TGD as it is towards end of 2024: part II

This article is the second part of the article trying to give a rough overall view about Topological Geometrodynamics (TGD) as it is towards the end of 2024. Various views about TGD and their relationship are discussed at the general level. In the first part of the article the geometric and number theoretic visions of TGD were discussed.

In the first part of the article the two visions of TGD: physics as geometry and physics as number theory were discussed. The second part is devoted to the details of M⁸-H duality relating these two visions, to zero energy ontology (ZEO), and to a general view about scattering amplitudes.

Classical physics is coded either by the space-time surfaces of H or by 4-surfaces of M⁸ with Euclidean signature having associative normal space, which is metrically M⁴. M⁸-H duality as the analog of momentum-position duality relates geometric and number theoretic views. The pre-image of causal diamond cd, identified as the intersection of oppositely directed light-cones, at the level of M⁸ is a pair of half-light-cones. M⁸-H duality maps the points of cognitive representations as momenta of fermions with fixed mass m in M⁸ to hyperboloids of CD\subset H with light-cone proper time a= h_eff/m.

Holography can be realized in terms of 3-D data in both cases. In H the holographic dynamics is determined by generalized holomorphy leading to an explicit general expression for the preferred extremals, which are analogs of Bohr orbits for particles interpreted as 3-surfaces. At the level of M⁸ the dynamics is determined by associativity of the normal space.

Zero energy ontology (ZEO) emerges from the holography and means that instead of 3-surfaces as counterparts of particles their 4-D Bohr orbits, which are not completely deterministic, are the basic dynamical entities. Quantum states would be superpositions of these and this leads to a solution of the basic problem of the quantum measurement theory. It also leads also to a generalization of quantum measurement theory predicting that in the TGD counterpart of the ordinary state function reduction, the arrow of time changes.

A rather detailed connection with the number theoretic vision predicting a hierarchy of Planck constants labelling phases of the ordinary matter behaving like dark matter and ramified primes associated with polynomials determining space-time regions as labels of p-adic length scales. There has been progress also in the understanding of the scattering amplitudes and it is now possible to identify particle creation vertices as singularities of minimal surfaces associated with the partonic orbits and fermion lines at them. Also a connection with exotic smooth structures identifiable as the standard smooth structure with defects identified as vertices emerges.

See the article TGD as it is towards end of 2024: part II or the chapter with the same title.

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

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

TGD as it is towards end of 2024: part I

This article is the first part of the article, which tries to give a rough overall view about Topological Geometrodynamics (TGD) as it is towards the end of 2024. Various views about TGD and their relationship are discussed at the general level.

1. The first view generalizes Einstein's program for the geometrization of physics. Space-time surfaces are 4-surfaces in H=M⁴× CP₂ and general coordinate invariance leads to their identification as preferred extremals of an action principle satisfying holography. This implies zero energy ontology (ZEO) allowing to solve the basic paradox of quantum measurement theory.
2. Holography = holomorphy principle makes it possible to construct the general solution of field equations in terms of generalized analytic functions. This leads to two different views of the construction of space-time surfaces in H, which seem to be mutually consistent.
3. The entire quantum physics is geometrized in terms of the notion of "world of classical worlds" (WCW), which by its infinite dimension has a unique K\"ahler geometry. Holography = holomorphy vision leads to an explicit general solution of field equations in terms of generalized holomorphy and has induced a dramatic progress in the understanding of TGD.

Second vision reduces physics to number theory.

1. Classical number fields (reals, complex numbers, quaternions, and octonions) are central as also p-adic number fields and extensions of rationals. Octonions with number theoretic norm RE(o²) is metrically Minkowski space, having an interpretation as an analog of momentum space M⁸ for particles identified as 3-surfaces of H, serving as the arena of number theoretical physics.
2. Classical physics is coded either by the space-time surfaces of H or by 4-surfaces of M⁸ with Euclidean signature having associative normal space, which is metrically M⁴. M⁸-H duality as analog of momentum-position duality relates these views. The pre-image of CD at the level of M⁸ is a pair of half-light-cones. M⁸-H duality maps the points of cognitive representations as momenta of fermions with fixed mass m in M⁸ to hyperboloids of CD\subset H with light-cone proper time a= h_eff/m.

   Holography can be realized in terms of 3-D data in both cases. In H the holographic dynamics is determined by generalized holomorphy leading to an explicit general expression for the preferred extremals, which are analogs of Bohr orbits for particles interpreted as 3-surfaces. At the level of M⁸ the dynamics is determined by associativity. The 4-D analog of holomorphy implies a deep analogy with analytic functions of complex variables for which holography means that analytic function can be constructed using the data associated with its poles and cuts. Cuts are replaced by fermion lines defining the boundaries of string world sheets as counterparts of cuts.

3. Number theoretical physics means also p-adicization and adelization. This is possible in the number theoretical discretization of both the space-time surface and WCW implying an evolutionary hierarchy in which effective Planck constant identifiable in terms of the dimension of algebraic extension of the base field appearing in the coefficients of polynomials is central.

This summary was motivated by a progress in several aspects of TGD.

1. The notion of causal diamond (CD), central to zero energy ontology (ZEO), emerges as a prediction at the level of H. The moduli space of CDs has emerged as a new notion.
2. Galois confinement at the level of M⁸ is understood at the level of momentum space and is found to be necessary. Galois confinement implies that fermion momenta in suitable units are algebraic integers but integers for Galois singlets just as in the ordinary quantization for a particle in a box replaced by CD. Galois confinement could provide a universal mechanism for the formation of all bound states.
3. There has been progress in the understanding of the quantum measurement theory based on ZEO. From the point of view of cognition BSFRs would be like heureka moments and the sequence of SSFRs could correspond to an analysis, possibly having the decay of 3-surface to smaller 3-surfaces as a correlate.

In the first part of the article the two visions of TGD: physics as geometry and physics as number theory are discussed. The second part is devoted to M⁸-H duality relating these two visions, to zero energy ontology (ZEO), and to a general view about scattering amplitudes.

See the article TGD as it is towards end of 2024: part I or the chapter with the same title.

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

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

New support for the TGD based explanation for the origin of Moon

I have learned that the Moon is a rather mysterious object. The origin of the Moon is a mystery although the fact that its composition is the same as that of Earth gives hints; Moon is receding from us (cosmic recession velocity is 78 per cent of this velocity, which suggests that surplus recession velocity is due to the explosion) (see this) it seems that the Moon has effectively turned inside out; the faces of the Moon are very different; the latest mystery that I learned of, are the magnetic anomalies of the Moon. The TGD based view of the origin of the Moon combined with the TGD view of magnetic fields generalizing the Maxwellian view explains all these mysterious looking findings.

The magnetic field of the Moon (see the Wikipedia article) is mysterious. There are two ScienceAlert articles about the topic (see this and this). There is an article by Krawzynksi et al with the title "Possibility of Lunar Crustal Magnatism Producing Strong Crustal Magnetism" to be referred as Ketal (see this). The article by Hemingway and Tikoo with the title "Lunar Swirl Morphology Constrains the Geometry, Magnetization, and Origins of Lunar Magnetic Anomalies" to be referred as HT (see this) considers a model for the origin local magnetic anomalies of the Moon manifesting themselves as lunar swirls.

1. The magnetic anomalies of the Moon

1. The Moon has no global magnetic field but there are local rather strong magnetic fields. What puts bells ringing is that their ancient strengths according to HT are of the same order of magnitude as the strength of the Earth's magnetic field with a nominal value of B_E≈ .5 Gauss. Note that also Mars lacks long range magnetic field but has similar local anomalies so that Martian auroras are possible. The mechanism causing these fields might be the same.
2. The crustal fields are a surface phenomenon and it is implausible that they could be caused by the rotation of plasma in the core of the Moon. The crustal magnetic fields seem to be associated with the lunar swirls, which are light-colored and therefore reflecting regions observed already at the 16^th century. Reiner Gamma is a classical example of a lunar swirl illustrated by Fig 1. of this. The origin of the swirlds is a mystery and several mechanisms have been proposed besides the crustal magnetism.
3. Since Moon does not have a global magnetic field shielding it from the solar wind and cosmic rays, weathering is expected to occur and change the chemistry of the surface so that it becomes dark colored and ceases to be reflective. In lunar maria this darkening has been indeed observed. The lunar swirls are an exception and a possible explanation is that they involve a relatively strong local magnetic field, which does the same as the magnetic field of Earth, and shields them from the weathering effects. It is known that the swirls are accompanied by magnetic fields much stronger than might be expected. What is interesting is that the opposite face of the Moon is mostly light-colored. Does this mean that there is a global magnetic field taking care of the shielding.

The article HT discusses a mechanism for how exceptionally strong magnetization could be associated with the vertical lava tubes and what are called dikes. The name indicates that the dikes are parallel to the surface.

1. The radar evidence indicates that the surface of the Moon once contained a molten rock. This suggest a period of high temperature and volcanic activity billions of years ago. Using a model of lava cooling rates Krawczynski and his colleagues have examined how a titanium-iron oxide, a mineral known as ilmenite - abundant on the Moon and commonly found in volcanic rock - could have produced a magnetization. Their experiments demonstrate that under the right conditions, the slow cooling of ilmenite can stimulate grains of metallic iron and iron nickel alloys within the Moon's crust and upper mantle to produce a powerful magnetic field explaining the swirls.
2. The paleomagnetic analysis of the Apollo samples suggests that there was a global magnetic field during period ≈ 3.85-3.56 Ga (the conjectured Theia event would have occurred ≈ 4.5 Ga ago), which would have reached intensities .78+/- .43 Gauss. The order of magnitude for this field is the same as that for the Earth's recent magnetic field. At the landing site of Apollo 16 magnetic fields as strong as .327 × 10^-3 Gauss were detected. A further analysis suggests the possibility of crustal fields of order 10^-2 Gauss to be compared with the Earth's magnetic field of .5 Gauss.
3. The lunar swirls consist of bright and dark surface markings alternating in a scale of 1-5 km. If their origin is magnetic, also the crustal magnetic fields must vary in the same scale. The associated source structures, modellable as magnetic dipoles, should have the same length scale. The restricted volume of the source bodies should imply strong magnetization. 300 nT crustal fields (.3 × 10^-2 Gauss) are necessary to produce the swirl markings. The required rock magnetization would be higher than .5 A/m (note that 1 A/m corresponds to about 1.25× 10^-2 Gauss).

   The model assumes that below the surface there are vertical magnetic dipoles serving as sources of the local magnetic field. The swirls as light regions would be above the dipoles generating a vertical magnetic field. In the dark regions, the magnetic field would be weak and approximately tangential due the absence of magnetization.

4. A mechanism is needed to enhance the magnetization carrying capacity of the rocks. The proposal is that a heating associated with the magmatic activity would have thermodynamically altered the host rocks making possible magnetizations, which are by an order of magnitude stronger than those associated with the lunar mare basalts (the existence of which suggets that the surface was once in a magma state). The slow cooling would have enhanced the metal content of the rocks and magnetization would have formed a stable record of the ancient global magnetic field of the Moon.

2. The TGD based model for the magnetic field of the Moon

The above picture would conform with the TGD based model in which the face of the Moon opposite to us corresponds to the bottom of the ancient Earth's crust. It could have been at high enough temperature at the time of the explosion producing the Moon. The volcanic activity would have occurred in the Earth's crust and magnetization would be inherited from that period.

One can however wonder how the magnetized structures could have survived for such a long time. The magnetic fields generated by macroscopic currents in the core are unstable and their maintenance in the standard electrodynamics is a mystery to which TGD suggests a solution in terms of the monopole flux contribution of about 2B_E/5 to the Earth's magnetic field which is topologically stable (see this). If the TGD explanation for the origin of the Moon is correct, these stable monopole fluxes assignable with the ancient crust of the Earth should be present also in the recent Moon and could cause a strong magnetization.

The mysterious findings could be indeed understood in the TGD based model for the birth of the Moon as being due to an explosion throwing out the crust of Earth as a spherical shell which condensed to form the Moon.

1. The TGD based model for the magnetic field of the Earth (see this) predicts that the Earth's magnetic field is the sum of a Maxwellian contribution and monopole contribution, which is topologically stable. This part corresponds to monopole flux tubes reflecting the nontrivial topology of CP₂. The monopole flux tubes have a closed 2-surface as a cross section and, unlike ordinary Maxwellian magnetic fields, the monopole part requires no currents to generate it. This explains why the Earth's magnetic field is stable in conflict with prediction that it should decay rather rapidly. Also an explanation for magnetic fields in cosmic scales emerges.
2. The Moon's magnetic field is known to be a surface phenomenon and very probably does originate from the rotation of the Moon's core as the Earth's magnetic field is believed to originate. In TGD, the stable monopole part would induce the flow of charged matter generating Maxwellian magnetic field and magnetization would also take place.

   If the Moon was born in the explosion throwing out the crust of Earth, the recent magnetic field should correspond to the part of the Earth's magnetic field associated with the monopole magnetic flux tubes in the crust. The flux tubes must be closed, which suggests that the loops run along the outer boundaries of the crust somewhat like dipole flux and return back along the inner boundaries of the crust. Therefore they formed a magnetic bubble. I have proposed that the explosions of magnetic bubbles of this kind generated in the explosions of the Sun gave rise to the planets (see this and this).

3. After the explosion throwing out the expanding magnetic bubble, the closed monopole flux tubes could have suffered reconnections changing the topology. I have considered a model for the Sunspot cycle (see this) in terms of a decay and reversal of the magnetic field of Sun based on the mechanism in monopole flux tube loops forming a a magnetic bubble at the surface of the Sun split by reconnection to shorter monopole flux loops for which the reversal occurs easily and is followed by a reconnection back to long loops with opposite direction of the flux. This process is like death followed by decay and reincarnation and corresponds to a pair of "big" state function reductions (BSFRs) in the scale of the Sun. Actually biological death could involve a similar decay of the monopole flux tubes associated with the magnetic body of the organism and meaning reduction of quantum coherence.
4. The formation of the Moon would have started with an explosion in which a magnetic bubble with thickness of about R_E/20 ≈ 100 km, presumably the crust of the Earth, was thrown out. A hole in the bubble was formed and after that the bubble developed to a disk at a surface of possibly expanding sphere, which contracted in the tangential direction to form the Moon. The monopole flux tubes of the shell followed matter in the process. In the first approximation, the Moon would have been a disk. The radius of Moon is less than one third of that for the Earth so that monopole flux tube loops of the crust with length of 2π R_E had to contract by a factor of about 1/3 to give rise to similar flux tubes of Moon. This would have increased the density by a factor of order 9 if the Moon were a disk, which of course does not make sense.

5. If the mass density did not change appreciably, the spherical shell with a hole had to transform to a structure filling the volume of the Moon. One can try to imagine how this happened.
   1. The basic assumption is that the far side corresponds to the surface of the ancient Earth. Near side could correspond to the lower boundary of its crust. A weaker condition is that the near side and a large part of the interior correspond to magma formed in the explosion and in the gravitational collapse to form the Moon. There is indeed evidence that the near side of the Moon has been in a molten magma state. This suggests that the crust divided into a solid part and magma in the explosion, which liberated a lot of energy and heated the lower boundary of the crust.
   2. Part of the solid outer part of the disk gave rise to the far side of the Moon. When the spherical disk collapsed under its own gravitational attraction, some fraction of the solid outer part, which could not contract, formed an outwards directed spherical bulge whereas the magma formed an inwards directed bulge.
   3. The energy liberated in the gravitational collapse melted the remaining fraction of the spherical disk as it fused to the proto Moon. From R_M≈ R_E/3, the area of the far side of the Moon is roughly by a factor 1/18 smaller than the area of the spherical disk, which means that the radius of the part of disk forming the far side is about R_E/4 and somewhat smaller than R_M. Most of the spherical disk had to melt in the gravitational collapse. The thin crust of the near side was formed in the cooling process.
   This model applies also to the formation of planets. The proposal indeed is that the planets formed by a collapse of a spherical disk produced in the explosion of Sun (see this). Moons of other planets could have formed from ring-like structures by the gravitational collapse of a split ring.
6. The magnitude of the dark monopole flux for Earth is about B_M =2B_E/5 ≈ .2 Gauss for the nominal value B_E=.5 Gauss. The monopole flux for the long loops is tangential but if reconnection occurs there are portions with length ΔR  inside  which the flux is vertical and connects the upper and lower boundaries of the  layer. Note  that in the TGD inspired quantum hydrodynamics  also dark Z⁰ magnetic fields associated with hydrodynamic flows  are possible and could be important in superfluidity (see this).
7. As already noticed, the far side of the Moon, which would correspond to the surface of the ancient Earth, is light-colored, which suggests that the monopole magnetic fields might be global and tangential at the far side. If so, the reconnection of the monopole flux tubes have not taken place at the far side. If magnetic anomalies are absent at the far side, the monopole part of the magnetic field should have taken care of the shielding by capturing the ions of the solar wind and cosmic rays as I have proposed. The dark monopole flux tubes play a key role in the TGD based model for the terrestrial life and this raises the question whether life could be possible also in the Moon, perhaps in its interior.

See the article Moon is mysterious or the chapter Magnetic Bubbles in TGD Universe: Part I.

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

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

Holography=holomorphy vision in relation to quantum criticality, hierarchy of Planck constants, and M8-H duality

Holography = holomorphy vision generalizes the realization of quantum criticality in terms of conformal invariance. Holography = holomorphy vision provides a general explicit solution to the field equations determining space-time surfaces as minimal surfaces X⁴⊂ H=M⁴× CP₂. For the first option the space-time surfaces are roots of two generalized analytic functions P₁,P₂ defined in H . For the second option single analytic generalized analytic function defines X⁴ as its root and as the base space of 6-D twistor twistor-surface X⁶ in the twistor bundle T(H)=T(M⁴)× TCP₂) identified as a zero section.

By holography, the space-time surfaces correspond to not completely deterministic orbits of particles as 3-surfaces and are thus analogous to Bohr orbits. This implies zero energy ontology (ZEO) and to the view of quantum TGD as wave mechanics in the space of these Bohr orbits located inside a causal diamond (CD), which form a causal hierarchy. Also the consruction of vertices for particle reactions has evolved dramatically during the last year and one can assign the vertices to partonic 2-surfaces.

M⁸-H duality is a second key principle of TGD. M⁸-H duality can be seen a number theoretic analog for momentum-position duality and brings in mind Langlands duality. M⁸ can be identified as octonions when the number-theoretic Minkowski norm is defined as Re(o²). The quaternionic normal space N(y) of y∈ Y⁴⊂ M⁸ having a 2-D commutative complex sub-space is mapped to a point of CP₂. Y⁴ has Euclidian signature with respect to Re(o²). The points y∈ Y⁴ are lifted by a multiplication with a co-quaternionic unit to points of the quaternionic normal space N(y) and mapped to M⁴⊂ H inversion.

This article discusses the relationship of the holography = holomorphy vision with the number theoretic vision predicting a hierarchy h_eff=nh₀ of effective Planck constants such that n corresponds to the dimension for an extension rationals (or extension F of rationals). How could this hierarchy follow from the recent view of M⁸-H duality? Both realizations of holography = holomorphy vision assume that the polynomials involved have coefficients in an extension F of rationals Partonic 2-surfaces would represent a stronger form of quantum criticality than the generalized holomorphy: one could say islands of algebraic extensions F from the ocean of complex numbers are selected. For the P option, the fermionic lines would be roots of P and dP/dz inducing an extension of F in the twistor sphere. Adelic physics would emerge at quantum criticality and scattering amplitudes would become number-theoretically universal. In particular, the hierarchy of Planck constants and the identification of p-adic primes as ramified primes would emerge as a prediction.

Also a generalization of the theory of analytic functions to the 4-D situation is suggestive. The poles of cuts of analytic functions would correspond to the 2-D partonic surfaces as vertices at which holomorphy fails and 2-D string worlds sheets could correspond to the cuts. This provides a general view of the breaking of the generalized conformal symmetries and their super counterparts as a necessary condition for the non-triviality of the scattering amplitudes.

See the artice Holography = holomorphy vision in relation to quantum criticality, hierarchy of Planck constants, and M⁸-H duality or the chapter with the same title.

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

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

About the origin of multicellularity in the TGD Universe

A living organism consists of cells that are almost identical and contain DNA that is the same for all of them but expresses itself in different ways. This genetic holography is a fundamental property of living organisms. Where does it originate?

Dark DNA associated with magnetic flux tubes is one or the basic predictions of the TGD inspired biology. One can say that the magnetic body controls the ordinary biomatter and dictates its development. Could one have a structure that would consist of a huge number of almost identical copies of dark DNA forming a quantum coherent unit inducing the coherence of ordinary biomatter? Could this structure induce the self-organization of the ordinary DNA and the cell containing it.

Could one understand this by using the TGD based spacetime concept. There are two cases to be considered. The general option is that f_i are analytic functions of 3 complex coordinates and 1 hypercomplex (light-like) coordinate of H and (f₁,f₂)=(0,0) defines the space-time surface.

A simpler option is that f_i are polynomials P_i with rational or even algebraic coefficients. Evolution as an increase of number theoretic complexity (see this) suggest that polynomials with rational coefficients emerged first in the evolution.

1. For the general option (f₁,f₂), the extension of rationals could emerge as follows. Assume 2-D singularity X²_i at a particular light-like partonic orbit (m_i such orbits for f_i) defining a X²_i as a root of f_i. If f₂ (f₁ ) is restricted to X²₁ resp. X²₂ is a polynomial P²_i with algebraic coefficients, it has m₂ resp. m₁ discrete roots, which are in an algebraic extension of rationals with dimension m₂ resp. m₁. Note that m₂ can depend on X²_i. Only a single extension appears for a given root and can depend on it. The identification of h_eff=n_ih₀ looks natural and would mean that h_eff is a local property characterizing a particular interaction vertex. Note that it is possible that the coefficients of the resulting polynomial are algebraic numbers.

   For the polynomial option (f₁,f₂)=(P₁,P₂), the argument is essentially the same except that now the number of roots of P₁ resp. P₂ does not depend on X²₂ resp. X²₁. The dimension n₁ resp. n₂ of the extension however depends on X²₂ resp. X²₁ since the coefficients of P₁ resp. P₂ depend on it.

2. The proposal of the number theoretic vision of TGD is that the effective Planck constant is given by h_eff=nh₀, h₀<h is the minimal value of h_eff and n corresponds to the dimension of the algebraic extension of rationals. As noticed, n would depend on the roots considered and in principle m=m₁m₂ values are possible. This identification looks natural since the field of rationals is replaced with its extension and n defines an algebraic dimension of the extension. n=m₁m₂ can be also considered. For the general option, the degree of the polynomial P₁ can depend on a particular root X²₂ of f₂ .
3. The dimension n_E of the extension depends on the polynomial and typically seems to increase with an exponential rate with the degree of the polynomials. If the Galois group is the permutation group S_m it has m! elements. If it is a cyclic group Z_m, it has m elements.

For the original view of M⁸-H duality, single polynomial P of complex variable with rational coefficients determined the boundary data of associative holography (see this, (see this, and this). The iteration of P was proposed as an evolutionary process leading to chaos (see this) and led to an exponential increase of the degree of the iterated polynomial as powers m^k of the degree m of P and to a similar increases of the dimension of its algebraic extension.

This might generalize to the recent situation (see this) if the iteration of polynomials P₁ resp. P₂ at the partonic 2-surface X²₂ resp. X²₁ defining holographic data makes sense and therefore induces a similar evolutionary process by holography. This could give rise to a transition to chaos at X²_i making itself manifest as the exponential increase in the number of roots and degree of extension of rationals and h_eff. One can consider the situation also from a more restricted point of view provided by the structure of H.

1. The space-time surface in H=M⁴× CP₂ can be many-sheeted in the sense that CP₂ coordinates are m₁-valued functions of M⁴ coordinates. Already this means deviation from the standard quantum field theories. This generates a m₁-sheeted quantum coherent structure not encountered in QFTs. Anyons could be the basic example in condensed matter physics (see this). m₁ is not very large in this case since CP₂ has extremely small size (about 10⁴ Planck lengths) and one would expect that the number of sheets cannot be too large.
2. M⁴ and CP₂ can change the roles: M⁴ coordinates define the fields and CP₂ takes the role of the space-time. M⁴ coordinates could be m₂ valued functions of CP₂ coordinates: this would give a quantum coherent system acting as a unit consisting of a very large number m₂ of almost identical copies at different positions in M⁴. The reason is that there is a lot of room in M⁴. These regions could correspond to monopole flux tubes forming a bundle and also to almost identical basic units.

   If m_i corresponds to the degree of a polynomial, quite high degrees are required. The iteration of polynomials would mean an exponential increase in powers d^k of the degree d of the iterated polynomial P and a transition to chaos. For a polynomial of degree d=2 one would obtain a hierarchy m=2^k.

3. Lattice like systems would be a basic candidate for this kind of system with repeating units. The lattice could be also realized at the level of the field body (magnetic body) as a hyperbolic tessellation. The fundamental realization of the genetic code would rely on a completely unique hyperbolic tessellation known as icosa tetrahedral tessellation involving tetrahedron, octahedron and icosahedron as the basic units (see this and this). This tessellation could define a universal genetic code extending far beyond the chemical life and having several realizations also in ordinary biology.
4. The number of neurons in the brain is estimated to be about 86 billions: 10¹²≈ 2⁴⁰. If cell replications correspond to an iteration of a polynomial of degree 2, morphogenesis involves 40 replications. Human fetal cells replicate 50-70 times. Could the m almost copies of the basic system define a region of M⁴ corresponding to genes and cells? Could our body and brain be this kind of quantum coherent system with a very large number of almost copies of the same basic system. The basic units would be analogs of monads of Leibniz and form a polymonad. They could quantum entangle and interact.
5. If n=h_eff/h₀ corresponds to the dimension n_E of the extension, it could be of the order 10¹⁴ or even larger for the gravitational magnetic body (MB). The MB could be associated with the Earth or even of the Sun: the characteristic Compton length would be about .5 cm for the Earth and half of the Earth radius for the Sun).

Could this give a recipe for building geometric and topological models for living organisms? Take sufficiently high degree polynomials f₁ and f₂ and find the corresponding 4-surface from the condition that they vanish. Holography=holomorphy vision would also give a model for the classical time evolution of this system as classical, and not completely deterministic realization of behaviors and functions. Also a quantum variant of computationalistic view emerges.

See the article TGD view about water memory and the notion of morphogenetic field or the chapter with the same title.

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

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

See the article TGD view about water memory and the notion of morphogenetic field or the chapter with the same title.

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

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

About the generation of matter-antimatter asymmetry in the TGD Universe?

I have developed a rather detailed view of interaction vertices (see this). Everything boils down to the question of what the creation of a fermion-antifermion pair is in TGD. Since bosonic fields are not primary fields (bosons are bound states of fermions and antifermions), the usual view about generation of fermion antifermion pairs does not work as such and the naive conclusion seems to be that fermions and antifermions are separately conserved.

Holography=holomorphy identification leading to an explicit general solution of field equations defining space-time surfaces as minimal surfaces with 2-D singularities at which the minimal surface property fails, is the starting point. A generalized holomorphism, which maps H to itself, is characterized by a generalized analyticity, in particular by a hyper-complex analyticity. The analytic function from H to H in the generalized sense depends on the light-like coordinate or its dual ( say -t+z and t+z in the simplest case) and the 3 remaining complex coordinates of H=M⁴/ti,esCP₂.

Let's take two such functions, f₁ and f₂, and set them to zero. We get a 4-D space-time surface that is a holomorphic minimal surface with 2-D singularities at which the minimal surface property and holomorphy fails. Singularities are analogs of poles. Also the analogs of cuts can be considered and would look like string world sheets: they would be analogous to a positive real axis along which complex function z^(i/n) has discontinuity unless one replaces the complex plane with its n-fold covering. The singularities correspond to vertices. and the fundamental vertex corresponds to a creation of fermion-antifermion pair.

There are at least two types of holomorphy in the hypercomplete sense, corresponding to analyticity with respect to -t+z or t+z as a light-like coordinate defining the analogs of complex coordinates z and its conjugate. Also CP₂ complex coordinates could be conjugated.

These two kinds of analyticities would naturally correspond to fermions and to antifermions identified as time-reflected (CP reflected) fermions. This time reflection transforms fermion to antifermion. This is not the reversal of the arrow of time occurring in a "big" state function reduction (BSFR) as TGD counterpart of what occurs in quantum measurement, which corresponds to interchange of the roles of the fermionic creation and annihilation operators.

When a fermion pair, which can also form a boson as a bound state, is created, the partonic 2-surface to which the fermion line is assigned, turns back in time. At the vertex, where this occurs, neither of these two analyticities applies: holomorphy and the minimal surface property are violated because at the vertex the type of analyticity changes.

Now comes the crucial observation: the number theoretic vision of TGD predicts that quantum coherence is possible in macroscopic and even astrophysical and cosmological scales and corresponds to the existence of arbitrarily large connected space-time regions acting as quantum coherence regions: field bodies as counterparts of Maxwellian fields can indeed be arbitrarily large.

For a given region of this kind one must choose the same kind of generalized analyticity, say -t+z or t+z even at very long scales. Only fermions or antifermions but not both are possible for this kind of space-time sheets! Does this solve the mystery of matter-antimatter asymmetry and does its presence demonstrate that quantum coherence is possible even in cosmological scales?

See the article What gravitons are and could one detect them in TGD Universe? or the chapter with the same title.

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

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

Comparing TGD- and QFT based descriptions of particle interactions

Marko Manninen made interesting questions related to the relationship between TGD and quantum field theories (QFTs). In the following, I will try to summarize an overview of this relationship in the recent view about quantum TGD. I have developed the latest view of quantum TGD in various articles (see this, this, this, this, this, and this).

Differences between QFT and quantum TGD

Several key ideas related to quantum TGD distinguish between TGD and QFTs.

1. The basic problem of QFT is that it involves only an algebraic description of particles. An explicit geometric and topological description is missing but is implicitly present since the algebraic structure of QFTs expresses the point-like character of the particles via commutation and anticommutation relations for the quantum fields assigned to the particles.

   In the string models, the point-like particle is replaced by a string, and in the string field theory, the quantum field Ψ(x) is replaced by the stringy quantum field Ψ(string), where the "string" corresponds a point in the infinite-D space of string configurations.

   In TGD, the quantum field Ψ(x) is replaced by a formally classical spinor field Ψ (Bohr orbit). The 4-D Bohr orbits are preferred extremals of classical action satisfying holography forced by general coordinate invariance without path integral and represent points of the "world of classical worlds" (WCW). The components of Ψ correspond to multi-fermion states, which are pairs of ordinary 3-D many-fermion states at the boundaries of causal diamond (CD).

   The gamma matrices of the WCW spinor structure are linear combinations of fermionic oscillator operators for the second quantized free spinor field of H. They anticommute to the WCW metric, which is uniquely determined by the maximal isometries for WCW guaranteeing the existence of the spinor connection. Physics is unique from its existence, as implied also by the twistor lift and number theoretic vision and of course, by the standard model symmetries and fields.

2. In TGD, the notion of a classical particle as a 3-surface moving along 4-D "Bohr orbit" as the counterpart of world-line and string world sheet is an exact aspect of quantum theory at the fundamental level. The notions of classical 3-space and particle are unified. This is not the case in QFT and the notion of a Bohr orbit does not exist in QFTs. TGD view of course conforms with the empirical reality: particle physics is much more than measuring of the correlation functions for quantum fields.

   Quantum TGD is a generalization of wave mechanics defined in the space of Bohr orbits. The Bohr orbit corresponds to holography realized as a generalized holomorphy generalizing 2-D complex structure to its 4-D counterpart, which I call Hamilton-Jacobi structures (see this). Classical physics becomes an exact part of quantum physics in the sense that Bohr orbits are solutions of classical field equations as analogs of complex 4-surfaces in complex M⁴×CP₂ defined as roots of two generalized complex functions. The space of these 4-D Bohr orbits gives the WCW (see this), which corresponds to the configuration space of an electron in ordinary wave mechanics.

3. The spinor fields of H are needed to define the spinor structure in WCW. The spinor fields of H are the free spinor fields in H coupling to its spinor connection of H. The Dirac equation can be solved exactly and second quantization is trivial.

   This determines the fermionic propagators in H and induces them at the space-time surfaces. The propagation of fermions is thus trivialized. All that remains is to identify the vertices. But there is also a problem: how to avoid the separate conservation of fermion and antifermion numbers. This will be discussed below.

4. At the fermion level, all elementary particles, including bosons, can be said to be made up of fermions and antifermions, which at the basic level correspond to light-like world lines on 3-D parton trajectories, which are the light-like 3-D interfaces of Minkowski spacetime sheets and the wormhole contacts connecting them.

   The light-like world lines of fermions are boundaries of 2-D string world sheets and they connect the 3-D light-like partonic orbits bounding different 4-D wormhole contacts to each other. The 2-D surfaces are analogues of the strings of the string models.

5. In TGD, classical boson fields are induced fields and no attempt is made to quantize them. Bosons as elementary particles are bound states of fermions and antifermions. This is extraordinarily elegant since the expressions of the induced gauge fields in terms of embedding space coordinates and their gradients are extremely non-linear as also the action principle. This makes standard quantization of classical boson fields using path integral or operator formalism a hopeless task.

   There is however a problem: how to describe the creation of a pair of fermions and, in a special case, the corresponding bosons, when there are no primary boson fields? Can one avoid the separate conservation of the fermion and the antifermion numbers?

Description of interactions in TGD

Many-particle interactions have two aspects: the classical geometric description, which QFTs do not allow, and the description in terms of fermions (bosons do not appear as primary quantum fields in TGD).

1. At the classical level, particle reactions correspond to topological reactions, where the 3-surface breaks, for example, into two. This is exactly what we see in particle experiments quite concretely. For instance, a closed monopole flux tube representing an elementary particle decomposes to two in a 3-particle vertex.

   There is field-particle duality realized geometrically. The minimal surface as a holomorphic solution of the field equations defines the generalization of the light-like world line of a massless particle as a Bohr orbit as a 4-surface. The equations of the minimal surface in turn state the vanishing of the generalized acceleration of a 3-D particle identified as 3-surface.

   At the field level, minimal surfaces satisfy the analogs of the field equations of a massless free field. They are valid everywhere except at 2-D singularities associated with 3-D light-like parton trajectories. At singularities the minimal surface equation fails since the generalized acceleration becomes infinite rather than vanishing. The analog of the Brownian particle experiences acceleration: there is an "edge" on the track.

   At singularities, the field equations of the whole action are valid, but are not separately true for various parts of the action. Generalized holomorphy breaks down. These 2-D singularities are completely analogous to the poles of an analytic function in 2-D case and there is analogy with the 2-D electrostatics, where the poles of analytic function correspond to point charges and cuts to line charges.

   This gives the TGD counterparts of Einstein's equations, analogs of geodesic equations, and also the analogy Newton's F=ma. Everything interesting is localized at 2-D singularities defining the vertices. The generalized 8-D acceleration H^k defined by the trace of the second fundamental form, is localized on these 2-D parton surfaces, vertices. One has a generalization of Brownian motion for a particle-like object defined by a partonic 2-surface or equivalently for a particle as 3-surface. Intriguingly, Brownian motion has been known for a century and Einstein wrote his first paper after his thesis about Brownian motion!

   Singularities correspond to sources of fermion fields and are associated with various conserved fermion currents: just like in QFTs. For a given spacetime surface, the source- vertex - is a discrete set of 2-D partonic surface just as charges correspond to poles of analytic function in 2-D electrostatics.

   At the classical level, the 2-D singularities of the minimal surfaces therefore correspond to vertices and are localized to the light-like paths of parton surfaces where the generalized holomorphy breaks down and the generalized acceleration H^k is there non-vanishing and infinite.

Description of the interaction vertices

1. How to get the TGD counterparts of the QFT vertices?

   Vertices typically contain a fermion and an antifermion and the gauge potential, which is second quantized. Now, classical gauge potentials are not second quantized. How to obtain the basic gauge theory vertices?

   This is where the standard approximation of QFTs helps intuition: replace the quantized boson field with a classical one. This gives the vertex corresponding to the creation of a pair of fermions. Thanks to that, only the fermion and the sum of the antifermion numbers are conserved and the theory does not reduce to a free field theory. One should be able to do the same now. However, the precise formulation of this vision is far from trivial.

2. The modified Dirac action should give elementary particle vertices for a given Bohr trajectory.

   There are two options:

   1. Modified gammas are defined as contractions of ordinary gamma matrices of H with the canonical momentum currents associated with the classical action defining the space-time surface. Supersymmetry is now exact: besides color and Poincare super generators there is an infinite number of conserved super symplectic generators and infinitesimal generalized superholomorphisms.

      This option does not work: the modified Dirac equation implies that the Dirac action and also vertices vanish identically. Although one has partonic 2-surfaces as singularities of minimal surfaces defining vertices, the theory is trivial because the usual perturbation theory does not work.

   2. Modified gamma matrices are replaced by the induced gamma matrices defined by the volume term (cosmological term of the classical action). Supersymmetry is broken but only at the 2-D vertices. The anticommutator of the induced gammas gives the induced metric. This is not true for the modified gammas defined by the entire action: in this case the anticommutators are rather complex, being bilinear in the canonical momentum currents. Is it possible to have a non-trivial theory despite the breaking of supersymmetry at vertices or or does the supersymmetry breaking make possible a non-trivial theory? This seems to be the case.
      1. In 2-D vertices, the generalized acceleration field H^k is proportional to the 2-D delta function and gives rise to the graviton and Higgs vertices. One obtains also the vertices related to gauge bosons from the coupling of the induced spinor field to induced spinor connection. Only the couplings to electroweak gauge potentials and U(1) K&aum;hler gauge potential of M⁴ are obtained. The failure of the generalized holomorphy is absolutely essential.
      2. Color degrees of freedom are completely analogous to translational degrees of freedom since color quantum numbers are not spin-like in TGD. Strong interactions are vectorial and correspond to Kähler gauge potentials.
      3. Generalized Brownian motion gives the vertices. One obtains the equivalents of Einstein's and Newton's equations at the vertices. The M⁴ part M^k of the generalized acceleration is related to the gravitons and the CP₂ part S^k to the Higgs field. Spin J=2 for graviton is due to the rotational motion of the closed monopole flux tube associated with the gravitation giving an additional unit of spin besides the spin of H^k, which is S=1.
   3. Consider now the description of fermion pair creation.
      1. Intuitively, the creation of a fermion pair (and thus also a boson) corresponds to the fermion turning backwards in time. At the level of the geometry of the space-time surface, this corresponds to the partonic 2-surface turning backwards in time, and the same happens to the corresponding fermion line. Turning back in time means that effectively the fermion current is not conserved: if one does not take into account that the parton surface turns in the other direction of time, the fermion disappears effectively and the current must has a singular divergence. This is what the divergence of the generalized acceleration means.
      2. This implies that the separate conservation is lost for fermion and antifermion numbers. This means breaking of supersymmetry, of masslessness, of generalized holomorphy and also the generation of the analog of Higgs vacuum excitation as CP₂ part S^k of the generalized acceleration H^k. The Higgs vacuum expectation is only at the vertices. But this is exactly what is actually wanted! No separate symmetry breaking mechanism is needed!
      3. The failure of the generalized holomorphy at the 2-D vertex means that the holomorphic partonic orbit turns at the singularity to an antiholomorphic one. For the annihilation vertex it could occur only for the hypercomplex part of the generalized complex structure.
      4. Remarkably, the states associated with connected 4-surfaces consist of either fermions or antifermions but not both. This explains matter antimatter asymmetry if quantum coherence is possible in arbitrarily long scales. In TGD, space-time surfaces decompose to regions containing either matter or antimatter and, by the presence of quantum coherence even in cosmological scales, these regions can be very large. The quantum coherence in large scales is implied by the number theoretic vision predicting a hierarchy of Planck constants labelling phases of ordinary matter behaving like dark matter (see for instance this).
      5. What is the precise mathematical formulation of this vision? This is where a completely unique feature of 4-dimensional manifolds comes in: they allow exotic smooth structures. Exotic smooth structure is the standard smooth structure with lower-dimensional defects. In TGD, the defects correspond in TGD to 2-D parton vertices as "edges" of Brownian motion. In the exotic smooth structure, the edge disappears and everything is soft. Pair creation and non-trivial theory is possible only in dimension D=4 (see this and this ).

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

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