Friday, September 20, 2019

The details of the genetic code in the model based on bio-harmony

TGD suggests several realizations of music harmonies in terms of Hamiltonian cycles representing the notes of music scale, most naturally 12-note scale represented as vertices of the graph used. The most plausible realization of the harmony is as icosahedral harmony (see this and this).

  1. Icosahedron (see this) has 12 vertices and Hamiltonian cycle as a representation of 12-note scale would go through all vertices such that two nearest vertices along the cycle would differ by quint (frequency scaling by factor 3/2 modulo octave equivalene). Icosahedron allows a large number of inequivalent Hamiltonian cycles and thus harmonies characterized by the subgroup of icosahedral group leaving the cycle invariant. This group can be Z6, Z4, or Z2 which acts either as reflection group or corresponds to a rotation by π.

  2. The fusion of 3 icosahedral harmonies with symmetry groups Z6, Z4 and Z2 gives 20+20+20=60 3-chords and 3+1 + 5 + 10 =19 orbits of these under symmetry group and almost vertebrate genetic code when 3-chords are identified as analogs of DNA codons and their orbits as amino-acids. One obtains counterparts of 60 DNA codons and 3+1 + 5 + 10 =19 amino-acids so that 4 DNA codons and 1 amino-acid are missing.

  3. The problem disappears if one adds tetrahedral harmony with 4 codons as faces of tetrahedron and 1 amino-acid as the orbit of the face of tetrahedron. One obtains 64 analogs of DNA codons and 20 analogs of amino-acids. I call this harmony bio-harmony. The predicted number of DNA codons coding for given amino-acid is the number of triangles at the orbit of given triangle and the numbers are those for genetic code.

  4. How to realize the fusion of harmonies? Perhaps the simplest realization that I have found hitherto is based on union of tetrahedron of 3 icosahedrons obtained by gluing tetrahedron to icosahedron along its face which is triangle. The precise geometric interpretation of this realization has been however missing and I have considered several variants. I have proposed that the model could explain the two additional amino-acids Pyl and Sec appearing in Nature.

    There is also a slight breaking of symmetries: ile 4-plet breaks into ile triplet and met singlet and trp double breaks into stop and trp also leu 4-plet can break in leu triplet and ser singlet (see this). This symmetry breaking should be understood.

The following argument suggests a more detailed solution of these problems than proposed earlier.
  1. The copies of icosahedron would differ by a rotation by multiples of 2π/3 (Z3) around axis through the common triangular face. This face unlike the other faces remains un-affected. Also tetrahedron remains un-affected so that it is counted only once.

    If the 3 copies of the icosahedral common face are counted as separate (this is important!), one obtains 20+20+20 faces from icosahedron. If also tetrahedral shared faces is counted as separate, tetrahedron gives 4 faces: 64 codons altogether as required. One obtains 19 orbits from the 3 icosahedra and 1 orbit from tetrahedron: 20 orbits as counterparts of amino-acids altogether.

  2. But can one really counter the 4 common faces as separate? One must do so. Could these faces be interpreted as somehow special codons? Maybe as stop codons or start codons for the vertebrate genetic code which also corresponds to the realization of DNA, RNA ,tRNA, and amino-acids as dark proton triplets so that DNA sequences would correspond to dark proton sequences. Could the shared codons be assigned with various modifications of the vertebrate code involving also exotic amino-acids Pyl and Sec.

  3. Consider first the tetrahedral face. If the common face is removed from the 4-face orbit of tetrahedron, the orbit has only 3 faces and correspond to an amino-acid coded by 3 DNA codons. ile is the only such amino-acid and the interpretation could be that one ile corresponds to the 3 tetrahedral faces and met acting as start codon to the fourth shared face.

  4. Also 3 icosahedral amino-acids corresponding to orbits containing the shared face can lose 1 codon each. To nake this more concrete, one can look for the deviations from the vertebrate code.

    1. There are 10 doublets if the doublet UAA, UAG acting as stop codons is counted as doublet coding for stop regarded formally as amino-acid.

    2. The second member in the doublet UGA, UGG coding for tyr in code table could correspond to a common face and act as a stop codon.

    3. For the modifications of genetic code UAG coding for stop can code for Pyl and UGA coding for stop can also code for Sec. UGA can also code for trp so that there would not be any symmetry breaking in this case. Could UAG and UGA correspond to common faces for two icosahedra?

    4. There is also third icosahedral shared face. CUG coding for leu can also code for ser. Could this correspond to the third exceptional codon associated with the icosahedral part of the code?

  5. If the answers to the questions are affirmative, all basic deviations from the vertebrate code can be understood. The translation of the codons associated with shared face would be unstable for some reason.

    1. 3-chord representation is more fundamental than the chemical one. This could mean that the chords associated with the shared faces are very near to each other so that the correspondence between 3-chord representation and chemical representation of codons becomes unstable if based on triple resonance.

    2. The proposal has indeed been that the 13th vertex implied by tetrahedron corresponds to a note very near to one of the notes of 12-note scale - this note is necessary since the 12-note scale defined by quints gives 12th note slightly more than octave under octave equivalence as discovered already by Pythagoras.

      If this picture is correct, the symmetry breaking of the genetic code would be due to the presence of the face common to icosahedron and tetrahedron and reflect the problem discovered already by Pythagoras. The rational number based Pythagorean scale defined by quints is special: people with absolute pitch prefer it over the well-tempered scale involving powers of irrational number 21/12 requiring extension of rationals.

See the article An Overall View about Models of Genetic Code and Bio-harmony or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Monday, September 16, 2019

How the new view about solar fusion forces to change the ordinary views about nuclear physics?

First a general comment about nuclear physics, which applies with appropriate modifications also to the evolution of theoretical particle physics (or lack of it) after the emergence of standard model followed by GUTS and superstring models.

  1. One can see the standard nuclear physics as a tragic Odysseia due to the stubborn sticking to the naive length scale reductionism. All began with the modelling of nucleons as point like particles inspired by the successes of atomic physics. It turned out that the model for nucleons as point like particles failed and we still do not understand low energy nuclear physics. The wave-mechanical potential models and QFT models assuming the notion of point-like nucleon led to an inflation of nuclear models each of them explaining some aspects of nuclei but a real theory is still missing.

  2. Dark nuclear physics was originally suggested in TGD framework to explain "cold fusion" and later conjectured to allow the understanding of pre-stellar evolution as a step-wise process leading to the gradual heating of matter leading to nuclear fusion. The model relies on nuclear strings and their dark variants as dark nuclear matter. In this article it is argued that this picture leads to a realistic model of nuclear fusion and of stellar core and perhaps entire stellar interior as a dark spaghetti like structure. Ironically, "cold fusion" researchers regarded for decades as the pariahs of physics community, would show the path to follow.

The proposed model involves several new deep ideas inspired by the fusion of general TGD based visions about nuclear physics on one hand and about the formation of galaxies, stars, and planets on the other hand. Behind both visions is the notion about fractal hierarchy of flux tubes formed from cosmic strings by gradual thickening during the cosmic evolution. A further important piece is ZEO based view about quantum state and quantum measurement forcing to modify ordinary quantum mechanical description.
  1. The idea is that Sun and its Kähler magnetic field form a sub-tangle of the galactic tangle associated with a long cosmic string and extending outside Sun, and perhaps including also planets as sub-tangles. This can be made more precise by assuming that total mass of the straight cosmic string portion involved equals to the total mass of the system considered. The estimate from the diameter of Sun suggests that the total mass is few times the solar mass. This model connects closely with the problem of cosmological constant solved by the twistor lift of TGD and solar physics can be associated with one particular value of length scale dependent cosmological constant: also this idea forced by TGD is revolutionary.

  2. Quantum classical correspondence stating that quantum states are superpositions of Bohr orbit like preferred extremals challenges the idea about tunnelling as an essential element of nuclear physics. The first option is that BSFR - identified as ordinary → dark phase transition increasing the value of heff and involving time reversal followed by its reversal - allows wave-mechanical tunnelling as an approximate description. An alternative realization encouraged by M8-H duality would be as SSFR involving no time reversal but discontinuity at the level of space-time development involving TGD counterparts of branes. This option resonates with the idea about sequence of SRFFs as TGD counterpart of a unitary time evolution suggested by the wave mechanical model. In any case, both TGD view about dark matter and ZEO would become part of nuclear physics, and mean giving up standard ontology and standard wave mechanics as a description of nuclei.

    It would not be surprising if similar view about tunnelling could apply also to particle reactions and I have proposed that dark variants of nuclei of M89 hadron physics as scaled variant of ordinary M107 hadron physics have made themselves visible via the observed (but neglected) bumps with masses obtained by scaling up the masses of ordinary mesons by factor 512. Tunnelling would be now from ordinary hadron physics to dark M89 hadron physics.

  3. When one has paradox, one knows that something is wrong with the basic conceptualization. The presence of dark variants of nuclei makes itself directly visible via the conflict between metallicities deduced from spectroscopy and meteorite abundances and those derived from helio-seismology and solar neutrino physics. Besides ordinary nuclei also their dark variants would present and contribute to metallicity in the solar interior.

See the article Solar metallicity problem from TGD perspective or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

New predictions from the flux tube model of galaxies

The proposed solution of the abundance problem of solar models leads to a much more detailed view about
the formation of stars as flux tube tangles. The model allows to relate to radius of the Sun to its mass assuming that Sun has been produced by a thickening of a straight portion of a cosmic string. This
I have proposed that this general vision applies also to the formation of spiral galaxies. This can be tested in the case of Milky Way at order of magnitude level.

  1. The mass M(gal) of the Milky way is estimated to be in the range [.8,4.5]× 1012M(Sun). For a string with maximal string tension this would correspond to a direct string portion with length L(gal)= M(gal)/R(Sun)= M(gal)/M(Sun).
    In fact, this stringy mass formula is known to hold for quite a many astrophysical objects as I learned decades ago in a particle physics conference - in good old times times particle physics conferences allowed non-main-stream talks during the last conference day. This gives the estimate L(gal)∈ [.6,3.3]× 105 ly. The radius Rgal of galaxy is estimated to be in the range [.75,1.0]× 105 ly. The length of string within galactic radius would satisfy Lgal=[.8,3.3]Rgal. The estimate excludes the lower bound. For the upper bound the one has Lgal ∼ 3.3 × Rgal.

    The thickness of the Milky Way is about 2× 103 ly which suggests that the portion of long string making galaxy is soaked up to the galactic plane..

  2. The supermassive blackhole in the galactic center is estimated to have mass M(BH)=4× 106× M(Sun). By scaling this would correspond to a straight cosmic string portion with length LBH∼ .1 ly. The size of the galactic blackhole (see this) is RS,BH∼ 4.4× 10-5 ly giving RS,BH/Lgal∼ 4.4× 10-4. One has Tmax∼ 10-6/G and blackhole corresponds effectively to a string with tension TG∼ 1/2G and length RS,BH so that the ratio would be RS,BH/Lgal ∼ 2G/Tmax∼ 2× 10-6. The straight string with length LBH would have been compressed to a volume of Schwartchild radius RB,S∼ 2-11LBH.

  3. Could the spiral structure of spiral galaxies involving several spiral correspond to a rotating cosmic string thickened to a flux tube? The original model for the spiral structure as a cosmic string at rest in in Robertson-Walker coordinates and seemingly rotating in linear Minkowski coordinates failed since it predicted too weak spiralling. The observed spiral structure could however corresponds to a thickened dark flux tube with lower string tension and longer length.

    If so the length of the original spiral should be about Lgal=3.3× Rgal. Perhaps the primordial configuration of the dark flux tube could be modelled as a cosmic string solution at rest in Robertson-Walker coordinates, which then thickened and gained length becoming more spiral.

  4. For elliptic galaxies (see this) the sizes vary in the range [3× 103, 7× 105] ly (roughly 2 orders of magnitude) and masses in the range [105,1013] ly (8 orders of magnitude!) so that linear relationship between size and mass is excluded. The length L(gal) of the original straight string would be in the range [10-8,7.4× 105] ly giving Lgal∈ [.3× 10-6,1.0]× Rgal. Thus elliptical cannot correspond to cosmic strings. At the upper limit elliptic galaxy could correspond to straight cosmic string and the visible matter would not come from the decay of the cosmic string. This estimate conforms with the earlier proposal that only spiral galaxies correspond to cosmic strings.

See the article Solar metallicity problem from TGD perspective or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Friday, September 13, 2019

Solar metallicity problem from TGD perspective

For ten years ago it was thought that Sun is a well-understood system but more precise computations demonstrated a problem. The metallicities deduced from spectroscopic data deviate strongly from those deduced from helio-seismology and solar neutrino data as described in the Annual Review of Astronomy and Astrophysics by Martin Asplund et al, who were pioneers modelling solar surface as 3-D structure rather than idealizing it with 2-D structure.

The abundances used are determined from meteorites and these estimates are more accurate and are consistent with the values determined by Asplund et al and used also to extrapolate the metallicities in core.

  1. The metallicity of Sun deduced from spectroscopy by Asplund et al would be 1.3 per cent whereas the older model and also helio-seismology give 1.8 per cent metallicity. Is the metallicity indeed 1.3 per cent using standard model to extrapolate the spectroscopic data at surface? Or is it 1.8 per cent deeper in the interior in which case the extrapolation used to deduce metallicity in the interior would not be realistic.

  2. There are also other discrepancies. The height of convective zone at which radiative energy transfer is replaced with convection is given by RCZ= .724R. The predicted He abundance at surface is Ysurf=.231. These values are in conflict with RCZ= .713R and Ysurf=.248 deduced from helio-seismological data. Also density and sound velocity profiles deviate from those deduced from the helio-seismology. The earlier model approximating solar surface as 2-D structure is in excellent accordance with the helio-seismological data.

Dark matter identified as heff=nh0 phases has become key player in TGD inspired new physics being now a crucial element of TGD based view about living matter. Dark nuclear fusion is proposed to provide the new physics allowing to understand "cold fusion". In the following it will be found that dark matter in TGD associated with solar core could provide an elegant solution also to the solar metallicity problem.

In TGD classical physics is an exact part of quantum physics. The tunnelling phenomenon essential for nuclear physics based model of solar nuclear fusion would correspond in TGD to a state function reduction creating a phase consisting of dark nuclei which can fuse without tunnelling due to the reduction of the binding energy scale. State function reduction to ordinary phase leads to the final state of the reaction. In ZEO "big" (ordinary) state function reduction would reverse the arrow of time so that if tunnelling phenomenon is assignable to "big" state function reduction rather than TGD counterpart of "weak" measurement, ZEO would make possible nuclear fusion.

The missing nuclear matter inside core would be dark variants of nuclei associated with dark flux tubes. This would explain the conflict between the metallicities deduced from spectroscopic and meteoritic data on one hand and those deduced from helio-seismic data. The reason is that sound waves and photons in the core couple to both ordinary and dark matter so that helio-seismology gives metallicities as sums of ordinary and dark metallicities. Using the estimate for the thickness of the dark flux tube coming from the TGD based model of "cold fusion", one can estimate the length of dark flux tube inside solar core and it turns out to fill about 30 per cent of its volume.

One can relate the model also to the model for the formation of galaxies, stars, and planets as tangles assignable to cosmic strings thickened to flux tubes implying the decay of their Kähler magnetic energy to ordinary matter in analogy with the decay of inflaton field and nice quantitative estimates follow. Also a connection with twistor lift of TGD predicting hierarchy of cosmological constants emerges and the radius of solar core turns out to corresponds to the value of cosmological constant implied by the amount of missing matter identified as dark matter at flux tubes.

See the article Solar metallicity problem from TGD perspective or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Thursday, September 05, 2019

The problem with SUSY

SUSY is the basic problem of modern physics. Or rather its mis-interpretation forcing Majorana spinors and loss of fermion number conservation. Most importantly, there is no sign about SUSY in this sense at LHC but it seems that the message is still not received.

SUSY has been also a problem of TGD for decades. TGD forces a huge extension of super-conformal invariance by replacing 2-D surfaces with 3-D light-like surfaces. The extended super-conformal and super-symplectic symmetries characterize also the light-cone boundary of 4-D Minkowski space with points replaced with CP2 making the dimension of M4 unique. M4×CP2 is also forced by the existence of twistor lift of TGD.

TGD SUSY must conserve fermion number. How? TGD allows separate conservation of baryon and lepton number and the idea was that right-handed neutrino generates the least broken SUSY as N=2 supersymmetry. The idea was wrong.

The generalization of super-space geometry to a super-geometry of sub-manifolds led to a beatiful generalization of super-imbedding space in which coordinates are hermitian and their superparts are sums of monomials of quarks and antiquarks with vanishing quark number. Leptons are also possible but they are not necessary and actually excluded by SO(1,7) triality.

This also led to a generalization of second quantization of quark field implying SUSY as side product: in particular, theta parameters are replaced with oscillator operators for sparticles are created by local composites of quark oscillator operators. This is something totally new. Number theoretical vision plays crucial role in the picture.

Quark super spinors can have similar structure and the monomials in their expansion possess same electroweak quantum numbers as quarks. Leptons can be regarded as local composites of 3 antiquarks and thus spartners of quarks. SUSY would have been staring us directly to eyes for almost a century! Matter antimatter asymmetry is generated because small CP breaking predicted by TGD favors local composites of quarks as leptons over non-local composites for antibaryons. Standard model spectrum is predicted apart from pseudoscalar for which there exist indications from LHC at correct mass.

The outcome is an explicit proposal for S-matrix: S-matrix would be given by super-variant of the exponential of action defining the super-Kaehler function of "world of classical worlds" (WCW) generalized to super-Kaehler geometry. The construction of unitary S-matrix has been second challenge of TGD for decades. Rather precisely 41 years after the emergence of the basic idea of TGD I can say that it is done now!

Against this background it is somewhat frustrating to see colleagues busily planning new super-collider to test a wrong vision about SUSY already known to be excluded by LHC findings.

For details see the article Recent view about SUSY in TGD Universe .

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Tuesday, September 03, 2019

Do hydrogels learn in presence of irradiation and heating?

A research group in Aalto yliopisto led by professor Olli Ikkala has published an interesting article with title " Programmable responsive hydrogels inspired by classical conditioning algorithm". What is observed
that a system consisting of hydrogel and Gold nanoparticles can get conditioned when it is heated in the presence of irradiation at blue and red wavelengths. Conditioning means that the system melting under heating learns to melt in the presence of only irradiation. The experimenters assume that the Gold nanoparticles forming chains during heating serve as a memory element in the learning.

The TGD based quantum model for the conditioning of hydrogel system relies on TGD inspired general model of living systems extended recently to a model of quantum self-organization in which energy feed serving as metabolic energy feed induces generation of dark matter as heff=nh0 phases of ordinary matter at the magnetic body of the system. In number theoretic vision the presence of these phases correspond to higher algebraic complexity and higher "IQ".

The light signal would generate Pollack effect, which in TGD framework means transfer of protons from photo-acids to dark heff=nh0 protons at magnetic flux tubes parallel to nanoparticle chains. The "IQ" of the system or its magnetic body characterized by heff would increase and it would become able to self-organize. The energy from the heating would be stored to the nanoparticle chains taking the role of proteins as energy storage. Melting would be a self-organization process increasing complexity, and in absence of heating (and perhaps even in its presence) the gel phase would receive the energy needed from the nanoparticle chains. The conditioning in this sense would not be a passive mechanical response. The system would be macroscopic quantum system, and the energy feed would make possible for it to evolve to a higher level of complexity and conscious intelligence.

See the article Do hydrogels learn in presence of irradiation and heating? or the chapter Life-like properties observed in a very simple systems.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Monday, September 02, 2019

Has LIGO observed gravitational echoes in 21 minute time scale?

LIGO has observed for few days ago two gravitonal waves with a time lapse of 21 minutes in the same direction (see this). The events are christened as S190828j and S190828l. This suggests that the signals coule orginate from same event. Gravitational lense effect could be one explanation.

TGD suggests an alternative explanation based on the notion of gravitational flux tubes. Magnetic flux tubes, in particular gravitational flux ones, form loops. The later signal could have spent 21 minutes by rotating around this kind of loop. This rotation can occur several times but the intensity of signal is expected to diminish exponentially if only a constant fraction remains in loop at each turn.

This sticking of radiation inside magnetic loops predicting echo like phenomenon is a general prediction of TGD and I have considered the possible occurrence of this phenomenon for cosmic gamma rays arriving in solar solar system in a model for solar cycle.

This kind of repetition of the signal has been observed already earlier for gravitational waves and has been dubbed "blackhole echoes" (see this) but in a time scale of .1 seconds (fundamental bio-rhythm by the way). For possible TGD based explanations of blackhole echoes see this and this.

The two time scales differ by four orders of magnitude but one cannot exclude same explanation. With light velocity Earth sized loop would correspond to a time lapse of about .1 seconds. Light travels in 21 minutes over a distance of 378 million kilometers to be compared with astronomical unit AU = 150 million kilometers defining the distance of Earth from Sun. Therefore loops in the scale of Earth's orbit around Sun could be involved and perhaps associated with the magnetic body of the collapsed system. .1 seconds defining the time scale for the blackhole echoes in turn corresponds to a circumference of order Earth circumference.

See the article TGD View about Quasars or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Saturday, August 31, 2019

But can one calculate anything?

During this summer a dramatic progress in the understanding of SUSY in TGD framework occurred. As a consequence, there is now a rather concrete procedure for constructing S-matrix and precise formulation of quantum criticality. The simplest formulation of TGD involves only quarks, and leptons can be seen as local 3-quark composites - spartners of quarks.

The process from an idea to a precise mathematical theory took 41 years and forced a generalization of entire physics so that it is number theoretically universal and describes also the correlates of cognition. Consciousness theory becomes part of TGD based physics. Quantum physics is generalized considerably: hierarchy of Planck constants is one implication. Second deep implication is zero energy ontology (ZEO) allowing solve the basic paradox of quantum measurement theory.

In TGD framework M8-H duality allows to geometrize the notion of super-twistor in the sense that at the level of M8 different components of super-field correspond to components of super-octonion each of which corresponds to a space-time surfaces satisfying minimal surface equations with string world sheets as singularities - this is geometric counterpart for masslessness.

New view about SUSY

The progress in understanding of M8-H duality throws also light to the problem whether SUSY is realized in TGD and what SUSY breaking cold mean. It is now rather clear that sparticles are predicted and SUSY remains exact but that p-adic thermodynamics causes thermal massivation: unlike Higgs mechanism, this massivation mechanism is universal and has nothing to do with dynamics. This is due to the fact that zero energy states are superpositions of states with different masses. The selection of p-adic prime characterizing the sparticle causes the mass splitting between members of super-multiplets although the mass formula is same for all of them. Super-octonion components of polynomials have different orders so that also the extension of rational assignable to them is different and therefore also the ramified primes so that p-adic prime as one them can be different for the members of SUSY multiplet and mass splitting is obtained.

The question how to realize super-field formalism at the level of H=M4× CP2 led to a dramatic progress in the identification of elementary particles and SUSY dynamics. The most surprising outcome was the possibility to interpret leptons and corresponding neutrinos as local 3-quark composites with quantum numbers of anti-proton and anti-neutron. Leptons belong to the same super-multiplet as quarks and are antiparticles of neutron and proton as far quantum numbers are consided. One implication is the understanding of matter-antimatter asymmetry. Also bosons can be interpreted as local composites of quark and anti-quark.

Hadrons and perhaps also hadronic gluons would still correspond to the analog of monopole phase in QFTs. Homology charge could appear as a space-time correlate for color at space-time level and explain color confinement. Also color octet variants of weak bosons, Higgs, and Higgs like particle and the predicted new pseudo-scalar are predicted. They could explain the successes of conserved vector current hypothesis (CVC) and partially conserved axial current hypothesis (PCAC).

One ends up with an improved understanding of quantum criticality and the relation between its descriptions at M8 level and H-level. Polynomials describing a hierarchy of dark matters describe also a hierarchy of criticalities and one can identify inclusion hierarchies as sub-hierarchies formed by functional composition of polynomials: the criticality is criticality for the polynomials interpreted as p-adic polynomials in O(p)=0 approximation meaning the presence of multiple roots in this approximation.

Connection of SUSY and second quantization

The linear combinations monomials of theta parameters appearing in super-fields are replaced in case of hermitian H super coordinates consisting of combinations of monomials with vanishing quark number. For super-spinors of H the monomials carry odd quark number. Monomials of theta parameters are replaced by local monomials of quark oscillator operators labelled besides spin and weak isospin also by points of cognitive representation with imbedding space coordinates in an extension of rationals defining the adele. Discretization allows anti-commutators which are Kronecker deltas rather than delta functions. If continuum limit makes sense, normal ordering must be assumed to avoid delta functions at zero coming from the contractions.

The monomials (not only the coefficients appearing in them) are solved from generalized classical field equations and are linearly related to the monomials at boundary of CD playing the role of quantum fields and classical field equations determine the analogs of propagators.

The Wick contractions of quark-antiquark monomials appearing in the expansion of super-coordinate of H could define the analog of radiative corrections in discrete approach. M8-H duality and number theoretic vision require that the number of non-vanishing Wick contractions is finite. The number of contractions is bounded by the finite number of points in cognitive representation and increases with the degree of the octonionic polynomial and gives rise to a discrete coupling constant evolution parameterized by the extensions of rationals. The polynomial composition hierarchies correspond to inclusion hierarchies for isomorphic sub-algebras of super-symplectic algebra having interpretation in terms of inclusions of hyper-finite factors of type II1.

Quark oscillator operators in cognitive representation correspond to quark field q. Only terms with quark number 1 appear in q and leptons emerge in Kähler action as local 3-quark composites. Internal consistency requires that q must be the super-spinor field satisfying super Dirac equation. This leads to a self-referential condition qs=q identifying q and its super-counterpart qs. The condition has interpretation in terms of a fixed point of iteration and expression of quantum criticality. The coefficients of various terms in q analogous to coupling constants can be fixed from this condition so that one obtains discrete number theoretical coupling constant evolution. The basic equations are quantum criticality condition q=qs, Dα,sΓαs=0 coming from Kähler action, and the super-Dirac equation Dsq=0.

Could the exponent of super-Kähler function as vacuum functional define S-matrix as its matrix elements

Consider first the key ideas - some of them new - formulated as questions.

  1. Could one see SUSY in TGD sense as a counterpart for the quantization in the sense of QFT so that oscillator operators replace theta parameters and would become fermionic oscillator operators labelled by spin and electroweak spin - as proposed originally - and by selected points of 3-surface of light-cone boundary with imbedding space coordinates in extension of rationals? One would have analog of fermion field in lattice identified as a number theoretic cognitive representation for given extension of rationals. The new thing would be allowance of local composites of oscillator operators having interpretation in terms of analogs for the components of super-field.

    SUSY in TGD sense would be realized by allowing local composites of oscillator operators containing 4+4 quark oscillator operators at most. At continuum limit normal ordering would produce delta functions at origin unless one assumes normal ordering from beginning. For cognitive representations one would have only Kronecker deltas and one can consider the possibility that normal ordering is not present. The vanishing of normal ordering terms above some number of them suggested to be the dimension for the extension of rationals would give rise to a discrete coupling constant evolution due to the contractions of fermionic oscillator operators.

  2. What is dynamical in the superpositions of oscillator operator monomials? Are the coefficients dynamical? Or are the oscillator operators themselves dynamical - this would mean a QFT type reduction to single particle level? The latter option seems to be correct. Oscillator operators are labelled by points of cognitive representation and in continuum case define an analog of quantum spinor field, call it q. This suggests that this field satisfies the super counter part of modified Dirac equation and must involve also super part formed from the monomials of q and qbar. This however requires the replacement of q with qs in super-Dirac operator and super-coordinates hs and one ends up with an iteration q→ qs→ ...

    The only solution to the paradoxical situation is that one has self-referential equation q=qs having interpretation in terms of quantum criticality fixing the coefficients of terms in q=qs and in H super-coordinate hs interpreted as coupling constants so that a discrete coupling constant evolution as function of extension of rationals follows. Also super-Dirac equation Dsqs=0 and field equations Ds,αΓα,s=0 for Kähler action guaranteeing hermiticity are satisfied.

  3. Could one interpret the time reversal operation taking creation- and annihilation operators to each other as time reflection permuting the points at the opposite boundaries of CD? The positive resp. negative energy parts of zero energy states would be created by creation resp. annihilation operators from respective vacuums assigned to the opposite boundaries of CD.

  4. Could one regard preferred extremal regarded as 4-surface in super imbedding space parameterized by the hermitian imbedding coordinates plus the coefficients of the monomials of quarks and antiquarks with vanishing quark number, whose time evolution follows from dimensionally reduced 6-D super-Kähler action? Could one assume similar interpretation for super spinors consisting of monomials with odd quark number and appearing in super-Dirac action?

  5. In WKB approximation the exponent of action defines wave function. In QFTs path integral is defined by an exponent of action and scattering operator can be formally defined as action exponential. Could the matrix elements for the exponent of the super counterpart of Kähler function plus super Dirac action between states at opposite boundaries of CD between positive and negative energy parts of zero energy states define S-matrix? Could the positive and negative energy parts of zero energy states be identified as many particles states formed from the monomials associated with imbedding space super-coordinates and super-spinors?

  6. Could the construction of S-matrix elements as matrix elements of super-action exponential reduce to classical theory? Super-field monomials in the interior of CD would be linear superpositions of super-field monomials at boundary of CD. Note that oscillator operator monomials rather than their coefficients would be the basic entities and the dynamics would reduce to that for oscillator operators as in QFTs. The analogs of propagators would relate the monomials to those at boundary ly to the monomials at the boundary of CD, and would be determined by classical field equations so that in this sense everything would be classical. Note however that the fixed point condition q=qs and super counterpart of modified Dirac equation are non-linear.

    Vertices would be defined by monomials appearing in super-coordinate and super-spinor field appearing in terms of those at boundary of CD. If two vertices at interior points x and y of CD are connected there is line leading from x to a point z at boundary of CD and back to y and one would have sum over points z in cognitive representation. This applies also to self energy corrections with x=y. At the possibly existing continuum limit integral would smoothen the delta function singularities and in presence of normal ordering at continuum would eliminate them.

    In the expressions for the elements of S-matrix annihilation operators appearing in the monomials would be connected to the passive boundary P of CD and creation operators to the active boundary. If no partonic 2-surfaces appear as topological vertices in the interior of CD, this would give trivial S-matrix!

    M8-H duality however predicts the existence of brane like entities as universal 6-D surfaces as solutions of equations determining space-time surfaces. Their M4 projection is t=rn hyperplane, where rn corresponds to a root of a real polynomial with algebraic coefficients giving rise to octonion polynomial, and is mapped to similar surface in H. 4-D space-time surfaces representing incoming and outgoing lines would meet along their ends at these partonic 2-surfaces.

    Partonic 2-surfaces at these hyper-surfaces would contain ordinary vertices as points in cognitive representation. Given vertex would have at most 4+4 incoming and outgoing lines assignable to the monomial defining the vertex. This picture resembles strongly the picture suggested by twistor Grassmannian approach. In particular the number of vertices is finite and their seems to be no superposition over different diagrams. In this proposal, the lines connecting vertices would correspond to 1-D singularities of the space-time surfaces as minimal surfaces in H. Also stringy singularities can be considered but also these should be discretized.

    By fixing the set of monomials possibly defining orthonormal state basis at both boundaries one would obtain given S-matrix element. S-matrix elements would be matrix elements of the super-action exponential between states formed by monomials of quark oscillator operators. Also entanglement between the monomials defining initial and finals states can be allowed. Note that this in principle allows also initial and final states not expressible using monomials but that monomials are natural building bricks as analogs of field operators in QFTs.

  7. The monomials associated with imbedding space coordinates are imbedding space vectors constructible from Dirac currents (left- or right-handed) with oscillator operators replacing the induced spinor field and its conjugate. The proposed rules for constructing S-matrix would give also scattering amplitudes with odd quark number at boundaries of CD. Could the super counterpart of the bosonic action (super Kähler function) be all that is needed to construct the S-matrix?

    In fact, classically Dirac action vanishes on mass shell: if this is true also for super-Dirac action then the addition of Dirac action would not be needed. The super-Taylor expansion of super- Kähler action gives rise to the analogs of perturbation theoretic interaction terms so that one has perturbation theory without perturbation theory as Wheeler might state it. The detailed study of the structure of the monomials appearing in the super-Kähler action shows that they have interpretation as currents assignable to gauge bosons and scalar and pseudo-scalar Higgs.

    Super Dirac action is however needed. Super-Dirac equation for q and Dα,sΓαs=0 allow to reduce ordinary divergences ∂αjα of fermionic currents appearing in super-Kähler action to commutators [Aα,sjα]. Therefore no information about q at nearby points is needed and one avoids lattice discretization: cognitive representation is enough.

  8. Topological vertices represent discontinuities of the space-time surface bringing strongly in mind the non-determinism of quantum measurement, and one can ask whether the 3-branes and associated partonic 2-surfaces. Could the state function reductions analogous to weak measurements correspond to these discontinuities? Ordinary state function reductions would change the arrow of time and the roles of active and passive boundaries of CD. In TGD inspired theory of consciousness these time values would correspond to "very special moments" in the life of self.

But can one calculate anything?

The path from precise formulation to concere calculations is long since TGD is much more ambitious approach than the usual models based on action and Feynman rules. One can ask whether the information needed to make calculations is in principle available in number theoretic approach based on cognitive representations.

  1. The condition that super-Dirac equation is satisfied would remove the need to have a lattice and cognitive representation would be enough. If the condition ∂αq=0 holds true, the situation simplifies even more but this condition is not essential. The condition that the points of the cognitive representation assignable to quark oscillator operators correspond to singularities of space-time surface at which several space-time sheets intersect, would make the identification of these points of cognitive representation easier. Note that the notion of singular point makes sense also at the continuum limit giving cognitive representation even in this case in terms of possibly transcendental roots of octonion analytic functions.

    If the singular points correspond to solution to 4 polynomial conditions on octonionic polynomials besides the 4 conditions giving rise to the space-time surfaces. The intersections for two branches representing two roots of polynomial equation for space-time surface indeed involve 4 additional polynomial conditions so that the points would have coordinates in an extension of rationals, which is however larger than for the roots t=rn. One could of course consider an additional condition requiring that the points belong to the extension defined by rn but this seems un-necessary.

    The octonionic coordinates used at M8-side are unique apart from a translation of real coordinate and value of the radial light-like coordinate t=rn corresponds to a root of the polynomial defining the octonionic polynomial as its algebraic continuation. At this plane the space-time surfaces corresponding to polynomials defining external particles as space-time surfaces would intersect at partonic 2-surfaces containing the shared singular points defined as intersections.

  2. The identification of cognitive representations goes beyond the recent knowhow in algebraic geometry. I have considered this problem in light of some recent number theoretic ideas. If the preferred extremals are images of octonionic polynomial surfaces and M8-H duality the situation improves, and one might hope of having explicit representation of the images surfaces in H-side as minimal surfaces defined by polynomials.

See the article Recent View about SUSY in TGD Universe or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.