Monday, June 29, 2020

John Baez about Noether's theorem in algebraic approach

John Baez gives very nice summary (see this of the triple of states, observables, and generators of symmetries from purely algebraic point of view. Jordan Banach algebra with commutative product A*B= (AB+BA)/2 would play the role of observables. The operators correspond to symmetry generators commuting with observables and the unitary evolutions generated by them in this algebra are trivial. One could say that this defines analogy of Noether's theorem usually deduced for the symmetries of action principle.

To me the weakness of the algebraic approach is that it says very little about the dynamics- it woul be just unitary evolution generation by some generator of symmetry. Second problem is mentioned at the end of the posting is how classical relates to quantal. And there is nothing about quantum measurement problem so that basically an attempt to reproduce wave mechanics using operators is in question.

My own view - zero energy ontology (ZEO) - goes much beyond quantum mechanics of simple systems.

1. The basic problem of quantum measurement theory is the starting point. The notion of quantum state is modified. In wave mechanics and quantum field theories it is based on initial value problem in configuration space (space of positions for particle). Initial state is wave function - a superposition of possible initial values in configuration space. Time evolution is formulated in terms of unitary evolution defined by exponential of Hamiltonian and reduces to Schroedinger equation.

What happens in quantum measurements is not consistent with this time evolution. This is the problem.

1. In TGD one replaces initial value problem with a boundary value problem with boundary corresponding to values at times t1 and t2 (this is a simplification).
2. One defines states as superpositions of deterministic classical time evolutions - preferred extremals - analogous to Bohr orbits having the property that boundary value problem is equivalent to initial value problem. Once on knows configuration at t1, one knows it at t2.
3. Quantum states are superpositions of these preferred extremals from t1 to t2 and quantum jump replaces this kind of superposition with a new one. I call this approach zero energy ontology (ZEO).
4. The basic problem of quantum measurement theory disappears since there are two causalities: that of quantum jump and classical causality, and there is no need to break the deterministic time evolution analogous to that given by Schroedinger equation in quantum jump. There are two times: experienced time as sequence of quantum jumps and geometric time. One ends up with a theory of consciousness without moment of consciousness identifiable as state function reduction. Also a ZEO theory of self-organization emerges. This is of course only the basic idea. For instance, one must understand how correlation between experienced time and geometric time emerges.
5. One the many implications of ZEO is new view about quantum tunnelling: it must have classical time evolution as quantum correlate. This leads to a new view about tunnelling in nuclear reactions relying essentially on the change of the arrow of time in ordinary state function reduction. Just today I received link telling about strange phenomenon occurring in what is believe to be ordinary electron tunnelling. The electron getting through the barrier radiates energy which increases with the height of the barrier. I discusse the ZEO based explanation in a related posting New support for TGD view about quantum tunnelling .

How to realize this picture and how unique is it? Here one must leave the realm of wave mechanics.

1. Loosely speaking, in TGD framework point-like particle is replaced with 3-D surface in M4× CP2 and its orbit as preferred extremal of action principle, whatever it might be, defines space-time region. A generalization of string model is in question. Also a generalization of general relativity solving its problem due to the loss of Poincare symmetries is in question.
2. This leads to a generalization of Einstein's geometrization program: replace configuration space with the "world of classical worlds" (WCW) and give it Kaehler geometry to realize geometrization of quantum theory. Points of WCW are 3-surfaces or equivalently 4-surfaces: this reduces holography and reduces it to general coordinate invariance.

WCW spinor fields would represent physical states as "wave functions". Configuration space gamma matrices would be superpositions of fermionic oscillator operators so tha also fermions are geometrized. The mere existence of WCW Kaehler geometry requires maximal isometries and this fixes TGD highly uniquely. Freed realize the uniqueness for loop spaces.

How to realize the crucial preferred extremal property making initial value problem equivalent with boundary value problem?
1. Here the maximal isometry group of WCW enter the game. The symmetry algebra is replaced with an analog of infinite-D symplectic algebra acting as isometries of WCW induced from symplectic transformations at delta M4+xCP2 labelled by integer valued conformal weights assignable to the radial light-like coordinate of light-cone bounary δM4+ defining second boundary of causal diamond cd identified as the intersection of future and past directed light-cones.
2. The crucial point is that this algebra - call it A - has fractal hierarchy of sub-algebras An with conformal weights coming as multiples of n=1,2,... which very probably corresponds to a hierarchy of hyper-finite factors of type II1 forming inclusion hierarchies labelled by sequences ...n1 divides n2 divides....
3. Infinite-D sub-algebra An appears would have vanishing classical Noether charges in the class of preferred extremals associated with An. Also [An,A] would have the same property.This is like posing the condition that analogous sub-algebra of say Kac-Moody algebra annilates physical states. The space-time surfaces in question would be minimal surfaces satisfying the additional condition that they extremize also what I call Kaehler action, and being analogous to Maxwell action.

See the article Some comments related to Zero Energy Ontology (ZEO).

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

New support for TGD view about quantum tunnelling

There is a popular article in Phys.org describing a highly interesting finding made by condensed matter physicist Doug Natelson and his colleagues at Rice and the University of Colorado Boulder. Light emission associated with the tunnelling of electrons through a nano-scaled potential barrier between Gold electrodes has been observed. The intensity of emission is larger by factor 10,000 than predicted and this suggests new physics.

By definition tunnelling means that electrons get through the potential barrier without getting energy - classical picture would require this. Now it however seems that electron receives energy and the higher the barrier the larger energy is needed. Should one challenge the notion of tunnelling? Could it have a classical counterpart?

In TGD framework all quantum phenomena should have classical counterparts, also tunnelling. Therefore the tunnelling electron would actually get energy to get over the potential barrier classically.

1. In zero energy ontology (ZEO) solving the basic problem of quantum measurement theory quantum states are superpositions of deterministic classical time evolutions, preferred extremals. Classical physics is exact part of quantum theory. The key prediction is that in ordinary, "big", state function reductions (BSFRs) the arrow of time is changed. In small SFRs (SSFRs) - analogs of weak measurements - this does not happen. ZEO leads to a theory of self-organization in which energy feed to self-organizing system corresponds to dissipation for time reversed state associated with dissipating system.
2. Tunnelling could mean in TGD framework that electrons make a BSFR reversing the arrow of time. In time reversed state they dissipate in reverse time direction: in standard time direction of observer they receive energy in standard time direction allowing them to get over the potential barrier. In the second BSFR establishing the original arrow of time they would liberate the energy and the higher the barrier, the larger the liberated energy and the brighter the light emission. The tunnelling in this sense would be already self-organization phenomenon involving BSFR. This dynamical tunnelling does not exclude the analog of wave-mechanical tunnelling as a non-dynamical process. The dynamical tunnelling could actually lead to asymptotic states in which particle is at both sides of the forbidden region or even in forbidden region but with extraction of classical energy from environment so high that this is not forbidden anymore.
For ZEO see the article Some comments related to Zero Energy Ontology (ZEO). For TGD view about tunnelling in nuclear reactions see Solar Metallicity Problem from TGD Perspective

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

Saturday, June 27, 2020

New ideas about the transition to ferromagnetic phase

I received a link to a highly interesting article about ferromagnetism. According to the article, Yi Li, a physicist working at John Hopkins University and his two graduate students, Eric Bobrow and Keaton Stubis, seem to have made a considerable progress in understanding how the system of electron spins in lattice ends up to a ferromagnetic sate. This ferromagnetism is known as itinerant ferromagnetism and involves vacancies, sites without electron, which can be moved freely without affecting the energy of the state.

1. The ideas related to the work of Li et al

The problem considered by Li et al how the ferromagnetic state could emerge from an arbitrary state with some numbers of spin up and spin down states at lattice sites connected by edges.

1. Permutation of electrons with same spin leave the ferromagnetic state invariant and does not cost energy while permutations in arbitrary configuration can do so.
2. Li et al considered a simple 4× 4 lattice with single vacancy and noticed a connection with so called 15-puzzle involving 15 tiles and single vacancy with neighboring tiles of vacancy able to move to its position. The observation is following. If one has spin lattice containing single vacancy, one can number the sites by a number running from 1 to N (now 15) in arbitrary manner. If so called connectedness condition holds true one can realize any permutation of these numbers. This means that 15-puzzle has always a solution. In particular, one can arrange the situation that the numbers form an ordered sequence from 1 to N so that numbers n and n+1 are nearest neighbors.

The result found by Li et al first for 2-D 4× 4 lattice with single vacancy generalizes to lattices, which are non-separable in the sense the removal of a lattice site does not separate any pair of spins - they are still connected by an edge-loop.

3. The curve solving the 15-puzzle goes through all points of the 4× 4 lattice and is generally known as Hamiltonian curve. It becomes Hamiltonian cycle if the numbers 1 and N are nearest neighbors.
4. The basic problem of this approach is that the theorem is true only for single vacancy and does not allow generalization to a larger number of vacancies. It is however known that ferromagnetism is possible up to fraction 1/3 for vacancies. The challenge is to generalize the result of Li et al.

2. Some reasons to get interested

In TGD framework there are good reasons of getting interested on these results.

1. The result of Li et al states that ferromagnetic phase transition might be understood in terms of shifting of lattice vacancy if the lattice with single defect allows deformaton of any configuration of spin labelled by numbers n running from 1 to N to a closed curve connecting nearest neighbors along which n increases. Could there be a connection with Hamiltonian curves making sense for lattice like structures (actually all graphs)? Could Hamiltonian curve have some deeper physical meaning or is it only an auxialiary notion useful for representing the possibility to realize all points of the lattice with vacancy by shifting it suitably?

Hamiltonian curve connects neighboring points of a lattice and goes through all points without self-intersections. Icosahedral geometry appears in biology and one can ask whether this kind of cycles could be actually realized physically - say as flux tubes at icosahedron and tetrahedron, which play key role in TGD inspired biology. Flux tube are actually fundamental objects in TGD Universe in all scales. For instance, final states of stars could correspond to flux tube spaghettis consisting of single volume filling flux tube (see this).

2. If the Hamiltonian cycle is something physical it could correspond to flux tube. The notion of magnetic flux tube central in TGD might allow application to ferromagnetism. TGD predicts two kinds of flux tubes: Maxwellian ones and monopole flux tubes with magnetic fields requiring no currents to generate them: they are not not allowed by Maxwell's theory.

The preservation of the Earth's magnetic field predicted to decay rather rapidly as currents generating it dissipate supports the view that it contains monopole flux part which from biological input would correspond to endogenous magnetic field Bend, which is a fraction 2/5 about the nominal value of BE=.5 Gauss. The presence of magnetic fields in cosmological scales is also a mystery finding a solution in terms of monopole flux tubes.

3. Monopole flux tubes must be closed. Closed non-intersecting flux tubes connecting nearest neighbors in lattice would correspond to Hamiltonian cycles. In TGD inspired biology Hamiltonian cycles associated with icosahedron and tetrahedron provide a realization of the vertebrate genetic code (see this) but it is still somewhat of mystery why the points of icosahedron and tetrahedron, which are lattices (tesselations) at sphere, would be connected by a curve. Quantum classical correspondence suggests that magnetization corresponds to flux tubes connecting magnetic dipoles as formal analogs of monopole-antimonopole pairs. Could magnetic flux tubes provide a concrete realizaon for these Hamiltonan cycles?
4. Closed monopole flux tubes seem to be unrealistic for the description of ferromagnetism, which suggests the presence of N parallel flux tubes carrying magnetization M and defining a braid connecting opposite ends of ferromagnet. The monopole fluxes could arrive as single flux along parallel space-time sheet carrying field H defined by single thick flux tube. Test particle would experience B= M+H.

The following considerations are not much more than first impressions and probably require updating.

3. TGD based view

Flux tubes are the new element of condensed matter physics predicted by TGD. Could they provide insights into ferromagnetism?

3.1 Starting from text book picture about ferromagnetism

To develop TG view about ferromagnetism it is best to start from the text book picture.

1. In the standard model of ferromagnetism one assumes the presence of field B identified as sum B= M+H of magnetization and field H equal to B outside the magnet. M is due to magnetic dipoles besides magnetic field B and the interaction of spins with H is important. B is usually regarded as the fundamental field M and and H appear as auxiliary notions and their relation to B requires a model for the system: typically H, B, and B are assumed to be linearly related.
2. The field M could be naturally assigned with a flux tube connecting the spins - perhaps at nearest neighbor lattice points. What about H? In standard model H and B are parallel for the ferromagnetic configuration. If B is assigned with the flux tube connecting the magnetic moments and B is parallel to H, this would suggest a flux tube consisting of long straight portions parallel to each other.

In the many-sheeted space-time of TGD M and B can reside at different space-time sheets, which are parallel so that they are on top of each other in M4×CP2. The decomposition to sum would have representation as a set theoretic union.

The test particle would experience the sum of the magnetic fields associated with the two sheets. Could M and H as the return flux associated with M and superpositing with the external contribution to H correspond to these two space-time sheets so that particle would experience their sum B=M+H? If so, ferromagnetism could be seen as a direct signature of many-sheeted space-time.

3.2 Could also monopole flux tubes be important?

There is still one important aspect related to the TGD view about magnetic field which might play important role. TGD predicts two kind of flux tubes. The first kind of flux tubes could be called Maxwellian and the corresponding magnetic fields require current to generate them. There are also flux tubes having closed cross section and carrying monopole fluxes. No currents are required to generate corresponding magnetic fields. Could also these flux tubes having no current as sources be present? This would mean new physics.

1. The first thing to notice is that the interpretation of magnetization M is as a magnetic field generated by magnetic moments. The usual interpretation is that spins are analogous to magnetic moments created by currents consisting of rotating charge. Now there is no such rotating charge. Second interpretation is as magnetic moments identifiable as infinitesimal monopole pairs.
2. Could one think that the flux tubes containing sequence magnetic moments correspond to monopole flux and that closing this loop could give rise to monopole magnetic field? Ordinary Maxwellian part could be also present and have current as source. How M and H would relate to these. Could M correspond to the monopole part and H the Maxwellian part?

Are spins necessary for the existence of a monopole flux tube? Could quantum classical correspondence require this? Could dark charged matter assigned with the monopole flux tubes correspond to the magnetic moments of say dark valence electrons with non-standard value of h so that M would be represented by monopole flux tubes classically? If the return flux represented by H is absent, flux tube must give rise to a Hamiltonian cycle. If H is present, it would be enough to have flux tubes representing N braid strands fusing to single monopole flux carrying the return flux.

Formation of a flux loop defining Hamiltonian cycle would be a new kind of phenomenon analogous to spontaneous magnetization requiring no external field H. Spontaneous magnetization would be however something different. A trivial braid consisting of N parallel strands representing M and parallel to it locally with return flux arriving along single large flux tube carrying H would be formed in ferromagnetic transition and also in spontaneous magnetization.

3.3 Bringing in thermodynamics

One can try to make this more concrete by bringing in thermodynamics.

1. Assume that there exists single flux tube - connecting all the lattice points (magnetic moments) or possibly N flux tubes parallel to local magnetization M and giving rise to a braid like structure representing the topology of flux lines of M connecting opposite boundaries of magnet.
2. In the general case, the points of the lattice could be connected by a flux tubes connecting points, which need not be nearest neighbors. The first guess is that the magnetic interaction energy of spins at the ends of the flux tube portion connecting them decreases with the distance between spins. There sould be also magnetic energy associated with the field H at the space-time sheet carrying the return flux. Thermodynamics would bring in entropy and free energy F=E-TS would be mimimized. Entropy maximization would favor long random flux tubes and energy minimization short flux tubes.

One expects that flux tube has free energy F increasing with flux tube length. If one does not allow self-intersections - as suggested by repulsive Coulomb interaction and Fermi statistics - the flux tube could be either Hamiltonian cycle or consist of analogs of braid strands: in the case of ferromagnetism the strands would be parallel to each other. The interacton energy would be same for all Hamiltonian cycles if determined by nearest neighbour interactions.

3. In the general case with lattice replaced by graph one expects that a large number of Hamiltonian cycles not related by rotation to each other exists so that one would have large number of states with same minimum energy. Could this somehow correspond to spin glass state allowing large number of degenerate states? The flux tube need not be closed. In ferromagnetic configuration this would be the case.
4. How would the assignment of spin direction to the lattice points affect the situation? Could the numbers N+ and N- of spin up and spin down electrons determine the flux tube configuration by (Gibbs) energy minimization?

3.4 Could 2-braid describe the transition to ferromagnetism?

In the work of Li et al discussed in the article, the permutations of lattice points are induced by moving the vacancy around. This picture inspired the considerations above but is too limited. In fact the work of Li et al only directed attention to Hamiltonian cycles and braids formed by the non-closed analogs.

1. TGD picture brings in mind braid-knot connection. One can replace braid assiciated with M with a knot by connecting the magnetic moments at the opposite ends of the braid by trands of a trivial braid at parallel space-time sheet. This trivial braid would carry the return flux having interpretation in terms of field H.

The flux tubes of trivial braid could also fuse to single thicker flux tube carrying the total return flux associated with M. This would conform with the idea that H provides a description of the system in longer length scale being analogous to a smoothed out total magnetic field acting as self-consistent background.

This stimulates a critical question. Could one assume that only H assignable to big flux tube has constant direction and magnitude and that M is represented as flux tubes connecting dipoles can in principle correspond to any permutation of atoms. For this option the spontaneous magnetization would correspond to a superposition of different configurations with same weights and would be invariant under permutations as in the argument of Li et al involving no flux tubes. This option does not look attractive.

2. What braid picture allows to say about the transition to ferromagnetism? Could the transition be realized by deforming the flux tubes associated with M and forming a non-trivial braid be induced by permutation of the lattice points taking the non-trivial braid to trivial one? This would be like opening the braid. The lattice points in the initial and final state would correspond to the ends of a dynamical evolution. The permutation would be realized as a time-like braiding with braid strands in time direction.

Mathematically braid group corresponds to the covering group of permutation group and quantum group representations correspond to the representations of braid groups. The description of the transition could provide a new application of quantum groups.

The description as time-like braiding is not however complete since there isan additional structure involved: the flux tubes connecting the magnetic dipoles in lattice and defining a braid or even more complex configuration having flux tube connections between non-neighboring poins.
1. If there is no return flux assignable to H, M corresponds to a closed flux tube carrying monopole flux the dynamical time-like dynamcial braiding would lead to a Hamiltonian cycle in this case and the number of final state configurations would be finite, there is degeneracy. Could spin glass phase correspond to this situation?
2. In ferromagnetism final state would contain N parallel strands carrying the monopole flux assignable to M and the return flux H would arrive along parallel thick flux tube. In general configuration these strands can be braided. The transition to ferromagnetism would represent time-like braiding of an ordinary 3-D braiding of flux tube strands connecting the opposite boundaries of ferromagnetic. In the initial state braid would be non-trivial and the flux tubes of braid would not have minimal length and minimum energy. In the the final ferromagnetic state braid would be trivial with parallel flux tubes.

Mathematically this process would correspond to what is called 2-braiding: I have proposed that 2-braidings are important in TGD inspired biology as a topological description of dynamical processes. An interesting interpretation is as a topological analog for problem solving. I have also proposed that in biosystems topological quantum computation programs are represented as this kind 2-braidings for flux tubes (see this and this.)

Ferromagnetism would correspond to an opening of a non-trivial braid. If the return flux arrives along flux tubes this is possible smoothly only if the knot defined in this manner is trivial. To achieve opening, the 2-braiding must involve reconnections, which correspond to cutting the knot strand and reconnecting the pieces in new manner: this is how Alexander opened his knot. Fermi statistics and repulsive Coulomb interaction do not fabour this mechanism. If the return flux arrives along single flux tube, the opening could correspond to a smooth deformation without reconnections transferring the braidedness to the parallel space-time sheet, where it is "neutralized" by fusing the flux tubes to single flux tube.

See the article TGD based view about ferromagnetism or the chapter Quantum criticality and dark matter: part I.

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

Thursday, June 25, 2020

Planning to move to Mars?: have you asked the opinion of your magnetic body?

There has been a lot of talk about the colonization of Mars. I was asked about possible problems involved. In the following I consider the question only from TGD point of view about life.

The new element of TGD based quantum biology is the notion of magnetic body carrying dark matter as phases of ordinary matter labelled by the effective value heff =nh0 of Planck constant (h= 6h0 is a good guess). MB is the boss controlling ordinary matter by using dark photons transforming to ordinary photon and vice versa. Actually one has hierarchy of values of heff and master slave hierarchy.

Cyclotron frequencies and generalize Josephson frequencies are in a central roles in the communications between MB and biological body relying on frequency and energy resonance if the values of heff are identical or only energy resonance if they are not same.

Frequencies are thus fundamental for the control level of life and the question is whether MB can adapt and change its frequency spectrum. Also Schumann frequency is known to be important for life and in Mars its scae 2 times that in Earth. These problems might make already the travel to Mars dangerous to say nothing about living there.

1. Cyclotron frequencies and Schumann frequencies

On basis of findings done already at seventies about unexpected effects of ELF em fields on vertebrate brain, EEG resonances seem correspond to cyclotron frequencies in endogeneous magnetic field of Bend=.2 Gauss which is 2/5 of the nominal value BE=.5 Gauss of Earth's magnetic field. A possible interpretation is that Bend corresponds to field strength for the monopole flux part of BE. It is stable against dissipation since no current is needed to create it unlike ordinary Maxwellian magnetic field. This explains why BE has not decayed away long time ago by the decay of the current creating it.

Cyclotron frequencies are in a key role in TGD based quantum biology. The value spectrum of heff determining bio-photon energies is in central role: dark photon energies are given by E=heff×f.

The cyclotron energy spectrum in Earth would be in bio-photon range (visible and UV) and same for all charged particles - there is no dependence on mass of charged particle since one has hbareff= hbargr= GMm/v0 at gravitational flux tubes. hbargr= GMm/v0 is gravitational Planck constant fundamental introduced first by Nottale. v0 is velocity parameter with value about v0/c= 2-11 in the case of Sun. It might relate to the parameters of the system but could be also almost universal parameter (it is near electron proton mass ratio).

The cyclotron energy spectrum in Earth would be in bio-photon range and same for all charged partcles - there is no dependence on mass since one has hbareff= hbargr= GMm/v0 at what I call gravitational flux tubes. This would make possible the control of molecular transitions by transforming dark photons to visible and UV photons first. MB could control molecular transitions and act as "boss". Quite generally, the layers of MB with different values of heff would form a master slave hierarchy with ordinary biomatter at bottom.

Schumann resonances correspond to resonance frequencies for the radiation fields of Earth. In the approximation that the conductivity of the medium is infinite, the wavelengths depend on the radius RE of Earth only. Lowest wavelength corresponds in good approximation to lambda= 2πRE. Lowest Schumann frequency has nominal value 7.8 Hz. These frequencies are in EEG range.

Remark: I remember having read in some source that 10 Hz frequency corresponding to alpha band in brain is needed in space and is produced artificially. I could not find the original source.

2. Coupling between Schumann frequencies and cyclotron frequencies

1. Schumann frequencies are is in the range of EEG frequencies, which strongly suggests resonant coupling with the cyclotron transitions corresponding to frequencies near to Schumann frequencies.
2. For instance, the cyclotron frequency 7.5 Hz for K+ ions is near to the nominal value 7.8 Hz of the lowest Schumann frequency. K+ currents through K+ channels cell membrane are known to be important for what happens in anesthesia and when organism falls asleep and thus for consciousness.

What happens when organism falls asleep is of course not quite clear. The membrane voltage increases and nerve pulse generation becomes improbable. Metabolism continues and this means consciousness in TG framework but not sensory and motor consciousness. Could the apparent loss of consciousness mean fusion of self with higher level self representing the collecting consciousness of magnetic Mother Gaia? Could it be that during sleep we could form a larger collective consciousness representing "human condition" (see this) .

3. Irradiaton by frequency around 7 Hz is also necessary in the experiments of HIV Nobelist Montagnier involve remote DNA replication (see this) and this) .
4. Irradiation by Schumann frequency around 60 Hz appears leads to healing of cancer cell population in lab (see this) .
5. Also higher Schumann resonances could represent resonant coupling between Mother Gaia and brains of vertebrates allowing realization of levels of collective consciousness. Coherent collective gene expression in various scales (organ, organism, group, population,...) is one of the possible testable implication.
3. Possible problems related to different values of magnetic fields in Earth and Mars

The first problem relates to the magnetic field of Mars. Martian magnetic field has been believed to be weak but the recent findings suggests that it might be present and order of magnitude as in Earth (see this) .

1. TGD based view about life requires Bend which could be monopole part of BE. Flux tubes can carry monopole flux or not - the latter option is realized in the case of ordinary Maxwellian magnetic field. Monopole flux would be biologically important and it would be in Earth 2/5 of total flux. It could be of course different in Mars. Different value of Bend in Mars could lead to severe difficulties since cyclotron frequencies are in a key role in TGD based quantum biology.
2. Also the value spectrum of hbargr= GMm/v0 determining biophoton energies is in central role: one has for energies E=heff×f. In Mars the mass parameter M and velocity parameter v0 in hgr could be different and this could have fatal consequences for bio-control by the magnetic body of earthly biosystems.
1. In the case of Earth M seems to correspond to the mass of "inner inner core" of Earth. If M is same for Mars (planets would be like atoms having varying number of identical "shells"), then it would be enough to have same value B/v0 to keep the spectrum same. The value of B is under control by varying the flux tube thickness and this control mechanism would be used in terrestial biology.
2. The optimistic guess is that v0 has universal spectrum and the value is same in Mars and Earth. A not necessarily good guess is that the velocity v0 scales like the rotation frequency of planet. Here we would have good luck! The Martian day has essentially the same length as day in Earth. The value of v0 would be same! In this case hbargr could be the same! Sounds too optimistic!
3. One can imagine also second option. The strength of monopole flux magnetic field Bend =.2 Gauss at flux tubes is free parameter and magnetic bodies could vary its value by varying thickness of flux tubes: this would be a basic biological control mechanism for instance in water and allow water to develop antennas sensitive to the cyclotron frequencies of invader molecules and detect them: this would be water memory and basis of the immune system.

4. Problems related to different Schumann frequencies

Schumann frequency scale in Mars is roughly twice that in Earth since radius is about 2 times smaller: around 15.6 Hz instead of 7.8 Hz for lowest Schumann frequency. Schumann frequency is known to be important for us although this is not well-understood. This might cause severe problems.

The scaling of Schumann frequencies by 2 might cause bad complications but we do not really understand their effect even in Earth.

We of course do not have thorough knowledge about B in Mars. There are auroras but earlier it was believed that magnetic field in large scales is very small.The latest findings suggest in TGD framework that there is monopole flux tube part of order Bend (see this).

One can consider a possible solution to Schumann frequency doubling problem. Scale also the cyclotron frequencies by factor 2!

1. At Earth the lowest Schumann frequency 7.8 Hz corresponds to cyclotron frequency 7.5 Hz of K+ ion quite well and this might make possible resonant coupling between Mother Gaia and magnetic bodies of living system. If the Bend=.2 Gauss is scaled by factor 2 in Mars to about .4 Gauss, cyclotron frequency scale doubles and one would have 15 Hz cyclotron frequency for K+ not far from Schumann.
2. Could human MB scale up the cyclotron frequencies by factor 2? Note that TGD based model for Cambrian explosion (see this) predicts that Schumann frequency doubled in geologically short time scale (extremely long from human perspective). The lifeforms that would have bursted from underground oceans to the surface survived this transition. One could test what happens if artificially produced counterpart of Schumann frequency is doubled in artificial conditions using Faraday cage.

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

Wednesday, June 24, 2020

Schumann resonances and consciousness

The role of Schumann resonances for consciousness is highly interesting in TGD framework, where the notion of magnetic body (MB) carrying dark matter as heff=n*h0 phases is in key role. MB would control biological body by dark photons - probably at cyclotron frequencies - and receive information as dark generalized Josephson radiation from cell membrane with the modulation of membrane potential inducing frequency modulation coding for the sensory data (see this ) .

On basis of findings done already at seventies about unexpected effects of ELF em fields on vertebrate brain, EEG resonances seem correspond to cyclotron frequencies in endogeneous magnetic field of Bend=.2 Gauss which is 2/5 of the nominal value BE=.5 Gauss of Earth's magnetic field. A possible interpretation is that Bend corresponds to field strength for the monopole flux part of BE. It is stable against dissipation since no current is needed to create it unlike ordinary Maxwellian magnetic field. This explains why BE has not decayed away long time ago by the decay of the current creating it.

Cyclotron frequencies are in a key role in TGD based quantum biology. The value spectrum of heff determining bio-photon energies is in central role: dark photon energies are given by E=heff*f. The cyclotron energy spectrum in Earth would be in bio-photon range (visible and UV) and same for all charged particles - no dependence on mass since one has hbareff= hbargr= GMm/v0 at what I call gravitational flux tubes. This would make possible for MB to control molecular transitions and act as "boss". Quite generally, the layers of MB with different values of heff would form a master slave hierarchy with ordinary biomatter at bottom.

1. Schumann frequencies are is in the range of EEG frequencies, which strongly suggests resonant coupling with the cyclotron transitions corresponding to frequencies near to Schumann frequencies.
2. For instance, the cyclotron frequency 7.5 Hz for K+ ions is near to the nominal value 7.8 Hz of the lowest Schumann frequency. K+ currents through K+ channels cell membrane are known to be important for what happens in anesthesia and when organism falls asleep and thus for consciousness.

What happens when organism falls asleep is of course not quite clear. The membrane voltage increases and nerve pulse generation becomes improbable. Metabolism continues and this means consciousness in TG framework but not sensory and motor consciousness. Could the apparent loss of consciousness mean fusion of self with higher level self representing the collecting consciousness of magnetic Mother Gaia? Could it be that during sleep we could form a larger collective consciousness representing "human condition" (see this) .

3. Irradiaton by frequency around 7 Hz is also necessary in the experiments of HIV Nobelist Montagnier involve remote DNA replication. See (see this and this) .
4. Irradiation by Schumann frequency around 60 Hz appears leads to healing of cancer cell population in lab (see this) .
5. Also higher Schumann resonances could represent resonant coupling between Mother Gaia and brains of vertebrates allowing realization of levels of collective consciousness. Coherent collective gene expression in various scales (organ, organism, group, population,...) is one of the possible testable implications.

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

Monday, June 22, 2020

Has Pluto experienced a period of fast expansion?

The surprise of this morning was that Pluto seems to have experienced a rapid expansion and might have had underground oceans.

Actually not a surprise if one lives in TGD Universe. TGD predicts that all astrophysical objects, including planets, might have experienced this kind of periods. In general relativity these periods would correspond smooth cosmological expansion of this objects: the absence this smooth expansion is indeed an enigma - astrophysical objects co-move with expanion in large scale but do not expand themselves. Earth would have experienced this expansion during Cambrian explosion.

The underground oceans would make possible evolution of life without problems from cosmic rays and meteorites and radioactivity (for instance) could provide the metabolic energy.

This would solve Fermi's paradox: most planets could be carriers of intraplanetary invisible-to-outsider life waiting the next fast expansion and burst to the surface of the planet.

See http://tgdtheory.fi/public_html/articles/expearth.pdf .

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

Sunday, June 21, 2020

New evidence for two values of Hubble constant

Sabine Hossenfelder tells about a further evidence for the Hubble constant discrepancy and refers to this popular article discussing new kind of local measurements independent of other measurements also giving 10 per cent higher value than the value deduce from CMB measurements in much longer scales.

In the many-sheeted cosmology of TGD the cosmic parameters are length scale dependent. The natural guess is that the two measurements could correspond to sHubble constant in different p-adic length scales. The naive dimensional guess based inspired by the p-adic length scale hypothesis would suggest that Hubble constant could come as half octaves. The ratio of the two values is however around .1 much nearer to unity than 2-1/2≈.71.

Some time there was a popular article about a possible explanation of Hubble constant discrepancy (see this). The article told about a proposal of Lucas Lombriser discussed in the article Consistency of the local Hubble constant with the cosmic microwave background (see this) for an explanation of this discrepancy. The proposal is that the local region around our galaxy having size of order few hundred Mly - this is the scale of the large voids forming a honeycomb like structure containing galaxies at their boundaries - has average density of the matter 1/2 of that elsewhere.

This would fit nicely with the TGD picture. I do not bother to repeat the argument given in the earlier blog posting. The point is that in TGD the string tension of magnetic flux tubes comes by p-adic length scale hypothesis in powers of 2. The density of matter is generated by the decay of matter and energy of cosmic string like objects to ordinary matter as they thicken to flux tubes. If the string tension of local void is by a factor 1/2 smaller than for a typical void, the density of ordinary matter would be smaller by this factor. One could say that local void would be a forerunner in cosmic evolution. This is of cousre highly interesting from the persperctive of biological evolution, in particular Fermi paradox).

To sum up, this model would rely on the prediction that there are two kinds of flux tubes and that the cosmic evolution proceeds by phase transitions increasing p-adic length scale by half octave reducing the energy density by factor 1/2 at flux tubes. The local void would be one step further in cosmic evolution as compared to a typical void.

See the article The problem of two Hubble constants or the chapter More about TGD and Cosmology.

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

Thursday, June 18, 2020

How are the space-time surfaces assignable to the opposite boundaries of CD glued together?

How do the solutions assignable to the opposite boundaries of CD relate to each other?

Causal diamond identified basically as intersection of future and past directed light-cones is basic notion in zero energy ontology. It has 4-D variant cd4, 8-D Minkowski variant cd8, and H=M4× CP2-variant CD= cd4× CP2.

Space-time surfaces in M picture are defined as roots for "real" or "imaginary" part of complexified octonion valued polynomial obtained by algebraically continuing first real valued polynomial with rational coefficients to complex valued polynomial by replacing real argument with complex argument $z$ commuting with octonions and then perfoming continuation to complexified octonion valued polynomial. One can continue at first step to a polynomial of z or of z** and complex conjugation has interpretation as particle- antiparticle conjugation.

To construct the solutions to the polynomial equations one must consider the equations near both boundaries of CD and glue them together smoothly. I have not consider this problem earlier. In principle, the polynomials associated with them could be different in the general formulation discussed in but they could be also same (see this and this). How are the solutions associated with opposite boundaries of CD glued together in a continuous manner?

1. The polynomials assignable to the opposite boundaries of CD are allowed to be polynomials of o resp. (o-T): here T is the distance between the tips of CD.
2. CD brings in mind the realization of conformal invariance at sphere: the two hemispheres correspond to powers of z and 1/z: the condition z*= 1/z at unit circle is essential and there is no real conjugation. How the sphere is replaced with 8-D CD which is also complexified. The absence of conjugation looks natural also now: could CD contain a 3-surface analogous to the unit circle of sphere at which the analog of z*= 1/z holds true? If so, one has P(o,z)=P(1/o,z) and the solutions representing roots fo P(o,z) and P(1/o,z) can be glued together.

Note that 1/o can be expressed as o*/oo* when the Minkowskian norm squared oo* is non-vanishing and one has polynomial equation also now. This condition is true outside the boundary of 8-D light-cone, in particular near the upper boundary of CD.

The counter part for the length squared of octonion in Minkowskian signature is light-one proper time coordinate a2=t2-r2 for M8+. Replacing o which scaled dimensionless variable o1= o/(T/2) the gluing take place along a=T/2 hyperboloid.

One has algebraic holomorphy with respect to o but also anti-holomorphy with respect to o is possible. What could these two options correspond to? Could the space-time surfaces assignable to self and its time-reversal relate by octonionic conjugation o→o* relating two Fock vacuums annihilated by fermionic annihilation resp. creation operators?

See the article About M8-H-duality, p-adic length scale hypothesis and dark matter hierarchy or the chapter Zero energy ontology and matrices.

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

Indications for an axionlike state in mass range 1-7 keV from XENON

There was a popular article about bump claimed by XENON group and suggesting the existence of an axion-like state with mass in the range 1-7 keV. Also Jester discusses the evidence for the claimed bump.

Originally XENON searched evidence for WIMPs - weak interacting very massive particles. They would have made themselves visible via scattering from ZENON nuclei. Nothing was found.

Second candidate for dark matter particles are very light axions, which could be produced copiously in Sun. They would not have any detectable effect on heavy XENON atom but they could scatter from electrons and ionize XENON atom. The figure in the posting of Jester summarizes the energy spectrum of the observed ionization events. The figure shows approximately constant backgroud below 30 keV down to 1 keV below which it drops abruptly suggesting threshold. There are also indications for a peak around 1-2 keV. There is 3.5 sigma excess of events in the range 1-7 keV.

The mass of the dark particle candidate is in the range 1-7 keV. TGD allows to imagine several options but for all of them one would have analog of pion as dark matter candidate.

1. TGD Universe is fractal and this predicts p-adically scaled variants of hadron physics and electroweak physics. Mass squared scales would come as powers of 2. Mersenne primes and Gaussian Mersennes define especially promising candidates.
1. M89 hadron physics (see this) would be scaled up variant of ordinary hadron physics (M107) and would make itself visible at LHC. The masses of M89 hadrons would be scaled up by factor 512 from those of ordinary hadrons. There is evidence for bumps with predicted masses and the original proposal as Higgs did not work and they were forgotten. The mesons of this physics would be dark with heff/h0=n≈ 512 so that the Compton lengths would be those of ordinary mesons and they would appear at quantum criticality for what was expected to be de-confiment phase transition.
2. There are indications for the particles of these physics having mass scaled by a power of 2 from that for say ordinary meson. Could the particle be a scaled down pion of some kind. There are actually several candidates for scaled variants of pion. There is evidence for so called X boson with mass around 16-17 MeV proposed to be spin 1 bosonof a fifth force (see this). In TGD framework the identification as pion-like state is more natural and provides new insights on the relation between weak and strong interactions (see this). There is also quite recent evidence for pionlike exotic particle with mass not far from that of pion showing itself in the decays of long-lived kaon: there is actually evidence for scaled variants of pion also from earlier experiments. These pieces of evidence are discussed from TGD point of view in (see this) (see this).
3. In biologically important length scales there are as many as 4 Gaussian Mersenne MG,n=(1+i)n-1 with n=151, 157, 163, 167 defining p-adic length scales in the range 10 nm (cell membrane thickness) and 2.5 μm (cell nucleus size) and might involve scaled variants of hadron and electroweak physics.
p-Adic length scale hypothesis also allows the possibility of p-adically scaled variants of leptons and quarks with mass scaled down or up by a power of 2 and there are some indications for this kind of states. For instance, the claimed axion-like state could be a scaled down pion as bound state of scaled down quarks.
2. Heavy ion collisions near Coulomb wall gave already around seventies indicatons for a pion-like state of mass 1 MeV decaying to electron positron pair. TGD inspred interpretation (see this) was in terms of electropion identified as bound state of color octet electrons. TGD view about color indeed allows colored excitation of leptons since color is not spin-like but angular momentum like quantum number assignable to CP2 color partial waves. Later evidence for muon and tau analogs of this state has emerged. The decay widths of weak bosons do not allow color octet ptons in MeV scale and this forced the interpretation that they are dark in some sense and appear ony at quantum criticality - now at collision energies around Coulomb wall.

Leptopion could also be color bound state of quark and antiquark. As noticed, there is evidence for several bound states of this kind.

3. The TGD based model for "cold fusion" (see this, this), and this) led to a new view about nuclear physics (see this) in which dark nuclei appear also as intermediate states of ordinary nuclear reactions. Dark nuclei as nuclear string with distance of about electron Compton length would be crucial for "cold fusion" and would have dark nuclear binding energies in few keV range.

Remarkably, they would have scaled down dark nuclear binding energies in few keV range. This because the binding energy scale of ordinary nuclear physics about 7 MeV would be scaled down by the ratio 2-10≈ 10-3 of the p-adic length scales of proton and electron labelled by k=107 and k=127 to a value about 7 keV, which represents the upper end of the range 1-7 keV. There is also evidence that X ray emission with energies of this order of magnitude from Sun affects nuclear decay rates at Earth.

The pion-like particles could be indeed dark in TGD sense (ordinary particle but with heff=n× h0>h). Could the axion candidate be scaled down variant of electro-pion with mass 1 MeV with k=127: if the mass of electro-pion scales down like the nuclear binding energy, the scaling k=107 → 127 would take the mass of electro-pion to 1 keV. Also scaled down pion formed by quarks could be in question.

To sum up, the recent axion-like state would add only one item to the increasing list of pion-like states allowed by p-adic fractality of TGD Universe. One might hope that despite their high specialization colleagues could eventually wake-up to see the big picture behind the anomalies.

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

Monday, June 15, 2020

Skyrmions in TGD

I received a link to an article telling about research carried out for condensed matter skyrmions, which are very general condensed matter quasiparticles. They were found to replicate like DNA and cells. I realized that I have not clarified myself the possibility of skyrmions on TGD world and decided to clarify my thoughts.

What skyrmions are?

Consider first what skyrmions are.

1. Skyrmions are topological entities. One has some order parameter having values in some compact space S. This parameter is defined in say 3-ball such that the parameter is constant at the boundary meaning that one has effectively 3-sphere. If the 3rd homotopy group of S characterizing topology equivalence classes of maps from 3-sphere to S is non-trivial, you get soliton-llike entities, stable field configurations not deformable to trivial ones (constant value). Skyrmions can be assigned to space S which is coset space SU(2)L× SU(2)R/SU(2)V, essentially S3 and are labelled by conserved integer-valued topological quantum number.

One can imagine variants of this. For instance, one can replace 3-ball with disk. SO(3)=S3 with 2-sphere S2. The example considered in the article corresponds to discretized situation in which one has magnetic dipoles/spins at points of say discretized disk such that spins have same direction about boundary circle. The distribution of directions of spin can give rise to skyrmion-like entity. Second option is distribution of molecules which do not have symmetry axis so that as rigid bodies the space of their orientations is discretized version of SO(3). The field would be the orientation of a molecule of lattice and one has also now discrete analogs of skyrmions.

2. More generally, skyrmions emerge naturally in old-fashioned hadron physics, where SU(2)L× SU(2)R/SU(2)V involves left-handed, right-handed and vectorial (diagonal) subgroups of SO(4)=SU(2)L× SU(2)R. The realization would be in terms of 4-component field (π,σ), where π is charged pion with 3 components - axial vector - and σ which is scalar. The additional constraint π•π +σ2= constant defines 3-sphere so that one has field with values in S3. There are models assigning this kind of skyrmion with nucleon, atomic nuclei, and also in the bag model of hadrons bag can be thought of as a hole inside skyrmion. These models seem to have something to do with reality so that a natural question is whether skyrmions might appear in TGD.

Skyrmion number as winding number

In TGD framework one can regard space-time as 4-surface in either octonionic M8c - "c" refers here to complexification by an imaginary unit i commuting with octonions- or in M4× CP2. For the solution surfaces M8 has natural decomposition M8=M2× E6 and E6 has SO(6) as isometry group containing subgroup SU(3) having automorphisms of octonions as subgroup leaving M2 invariant. SO(6)=SU(4) contains SU(3) as subgroup, which has interpretation as isometries of CP2 and counterpart of color gauge group. This supports M8-H duality, whose most recent form is discussed here.

The map S3→ S3 defining skyrmion could be taken as a phenomenological consequence of M8-H duality implying the old-fashioned description of hadrons involving broken SO(4) symmetry (PCAC) and unbroken symmetry for diagonal group SO(3)V (CCV). The analog of (π,σ) field could correspond to a B-E condensate of pions (π,σ).

The obvious question is whether the map S3→ S3 defining skyrmion could have a deeper interpretation in TGD framework. I failed to find any elegant formulation. One could however generalize and ask whether skyrmion like entities characterize by winding number are predicted by basic TGD.

1. In the models of nucleon and nuclei the interpretation of conserved topological skyrmion number is as baryon number. This number should correspond to the homotopy class of the map in question, essentially winding number. For polynomials of complex number degree corresponds to winding number. Could the degree n=heff/h0 of polynomial P having interpretation as effective Planck constant and measure of complexity - kind of number theoretic IQ - be identifiable as skyrmion number? Could it be interpreted as baryon number too?
2. For leptons regarded as local 3 anti-quark composites in TGD based view about SUSY \citesusyTGD the same interpretation would make sense. It seems however that the winding number must have both signs. Degree is n is however non-negative.

Here complexification of M8 to M8c is essential. One an allow both holomorphic and anti-holomorphic continuations of real polynomials P (with rational coefficients) using complexification defined by commutative imaginary unit i in M8c so that one has polynomials P(z) resp. P(z*) in turn algebraically continued to complexified octonionic polynomials P(z,o) resp. P(z*,o).

Particles resp. antiparticles would correspond to the roots of octonionic polynomial P(z,o) resp. P(z*,o) meaning space-time geometrization of the particle-antiparticle dichotomy and would be conjugates of each other. This could give a nice physical interpretation to the somewhat mysterious complex roots of P.

More detailed formulation

To make this formulation more detailed on must ask how 4-D space-time surfaces correspond to 8-D "roots" for the "imaginary" ("real") part of complexified octonionic polynomial as surfaces in M8c.

1. Equations state the simultaneous vanishing of the 4 components of complexified quaternion valued polynomial having degree n and with coefficients depending on the components of Oc, which are regarded as complex numbers x+iy, where i commutes with octonionic units. The coefficients of polynomials depend on complex coordinates associated with non-vanishing "real" ("imaginary") part of the Oc valued polynomial.
2. To get perspective, one can compare the situation with that in catastrophe theory in which one considers roots for the gradient of potential function of behavior variables xi. Potential function is polynomial having control variables as parameters. Now behavior variable correspond "imaginary" ("real") part and control variables to "real" ("imaginary") of octonionic polynomial.

For a polynomial with real coefficients the solution divides to regions in which some roots are real and some roots are complex. In the case of cusp catastrophe one has cusp region with 3-D region of the parameter defined by behavior variable x and 2 control parameters with 3 real roots, the region in which one has one real root. The boundaries for the projection of 3-sheeted cusp to the plane defined by control variables correspond to degeneration of two complex roots to one real root.

In the recent case it is not clear whether one cannot require the M8c coordinates for space-time surface to be real but to be in M8=M1+iE7 .

3. Allowing complex roots gives 8-D space-time surfaces. How to obtain real 4-D space-time surfaces?
1. One could project space-time surfaces to real M8=M1+iE7 to obtain 4-D real space-time surfaces. In time direction the real part of root is accepted and is same for the root and its conjugate. For E7 this would mean that imaginary part is picked up.
2. If one allows only real roots, the complex conjugation proposed to relate fermions and anti-fermions would be lost.
4. One can select for 4 complex M8c coordinates Xk of the surface and the remaining 4 coordinates Yk can be formally solved as roots of n:th degree polynomial with dynamical coefficients depending on Xk and the remaining Yk. This is expected to give rise to preferred extremals with varying dimension of M4 and CP2 projections.
5. It seems that all roots must be complex.
1. The holomorphy of the polynomials with respect to the complex M8c coordinates implies that the coefficients are complex in the generic point M8c. If so, all 4 roots are in general complex but do not appear as conjugate pairs. The naive guess is that the maximal number of solutions would be n4 for a given choice of M8 coordinates solved as roots. An open question is whether one can select subset of roots and what happens at t=rn surfaces: could different solutions be glued together at them.
2. Just for completeness one can consider also the case that the dynamical coefficients are real - this is true in the E8 sector and whether it has physical meaning is not clear. In this case the roots come as real roots and pairs formed by complex root and its conjugate. The solution surface can be divided into regions depending on the character of 4 roots. The n roots consist of complex root pairs and real roots. The members or complex root pairs are mapped to same point in E8.
Could skyrmions in TGD sense replicate?

What about the observation that condensed matter skyrmions replicate? Could this have analog at fundamental level?

1. The assignment of conserved topological quantum number to the skyrmion is not consistent with replication unless the skyrmion numbers of outgoing states sum up to that of the initial state. If the system is open one can circumvent this objection. The replication would be like replication of DNA in which nucleotides of new DNA strands are brought to the system to form new strands.
2. It would be fascinating if all skyrmions would correspond to space-time surfaces at fundamental M8 level. If so, skyrmion property also in magnetic sense would be induced by from a deeper geometric skyrmion property of the MB of the system. The openness of the system would be essential to guarantee conservation of baryon number. Here the fact that leptons and baryons have opposite baryon numbers helps in TGD framework. Note also ordinary DNA replication could correspond to replication of MB and thus of skyrmion sequences.
See the article About M8-H-duality, p-adic length scale hypothesis and dark matter hierarchy or the chapter Zero energy ontology and matrices.

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

Friday, June 12, 2020

Basic organic molecules appear much before they should

Nikolina Benedikovic gave a link to an interesting article about the appearance of pre-biotic molecules in stellar nurseries at time when star formation had not yet begun. The article told about the work of astrophysicists Yancy Shirley and Samantha Scibelli published in Astrophysical Journal with the title "Prevalence of Complex Organic Molecules in Starless and Prestellar Cores within the Taurus Molecular Cloud".

The stellar nursery studied consisted of 31 starless cores scattered throughout a star-forming region in Taurus molecular cloud located about 440 ly from Earth. The molecules studied were methanol (CH3OH) and acetaldehyde (CH3CHO). These molecules serving as building bricks for chemical life were found to be much more ubiquitous than expected and present hundreds of thousands of years before star formation actually began.

The molecular evolution producing complex organic molecules requires the analog of metabolic energy feed: the temperature should be quite high - measured using molecular binding energies as scale - about few eV typically. The existing theories assume that proto-stars - stars in the process of formation produced the heating necessary for the formation of these molecules. But in regions without proto-stars the temperature of gas has been quite to provide the heating. Where did the energy needed for local heating come from?

TGD based view about "cold fusion" - which during last years has been getting rid of the label of pseudoscience - is that it is induced by what I call dark nuclear fusion possible at low temperatures (see this). Dark matter corresponds in TGD Universe to ordinary particles but with non-standard value of Planck constant (or cautiously effective Planck constant) heff=nh0 larger than h.

Dark nuclei formed as dark proton sequences at magnetic flux tubes with protons having distance about electron Compton length and scaled down nuclear binding energy in eV scale would have formed at temperatures of order eV or even lower. They would have spontaneously decayed to ordinary nuclei liberating energy of order nuclear binding energy, which is in MeV scale. The dark nuclei can actually occur in several scales but could transform sequentially to ordinary nuclei. The liberated nuclear binding energy would have heated the gas locally. This kind of regions would have served as pre-stellar objects leading to protostars and eventually stellar cores as ordinary nuclear reactions would have started.

TGD predicts that also ordinary nuclear reactions are initiated by phase transitions generating dark nuclei as intermediate states: this would be the counterpart for quantum tunnelling assumed to take place in ordinary nuclear reactions and allow them to already occur at collision energies about 1/100 lower than classical considerations would allow (see for instance this and this).

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.

Thursday, June 11, 2020

TGD interpretation of new experimental results about the mechanism of anesthesia

I received a link to a highly interesting popular article with title Century-Old Scientific Debate Settled: Anesthesia’s Effect on Consciousness Solved). The article tells about a study from Scripps Research published in the Proceedings of the National Academies of Sciences (PNAS) . The paper Studies on the mechanism of general anesthesia has appeared in PNAS. In addition to Lerner and Hansen, the authors are Mahmud Arif Pavel, E. Nicholas Petersen and Hao Wang, all of Scripps Research.

I have pondered possible mechanism of anesthesia in TGD framework several times earlier (see this and this) and it is interesting to see whether the findings allow to make earlier insights more detailed or even develop new ones.

What was observed

According to the popular article the discovery by chemist Richard Lerner, MD, and molecular biologist Scott Hansen, PhD, settles a century-old scientific debate about whether anesthetics act directly on cell-membrane gates called ion channels, or do they somehow act on the membrane to signal cell changes in a new and unexpected way. The conclusion of the researcheres is that anesthetic action is a two-step process that begins in the membrane. The anesthetics perturb ordered lipid clusters within the cell membrane known as "lipid rafts" to initiate the signal. There are two kinds of clusters involved and known with names GM1 and PIP2.

What was observed was following.

• A shift in the GM1 cluster’s organization, a shift from a tightly packed ball to a disrupted mess occurred first As GM1 grew disordered, it spilled its contents, among them, an enzyme called phospholipase D2 (PLD2). Melting is a good analog for what happens. Gel-to-sol transition in cytoplasm is second analogy.
• PLD2 moved like a billiard ball away from its GM1 home and over to a different, less-preferred lipid cluster called PIP2.
• This activates key molecules within PIP2 clusters, TREK1 potassium ion channels and their lipid activator, phosphatidic acid (PA) are among them. The activation of TREK1 potassium channels releases potassium hyper-polarizing the nerve and it makes it more difficult to fire. Nerve pulse generation rate becomes low and leads to a loss of consciousness - at least in clinical sense. Something analogous to this could happen when one falls in sleep.
In the article I try to understand in the framework provided by TGD inspired model of cell membrane and nerve pulse (see this), compare these findings to TGD inspired views about anesthesia based on hyperpolarization, and also try to build a bridge from TGD description provided by a generalization of thermodynamics forced by zero energy ontology (ZEO) predicting that in ordinary state function reduction the arrow of time changes (see this and this).

See the article TGD interpretation of new experimental results about the mechanism of anesthesia or the chapter TGD Inspired Model for Nerve Pulse.

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

Monday, June 08, 2020

How the life cycle of self could corresponds to a transition to chaos as iteration of polynomial?

I have discussed how the evolution of self by BSFRs could correspond to a transition to chaos as iteration of the polynomial defining the space-time surface. The proposed picture was that the evolution by SSFRs corresponds to iteration of a polynomial P assignable to the active boundary of CD. This would predict a continual increase of the degree of the polynomial involved. This is however only one possibility to interpret the evolution of self as iteration leading to chaos.
1. One could argue that the polynomial Pnk= Pn∘....∘ Pn associated with the active boundary remains the same during SSFRs as long as possible. This because the increase of degree from nk to n(k+1) in Pnk→ Pnk∘ Pn increases heff by factor (k+1)/k so that the metabolic feed needed to preserve the value of heff increases.

Rather, when all roots of the polynomials P assignable to the active boundary of CD are revealed in the gradual increase of CD preserving Pnk, the transition Pnk→ Pnk∘ Pn could occur provided the metabolic resources allow this. Otherwise BSFR occurs and self dies and re-incarnates. The idea that BSFR occurs when metabolic resources are not available is very natural for this option.

2. Could Pnk→ Pnk∘ Pn occur only in BSFRs so that the degree n of P would be preserved during single life cycle of self - that n can increase only in BSFRs was indeed the original guess.
See the articles Could quantum randomness have something to do with classical chaos? and When does "big" state function reduction as universal death and re-incarnation with reversed arrow of time take place?.

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

Sunday, June 07, 2020

Conscious problem solving and quantum counterpart of computationalism in TGD

This posting was inspired by discussion with Bruno Marchal about his article with title "Do the laws of physics apply to the mind?".

I do not go to the discussion itself here. Bruno Marchal is a representative of computationalism, which might be called idealistic and Bruno believes that physics follows from computationalism. The somewhat mystical and certainly inspiring notion of self-reference is believed to lead to consciousness.

I do not share this view. I do not share this view. The gist of the posting comes towards end where I describe how computationalism generalizes to quantum computationalism in TGD generalizing also the notion of quantum computation. What conscious problem solving is? This is the question to be discussed.

Start from a problem

The basic problem in consciousness theories is that people do not have ability to leave this division to idealistic, materialistic and dualistic camps. Each of these approaches fails but people put themelves into one of these big and safe boxes. One should be able to see the biggest picture but this is surprisingly difficult.

To get out of the box, one should start from the problems rather than text book wisdom.

1. Idealism and materialism are only mirror images of each other and their problems mirror each other: for materialism mind as illusion and for idealism matter is illusion. Computationalism must select between these options two.
2. If dualist wants to cope with what we know, he ends up with materialism or idealism. In dualism the identification of mind and matter as separate substances is in conflict with the fact that mind is about something, it is does just exist as matter. Substance aspect of mind has however analog in TGD: p-adic space-time sheets serve as correlates for cognition, as thought bubbles but are not conscious thoughts.
To make progress, one must have some ideas about what conscious experience rather than deciding what it is in terms of something already known. One must identify the differences between conscious existence and physical existence in the classical sense.

Some of them are the "aboutness" property, the division of the world to me and external world in conscious experience, and experience (at least) of free will. How to realize free will without regarding laws of physics as illusion created by mind or denying it? Here the physical mystery of state function reduction comes in rescue. From this one must begin. When one does this one eventually ends up with what I call zero energy ontology, ZEO.

TGD view about life and consciousness

First my view about the relationship of physics and conscious experience.

1. In TGD Universe minds do not reduce to the properties of physical system. Nor is the reduction of physics to dynamics of mind possible. As a matter of fact, there are no minds as logical or any other kind of entities - entities are not about something as conscious experience is: the belief to mind as entity is the failure leading to the problems with free will and extremes the denial of mind or matter except as illusion. Mountains of literature has been written in order to put this problem under the rug. But in vain: black remains black and does not transform to white.

Denialism is never a solution of a problem, it is essentially an attempt to get rid of cognitive dissonance. It is better to start from a problem or rather - paradox in the recent case. Nor can physics in its recent form explain conscious mind.

State function reduction (SFR) is the black sheep of quantum theory and by its non-determinism rather obvious candidate for moment of consciousness and act of free will.

2. Conscious experience could therefore be in quantum jump between quantum states - SFR. The occurrence of SFR is a physical fact and with proper generalization of ontology to what I call zero energy ontology (ZEO) one can solve the problem of free will and basic paradox of quantum measurement theory.

The realization of this simple modification of ontology to ZEO - conscious existence is in change as physiologists realized long time ago - seems to be extremely difficult for both physicists and philosophers. They cannot overcome the boundaries posed by the dogmatism taught to them - be it materialism, idealism or dualism. Again and agan I find that the proposals proposing a new brave theory of conciousness remain in one of these three boxes.

3. TGD based physics involves several dynamics. The deterministic of classical dynamics as field equations for space-time surfaces is exact part of quantum theory in ZEO. The deterministic dynamics of second quantized induced spinor fields at space-time surfaces. Quantum states in ZEO are just superpositions of these deterministic time evolutions. And also that of spinor fields of "world of classical worlds" (WCW) describing the quantum states of the Universe.

There is also the dynamics of conscious experience. Sequence of unitary time evolutions and "small" SFRs following each of them defines sensory experience and all that accompanies it. There are also "big" (ordinary) SFRs (BSFRs) meaning the death of conscious entity and its re-incarnation with opposite arrow of time: this in universal sense - not only biologically. ZEO gives also rise to self-organization forced by generalization of second law assuming that the hierarchy of effective Planck constants predicted by number theoretical TGD is accepted.

4. Boolean logic, which is often raised by comptationalistic to a fundamental role, can be understood in terms of fermions: this is a further new element provided by TGD and provides interpretation for anti-commutation relations of fermions having also purely geometric interpretation at the level WCW in terms of the spinor structure of WCW. The truth preserving dynamics of logic is determined by modified Dirac equation at space-time level and means infinite number of conservations laws representing physics laws. ZEO is essential again: state pairs as zero energy states correspond to initial and final state connected by the dynamics described.
5. In TGD framework correlates of cognition become part of what physics described using p-adic number fields and adelic physics. Number theoretical universality is the basic principle and formally p-adic physics obeys same field equations as real number based physics correlates for for sensory perception. Cognition is universal and present already for elementary particles and has deep implications for physics itself: the success of p-adic mass calculations is one example of this.

p-Adic space-time sheets as correlates of cognition are "thought bubbles" - the mind stuff of Descartes but not conscious as such.

The intersection of sensory and cognitive - cognitive representation - consists of the points of space-time surface with imbedding space coordinates in the extension of rationals defining the adele in question. This hierarchy corresponds to evolutionary hierarchy with increasing algebraic complexity characterized partially by the dimension n of extension having interpretation in terms of effective Planck constant and labelling ordinary phases of matter behaving like dark matter.

Cognitive representations are unique and in the generic case finite and basic stuff in number theory, which becomes part of quantum physics in TGD.

6. At deeper level p-adic physics and hierarchy of effective Planck constants heff= nh0 labelling phases of ordinary matter behaving like dark matter follow both from number theoretic vision and servng as macroscopically quantum coherent masters controlling ordinary matter and explaining its coherence as induced coherence. n corresponds to the degree of polynomial determining space-time region as algebraic surface in octonionic M8 and mapped to H=M4× CP2 by M8-H duality. Dark matter in this sense is absolutely essential for understanding of living matter in TGD framework.

ZEO gives also the dynamics self-organization in ZEO implied by dissipation in reverse arrow of time solely so that nothing new is needed besides generalized thermodynamics. It also explains the necessity of energy feed: it is required to increase the value of heff (meaning increase of algebraic complexity as "IQ") and is implied by dissipation with reversed arrow of time. The laws of self-organization essentially analogs of traffic rules based on useful conventions obeyed only in statistical sese. This dynamics Wolfram fatally confuses with the fundamental dynamics.

See this .

The relationship of TGD view about consciousness to computationalism

Computationalism is one of the failed approaches to consciousness - it cannot cope with free will for instance. It however contains an essential aspect which is correct: the idea of deterministic program leading from A to B.

Problem solving can regarded as attempt to find this program. You fix A as initial data and try to find a program leading from A to a final state characterized by data B. The program has duration T and can be very long and it is not clear whether it exists at all. You try again and again and eventually you might find it. In the real conscious problem solving this process means making guesses so that the process cannot be deterministic.

What does this view about problem solving correspond to in ZEO? We have states A and B represented as quantum states and we try to find quantum analog of classical program leading from A to B in some time T which can be varied.

1. A and B are realized as superpositions of 3-surfaces and fermionic states at them - located at time values t=0 and t=T. T can vary. Can we find by varying T a (superposition of) deterministic time evolution(s) - preferred extremal(s) (PE) - connecting A and B?

In ZEO and for fixed A and T PE in general does not exist. In ideal situation (infinite measurement resolution) and for given A and T, B is unique if it exists at all. One has analog of Bohr orbit and the quantum analog of classical program as the superposition of Bohr orbits starting from A and hopefully leading to B as a solution of the problem.

Remark: These superpositions can be regarded as counterparts of functions in biology and behaviors in neuroscience. The big difference to standard physics is that time=constant snapshot in time evolution of say bio-system is replaced with quantum superposition of very special time evolutions - PEs. Darwinian selection of also behaviors in biology correlates strongly with this.

2. So: given A and B, we try to find a value of T for which superposition of PEs from A to B exists. This would be the quantum program leading from A to B, and solving our problem.

Actually, not only ours, universe is full of conscious entities solving problems at various levels of self hierarchy. This takes place by a sequences of "small" SFRs (SSFRs, weak measurements) increasing T in statistical sense and replacing the state at B with a new one determined by state A for given value of T. At the level of conscious experience this is sensory perception and all that which is associated with it.

Finding the solution is analogous to the halting of quantum Turing machine by ordinary state function reduction, which corresponds in ZEO to a "big" (ordinary) SFR (BSFR). This would mean death in universal sense and reincarnation with reversed arrow of time in ZEO? Or is BSFR and death failure to solve the problem? I cannot answer. Remark: The notion of self-reference is replaced with much more concrete notion of becoming conscious of what one was conscious of before SSFR. SSFR indeed gives rise to conscious eperience and one avoids the infinite regress associated with genuine self-reference. As an additional bonus one obtains evolution since the extension of rationals characterizing space-time surfaces can increase meaning higher level of consciousness. At the limit algebraic numbers the cognitive representation is a dense subset of space-time surface.

3. Also finite measurement resolution and discreteness characterizing computation emerge from number theory. To be a solution classically means that the 3-surface(s) representing B to have fixed discrete cognitive representation given by finite number of imbedding space points in the extension of rationals defining the adele. Quantally, quantum superpositions of these points with fixed quantum numbers represent the desired final state. Also Boolean logic emerges at fundamental level as square root of Kähler geometry one might say. Many-fermion state basis defines a Boolean algebra and time evolution for induced spinors is analogous to truth preserving Boolean map in which truths code for infinite number of conservation laws associated with symmetries of WCW.
4. How to find the possibly existing solution at given step (unitary evolution plus SSFR) with t=T? One performs cognitive quantum measurements at each step represented by SSFR. They reduce to cascades of quantum measurements for the states in the group algebra of Galois group - call it Gal - of Galois extension considered.

Gal has hierarchical decomposition to inclusion hierarchy of normal subgroups implying the representation of states in group algebra of Gal as entangled states in the tensor product of the group algebras of normal sub-groups of Gal. The hope is that this Galois cascade of SFRs produces desired state as an outcome and one can shout "Eureka!".

See the article The dynamics of SSFRs as quantum measurement cascades in the group algebra of Galois group or the chapter Zero Energy Ontology and Matrices .

Monday, June 01, 2020

New resuls on M8-H duality

M8-H duality (H=M4× CP2) has taken a central role in TGD framework. M8-H duality allows to identify space-time regions as "roots" of octonionic polynomials P in complexified M8 - M8c - or as minimal surfaces in H=M4× CP2 having 2-D singularities.

Remark:Oc,Hc,Cc,Rc will be used in the sequel for complexifications of octonions, quaternions, etc.. number fields using commuting imaginary unit i appearing naturally via the roots of real polynomials.

Space-time as algebraic surface in M8c regarded complexified octonions

The octonionic polynomial giving rise to space-time surface as its "root" is obtained from ordinary real polynomial P with rational coefficients by algebraic continuation. The conjecture is that the identification in terms of roots of polynomials of even real analytic functions guarantees associativity and one can formulate this as rather convincing argument. Space-time surface X4c is identified as a 4-D root for a Hc-valued "imaginary" or "real" part of Oc valued polynomial obtained as an Oc continuation of a real polynomial P with rational coefficients, which can be chosen to be integers. These options correspond to complexified-quaternionic tangent- or normal spaces. For P(x)= xn+.. ordinary roots are algebraic integers. The real 4-D space-time surface is projection of this surface from M8c to M8. One could drop the subscripts "c" but in the sequel they will be kept.

M4c appears as a special solution for any polynomial P. M4c seems to be like a universal reference solution with which to compare other solutions.

One obtains also brane-like 6-surfaces as 6-spheres as universal solutions. They have M4 projection, which is a piece of hyper-surface for which Minkowski time as time coordinate of CD corresponds to a root t=rn of P. For monic polynomials these time values are algebraic integers and Galois group permutes them.

One cannot exclude rational functions or even real analytic functions in the sense that Taylor coefficients are octonionically real (proportional to octonionic real unit). Number theoretical vision - adelic physics suggests that polynomial coefficients are rational or perhaps in extensions of rationals. The real coefficients could in principle be replaced with complex numbers a+ib, where i commutes with the octonionic units and defines complexifiation of octonions. i appears also in the roots defining complex extensions of rationals.

Brane-like solutions

One obtains also 6-D brane-like solutions to the equations.

1. In general the zero loci for imaginary or real part are 4-D but the 7-D light-cone δ M8+ of M8 with tip at the origin of coordinates is an exception. At δ M8+ the octonionic coordinate o is light-like and one can write o= re, where 8-D time coordinate and radial coordinate are related by t=r and one has e=(1+er)/\sqrt2 such that one as e2=e.

Polynomial P(o) can be written at δ M8+ as P(o)=P(r)e and its roots correspond to 6-spheres S6 represented as surfaces tM=t= rN, rM= \sqrtrN2-rE2≤ rN, rE≤ rN, where the value of Minkowski time t=r=rN is a root of P(r) and rM denotes radial Minkowski coordinate. The points with distance rM from origin of t=rN ball of M4 has as fiber 3-sphere with radius r =\sqrtrN2-rE2. At the boundary of S3 contracts to a point.

2. These 6-spheres are analogous to 6-D branes in that the 4-D solutions would intersect them in the generic case along 2-D surfaces X2. The boundaries rM=rN of balls belong to the boundary of M4 light-cone. In this case the intersection would be that of 4-D and 3-D surface, and empty in the generic case (it is however quite not clear whether topological notion of "genericity" applies to octonionic polynomials with very special symmetry properties).
3. The 6-spheres tM=rN would be very special. At these 6-spheres the 4-D space-time surfaces X4 as usual roots of P(o) could meet. Brane picture suggests that the 4-D solutions connect the 6-D branes with different values of rn.

The basic assumption has been that particle vertices are 2-D partonic 2-surfaces and light-like 3-D surfaces - partonic orbits identified as boundaries between Minkowskian and Euclidian regions of space-time surface in the induced metric (at least at H level) - meet along their 2-D ends X2 at these partonic 2-surfaces. This would generalize the vertices of ordinary Feynman diagrams. Obviously this would make the definition of the generalized vertices mathematically elegant and simple.

Note that this does not require that space-time surfaces X4 meet along 3-D surfaces at S6. The interpretation of the times tn as moments of phase transition like phenomena is suggestive. ZEO based theory of consciousness suggests interpretation as moments for state function reductions analogous to weak measurements ad giving rise to the flow of experienced time.

4. One could perhaps interpret the free selection of 2-D partonic surfaces at the 6-D roots as initial data fixing the 4-D roots of polynomials. This would give precise content to strong form of holography (SH), which is one of the central ideas of TGD and strengthens the 3-D holography coded by ZEO alone in the sense that pairs of 3-surfaces at boundaries of CD define unique preferred extremals. The reduction to 2-D holography would be due to preferred extremal property realizing the huge symplectic symmetries and making M8-H duality possible as also classical twistor lift.

I have also considered the possibility that 2-D string world sheets in M8 could correspond to intersections X4∩ S6? This is not possible since time coordinate tM constant at the roots and varies at string world sheets.

Note that the compexification of M8 (or equivalently octonionic E8) allows to consider also different variants for the signature of the 6-D roots and hyperbolic spaces would appear for (ε1, εi,..,ε8), epsiloni=+/- 1 signatures. Their physical interpretation - if any - remains open at this moment.

5. The universal 6-D brane-like solutions S6c have also lower-D counterparts. The condition determining X2 states that the Cc-valued "real" or "imaginary" for the non-vanishing Qc-valued "real" or "imaginary" for P vanishes. This condition allows universal brane-like solution as a restriction of Oc to M4c (that is CDc) and corresponds to the complexified time=constant hyperplanes defined by the roots t=rn of P defining "special moments in the life of self" assignable to CD. The condition for reality in Rc sense in turn gives roots of t=rn a hyper-surfaces in M2c.
Explicit realization of M8-H duality

M8-H duality allows to map space-time surfaces in M8 to H so that one has two equivalent descriptions for the space-time surfaces as algebraic surfaces in M8 and as minimal surfaces with 2-D singularities in H satisfying an infinite number of additional conditions stating vanishing of Noether charges for super-symplectic algebra actings as isometries for the "world of classical worlds" (WCW). Twistor lift allows variants of this duality. M8H duality predicts that space-time surfaces form a hierarchy induced by the hierarchy of extensions of rationals defining an evolutionary hierarchy. This forms the basis for the number theoretical vision about TGD.

M8-H duality makes sense under 2 additional assumptions to be considered in the following more explicitly than in earlier discussions.

1. Associativity condition for tangent-/normal space is the first essential condition for the existence of M8-H duality and means that tangent - or normal space is quaternionic.
2. Also second condition must be satisfied. The tangent space of space-time surface and thus space-time surface itself must contain a preferred M2c⊂ M4c or more generally, an integrable distribution of tangent spaces M2c(x) and similar distribution of their complements E2c(x). The string world sheet like entity defined by this distribution is 2-D surface X2c⊂ X4c in Rc sense. E2c(x) would correspond to partonic 2-surface.

One can imagine two realizations for this condition.

Option I: Global option states that the distributions M2c(x) and E2c(x) define slicing of X4c.

Option II: Only discrete set of 2-surfaces satisfying the conditions exist, they are mapped to H, and strong form of holography (SH) applied in H allows to deduce space-time surfaces in H. This would be the minimal option.

How these conditions would be realized?

1. The basic observation is that X2c can be fixed by posing to the non-vanishing Hc-valued part of octonionic polynomial P condition that the Cc valued "real" or "imaginary" part in Cc sense for P vanishes. M2c would be the simplest solution but also more general complex sub-manifolds X2c⊂ M4c are possible. This condition allows only a discrete set of 2-surfaces as its solutions so that it works only for Option II.

These surfaces would be like the families of curves in complex plane defined by u=0 an v= 0 curves of analytic function f(z)= u+iv. One should have family of polynomials differing by a constant term, which should be real so that v=0 surfaces would form a discrete set.

2. One can generalize this condition so that it selects 1-D surface in X2c. By assuming that Rc-valued "real" or "imaginary" part of quaternionic part of P at this 2-surface vanishes. one obtains preferred M1c or E1c containing octonionic real and preferred imaginary unit or distribution of the imaginary unit having interpretation as complexified string. Together these kind 1-D surfaces in Rc sense would define local quantization axis of energy and spin. The outcome would be a realization of the hierarchy Rc→ Cc→ Hc→ Oc realized as surfaces.

This option could be made possible by SH. SH states that preferred extremals are determined by data at 2-D surfaces of X4. Even if the conditions defining X2c have only a discrete set of solutions, SH at the level of H could allow to deduce the preferred extremals from the data provided by the images of these 2-surfaces under M8-H duality. Associativity and existence of M2(x) would be required only at the 2-D surfaces.

3. I have proposed that physical string world sheets and partonic 2-surfaces appear as singularities and correspond to 2-D folds of space-time surfaces at which the dimension of the quaternionic tangent space degenerates from 4 to 2. This interpretation is consistent with a book like structure with 2-pages. Also 1-D real and imaginary manifolds could be interpreted as folds or equivalently books with 2 pages.

For the singular surfaces the dimension quaternionic tangent or normal space would reduce from 4 to 2 and it is not possible to assign CP2 point to the tangent space. This does not of course preclude the singular surfaces and they could be analogous to poles of analytic function. Light-like orbits of partonic 2-surfaces would in turn correspond to cuts.

Does M8-H duality relate hadron physics at high and low energies?

During the writing of this article I realized that M8-H duality has very nice interpretation in terms of symmetries. For H=M4× CP2 the isometries correspond to Poincare symmetries and color SU(3) plus electroweak symmetries as holonomies of CP2. For octonionic M8 the subgroup SU(3) ⊂ G2 is the sub-group of octonionic automorphisms leaving fixed octonionic imaginary unit invariant - this is essential for M8-H duality. SU(3) is also subgroup of SO(6)== SU(4) acting as rotation on M8= M2× E6. The subgroup of the holonomy group of SO(4) for E4 factor of M8= M4× E4 is SU(2)× U(1) and corresponds to electroweak symmetries. One can say that at the level of M8 one has symmetry breaking from SO(6) to SU(3) and from SO(4)= SU(2)× SO(3) to U(2).

This interpretation gives a justification for the earlier proposal that the descriptions provided by the old-fashioned low energy hadron physics assuming SU(2)L× SU(2)R and acting acting as covering group for isometries SO(4) of E4 and by high energy hadron physics relying on color group SU(3) are dual to each other.

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