contribution to baryonic masses

Quarks explain only a small fraction of the baryon mass and that there is an additional contribution which in a good approximation does not depend on baryon. This contribution should correspond to the non-perturbative aspects of QCD.

A possible identification of this contribution is in terms of super-canonical gluons predicted by TGD. Baryonic space-time sheet with k=107 would contain a many-particle state of super-canonical gluons with net conformal weight of 16 units. This leads to a model of baryons masses in which masses are predicted with an accuracy better than 1 per cent. Super-canonical gluons also provide a possible solution to the spin puzzle of proton. One ends up also to a prediction α_s =α_K=1/4 at hadronic space-time sheet.

Hadronic string model provides a phenomenological description of non-perturbative aspects of QCD and a connection with the hadronic string model indeed emerges. Hadronic string tension is predicted correctly from the additivity of mass squared for J= bound states of super-canonical quanta. If the topological mixing for super-canonical bosons is equal to that for U type quarks then a 3-particle state formed by super-canonical quanta from the first generation and 1 quantum from the second generation would define baryonic ground state with 16 units of conformal weight.

In the case of mesons pion could contain super-canonical boson of first generation preventing the large negative contribution of the color magnetic spin-spin interaction to make pion a tachyon. For heavier bosons super-canonical boson need not to be assumed. The preferred role of pion would relate to the fact that its mass scale is below QCD Λ.

4. Description of color magnetic spin-spin splitting in terms of conformal weight

Conformal weight replaces energy as the basic variable but group theoretical structure of color magnetic contribution to the conformal weight associated with hadronic space-time sheet ($k=107$) is same as in case of energy. The predictions for the masses of mesons are not so good than for baryons, and one might criticize the application of the format of perturbative QCD in an essentially non-perturbative situation.

The comparison of the super-canonical conformal weights associated with spin 0 and spin 1 states and spin 1/2 and spin 3/2 states shows that the different masses of these states could be understood in terms of the super-canonical particle contents of the state correlating with the total quark spin. The resulting model allows excellent predictions also for the meson masses and implies that only pion and kaon can be regarded as Goldstone boson like states. The model based on spin-spin splittings is consistent with the model.

To sum up, the model provides an excellent understanding of baryon and meson masses. This success is highly non-trivial since the fit involves only the integers characterizing the p-adic length scales of quarks and the integers characterizing color magnetic spin-spin splitting plus p-adic thermodynamics and topological mixing for super-canonical gluons. The next challenge would be to predict the correlation of hadron spin with super-canonical particle content in the case of long-lived hadrons.

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

Number theretic braids and flow lines of Kähler magnetic field

The precise identification of the number theoretic braiding has been one of problems in the definition of quantum TGD. The reading of Kea's post mentioning discretized cobordism lead to realize that the identification which I considered first and then gave up is indeed the correct one (for number theoretic braids see this).

The basic point is that in order to achieve quantum classical correspondence, the braiding should be defined by the Chern-Simons action defining the quantum dynamics at parton level. The dual of the induced Kähler form defines a conserved topological current at light-like 3-surface whose flow lines are field lines of the Kähler magnetic field in the light like direction. This current and infinite family of analogous currents plays also a key role in the construction of extremals of Kähler action (see this). Number theoretical braid (tangle if magnetic lines can turn around) would correspond to the unique orbit for the points of the number theoretic braid at the initial partonic 2-surface. The points of the braid would be algebraic only in suitably chosen discrete time slices but this would not lead to a loss of uniqueness. Hence also cobordism can be said to become discrete in number theoretical sense.

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 M⁴_+/-× CP₂ (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 M₁₀₇ 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 CP₂.

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_p², 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.

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

   M²=M₀²J ,

   M₀²≈ .9 GeV².

   A more general formula is M²=kn.

2. 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 M²=M₀²n, M₀²=n₀m₁₀₇². The angular momentum of the building blocks has some spectrum fixed by Virasoro conditions. If the basic building block has angular momentum J₀ and mass squared M₀², one obtains M²= M₀² J, k=M₀², J= nJ₀. The values of n are even in old fashioned string model for a Regge trajectory with a fixed parity. J₀=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.
3. g=1 super-canonical gluons has mass squared

   M₀²= 9m₁₀₇².

   The bound states of super-canonical bosons with net mass squared M₀²= 16m₁₀₇²

   are responsible for the ground state mass of baryons in the model predicting baryon masses with few per cent accuracy. The value of M₀² is .88 GeV² to be compared with its nominal value .9 GeV² so that also hadronic string tension is predicted correctly!

This picture allows also to consider a possible mechanism explaining spin puzzle of proton and I have already earlier considered an explanation in terms of super-canonical spin (see this) assuming that the state is a superposition of ordinary (J=0,J_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.

Progress in the understanding of baryon masses

In the previous posting I explained the progress made in understanding of mesonic masses basically due to the realization how the Chern-Simons coupling k determines Kähler coupling strength and p-adic temperature discussed in still earlier posting.

Today I took a more precise look at the baryonic masses. It the case of scalar mesons quarks give the dominating contribution to the meson mass. This is not true for spin 1/2 baryons and the dominating contribution must have some other origin. The identification of this contribution has remained a challenge for years.

A realization of a simple numerical co-incidence related to the p-adic mass squared unit led to an identification of this contribution in terms of states created by purely bosonic generators of super-canonical algebra and having as a space-time correlate CP₂ type vacuum extremals topologically condensed at k=107 space-time sheet (or having this space-time sheet as field body). Proton and neutron masses are predicted with .5 per cent accuracy and Δ-N mass splitting with .2 per cent accuracy. A further outcome is a possible solution to the spin puzzle of proton.

1. Does k=107 hadronic space-time sheet give the large contribution to baryon mass?

In the sigma model for baryons the dominating contribution to the mass of baryon results as a vacuum expectation value of scalar field and mesons are analogous to Goldstone bosons whose masses are basically due to the masses of light quarks.

This would suggest that k=107 gluonic/hadronic space-time sheet gives a large contribution to the mass squared of baryon. p-Adic thermodynamics allows to expect that the contribution to the mass squared is in good approximation of form

Δm²= nm²(107),

where m²(107) is the minimum possible p-adic mass mass squared and n a positive integer. One has m(107)=2¹⁰m(127)= 2¹⁰m_e5^1/2=233.55 MeV for Y_e=0 favored by the top quark mass.

1. n=11 predicts (m(n),m(p))=(944.5, 939.3) MeV: the actual masses are (m(n),m(p)=(939.6,938.3) MeV. Coulombic repulsion between u quarks could reduce the p-n difference to a realistic value.

2. λ-n mass splitting would be 184.7 MeV for k(s)=111 to be compared with the real difference which is 176.0 MeV. Note however that color magnetic spin-spin splitting requires that the ground state mass squared is larger than 11m₀²(107).

2. What is responsible for the large ground state mass of the baryon?

The observations made above do not leave much room for alternative models. The basic problem is the identification of the large contribution to the mass squared coming from the hadronic space-time sheet with k=107. This contribution could have the energy of color fields as a space-time correlate.

1. The assignment of the energy to the vacuum expectation value of sigma boson does not look very promising since the very of existence sigma boson is questionable and it does not relate naturally to classical color gauge fields. More generally, since no gauge symmetry breaking is involved, the counterpart of Higgs mechanism as a development of a coherent state of scalar bosons does not look like a plausible idea.

2. One can however consider the possibility of Bose-Einstein condensate or of a more general many-particle state of massive bosons possibly carrying color quantum numbers. A many-boson state of exotic bosons at k=107 space-time sheet having net mass squared

   m²=nm₀²(107), n=∑_i n_i

   could explain the baryonic ground state mass. Note that the possible values of n_i are predicted by p-adic thermodynamics with T_p=1.

3. Glueballs cannot be in question

Glueballs (see this and this) define the first candidate for the exotic boson in question. There are however several objections against this idea.

1. QCD predicts that lightest glue-balls consisting of two gluons have J^PC= 0⁺⁺ and 2⁺⁺ and have mass about 1650 MeV. If one takes QCD seriously, one must exclude this option. One can also argue that light glue balls should have been observed long ago and wonder why their Bose-Einstein condensate is not associated with mesons.

2. There are also theoretical objections in TGD framework.
   • Can one really apply p-adic thermodynamics to the bound states of gluons? Even if this is possible, can one assume the p-adic temperature T_p=1 for them if it is quite generally T_p=1/26 for gauge bosons consisting of fermion-antifermion pairs (see this).

   • Baryons are fermions and one can argue that they must correspond to single space-time sheet rather than a pair of positive and negative energy space-time sheets required by the glueball Bose-Einstein condensate realized as wormhole contacts connecting these space-time sheets.

4. Do exotic colored bosons give rise to the ground state mass of baryon?

The objections listed above lead to an identification of bosons responsible for the ground state mass, which looks much more promising.

1. TGD predicts exotic bosons, which can be regarded as super-conformal partners of fermions created by the purely bosonic part of super-canonical algebra, whose generators belong to the representations of the color group and 3-D rotation group but have vanishing electro-weak quantum numbers. Their spin is analogous to orbital angular momentum whereas the spin of ordinary gauge bosons reduces to fermionic spin. Thus an additional bonus is a possible solution to the spin puzzle of proton.

2. Exotic bosons are single-sheeted structures meaning that they correspond to a single wormhole throat associated with a CP₂ type vacuum extremal and would thus be absent in the meson sector as required. T_p=1 would characterize these bosons by super-conformal symmetry. The only contribution to the mass would come from the genus and g=0 state would be massless so that these bosons cannot condense on the ground state unless they suffer topological mixing with higher genera and become massive in this manner. g=1 glueball would have mass squared 9m₀²(k) which is smaller than 11m₀². For a ground state containing two g=1 exotic bosons, one would have ground state mass squared 18m₀² corresponding to (m(n),m(p))=(1160.8,1155.6) MeV. Electromagnetic Coulomb interaction energy can reduce the p-n mass splitting to a realistic value.

3. Color magnetic spin-spin splitting for baryons gives a test for this hypothesis. The splitting of the conformal weight is by group theoretic arguments of the same general form as that of color magnetic energy and given by (m²(N),m²(Δ))= (18m₀²-X, 18m₀²+X) in absence of topological mixing. n=11 for nucleon mass implies X=7 and m(Δ) =5m₀(107)= 1338 MeV to be compared with the actual mass m(Δ)= 1232 MeV. The prediction is too large by about 8.6 per cent. If one allows topological mixing one can have m²=8m₀² instead of 9m₀². This gives m(Δ)=1240 MeV so that the error is only .6 per cent. The mass of topologically mixed exotic boson would be 660.6 MeV and equals m₀²(104). Amusingly k=104 happens to corresponds to the inverse of α_K for gauge bosons.

4. In the simplest situation a two-particle state of these exotic bosons could be responsible for the ground state mass of baryon. Also the baryonic spin puzzle caused by the fact that quarks give only a small contribution to the spin of baryons, could find a natural solution since these bosons could give to the spin of baryon an angular momentum like contribution having nothing to do with the angular momentum of quarks.

5. The large value of the Kähler coupling strength α_K=1/4 would characterize the hadronic space-time sheet as opposed to α_K=1/104 assignable to the gauge boson space-time sheets. This would make the color gauge coupling characterizing their interactions strong. This would be a precise articulation for what the generation of the hadronic space-time sheet in the phase transition to a non-perturbative phase of QCD really means.

6. The identification would also lead to a physical interpretation of super(-conformal) symmetries. It must be emphasized the super-canonical generators do not create ordinary fermions so that ordinary gauge bosons need not have super-conformal partners. One can of course imagine that also ordinary gauge bosons could have super-partners obtained by assuming that one wormhole throat (or both of them) is purely bosonic. If both wormhole throats are purely bosonic Higgs mechanism would leave the state essentially massless unless p-adic thermal stability allows T_p=1. Color confinement could be responsible for the stability. For super-partners having fermion number Higgs mechanism would make this kind of state massive unless the quantum numbers are those of a right handed neutrino.

7. The importance of the result is that it becomes possible to derive general mass formulas also for the baryons of scaled up copies of QCD possibly associated with various Mersenne primes and Gaussian Mersennes. In particular, the mass formulas for "electro-baryons" and "muon-baryons" can be deduced (see this)

For more details about p-adic mass calculations of elementary particle masses see the chapter p-Adic mass calculations: elementary particle masses of the book "p-Adic Length Scale Hierarchy and Dark Matter Hierarchy". The chapter p-Adic mass calculations: hadron masses describes the model for hadronic masses. The chapter p-Adic mass calculations: New Physics explains the new view about Kähler coupling strength.

The model for hadron masses revisited

The blog of Tommaso Dorigo contains two postings which served as a partial stimulus to reconsider the model of hadron masses. The first posting is The top quark mass measured from its production rate and tells about new high precision determination of top quark mass reducing its value to the most probale value 169.1 GeV in allowed interval 164.7-175.5 GeV. Second posting Rumsfeld hadrons tells about "crackpottish" finding that the mass of B_c meson is in an excellent approximation average of the mass of Ψ and Υ mesons. TGD based model for hadron masses allows to understand this finding.

1. Motivations

There were several motivations for looking again the p-adic mass calculations for quarks and hadrons.

1. If one takes seriously the prediction that p-adic temperature is T_p=1 for fermions and T_p=1/26 for gauge bosons as suggested by the considerations of blog posting (see also this), and accepts the picture about fermions as topologically condensed CP₂ type vacuum extremals with single light-like wormhole throat and gauge bosons and Higgs boson as wormhole contacts with two light-like wormhole throats and connecting space-time sheets with opposite time orientation and energy, one is led to the conclusion that although fermions can couple to Higgs, Higgs vacuum expectation value must vanish for fermions. One must check whether it is indeed possible to understand the fermion masses from p-adic thermodynamics without Higgs contribution. This turns out to be the case. This also means that the coupling of fermions to Higgs can be arbitrarily small, which could explain why Higgs has not been detected.

2. There has been some problems in understanding top quark mass in TGD framework. Depending on the selection of p-adic prime p≈ 2^k characterizing top quark the mass is too high or too low by about 15-20 per cent. This problem had a trivial resolution: it was due to a calculational error due to inclusion of only the topological contribution depending on the genus of partonic 2-surface. The positive surprise was that the maximal value for CP₂ mass corresponding to the vanishing of second order correction to electron mass and maximal value of the second order contribution to top mass predicts exactly the recent best value 169.1 GeV of top mass. This in turn allows to clean up uncertainties in the model of hadron masses.

2. The model for hadron masses

The basic assumptions in the model of hadron masses are following.

1. Quarks are characterized by two kinds of masses: current quark masses assignable to free quarks and constituent quark masses assignable to bound state quarks (see this). This can be understood if the integer k_q characterizing the p-adic length scale of quark is different for free quarks and bound quarks so that bound state quarks are much heavier than free quarks. A further generalization is that the value of k can depend on hadron. This leads to an elegant model explaining meson and baryon masses within few percent. The model becomes more precise from the fixing of the CP₂ mass scale from top mass (note that top quark is always free since toponium does not exist). This predicts several copies of various quarks and there is evidence for three copies of top corresponding to the values k_t=95,94,93. Also current quarks u and d can correspond to several values of k.

2. The lowest mass mesonic formula is extremely simple. If the quarks characterized by same p-adic prime, their conformal weights and thus mass squared are additive:

   m²_B = m²_q1+ m²_q2.

   If the p-adic primes labelling quarks are different masses are additive m_B = m_q1+ m_q2.

   This formula generalizes in an obvious manner to the case of baryons.

   Thus apart from effects like color magnetic spin-spin splitting describable p-adically for diagonal mesons and in terms of color magnetic interaction energy in case of nondiagonal mesons, basic effect of binding is modification of the integer k labelling the quark.

3. The formula produces the masses of mesons and also baryons with few per cent accuracy. There are however some exceptions.
   1. The mass of η' meson becomes slightly too large. In case of η' negative color binding conformal weight can reduce the mass. Also mixing with two gluon gluonium can save the situation.

   2. Some light non-diagonal mesons such as K mesons have also slightly too large mass. In this case negative color binding energy can save the situation.

2. An example about how mesonic mass formulas work.

The mass formulas allow to understand why the "crackpottish" mass formula for B_c holds true.

The mass of the B_c meson (bound state of b and c quark and antiquark) has been measured with a precision by CDF (see the blog posting by Tommaso Dorigo) and is found to be M(B_c)=6276.5+/- 4.8 MeV. Dorigo notices that there is a strange "crackpottian" co-incidence involved. Take the masses of the fundamental mesons made of c anti-c (Ψ) and b anti-b (Υ), add them, and divide by two. The value of mass turns out to be 6278.6 MeV, less than one part per mille away from the B_c mass!

The general p-adic mass formulas and the dependence of k_qon hadron explain the co-incidence. The mass of B_c is given as

m(B_c)= m(c,k_c(B_c))+ m(b,k_b(B_c)),

whereas the masses of Ψ and Υ are given by

m( Ψ)= 2^1/2m(c,k_Ψ)

and

m(Υ)= 2^1/2m(b,k_Υ).

Assuming k_c(B_c)= k_c(Ψ) and k_b(B_c)= k_b(Υ) would give m(B_c)= 2^-1/2[m( Ψ)+m( Υ)] which is by a factor 2^1/2 higher than the prediction of the "crackpot" formula. k_c(B_c)= k_c( Ψ)+1 and k_b(B_c)= k_b( Υ)+1 however gives the correct result.

As such the formula makes sense but the one part per mille accuracy must be an accident in TGD framework.

1. The predictions for Ψ and Υ masses are too small by 2 resp. 5 per cent in the model assuming no effective scaling down of CP₂ mass.

2. The formula makes sense if the quarks are effectively free inside hadrons and the only effect of the binding is the change of the mass scale of the quark. This makes sense if the contribution of the color interactions, in particular color magnetic spin-spin splitting, to the heavy meson masses are small enough. Ψ and η_c have spins 1 and 0 and their masses differ by 3.7 per cent (m(η_c)=2980 MeV and m(Ψ)= 3096.9 MeV) so that color magnetic spin-spin splitting is measured using per cent as natural unit.

For more details about p-adic mass calculations of elementary particle masses see the chapter p-Adic mass calculations: elementary particle masses of the book "p-Adic Length Scale Hierarchy and Dark Matter Hierarchy". The chapter p-Adic mass calculations: hadron masses describes the model for hadronic masses.

Still about Witten's new ideas

Witten's paper on 3-D quantum gravity is now in the web. Here is the abstract.

We consider the problem of identifying the CFT's that may be dual to pure gravity in three dimensions with negative cosmological constant. The c-theorem indicates that three-dimensional pure gravity is consistent only at certain values of the coupling constant, and the relation to Chern-Simons gauge theory hints that these may be the values at which the dual CFT can be holomorphically factorized. If so, and one takes at face value the minimum mass of a BTZ black hole, then the energy spectrum of three-dimensional gravity with negative cosmological constant can be determined exactly. At the most negative possible value of the cosmological constant, the dual CFT is very likely the monster theory of Frenkel, Lepowsky, and Meurman. The monster theory may be the first in a discrete series of CFT's that are dual to three-dimensional gravity. The partition function of the second theory in the sequence can be determined on a hyperelliptic Riemann surface of any genus. We also make a similar analysis of supergravity.

1. Key ideas

The key observations are following.

1. In 3-D case Einstein-Hilbert action can be written as Chern-Simons action for gauge group SO(2,2) by combining vielbein connection form with vielbein (this for negative cosmological constant).

2. Although 3-D gravity is trivial dynamically, it allows so called BTZ blackholes for negative cosmological constant. These blackholes have a huge degeneracy of states and the idea is that this degeneracy could correspond to primary fields of QFT defined by Chern-Simons action.

3. The Virasoro algebra decomposes into direct sum of left and right algebras corresponding to left and right movers. If gravitational Chern-Simons action is assumed to vanish, one has left-right symmetry and k=k_L=k_R=k and c=24k. Holomorphic factorization of the partition function into a product occurring for c_L= c_L=24k makes the model calculable and the partition function can be expressed as power series using the well known modular invariant J-function j(q)= 1/q+ 19688q+... appearing in number theory and associated with the Monster group. This allows to identify blackhole states in terms of primary fields belonging to the representations of the Monster group.

There is a critical discussion of by Jacques Distler criticizing the holomorphic factorization assumption as unphysical at the classical limit when k is large. To me the condition k_L=k_R looks very natural. Distler also argues that this kind of approach probably fails if blackholes are actually absent since the condition that there is a gap 0<h<k+1 in the spectrum of states is too strong so that one would be no pure 3-D quantum gravitation in the proposed sense. For k=1 there exists precisely one CFT for which Kac-Moody symmetry is absent so that absence of gravitons is realized in this sense.

2. Objections

Witten considers also some objections against the equivalence of 3-D quantum gravity with Chern-Simons gauge theory.

1. Witten makes a comment about the invertibility of the vielbein as a condition for perturbative well-definedness since in the gauge theory formulation the gauge potential (ω,e/l)=(0,0) (l is the length scale defining cosmological constant) is legitimate whereas the perturbation theory is performed around a solution for which e is invertible (3-metric is non-degenerate). Exactly this would occur in TGD by effective metric 2-dimensionality of lightlike 3-surfaces.

2. Witten also notices that in quantum gravity description using path integral one expects sum over all topologies whereas in gauge theory description no such sum is necessary. In TGD framework where the generalization of S-matrix defines time like entanglement coefficients between positive and negative energy parts of zero energy state there is no need to sum over intermediate topologies.

Witten comments also the case with vanishing cosmological constant which is of special interest from the point of view of TGD. The objection is that since no particles exist there is no S-matrix. The situation changes in TGD framework where lightlike 3-surfaces define particles instead of 2-geometries and the 3-D states define generalized S-matrices as or briefly Matrices as "square roots" of density matrix.

The formula for Kac Moody central charge in absence of gravitational Chern-Simons term is of form k = l/16G, l =sqrt(-1/λ). For λ→0 l and thus k infinite and the theory would become classical. Also c= 24k would diverge at this limit meaning classicality in Virasoro degrees of freedom. The number of "blackhole states" would become infinite. Note however that both curvature scalar and gravitational Chern-Simons term vanish at this limit if the metric corresponds to that of a lightlike surface.

3. Comparison with TGD

It is interesting to look the situation from the TGD point of view since usually this kind of comparisons bring new insights to TGD.

1. I have already told about strong analogies between this theory and quantum TGD as an almost topological QFT for lightlike 3-surfaces as basic objects with extended conformal invariance (see this and this). There are of course also deep differences. In Witten's theory 3-D space-time is not a surface. In TGD one has light-like 3-surfaces which correspond to solutions of Einstein's equations with degenerate metric for vanishing cosmological constant. Thus light-likeness characterizing a property as a 3-surface forces vacuum Einstein's equations characterizing purely geometric property. Dynamical variables in TGD framework correspond to second quantized free induced spinor field and deformations of lightlike 3-surface.

2. In the case of a degenerate metric with vanishing cosmological constant, the vielbein group would reduce to SO(2) and extension by vielbein would reduce to trivial extension. What makes this interesting is that induced Kähler form corresponds to U(1), which strictly speaking does not define a gauge symmetry but spin glass type dynamical symmetry acting as gauge transformations only for vacuum extremals. It is just the classical gravitation which breaks the gauge symmetry character of U(1). This picture conforms nicely with quantum criticality of TGD Universe.

3. Partition function can be determined for hyperelliptic 2-surfaces of any genus (all g<3 surfaces are hyperelliptic). In TGD framework hyper-elliptic surfaces should correspond to fermions and gauge bosons having identification as fermion-antifermion bound states and hyperellipticity implied by the generalization of the imbedding space concept and quantum classical correspondence implies the vanishing of the elementary particle vacuum functionals for g>2 so that only three fermion and gauge boson families are possible.

4. An interesting question is whether one should add to TGD 3-D gravitational action and corresponding Chern-Simons term. The curvature scalar part vanishes identically by the metric 2-dimensionality. For the same reason also Chern-Simons term should vanish, at least in a suitable gauge. If not, gravitational Chern-Simons would remove the vacuum degeneracy crucial for TGD and spoil also the almost topological QFT property of quantum TGD.

4. Does the quantization of Kähler coupling strength reduce to the quantization of Chern-Simons coupling?

Usually this kind of comparisons have yielded new insights also into TGD. This was the case also now. The conjectured quantization of the coupling constant guaranteing so called holomorphic factorization is implied by the integer valuedness of the Chern-Simons coupling strength k. As Witten explains, this follows from the quantization of the first Chern-Simons class for closed 4-manifolds plus the requirement that the phase defined by Chern-Simons action equals to 1 for a boundaryless 4-manifold obtained by gluieing together to 4-manifolds along their boundaries.

The quantization argument seems to generalize to the case of TGD. What is clear that this quantization should closely relate to the quantization of Kähler coupling strength appearing in the 4-D Kähler action defining Kähler function for the world of classical worlds and conjectured to result as a Dirac determinant. The argument leading leading to an extremely simple formula for Kähler coupling strength as α_K =1/4k and allowing to identify p-adic temperature as T_p=1/k with k =26 for bosons and k=1 for fermions, is discussed in separate posting.

Does the quantization of Kähler coupling strength reduce to the quantization of Chern-Simons coupling at partonic level?

Kähler coupling strength associated with Kähler action (Maxwell action for the induced Kähler form) is the only coupling constant parameter in quantum TGD, and its value (or values) is in principle fixed by the condition of quantum criticality since Kähler coupling strength is completely analogous to critical temperature. The quantum TGD at parton level reduces to almost topological QFT for light-like 3-surfaces. This almost TQFT involves Abelian Chern-Simons action for the induced Kähler form.

This raises the question whether the integer valued quantization of the Chern-Simons coupling k could predict the values of the Kähler coupling strength. I considered this kind of possibility already for more than 15 years ago but only the reading of the introduction of the recent paper by Witten about his new approach to 3-D quantum gravity led to the discovery of a childishly simple argument that the inverse of Kähler coupling strength could indeed be proportional to the integer valued Chern-Simons coupling k: 1/α_K=4k if all factors are correct. k=26 is forced by the comparison with some physical input. Also p-adic temperature could be identified as T_p=1/k.

1. Quantization of Chern-Simons coupling strength

For Chern-Simons action the quantization of the coupling constant guaranteing so called holomorphic factorization is implied by the integer valuedness of the Chern-Simons coupling strength k. As Witten explains, this follows from the quantization of the first Chern-Simons class for closed 4-manifolds plus the requirement that the phase defined by Chern-Simons action equals to 1 for a boundaryless 4-manifold obtained by gluing together two 4-manifolds along their boundaries. As explained by Witten in his paper, one can consider also "anyonic" situation in which k has spectrum Z/n² for n-fold covering of the gauge group and in dark matter sector one can consider this kind of quantization.

2. Formula for Kähler coupling strength

The quantization argument for k seems to generalize to the case of TGD. What is clear that this quantization should closely relate to the quantization of the Kähler coupling strength appearing in the 4-D Kähler action defining Kähler function for the world of classical worlds and conjectured to result as a Dirac determinant. The conjecture has been that g_K² has only single value. With some physical input one can make educated guesses about this value. The connection with the quantization of Chern-Simons coupling would however suggest a spectrum of values. This spectrum is easy to guess.

1. The U(1) counterpart of Chern-Simons action is obtained as the analog of the "instanton" density obtained from Maxwell action by replacing J wedge *J with J wedge J. This looks natural since for self dual J associated with CP₂ extremals Maxwell action reduces to instanton density and therefore to Chern-Simons term. Also the interpretation as Chern-Simons action associated with the classical SU(3) color gauge field defined by Killing vector fields of CP₂ and having Abelian holonomy is possible. Note however that instanton density is multiplied by imaginary unit in the action exponential of path integral. One should find justification for this "Wick rotation" not changing the value of coupling strength and later this kind of justification will be proposed.

2. Wick rotation argument suggests the correspondence k/4π = 1/4g_K² between Chern-Simons coupling strength and the Kähler coupling strength g_K appearing in 4-D Kähler action. This would give

   g_K²=π/k .

   The spectrum of 1/α_K would be integer valued

   1/α_K=4k.

   The result is very nice from the point of number theoretic vision since the powers of α_K appearing in perturbative expansions would be rational numbers (ironically, radiative corrections might vanish but this might happen only for these rational values of α_K!).

3. It is interesting to compare the prediction with the experimental constraints on the value of α_K. The basic empirical input is that electroweak U(1) coupling strength reduces to Kähler coupling at electron length scale (see this). This gives α_K= α_U(1)(M₁₂₇)≈ 104.1867, which corresponds to k=26.0467. k=26 would give α_K= 104: the deviation would be only .2 per cent and one would obtain exact prediction for α_U(1)(M₁₂₇)! This would explain why the inverse of the fine structure constant is so near to 137 but not quite. Amusingly, k=26 is the critical space-time dimension of the bosonic string model. Also the conjectured formula for the gravitational constant in terms of α_K and p-adic prime p involves all primes smaller than 26 (see this).

4. Note however that if k is allowed to have values in Z/n², the strongest possible coupling strength is scaled to n²/4 if hbar is not scaled: already for n=2 the resulting perturbative expansion might fail to converge. In the scalings of hbar associated with M⁴ degrees of freedom hbar however scales as 1/n² so that the spectrum of α_K would remain invariant.

3. Justification for Wick rotation

It is not too difficult to believe to the formula 1/α_K =qk, q some rational. q=4 however requires a justification for the Wick rotation bringing the imaginary unit to Chern-Simons action exponential lacking from Kähler function exponential.

In this kind of situation one might hope that an additional symmetry might come in rescue. The guess is that number theoretic vision could justify this symmetry.

1. To see what this symmetry might be consider the generalization of the Montonen-Olive duality obtained by combining theta angle and gauge coupling to single complex number via the formula

   τ= θ/2π+i4π/g².

   What this means in the recent case that for CP₂ type vacuum extremals (see this) Kähler action and instanton term reduce by self duality to Kähler action obtained by the replacement g² with -iτ/4π. The first duality τ→τ+1 corresponds to the periodicity of the theta angle. Second duality τ→-1/τ corresponds to the generalization of Montonen-Olive duality α→ 1/α. These dualities are definitely not symmetries of the theory in the recent case.

2. Despite the failure of dualities, it is interesting to write the formula for τ in the case of Chern-Simons theory assuming g_K²=π/k with k>0 holding true for Kac-Moody representations. What one obtains is

   τ= 4k(1-i).

   The allowed values of τ are integer spaced along a line whose direction angle corresponds to the phase exp(i2π/n), n=4. The transformations τ→ τ+ 4(1-i) generate a dynamical symmetry and as Lorentz transformations define a subgroup of the group E² leaving invariant light-like momentum (this brings in mind quantum criticality!). One should understand what is so special in this line.

3. This formula conforms with the number theoretic vision suggesting that the allowed values of τ belong to an integer spaced lattice. Indeed, if one requires that the phase angles are proportional to vectors with rational components then only phase angles associated with orthogonal triangles with short sides having integer valued lengths m and n are possible. The additional condition that the phase angles correspond to roots of unity! This leaves only m=n and m=-n>0 into consideration so that one would have τ= n(1-i) from k>0.

4. Notice that theta angle is a multiple of 8kπ so that a trivial strong CP breaking results and no QCD axion is needed (this of one takes seriously the equivalence of Kähler action to the classical color YM action).

4. Is p-adicization needed and possible only in 3-D sense?

The action of CP₂ type extremal is given as S=π/8α_K= kπ/2. Therefore the exponent of Kähler action appearing in the vacuum functional would be exp(kπ) known to be a transcendental number (Gelfond's constant). Also its powers are transcendental. If one wants to p-adicize also in 4-D sense, this raises a problem.

Before considering this problem, consider first the 4-D p-adicization more generally.

1. The definition of Kähler action and Kähler function in p-adic case can be obtained only by algebraic continuation from the real case since no satisfactory definition of p-adic definite integral exists. These difficulties are even more serious at the level of configuration space unless algebraic continuation allows to reduce everything to real context. If TGD is integrable theory in the sense that functional integral over 3-surfaces reduces to calculable functional integrals around the maxima of Kähler function, one might dream of achieving the algebraic continuation of real formulas. Note however that for lightlike 3-surface the restriction to a category of algebraic surfaces essential for the re-interpretation of real equations of 3-surface as p-adic equations. It is far from clear whether also preferred extremals of Kähler action have this property.

2. Is 4-D p-adicization the really needed? The extension of light-like partonic 3-surfaces to 4-D space-time surfaces brings in classical dynamical variables necessary for quantum measurement theory. p-Adic physics defines correlates for cognition and intentionality. One can argue that these are not quantum measured in the conventional sense so that 4-D p-adic space-time sheets would not be needed at all. The p-adic variant for the exponent of Chern-Simons action can make sense using a finite-D algebraic extension defined by q=exp(i2π/n) and restricting the allowed lightlike partonic 3-surfaces so that the exponent of Chern-Simons form belongs to this extension of p-adic numbers. This restriction is very natural from the point of view of dark matter hierarchy involving extensions of p-adics by quantum phase q.

If one remains optimistic and wants to p-adicize also in 4-D sense, the transcendental value of the vacuum functional for CP₂ type vacuum extremals poses a problem (not the only one since the p-adic norm of the exponent of Kähler action can become completely unpredictable).

1. One can also consider extending p-adic numbers by introducing exp(π) and its powers and possibly also π. This would make the extension of p-adics infinite-dimensional which does not conform with the basic ideas about cognition. Note that e^p is not p-adic transcendental so that extension of p-adics by powers e is finite-dimensional and if p-adics are first extended by powers of π then further extension by exp(π) is p-dimensional.

2. A more tricky manner to overcome the problem posed by the CP₂ extremals is to notice CP₂ type extremals are necessarily deformed and contain a hole corresponding to the lightlike 3-surface or several of them. This would reduce the value of Kähler action and one could argue that the allowed p-adic deformations are such that the exponent of Kähler action is a p-adic number in a finite extension of p-adics. This option does not look promising.

5. Is the p-adic temperature proportional to the Kähler coupling strength?

Kähler coupling strength would have the same spectrum as p-adic temperature T_p apart from a multiplicative factor. The identification T_p=1/k is indeed very natural since also g_K² is a temperature like parameter. The simplest guess is

T_p= 1/k.

Also gauge couplings strengths are expected to be proportional to g_K² and thus to 1/k apart from a factor characterizing p-adic coupling constant evolution. That all basic parameters of theory would have simple expressions in terms of k would be very nice from the point of view quantum classical correspondence.

If U(1) coupling constant strength at electron length scales equals α_K=1/104, this would give 1/T_p≈ 1/26. This means that photon, graviton, and gluons would be massless in an excellent approximation for say p=M₈₉, which characterizes electroweak gauge bosons receiving their masses from their coupling to Higgs boson. For fermions one has T_p=1 so that fermionic lightlike wormhole throats would correspond to the strongest possible coupling strength α_K=1/4 whereas gauge bosons identified as pairs of light-like wormhole throats associated with wormhole contacts would correspond to α_K=1/104. Perhaps T_p=1/26 is the highest p-adic temperature at which gauge boson wormhole contacts are stable against splitting to fermion-antifermion pair. Fermions and possible exotic bosons created by bosonic generators of super-canonical algebra would correspond to single wormhole throat and could also naturally correspond to the maximal value of p-adic temperature since there is nothing to which they can decay.

A fascinating problem is whether k=26 defines internally consistent conformal field theory and is there something very special in it. Also the thermal stability argument for gauge bosons should be checked.

What could go wrong with this picture? The different value for the fermionic and bosonic α_K makes sense only if the 4-D space-time sheets associated with fermions and bosons can be regarded as disjoint space-time regions. Gauge bosons correspond to wormhole contacts connecting (deformed pieces of CP₂ type extremal) positive and negative energy space-time sheets whereas fermions would correspond to deformed CP₂ type extremal glued to single space-time sheet having either positive or negative energy. These space-time sheets should make contact only in interaction vertices of the generalized Feynman diagrams, where partonic 3-surfaces are glued together along their ends. If this gluing together occurs only in these vertices, fermionic and bosonic space-time sheets are disjoint. For stringy diagrams this picture would fail.

To sum up, the resulting overall vision seems to be internally consistent and is consistent with generalized Feynman graphics, predicts exactly the spectrum of α_K, allows to identify the inverse of p-adic temperature with k, allows to understand the differences between fermionic and bosonic massivation, and reduces Wick rotation to a number theoretic symmetry. One might hope that the additional objections (to be found sooner or later!) could allow to develop a more detailed picture.

For more details see the chapter Is it Possible to Understand Coupling Constant Evolution at Space-Time Level? of "Towards S-matrix".

Schwartschild horizon for a rotating blackhole like object as a 3-D lightlike surface defining wormhole throat

The metric determinant at Schwartschild radius is non-vanishing. This does not quite conform with the interpretation as an analog of a light-like partonic 3-surface identifiable as a wormhole throat for which the determinant of the induced 4-metric vanishes and at which the signature of the induced metric changes from Minkowskian to Euclidian.

An interesting question is what happens if one makes the vacuum extremal representing imbedding of Schwartshild metric a rotating solution by a very simple replacement Φ→ Φ+nΦ, where Φ is the angle angle coordinate of homologically trivial geodesic sphere S² for the simplest vacuum extremals, and Φ the angle coordinate of M⁴ spherical coordinates. It turns out that Schwartschild horizon is transformed to a surface at which det(g₄) vanishes so that the interpretation as a wormhole throat makes sense. If one assumes that black hole horizon is analogous to a wormhole contact, only rotating black hole like structures with quantized angular momentum are possible in TGD Universe.

For details see the chapter TGD and GRT of "Classical Physics in Many-Sheeted Space-time".

Strings 2007 is not only about strings

Peter Woit and Kea commented Witten's talk in Strings 2007 about his new ideas related to 3-D quantum theory of gravity having very little to do with strings.

The little I know about LQG is that it is 3+1-D theory with 3-geometries as basic objects expressed in terms of loop variables. Witten considers 3-D theory with Chern Simons action: in this case 2-geometries would be the basic dynamical objects. Witten himself made clear that he has no idea about how to generalize the theory to 4-D context.

A connection to 4-D theory in TGD framework is obtained if one brings in holography, replaces 3-metrics with light-like 3-surfaces (light-likeness constraint is possible by 4-D general coordinate invariance), and accepts the new view about S-matrix implied by the zero energy ontology.

• Light-like 3-surfaces can be regarded as solutions vacuum Einstein equations with vanishing cosmological constant (Witten considers solutions with non-vanishing cosmological constant). The effective 2-D character of the induced metric is what makes this possible.

• Zero energy ontology is also an essential element: quantum states of 3-D theory in zero energy ontology correspond to generalized S-matrices: Matrix or M-matrix might be a proper term. Matrix is a "complex square root" of density matrix -matrix valued generalization of Schrödinger amplitude - defining time like entanglement coefficients. Its "phase" is unitary matrix and might be rather universal. Matrix is a functor from the category of Feyman cobordisms and matrices have groupoid like structure.

• Theory becomes genuinely 4-D because S-matrix is not universal anymore but characterizes zero energy states.

• 4-D holography is obtained via the Kähler metric of the world of classical worlds assigning to light-like 3-surface a preferred extremal of Kähler action as the analog of Bohr orbit containing 3-D lightlike surfaces as submanifolds (analogs of blackhole horizons and lightlike boundaries). Interiors of 4-D space-time sheets corresponds to zero modes of the metric and to the classical variables of quantum measurement theory (quantum classical correspondence). The conjecture is that Dirac determinant for the modified Dirac action associated with partonic 3-surfaces defines the vacuum functional as the exponent of Kähler function with Kähler coupling strength fixed completely as the analog of critical temperature so that everything reduces to almost topological QFT.

• The counterpart of ordinary S-matrix is between zero energy states. I call it U-matrix. It has nothing to do with particle reactions. It is crucial for understanding consciousness via moment of consciousness as quantum jump identification.

For more details see the chapter Construction of Quantum Theory: S-matrix of "Towards S-matrix".

New Developments in TGD and Their Implications for TGD Inspired Theory of Consciousness

There have been quite an impressive development in the understanding of quantum TGD at the basic level, and the interaction of the new ideas with TGD inspired theory of conciousness and model of quantum biology will be a fascinating adventure. The first task was to write a chapter summarizing the updatings and develop a systematic overall view about the consequences for TGD inspired theory of consciousness. I glue the abstract of the chapter here.

The conflict between the non-determinism of state function reduction and determinism of time evolution of Schrödinger equation is serious enough a problem to motivate the attempt to extend physics to a theory of consciousness by raising the observer from an outsider to a key notion also at the level of physical theory. Further motivations come from the failure of the materialistic and reductionistic dogmas in attempts to understand consciousness in neuroscience context.

There are reasons to doubt that standard quantum physics could be enough to achieve this goal and the new physics predicted by TGD is indeed central in the proposed theory. The developments in quantum TGD during last years have led to a fusion of real and p-adic physics by using generalization of number concept, to the realization of the crucial role of hyper-finite factors of type II₁ for quantum TGD, to the generalization of the imbedding space implying hierarchy of quantized values of Planck constant, to so called zero energy ontology, and to the reduction of quantum TGD to parton level with parton understand as 2-D surface whose orbit is light-like 3-surface, and to the realization that quantum TGD can be formulated as almost topological quantum field theory using category theoretical framework.

These developments have considerably simplified the conceptual framework behind both TGD and TGD inspired theory of consciousness and provided justification for various concepts of consciousness theory deduced earlier from quantum classical correspondence and properties of many-sheeted space-time.

The notions of quantum jump and self can be unified in the recent formulation of TGD relying on dark matter hierarchy characterized by increasing values of Planck constant. Negentropy Maximization Principle serves as a basic variational principle for the dynamics of quantum jump and must be modified to the case of hyper-finite factors of type II₁ The new view about the relation of geometric and subjective time together with zero energy ontology leads to a new view about memory and intentional action. The quantum measurement theory based on finite measurement resolution and realized in terms of hyper-finite factors of type II₁ justifies the notions of sharing of mental images and stereo-consciousness deduced earlier on basis of quantum classical correspondence. A new element is finite resolution of quantum measurement and cognitive and sensory experience. Qualia reduce to quantum number increments associated with quantum jump. Self-referentiality of consciousness can be understood from quantum classical correspondence implying a symbolic representation of contents of consciousness at space-time level updated in each quantum jump. p-Adic physics provides space-time correlates for cognition and intentionality.

For details see the new chapter New Developments in TGD and Their Implications for TGD Inspired Theory of Consciousness.

Declaration of Academic Freedom

The recent crackpot hunting activities have their comic aspects but what looks comedy from a safe distance, is a tragedy when seen from nearby. A public defame literally in a global scale can produce a lot of suffering to their victims and their friends and relatives. The young crackpot hunters seems to be quite blind to this human aspect.

Many western intellectuals accept Physical Integrity as a basic value. For some reason these people however see nothing bad in the violation of what might be called Intellectual Integrity or Emotional Integrity. The events in comment sections of some blogs indeed bring in mind a story about how primitive tribes treated the individuals who broke the taboo: the taboo breaker was literally torn in pieces in a bloody orgy.

The crackpot hunting is of course only a tip of iceberg. The censorship applied by so called respected journals and by electronic archives such as arXiv.org plus academic discrimination prevents very effectively the communitation of new ideas.

There is an organization known as Archive Freedom founded for few years ago by the victims of these activities. It has also electronic archive to which people censored out of Archiv.org and unable to publish in so called respected journals can post their papers.

Kea had added to her blog a piece of the Declaration of Academic Freedom to her blog. I think that this piece of text deserves to be added also here.

Article 2: Who is a scientist

A scientist is any person who does science. Any person who collaborates with a scientist in developing and propounding ideas and data in research or application is also a scientist. The holding of a formal qualification is not a prerequisite for a person to be a scientist.

Article 4: Freedom of choice of research theme

Many scientists working for higher research degrees or in other research programmes at academic institutions such as universities and colleges of advanced study, are prevented from working upon a research theme of their own choice by senior academic and/or administrative officials, not for lack of support facilities but instead because the academic hierarchy and/or other officials simply do not approve of the line of inquiry owing to its potential to upset mainstream dogma, favoured theories, or the funding of other projects that might be discredited by the proposed research. The authority of the orthodox majority is quite often evoked to scuttle a research project so that authority and budgets are not upset. This commonplace practice is a deliberate obstruction to free scientific thought, is unscientific in the extreme, and is criminal. It cannot be tolerated.

A scientist working for any academic institution, authority or agency, is to be completely free as to choice of a research theme, limited only by the material support and intellectual skills able to be offered by the educational institution, agency or authority. If a scientist carries out research as a member of a collaborative group, the research directors and team leaders shall be limited to advisory and consulting roles in relation to choice of a relevant research theme by a scientist in the group.

Article 8: Freedom to publish scientific results:

A deplorable censorship of scientific papers has now become the standard practice of the editorial boards of major journals and electronic archives, and their bands of alleged expert referees. The referees are for the most part protected by anonymity so that an author cannot verify their alleged expertise. Papers are now routinely rejected if the author disagrees with or contradicts preferred theory and the mainstream orthodoxy. Many papers are now rejected automatically by virtue of the appearance in the author list of a particular scientist who has not found favour with the editors, the referees, or other expert censors, without any regard whatsoever for the contents of the paper. There is a blacklisting of dissenting scientists and this list is ommunicated between participating editorial boards. This all amounts to gross bias and a culpable suppression of free thinking, and are to be condemned by the international scientific community.

All scientists shall have the right to present their scientific research results, in whole or in part, at relevant scientific conferences, and to publish the same in printed scientific journals, electronic archives, and any other media. No scientist shall have their papers or reports rejected when submitted for publication in scientific journals, electronic archives, or other media, simply because their work questions current majority opinion, conflicts with the views of an editorial board, undermines the bases of other current or planned research projects by other scientists, is in conflict with any political dogma or religious creed, or the personal opinion of another, and no scientist shall be blacklisted or otherwise censured and prevented from publication by any other person whomsoever. No scientist shall block, modify, or otherwise interfere with the publication of a scientist's work in the promise of any presents or other bribes whatsoever.

P.S. In n-Category Cafe there is a nice posting of David Corfield expressing what science is at best: a spiritual endeavour rather than fight for academic positions.

S-matrix as a functor and the groupoid like structure formed by S-matrices

In zero energy ontology S-matrix can be seen as a functor from the category of Feynman cobordisms to the category of operators. S-matrix can be identified as a "complex square root" of the positive energy density matrix S= ρ^1/2₊S₀, where S₀ is a unitary matrix and ρ₊ is the density matrix for positive energy part of the zero energy state. Obviously one has SS^*=ρ₊. S^*S=ρ_- gives the density matrix for negative energy part of zero energy state. Clearly, S-matrix can be seen as a matrix valued generalization of Schrödinger amplitude. Note that the "indices" of the S-matrices correspond to configuration space spinors (fermions and their bound states giving rise to gauge bosons and gravitons) and to configuration space degrees of freedom (world of classical worlds). For hyper-finite factor of II₁ it is not strictly speaking possible to speak about indices since the matrix elements are traces of the S-matrix multiplied by projection operators to infinite-dimensional subspaces from right and left.

The functor property of S-matrices implies that they form a multiplicative structure analogous but not identical to groupoid. Groupoid has associative product and there exist always right and left inverses and identity in the sense that ff^-1 and f^-1f are defined but not identical in general, and one has fgg^-1=f and f^-1fg= g.

The reason for the groupoid like property is that S-matrix is a map between state spaces associated with initial and final sets of partonic surfaces and these state spaces are different so that inverse must be replaced with right and left inverse. The defining conditions for the groupoid are however replaced with more general ones. Associativity holds also now but the role of inverse is taken by hermitian conjugate. Thus one has the conditions fgg^*=fρ_{g,+} and f^*fg= ρ_f,-g, and the conditions ff^*=ρ₊ and f^*f=ρ_- are satisfied. Here ρ_f+/- is density matrix associated with positive/negative energy parts of zero energy state. If the inverses of the density matrices exist, groupoid axioms hold true since f^-1_L=f^*ρ_f,+^-1 satisfies ff^-1_L= Id₊ and f_R^-1=ρ_f,-^-1f^* satisfies f^-1_Rf= Id_-.

There are good reasons to believe that also tensor product of its appropriate generalization to the analog of co-product makes sense with non-triviality characterizing the interaction between the systems of the tensor product. If so, the S-matrices would form very beautiful mathematical structure bringing in mind the corresponding structures for 2-tangles and N-tangles. Knowing how incredibly powerful the group like structures have been in physics one has good reasons to hope that groupoid like structure might help to deduce a lot of information about the quantum dynamics of TGD.

A word about nomenclature is in order. S has strong associations to unitarity and it might be appropriate to replace S with some other letter. The interpretation of S-matrix as a generalized Schrödinger amplitude would suggest Ψ-matrix. Since the interaction with Kea's M-theory blog (with M denoting Monad or Motif in this context) helped to realize the connection with density matrix, also M-matrix might work. S-matrix as a functor from the category of Feynman cobordisms in turn suggests C or F. Or could just Matrix denoted by M in formulas be enough? Certainly it would inspire feeling of awe but create associations with M-theory in the stringy sense of the word but wouldn't it be fair if stringy M-theory could leave at least some trace to physics;-)!

For details see the chapter Construction of Quantum Theory: S-matrix of "Towards S-matrix".

It is time for crackpot hunting

Summer time seems to be especially active period for young crackpot hunters and the best season has begun. Only a few days ago Lubos revealed that Tommaso Dorigo is a "small crackpot". The scientific justification was that Tommaso had visited several times in Peter Woit's Not-Even-Wrong (Peter Woit is the "black crackpot" to be distinguished from Lee Smolin, the "blue crackpot"). As Tommaso noticed, the fact that Lubos knew about these visits, revealed that also Lubos had visited the same place, from which Lubos himself can make the obvious conclusion and come out of closet. Note that Lubos has also a special message for anyone trying to get to the blog of Lubos from Not-Even-Wrong.

For some time ago I saw commented Sean Carroll's idea that cosmology explains second law. Lubos Motl had already commented the idea and quite correctly pointed out the most obvious flaw in Sean's thinking, namely the failure to realize that cosmological time scale is totally different than the time scale of second law in laboratory environment. Lubos also concluded that Sean Carroll is a crackpot. What I did in the posting Second law and cosmology was some analysis of basic flaws in the thinking of not only Carroll and Lubos but entire community basically due to the refusal to return to roots and concentrate seriously on the fundamental conceptual problems instead of crackpot hunting. I commented the sad situation also from the point of view of particle physics in the posting About landscape: what's the real problem? .

I did not call anyone crackpot since I would regard scientific argumentation based on ad hominem attacks as a criterion number one for crackpot-ness if crackpot hunting were my hobby. Only the contents matter and if someone lacks the substance needed for a serious discussion, he (as a rule he and almost as a rule from US) should listen and learn rather than argue.

Both these young fellows seem to lack the courage and arguments to attack directly against me: this would be also against the politics of total silence about serious competitors of super string hegemony. However, a comment from someone outside Harward, US, and even academy showing a lack of deep unjustified respect towards anything declared by someone from the academic heights of Sean Carrol or Lubos Motl was too much for the vanity of these young bloggers and empire stroke back indirectly this morning. Sean Carroll devoted his posting to the identification criteria of alternative scientists and also Lubos used opportunity and emphasize that "alternative scientist", the characterization that he uses about me in his article to Physicists category of Wikipedia, is his polite expression for a crackpot. Lubos forgot that he himself is very alternative climate scientist.

Amusingly, someone in the comment section of Lubos noticed Lubos had deemed also Carroll as a crackpot in his comments about Carroll's cosmic explanation of second law. Lubos indeed admitted that poor Sean, the discoverer of the new brilliant crackpot identification criteria, becomes the first victim of his own method. If my proposal that ad hominem attacks as a substitute for a scientific argumentation is also basic signature of crackpot-ness is accepted then both these crackpot hunters suffer the same fate. Perhaps scientists using so much valuable time to the hunting of crackpots deserve this.

P.S. Could one consider some kind of mind police of theoretical physics analogous to KGB in Soviet Union, where professionals would take care of crackpot hunting so that super string theorists could concentrate on deducing new predictions from M-theory;-)?

Empirical support for TGD based model of long term memories

Quite recently I learned about empirical support for the TGD inspired model of long term memories. Experiments with mice have shown that loss of long-term memory can be reversed with drugs that seem to trigger the rewiring of brain cells. The findings suggest ways of treating dementia and other neuro-degenerative diseases that are associated with impaired learning and memory loss, says Li-Huei Tsai of the Massachussetts Institute of Technology. Tsai and her team studied mice genetically engineered to express a protein called p25 when their diet contained an antibiotic. The protein has been implicated in some neuro-degenerative diseases.

The mice were first placed in a tank of water and trained to find their way to a platform just below the surface. Next, the team ensured that the task was stored in the mice's long-term memory by waiting for a few weeks. Then they induced the mice to produce p25, leading to loss of neurons, learning ability and memory. When the mice were replaced in an environment rich of various stimuli the memories were restored.

The first interpretation is that the memories are stored in the ordinary sense but not to synaptic contacts but somewhere else, say to RNA inside cell nuclei known now to be coded in large quantities by the intronic portion of DNA (see the recent article in New Scientist). Stable storage of memories in the conventional sense of the word seems however to require that these RNA molecules remain in the nucleus which need not make sense.

TGD based model of long term memories for which zero energy ontology provides a justification, provides an alternative explanation. The basic ideas are following.

• Long term sensory or episodal memories making possible memory feats correspond to either sharing of mental images of the geometric past by time-like entanglement.

• This storage mechanism is not very efficient and a more efficient mechanism would be based on communication with geometric past. Memory recall would be represented as a signal sent from magnetic body at appropriate level of onionlike hierarchy of magnetic bodies to the brain of the geometric past and realized as phase conjugate negative energy dark photons. Memory would be communicated to the geometric future by using analogous positive energy signal. This time mirror mechanism is also the key mechanism of remote metabolism and mechanism of intentional action and explains the findings of Libet about strange time delays of consciousness.

• The problems due to the extremely low photon energies are circumvented if photons are dark and thus correspond to so large value of Planck constant that their energies are above thermal energy at room temperature. EEG represents only a small portion of frequencies involved and corresponds to time scale which is fraction of second. Much longer time scales are involved with what we are used to call long term memories.

The model makes un-necessary explicit storage mechanisms and memories as such are intact in the geometric past apart from the possible changes induced by quantum jumps and only the ability to recall them is lost as a consequence of treatment and it would be this ability which is restored in the stimulus rich environment. This option allows the representation of memories in terms of RNA without requiring the stability of RNA molecules. Also the long microtobuli associated with axons are excellent candidates for providing representations of memories.

In fact, almost any quantum dynamical event of the geometric past in the living body could serve as a memory storage. Nerve pulse patterns would serve only as symbolic "digital" representations of memories. This representation would be much more economic and than the representation as sensory memories localizable at the level of primary sensory organs.

Introns transcribed to RNA inside cell nuclei

The last issue of New Scientist contains an article about the discovery that only roughly one half of DNA expresses itself as aminoacid sequences. The article is published in Nature (thanks for Doug for the link). The Encyclopedia of DNA Elements (ENCODE) project has quantified RNA transcription patterns and found that while the "standard" RNA copy of a gene gets translated into a protein as expected, for each copy of a gene cells also make RNA copies of many other sections of DNA. In particular, intron portions ("junk DNA", the portion of which increases as one climbs up in evolutionary hierarchy) are transcribed to RNA in large amounts. What is also interesting that the RNA fragments correspond to pieces from several genes which raises the question whether there is some fundamental unit smaller than gene.

In particular, intron portions ("junk DNA", the portion of which increases as one climbs up in evolutionary hierarchy) are transcribed to RNA in large amounts. What is also interesting that the RNA fragments correspond to pieces from several genes which raises the question whether there is some fundamental unit smaller than gene.

None of the extra RNA fragments gets translated into proteins, so the race is on to discover just what their function is. TGD proposal is that it gets braided and performs a lot of topological quantum computation (see this). Topologically quantum computing RNA fits nicely with replicating number theoretic braids associated with light-like orbits of partonic 2-surfaces and with their spatial "printed text" representations as linked and knotted partonic 2-surfaces giving braids as a special case (see this). An interesting question is how printing and reading could take place. Is it something comparable to what occurs when we read consciously? Is the biological portion of our conscious life identifiable with this reading process accompanied by copying by cell replication and as secondary printing using aminoacid sequences?

This picture conforms with TGD view about pre-biotic evolution. Plasmoids [1], which are known to share many basic characteristics assigned with life, came first: high temperatures are not a problem in TGD Universe since given frequency corresponds to energy above thermal energy for large enough value of hbar. Plasmoids were followed by RNA, and DNA and aminoacid sequences emerged only after the fusion of 1- and 2-letter codes fusing to the recent 3-letter code. The cross like structure of tRNA molecules carries clear signatures supporting this vision. RNA would be still responsible for roughly half of intracellular life and perhaps for the core of "intelligent life".

I have also proposed that this expression uses memetic code which would correspond to Mersenne M₁₂₇=2¹²⁷-1 with 2 ¹²⁶ codons whereas ordinary genetic code would correspond to M₇=2⁷-1 with 2⁶ codons. Memetic codons in DNA representations would consist of sequences of 21 ordinary codons. Also representations in terms of field patterns with duration of .1 seconds (secondary p-adic time scale associated with M ₁₂₇ defining a fundamental biorhythm) can be considered.

A hypothesis worth of killing would be that the DNA coding for RNA has memetic codons scattered around genome as basic units. It is interesting to see whether the structure of DNA could give any hints that memetic codon appears as a basic unit.

1. In a "relaxed" double-helical segment of DNA, the two strands twist around the helical axis once every 10.4 base pairs of sequence. 21 genetic codons correspond 63 base pairs whereas 6 full twists would correspond to 62.4 base pairs.

2. Nucleosomes are fundamental repeating units in eukaryotic chromatin possessing what is known as 10 nm beads-on-string structure. They repeat roughly every 200 base pairs: integer number of genetic codons would suggest 201 base pairs. 3 memetic codons makes 189 base pairs. Could this mean that only a fraction p≈ 12/201, which happens to be of same order of magnitude as the portion of introns in human genome, consists of ordinary codons? Inside nucleosomes the distance between neighboring contacts between histone and DNA is about 10 nm, the p-adic length scale L(151) associated with the Gaussian Mersenne (1+i)¹⁵¹-1 characterizing also cell membrane thickness and the size of nucleosomes. This length corresponds to 10 codons so that there would be two contacts per single memetic codon in a reasonable approximation. In the example of Wikipedia nucleosome corresponds to about 146=126+20 base pairs: 147 base pairs would make 2 memetic codons and 7 genetic codons.

   The remaining 54 base pairs between histone units + 3 ordinary codons from histone unit would make single memetic codon. That only single memetic codon is between histone units and part of the memetic codon overlaps with histone containing unit conforms with the finding that chromatin accessibility and histone modification patterns are highly predictive of both the presence and activity of transcription start sites. This would leave 4 genetic codons and 201 base pairs could decompose as memetic codon+2 genetic codons+memetic codon+2 genetic codons. The simplest possibility is however that memetic codons are between histone units and histone units consist of genetic codons. Note that memetic codons could be transcribed without the straightening of histone unit occurring during the transcription leading to protein coding. Note that prokaryote genome lacks the histone units so that the transition from prokaryotes to eukaryotes would mean the emergence of memetic code.

[1] E. Lozneanu and M. Sanduloviciu (2003), Minimal-cell system created in laboratory by self-organization, Chaos, Solitons and Fractals, Volume 18, Issue 2, October, p. 335. See also Plasma blobs hint at new form of life, New Scientist vol. 179 issue 2413 - 20 September 2003, page 16.

For background see the chapter Topological Quantum Computation in TGD Universe of "TGD as a Generalized Number Theory" and the chapter Pre-biotic Evolution in Many-Sheeted Space-Time of "Genes and Memes".

In what sense tangles are realized in TGD Universe?

Kea gave a link to a highly interesting article of Kauffman and Lambropoulou about rational 2-tangles having commutative sum and product allowing to map them to rationals. The illustrations of the article are beautiful and make it easy to get the gist of various ideas. The theorem of the article states that equivalent rational tangles giving trivial tangle in the product correspond to subsequent Farey numbers a/b and c/d satisfying ad-bc=+/-1 so that the pair defines element of the modular group SL(2,Z).

1. The basic observation is that 2-tangles are 2-tangles in both "s- and t-channels". Product and sum can be defined for all tangles but only in the case of 2-tangles the sum, which in this case reduces to product in t-channel obtained by putting tangles in series, gives 2-tangle. The sum of M- and N-tangles is M+N-2-tangle and combines various N-tangles to a monoidal structure. Tensor product like operation giving M+N tangle looks to me physically more natural than the sum.

2. The reason why general 2-tangles are non-commutative although 2-braids obviously commute is that 2-tangles can be regarded as sequences of N-tangles with 2-tangles appearing only as the initial and final state: N is actually even for intermediate states. Since N>2-braid groups are non-commutative, non-commutativity results. It would be interesting to know whether braid group representations have been used to construct representations of N-tangles.

The article stimulated the question in what senses N-tangles could be obtained in TGD framework.

1. Tangles as number theoretic braids?

The strands of number theoretical N-braids correspond to roots of N:th order polynomial and if one allows time evolutions of partonic 2-surface leading to the disappearance or appearance of real roots N-tangles become possible. This however means continuous evolution of roots so that the coefficients of polynomials defining the partonic 2-surface can be rational only in initial and final state but not in all intermediate "virtual" states.

2. Tangles as tangled partonic 2-surfaces?

Tangles could appear in TGD also in second manner.

• Partonic 2-surfaces are sub-manifolds of a 3-D section of space-time surface. If partonic 2-surfaces have genus g>0 the handles can become knotted and linked and one obtains besides ordinary knots and links more general knots and links in which circle is replaced by figure eight and its generalizations obtained by adding more circles (eyeglasses for N-eyed creatures;-)).

• Since these 2-surfaces are space-like, the resulting structures are indeed tangles rather than only braids. Tangles made of strands with fixed ends would result by allowing spherical partons elongate to long strands with fixed ends. DNA tangles would the basic example, and are discussed also in the article. DNA sequences to which I have speculatively assigned invisible (dark) braid structures might be seen in this context as space-like "written language representations" of genetic programs represented as number theoretic braids.

For details see the chapter Hyper-Finite Factors and Construction of S-Matrix of "Towards S-Matrix".

Farey sequences, Riemann Hypothesis, and Platonia as the best possible world

Kea has mentioned Farey sequences in her blog couple of times (see this and this).

Some basic facts about Farey sequences demonstrate that they are very interesting also from TGD point of view.

1. Farey sequence F_N is defined as the set of rationals 0< q= m/n≤1 satisfying the conditions n≤ N ordered in an increasing sequence.

2. Two subsequent terms a/b and c/d in F_N satisfy the condition ad-bc=1 and thus define and element of the modular group SL(2,Z).

3. The number of terms in Farey sequence is given by

   F(N) =F(N-1)+ φ(N-1).

   Here φ(n) is Euler's totient function giving the number of divisors of n. For primes one has φ(p)=1 so that in the transition from p to p+1 the length of Farey sequence increases by one unit by the addition of q=1/(p+1) to the sequence.

The members of Farey sequence F_N are in one-one correspondence with the set of quantum phases q_n=exp(i2π/n) , 0≤ n≤ N. This suggests a close connection with the hierarchy of Jones inclusions, quantum groups, and in TGD context with quantum measurement theory with finite measurement resolutiona nd the hierarchy of Planck constants involving the generalization of imbedding space. Also the recent TGD inspired ideas about the hierachy of subgroups of rational modular group with subgroups labelled by N and in direct correspondence with the hierarchy of quantum critical phases would naturally relate to Farey sequence.

1. Riemann Hypothesis and Farey sequences

Farey sequences are used in two equivalent formulations of the Riemann hypothesis. Suppose the terms of F_N are a_n,N, 0 < n≤ m_N. Define d_n,N = a_n,N - n /m_N: in other words d_n,N is the difference between the n:th term of the N:th Farey sequence, and the n:th member of a set of the same number of points, distributed evenly on the unit interval. Franel and Landau proved that both of the two statements

• ∑_n=1,...,mNd_n,N =O(N^r) for any r>1/2.

• ∑_n=1,...,mN d_n,N² =O(N^r) for any r>1.

are equivalent with Riemann hypothesis.

One can say that RH would guarantee that the numbers of Farey sequence provide the best possible approximate representation for the evenly distributed rational numbers n/m_N.

2. Farey sequences and TGD

Farey sequences seem to relate very closely to TGD.

1. The rationals in the Farey sequence can be mapped to the roots of unity by the map q→exp(i2π q). The numbers 1/m_N are in turn mapped to the numbers exp(i2π/m_N), which are also roots of unity. The statement would be that the algebraic phases defined by Farey sequence give the best possible approximate representation for the phases exp(in2π/m_N) with evenly distributed phase angle.

2. In TGD framework the phase factors defined by F_N corresponds to the set of quantum phases corresponding to Jones inclusions labelled by q=exp(i2π/n), n≤ N, and thus to the N lowest levels of dark matter hierarchy. There are actually two hierarchies corresponding to M⁴ and CP₂ degrees of freedom and the Planck constant appearing in Schrödinger equation corresponds to the ratio n_a/n_b defining quantum phases in these degrees of freedom. Z_{na× nb} appears as a conformal symmetry of "dark" partonic 2-surfaces and with very general assumptions this implies that there are only three fermion families in TGD Universe.

3. The fusion of physics associated with various number fields to single coherent whole requires algebraic universality. In particular, the roots of unity, which are complex algebraic numbers, should define approximations to continuum of phase factors. At least the S-matrix associated with p-adic-to-real transitions and more generally p₁ → p₂ transitions between states for which the partonic space-time sheets are p₁- resp. p₂-adic can involve only this kind of algebraic phases. One can also say that cognitive representations can involve only algebraic phases and algebraic numbers in general. For real-to-real transitions and real-to-padic transitions U-matrix might be non-algebraic or obtained by analytic continuation of algebraic U-matrix. S-matrix is by definition diagonal with respect to number field and similar continuation principle might apply also in this case.

4. The subgroups of the hierarchy of subgroups of the modular group with rational matrix elements are labelled by integer N and relate naturally to the hierarchy of Farey sequences. The hierarchy of quantum critical phases is labelled by integers N with quantum phase transitions occuring only between phases for which the smaller integer divides the larger one.

5. The 2-tangles known as rational tangles form are characterized by a rational number a/b (for detailed definitions see the article of Kaufmann and Lambropoulou). According to the result of the same article, two rational tangles labelled by a/b and c/d and possessing commutative sum and product combine to form an unknot if and only if a/b and c/d are two subsequent Farey numbers and therefore satisfy ad-bc=+/-1. An interesting question is whether the result somehow generalizes to the case of N-tangles and whether this generalization relates to the hierarchy of subgroups of the rational modular group obtained by replacing the generator τ→τ+1 with τ→ τ+1/N.

3. Interpretation of RH in TGD framework

Number theoretic universality of physics suggests an interpretation for the Riemann hypothesis in TGD framework. RH would be equivalent to the statement that the Farey numbers provide best possible approximation to the set of rationals k/m_N. Or to the statement that the roots of unity contained by F_N define the best possible approximation for the roots of unity defined as exp(ik2π/m_N) with evenly spaced phase angles. The roots of unity allowed by the lowest N levels of the hierarchy of Jones inclusions allows the best possible approximate representation for algebraic phases represented exactly at m_N:th level of hierarchy.

A stronger statement would be that the Platonia where RH holds true would be the best possible world in the sense that algebraic physics behind the cognitive representations would allow the best possible approximation hierarchy for the continuum physics (both for numbers in unit interval and for phases on unit circle). Platonia with RH would be cognitive paradise;-).

One could see this also from different view point. "Platonia as the cognitively best possible world" could be taken as the "axiom of all axioms": a kind of fundamental variational principle of mathematics. Among other things it would allow to conclude that RH is true: RH must hold true either as a theorem following from some axiomatics or as an axiom in itself.

For details see the chapter Hyper-Finite Factors and Construction of S-Matrix of "Towards S-Matrix".