How detailed the quantum classical correspondence can be?

Can the dynamics defined by preferred extremals of Kähler action be dissipative in some sense? The generation of the arrow of time has a nice realization in zero energy ontology as a choice of well-defined particle numbers and other quantum numbers at the "lower" end of CD. By quantum classical correspondence this should have a space-time correlate. Gradient dynamics is a highly phenomenological realization of the dissipative dynamics and one must try to identify a microscopic variant of dissipation in terms of entropy growth of some kind. If the arrow of time and dissipation has space-time correlate, there are hopes about the identification of this kind of correlate.

Quantum classical correspondence has been perhaps the most useful guiding principle in the construction of quantum TGD. What is says that not only quantum numbers but also quantum jump sequences should have space-time correlates: about this the failure of strict determinism of Kähler action gives good hopes. Even the quantum superposition - at least for certain situations - might have space-time correlates.

1. Measurement interaction term in the modified Dirac action at the upper end of CD indeed defines a coupling to the classical dynamics kenociteallb/Dirac in a very delicate manner. This kind of measurement interaction is indeed basic element of quantum TGD. Also the color and charges and angular momentum associated with the Hamiltonians at point of braids could couple to the dynamics via the boundary conditions.

2. The braid strand with a given Hamiltonian could obey Hamiltonian equations of motion: this would give rise to a skeleton of space-time defined by braid strands possibly continued to string world sheets and would provided different realization of quantum classical correspondence. Symplectic tringulation suggests by the symplectic QFT proposed to describe physics in zero modes would add to the skeleton edges connecting string ends continued to 2-D sheets in the interior of space-time.

3. Quantum TGD can be regarded as a square root of thermodynamics in well-defined sense. Could it be possible to couple the Hermitian square root of density matrix appearing in M-matric and characterizing zero energy state thermally to the geometry of space-time sheets by coupling it to the classical dynamical via boundary conditions depending on its eigenvalues? The necessity to choose single eigenvalue spoils the attempt and one obtains only a representation for single measurement outcome. It seems that one can achieve only a representation of the ensemble at space-time level consisting of space-time sheets representing various outcomes of measurement. This ensemble would be realized as ensemble of sub-CDs for a given CD.

4. One can pose even more ambigious question: could quantum superposition of WCW spinor fields have a space-time correlate in the sense that all space-time surfaces in the superposition would carry information about the superposition itself? Obviously this would mean self-referentiality via quantum-classical feedback.

The following discussion concentrates on possible space-time correlates for the quantum superposition of WCW spinor fields and for the arrow of time.

1. It seems difficult to imagine space-time correlate for the quantum superposition of final states with varying quantum numbers since these states correspond to quantum superpositions of different space-time surfaces. How could one code information about quantum superposition of space-time surfaces to the space-time surfaces appearing in the superposition? This kind of self-referentiality seems to be necessary if one requires that various quantum numbers characterizing the superposition (say momentum) couple via boundary conditions to the space-time dynamics.

2. The failure of non-determinism of quantum dynamics is behind dissipation and strict determinism fails for Kähler action. This gives hopes that the dynamics induces also arrow of time. Energy non-conservation is of course excluded and one should be able to identify a measure of entropy and the analog of second law of thermodynamics telling what happens at for preferred extremals when the situation becomes non-deterministic. The vertices of generalized Feynman graphs are natural places were non-determinism emerges as are also sub-CDs. Naive physical intuition would suggest that dissipation means generation of entropy: the vertices would favor decay of particles rather than their spontaneous assembly. The analog of blackhole entropy assignable to partonic 2-surfaces might allow to characterize this quantatively. The symplectic area of partonic 2-surface could be a symplectic invariant of this kind.

3. Could the mysterious branching of partonic 2-surfaces -obviously analogous to even more mysterious branching of quantum state in many worlds interpretation of quantum mechanics- assigned to the multivalued character of the correspondence between canonical momentum densities and time derivatives of H coordinates allow to understand how the arrow of time is represented at space-time level? Recall that this brancing is what implies the effective hierarchy of Planck constants as integer multiples of its minimal value absolutely crucial for the application of TGD in biology and consciousness and to the understanding of dark matter as large hbar phases
   1. This branching would effectively replace CD with its singular covering with number of branches dependin on space-time region. The relative homology with respect to the upper boundary of CD (so that the branches of the trees would effectively meet there) could define the analog of Floer homology with various paths defined by the orbits of partonic 2-surfaces along lines of generalize Feynman diagram defining the first homology group. Typically tree like structures would be involved with the ends of the tree at the upper boundary of CD effectively identified.

   2. This branching could serve as a representation for the branching of quantum state to a superposition of eigenstates of measured quantum observables. If this is the case, the various branches to which partonic 2-surface decays at partonic 2-surface would more or less relate to quantum superposition of final states in particle reaction. The number of branches would be finite by finite measurement resolution. For a given choice of the arrow of geometric time the partonic surface would not fuse back at the upper end of CD.

   3. Rather paradoxically, the space-time correlate for the dissipation would reduce the dissipation by increasing the effective value of hbar: the interpretation would be however in terms of dark matter identified in terms of large hbar phase. In the same manner dissipation would be accompanied by evolution since the increase of hbar naturally implies formation of macroscopically quantum coherent states. The space-time representation of dissipation would compensate the increase of entropy at the ensemble level.

   4. The geometric representation of quantum superposition might take place only in the intersection of real and p-adic worlds and have interpretation in terms of cognitive representations. In the intersection one can also have a generalization of second law kenociteallb/nmpc in which the generation of genuine negentropy in some space-time regions via the build up of cognitive representation compensated by the generation of entropy at other space-time regions. The entropy generating behavior of living matter conforms with this modification of the second law. The negentropy measure in question relies on the replacement of logarithms of probabilities with logarithms of their p-adic norms and works for rational probabilities and also their algebraic variants for finite-dimensional algebraic extensions of rationals.

   5. Each state in the superposition of WCW quantum states would contain this representation as its space-time correlate realizing self-referentiality at quantum level in the intersection of real and p-adic worlds. Also the state function reduced members of ensemble could contain this cognitive representation at space-time level. Essentially quantum memory making possible self-referential linguistic representation of quantum state in terms of space-time geometry and topology would be in question. The formulas written by mathematicians would define similar map from quantum level to the space-time level making possible to "see" one's thoughts.

For more details see the new chapter Infinite Primes and Motives of "Physics as Generalized Number Theory" or the article with same title.

Floer homology and TGD

TGD can be seen as almost topological quantum field theory. This could have served as a motivation for spending most of last months to the attempt to learn some of the mathematics related to various kind of homologies and cohomologies. The decisive stimulus came from the attempt to understand the basic ideas of motivic cohomology. I am not a specialist and do not have any ambition or abilities to become such. My goals is to see whether these ideas could be applied in quantum TGD.

Documentation is the best manner to develop ideas and the learning process has materialized as a new chapter entitled Infinite Primes and Motives of "Physics as Generalized Number Theory". It soon became clear that much of the mathematics needed by TGD has existed for decades and developing all the time. The difficult task is to understand the essentials of this mathematics and translate to the language that I talk and understand. Also generalization is unavoidable. Those who think that new physics can be done by taking math as such are wasting their time.

Among another things I have been learning about various cohomologies and homologies - about quantum cohomology, about Floer homology and topological string theories, about Gromov-Witten invariants,... It would be very naive to think that these notions would work as such in TGD framework. It looks however very plausible that the their generalizations to TGD exist, and could be very useful in the more detailed formulation of quantum TGD. The crucially important notion is finite measurement resolution making everything almost topological and highly number theoretic. In this brain-stormy spirit I have even become a proud father of my own pet homology, which I have christened as braided Galois homology. It is based on the correspondence between infinite primes and polynomials of several variables and is formulated in braided group algebras with braidings realized as symplectic flows and generalizing somewhat the usual notion of homology meaning that the square of boundary operation gives something in commutator group reducing to unit element of ordinary homology only in the factor group obtained by dividing with the commutator group.

Floer homology in its original form replaces Morse function in symplectic manifold M in the loop space LM of M. The loops can be seen as homotopies of Hamiltonians and paths in loops space describe cylinders in M. With an appropriate choice of symplectic action these cylinderes can be regarded as (pseudo-)holomorphic surface completely analogous to string orbits. By combining Floer's theory with Witten's discovery about the connection between the Morse theory and supersymmetry one ends up with topological QFTs as a manner to formulate Floer homology and various variants of this notion- in particular topological QFTs characterizing topology of three-manifolds.

This kind of learning periods are very useful as a rule since they allow to improve bird's eye of view about TGD and its problems. The understanding of both quantum TGD and its classical counterpart is still far from from comprehensive.

For instance, the view about the physical and mathematical roles of Kähler actions for Euclidian and Minkowskian space-time regions is far from clear. Do they provide dual descriptions as suggested or are both needed? Kähler action for preferred extremal in Euclidian regions defines naturally positive definite Kähler function. But can one regard the Kähler action in Minkowskian regions as equivalent definition for Kähler function or should one regard it as imaginary as the presence of square root of metric determinant would suggest? What could be the interpretation in this case? The basic ideas about Floer homology suggest and answer to these questions.

1. Since quantum fluctuating WCW degrees of freedom correspond to a symmetric space assignable to the symplectic group in TGD framework symplectic geometry is of special interest from TGD point of view. Floer homology is indeed about symplectic geometry as also Gromov-Witten invariants and topological string theories developed for the purpose of calculating these invariants. Hence the question whether Floer homology could have a generalization to TGD framework is highly relevant.

2. As such Floer homology for M⁴× CP₂ is deadly boring since it reduces to ordinary singular homology. The correspondence between canonical momentum densities of Kähler action and time derivatives of imbedding space coordinates is however one-to-many- and inspires the replacement of the imbedding space with its singular covering with different space-time regions corresponding to different number of sheets for the covering. The effective hierarchy of Planck constants emerges as a result. The homology in WCW could be mapped to the homology of this structure just as the homology of loop space of M is mapped to that of M in Floer theory.

3. The obvious question is how to generalize Floer homology to TGD framework and the obvious guess is that Kähler action for preferred extremals must take the role of symplectic action for pseudo-holomorphic surfaces which could in fact be replaced with hyper-quaternionic space-time surfaces containing string world sheets whose ends defined braid strands carrying quantum numbers and which intersect partonic 2-surfaces at the future and past light-like boundaries of CDs. This actually suggests an obvious generalization for quantum cohomology based on quantal notion of intersection: partonic surfaces intersect if there exist a string world sheets connecting them. Fuzzy intersection has interpretation in terms of causal dependence: by effective 2-dimensionality this causal dependence is along light-like 3-surfaces and along space-like 3-surfaces at the boundaries of CDs. The notion of quantum intersection is so beautiful that one an almost forgive for the theoricians who have begun to take seriously the idea about branes connected by strings.

4. The question providing the new insight is simple. Could Kähler function allow to define Morse theory? The answer is negative. Kähler metric must be positive definite so that the Hessian associated with it in quantum fluctuating degrees of freedom must have positive signature: no saddle points are possible in quantum fluctuating degrees of freedom although in zero modes they are allowed. Second counter argument is that quantum Morse theory is based on path integral rather than functional integral.

   How could one circumvent this difficulty? Could Kähler action in Minkowskian regions- naturally imaginary by negative sign of metric determinant- give an imaginary contribution to the vacuum functional and define Morse function so that both Kähler and Morse would find a prominent role in the world order of TGD? Maybe! The presence of Kähler function and Morse function in the vacuum functional would give much more direct connection with the path integral approach and Kähler function would also make path integral well-defined since one integrates only over preferred extremals of Kähler action for which Kähler action reduces to Chern-Simons term coming from Minkowskian region and contribution from Euclidian region (generalized Feynman graph).

Should one assume that the reduction to Chern-Simons terms occurs for the preferred extremals in both Minkowskian and Euclidian regions or only in Minkowskian regions?

1. All arguments for this have been represented for Minkowskian regions involve local light-like momentum direction which does not make sense in the Euclidian regions. This does not however kill the argument: one can have non-trivial solutions of Laplacian equation in the region of CP₂ bounded by wormhole throats: for CP₂ itself only covariantly constant right-handed neutrino represents this kind of solution and at the same time supersymmetry. In the general case solutions of Laplacian represent broken super-symmetries and should be in one-one correspondences with the solutions of the modified Dirac equation. The interpretation for the counterparts of momentum and polarization would be in terms of classical representation of color quantum numbers.

   If the reduction occurs in Euclidian regions, it gives in the case of CP₂ two 3-D terms corresponding to two 3-D gluing regions for three coordinate patches needed to define coordinates and spinor connection for CP₂ so that one would have two Chern-Simons terms. Without any other contributions the first term would be identical with that from Minkowskian region apart from imaginary unit. Second Chern-Simons term would be however independent of this. For wormhole contacts the two terms could be assigned with opposite wormhole throats and would be identical with their Minkowskian cousins from imaginary unit. This looks a little bit strange.

2. There is however a very delicate issue involved. Quantum classical correspondence requires that the quantum numbers of partonic states must be coded to the space-time geometry, and this is achieved by adding to the action a measurement interaction term which reduces to what is almost a gauge term present only in Chern-Simons-Dirac equation but not at space-time interior. This term would represent a coupling to Poincare quantum numbers at the Minkowskian side and to color and electro-weak quantum numbers at CP₂ side. Therefore the net Chern-Simons contributions and would be different.

3. There is also a very beautiful argument stating that Dirac determinant for Chern-Simons-Dirac action equals to Kähler function, which would be lost if Euclidian regions would not obey holography. The argument obviously generalizes and applies to both Morse and Kähler function.

In any case, it is still too early to give up the possibility that these two parts of Kähler action (real and positive- imaginary) provide dual descriptions as functional integral and path integral: Wick rotations is what comes in mind. Certainly, the rigorous definition of the path integral would be as difficult -should one say hopeless- as in ordinary QFT.

Floer homology and Gromov-Witten invariants provide also other insights about quantum TGD. For more details see the new chapter Infinite Primes and Motives or the article with same title.

Black holes at LHC? Or maybe just scaled up bottonium?

The latest Tommaso Dorigo's posting has a rather provocative title: The Plot Of The Week - A Black Hole Candidate. Some theories inspired by string theories predict micro black holes at LHC. Micro blackholes have been proposed as explanation for certain exotic cosmic ray events such as Centauros, which however seem to have standard physics explanation.

Without being a specialist one could expect that evaporating black hole would be in many respects analogous to quark gluon plasma phase decaying to elementary particles producing jets. Or any particle like system, which has forgot all information about colliding particles which created it- say the information about the scattering plane of partons leading to the jets as a final state and reflecting itself as the coplanarity of the jets. If the information about initial state is lost, one would expect more or less spherical jet distribution. The variable used as in the study is sum of transverse energies for jets emerging from same point and having at least 50 GeV transverse energy. QCD predicts that this kind of events should be rather scarce and if they are present, one can seriously consider the possibility of new physics.

The LHC document containing the sensational proposal is titled Search for Black Holes in pp collisions at sqrt(s) = 7 TeV and has the following abstract:

An update on a search for microscopic black hole production in pp collisions at a center-of-mass energy of 7 TeV by the CMS experiment at the LHC is presented using a 2011 data sample corresponding to an integrated luminosity of 190 pb⁻¹. This corresponds to a six-fold increase in statistics compared to the original search based on 2010 data. Events with large total transverse energy have been analyzed for the presence of multiple energetic jets, leptons, and photons, typical of a signal from an evaporating black hole. A good agreement with the expected standard model backgrounds, dominated by QCD multijet production, has been observed for various multiplicities of the final state. Stringent model-independent limits on new physics production in high-multiplicity energetic final states have been set, along with model-specific lim- its on semi-classical black hole masses in the 4-5 TeV range for a variety of model parameters. This update extends substantially the sensitivity of the 2010 analysis.

The abstract would suggest that nothing special has been found but in sharp contrast with this the article mentions black hole candidate decaying to 10 jets with total transverse energy S_T. The event is illustrated in the figure 3 of the article. The large number of jets emanating from single point would suggest a single object decaying producing the jets.

Personally I cannot take black holes as an explanation of the event seriously. What can I offer instead? p-Adic mass calculations rely on p-adic thermodynamics and this inspires obvious questions. What p-adic cooling and heating processes could mean? Can one speak about p-adic hot spots? What p-adic overheating and over-cooling could mean? Could the octaves of pions and possibly other mesons explaining several anomalous findings including CDF bump correspond to unstable over-heated hadrons for which the p-adic prime near power of two is smaller than normally and p-adic mass scale is correspondingly scaled up by a power of two?

The best manner to learn is by excluding various alternative explanations for the 10 jet event.

1. M₈₉ variants of QCD jets are excluded both because their production requires higher energies and because their number would be small. The first QCD three-jets were observed around 1979. q-qbar-g three-jet was in question and it was detected in e⁺ e^- collision with cm energy about 7 GeV. The naive scaling by factor 512 would suggest that something like 5.6 TeV cm energy is needed to observed M₈₉ parton jets. The recent energy is 7 TeV so that there are hopes of observing M₈₉ three- jets in decays of heavy M₈₉. For instance, the decays of charmonium and bottonium of M₈₉ physics to three gluons or two-gluons and photon would create three-jets.

2. Ordinary quark gluon plasma is excluded since in a sufficiently large volume of quark gluon plasma so called jet quenching occurs so that jets have small transverse energies. This would be due to the dissipation of energy in the dense quark gluon plasma. Also ordinary QCD jets are predicted to be rare at these transverse energies: this is of course the very idea of how black hole evaporation might be observed. Creation of quark gluon plasma of M₈₉ hadron physics cannot be in question since ordinary quark gluon plasma was created in p-anti-p collision with cm energy of few TeV so that something like 512 TeV of cm energy might be needed!

3. Could the decay correspond to a decay of a blob of M₈₉ hadronic phase to M₁₀₇ hadrons? How this process could take place? I proposed for about 15 years ago see (this) that the transition from M₈₉ hadron physics to M₁₀₇ hadron physics might take place as a p-adic cooling via a cascade like process via highly unstable intermediate hadron physics. The p-adic temperature is quantized and given by T_p=n/log(p)≈ n/klog(2) for p≈ 2^k and p-adic cooling process would proceed in a step-wise manner as k→ k+2→ k+4+... Also k → k+1→ k+2+.. with mass scale reduced in powers of square root of 2 can be considered. If only octaves are allowed, the p-adic prime characterizing the hadronic space-time sheets and quark mass scale could decrease in nine steps from M₈₉ mass scale proportional to 2^-89/2 octave by octave down to the hadronic mass scale proportional 2^-107/2 as k=89→ 91→ 93...→ 107. At each step the mass in the propagator of the particle would be changed. In particular on mass shell particles would become off mass shell particles which could decay.

   At quark level the cooling process would naturally stop when the value of k corresponds to that characterizing the quark. For instance b quark one has k(b)=103 so that 7 steps would be involved. This would mean the decay of M₈₉ hadrons to highly unstable intermediate states corresponding to k=91,93,...,107. At every step states almost at rest could be produced and the final decay would produce large number of jets and the outcome would resemble the spectrum blackhole evaporation. Note that for u,d,s quarks one has k=113 characterizing also nuclei and muon which would mean that valence quark space-time sheets of lightest hadrons would be cooler than hadronic space-time sheet, which could be heated by sea partons. Note also that quantum superposition of phases with several p-adic temperatures can be considered in zero energy ontology.

   This is of course just a proposal and might not be the real mechanism. If M₈₉ hadrons are dark in TGD sense as the TGD based explanation of CDF-D0 discrepancy suggests, also the transformation changing the value of Planck constant is involved.

4. This picture does not make sense in the model explaining DAMA observations and DAMA-Xenon100 anomaly, CDF bump (see this) and two and half year old CDF anomaly (see this) . The model involves creation of second octave of M₈₉ pions decaying in stepwise manner. A natural interpretation of p-adic octaves of pions is in terms of a creation of over-heated unstable hadronic space-time sheet having k=85 instead of k=89 and p-adically cooling down to relatively thermally stable M₈₉ sheet and containing light mesons and electroweak bosons. If so then the production of CDF bump would correspond to a creation of hadronic space-time sheet with p-adic temperature corresponding to k=85 cooling by the decay to k=87 pions in turn decaying to k=89. After this the decay to M₁₀₇ hadrons and other particles would take place.

Consider now whether the 10 jet event could be understood as a creation of a p-adic hot spot perhaps assignable to some heavy meson of M₈₉ physics. For current quarks the p-adic primes can be much large so that in the case of u and d quark the masses can be in 10 MeV range (which together with detailed model for light hadrons supports the view that quarks can appear at several p-adic temperatures).

1. According to p-adic mass calculations ordinary charmed quark corresponds to k=104=107-3 and that of bottom quark to k=103=107-4, which is prime and correspond to the second octave of M₁₀₇ mass scale assignable to the highest state of pion cascade. By naive scaling M₈₉ charmonium states (Ψ would correspond to k=89-3=86 with mass of about 1.55 TeV by direct scaling. k=89-4=85 would give mass about 3.1 GeV and there is slight evidence for a resonance around 3.3 TeV perhaps identifiable as charmonium. Υ (bottonium) consisting of bbar pair correspond to k=89-4=85 just like the second octave of M₈₉ pion. The mass of M₈₉ Υ meson would be about 4.8 TeV for k=85. k=83 one obtains 9.6 TeV, which exceeds the total cm energy 7 TeV.

2. Intriguingly, k=85 for the bottom quark and for first octave of charmonium would correspond to the second octave of M₈₉ pion. Could it be that the hadronic space-time sheet of Υ is heated to the p-adic temperature of the bottom quark and then cools down in a stepwise manner? If so, the decay of Υ could proceed by the decay to higher octaves of light M₈₉ mesons in a process involving two steps and could produce a large number jets.

3. For the decay of ordinary Υ meson 81.7 per cent of the decays take place via ggg state. In the recent case they would create three M₈₉ parton jets producing relativistic M₈₉ hadrons. 2.2 per cent of decays take place via γ gg state producing virtual photon plus M₈₉ hadrons. The total energies of the three jets would be about 1.6 TeV each and much higher than the energies of QCD jets so that this kind of jets would serve as a clearcut signature of M₈₉ hadron physics and its bottom quark. Note that there already exists slight evidence for charmonium state. Recall that the total transverse energy of the 10 jet event was about 1 TeV.

   Also direct decays to M₈₉ hadrons take place. η' +anything- presumably favored by the large contribution of bbar state in η' - corresponds to 2.9 per cent branching ratio for ordinary hadrons. If second octaves of η' and other hadrons appear in the hadron state, the decay product could be nearly at rest and large number of M₈₉ would result in the p-adic cooling process (the naive scaling of η' mass gives .5 TeV and second octave would correspond to 2 TeV.

4. If two octave p-adic over-heating is dynamically favored, one must also consider the first octave of of scaled variant of J/Ψ state with mass 3.1 GeV scaled up to 3.1 TeV for the first octave. The dominating hadronic final state in the decay of J/Ψ is ρ^+/-π^-/+ with branching ratio of 1.7 per cent. The branching fractions of ωπ⁺π⁺π^-π^-, ωπ⁺π^-π⁰, and ωπ⁺π⁺pi^- are 8.5× 10^-3 4.0× 10^-3, and 8.6× 10^-3 respectively. The second octaves for the masses of ρ and π would be 1.3 TeV and .6 TeV giving net mass of 1.9 TeV so that these mesons would be relativistic if charmonium state with mass around 3.3 TeV is in question. If the two mesons decay by cooling, one would obtain two jets decaying two jets. Since the original mesons are relativistic one would probably obtain two wide jets decomposing to sub-jets. This would not give the desired fireball like outcome.

   The decays ωπ⁺π⁺π^-π^- (see Particle Data Tables would produce five mesons, which are second octaves of M₈₉ mesons. The rest masses of M₈₉ mesons would in this case give total rest mass of 3.5 TeV. In this kind of decay -if kinematically possible- the hadrons would be nearly at rest. They would decay further to lower octaves almost at rest. These states in turn would decay to ordinary quark pairs and electroweak bosons producing a large number of jets and black hole like signatures might be obtained. If the process proceeds more slowly from M₈₉ level, the visible jets would correspond to M₈₉ hadrons decaying to ordinary hadrons. Their transverse energies would be very high.

To sum up, a possible interpretation for the 10-jet event in TGD framework would be as p-adic hot spot produced in collision created by the overheating of M₈₉ hadronic space-time sheets by the presence of bottonium or possibly charmonium state. The general signature of M₈₉ hadron physics is jets which are much more energetic than QCD jets and that data indicate their presence.

For more about new physics predicted by TGD see the chapter New Particle Physics Predicted by TGD: Part I of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy". For reader's convenience I have added a short pdf article Is the new boson reported by CDF pion of M89 hadron physics? at my homepage.

Could Gromov-Witten invariants and braided Galois homology together allow to construct WCW spinor fields?

The challenge of TGD is to understand the structure of WCW spinor fields both in the zero modes which correspond to symplectically invariant degrees of freedom not contributing to the WCW Kähler metric and in quantum fluctuating degrees of freedom parametrized by the symplectic group of δ M⁴_+/-× CP₂. Basically the challenge is is to understand the symplectic (or more precisely, contact geometry of δ M⁴_+/-× CP₂. It seems that mathematicians and mathematical physicists (Gromov, Witten, Floer, and so on) have developed refined concepts for dealing with this problem.

One can develop good arguments suggesting that an appropriate generalization of Gromov-Witten invariants to covariants combined with braid Galois homology discussed in previous posting could allow do construct WCW spinor fields and at the same time M-matrices defining the rows of the unitary U-matrix between zero energy states. Finite measurement resolution would be the magic notion making everything calculable.

In the proposed framework the view about construction of WCW spinor fields would be roughly following.

1. One can distinguish between WCW "orbital" degrees of freedom and fermionic degrees of freedom and in the case of WCW degrees of freedom also between zero modes and quantum fluctuating degrees of freedom. Zero modes correspond essentially to the non-local symplectic invariants assignable to the projections of the δ M⁴_+/- and CP₂ Kähler forms to the space-time surface. Quantum fluctuating degrees of freedom correspond to the symplectic algebra in the basis defined by Hamiltonians belonging to the irreps of rotation group and color group.

2. At the level of partonic 2-surfaces finite measurement resolution leads to discretization in terms of braid ends and symplectic triangulation. At the level of WCW discretization replaces symplectic group with its discrete subgroup. This discrete subgroup must result as a coset space defined by the subgroup of symplectic group acting as Galois group in the set of braid points and its normal subgroup leaving them invariant. The group algebra of this discrete subgroup of symplectic group would have interpretation in terms of braided Galois cohomology. This picture provides an elegant realization for finite measurement resolution and there is also a connection with the realization of finite measurement resolution using categorification.

3. The generating function for Gromow Witten invariants would define an excellent candidate for the part of WCW spinor field defining on zero modes only. The generalization of Gromov-Witten invariants to n-point functions defined by Hamiltonians of δ M⁴_+/-× CP₂ are symplectic invariants if net δ M⁴_+/-× CP₂ quantum numbers vanish. The most general definition assumes that the vanishing of quantum numbers occurs only for zero energy states having disjoint unions of partonic 2-surfaces at the boundaries of CDs as geometric correlate. A close analogy to the topological string theory of type A emerges. This seems puzzling since in the twistorial approach to N=4 SUSY however topological stringy theory of type B emerges. The celebrated mirror symmetry relating Calabi-Yau-manifolds means that topological string theory of type A is mapped to that of type B in the mirror transformation. The proposal is that the two formulations of TGD in terms of M⁴× CP₂ on one hand and CP₃ × CP₃ on one hand are related in the similar manner so that the analog of topological string theory of type B would apply in the latter representation of quantum TGD.

4. The proposed generalized homology theory involving braided Galois group and symplectic group of δ M⁴_+/-× CP₂ would realize the "almost" in TGD as almost topological QFT in finite measurement resolution replacing symplectic group with its discretized version. This algebra would relate to the quantum fluctuating degrees of freedom. The braids would carry only fermion number and there would be no Hamiltonians attached with them. The braided Galois homology could define in the more general situation invariants of symplectic isotopies.

5. One should also add four-momenta and twistors to this picture. The separation of dynamical fermionic and sup-symplectic degrees of freedom suggesets that the Fourier transforms for amplitudes containing the fermionic braid end points as arguments define twistorial amplitudes. The representations of light-like momenta using twistors would lead to a generalization of the twistor formalism. At zero momentum limit one would obtain symplectic QFT with states characterized by collections of Hamiltonians and their super-counterparts.

For details see the new chapter Motives and Infinite Primes of "TGD as a Generalized Number Theory" or the article with same title.

Categorification and finite measurement resolution

I read a very stimulating article by John Baez with title Categorification about the basic ideas behind a process called categorification. The process starts from sets consisting of elements. In the following I describe the basic ideas and propose how categorification could be applied to realize the notion of finite measurement resolution in TGD framework.

What categorification is?

In categorification sets are replaced with categories and elements of sets are replaced with objects. Equations between elements are replaced with isomorphisms between objects: the right and left hand sides of equations are not the same thing but only related by an isomorphism so that they are not tautologies anymore. Functions between sets are replaced with functors between categories taking objects to objects and morphisms to morphisms and respecting the composition of morphisms. Equations between functions are replaced with natural isomorphisms between functors, which must satisfy certain coherence laws representable in terms of commuting diagrams expressing conditions such as commutativity and associativity.

The isomorphism between objects represents equation between elements of set replaces identity. What about isomorphisms themselves? Should also these be defined only up to an isomorphism of isomorphism? And what about functors? Should one continue this replacement ad infinitum to obtain a hierarchy of what might be called n-categories, for which the process stops after n:th level. This rather fuzzy buisiness is what mathematicians like John Baez are actually doing.

Why categorification?

There are good motivations for the categofication. Consider the fact that natural numbers. Mathematically oriented person would think number '3' in terms of an abstract set theoretic axiomatization of natural numbers. One could also identify numbers as a series of digits. In the real life the representations of three-ness are more concrete involving many kinds of associations. For child '3' could correspond to three fingers. For a mystic it could correspond to holy trinity. For a Christian "faith,hope,love". All these representations are isomorphic representation of threeness but as real life objects three sheeps and three cows are not identical.

We have however performed what might be called decategorification: that is forgitten that the isomorphic objects are not equal. Decatecorification was of course a stroke of mathematical genius with enormous practical implications: our information society represents all kinds of things in terms of numbers and simulates successfully the real world using only bit sequences. The dark side is that treating people as mere numbers can lead to a rather cold society.

Equally brilliant stroke of mathematical genius is the realization that isomorphic objects are not equal. Decategorization means a loss of information. Categorification brings back this information by bringing in consistency conditions known as coherence laws and finding these laws is the hard part of categorization meaning discovery of new mathematics. For instance, for braid groups commutativity modulo isomorphisms defines a highly non-trivial coherence law leading to an extremely powerful notion of quantum group having among other things applications in topological quantum compuatation.

No-one would have proposed categorification unless it were demanded by practical needs of mathematics. In many mathematical applications it is obvious that isomorphism does not mean identity. For instance, in homotopy theory all paths deformable to each other in continuous manner are homotopy equivalent but not identical. Isomorphism is now homotopy. These paths can be connected and form a groupoid. The outcome of the groupoid operation is determined up to homotopy. The deformations of closed path starting from a given point modulo homotopies form homotopy group and one can interpret the elements of homotopy group as copies of the point which are isomorphic. The replacement of the space with its universal covering makes this distinction explicit. One can form homotopies of homotopies and continue this process ad infinitum and obtain in this manner homotopy groups as characterizes of the topology of the space.

Cateforification as a manner describe finite measurement resolution?

In quantum physics gauge equivalence represents a standard example about equivalence modulo isomorphisms which are now gauge transformations. There is a practical strategy to treat the situation: perform a gauge choice by picking up one representative amongst infinitely many isomorphic objects. At the level of natural numbers a very convenient gauge fixing would correspond the representation of natural number as a sequence of decimal digits rather than image of three cows.

In TGD framework a excellent motivation for categorification is the need to find an elegant mathematical realization for the notion of finite measurement resolution. Finite measurement resolutions (or cognitive resolutions) at various levels of information transfer hierarchy imply accumulation of uncertainties. Consider as a concrete example uncertainty in the determination of basic parameters of a mathematical model. This uncertainty is reflected to final outcome as via a long sequence of mathematical maps and additional uncertainties are produced by the approximations at each step of this process.

How could onbe describe the finite measurement resolution elegantly in TGD Universe? Categorification suggests a natural method. The points equivalent with measurement resolution are isomorphic with each other. A natural guess inspired by gauge theories is that one should perform a gauge choice as an analog of decategorification. This allows also to avoid continuum of objects connected by arrows: reader can easily imagine what a mess results when one tries to do this;-)!

1. At space-time level gauge choice means discretization of partonic 2-surfaces replacing them with a discrete set points serving as representatives of equivalence classes of points equivalent under finite measurement resolution. An especially interesting choice of points is as rational points or algebraic numbers and emerges naturally in p-adicization process. One can also introduce what I have called symplectic triangulation of partonic 2-surfaces with the nodes of the triangulation representing the discretization and carrying quantum numbers of various kinds.

2. At the level of "world classical worlds" (WCW) this means the replacement of the sub-group if the symplectic group of δ M⁴× CP₂ -call it G- permuting the points of the symplectic triangulation with its discrete subgroup obtained as a factor group G/H, where H is a normal subgroup of G leaving the points of the symplectic triangulation fixed. One can also consider subgroups of the permutation group for the points of the triangulation. One can also consider flows with these properties to get braided variant of G/H. It would seem that one cannot regard the points of triangulation as isomorphic in the category theoretical sense. This because, one can have quantum superpositions of states located at these points and the factor group acts as the analog of isometry group. One can also have many-particle states with quantum numbers at several points. The possibility to assign quantum numbers to a given point becomes the physical counterpart for the axiom of choice. What is so fantastic is that finite measurement resolution leads to a replacement of the infinite-dimensional world of classical points with a discrete structure. Therefore operation like integration over entire "world of classical worlds" is replaced with a discrete sum. This makes things much easier- believe or not - and if not try yourself;-).

3. What suggests itself strongly is a hierarchy of n-categories as a proper description for the finite measurement resolution. The increase of measurement resolution means increase for the number of braid points. One has also braids of braids of braids structure implied by the possibility to map infinite primes, integers, and rationals to rational functions of several variables and the conjecture possibility to represent the hierarchy of Galois groups involved as symplectic flows. If so the hierarchy of n-categories would correspond to the hierarchy of infinite primes having also interpretation in terms of repeated second quantization of an arithmetic SUSY such that many particle states of previous level become single particle states of the next level.

The finite measurement resolution has also a representation in terms of inclusions of hyperfinite factors of type II₁ about which the Clifford algebra generated by the gamma matrices of WCW represents an example.

1. The included algebra represents finite measurement resolution in the sense that its action generates states which are cannot be distinguished from each other within measurement resolution used. The natural conjecture is that this indistuinguishability corresponds to a gauge invariance for some gauge group and that TGD Universe is analogous to Turing machine in that almost any gauge group can be represented in terms of finite measurement resolution.

2. Second natural conjecture inspired by the fact that symplectic groups have enormous representabive power is that these gauge symmetries allow representation as subgroups of the symplectic group of δ M⁴× CP₂. A nice article about universality of symplectic groups is the article The symplectification of science by Mark. J. Gotay.

3. An interesting question is whether there exists a finite-dimensional space, whose symplecto-morphisms would allow a representation of any gauge group (or of all possible Galois groups as factor groups) and whether δ M⁴× CP₂ could be a space of this kind with the smallest possible dimension.

Arrows are not all that is needed

There have been proposals that categories could be fundamental and space-time, symmetries, and particles could emerge in some sense. Personally I do not find this idea sound.

1. Categories consist of discrete objects and on basis of above arguments provide indispensable tool for physicist and consciousness theorist since both measurement resolution and cognitive resolution are always finite. In fact, finite resolution is not at all a negative thing since it forces abstraction process by forming equivalence classes from objects not distinguishable from each other. Written language is one of the victories of abstraction process: just a sequence of letters "human" becomes are representation for entire species. It would be however nonsense to assume that the world is actually discrete. Practically all physics would be lost and only manner to get it would be by effectively replacing the discrete structures with continuum. Mathematics would suffer the same fate and there would be very little use for category theory.

2. I find also the idea of throwing away group theory as very weird. Isomorphisms between objects form groups and are the corner stone of category theory: why should one throw them away? 90 per cent of recent day quantum physics is group theory and the above arguments suggest that category theory is natural in the description of finite measurement resolution reducing the infinite-dimensional groups involved to discrete groups and giving also a profound connection with number theory. Without symmetries we do not have observables which in quantum theory correspond to Lie algebras for continuous groups. As a matter fact, in TGD framework the role of symmetries is taken to extreme: zero energy states correspond to Lie algebra for an infinite-dimensional Yangian. The world of quantum worlds is Lie algebra.

3. It has been also suggested that so called associahedrons emerging in n-category theory could replace space-time and space as fundamental objects. Associahedrons are polygons used to represent geometrically associativity or its weaker form modulo isomorphism for the products of n objects bracketed in all possible manners. The polygon defines a hierarchy containing sub-polygons as its edges containing.... Associativity states the isomorphy of these polygons. According to John Baez associahedrons indeed allow a beautiful geometric realization of the coherence laws.

   One must however not forget that the very notion of associahedron is an auxiliary tool which assumes the notion of Euclidian space so that the claim about emergence of space from category theory is an illusion just as the claims that continuous space-time can emerge from a discrete lattice at infrared limit.

   One should also remember that the notion of n-category has its roots in homotopy theory which describes topological invariants of various spaces. Only non-sense with arrows remains if one throws away all those structures whose description has motivated the development of the category theoretical approach. This kind of emergence is also in conflict with the very idea of categorification since it would identify the isomorphic points- say points of continuum equivalent within measuremet resolution- to get discrete structure and then conclude that this discrete structure is all that exists.

A TGD based explanation for CDF-D0 discrepancy concerning 150 GeV bump

In the latest posting I wrote about beautiful model providing a unified description of the bumps observed recently in D0 and CDF. Yesterday came the fold show. The new results from D0 do not support CDF bump (see Lubos, Jester, and Tommaso).

This shows only that either CDF or D0 is wrong, not that CDF is wrong as some of us suddenly want to believe. It is very difficult to remain rational in this kind of matters: believer in CDF bump transforms suddenly to a non-believer when it turns out that his theory fails to explain it;-).

My own tentative interpretation -not a belief- relies on bigger picture provided by TGD and is that both 150 GeV, 300 GeV, and 325 GeV resonances are there and have interpretations in terms of π and it p-adic octave,ρ, and ω of M₈₉ hadron physics. I could of course be wrong. LHC will be the ultimate jury.

In any case, neither CDF and D0 are cheating and one should explain the discrepancy rationally. Resonaances mentions different estimates for QCD background as a possible explanation. What one could say about this in TGD framework?

1. There is long history of this kind of forgotten discoveries having same interpretation in TGD framework. Always pionlike states-possibly coherent state of them- would have been produced in strong non-orthogonal magnetic and electric fields of the colliding charges and most pion-like states predicted to be almost at rest in cm frame. Electropions were observed already at seventies in the collisions of heavy nuclei at energies near Coulomb wall, resonances having interpretation as mu-pions about three years ago, tau-pions detected by CDF for two and half years ago with refutation coming from D0, now DAMA and Cogent observed dark matter candidate having explanation in terms of tau-pion in TGD framework but Xenon100 found nothing (in this case on can understand the discrepancy in TGD framework). The octaves of M₈₉ pions would represent the last episode of this strange history. In the previous posting universality of the production mechanism forced to made the proposal that also the collisions of ordinary nuclei could generate octaves of ordinary pions. They have not been observed and as I proposed this might due to the peculiarity of the production mechanism.

   What could be a common denominator for this strange sequence of almost discoveries? Light colored excitations of leptons can be of course be argued to be non-existent because intermediate boson decay widths do not allow them but it is difficult to believe that his would have been the sole reason for not taking leptopions seriously.

2. Could the generation of a pionic coherent state as a critical phenomenon very sensitive to the detailed values of the dynamical parameters, say the precise cm energies of the colliding beams? For leptopions a phase transition generating dark colored variants of leptons (dark in the sense having non-standard value of Planck constant) would indeed take place so that criticality might make sense. Could also M₈₉ quarks be dark or colored excitations of ordinary quarks which are dark? Could the M₁₀₇→ M₈₉ phase transition take place only near criticality? This alone does not seem to be enough however.

3. The peculiarity of the production mechanism is that the pion like states are produced mostly at rest in cm frame of the colliding charges. Suppose that the cm frame for the colliding charged particles is not quite the lab frame in D0. Since most dark pions are produced nearly at rest in the cm frame, they could in this kind of situation leave the detector before decaying to ordinary particles: they would behave just like dark matter is expected to behave and would not be detected! The only signature would be missing energy. This would also predict that dark octaves of ordinary pions would not be detected in experiments using target which is at rest in lab frame. Just asking of course;-).

4. This mechanism is actually quite general. Dark matter particles decaying to ordinary matter and having long lifetime remain undetected if they move with high enough velocity with respect to laboratory. Long lifetime would be partially due to the large value of hbar and relativistic with respect to laboratory velocities also time dilation would increases the lifetime. Dark matter particles could be detected only as a missing energy not identifiable in terms of neutrinos. A special attention should be directed to state candidates which are nearly at rest in laboratory.

An example from ordinary hadron physics is the production of pions and their octaves in the strong electric and magnetic field of nuclei colliding with a target at rest in lab. The lifetime of neutral pion is about 10^-8 seconds and scaled up for large hbar and by time dilation when the colliding nucleons have relativistic energies. Therefore the dark pion might leave the measurement volume before decay to two gammas when the the target is at rest in laboratory. It is not even clear whether the gammas need to have standard value of Planck constant.

For the second octave of M₈₉ pion the lifetime would be scaled down by the ratio of masses giving a factor 2¹¹ and lifetime of order .5× 10^-11 seconds. Large hbar would scale up the lifetime. For non-relativistic relativistic velocities the distance travelled before the decay to gamma pair would L=(hbar/hbar₀)× (v/c)× 1.1 mm.

If also the gamma pair is dark, the detection would require even larger volume. TGD suggests strongly that also photons have a small mass which they obtain by eating the remaining component of Higgs a la TGD (transforming like 1+3 under vectorial weak SU(2)). If photon mass defines the upper bound for the rate for the transformation to ordinary photons, dark photons would remain undetected. For more about new physics predicted by TGD see the chapter New Particle Physics Predicted by TGD: Part I of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy". For reader's convenience I have added a short pdf article Is the new boson reported by CDF pion of M89 hadron physics? at my homepage.

The simplest identification of the 150 GeV resonance in TGD framework

The picture about CDF resonance has become (see the postings Theorists vs. the CDF bump and More details about the CDF bump by Jester. One of the results is that leptophobic Z' can explain only 60 per cent of the production rate. There is also evidence that Wjj corresponds to a resonance with mass slightly below 300 GeV as naturally predicted by technicolor models.

The simplest TGD based model indeed relies on the assumption that the entire Wjj corresponds to a resonance with mass slightly below 300 GeV for which there is some evidence. If one assume that only neutral pions are produced in strong non-orthogonal electric and magnetic fields of colliding proton and antiproton, the mother particle must be actually second octave of 147 GeV pion and have mass somewhat below 600 GeV producing in its possibly allowed strong decays pions which are almost at rest for kinematic reasons. Therefore the production mechanism could be exactly the same as proposed for two and one half year old CDF anomaly and for the explanation of DAMA events and DAMA-Xenon100 discrepancy,

1. This suggests that the mass of the mother resonance is in a good accuracy two times the mass of 150 GeV bump for which best estimate is 147+/- 5 GeV. This brings in mind the explanation for the two and half year old CDF anomaly in which tau-pions with masses coming as octaves of basic tau-pion played a key role (masses were in good approximation 2^k× m(π_τ), m(π_τ)≈ 2m_τ, k=1,2. The same mechanism would explain the discrepancy between the DAMA and Xenon100 experiments.

2. If this mechanism is at work now, the mass of the lowest M₈₉ pion should be around 73 GeV as the naivest scaling estimate gives. One can however consider first the option for which lightest M₈₉ has mass around 147 GeV so that the 300 GeV resonance could correspond to its first p-adic octave. This pion would decay to W and neutral M₈₉ pion with mass around 147 GeV in turn decaying to two jets. At quark level the simplest diagram would involve the emission of W and exchange of gluon of M₈₉ hadron physics. Also the decay to Z and charged pion is possible but in this case the decay of the final state could not take place via annihilation to gluon so that jet pair need not be produced.

3. One could also imagine the mother particle to be ρ meson of M₈₉ hadron physics with mass in a good approximation equal to pion mass. At the level of mathematics this option is very similar to the technicolor model of CDF bump based also on the decay of ρ meson. In this model the decays of π to heavy quarks have been assumed to dominate. In TGD framework the situation is different. If π consists of scaled up u and d quarks, the decays mediated by boson exchanges would produce light quarks. In the annihilation to quark pair by a box diagram involving two gluons and two quarks at edges the information about the quark content of pion is lost. The decays involving emission of Z boson the resulting pion would be charged and its decays by annihilation to gluon would be forbidden so that Wjj final states would dominate over Zjj final states as observed.

4. The strong decay of scaled up pion to charged and neutral pion are forbidden by parity conservation. The decay can however proceed by via the exchange of intermediate gauge boson as a virtual particle. The first quark would emit virtual W/Z boson and second quark the gluon of the hadron physics. Gluon would decay to a quark pair and second quark would absorb the virtual W boson so that a two-pion final state would be produced. The process would involve same vertices as the decay of ρ meson to W boson and pion. The proposed model of the two and one half year old CDF anomaly and the explanation of DAMA and Xenon100 experiments assumes cascade like decay of pion at given level of hierarchy to two pions at lower level of hierarchy and the mechanism of decay should be this.

Consider next the masses of the M₈₉ mesons. Naive scaling of the mass of ordinary pion gives mass about 71 GeV for M₈₉ pion. One can however argue that color magnetic spin-spin splitting need not obey scaling formula and that it becomes small because if is proportional to eB/m where B denotes typical value of color magnetic field and m quark mass scale which is now large. The mass of pion at the limit of vanishing color magnetic splitting given by m₀ could however obey the naive scaling.

1. For (ρ,π) system the QCD estimate for the color magnetic spin-spin splitting would be

   (m(ρ),m(π))= (m₀+3Δ/4,m₀-Δ/4) .

   p-Adic mass calculations are for mass squared rather than mass and the calculations for the mass splittings of mesons (see this) force to replace this formula with

   (m²(ρ),m²(π))= (m₀² +3Δ²/4,m₀²-Δ²/4) .

   The masses of ρ and ω are very near to each other: (m(ρ),m(ω)=(.770,.782) GeV and obey the same mass formula in good approximation. The same is expected to hold true also for M₈₉.

2. One obtains for the parameters Δ and m₀ the formulas

   Δ= [mⁿ(ρ)-mⁿ(π)]^1/n , m₀= [(m²(ρ)+3m(π)²)/4]^1/n .

   Here n=1 corresponds to ordinary QCD and n=2 to p-adic mass calculations.

3. Assuming that m₀ experiences an exact scaling by a factor 512, one can deduce the value of the parameter Δ from the mass 147 GeV of M₈₉ pion and therefore predict the mass of ρ₈₉. The results are following

   m₀=152.3 GeV , Δ= 21.3 GeV , m(ρ₈₉)=168.28 GeV

   for QCD model for spin-spin splitting and

   m₀=206.7 GeV , Δ= 290.5 GeV , m(ρ₈₉)=325.6 GeV .

   for TGD model for spin-spin splitting.

4. Rather remarkably, there are indications from D0 for charged and from CDF for neutral resonances with masses around 325 GeV such that the neutral one is split by .2 GeV: the splitting could correspond to ρ-ω mass splitting. Hence one obtains support for both M₈₉ hadron physics and p-adic formulas for color magnetic spin-spin splitting. Note that the result excludes also the interpretation of the nearly 300 GeV resonance as ρ₈₉ in TGD framework.

5. This scenario allows to make estimates also for the masses other resonances and naive scaling argument is expected to improve as the mass increases. For (K₈₉,K^*₈₉) system this would predict mass m(K₈₉)>256 GeV and m(K^*₈₉)<456.7 GeV.

The nasty question is why the octaves of pion are not realized as a resonances in ordinary hadron physics. If they were there, their decays to ordinary pion pairs by this mechanism would very slow.

1. Could it be that also ordinary pion has these octaves but are not produced by ordinary strong interactions in nucleon collisions since the nucleons do not contain the p-adically scaled up quarks fusing to form the higher octave of the pion. Also the fusion rate for two pions to higher octave of pion would be rather small by parity breaking requiring weak interactions.

2. The production mechanism for the octaves of ordinary pions, for M₈₉ pions in the collisions of ordinary nucleons, and for leptohadrons would be universal, namely the collision of charged particles with cm kinetic energy above the octave of pion. The presence of strong non-orthogonal electric and magnetic fields varying considerably in the time scale defined by the Compton time of the pion is necessary since the interaction Lagrangian density is essentially the product of the abelian instanton density and pion field. In fact, in technicolor article it is mentioned that 300 GeV particle candidate is indeed created at rest in Tevatron lab -in other words in the cm system of colliding proton and antiproton beams.

3. The question is whether the preoduction of the octaves of scaled up pions could have been missed in proton-proton and proton antiproton collisions due to the very peculiar kinematics: pions would be created almost at rest in cm system (see this). Whether or not this is the case should be easy to test. For a theorists this kind of scenario does not look impossible but at the era of LHC it would require a diplomatic genius and authority of Witten to persuade experimentalists to check whether low energy collisions of protons produce octaves of pions!

There is also the question about the general production mechanisms for M₈₉ hadrons.

1. Besides the production of scalar mesons in strong non-orthogonal magnetic and electric fields also the production via annihilation of quark pairs to photon and weak bosons in turn decaying to the quarks of M₈₉ hadron physics serves as a possible production mechanism. These production mechanisms do not give much hopes about the production of nucleons of M₈₉ physics.

2. If ordinary gluons couple to M₈₉ quarks, also the production via fusion to gluons is possible. If the transition from M₁₀₇ hadron physics corresponds to a phase transition transforming M₁₀₇ hadronic space-time sheets/gluons to M₈₉ space-time sheets/gluons, M₁₀₇ gluons do not couple directly to M₈₉ gluons. In this case however color spin glass phase for M₁₀₇ gluons could decay to M₈₉ gluons in turn producing also M₈₉ nucleons. Recall that naive scalings for M₈₉ nucleon the mass 481 GeV. The actual mass is expected to be higher but below the scaled up Δ resonance mass predicted to be below 631 GeV.

For more about new physics predicted by TGD see the chapter New Particle Physics Predicted by TGD: Part I of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy". For reader's convenience I have added a short pdf article Is the new boson reported by CDF pion of M89 hadron physics? at my homepage.

Beltrami flows, symplectic invariance and gauge and gravitational interactions

One of the most recent observations made by people working with twistors is the finding of Monteiro and O'Connell described in the preprint The Kinematic Algebra From the Self-Dual Sector . The claim is that one can obtain supergravity amplitudes by replacing the color factors with kinematic factors which obey formally 2-D symplectic algebra defined by the plane defined by light-like momentum direction and complexified variable in the plane defined by polarizations. One could say that momentum and polarization dependent kinematic factors are in exactly the same role as the factors coming from Yang-Mills couplings. Unfortunately, the symplectic algebra looks rather formal object since the first coordinate is light-like coordinate and second coordinate complex transverse coordinate. It could make sense only in the complexification of Minkowski space.

In any case, this would suggest that the gravitational gauge group (to be distinguished from diffeomorphisms) is symplectic group of some kind having enormous representative power as we know from the fact that the symmetries of practically any physical system are realized in terms of symplectic transformations. According to the authors of kenocitebthe/kinealgebra one can identify the Lie algebra of symplectic group of sphere with that of SU(N) at large N limit in suitable basis. What makes this interesting is that at large N limit non-planar diagrams which are the problem of twistor Grassmann approach vanish: this is old result of t'Hooft, which initiated the developments leading to AdS/CFT correspondence.

The symplectic group of δ M⁴_+/-× CP₂ is the isometry algebra of WCW and I have proposed that the effective replacement of gauge group with this group implies the vanishing of non-planar diagrams (see this). The extension of SYM to a theory of also gravitation in TGD framework could make Yangian symmetry exact, resolve the infrared divergences, and the problems caused by non-planar diagrams. It would also imply stringy picture in finite measurement resolution. Also the the construction of the non-commutative homology and cohomology in TGD framework led to the lifting of Galois group algebras to their braided variants realized as symplectic flows and to the conjecture that in finite measurement resolution the cohomology obtained in this manner represents WCW ("world of classical worlds") spinor fields (or at least something very essential about them) [see this].

It is however difficult to understand how one could generalize the symplectic structure so that also symplectic transformations involving light-like coordinate and complex coordinate of the partonic 2-surface would make sense in some sense. In fact, a more natural interpretation for the kinematic algebra would in terms of volume preserving flows which are also Beltrami flows (see for instance this). This gives a connection with quantum TGD since Beltrami flows define a basic dynamical symmetry for the preferred extremals of Kähler action which might be called Maxwellian phase.

1. Classical TGD is defined by Kähler action which is the analog of Maxwell action with Maxwell field expressed as the projection of CP₂ Kähler form. The field equations are extremely non-linear and only the second topological half of Maxwell equations is satisfied. The remaining equations state conservation laws for various isometry currents. Actually much more general conservation laws are obtained.

2. As a special case one obtains solutions analogous to those for Maxwell equations but there are also other objects such as CP₂ type vacuum extremals providing correlates for elementary particles and string like objects: for these solutions it does not make sense to speak about QFT in Minkowski space-time. For the Maxwell like solutions linear superposition is lost but a superposition holds true for solutions with the same local direction of polarization and massless four-momentum. This is a very quantal outcome (in accordance with quantum classical correspondence) since also in quantum measurement one obtains final state with fixed polarization and momentum. So called massless extremals (topological light rays) analogous to wave guides containing laser beam and its phase conjugate are solutions of this kind. The solutions are very interesting since no dispersion occurs so that wave packet preserves its form and the radiation is precisely targeted.

3. Maxwellian preferred extremals decompose in Minkowskian space-time regions to regions that can be regarded as classical space-time correlates for massless particles. Massless particles are characterized by polarization direction and light-like momentum direction. Now these directions can depend on position and are characterized by gradients of two scalar functions Φ and Ψ. Φ defines light-like momentum direction and the square of the gradient of Φ in Minkowski metric must vanish. Ψ defines polarization direction and its gradient is orthogonal to the gradient of Φ since polarization is orthogonal to momentum.

4. The flow has the additional property that the coordinate associated with the flow lines integrates to a global coordinate. Beltrami flow is the term used by mathematicians. Beltrami property means that the condition j kenowedge dj =0 is satisfied. In other words, tjhe current is in the plane defined by its exterior derivative. The above representation obviously guarantees this. Beltrami property allows to assign order parameter to the flow depending only the parameter varying along flow line.

   This is essential for the hydrodynamical interpretation of the preferred extremals which relies on the idea that varies conservation laws hold along flow lines. For instance, super-conducting phase requires this kind of flow and velocity along flow line is gradient of the order parameter. The breakdown of super-conductivity would mean topologically the loss of the Beltrami flow property. One might say that the space-time sheets in TGD Universe represent analogs of supra flow and this property is spoiled only by the finite size of the sheets. This strongly suggests that the space-time sheets correspond to perfect fluid flows with very low viscosity to entropy ratio and one application is to the observed perfect flow behavior of quark gluon plasma.

5. The current J=Φ∇ Ψ has vanishing divergence if besides the orthogonality of the gradients the functions Ψ and Φ satisfy massless d'Alembert equation. This is natural for massless field modes and when these functions represent constant wave vector and polarization also d'Alembert equations are satisfied. One can actually add to ∇Ψ a gradient of an arbitrary function of Φ this corresponds to U(1) gauge invariance and the addition to the polarization vector a vector parallel to light-like four-momentum. One can replace Φ by any function of Φ so that one has Abelian Lie algebra analogous to U(1) gauge algebra restricted to functions depending on Φ only.

The general Beltrami flow gives as a special case the kinetic flow associated by Monteiro and O'Connell with plane waves. For ordinary plane wave with constant direction of momentum vector and polarization vector one could take Φ =cos(φ), φ=kkenocdot m and Ψ = εkenocdot m. This would give a real flow. The kinematical factor in SYM diagrams corresponds to a complexified flow Φ =exp(iφ) and Ψ= φ+ w, where w is complex coordinate for polarization plane or more naturally, complexificaton of the coordinate in polarization direction. The flow is not unique since gauge invariance allows to modify φ term. The complexified flow is volume preserving only in the formal algebraic sense and satisfies the analog of Beltrami condition only in Dolbeault cohomology where d is identified as complex exterior derivative (df=df/dzdz for holomorphic functions). In ordinary cohomology it fails. This formal complex flow of course does not define a real diffeomorphism at space-time level: one should replace Minkowski space with its complexification to get a genuine flow.

The finding of Monteiro and O'Connell encourages to think that the proposed more general Abelian algebra pops up also in non-Abelian YM theories. Discretization by braids would actually select single polarization and momentum direction. If the volume preserving Beltrami flows characterize the basic building bricks of radiation solutions of both general relativity and YM theories, it would not be surprising if the kinematic Lie algebra generators would appear in the vertices of YM theory and replace color factors in the transition from YM theory to general relativity. In TGD framework the construction of vertices at partonic two-surfaces would define local kinematic factors as effectively constant ones.

For background see the chapter Basic Extremals of Kähler Action of "Physics in Many-Sheeted Space-time".

Zero energy ontology, polylogarithms, and hyperbolic 3-manifolds

While performing web searches for twistors and motives I have begun to realize that Russian mathematicians have been building the mathematics needed by quantum TGD for decades while realizing the great visions of Grothendieck. Maybe I am also beginning to vaguely grasp something about the connection of Grassmannian twistor approach to the motivic integrals.

Hyperbolic 3-manifolds and zero energy ontology

The latest finding was the article Volumes of hyperbolic manifolds and mixed Tate motives by Goncharov- one of the great Russian mathematicians involved with the drama. The article is about polylogarithms emerging in twistor calculations and their relationship to the volumes of hyperbolic n-manifolds. I do not of course understand anything about the jargon of the article: it is written by a specialist for specialists and I can only try to understand the general notions and the possible meaning of the results from TGD point of view.

Hyperbolic n-manifolds are n-manifolds equipped with complete Riemann metric having constant sectional curvature equal to -1 (with a suitable choice of length unit) and therefore obeying Einstein's equations with cosmological constant. They are obtained as coset spaces on proper-time constant hyperboloids of n+1-dimensional Minkowski space by dividing by the action of discrete subgroup of SO(n,1), whose action defines a lattice like structure on the hyperboloid. What is remarkable is that the volumes of these closed spaces are homotopy invariants in a well-defined sense.

What is even more remarkable that hyperbolic 3-manifolds are completely exceptional in that there are very many of them. The complements of knots and links in 3-sphere are often cusped hyperbolic 3-manifolds (having therefore tori as boundaries). Also Haken manifolds are hyperbolic. Says Wikipedia:

According to Thurston's geometrization conjecture, proved by Perelman, any closed, irreducible, atoroidal 3-manifold with infinite fundamental group is hyperbolic. There is an analogous statement for 3-manifolds with boundary.

Therefore there are very many hyperbolic 3-manifolds.

The geometrization conjecture of Thurston allows to see hyperbolic 3-manifolds in a wider framework. The theorem states that compact 3-manifolds can be decomposed canonically into sub-manifolds that have geometric structures. It was Perelman who sketched the proof of the conjecture. The prime decomposition with respect to connected sum reduces the problem to the classification of prime 3-manifolds and geometrization conjecture states that closed 3-manifold can be cut along tori such that the interior of each piece has a geometric structure with finite volume serving as a topological invariant. There are 8 possible geometric structures in dimension three and they are characterized by the isometry group of the geometry and the isotropy group of point.

Important is also the behavior under Ricci flow ∂_tg_ij= -2R_ij: here t is not space-time coordinate but a parameter of homotopy. If I have understood correctly, Ricci flow is a dissipative flow gradually polishing the metric for a particular region of 3-manifold to one of the 8 highly symmetric local metrics defining topological invariants. This conforms with the general vision about dissipation as source of maximal symmetries. For compact n-manifolds the normalized Ricci flow ∂_tg_ij= -2R_ij +(2/n)Rg_ij preserving the volume makes sense. Interestingly, for n=4 the right hand side is Einstein tensor so that the solutions of vacuum Einstein's equations in dimension four are fixed points of normalized Ricci flow. Ricci flow expands the negatively curved regions and contracts the positively curved regions of space-time time. Hyperbolic geometries represent one these 8 geometries and for the Ricci flow is expanding. The outcome is amazingly simple and gives also support for the idea that the preferred extremals of Kähler action could represent maximally symmetries 4-geometries defining topological invariants: the preferred extremals would be maximally symmetric representatives with a given topology or algebraic geometry.

The volume spectrum for hyperbolic 3-manifolds forms a countable set which is however not discrete: the statement that one can assign to them ordinal ω^ω does not have any obvious meaning for the man of the street;-). What comes into my simple mind is that p-adic integers and more generally, profinite spaces with infinite number of points, might be something similar: one can enumerate them by infinitely long sequences of pinary digits so that they are countable (I do not know whether also infinite p-adic primes must be allowed and whether they could somehow correspond the hierarchy of infinite ordinals). They are totally disconnected in real sense but do not form a discrete set since since can connect any two points by a p-adically continuous curve.

What makes twistor people excited is that the polylogarithms emerging from twistor integrals (see this and this) seem to be expressible in terms of the volumes of hyperbolic manifolds. What fascinates me is that the polylogarithms in question make sense also p-adically and that the moduli spaces for causal diamonds -or rather, for the double light-cones associated with their M⁴ projections with second tip fixed - are naturally lattices of the 3-dimensional hyperbolic space defined by all positions of the second tip and 3-dimensional hyperbolic spaces are the most interesting ones! In the intersection of the real and p-adic worlds both algebraic universality and finite measurement resolution require number theoretic discretization so that the 3-volume volume could be quantized in discrete manner.

For n=3 the group defining the lattice is a discrete subgroup of the group of SO(3,1) which equals to PSL(2,C) obtained by identifying SL(2,C) matrices with opposite sign. The divisor group defining the lattice and hyperbolic spaces as its lattice cell is therefore a subgroup of PSL(2,Z_c), where Z_c denotes complex integers. Recall that PSL(2,Z_c) acts also in complex plane (and therefore on partonic 2-surfaces) as discrete Möbius transformations whereas PSL(2,Z) correspond to 3-braid group. Reader is perhaps familiar with fractal like orbits of points of plane under iterated Möbius transformations. The lattice cell of this lattice obtained by identifying symmetry related points defines hyperbolic 3-manifolds. Therefore zero energy ontology realizes directly the hyperboliic manifolds whose volumes should somehow represent the poly-logarithms.

The volumes are topological invariants in the sense that homeomorphism does not affect the volume of the space in question if it is given hyperbolic metric. The spectrum of volumes is said to be highly transcendental. In the intersection of real and p-adic worlds only algebraic volumes are possible unless one allows extension by say finite number of roots of e (e^p is p-adic number). The p-adic existence of polylogarithms suggests that also p-adic variants of hyperbolic spaces make sense and that one can assign to them volume as topological invariant although the notion of ordinary volume integral is problematic. In fact, hyperbolic spaces are symmetric spaces and the general arguments that I have developed earlier allow to imagine what the p-adic variants of real symmetric spaces could be.

What about AdS counterparts of hyperbolic manifolds?

Not surprisingly, also AdS-CFT enthusiasts would like to have similar invariants for for AdS (Minkowskian analog of hyperbolic space) and even dS (Minkowskian analog of sphere). Mitchell Porter gives a link to the talk of Maldacena. The expected non-compactness of these spaces implies infinite volume and this problem should be circumvented somehow.

Maybe the preferred role of hyperbolic spaces over AdS and dS might finally select between TGD and M-theory like approach. This would simplify matters enormously since 10-dimensional holography would reduce to 4-dimensional one and would have a direct connection with physics as we have used to know it. For condensed matter physicists expected to say something interesting about this real world already the complexities of 3-D world represent a tough enough challenge and the formulation of the problems in terms of 10-dimensional blackholes migh be too much;-).

For more details see the new chapter Infinite Primes and Motives or the article with same title.

ATLAS continues to exclude standard SUSY

ATLAS has tightened also the limits on SUSY parameters. Gluino mass should be above 1 GeV and squark masses above 725 GeV. Lubos does not give up and puts his remaining hopes on what he calls Indian supersymmetric island. Everyone has his own background and with my background it is clear that standard SUSY with R-parity invariance is dead.

Even if standard SUSY managed to survive in some corner of the parameter space it would not solve the problems which (at least could have) motivated its proposal (say anomalous g-2 for muon and little Higgs problem). Standard SUSY is already now a mere trouble maker. Lubos did not have this in mind when he compared standard SUSY to Osama Bin Laden lurking in some cave: probably Lubos is forced to take the analogy with Osama to its bitter end.

It might be interesting for a SUSY professional to imagine what the situation could be if the SUSY to be tested at LHC would be TGD variant of it it. Right-handed neutrino would be the generator of the weakest broken supersymmetry and R-parity invariance would be broken by its mixing with left handed neutrino so that the decays sparticle → particle +neutrino (perhaps also to particle+ lepton) would serve as the signatures of sparticles. What would be the bounds of squark, gluino and slepton masses and on the couplings determining mixing rate and decay rate in this kind of situation? Neutrino mass scale would be the characterizer of mixing rate so that one could expect relatively long life times of spartners (here I should be very cautious: p-adic scaling of neutrino mass scale by a multiple of half-octave is possible always).

One could have ended TGD variant of SUSY from a real physical problem of standard model. What is the physical role of the mysterious right handed neutrino not belonging to the standard model spectrum? Is it there or is it not? Neutrino massivation forces to answer "Yes". But why it is so different? Why does it have only gravitational interactions and why it is mixed with left handed neutrino? What causes this mixing and at the same time massivation? How this mixing relates to the more general problem of particle massivation? Unfortunately, M-theorists had enough to do with their landscape and brane constructions and did not have time for details like tiny little right-handed neutrino. In this mental frame theoreticians relying on problem-based approach are crackpot-slash- geniuses as Lubos would put it.

New Data from ATLAS

Lubos commented last ATLAS release about dijet production. There is something which one might interpret as the presence of resonances above 3.3 TeV [see Fig. 2) of the article]. "With a modest does of optimism" Lubos manages to see this as a conformation of a stringy prediction. According to always-so-optimistic Lubos 150 GeV bump interpreted as Z' boson suffering leptophoby was the first success of string model;-). Lubos of course does not forget to tell that crackpots certainly get excited of this;-). Maybe he is right.

In a very optimistic mood I could believe that a new hadron physics is being discovered (150 GeV boson could be identified as charged pion and 325 GeV bumps could allow interpretation as kaons). With this almost killer dose of optimism the natural question is whether this extremely slight indication about new physics might have interpretation as a scaled up J/Psi and various other charmonium states above it giving rise to what is not single very wide bump to a family of several resonances in the range 3-4 TeV by scaling the 3-4 GeV range for charmonium resonances. For instance, J/Psi decay width is very small, about .1 MeV, which is about .3×10^-4 of the mass of J/Psi. In the recent case direct scaling would give decay of about 300 MeV for the counterpart of J/Psi if the decay is also now slow for kinematic reasons. For other charmonium resonances the widths are measurement in per cents meaning in the recent case width of order of magnitude 30 GeV: this estimate looks more reasonable as the first estimate.

One can also now perform naive scalings. J/Psi has mass of about 3 GeV. If the scaling of ordinary pion mass from .14 GeV indeed gives something like 145 GeV then one can be very naive and apply the same scaling factor of about 1030 to get the scaled up J/Psi with mass of order 3.1 TeV. The better way to understand the situation is to assume that color-magnetic spin spin splitting is small also for M₈₉ charmonium states and apply naive scaling to the mass of Ψ/J to get a lower bound for the mass of its M₈₉ counterpart. This would give mass of 1.55 TeV which is by a factor 1/2 too small. p-Adic mass calculations lead to the conclusion that c quark is characterized by p≈ 2^k, k=104. Naive scaling would give k=104-18= 86 and 1.55 TeV mass for Ψ/J. Nothing however exludes k=84 and the lower bound 3.1 TGD for the mass of Ψ/J. Since color magnetic spin-spin splitting is smaller for M₈₉ pion, same is expected to be true also for charmonium states so that the mass might well be around 3.3 TeV.

To sum up, I hope that no-one really thinks that I am taking this seriously. As a theoretical physicist working in conctact with the observed reality I just cannot avoid the temptation to these silly games games with numbers. There are also many lighter scalar mesons such as η, η' and corresponding vector mesons such as ρ and ω, K* and also these should eventually introduce themselves to the blog audience.

Quantum TGD as a generalized homology theory

Last days have meant a considerable progress in the understanding the homology and cohomology theories assignable to infinite primes and at the same time to algebraic surfaces defined by the corresponding rational functions. As a matter fact, the construction of quantum states in TGD Universe reduces by the finiteness of measurement resolution to the construction of what might be seen as an analog of homotopy and cohomotopy theory inducing homology and cohomology theories in the sense of algebraic geometry. The elements of homology groups have direct interpretation as quantum states of a supersymmetric quantum field theory.

This picture means a breakthrough in the understanding of quantum TGD as a rigorous mathematical theory able to contribute also to mathematics itself and is in accordance with the visions about quantum TGD as almost topological QFT, physics as as an infinite-D geometry, and physics as generalized number theory. The overall important "almost" means that one has algebraic geometry rather than mere topology. I attach a piece of abstract of the new chapter containing the formulation. The entire abstract appears in earlier posting.

In algebraic geometry the notion of variety defined by algebraic equation is very general: all number fields are allowed. One of the challenges is to define the counterparts of homology and cohomology groups for them. The notion of cohomology giving rise also to homology if Poincare duality holds true is central. The number of various cohomology theories has inflated and one of the basic challenges to find a sufficiently general approach allowing to interpret various cohomology theories as variations of the same motive as Grothendieck, who is the pioneer of the field responsible for many of the basic notions and visions, expressed it.

Cohomology requires a definition of integral for forms for all number fields. In p-adic context the lack of well-ordering of p-adic numbers implies difficulties both in homology and cohomology since the notion of boundary does not exist in topological sense. The notion of definite integral is problematic for the same reason. This has led to a proposal of reducing integration to Fourier analysis working for symmetric spaces but requiring algebraic extensions of p-adic numbers and an appropriate definition of the p-adic symmetric space. The definition is not unique and the interpretation is in terms of the varying measurement resolution.

The notion of infinite prime has gradually turned out to be more and more important for quantum TGD. Infinite primes, integers, and rationals form a hierarchy completely analogous to a hierarchy of second quantization for a super-symmetric arithmetic quantum field theory. The simplest infinite primes representing elementary particles at given level are in one-one correspondence with many-particle states of the previous level. More complex infinite primes have interpretation in terms of bound states.

1. What makes infinite primes interesting from the point of view of algebraic geometry is that infinite primes, integers and rationals at the n:th level of the hierarchy are in 1-1 correspondence with rational functions of n arguments. One can solve the roots of associated polynomials and perform a root decomposition of infinite primes at various levels of the hierarchy and assign to them Galois groups acting as automorphisms of the field extensions of polynomials defined by the roots coming as restrictions of the basic polynomial to planes x_n=0, x_n=x_n-1=0, etc...

2. These Galois groups are suggested to define non-commutative generalization of homotopy and homology theories and non-linear boundary operation for which a geometric interpretation in terms of the restriction to lower-dimensional plane is proposed. The Galois group G_k would be analogous to the relative homology group relative to the plane x_k-1=0 representing boundary and makes sense for all number fields also geometrically. One can ask whether the invariance of the complex of groups under the permutations of the orders of variables in the reduction process is necessary. Physical interpretation suggests that this is not the case and that all the groups obtained by the permutations are needed for a full description.

3. The algebraic counterpart of boundary map would map the elements of G_k identified as analog of homotopy group to the commutator group [G_k-2,G_k-2] and therefore to the unit element of the abelianized group defining cohomology group. In order to obtains something analogous to the ordinary homology and cohomology groups one must however replaces Galois groups by their group algebras with values in some field or ring. This allows to define the analogs of homotopy and homology groups as their abelianizations. Cohomotopy, and cohomology would emerge as duals of homotopy and homology in the dual of the group algebra.

4. That the algebraic representation of the boundary operation is not expected to be unique turns into blessing when on keeps the TGD as almost topological QFT vision as the guide line. One can include all boundary homomorphisms subject to the condition that the anticommutator δⁱ_kδ^j_k-1+δ^j_kδⁱ_k-1 maps to the group algebra of the commutator group [G_k-2,G_k-2]. By adding dual generators one obtains what looks like a generalization of anticommutative fermionic algebra and what comes in mind is the spectrum of quantum states of a SUSY algebra spanned by bosonic states realized as group algebra elements and fermionic states realized in terms of homotopy and cohomotopy and in abelianized version in terms of homology and cohomology. Galois group action allows to organize quantum states into multiplets of Galois groups acting as symmetry groups of physics. Poincare duality would map fermionic creation operators to annihilation operators and vice versa and the counterpart of pairing of k:th and n-k:th homology groups would be inner product analogous to that given by Grassmann integration.The interpretation in terms of fermions turns however to be wrong and the more appropriate interpretation is in terms of Dolbeault cohomology applying to forms with homomorphic and antiholomorphic indices.

5. The intuitive idea that the Galois group is analogous to 1-D homotopy group which is the only non-commutative homotopy group, the structure of infinite primes analogous to the braids of braids of braids of ... structure, the fact that Galois group is a subgroup of permutation group, and the possibility to lift permutation group to a braid group suggests a representation as flows of 2-D plane with punctures giving a direct connection with topological quantum field theories for braids, knots and links. The natural assumption is that the flows are induced from transformations of the symplectic group acting on δ M²_+/-× CP₂ representing quantum fluctuating degrees of freedom associated with WCW ("world of classical worlds"). Discretization of WCW and cutoff in the number of fermion modes would be due to the finite measurement resolution. The outcome would be rather far reaching: finite measurement resolution would allow to construct WCW spinor fields explicitly using the machinery of number theory and algebraic geometry.

6. A connection with operads is highly suggestive. What is nice from TGD perspective is that the non-commutative generalization homology and homotopy has direct connection to the basic structure of quantum TGD almost topological quantum theory where braids are basic objects and also to hyper-finite factors of type II₁. This notion of Galois group makes sense only for the algebraic varieties for which coefficient field is algebraic extension of some number field. Braid group approach however allows to generalize the approach to completely general polynomials since the braid group make sense also when the ends points for the braid are not algebraic points (roots of the polynomial).

This construction would realize the number theoretical, algebraic geometrical, and topological content in the construction of quantum states in TGD framework in accordance with TGD as almost TQFT philosophy, TGD as an infinite-D geometry, and TGD as a generalized number theory visions.

For more details see the new chapter Infinite Primes and Motives of "TGD as Generalized Number Theory" or the article with same title.