How could strong interactions emerge at the level of scattering amplitudes?

The above considerations are dangerous in that the intuitive QFT based thinking based is applied in TGD context where all interactions reduced to the dynamics of 3-surfaces and fields are geometrized by reducing them to the induced geometry at the level of space-time surface. Quantum field theory limit is obtained as an approximation and the applications of its notions at the fundamental level might be dangerous. In any case, it seems that only electroweak gauge potentials appear in the fermionic vertices and this might be a problem.

1. By holography perturbation series is not needed in TGD. Scattering amplitudes are sums of amplitudes associated with Bohr orbits, which are not completely deterministic: there is no path integral. Whether path integral could be an approximate approximation for this sum under some conditions is an interesting question.
2. It is best to start from a concrete problem. Is pair creation possible in TGD? The problem is that fermion and antifermion numbers are separately conserved for the most obvious proposals for scattering amplitudes. This essentially due to the fact that gauge bosons correspond to fermion-antifermion pairs. Intuitively, fermion pair creation means that fermion turns backwards in time. If one considers fermions in classical background fields this turning back corresponds to a 2-particle vertex. Could pair creation in classical fields be a fundamental process rather than a mere approximation in the TGD framework. This would conform with the vision that classical physics is an exact part of quantum physics.

   The turning back in time means a sharp corner of the fermion line, which is light-like elsewhere. M⁴ time coordinate has a discontinuous derivative with respect to the internal time coordinate of the line. I have propoeed (see this and this) that this kind of singularities are associated with vertices involving pair creation and that they correspond to local defects making the differentiable structure of X⁴ exotic. The basic problem of GRT would become a victory in the TGD framework and also mean that pair creation is possible only in 4-D space-time.

One can imagine two kinds of turning backs in time.

1. The turning back in time could occur for a 3-D surface such as monopole flux tube and induce the same process the string world sheets associated with the flux tubes and for the ends of the string world sheets as fermion lines ending at the 3-D light-like orbits of partonic 2-surfaces.
2. In the fusion of two 2-sheeted monopole flux tubes along their "ends" identifiable as partonic 2-surfaces wormhole contacts, the ends would fuse instantaneously (this process is analogous to "join along boundaries". The time reversal of this process would correspond to the splitting of the monopole flux tube inducing a turning back in time for a partonic 2-surface and for fermionic lines as boundaries of string world sheets at the partonic 2-surface.

   This would be analogous to a creation of a fermion pair in a classical induced gauge field, which is electroweak. The same would occur for the partonic 2-surfaces as opposite wormhole throats and for the string world sheets having light-like boundaries at the orbits of partonic 2-suraces.

3. The light-like orbit of a partonic 2-surface contains fermionic lines as light-like boundaries of string world sheets. A good guess is that the singularity is a cusp catastrophe so that the surface turns back in time in exactly the opposite direction. One would have an infinitely sharp knife edge.

What one can say about the scattering amplitudes on the basis of this picture? Can one obtain the analog for the 2-vertex describing a creation of a fermion pair in a classical external field?

1. The action for a geometric object of a given dimension defines modified gamma matrices in terms of canonical momentum currents as Γ^α= T^αkΓ_k, T^α_k= ∂ L/∂(∂_α h^k). By hermiticity, the covariant divergence D_αΓ^α of the vector defined by modified gamma matrices must vanish. This is true if the field equations are satisfied. This implies supersymmetry between fermionic and bosonic degrees of freedom.

   For space-time surfaces, the action is Kähler action plus volume term. For the 3-D light-partonic orbits one has Chern-Simons-Kähler action. For string world sheets one has area action plus the analog of Kähler magnetic flux. For the light-boundaries of string world sheets defining fermion lines one has the integral ∫ A_μdx^μ. The induced spinors are restrictions of the second quantized spinors fields of H=M⁴× CP₂ and the argument is that the modified Dirac equation holds true everywhere, except possibly at the turning points.

2. Consider now the interaction part of the action defining the fermionic vertices. The basic problem is that the entire modified Dirac action density is present and vanishes if the modified Dirac equation holds true everywhere. In perturbative QFT, one separates the interaction term from the action and obtains essentially ΨbarΓ^α D_αΨ. This is not possible now.

   The key observation is that the modified Dirac equation could fail at the turning points! QFT vertices would have purely geometric interpretation. The gamma matrices appearing in the modified Dirac action would be continuous but at the singularity the derivative ∂_μΨ= ∂_μm^k ∂_kΨ of the induced free second quantized spinor field of H would become discontinuous. For a Fourier mode with momentum p^k, one obtains

   ∂_μΨ_p= p_k∂_μ m^kΨ_p == p_μΨ_p.

   This derivative changes sign in the blade singularity. At the singularity one can define this derivative as an average and this leaves from the action Ψbar Γ^α D_αΨ only the term ΨbarΓ^α A_αΨ. This is just the interaction part of the action!

3. This argument can be applied to singularities of various dimensions. For D=3, the action contains the modified gamma matrices for the Kähler action plus volume term. For D=2, Chern-Simons-Kähler action defines the modified gamma matrices. For string world sheets the action could be induced from area action plus Kähler magnetic flux. For fermion lines from the 1-D action for fermion in induced gauge potential so that standard QFT result would be obtained in this case.

How does this picture relate to perturbative QFT?

1. The first thing to notice is that in the TGD framework gauge couplings do not appear at all in the interaction vertices. The induced gauge potentials do not correspond to A but to gA. The couplings emerge only at the level of scattering amplitudes when one goes to the QFT limit. Only the Kähler coupling strength and cosmological constant appear in the action.
2. The basic implication is that only the electroweak gauge potentials appear in the vertices. This conforms with the dangerous looking intuition that also strong interactions can be described in terms of electroweak vertices but this is of course a potential killer prediction. One should be able to show that the presence of WCW degrees of freedom taken into account minimally in terms of fermionic color partial waves in CP₂ predicts strong interactions and predicts the value of α_s. Note that the restriction of spinor harmonics of CP₂ to a homologically non-trivial geodesic sphere gives U(2) partial waves with the same quantum numbers as SU(3) color partial waves have.
3. TGD approach differs dramatically from the perturbative QFT. Since 1/α_s appears in the vertex, the increase of h_eff in the vertex increases it: just the opposite occurs in the perturbative QFT! This seems to be in conflict with QFT intuition suggesting a perturbation series in α_s ∝ 1/ℏ_eff. The explanation is that 1/α_K appears as a coupling parameter instead of α_s.

   This reminds of the electric-magnetic duality between perturbative and non-perturbative phases of gauge theories, where magnetic coupling strength is proportional to the inverse of the electric coupling strength. The description in terms of monopole flux tubes is therefore analogous to the description in terms of magnetic monopoles in the QFT framework. In TGD, it is the only natural description at the fundamental level. The decrease of α_K by increase of h_eff would indeed correspond to the QFT type description reduction of α_s.

   Could the description based on Maxwellian non-monopole flux tubes correspond to the usual perturbative phase without magnetic monopoles? In the Maxwellian phase there is huge vacuum degeneracy due to the presence of vacuum extremals with a vanishing Kähler form at the limit of vanishing volume action. Could this degeneracy allow path integral as a practical approximation at QFT limit.

4. h_eff/h₀ = n is proposed to correspond to the dimension of algebraic extension of rationals associated with the space-time surface and serve as a measure for algebraic complexity. The increase of algebraic complexity of the space-time region defining the strong interaction volume would also make interactions strong. In TGD, the fundamental coupling strength would be α_K and the increase of α_K for ordinary value of h would force the increase of h. This should happen below the electroweak scale or at least the confinement scales and make perturbation theory describing strong interactions possible. This description would involve monopole flux tubes and their reconnections.

   See the article About Platonization of Nuclear String Model and of Model of Atoms or the chapter with the same title.

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

   For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

About the identification of the Schrödinger galaxy

The latest mystery related created by the observations of James Webb (see this and this).

It has been found that the determination of the redshift 1+ z = a_now/a_emit gives two possible space-time positions for the Schrödinger galaxy CEERS-1749 a_now resp. a_emit corresponds to the scale factor for the recent cosmology resp. cosmology when the radiation was emitted. Note that for not too large distances the recession velocity β satisfies the Hubble law β= HD. The nickname "Schrödinger galaxy" comes from the impression that the same galaxy could have existed in two different times in the same direction.

Accordingly, CHEERs allows two alternative identifications: either as an exceptionally luminous galaxy with z≈ 17 or as a galaxy with exceptionally low luminosity with z≈ 5. Both these identifications challenge the standard view about galaxy formation based on Λ CDM cosmology.

1. The first interpretation is that CHEERS is very luminous, much more luminous than the standard cosmology would suggest, and has the redshift z≈ 17, which corresponds to light with the age of 13.6 billion years. The Universe was at the moment of emission t_emit=220 million years old.

   In the TGD framework, the puzzlingly high luminosity might be understood in terms of a cosmic web of monopole flux tubes guiding the radiation along the flux tubes. This would also make it possible to understand other similar galaxies with a high value of z but would not explain their very long evolutionary ages and sizes. Here the zero energy ontology (ZEO) of TGD could come in rescue (see this, this and this).

2. Another analysis suggest that the environment of the CHEERS contains galaxies with redshift z≈ 5. The mundane explanation would be that CHEERS is an exceptionally dusty/quenched galaxy with the redshift z≈ 5 for which light would be 12.5 billion years old.

   Could TGD explain the exceptionally low luminosity of z≈ 5 galaxy? Zero energy ontology (ZEO) and the TGD view of dark matter and energy predict that also galaxies should make "big" state function reductions (BSFRs) in astrophysical scales. In BSFRs the arrow of time changes so that the galaxy would become invisible since the classical signals from it would propagate to the geometric past. This might explain the passive periods of galaxies quite generally and the existence of galaxies older than the Universe. Could the z≈ 5 galaxy be in this passive phase with a reversed arrow of time so that the radiation from it would be exceptionally weak.

TGD seems to be consistent with both explanations. To make the situation even more confusing, one can ask whether two distinct galaxies at the same light of sight could be involved. This kind of assumption seems to be unnecessary but one can try to defend this question in the TGD framework.

1. In the TGD framework space-times are 4-surfaces in M⁴× CP₂. A good approximation is as an Einsteinian 4-surface, which by definition has a 4-D M⁴ projection. The scale factor a corresponds to the light-cone proper time assignable to the causal diamond CD with which the space-time surface is associated. a is a very convenient coordinate since it has a simple geometrical interpretation at the level of embedding space M⁴× CP₂. The cosmic time t assignable to the space-time surface is expressible as t(a).
2. Astrophysical objects, in particular galaxies, can form comoving tessellations (lattice-like structures) of the hyperbolic space H³, which corresponds to a=constant, and thus t(a) constant surfaces. The tessellation of H³ is expanding with cosmic time a and the values of the hyperbolic angle η and spatial direction angles for the points of the tessellation do not depend on the value of a. The direction angles and hyperbolic angle for the points of the tessellation are quantized in analogy with the angles characterizing the points of a Platonic solid and this gives rise to a quantized redshift.

   A tessellation for stars making possible gravitational diffraction and therefore channelling and amplification of gravitational radiation in discrete directions, could explain the recently observed gravitational hum (see this).

   These tessellations could also explain the mysterious God's fingers, discovered by Halton Arp, as sequences of identical look stars or galaxies of hyperbolic tessellations along the line of sight (see this and this. Maybe something similar is involved now.

This raises two questions.

1. Could two similar galaxies at the same line of sight be behind Schrödinger galaxy and correspond to the points of scaled versions of the tessellation of H³ having therefore different values of a and hyperbolic angle η? The spatial directions characterized by direction angle would be the same. Could one think that the tessellation consists of similar galaxies in the same way as lattices in condensed matter physics? The proposed explanation for the recently observed gravitational hum indeed assumes tessellation form by stars and most stars are very similar to our Sun (see this).

   The obvious question is whether also the neighbours of the z≈ 5 galaxy belong to the scaled up tessellation. The scaling factor between these two tessellations would be a₅/a₁₇= 17/5. Could it be that the resolution does allow to distinguish the neighbors of the z≈ 17 galaxy from each other so that they would be seen as a single galaxy with an exceptionally high luminosity? Or could it be that the z≈ 5 galaxy is in a passive phase with a reversed arrow of time and does not create any detectable signal so that the signal is due to z≈ 17 galaxy.

2. Could one even think that the values of hyperbolic angles are the same for the two galaxies in which case the z≈ 5 galaxy could correspond to z≈ 17 galaxy but in the passive phase with an opposite arrow of time? The ages of most galaxies are between 10 and 13.6 billion years so that this option deserves to be excluded. Could the hyperbolic tessellation explain why two similar galaxies could exist at the same line of sight in a 4-dimensional sense?

   This option is attractive but is actually easy to exclude. The light arriving from the galaxies propagates along light-like geodesics. Suppose that a light-like geodesic connects the observer to the z≈ 17 galaxy. The position of the z≈ 5 galaxy would be obtained by scaling the H³ of the older galaxy by the ratio a(young)/a(old). Geometrically it is rather obvious that the geodesic connecting it to the observer cannot be lightlike but becomes space-like. If one approximates space-time with M⁴ this is completely obvious.

   For more detailed analysis, see the article TGD view of the paradoxical findings of the James Webb telescope or the chapter TGD View of the Engine Powering Jets from Active Galactic Nuclei.

   For a summary of earlier postings see Latest progress in TGD. For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Pollack effect as a universal energy transfer mechanism?

The proposal of the recent article Some New Aspects of the TGD Inspired Model of the Nerve Pulse is that nerve pulse generation relies on the flip-flop mechanism using the energy liberated in the reversal of Pollack effect at one side of cell membrane to induce Pollack effect at the opposite side. The liberate energy would be channelled along the pair of monopole flux tubes emerging by re-connection from two U-shaped monopole flux tubes. The flip-flop mechanism is highly analogous to a seesaw in which the gravitational binding energy at the first end of the seesaw is reduced and transforms to kinetic energy reducing gravitational binding energy at the second end of the seesaw.

All biochemical processes involve a transfer of metabolic energy. Could the flip-flop mechanism serve as a universal mechanism of energy transfer accompanying biochemical processes?

The first example is TGD based view of biocatalysis according to which a phase transition reduces the value of h_eff and thus the length for the monopole flux tube pair connecting the reactants liberates energy, which kicks the reactants over the potential energy wall and in this way increases dramatically the rate of the reaction. Also now, the liberated energy could propagate as dark photons along the flux tube pair raise the system above the reaction wall or at least reduce its height.

Also the ADP→ ADP process could involve the Pollack effect and its reversal. In this process 3 protons are believed to flow through the cell membrane and liberate energy given to the ADP so that the process ADP+P_i → ATP takes place. This system has been compared to an energy plant. This raises heretic questions. Does the flow of ordinary protons through the mitochondrial membrane really occur? Could the charge separation be also now between the cell interior and its magnetic body?

1. The protons believed to flow through the mitochondrial membrane would be in the initial situation gravitationally dark and generated by Pollack effect for which the energy would be provided as energy liberated by biomolecules in a process which could be a time reversal for its storage in photosynthesis.
2. The reverse Pollack effect inside the mitochondrial membrane could transform the dark protons to ordinary protons and liberate energy, which is carried through the membrane as dark photons to the opposite side. This would allow the high energy phosphate bond of ATP to form in the reaction ADP+P_i → ATP. According to the TGD proposal (see this and this), the liberated energy could be used to kick the proton to the gravitational monopole flux tube, which would have length of order Earth size scale so that gravitational potential energy would of the same order of magnitude as the metabolic energy quantum with a nominal value .5 eV. This dark proton would be the energy carrier in the mysterious high phosphate energy bond, which does not quite fit the framework of biochemistry.
3. ATP would donate the phosphate ion P^- for the target molecule, which would utilize this temporarily stored metabolic energy as the dark proton transforms to an ordinary one. Depending on the lifetime of the dark proton, this could occur as the target molecule receives P or later. In any case, this should involve the transformation P^-→ P_i. This could correspond to the transformation of the gravitationally dark proton to ordinary proton so that the charge separation giving rise to P^- would be between P_i and its magnetic body.

In the chemical storage of the metabolic energy in photosynthesis, ATP provides the energy for the biomolecule storing the energy. This process should be accompanied by the transformation of P^- to P_i. It is instructive to consider two options that come immediately into mind.

Option I: The realistic looking option is that the energy is stored as the energy of an ordinary chemical bond.

1. Hydrogen bond, which can form between a proton and other electronegative atoms such as O or N, is a natural candidate. Hydrogen bond indeed has an energy, which is of the order of metabolic energy quantum .5 eV. The simplest option is that the metabolic energy provided by the gravitational flux tube of ATP is liberated and used to generate a hydrogen bond of the protein. The dark gravitational flux tube loop would be nothing but a very long hydrogen bond.
2. For negatively charged molecules, the proton of a hydrogen bond could be gravitationally dark. For dark positively charged ions, some valence electrons could be gravitationally dark. In the electronic case the reduction of the gravitational binding energy would be roughly by a factor m_e/m_p∼ 2^-11 smaller and this leads to a proposal of electronic metabolic energy quantum (see this and this and this) for which there is some empirical support from the work of Adamatsky (see this.

Option II: The less realistic looking option is that the molecule stores the metabolic energy permanently as a gravitationally dark proton. The motivation for its detailed consideration is that it provides insights to the Pollack effect.

1. The dark proton associated with P^- should become a dark proton associated with the molecule. In this case the length of hydrogen bond would become very long, increasing the ability to store metabolic energy.

   The hydrogen bonded structure would be effectively negatively charged but this is just what happens in the EZ in Pollack effect! This supports the view that the Pollack effect for water basically involves the lengthening of the hydrogen bonds to U-shaped gravitational monopole flux tubes.

2. The Pollack effect requires a metabolic energy feed since the value of h_gr tends to decrease spontaneously. This suggests that the dark gravitational hydrogen bonds are not long-lived enough for the purpose of long term metabolic energy storage. Rather, they would naturally serve as a temporary metabolic energy storage needed in the transfer of metabolic energy. The temporary storage of the metabolic energy to ATP would be a quantum variant of the seesaw.
3. The first naive guess for the scale of the life-time of the gravitationally dark proton would be given as a gravitational Compton time determined by the gravitational Compton length Λ_gr= GM/β₀ =r_S(M)/2β₀. For the Earth with r_S∼ 1 cm, one has T_gr=1.5 × 10^-11 s corresponding to the energy .6 meV for the ordinary Planck constant and perhaps related to the miniature membrane potentials. For the Moon with mass M_M=.01M_E, this time is about T_gr∼ 1.5× 10^-13 ns. For the ordinary Planck constant, this time corresponds to energy of .07 eV and is not far from the energy assignable to the membrane potential. For the Sun, one would gravitational Compton length is one half of the Earth's radius, which gives T_gr= .02 s, which corresponds to 50 Hz EEG frequency.

   Note that the rotation frequency for the ATP synthase analogous to a power plant is around 300 Hz which is the cyclotron frequency of the proton in the endogenous magnetic field .2 Gauss interpreted in TGD as the strength of the monopole fluz part of the Earth's magnetic field.

See the article Some New Aspects of the TGD Inspired Model of the Nerve Pulse or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Could a new kind of period of exploration save our civilization from collapse?

Everyone knows Columbus and other heros from the period of exploration journeys when we learned the geography of our home planet. After Newton emerged planetary physics and we started to learn of our planetary system. Eventually astrophysics and cosmology emerged. Now the James Webb telescope is shattering our views about the foundations of cosmology and astrophysics and a profound revolution of the world of physics seems unavoidable. This revolution also relates to our view of time. Also in biology, brain science, and science of consciousness, we are at the verge of revolution. After half a century our recent physics based world view might be regarded as childish as the world of Flat-Earthers.

In a sharp contrast to this progress in science and technology, our civilization has fallen in a state of stagnation. Its materialism based world view has deprived us of ethics and morals and the highest goal of modern man is to consume even more. Success in our society means money, fame and power. The uncontrolled application of various technological breakthroughs have created lethal looking environmental problems and social structures are breaking down. It is quite possible that our civilization is doomed to collapse. This is nothing new and actually fits with the fact that all complex systems are born, flourish, and eventually die. It is good to remember that after the collapse of the Roman empire there was a period of stagnation of about 1500 years before the development of mathematics created in antiquity could continue. In fact, they almost discovered calculus and computers before the collapse.

In this kind of situation one can wonder whether a new exploration period could be possible and give a deeper meaning for the existence of our western civilization and make us something more than consumers? Could the striking findings of the James Webb telescope and equally striking discoveries from other branches of natural science, which do not gain attention of popular media, inspire a new period of exploration of our world allowing us to update our lethally wrong world view?

What could be the big question?

What could be the question catching the attention of adventurous minds in the future? What dark matter and energy really are? This is certainly one of the deepest mysteries of recent day science. Could the attempt to understand dark matter give meaning to the existence of the society?

Dark matter is indeed an excellent candidate for the problem of the next century. The mainstream science knows only of the existence of dark matter and energy. The particle physics inspired models have repeatedly failed the tests and also the halo model for galactic dark matter relying on particle physics models is in deep difficulties. Same can be said about the o MOND model, which denies the existence of dark matter and postulates that Newtonian gravitation fails for weak fields. This is a rather paradoxical looking assumption, which very few can consider seriously.

TGD answer to the big question

The TGD explanation of dark matter relies on a new view of both space-time and quantum theory. TGD predicts the existence of a dark matter hierarchy as phases of the ordinary matter labelled by the values h_eff of the (effective) Planck constant, which is a multiple of its minimal value. Dark matter would be simply ordinary matter in a phase with a nonstandard value of Planck constant. If the value of the h_eff is large enough, this phase of matter is quantum coherent in even macroscopic scales. This would explain the mysterious ability of living matter to behave coherently in macroscales, impossible to understand in the biology as nothing but chemistry approach. The quantum coherence of the dark matter would induce the ordinary coherence of biomatter.

This view also revolutionizes the views about elementary particle-, hadron-, nuclear-, atomic- and molecular physics. The same basic topological mechanisms appear in all these physics and a lot of new physics is predicted (see this ). The dark matter would reside at space-time sheets (a new notion forced by the TGD view of space-time) characterized by the value of h_eff. The value of h_eff would characterize algebraic complexity of the space-time sheet, which in turn is a natural measure for the capacity to represent conscious information. The h_eff hierarchy would define an evolutionary hierarchy.

The most natural candidates for the space-time sheet carrying dark matter would be what I call magnetic bodies. The TGD view of space-time predicts that the electromagnetic fields of a system define a kind of field body of the system as a well-defined geometric anatomy and having body parts, motor actions, etc... In particular, the magnetic body consisting of monopole flux tubes would serve as the "boss", controller of the system because its IQ characterized by the high value of h_eff would be high.

The predicted values of Planck constant are largest for the monopole flux tubes mediating quantum gravitation. This conforms with the facts that gravitation has infinite range, is un-screened and the fact that quantum coherence scale increases with h_eff. The highest values of Planck constant would be associated with gravitational monopole flux tubes of Earth, Moon, other planets, Sun, and even galaxies. The unavoidable prediction is that the magnetic bodies of these astrophysical objects could play a key role in the quantum biology of the Earth. Horoscopes make no sense but astrologers might have not been completely wrong. Hard science must rely on numbers and the number of numerical miracles supporting this view has been accumulating (see for instance this, this, this).

The field bodies as the target of the new period of exploration?

These considerations suggest that the new period of exploration could have the electromagnetic environment of the Earth as its target. What do the magnetic and electric bodies of Earth, planets, Sun, galaxy,... look like? How dow they interact? This would be also exploration of the inner world, not only ours: the prediction is that life and consciousness are universal. This is so because the h_eff hierarchy plays a central role in understanding of conscious experience and intelligence.

Could the predicted new atomic physics kill the Platonic vision?

The Platonic vision connecting hadron physics, nuclear physics and atomic physics predicts a lot of new atomic physics and this could turn out to be fatal. I hasten to confess that the following speculations reflect my rudimentary knowledge of details of atomic physics. The new conceptual element are flux tubes, which can be regarded as springs with mass and elastic constant (string tension).

The first question concerns electric fields in the flux tube picture.

1. If there are only flux tubes present, the electric fluxes must run along them (a more conservative option is that fluxes flow to a large space-time sheet). Perhaps the most natural interpretation is that the localization of electric fluxes to flux tubes induces a constraint force due to the space-time geometry, something completely new. If so, one can argue that the dynamics for the flux tubes carrying also electric flux automatically describes the repulsive Coulomb force subject to geometrodynamic constraints.

   An important implication is that the Hamiltonian cycles of j-blocks must reconnect to the Hamiltonian cycles of other j-blocks and to the nucleus. The Hamiltonian cycles of the entire atom must fuse to a single large cycle, which can be closed for a neutral atom, and would correspond to closed monopole flux tube starting from the atomic nucleus. Each charge along the cycle contributes to the electric flux flowing in the monopole flux tube.

   It has been proposed (see this) that molecular bonds could be interpreted as electric flux tubes. This proposal is discussed from TGD point of view in \cite{allb/qcritdark3}. If the atoms of the molecule are ionized the Hamiltonian cycles of atoms must be reconnect by U-shaped tentacles and ionic bonds would correspond to flux tubes and presumably all chemical bonds.

Consider next the mass of the flux tube.

1. Flux tubes connecting neighboring charges could be p-adically scaled electropions with mass smaller than the mass 1 MeV of electropions and would contribute to the mass of the atom. In the case of nuclei scaled hadronic pions between nucleons having mass of order MeV are replaced by p-adically scaled elctropions. Note that electropions have mass of 1 MeV. In the case of atoms, their scaled variants should have a considerably smaller mass, which would naively correspond to the atomic p-adic length scale and mass scale of 1-10 keV. Note that 10 keV would be the scale of proposed nuclear excitation energies supported by nuclear physics X-ray anomalies. One can argue that the mass corresponds to the atomic p-adic length scale L(137) as a natural length scale for the flux tube gives and would be of order m\sim keV.
2. One the other hand, one could argue that the mass should be very small because, to my best knowledge, standard atomic physics works very well. However, the additive contribution of these masses does not affect the electronic bound state energies but only the total mass of the system. I do not know whether anyone has studied the possible dependence of the total mass of atom on the number of electrons? Does it contain an additive contribution increasing by one unit at each step along the row of the periodic table as an additional flux tube appears to the Hamilton cycle. These contributions could be also interpreted as contributions of the repulsive interactions of electrons to the energy.

As in the case of nuclei, the atomic flux tubes would act as springs, i.e. harmonic oscillators. This predicts a spectrum of excited states with scale determined by the elastic constant k or equivalent ground state oscillation frequency ω₀.

1. If ω₀ is large enough, the excitation energies would be greater than the ionization energy and there would be no detectable effects. The naive argument that ω₀ corresponds to the atomic length scale L(137) as a natural length scale for the flux tube gives ω₀\sim 1 keV. This energy scale would be for light atoms with Z≤ 9 (Oxygen) larger than the ionization energy E= Z²× 13.7 eV so that photons causing excitation would cause ionization.
2. An equally naive scaling from nuclear scale to atomic scale would suggest that the value of ω₀ is scaled from ℏω₀= 1 MeV by the ratio L(113/L(137)=2^-12 of nuclear and atomic length scales to about ω₀=.25 keV. This is not far from the above estimate.
3. How to deal with atoms with a small number of electrons, in particular Helium with 2 electrons? j=2 j-blocks are special in the sense that they do not allow sub-Hamiltonian cycle. Could the flux tube connecting the electrons be absent in this case so that only the repulsive electronic contribution would be present? Note also that the repulsive interaction energy between electrons would be smaller than the attractive interaction energy of electrons for atoms with Z=2. If this picture is correct, new atomic physics would emerge when j-block contains more than 2 electrons.

   One can also consider the possibility that the coupling to photons is weak enough, perhaps by the condition that the photon must transform first to dark photon. The behavior of multi-electron atoms in a radiation field whose photons have a low energy must have been studied.

One could also imagine that the flux tubes form a h_eff≥ h quantum coherent state, in which there are n=h_eff/h flux tubes forming the sub-tessellation of Platonic tessellation for a given j-block with vertices connected by flux tubs. Here n would be the number electrons in the j-block. The excitation energy E= ℏ_effω₀ is scaled by ℏ_eff/ℏ=n.

1. If all flux tubes associated with atom were excited at once as a phase transition, the required excitation energy would be rather large for large enough n and the excitations by photons might be possible without ionizing the atom.
2. The atoms at the left end of the row are the problem for this option and more generally, the atoms at the left end of each j-blocks. One expects that the flux tube length depends on the value of the principal quantum number N labelling the rows since the size of Platonic solid must increase with n like n². Can one assume that the mass of the spring does not depend on the row? If the elastic constant k does not depend on the row, one could consider a simultaneous collective excitation of all flux tubes so that the binding energy could increase enough.

See the article About Platonization of Nuclear String Model and of Model of Atoms or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Moon and neuroscience

I have already suggested that the gravitational magnetic body of Moon could play key role in the model of nerve pulse. The following model for the commutations between neuronal membrane and gravitational magnetic body by cyclotron resonance demonstrates that the expectation was correct. Moon would play a key role in neuroscience! Conformistic colleague can hardly imagine a crazier sounding statement and can happily concluded that I am a crackpot after all!

Quantum gravitation favors the communications between cell membranes as dark Josephson junction and corresponding MBs carrying dark charged particles. The variations of the membrane voltage induce the modulation of the Josephson frequency and resonant receival induces a sequence of pulses coding the variations of the membrane potential to a sequence of pulses.

1. The cyclotron energies

   E_c= ℏ_gr ZeB/m= GMZeB/β₀= r_s/[2l_B² β₀]

   do not depend on the mass m of the charged particle and are therefore universal. Same is true for the gravitational Compton length L_gr =r_s/2β₀ of the particle (r_S denotes Schwartchild radius).

2. Josephson frequencies are given by ZeV/2\pi ℏ_gr and is inversely proportional to the mass of the charged particle. In the case of ions this means the 1/A-proportionality and ordering of Josephson frequency scales as subharmonics.
3. The frequency resonance condition f_J= E_J/h_gr = f_c= ZeB/m is equivalent to the energy resonance condition

   E_J=ZeV_mem= ℏ_gr f_c= r_s/[2l_B²β₀] = r_S/[2β₀]×eB/ℏ .

   This condition fixes the relation between the voltage of the Josephson junction and the strength B of the magnetic field.

   eB= ZeV_mem × 2Zβ₀/r_S .

   For V_mem= .05 V,Z=2, r_S= r_S,E= 1 cm and β₀=1, and using the fact that B=1 Tesla corresponds to magnetic length l_B= (ℏ/eB)^1/2=64 nm, this gives B= 184 nT.

   It came as a surprise that this field strength is about 2.3× 10^-3 weaker than the endogenous magnetic field .2× 10^-4 Tesla at the surface of Earth. The strengths of the magnetic fields outside the inner magnetosphere are of order nTesla. Does this mean that the EEG signals from the cell membrane are received by charged particles at the flux tubes of the magnetosphere for which the field is much weaker than at the surface of Earth. This is indeed proposed in the model of EEG.

   How could one get rid of the problem?

   1. The expression for B is proportional to β₀ ≤ 1 and to 1/r_S. For the Moon the mass is .01M_E so that the value of the B would be scale by factor 100 so that it would be by factor .92 weaker than the nominal value of B_end. As proposed already earlier, the gravitational MB of Moon could be involved with the dynamics of the cell membrane and the endogenous magnetic field of Blackman could be assignable to Moon!
   2. The proportionality of B to eV_mem allows us to consider the possibility that also DNA involves Josephson junctions. In fact, the TGD inspired model for the Comorosan effect assumes that biomolecules quite generally involve them. By a naive dimensional argument one expects that the value of ZeV is scaled up by factor of order 100 as one scales the membrane thickness 10 nm to 1 Angstrom. This would give B_end for the gravitational flux tubes of the Earth.
   The possibility of simultaneous frequency and energy resonance means universal cyclotron resonance irrespective of the mass of the charged particle. Josephson frequencies are however inversely proportional to the mass of the charged particle appearing both in the cell membrane and the receiving flux tube. The resonance mechanism therefore makes it possible to use the same information for receivers with different masses. Each of them generates a different sequence of pulses at times for which modulated Josephson frequency equals the cyclotron frequency defining a specific kind of information characterized by the scale defined by Josephson period. Electron mass, proton mass and ion masses define characteristic frequency scales. For B_end, the cyclotron frequencies are in EEG range for ions which also favours the Moon option.

   See the article Some New Aspects of the TGD Inspired Model of the Nerve Pulse or the chapter with the same title.

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

   For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

A model for the generation of nerve pulse based on Pollack effect

The following view of what might happen in the generation of nerve pulse is only one of the many variants that I have imagined during years and can be only defended as the simplest one found hitherto. In this model Pollack effect for water plays a key role and Hodgkin-Huxley model would be simply wrong. Ionic currents would not cause the nerve pulse but would be caused by it.

Background observations

Let us consider the following assumptions.

1. The fact that the sign of the membrane potential changes sign temporarily but preserves its magnitude, suggests that the charge densities associated with the interior and exterior are changed so that the voltage changes the sign. There are many ways to achieve this and one should identify the simplest mechanism.
2. Hodgkin-Huxley model for nerve pulse involves dissipation. Nerve pulse could be generated as the failure of gravitational quantum coherence. This could also make possible Ohmic currents between axonal interior (AI) and axonal exterior (NE) but this, and even the loss of quantum gravitational coherence, might not be necessary. This is mildly suggested by the model of nerve pulse based on Josephson junction in which the pulse corresponds to a temporary change of the direction of rotation for the analogs of gravitational penduli.
3. In the Hodgkin-Huxley model the notions of channels and pumps are of course central for the recent biology. There are however puzzling observations challenging these notions and suggesting that the currents between cell interior and exterior have quantum nature and are universal in the sense that they do not depend on the cell membrane at all. One of the pioneers in the field has been Gilbert Ling, who has devoted for more than three decades to the problem, developed ingenious experiments, and written several books about the topic. The introduction of the book "Gells and cells" of Gerald Pollack gives an excellent layman summary about the paradoxical experimental results. I have discussed these findings also from the TGD point of view (see this).
4. In the TGD framework Pollack effect (PE) could induce the membrane potential and PE and its reversal (RPE) could be important. In the model to be discussed this is the case and the model differs dramatically from the Hodgkin-Huxley model in that ionic currents do not cause the nerve pulse but is caused by it.

The model of nerve pulse based on Pollack effect and its reversal

The simplest model for the generation of the nerve pulse is based on PE and RPE. In the following I will talk about neuronal interior (NI) and neuronal exterior (NE).

1. Sol-gel phase transition is known to accompany nerve pulse. This suggests that PE and RPE are involved. PE transforms gel phase to sol phase and generates a negatively charged exclusion zone (EZ).

   The TGD based model for PE involves the transformation of protons of water molecules to dark protons at the MB of the system with a large size so that the region of water becomes negatively charged EZ and transforms to a gel phase generating a potential. Since the flux tubes of gravitational MB have much larger size than the system, the protons/ions are effectively lost from the system.

   This corresponds to a polarization but not in the usual sense. Rather, the ends of the dipole correspond to EZ and MB. The charge separation is not between NI and NE but between NI (NE) and its MB.

2. An open question is whether PE could generalize also to other positive biologically important atoms which would become dark ions assignable to MB and leave behind electrons.
3. PE can take place for the water in NI. The transfer of charges to MB could also occur for the axonal microtubules but this transfer might be involved with the control of cell membrane and neuronal membrane, for instance MT could control the generation of nerve pulse.
4. The simplest model for how PE and RPE could be involved with nerve pulse generation is as follows. Before nerve pulse the water in NI (near to membrane) forms a negatively charged EZ since dark protons are at the MB outside the system. The water in NE is in gel phase and neutral. The negative charge of EZ gives rise to the membrane potential and ionic charges could give only small corrections to it.
5. The dark protons tend to transform to ordinary protons. Metabolic energy feed is needed to kick them back to the MB. The nerve pulse is generated by the RPE by stopping the metabolic energy feed for a moment. This induces a RPE as BSFR. In RPE dark protons are transformed to ordinary ones and return to the neuronal interior and gel→sol phase transition is induced. RPE liberates free energy, which in turn induces PE in NE and a negatively charged EZ is generated there. The sign of the membrane potential changes. The system is a kind of flip-flop in which RPE induces PE.
6. The reconnection of U-shaped flux tubes at the two sides of the neuronal membrane to form a flux tube pairs connecting NI and NE and associated with the ionic channels and pumps acting as Josephson junctions, would make possible an almost dissipation free transfer of the energy liberated in RPE to the opposite side of the membrane. The transfer of the liberated energy as a radiation from NI to NE and from NE to NI takes place along flux tube pairs associated with different membrane proteins, that is channels and pumps, which would therefore be channels for radiation rather than ions. Ionic Ohmic currents could be caused by the reversal of the membrane potential rather than causing it.
7. Contrary to the original guess, the nerve pulse would involve 4 BSFRs, which correspond to RPE in NI reducing the membrane potential V_i to V= 0 and liberating energy generating PE in NE changing the sign of the membrane potential: V=0→ -V_i. This PE is followed by RPE taking V=-V_i to V=0 and liberating energy generating PE in NI so that V=0 is transformed to V= V_i and the situation is returned back to the original. The times for the occurrences of BSFRs and changes of the arrow of time correspond to V=0, V= -V_i, V=0 and V= V_i.
8. What could be the role of microtubules? Quantum critical dynamics of axonal microtubules would make them ideal control tools of the dynamics at the level of cell membrane, in particular controllers of the nerve pulse generation and conduction. An attractive assumption is that the gravitational MBs of microtubules carry dark charges. Also the MBs associated with the cell exterior and inner and outer lipid layers could carry dark charges. Due to the large size of gravitational flux tubes, the charges transferred to the MBs (at least the microtubular MB) are effectively outside the axonal interior (AI) and exterior (NE) so that the charges of NI and NE are affected. This could bring the membrane potential below the threshold for the generation of the nerve pulse by the proposed mechanism. MB would be the boss using microtubules as control tools and water would do the hard work.

See the article Some New Aspects of the TGD Inspired Model of the Nerve Pulse or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Some New Aspects of the TGD Inspired Model of the Nerve Pulse

During year 2023 a considerable progress in the understanding of the TGD inspired model of nerve pulses has taken place.

1. Nerve pulses relate closely to the communications from the cell membranes to the magnetic body (MB) of the system using dark, frequency modulated Josephson radiation inducing at MB a sequence of cyclotron resonances serving as control signals and eventually giving rise to nerve pulse patterns. This would generalize the "right brain signs-left brain talks" metaphor. Also the model of meV spikes appearing in preneural systems is discussed.
2. Quantum gravitation in the TGD sense a can assign the needed huge values of h_eff to the gravitational magnetic bodies. Quantum gravitational flux tubes assignable to the Sun, Earth, and perhaps also other planets and even the Moon could be highly relevant for the living cell and the brain.
3. The connection with microtubular level is considered and the transfer of charged particles between microtubules and very long gravitational flux tubes assignable to them allows to induce membrane oscillations and even nerve pulse.
4. Zero Energy Ontology (ZEO) and Negentropy Maximization Principle (NMP) could allow computers to become effectively living intelligent systems able to reach goals by an analog of trial and error process. This requires the failure of quantum statistical determinism. This is the case if the gravitational Compton time defining a lower bound for the gravitational quantum coherence time is longer than the clock period of the computer. MB would play a key role also in the case of living computers and dark Josephson radiation could serve as a communication tool. Superconducting computers have Josephson junctions as basic active elements and are more promising than transistor based computers.
5. Also the recent finding that the neuronal system is in a certain sense 11-dimensional is discussed in the TGD framework. The basic observation is that the 12-neutron system, with neurons assignable to the 12 vertices of icosahedron and defining 11-D simplex, could be involved. Icosahedron and tetrahedron appear also with the TGD based model of bioharmony serving also as a model of the genetic code.

See the article Some New Aspects of the TGD Inspired Model of the Nerve Pulse or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

New insights about Langlands duality

Gary Ehlenberg sent an URL of a very interesting QuantaMagazine article, which discusses a work related to Langlands program.

Langlands duality relates number theory and geometry. At the number theory side one has representations of Galois groups. On the geometry side one has automorphic forms associated with the representations of Lie groups. For instance, in coset spaces of hyperbolic 3-space H³ in the case of the Lorentz group.

The work could be highly interesting from the TGD perspective. In TGD, the M⁸-H duality generalizes momentum-position duality so that it applies to particles represented as 3-surfaces instead of points. M⁸-H duality also relates physics as number theory and physics as geometry. Much like Langlands duality. The problem is to understand M⁸-H duality as an analog of Langlands duality.

1. H=M⁴×CP₂ is the counterpart of position space and particle corresponds to 3-surface in H. Physics as (differential) geometry applies at this side.

   The orbit of 3-surface is a 4-D space-time surface in H and holography, forced by 4-D general coordinate invariance, implies that space-time surfaces are minimal surfaces irrespective of the action (general coordinate invariant and determined by induced geometry) . They would obey 4-D generalization of holomorphy and this would imply universality.

   These minimal surfaces are also solutions of a nonlinear geometrized version of massless field equations. Field-particle duality has a geometrized variant: minimal surface represents in its interior massless field propagation and as an orbit of 3-D particles the generalization of a light-like geodesic. Hence a connection with electromagnetism mentioned in the popular article, actually metric and all gauge fields of the standard model are geometrized by induction procedure for geometry.

2. M⁸, or rather its complexification M⁸_c (complexification is only with respect to mass squared as coordinate,not hyperbolic and other angles) corresponds to momentum space and here the orbit of point-like particle in momentum space is replaced with a 4-surface in M⁸, or actually its complexification M⁸_c.

   The 3-D initial data for a given extension of rationals could correspond to a union of hyperbolic 3-manifolds as a union of fundamental regions for a tessellation of H³ consistent with the extensions, a kind of hyperbolic crystal. These spaces relate closely to automorphic functions and L-functions.

   At the M⁸ side polynomials with rational coefficients determine partially the 3-D data associated with number theoretical holography at M⁸-side. The number theoretical dynamical principle states that the normal space of the space-time surface in the octonionic M⁸_c is associative and initial data correspond to 3-surfaces at mass shells H³_c ⊂ M⁴_c ⊂ M⁸_c determined by the roots of the polynomial.

3. M⁸-H duality maps the 4-surfaces in M⁸_c to space-time surfaces in H. At the M⁸ side one has polynomials. At the geometric H-side one has naturally the generalizations of periodic functions since Fourier analysis or its generalization is natural for massless fields which space-time surfaces geometrize. L-functions represent a typical example of generalized periodic functions. Are the space-time surfaces at H-side expressible in terms of modular function in H³?

Here one must stop and take a breath. There are reasons to be very cautious! The proposed general exact solution of space-time surfaces as preferred extremals realizing almost exact holography as analogs of Bohr orbits of 3-D surfaces representing particles relies on a generalization of 2-D holomorphy to its 4-D analog. The 4-D generalization of holomorphic functions (see for instance this) assignable to 4-surfaces in H do not correspond to modular forms in 3-D hyperbolic manifolds assignable to the fundamental regions of tessellations of hyperbolic 3-space H³ (analogs of lattice cells in E³). Fermionic holography reduces the description of fermion states as wave functions at the mass shells of H³ and their images in H under M⁸-H duality, which are also hyperbolic 3-spaces.

1. This brings the modular forms of H³ naturally into the picture. Single fermion states correspond to wave functions in H³ instead of E³ as in the standard framework replacing infinite-D representations of the Poincare group with those of SL(2,C). The modular forms defining the wave functions inside the fundamental region of tessellation of H³ are analogs of wave functions of a particle in a box satisfying periodic boundary conditions making the box effectively a torus. Now it is replaced with a hyperbolic 3-manifold. The periodicity conditions code invariance under a discrete subgroup Γ of SL(2,C) and mean that H³ = SL(2,C)/U(2) is replaced with the double coset space Γ\SL(2,C)/U(2).

   Number theoretical vision makes this picture more precise and suggests ideas about the implications of the TGD counterpart of the Langlands duality.

2. Number theoretical approach restricts complex numbers to an extension of rationals. The complex numbers defining the elements SL(2,C) and U(2,C) matrices are replaced with matrices in discrete subgroups SL(2,F) and U(2,F), where F is the extension of rationals associated with the polynomial P defining the number theoretical holography in M⁸ inducing holography in H by M⁸-H duality. The group Γ defining the periodic boundary conditions must consist of matrices in SL(2,F).
3. The modular forms in H³ as wave functions are labelled by parameters analogous to momenta in the case of E³: in the case of E³ they characterize infinite-D irreducible representations of SL(2,C) as covering group of SO(1,3) with partial waves labelled by angular momentum quantum numbers and spin and by the analog of angular momentum associated with the hyperbolic angle (known as rapidity in particle physics): infinitesimal Lorentz boost in the direction of spin axis.

   The irreps are characterized by the values of a complex valued Casimir element of SL(2,C) quadratic in 3 generators of SL(2,C) or equivalently by two real Casimir elements of SO(1,3). Physical intuition encourages the shy question whether the second Casimir operator could correspond to the complex mass squared value defining the mass shell in M⁸. It belongs to the extension of rationals considered as a root of P.

   The construction of the unitary irreps of SL(2,C) is discussed in Wikipedia article. The representations are characterized by half integer j₀=n/2 and imaginary real number j₁= iν.

   The values of j₀ and j₁ must be restricted to the extension of rationals associated with the polynomial P defining the number theoretic holography.

4. The Galois group of the extension acts on these quantum numbers. Angular momentum quantum numbers are quantized already without number theory and are integers but the action on the hyperbolic momentum is of special interest. The spectrum of hyperbolic angular momentum must consist of a union of orbits of the Galois group and one obtains Galois multiplets. The Galois group generates from an irrep with a given value of j₁ a multiplet of irreps.

   A good guess is that the Galois action is central for M⁸-H duality as a TGD analog of Langlands correspondence. The Galois group would act on the parameter space of modular forms in Γ\SL(2,F)/U(2,F), F and extension of complex rationals and give rise to multiplets formed by the irreps of SL(2,F).

To sum up, M⁸-H duality is a rather precisely defined notion.

1. At the M⁸ side one has polynomials and roots and at the H-side one has automorphic functions in H³ and "periods" are interpreted as quantum numbers. What came first in my mind was that understanding of M⁸ duality boils down to the question about how the 4-surfaces given by number theoretical holography as associativity of normal space relate to those given by holography (that is generalized holomorphy) in H.
2. However, it seems that the problem should be posed in the fermionic sector. Indeed, above I have interpreted the problem as a challenge to understand what constraints the Galois symmetry on M⁸ side poses on the quantum numbers of fermionic wave functions in hyperbolic manifolds associated with H³. I do not know how closely this problem relates to the problem that Ben-Zvi, Sakellaridis and Venkatesh have been working with.

See for instance the article Some New Ideas Related to Langlands Program viz TGD.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Do cosmic strings with large string tension exist?

There is some empirical support for cosmic strings with a rather large string tension from gravitational lensing. Cosmic string tension T and string deficit angle Δ θ for lensing related bia the formula Δ θ = 8π× TG if general relativity is assumed to be a good description. The value of TGD deduced from data is TG= .05 and is very large and corresponds to an angle deficit Δ θ≈ 1.

For the ordinary value of Planck constant, TGD predicts the value of TG has upper bound in the range 10^-7-10^-6. The flat velocity spectrum for distant stars around galaxies determines the value of TG: one has v²=2TG from Kepler law so that the value of TG is determined from the measured value of the velocity v. The value of TG can be also deduced from the energy density of cosmic string-like objects predicted by TGD and is consistent with this estimate. If one takes the empirical evidence for a large value of TGseriously one must ask whether TGD can explain the claimed finding.

Could a large value of h_eff solve the discrepancy? String tension T as the linear energy density of the cosmic string is determined by the sum of Kähler action and volume term. The contribution of Kähler action to T is proportional to 1/α_K = g_K²/4πℏ. If cosmic string represents dark matter in TGD sense, one must make the replacement ℏ→ ℏ_eff so that the Kähhler contribution to T is proportional to ℏ_eff/ℏ. If the two contributions are of same order of magnitude or Kähler contribution dominates, ℏ_eff/ℏ=n≈ 10⁵ would give the needed large value TG. The physical interpretation would be that cosmic string is an n-sheeted structure with each sheet giving the same contribution so that the value of T is scaled up by n≈ 10⁵.

The physical interpretation would be that the cosmic string is an n-sheeted structure with each sheet giving the same contribution so that the value of T is scaled up by n≈ 10⁵. There are two options. The n-sheetedness is with respect to M⁴ so that one has a n-fold covering of M⁴ or with respect to CP₂ in which case one quantum coherent structure consisting of n parallel flux tubes.

It is intereting to consider in more detail the quantum model for the particles in the gravitational field of cosmic string.

1. The gravitational field of a straight cosmic string behaves like 1/ρ as a function of the radial distance ρ from string, and Kepler's law predicts a constant velocity v²= 2TG for circular orbits irrespective of their radius. This explains the flat velocity spectrum of stars rotating around galaxies.
2. Nottale proposed that planetary orbits obey Bohr quantization for the value of gravitational Planck constant ℏ_gr= GMm/β₀ assignable to a pair of masses M and M associated with the gravitational flux tube mediating the gravitational interaction between M and m.
3. If the mass M corresponds to a cosmic string idealized as straight string with an infinite length, the definition of ℏ_gr is problematic since M diverges. Therefore the application of Nottale's quantization to a distant star rotating cosmic string is problematic.

   What is however clear that ℏ_gr should be proportional to m by Equivalence Principle and one should have ℏ_gr= GM_effm/β₀ for the cosmic string. M_eff= TL_eff, where L_eff is the effective length of the cosmic string is also a reasonable parametrization.

4. Kepler law does not tell anything about the value of the radius r of the circular orbit. If the value of ℏ_gr is fixed somehow, one can apply the Bohr quantization condition ∮ pdq= nh_gr of angular momentum to circular orbits to obtain vr= nGM_eff giving

   r_n=nr₁,
   r₁=r_S,eff/[2(TG)^1/2β₀].

   A reasonable guess is that β₀ and the rotation velocity v/c=(2TG)^1/2 have the same order of magnitude. v/c= xβ₀< 1 would give β₀= (2TG)^<1/2/x. The minimal value of the orbital radius would be r₁=r_S,eff/[2xβ₀²].

An interesting question relates to the size scale of the n-sheeted structure interpreted as a covering of CP₂ by parallel cosmic strings or flux tubes. The gravitational Compton length Λ_gr= r_{S,eff}/2β₀ could give an estimate for the size scale of this structure, which as flux tube bundle would be naturally 2-D. There would be about 10⁵ flux tubes per gravitational Compton area with scale Λ_gr.

See the article Magnetic Bubbles in TGD Universe: Part I or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Evidence for the TGD view of quasars

I learned of an extremely interesting discovery providing additional support for the TGD view of quasars and galaxy formation (see this). Here is the abstract of the article published in Nature.

Quasars feature gas swirling towards a supermassive black hole inhabiting a galactic centre. The disk accretion produces enormous amounts of radiation from optical to ultraviolet (UV) wavelengths. Extreme UV (EUV) emission, stemming from the energetic innermost disk regions, has critical implications for the production of broad emission lines in quasars, the origin of the correlation between linewidth and luminosity (or the Baldwin effect) and cosmic reionization.

Spectroscopic and photometric analyses have claimed that brighter quasars have on average redder EUV spectral energy distributions (SEDs), which may, however, have been affected by a severe EUV detection incompleteness bias.

Here, after controlling for this bias, we reveal a luminosity-independent universal average SED down to a rest frame of ≈ 500 Å for redshift z&asymp: 2 quasars over nearly two orders of magnitude in luminosity, contrary to the standard thin disk prediction and the Baldwin effect, which persists even after controlling for the bias.

Furthermore, we show that the intrinsic bias-free mean SED is redder in the EUV than previous mean quasar composite spectra, while the intrinsic bias-free median SED is even redder and is unexpectedly consistent with the simply truncated wind model prediction, suggesting prevalent winds in quasars and altered black hole growth. A microscopic atomic origin is probably responsible for both the universality and redness of the average SED. What does TGD say?

1. In the standard accretion disk theory inner luminosity is determined by the mass of the accretion disk entering into the blackchole. What is however found that the spectral energy distribution of light from quasar does not depend on the inner luminosity at all in the extreme UV (EUV) range! It can even decrease when the intrinsic luminosity increases! These paradoxical findings challenge the standard accretion disk theory.
2. TGD based view of quasars (see for instance this, this, this, and this) suggests an explanation of the anomaly. The galactic matter would be formed as dark energy and dark matter from a cosmic string like objects thickening to a monopole flux tube with smaller string tension emits dark particles transforming to ordinary matter forming the galaxy. Cosmic strings would be transversal to the galactic plane and the gravitational field created by their dark energy energy predicts the flat velocity spectrum of galaxies.
3. The flow of radiation from the thickened flux tube (rather than from the energy liberated as matter of the accretion disk falls into the blackhole) would give rise to the spectral energy distribution in EUV and the inner luminosity at longer wavelengths would be determined by the accretion disk emission. Also the article suggests that galactic wind explains the energy spectrum: galactic wind would correspond to this EUV radiation from the monopole flux tube. This energy spectrum would be universal in the sense that it would reflect only the properties of the thickening cosmic string and universality is indeed claimed.

The model of the quasar as a portion of a cosmic string thickened to a flux tube tangle and emitting dark energy and matter transforming to ordinary matter challenges the standard model as a blackhole. The outflowing matter would create an accretion disk as a kind of traffic jam and at least part of the luminosity of the accretion disk would be due to the heating of the accretion disk caused by the flow of the particles colliding with the accretion disk. Also now the gravitational field of the cosmic string and of the flux tangle associated with it is present and a natural classical expectation is that the matter in the accretion disk tends to flow back to the quasar.

In atomic physics the quantization prevents the fall of electron to atomic nucleus. Could the same happen now and prevent the fall of matter from the accretion disk back to the quasar.

1. One can argue that a realistic quantum model for the matter around quasar is based on the treatment of the flux tube tangle as spherically symmetric mass distribution with the mass of the blackhole assigned to the quasars. Indeed, the straight portion of cosmic strings gives a large contribution to the gravitational force only at large distances so that the contribution of the tangle dominates.
2. The mechanism preventing the fall of matter to blackholes would be identical with that in the case of atoms. Also in the accretion disk model, the angular momentum of rotating matter in the accretion disk tends to prevent the fall into the blackhole and the angular momentum must be transferred away.
3. The orbital radii would be given by the Nottale model for planetary orbits with r_n = n²a_gr, where a_gr=4π GM/β₀²= 2π r_S/β₀² is gravitational Bohr radius. The ratio M/M_Sun of the mass M of the quasar blackhole to solar mass is estimated to be in the range [10⁷,3× 10⁹] predicting that the Schwartschild radius r_S is in the range 3× 10⁷-10¹⁰ km. The radius of r_acc should be larger than a_gr: a_gr<a_acc. Note that the size of the accretion disk is in some cases estimated to be few light-days: 1 light-day ≈ 10¹⁰ km whereas the visible size of quasar is measured in light years.
4. The condition a_gr<r_acc gives the condition 2π/β₀²<r_acc/r_S giving for β₀ an upper bound in the range β₀⊂ [.02,.2]. The values of β₀ in this range are considerably larger than the value β₀≈ 2^-11 predicted by the Bohr model for the orbits of inner planets. Note that for the Earth the estimate for β₀ is β₀≈ 1.

See the article Magnetic Bubbles in TGD Universe: Part I or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

About uniform tessellations of hyperbolic, Euclidean, spherical space

It has become clear that 3-D hyperbolic tessellations are probably very important in the TGD framework. The 3-D hyperbolic space H³ is realized as a mass shell or cosmic time constant hyperboloid, and an interesting conjecture is that particles with a fixed value of mass squared are associated with the vertices of a hyperbolic tessellation. This would give rise to a quantization of momenta consistent with the number theoretic discretization. Hyperbolic tessellations could also appear in cosmological scales. So-called icosatetrahedral tessellation seems to provide a model of genetic code and DNA and suggests that genetic code is not restricted in biology but universal and realized in all scales Euclidean.

Uniform tilings/tessellations/honeycombs can be identified by their vertex configuration given a list n₁.n₂.n₃... for the numbers n of vertices of regular n-polygons associated with the vertex. Uniform tilings can be regular, meaning that they are both vertex- and edge-transitive. Quasi-regular tessellations are only vertex-transitive and semiregular tessellations are neither vertex- nor face-transitive. Non-regular tessellations give, for instance, Archimedean solids obtained from Platonic solids by operations like truncation. In this case, the vertices are obtained by symmetries from each other but not the faces, which need not be identical anymore.

There exists an extremely general construction of tessellation of hyperbolic, Euclidean, and spherical tessellations which works at least in 2- and 3-D cases known as Wyuthoff construction. In the 2-D case, this construction is based on the so-called Schwarz triangles associated with a fundamental region of tessellation in the 2-D case and so-called Goursat tetrahedrons in the 3-D case. A natural generalization is that in n-dimensional one has n-simplexes. One would have what topologists call triangulation, which is very special in the sense that it utilizes the symmetries of the tessellation. These very special simplices are also consistent with the number theoretical constraints in the angles between n-1-faces correspond to angles defined by the roots of unity.

In the 2-D case, the angles between the edges of the fundamental triangle are rational multiples of π so that the cosines and sines of the angle are algebraic numbers, which are natural for a tessellation whose points in natural coordinates (momenta) have components that are numbers in an algebraic extension of rationals. In the 2-D case, the fundamental triangle is obtained by drawing from center points of the 2-D unit cell, say a regular polygon, connecting it to its vertices. In the 3-D case, the same is done for the 3-D unit cells of the fundamental region. Note that the tessellation can have several different types of unit cells and this is indeed true in the case of icosatetrahedral tessellations.

2-dimensional case

In the 2-D case, the angles between the edges of the triangle are given as (1/p,1/r,1/s)-multiples of π. p, r, and s are the orders of discrete rotation groups assignable to the vertices. They are generated by the reflections s_i with respect to edges of the triangle in one-to-one correspondence with opposite vertices. They satisfy the conditions s_i²=1 as reflections and the reflections s_i and s_i+j, j>1, commute and s_i s_i+1 generates a rotation with respect to the third vertex of the triangles with order determined by one of the numbers p, r, s. The conditions can be summed up to s_i²=1 and (s_i s_j)^m_ij=1, m_ij=2 for j ≠ i+/-1 and m_ij> 2 for j=i+/-1.

The conditions can be expressed in a concise way by using Coxeter-Dynkin diagrams having 3 vertices connected by edges. For m_ij=2, there is no edge, and for m_ij> 2, there is an edge and a number telling the order of the cyclic group in question.

All these 3 spaces are constant curvature spaces with positive, vanishing, or negative curvature, which is reflected as properties of the angle sum of the geodesic Schwartz triangle (note that these spaces also occur in cosmology). In the spherical case, the sum is larger than π and one has 1/p+1/r+1/s≥1. In the Euclidean case, the sum of the angles of the Schwarz triangle is π, which gives the condition 1/p+1/r+1/s=1. In the hyperbolic case, the angle sum is smaller than π and one has 1/p+1/r+1/s/le;1. Note that in the hyperbolic plane, the angles of infinitely sized Schwartz triangle can vanish (ideal triangle).

For the 2-sphere, these conditions give only Platonic solids as regular (vertex- and face-transitive) tessellation (no overlap between triangles). For the plane, the non-compactness implies that the conditions are not so restrictive as for the sphere. The most symmetric tessellations are regular tessellations: they involve only one kind of polygon and are vertex-, edge-, and face-transitive. For the Euclidean plane, there are regular tessellations by triangles, squares, and hexagons. If one weakens the transitivity conditions to say vertex-transitivity, more tessellations are possible and involve different kinds of regular polygons.

The Wikipedia article about the uniform tilings of the hyperbolic plane gives a good overall view of the uniform tessellations of the hyperbolic plane. For the hyperbolic tessellations, the conditions are the least restrictive. Intuitively, this is due to the fact that the angle sum can be small, and this allows small angles between edges and more degree of freedom at vertices. For a spherical tessellation, the situation is just the opposite. Uniform tilings of hyperbolic plane H² are by definition vertex-transitive and have a constant distance between neighboring vertices. This condition is physically natural and would correspond to mechanical equilibrium in which vertices are connected by springs of the same string tension. Each symmetry (p, r, s) allows 7 uniform tilings characterized by Wythoff symbol or Coxeter diagram. These tiling, in general, contain several kinds of geodesic polygons. Families with r=2 (right triangle) contain hyperbolic regular tilings.

The 3-dimensional case

There is a Wikipedia article about the uniform tessellations/honeycombs in the 3-D case, obtained by Wyuthoff construction, is a generalization from the 2-D case. Schwarz triangle is replaced with Goursat tetrahedron, and reflections are now in tetrahedral faces opposite to the vertices of the tetrahedron so that there are 4 reflections si satisfying s_i²=1 and (s_i s_j)^m_ij=1, m_ij=2 for j ≠ i +/-1. The cyclic subgroups act as rotations of faces meeting at the edges, and the angles defining the cyclic groups are dihedral angles. There are 9 compact Coxeter groups, and they define uniform tessellations with a finite fundamental domain. What is interesting is that the cyclic subgroups involved do not have order larger than 5.

The conditions are expressible in terms of Coxeter-Dynkin diagram with 4 vertices. The 2-D conditions are satisfied for the Schwarz triangles defining the faces of the tetrahedron. Besides the angle parameters defining the triangular phases of the tetrahedron, there are angle parameters defining the angles between the faces. All these angles are rational multiples of π and define subgroups of the symmetries of the tessellation. What is so beautiful is that the construction is generalized to higher dimensions and is recursive/hierarchical.

The hyperbolic character of the geometry allows Schwarz triangles and Goursat tetrahedra which in Euclidian case would not be possible due to the condition that the edges have the same length and faces have the same area.

Could hyperbolic, Euclidean, and spherical tessellations be realized in TGD space-time

An interesting question is whether the hyperbolic, Euclidean, and spherical tessellations could be realized in the TGD framework as induced 3-D geometry or rather, as slicing of space-time surface by time parameter such that each slice represents hyperbolic, Euclidean or spherical geometry locally allowing the tessellation.

Hyperbolic tessellations can be realized on the cosmic time constant hyperboloids and Euclidean tessellations on the Minkowski time constant hyperplanes of M⁴ and possibly partially on 3-surfaces which have hyperbolic 3-space as M⁴ projection.

The question boils down to a construction of a model of Robertson-Walker cosmology for which the induced metric of a=constant 3-surface is that of H³, E³, or S³ corresponding to the cosmologies with subcritical, critical and overcritical mass densities. The metric of H³ is proportional to a² scale factor. The simplest ansatz is a geodesic circle at geodesic sphere S²⊂CP₂ with metric ds²= -R²dθ²-sin²(θ)dΦ². The ansatz (sin(θ)=a/a₀,Φ=f(r)) gives in Robertson-Walker coordinates the induced metric

ds²= [1- R² (dθ/da)²] da² -a² (1/(1+r²)+ (R/a₀)²(df/dr)²) dr² + r²dΩ²

This gives the flat metric of E³ if the condition

(df/dr)²= (a₀/R)² r²/(1+r²)

This condition is satisfied for all values of r.

For S³ metric one obtains the condition

(df/dr)²= (a₀/R)² 2r²/(1-r⁴)

r=1 corresponds to singularity. For r=1, one has r_M= ar= a, which gives t= 2^1/2a. One can construct the S³ by gluing together the hemispheres corresponding to the 2 roots for df/dr so that it seems that one obtains the tessellations. The divergence of df/dr tells that the half-spheres become orthogonal to H³ at the gluing points.

For both E³ and S³ option, the component g_aa of the induced metric is equal to

g_aa= 1-(R/a₀)² 1/(1-(a/a₀)²)

g_aa diverges at a=a₀ so that the cosmic time would run infinitely fast. g_aa changes sign for a=a₀ so that for a>a₀ the signature of the induced metric becomes Euclidean. Unless one allows Euclidean signature in long scales, one must assume a0. Note that the action defined as the sum of Kähler action and volume action. If S² corresponds to the homologically trivially geodesic sphere of CP₂, the action reduces to volume action for these surfaces. The densities of Noether currents for volume action vanish at a= a₀ since they are proportional to the factor (g_aa)^1/2g^aa and thus approach to zero like [1-(a/a₀)²]^1/2. This is true also for the contribution of Kähler action present for homologically non-trivial geodesic sphere of CP₂. Very probably, this surface is not a minimal surface although the volume is finite. This is suggested by the fact that the volume element increases in comparison to hyperbolic volume element giving rise to minimal volume increases as the parameter a increases.

See the chapter More about TGD and Cosmology.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

The realization of the notions of assembly and tensegrity in the TGD Universe

In the TGD framework one ends up with an amazingly simple engineering principle resembling so called assembly theory and applying to atoms, nuclei, and hadrons (see this). Since TGD Universe is fractal, this principle is expected to apply in all scales.

1. The considerations of the above article relate closely to the observation that j-block consisting of parts of electron of atoms or nucleon shells of nuclei with fixed value of total angular momentum j=l+/- 1/2 and l=9 (at least) correspond to Platonic solids for l≤ 5 in the sense that different angular momentum eigenstates correspond to the vertices of the Platonic solid. If one assumes the presence of a Hamiltonian cycle going through all V vertices of the Platonic solid as a tessellations of sphere, one has F-2 free edges (F is the number of faces) besides the V edges of the cycle and one can also add particles to the middle points of the free edges. In the proposed model of atomic nuclei, one would have neutrons at the vertices and protons at the middle points or vice versa. Also the larger values of l appearing in highly deformed nuclei can be treated in the same way. If the unit of angular momentum increases to h_eff=nh, also these states can be assigned a Platonic solid.
2. The space-time surfaces assignable to all atoms, nuclei, and hadrons can be constructed by connecting the electrons, nucleons, or quarks at the vertices of Platonic solid or at the middle points of the free edges with flux tubes serving as analogs of springs stabilizing the structure and having interpretation as analogs of mesons. Tensegrity is the appropriate notion here.
3. In the case of hadrons, the predictions of the resulting mass formulas are satisfied within a few percent. This involves the predictions of TGD based mass calculations for fermion masses based on p-adic thermodynamics. This leads to an interpretation of the non-perturbative aspects of strong interaction in terms of a dark variant of weak interactions for which perturbation theory converges! The basic problem of QCD disappears in the TGD Universe. The same would apply to nuclear strong interactions but meson-like particles would have different p-adic length scales.

   2 and electroweak symmetries to the holonomies of CP₂ so that a very close relationship between these interactions must exist. One can say that a unification of strong and weak interactions analogous to that provided by Maxwell electrodynamics for electric and magnetic fields takes place. For a given p-adic length scale (several fractally scaled variants of hadron physics are predicted) one can regard mesons as weak bosons predicted by TGD to have the entire spectrum of exotics. For this there is already support (see this, this and this). Ordinary hadron physics would correspond to dark weak interactions for p-adic length scale defined by Mersenne prime M₁₀₇ and weak interactions to hadron physics for M₈₉!

4. In the case of nuclei, the MeV scale for excitation energies is correctly predicted and also a new 10 keV scale supported by various anomalies of nuclear physics is predicted. Besides this also Z^0 force is predicted to be significant and atom-like structures involving and having size scale 10 nm, which is a fundamental scale in biology, are predicted.

   The j-blocks (angular momentum) consisting of energy degenerate states with 2j states have as space-time correlates Platonic solids with Hamiltonian cycle as a closed flux tube, nuclear string connecting the vertices of the solid.

5. In atomic physics the same picture applies, and led to a realization that in the standard model the repulsive classical interaction energy of electrons goes like Z⁴ whereas the interaction energy nucleus goes like Z²! The question is whether quantum mechanics can really guarantee the stability of many electron atoms or is this just an assumption. In the TGD framework, the flux tubes would stabilize the atoms with several electrons. This predicts new atomic physics related to the oscillations of the flux tubes which in nuclear physics give justification for the harmonic oscillator model of nucleus.

See the article Neil Gersching's vision of self-replicating robots from TGD point of view or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD. For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Vision about unification of strong and weak interactions

The considerations of the article About Platonization of Nuclear String Model and of Model of Atoms) inspire a unified vision about strong and weak interactions.

1. At the level of H=M⁴ × CP₂, the color group SU(3) acting as isometries of CP₂ would describe the perturbative aspects of color interaction and give rise to color confinement. The non-perturbative aspects of strong interactions would correspond to the holonomy group of CP₂ and weak interactions for weak bosons which are either dark or p-adically scaled variants of ordinary weak bosons and massless below the scaled up Compton length.

   The large value of h_eff would make the perturbation theory for these weak interactions convergent (see this). Strong isospin can be identified as weak isospin. Both p-adic and h_eff hierarchies of length scales are required in the proposed vision.

2. At the level of M⁸ = M⁴ × E⁴, SU(3) corresponds to a subgroup of octonionic automorphisms and U(2) could be identified as a subgroup of isometries leaving invariant the number theoretic inner product in E⁴. This inspired the proposal that strong isospin corresponds to U(2) and hadron-parton duality corresponds to M⁸-H duality basically.

This picture explains various poorly understood aspects of strong interactions.

1. In the good old times, when strong interactions were not yet "understood" and it was also possible to think instead of merely computing, strange connections between strong and weak interactions were observed. The already mentioned conserved vector current hypothesis (CVC) and partially conserved axial current hypothesis (PCAC) were formulated and successful quantitative predictions emerged.

   Strong isospin is equal to weak isospin for nucleons but heavier quarks did not fit the picture. (c,s) and (t,b) dublets were assigned quantum numbers such as strangeness and charm, and they are not quantum numbers of weak interactions.

   When perturbative QCD became the dominating science industry, low energy hadron physics was forgotten. Lattice QCD was thought to describe hadrons but the successes were rather meager. Lattice QCD has even mathematical problems such as the description of quarks and the strong CP problem which lead to postulate the existence of axions, which have not been found.

2. In TGD these connections can be understood elegantly.

   1. The topological description of family replication phenomenon implies that strangeness and charm are not fundamental quantum numbers and the identification of weak and strong isospins makes sense.
   2. Strong interactions in long length scales for hadrons become p-adically scaled dark weak interactions. The flux tubes correspond to possibly p-adically scaled mesons or equivalently weak bosons in a generalized sense predicted by the TGD based explanation of family replication phenomenon. Tensegrity is the basic construction principle for hadrons and nuclei and even atoms, for which color octet excitations of leptons define the counterparts of mesons.

Also the fractality inspired ideas related to p-adically scaled up variants of strong and weak interactions organize to a beautiful picture.

1. p-Adic fractality inspired the idea that both strong and interaction physics appear as p-adically scaled variants. In particular, M₈₉ hadron physics would be a p-adically scaled up version of the ordinary hadron physics assignable with M₁₀₇ and would correspond to the same p-adic length scale as weak bosons. Various forgotten anomalies support this proposal (see this and this).

   But why both weak and strong interaction physics with the same p-adic length scale (or actually scales)? Both weak bosons and mesons would be described as string-like entities. How can one distinguish between these?

2. There is no need for both! Weak bosons and their predicted exotic counterparts implied by the family replication phenomenon are nothing but the mesons of M₈₉ hadron physics. TGD explanation of the family replication phenomenon indeed predicts the analog of family replication phenomenon for weak bosons basically similar to that for mesons. From the known spectrum of mesons of ordinary mesons one can predict masses of both M₈₉ mesons, or equivalently the masses of ordinary and exotic weak bosons. There is already evidence for the dark counterparts of M₈₉ mesons with scaled up Compton length equal to that for M₁₀₇ mesons. Also M₈₉ baryons are predicted.

3. Higgs would be the counterpart of sigma meson. There is evidence of the pseudoscalar counterpart of Higgs identifiable as a counterpart of M₈₉ pion. Weak bosons would be counterparts of ρ meson. Also axial vector weak mesons are predicted as counterparts of ω.

   The exotic weak mesons as counterparts of kaon, charmed mesons, etc.., are predicted but their p-adic length scale is shorter. Also for these there is some evidence (see this and this). In particular, there are indications for the existence of Higgs-like states decaying into e-μ pair (see this). This particle might correspond to kaon, which is pseudoscalar rather than scalar. All masses can be predicted from hadron physics by scaling apart from the p-adic prime defining the mass scale and satisfying the p-adic length scale hypothesis.

See the article About Platonization of Nuclear String Model and of Model of Atoms or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.