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 2^1/12 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.

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 h_eff 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 R_gal of galaxy is estimated to be in the range [.75,1.0]× 10⁵ ly. The length of string within galactic radius would satisfy L_gal=[.8,3.3]R_gal. The estimate excludes the lower bound. For the upper bound the one has L_gal ∼ 3.3 × R_gal.
   

   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 L_BH∼ .1 ly. The size of the galactic blackhole (see this) is R_S,BH∼ 4.4× 10^-5 ly giving R_S,BH/L_gal∼ 4.4× 10^-4. One has T_max∼ 10^-6/G and blackhole corresponds effectively to a string with tension T_G∼ 1/2G and length R_S,BH so that the ratio would be R_S,BH/L_gal ∼ 2G/T_max∼ 2× 10^-6. The straight string with length L_BH would have been compressed to a volume of Schwartchild radius R_B,S∼ 2^-11L_BH.
   
   
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 L_gal=3.3× R_gal. 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 L_gal∈ [.3× 10^-6,1.0]× R_gal. 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.

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 R_CZ= .724R. The predicted He abundance at surface is Y_surf=.231. These values are in conflict with R_CZ= .713R and Y_surf=.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 h_eff=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.

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.

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 h_eff=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 h_eff=nh₀ protons at magnetic flux tubes parallel to nanoparticle chains. The "IQ" of the system or its magnetic body characterized by h_eff 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.

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.

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 q_s=q identifying q and its super-counterpart q_s. 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=q_s, D_α,sΓ^α_s=0 coming from Kähler action, and the super-Dirac equation D_sq=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 q_s in super-Dirac operator and super-coordinates h_s and one ends up with an iteration q→ q_s→ ...
   

   The only solution to the paradoxical situation is that one has self-referential equation q=q_s having interpretation in terms of quantum criticality fixing the coefficients of terms in q=q_s and in H super-coordinate h_s interpreted as coupling constants so that a discrete coupling constant evolution as function of extension of rationals follows. Also super-Dirac equation D_sq_s=0 and field equations D_s,αΓ^α,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=q_s 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=r_n hyperplane, where r_n 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 D_α,sΓ^α_s=0 allow to reduce ordinary divergences ∂_αj^α of fermionic currents appearing in super-Kähler action to commutators [A_α,sj^α]. 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=r_n. One could of course consider an additional condition requiring that the points belong to the extension defined by r_n 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=r_n 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.

The recent view about SUSY in TGD Universe

The progress in understanding of M⁸-H duality throws also light to the problem whether SUSY is realized in TGD and what SUSY breaking does 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.

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 hadronic gluons would still correspond to the analog of monopole phase in QFTs. Homology charge would appear as 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 the precise understanding of quantum criticality and understand 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 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 indeed 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.

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 q_s=q identifying q and its super-counterpart q_s. 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=q_s, D_α,sΓ^α_s=0 coming from Kähler action, and the super-Dirac equation D_sq=0.

One also ends up to the first completely concrete proposal for how to construct S-matrix directly from the solutions of super-Dirac equations and super-field equations for space-time super-surfaces. The idea inspired by WKB approximation is that the exponent of the super variant of Kähler function including also super-variant of Dirac action defines S-matrix elements as its matrix elements between the positive and negative energy parts of the zero energy states formed from the corresponding vacua at the two boundaries of CD annihilated by annihilation operators and resp. creation operators. The states would be created by the monomials appearing in the super-coordinates and super-spinor.

Super-Dirac action vanishes on-mass-shell. The proposed construction relying on ZEO allows however to get scattering amplitudes between all possible states using the exponential of super-Kähler action. Super-Dirac equation makes possible to express the derivatives of the quark oscillator operators (values of quark field at points of cognitive representation) so that one can use only the points of cognitive representation without introducing lattice discretization. Discrete coupling constant evolution follows from the fact that the contractions of oscillator operators occur at the boundary of CD and their number is limited by the finite number of points of cognitive representation.

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.

A visit to Baden-Baden

I participated a symposium held in Baden-Baden (31st International Conference on Systems Research, Informatics and Cybernetics – InterSymp 2019: Symposium on Causal and Anticipative Systems in Living Science, Biophysics, Quantum Mechanics, Relativity). The visit was pleasant: Baden-Baden is small town which has preserved its personal warm character.

I wrote a slightly extended version of the related article mentioning the recent experimental finding of Minev et al supporting zero energy ontology crucial for the approach. The following is the abstract of the article.

Topological Geometro-Dynamics (TGD) proposes a unification of fundamental interactions by identifying space-times as 4-surfaces in 8-D space H=M⁴× CP₂, whose geometry codes for standard model symmetries and geometrizes known fields. Point-like particle is replaced with 3-surface (3-space). One ends up with the notions of many-sheeted space-time and magnetic body (MB) central in TGD inspired quantum biology. p-Adic and adelic physics follows from the extension of physics to describe also the correlates of cognition and imagination. Adelic physics predicts a hierarchy of Planck constants labelling phases of ordinary matter interpreted as dark matter: the predicted quantum coherence in arbitrarily long scales is crucial for quantum biology. Quantum TGD replaces standard ontology with "zero energy ontology" (ZEO) replacing quantum state as time=constant snapshot with zero energy state (ZES) identified as a superposition of deterministic classical time evolutions - kind of quantum program.

See the article TGD inspired theory of consciousness and living systems.

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

Articles and other material related to TGD.

Could metaplectic group have some role in TGD framework?

Metaplectic group appears as a covering group of linear symplectic group Sp(2n,F) for any number field and its representations can be regarded as analogs of spinor representations of the rotation group. Since infinite-D symplectic group of δ M⁴₊× CP₂, where δ M⁴₊ is light-cone boundary, appears as an excellent candidate for the isometries of the "world of classical worlds" in zero energy ontology (ZEO), one can ask whether and how the notion of metaplectic group could generalize to TGD framework.

The condition for the existence of metaplectic structure is same as of the spinor structure and not met in the case of CP₂. One however expects that also the modified metaplectic structure exists if one couples spinors to an odd integer multiple of Kähler gauge potential. For triality 1 representation assignable to quarks one has n=1. The fact that the center of SU(3) is Z₃ suggests that metaplectic group for CP₂ is 3- or 6-fold covering of symplectic group instead of 2-fold covering.

Besides the ordinary representations of SL(2,C) also the possibly existing analogs of metaplectic representations of SL(2,C) = Sp(2,C) acting on wave functions in hyperbolic space H₃ represented as a²=t²-r² hyperbololoid of M⁴₊ are cosmologically interesting since the many-sheeted space-time in number theoretic vision allows quantum coherence in even cosmological scales and there are indications for periodic redshift suggests tesselations of H₃ analogous to lattices in E³ and defined by discrete subgroup of Sl(2,C). In this case one could require that only the subgroup SU(2) is represented projectively so that one would have an analogy with modular functions for discrete subgroup of SL(2,Z) would be represented in this manner.

See the chapter Recent View about K\"ahler Geometry and Spin Structure of WCW.

Phase transitions generating dark phases and sensory perception

The TGD based model for biological self-organization relies on the hierarchy h_eff=nh₀ of effective Planck constants labelling dark phases of ordinary particles residing at magnetic flux quanta. This model generalizes and suggests the replacement of non-equilibrium thermodynamics as basis of self-organization with its quantum variant based on dark matter hierarchy. The challenge is to formulate basic thermodynamical notions like work in terms TGD based quantum theory relying on zero energy ontology (ZEO).

The basic mechanism would be a phase transition creating dark matter phase as a Bose-Einstein condensate like state with particles having identical conserved quantum numbers. Conservation laws would force the ordinary matter to have opposite total charges. For instance, in the case of work one has momentum or angular momentum as a conserved charge. In the case of charge separation and high Tc superconductivity it would be em charge. Even color charges can correspond to conserved charges in TGD framework allowing scaled variants of strong interaction physics.

Basic biological functions involving the notion of work and also the formation of sensory percepts would rely on this mechanism. Also the ZEO based theory of consciousness predicting the change the arrow of time in ordinary state function reduction plays a central role and a model of nerve pulse is discussed as an example.

Sensory perception (time reversal of motor action could involve generation of coherent phases of dark matter carrying collective quantum numbers in 1-1 corresponds with the sensory qualia. This would represent a general charge separation process.

Consider first sensory capacitor model for color qualia.

1. The notion of QCD color as analog of ordinary visual colors was originally introduced as a joke since the algebra of color summation resembles that for the summation of QCD colors in tensor product. In TGD however the dark hierarchy (h_eff) and p-adic length scale hierarchy predict that scaled variants of QCD type physics are possible for arbitrarily large length scales. In cellular scales scaled up QCDs are predicted. In the length scale range between cell membrane thickness and nucleus size there are as many as 4 Gaussian Mersennes, which is a number theoretical miracle. They could label copies of QCDs with size scale for the analogs of hadrons given by the corresponding p-adic length scales. QCD type colors could correspond to perceived colors.
   
   
2. Gluons or quarks labelled by color charge characterizing particular color quale would flow between the plates of "capacitor" associated with the sensory receptor. The amount of particular color charge increases at the other plate giving rise to sensation of this particular color quale and its complement at the other plate - by color confinement also the same plate could also contain regions with complementary colors. This would explain why we see around a region of particular color a narrow boundary with complementary color.
   
   
3. The model for sensory perception as sequences of analogs of weak measurements suggest that the flow of color charges could induce color qualia. The prediction emerging from the structure of SU(3) color algebra would be four pairs of basic color and complement color: 3 ordinary pairs and white-black pair. They could correspond to particular changes of color quantum numbers and color quantum numbers of gluons. Also color mixing could be understood.
   
   
4. Photons are not coloured but gluons (and also quarks) are, and the latter and could be responsible for color sensation. How photon flux can generate a flow of color quantum numbers? The notion of induced gauge field -classical color gauge potentials would be projections of SU(3) Killing vectors - explains this.
   

   In TGD classical em field is sum of two terms induced Kähler form and neutral vectorial component of spinor curvature. Classical gluon field has components proportional to classical color Hamiltonian (function in CP₂ which can be said to have quantum numbers of gluon) and induced Kähler form. In general case any classical em field is accompanied by a classical color field.
   

   Photons are accompanied by classical em fields and therefore also by classical gluon fields at the fundamental level: this correspondence disappears at QFT limit unable to describe biology and sensory experience. The flow of photons to retina would be accompanied by classical em and color fields and therefore a flow of gluons. Also quark flow between the plates of sensory capacitor could generate the color qualia.
   
   

5. A simple model for the visual qualia is in terms of a phase transition transforming gluons of a scaled copy of QCD to ordinary gluons. Dark gluons would form a BE condensate and force a formation its shadow at the level of ordinary matter. This is a variant of sensory receptor as quantum capacitor. The plates of capacitor correspond to dark and ordinary phase and the analog of electric breakdown means formation of the dark phase. Cooper pairs of quarks with quantum numbers of gluon would be second option but gluons in TGD framework are actually this kind of pairs!!
   
   

See the article Quantum self-organization by h_eff changing phase transitions or the chapter General theory of qualia.

Zero energy ontology and quantum model for nerve pulse

In TGD based model of nerve pulse axonal membrane is generalized cylindrical Josephson junction defined by axonal membrane consisting of smaller Josephson junctions defined by membrane proteins.

1. A sequence of mathematical penduli along axon in rotation in the same direction is the mechanical analog. Oscillation frequency Ω transforming to a rotation frequency above critical value is proportional to the resting potential V. When V is overcritical, the pendulum starts to rotate instead of oscillating. The system should be near quantum criticality for the transformation of rotation to oscillation or vice versa.
   
   
2. During nerve pulse membrane potential and therefore also rotation frequency is reduced and changes sign and then returns back to the original value. The first guess is that at criticality there is a kick reducing the rotation frequency Ω and continuing to change its sign and then return it to original.
   
   

The basic condition is that resting state becomes critical at critical hyper-polarization. There are two options for the resting state.

1. According to the original model resting state can be regarded as a soliton sequence associated with the phase difference over the membrane. More concretely, the mathematical penduli rotate in same direction with phase difference between determining the propagation velocity of solitons. The rotation frequency is slightly above that for oscillation. There is a preferred direction along axon. This conforms with the reduction and change of sign of potential and thus of Ω.
   

   Problem: Hypo- rather than hyper-polarization should cause the nerve pulse as a transformation of rotation to oscillation. Something goes wrong.
   
   

2. Alternatively, the penduli almost rotate being near criticality for the rotation: the penduli almost reaches the vertical position at each oscillation as required by criticality. That hyper-polarization would cause the nerve pulse as propagating soliton conforms with this idea.
   

   Problem: Ω and thus V should increase rather than reduce and even change sign temporarily.
   
   

Neither option seems to work as such but the first option is more plausible as a starting point of an improved model.

The membrane potential changes sign suggesting quantum jump. Could zero energy ontology (ZEO) based view about quantum jump as "big" (ordinary) state function reduction (BSFR) help? Could nerve pulse correspond to BSFR?

1. Could BSFR occur changing temporarily the arrow of time in ZEO and induce nerve pulse. Could opposite BSFR take place after this in millisecond scale and establish the original arrow of time. Using the language of TGD inspired theory of consciousness a conscious entity, sub-self or mental image, would die and reincarnate with an opposite arrow of time, live for the duration of nerve pulse and then die and reincarnate with the original arrow of time. Nerve pulse would be a propagation of a temporary neuron death along the axon and would occur as neuron becomes hyper-polarized.
   
   
2. In the article about the recent findings of Minev et al related to quantum jump in atomic physics are discussed. ZEO predicting that the arrow of time is changed in BSFR. This would create the illusion that discontinuous quantum jumps correspond to a classical time evolution leading smoothly and deterministically to the final state.
   

   This because BSFR leads to a state with reversed arrow of time, which corresponds to a superposition of classical time evolutions leading from the final state to the geometric past and it this, which is observed.
   This would also explain why the removal of the irradiation inducing quantum jumps has no effect during the transition process and why a stimulation inducing opposite quantum jump can stop the process. Also the findings of Libet related to the active aspects of consciousness showing that neural activity seems to preceded volitional act can be understood in this framework without giving up the notion of free will.
   

   The first half of the nerve pulse would correspond to this apparent evolution to the time reversed final state with opposite membrane potential but actually being time reversed evolution from the final state. The second half of nerve pulse would correspond to opposite state function reduction establishing the original arrow of time. This model looks attractive but many details remain to be checked.
   
   

Why hyper-polarization should cause the temporary death of neuron or its subself?

1. Metabolic energy feed is needed to preserve the polarization of neuron since membrane potential tends to get reduced by second law stating that all gradients are bound to decrease. There should be some maximal polarization possible to preserve using the existing metabolic energy resources.
   
   
2. Does quantum jump to a state with opposite arrow of time happen as this limit is reached? Why? Could the metabolic energy feed stop causing the neuron to die to starvation? Why the death of neuron should happen so fast? Could the quantum criticality against the change of rotation to oscillation be the reason. When neuron cannot rotate anymore it would die immediately: the mental image "I am rotating" would die and reincarnate as its time reversal. Does the neuron feeded by metabolic energy become a provider metabolic energy during this period somewhat like dead organisms after their death. Can one conclude that this energy goes to some purpose inside neuron?
   
   

See the article Quantum self-organization by h_eff changing phase transitions or the chapter Quantum model for nerve pulse.

Quantum self-organization by heff changing phase transitions

The TGD based model for biological self-organization relies on the hierarchy h_eff=nh₀ of effective Planck constants labelling dark phases of ordinary particles residing at magnetic flux quanta. This model generalizes and suggests the replacement of non-equilibrium thermodynamics as basis of self-organization with its quantum variant based on dark matter hierarchy. The challenge is to formulate basic thermodynamical notions like work in terms TGD based quantum theory.

The basic mechanism would be a phase transition creating dark matter phase as a Bose-Einstein condensate like state with particles having identical conserved quantum numbers. Conservation laws would force the ordinary matter to have opposite total charges. For instance, in the case of work one has momentum or angular momentum as a conserved charge. In the case of charge separation and high Tc superconductivity it would be em charge. Even color charges can correspond to conserved charges in TGD framework allowing scaled variants of strong interaction physics.

Basic biological functions involving the notion of work and also the formation of sensory percepts would rely on this mechanism. Also the ZEO based theory of consciousness predicting the change the arrow of time in ordinary state function reduction plays a central role and a model of nerve pulse is discussed as an example.

See the article Quantum self-organization by h_eff changing phase transitions or the chapter Quantum Theory of Self-Organization.

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

Articles and other material related to TGD.

About the mathematics needed in TGD

The following is a comment from FB discussion. Since the answer developed to a summary of the mathematics needed in TGD, I decided to make it blog post.

I think that TGD is a "problem" for anyone in the sense that it is very difficult to get graps of what it really is. The reason is that TGD have been silenced for 4 decades - censorship in archives started around 93 or so and has had fatal consequences. The idea about the hegemony of M-theory estabilished by censorship was terribly silly.

TGD is an outcome of concentrated effort lasted more than 4 decades and involves 24 books. I guess that the minimum time to get some perspective to TGD is one year. For instance, I have worked with twistors for about 10 years and gradually begin to really understand the twistorialization of TGD. I am skeptic about communication of TGD without a long series of lectures and personal face to face discussions. This has not been possible, and now also my age poses strong limitations.

Concerning math, I am not interested in technical things such as producing a new p-adic variant of self-interacting phi⁴ scalar field theory. Usually mathematically oriented people do this kind of things in lack of wider physics perspective. I have a big developing a vision binding physics, consciousness, and biology, and I want to identify and even develop the mathematics needed if needed.

1. The math involves sub-manifold geometry intended to geometrize field theory by replacing field patterns with 4-surfaces using generalization of the induction procedure for metric, spinor connection, and spinors (also twistors). Kaehler geometry is in central role. Induction is something new for physics, but well-known for mathematicians.
   
   
2. The great vision involves geometrization of quantum theory in terms of infinite-D Kaehler geometry for the world of "classical world". Wheeler with his superspace was pioneer in general relativity and loop spaces are predecessors in string theory. Infinite-D symmetries as generalization of superstringy conformal symmetries are in pivotal role: they guarantee the existence of infinite-D Kaehler geomery.
   
   
3. Number theoretic vision is complementary to geometric vision and number theory including extensions of rationals, p-adic number fields and their extensions induced by extensions of rationals and classical number fields. Extending the physics to a description of also cognition is the great vision and brings in p-adic number fields, adeles, and predicts hierarchy of Planck constants characterizing dark matter in TGD sense. Cognitive representations are fundamental notion and provide a unique number theoretical discretization of classical and quantum physics. One powerful implication is discretization of coupling constant evolution forced solely by number theoretical universality.
   
   
4. The generalization of twistors to 8-D context to solve basic problem of standard twistor approach (only massless particles are allowed) is part of the vision. A new element is the replacement of space-time surfaces with their twistor spaces represented as 6-surfaces in 12-D twistor space of imbedding space. One powerful prediction is that M⁴×CP_2 implied by standard model is the only possible choice for the imbedding space besides M⁸ which is equivalent dual choice: the reason is that only the spaces M⁴, E⁴ and CP₂ allow twistor space with Kaehler structure. TGD is unique from its existence.
   
   
5. I have considered also categories occasionally. I tend to see them as tools to organize and excellent for bureaucratic challenges: they do not seem to code for core ideas. As a physicist I see interpretations of quantum theory as as unsuccessful attempts to get rid of the basic problem of the measurement theory: here zero energy ontology (ZEO) provides the great vision and leads to a theory of consciousness and quantum biology.
   

   The great idea is terribly simple: transform initial value problem to boundary value problem by replacing initial values at t= constant hyperplane with boundary values characterizing deterministic time evolution t₁→>t₂ and by replacing quantum states with superpositions of deterministic time evolutions.
   
   

The mathematics of TGD has not been found by going to math library but developing the physical vision and generalizing the existing mathematical ideas appropriately. I trust to physics based arguments much more than mathematical proofs involving all kinds of technical assumptions.

Those interested in material about TGD can find links from Material about TGD.

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

Articles and other material related to TGD.

What extensions of rationals could be winners in the fight for survival?

It would seem that the fight for survival is between extensions of rationals rather than individual primes p. Intuition suggests that survivors tend to have maximal number of ramified primes. These number theoretical speciei can live in the same extension - to "co-operate".

Before starting one must clarify for myself some basic facts about extensions of rationals.

1. Extension of rationals are defined by an irreducible polynomial with rational coefficients. The roots give n algebraic
   numbers which can be used as a basis to generate the numbers of extension ast their rational linear combinations. Any number of extension can be expressed as a root of an irreducible polynomial. Physically it is is of interest, that in octonionic picture infinite number of octonionic polynomials gives rise to space-time surface corresponding to the same extension of rationals.
   
   
2. One can define the notion of integer for extension. A precise definition identifies the integers as ideals. Any integer of extension are defined as a root of a monic polynomials P(x)=xⁿ+p_n-1x^n-1+...+p₀ with integer coefficients. In octonionic monic polynomials are subset of octonionic polynomials and it is not clear whether these polynomials could be all that is needed.
   
   
3. By definition ramified primes divide the discriminant D of the extension defined as the product D=∏_{i≠ j} (r_i-r_j) of differences of the roots of (irreducible) monic polynomial with integer coefficients defining the basis for the integers of extension. Discriminant has a geometric interpretation as volume squared for the fundamental domain of the lattice of integers of the extension so that at criticality this volume interpreted as p-adic number would become small for ramified primes an vanish in O(p) approximation. The extension is defined by a polynomial with rational coefficients and integers of extension are defined by monic polynomials with roots in the extension: this is not of course true for all monic polynomials polynomial (see this).
   
   
4. The scaling of the n-1-tuple of coefficients (p_n-1,.....,p₁) to (ap_n-1,a²p_n-1.....,aⁿp₀) scales the roots by a: x_n→ ax_n. If a is rational, the extension of rationals is not affected. In the case of monic polynomials this is true for integers k. This gives rational multiples of given root.
   

   One can decompose the parameter space for monic polynomials to subsets invariant under scalings by rational k≠ 0. Given subset can be labelled by a subset with vanishing coefficients {p_ik}. One can get rid of this degeneracy by fixing the first non-vanishing p_n-k to a non-vanishing value, say 1. When the first non-vanishing p_k differs from p₀, integers label the polynomials giving rise to roots in the same extension. If only p₀ is non-vanishing, only the scaling by powers kⁿ give rise to new polynomials and the number of polynomials giving rise to same extension is smaller than in other cases.
   

   Remark: For octonionic polynomials the scaling symmetry changes the space-time surface so that for generic polynomials the number of space-time surfaces giving rise to fixed extension is larger than for the special kind polynomials.
   
   

Could one gain some understanding about ramified primes by starting from quantum criticality? The following argument is poor man's argument and I can only hope that my modest technical understanding of number theory does not spoil it.

1. The basic idea is that for ramified primes the minimal monic polynomial with integer coefficients defining the basis for the integers of extension has multiple roots in O(p)=0 approximation, when p is ramified prime dividing the discriminant of the monic polynomial. Multiple roots in O(p)=0 approximation occur also for the irreducible polynomial defining the extension of rationals. This would correspond approximate quantum criticality in some p-adic sectors of adelic physics.
   
   
2. When 2 roots for an irreducible rational polynomial co-incide, the criticality is exact: this is true for polynomials of rationals, reals, and all p-adic number fields. One could use this property to construct polynomials with given primes as ramified primes. Assume that the extension allows an irreducible olynomial having decomposition into a product of monomials =x-r_i associated with roots and two roots r₁ and r₂ are identical: r₁=r₂ so that irreducibility is lost.
   

   The deformation of the degenerate roots of an irreducible polynomial giving rise to the extension of rationals in an analogous manner gives rise to a degeneracy in O(p)=0 approximation. The degenerate root r₁=r₂ can be scaled in such a manner that the deformation r₂=r₁(1+q)), q=m/n=O(p) is small also in real sense by selecting n>>m.
   

   If the polynomial with rational coefficients gives rise to degenerate roots, same must happen also for monic polynomials. Deform the monic polynomial by changing (r₁,r₂=r₁) to (r₁,r₁(1+r)), where integer r has decomposition r=∏_pi^k_i to powers of prime. In O(p)=0 approximation the roots r₁ and r₂ of the monic polynomial are still degenerate so that p_i represent ramified primes.
   

   If the number of p_i is large, one has high degree of ramification perhaps favored by p-adic evolution as increase of number theoretic co-operation. On the other hand, large p-adic primes are expected to correspond to high evolutionary level. Is there a competition between large ramified primes and number of ramified primes? Large h_eff/h₀=n in turn favors large dimension n for extension.
   
   

3. The condition that two roots of a polynomial co-incide means that both polynomial P(x) and its derivative dP/dx vanish at the roots. Polynomial P(x)= xⁿ +p_n-1x^n-1+..p₀ is parameterized by the coefficients which are rationals (integers) for irreducible (monic) polynomials. n-1-tuple of coefficients (p_n-1,.....,p₀) defines parameter space for the polynomials. The criticality condition holds true at integer points n-1-D surface of this parameter space analogous to cognitive representation.
   

   The condition that critical points correspond to rational (integer) values of parameters gives an additional condition selecting from the boundary a discrete set of points allowing ramification. Therefore there are strong conditions on the occurrence of ramification and only very special monic polynomials are selected.
   

   This suggests octonionic polynomials with rational or even integer coefficients, define strongly critical surfaces, whose p-adic deformations define p-adically critical surfaces defining an extension with ramified primes p. The condition that the number of rational critical points is non-vanishing or even large could be one prerequisite for number theoretical fitness.
   
   

4. There is a connection to catastrophe theory, where criticality defines the boundary of the region of the parameter space in which discontinuous catastrophic change can take place as replacement of roots of P(x) with different root. Catastrophe theory involves polynomials P(x) and their roots as well as criticality. Cusp catastrophe is the simplest non-trivial example of catastrophe surface with P(x)= x⁴/4-ax-bx²/2: in the interior of V-shaped curve in (a,b)-plane there are 3 roots to dP(x)=0, at the curve 2 solutions, and outside it 1 solution. Note that now the parameterization is different from that proposed above. The reason is that in catastrophe theory diffeo-invariance is the basic motivation whereas in M⁸ there are highly unique octonionic preferred coordinates.
   
   

If p-adic length scale hypothesis holds true, primes near powers of 2, prime powers, in particular Mersenne primes should be ramified primes. Unfortunately, this picture does not allow to say anything about why ramified primes near power of 2 could be interesting. Could the appearance of ramified primes somehow relate to a mechanism in which p=2 as a ramified prime would precede other primes in the evolution. p=2 is indeed exceptional prime and also defines the smallest p-adic length scale.

For instance, could one have two roots a and a+2^k near to each other 2-adically and could the deformation be small in the sense that it replaces 2^k with a product of primes near powers of 2: 2^k = ∏_i 2^k_i→ ∏_ip_i, p_i near 2^k_i? For the irreducible polynomial defining the extension of rationals, the deforming could be defined by a→ a+2^k could be replaced by a→ a+2^k/N such that 2^k/N is small also in real sense.

See the article Trying to understand why ramified primes are so special physically or the chapter TGD View about Coupling Constant Evolution.

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

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Trying to understand why ramified primes are so special physically

Ramified primes (see this and this) are special in the sense that their expression as a product of primes of extension contains higher than first powers and the number of primes of extension is smaller than the maximal number n defined by the dimension of the extension. The proposed interpretation of ramified primes is as p-adic primes characterizing space-time sheets assignable to elementary particles and even more general systems.

In the following Dedekind zeta functions (see this) as a generalization of Riemann zeta are studied to understand what makes them so special. Dedekind zeta function characterizes given extension of rationals and is defined by reducing the contribution from ramified reduced so that effectively powers of primes of extension are replaced with first powers.

If one uses the naive definition of zeta as analog of partition function and includes full powers P_i^e_i, the zeta function becomes a product of Dedekind zeta and a term consisting of a finite number of factors having poles at imaginary axis. This happens for zeta function and its fermionic analog having zeros along imaginary axis. The poles would naturally relate to Ramond and N-S boundary conditions of radial partial waves at light-like boundary of causal diamond CD. The additional factor could code for the physics associated with the ramified primes.

The intuitive feeling is that quantum criticality is what makes ramified primes so special. In O(p)=0 approximation the irreducible polynomial defining the extension of rationals indeed reduces to a polynomial in finite field F_p and has multiple roots for ramified prime, and one can deduce a concrete geometric interpretation for ramification as quantum criticality using M⁸-H duality.

See the article Trying to understand why ramified primes are so special physically or the chapter TGD View about Coupling Constant Evolution.

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

Articles and other material related to TGD.

Libet's paradoxical findings and strange findings about state function reduction in atomic scales

Perceiving is basically quantum measuring, More precisely, perceptions correspond to the counterparts of so called weak measurements in TGD (zero energy ontology) analogous to classical measurements. The observables measured in weak measurements are such that they commute with the observables whose eigenstate is the permanent part of self, the "soul". Big ( that is ordinary) state function reductions mean the death of self and its reincarnation with opposite arrow of time. This holds universally in all scales.

For the change of the arrow of time the recent findings gave direct support in atomic scales (see this). Effectively there is a deterministic process leading to the final state of reduction. This is an illusion: reduction produces superposition of deterministic classical time evolutions beginning from the final state but backwards in time of observer. Experimenters misinterpreted this as time evolution with standard arrow of time leading to the final state of reduction.

Also Libet's findings about active aspects of consciousness can be interpreted in ZEO along the same lines. The observation that the neural activity begins before conscious decision can be understood by saying that the act of free will as a big state function reduction changed the arrow of time for an appropriate subsystem of the system studied. Tte time reversed classical evolutions from the outcome of the volitional action were interpreted erratically as a time evolution leading to the conscious decision. A less precise manner to say this is that conscious decision (big state function reduction) sent a classical signal to geometric past with opposite arrow of time initiating neural activity. Libet's finding led physicalistic neuroscientists to conclude that free will is an illusion. The actual illusions were physicalism and the belief that arrow of time is always the same.

See the article Copenhagen interpretation dead: long live ZEO based quantum measurement theory! or the chapter Life and Death and Consciousness of "TGD based view about living matter and remote mental interactions: Part I".

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

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M8-H duality and twistor space counterparts of space-time surfaces

It seems that by identifying CP_3,h as the twistor space of M⁴, one could develop M⁸-H duality to a surprisingly detailed level from the conditions that the dimensional reduction guaranteed by the identification of the twistor spheres takes place and the extensions of rationals associated with the polynomials defining the space-time surfaces at M⁸- and twistor space sides are the same. The reason is that minimal surface conditions reduce to holomorphy meaning algebraic conditions involving first partial derivatives in analogy with algebraic conditions at M⁸ side but involving no derivatives.

1. The simplest identification of twistor spheres is by z₁=z₂ for the complex coordinates of the spheres. One can consider replacing z_i by its Möbius transform but by a coordinate change the condition reduces to z₁=z₂.
   
   
2. At M⁸ side one has either RE(P)=0 or IM(P)=0 for octonionic polynomial obtained as continuation of a real polynomial P with rational coefficients giving 4 conditions (RE/IM denotes real/imaginary part in quaternionic sense). The condition guarantees that tangent/normal space is associative.
   

   Since quaternion can be decomposed to a sum of two complex numbers: q= z₁ + Jz₂ RE(P)=0 correspond to the conditions Re(RE(P))=0 and Im(RE(P))=0. IM(P)=0 in turn reduces to the conditions Re(IM(P))=0 and Im(IM(P))=0.
   
   

3. The extensions of rationals defined by these polynomial conditions must be the same as at the octonionic side. Also algebraic points must be mapped to algebraic points so that cognitive representations are mapped to cognitive representations. The counterparts of both RE(P)=0 and IM(P)=0 should be satisfied for the polynomials at twistor side defining the same extension of rationals.
   
   
4. M⁸-H duality must map the complex coordinates z₁₁=Re(RE) and z₁₂=Im(RE) (z₂₁=Re(IM) and z₂₂=Im(IM)) at M⁸ side to complex coordinates u_i1 and u_i2 with u_i1(0)=0 and u_i2(0)=0 for i=1 or i=2, at twistor side.
   

   Roots must be mapped to roots in the same extension of rationals, and no new roots are allowed at the twistor side. Hence the map must be linear: u_i1= a_iz_i1+b_iz_i2 and u_i2= c_iz_i1+d_iz_i2 so that the map for given value of i is characterized by SL(2,Q) matrix (a_i,b_i;c_i,d_i).
   
   

5. These conditions do not yet specify the choices of the coordinates (u_i1,u_i2) at twistor side. At CP₂ side the complex coordinates would naturally correspond to Eguchi-Hanson complex coordinates (w₁,w₂) determined apart from color SU(3) rotation as a counterpart of SU(3) as sub-group of automorphisms of octonions.
   

   If the base space of the twistor space CP_3,h of M⁴ is identified as CP_2,h, the hyper-complex counterpart of CP₂, the analogs of complex coordinates would be (w₃,w₄) with w₃ hypercomplex and w₄ complex. A priori one could select the pair (u_i1,u_i2) as any pair (w_k(i),w_l(i)), k(i)≠ l(i). These choices should give different kinds of extremals: such as CP₂ type extremals, string like objects, massless extremals, and their deformations.
   
   

String world sheet singularities and world-line singularities as their light-like boundaries at the light-like orbits of partonic 2-surfaces are conjectured to characterize preferred extremals as surfaces of H at which there is a transfer of canonical momentum currents between Kähler and volume degrees of freedom so that the extremal is not simultaneously an extremal of both Kähler action and volume term as elsewhere. What could be the counteparts of these surfaces in M⁸?

1. The interpretation of the pre-images of these singularities in M⁸ should be number theoretic and related to the identification of quaternionic imaginary units. One must specify two non-parallel octonionic imaginary units e¹ and e² to determine the third one as their cross product e³=e¹× e². If e¹ and e² are parallel at a point of octonionic surface, the cross product vanishes and the dimension of the quaternionic tangent/normal space reduces from D=4 to D=2.
   
   
2. Could string world sheets/partonic 2-surfaces be images of 2-D surfaces in M⁸ at which this takes place? The parallelity of the tangent/normal vectors defining imaginary units e_i, i=1,2 states that the component of e₂ orthogonal to e₁ vanishes. This indeed gives 2 conditions in the space of quaternionic units. Effectively the 4-D space-time surface would degenerate into 2-D at string world sheets and partonic 2-surfacesa as their duals. Note that this condition makes sense in both Euclidian and Minkowskian regions.
   
   
3. Partonic orbits in turn would correspond surfaces at which the dimension reduces to D=3 by light-likeness - this condition involves signature in an essential manner - and string world sheets would have 1-D boundaries at partonic orbits.
   
   

See the article Twistors in TGD or the chapter TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, M⁸-H Duality, SUSY, and Twistors.

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

Articles and other material related to TGD.