Monday, November 14, 2022

Could the exotic smooth structures have a physical interpretation in the TGD framework?

The exotic smooth structures appearing for 4-D manifolds, even R4, challenges both the classical GRT and quantum GRT based on path integral approach. They could also cause problems in the TGD framework. I have already considered the possibility that the exotics could be eliminated by what could be called holography of smoothness meaning that the smooth structure for the boundary dictates it for the entire 4-surface. One can however argue that the atlas of charts for 3-D boundary cannot define uniquely the atlas of charges for 4-D surface.

In the TGD framework, exotic smooth structures could also have a physical interpretation. As noticed, the failure of the standard smooth structure can be thought to occur at a point set of dimension zero and correspond to a set of point defects in condensed matter physics. This could have a deep physical meaning.

  1. The space-time surfaces in H=M4× CP2 are images of 4-D surfaces of M8 by M8-H-duality. The proposal is that they reduce to minimal surfaces analogous to soap films spanned by frames. Regions of both Minkowskian and Euclidean signature are predicted and the latter correspond to wormhole contacts represented by CP2 type extremals. The boundary between the Minkowskian and Euclidean region is a light-like 3-surface representing the orbit of partonic 2-surface identified as wormhole throat carrying fermionic lines as boundaries of string world sheets connecting orbits of partonic 2-surfaces.
  2. These fermionic lines are counterparts of the lines of ordinary Feynman graphs, and have ends at the partonic 2-surfaces located at the light-like boundaries of CD and in the interior of the space-time surface. The partonic surfaces, actually a pair of them as opposite throats of wormhole contact, in the interior define topological vertices, at which light-like partonic orbits meet along their ends.
  3. These points should be somehow special. Number theoretically they should correspond points with coordinates in an extension of rationals for a polynomial P defining 4-surface in H and space-time surface in H by M8-H duality. What comes first in mind is that the throats touch each other at these points so that the distance between Minkowskian space-time sheets vanishes. This is analogous to singularities of Fermi surface encountered in topological condensed matter physics: the energy bands touch each other. In TGD, the partonic 2-surfaces at the mass shells of M4 defined by the roots of P are indeed analogs of Fermi surfaces at the level of M4⊂ M8, having interpretation as analog of momentum space.

    Could these points correspond to the defects of the standard smooth structure in X4? Note that the branching at the partonic 2-surface defining a topological vertex implies the local failure of the manifold property. Note that the vertices of an ordinary Feynman diagram imply that it is not a smooth 1-manifold.

  4. Could the interpretation be that the 4-manifold obtained by removing the partonic 2-surface has exotic smooth structure with the defect of ordinary smooth structure assignable to the partonic 2-surface at its end. The situation would be rather similar to that for the representation of exotic R4 as a surface in CP2 with the sphere at infinity removed (see this).
  5. The failure of the cosmic censorship would make possible a pair creation. As explained, the fermionic lines can indeed turn backwards in time by going through the wormhole throat and turn backwards in time. The above picture suggests that this turning occurs only at the singularities at which the partonic throats touch each other. The QFT analog would be as a local vertex for pair creation.
  6. If all fermions at a given boundary of CD have the same sign of energy, fermions which have returned back to the boundary of CD, should correspond to antifermions without a change in the sign of energy. This would make pair creation without fermionic 4-vertices possible.

    If only the total energy has a fixed sign at a given boundary of CD, the returned fermion could have a negative energy and correspond to an annihilation operator. This view is nearer to the QFT picture and the idea that physical states are Galois confined states of virtual fundamental fermions with momentum components, which are algebraic integers. One can also ask whether the reversal of the arrow of time for the fermionic lines could give rise to gravitational quantum computation as proposed here.

See the article Intersection form for 4-manifolds, knots and 2-knots, smooth exotics, and TGD or the chapter Does M8 H duality reduce classical TGD to octonionic algebraic geometry?: Part II.

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

Articles related to TGD.

Sunday, November 13, 2022

Could the existence of exotic smooth structures pose problems for TGD?

The article of Gabor Etesi (see this) gives a good idea about the physical significance of the existence of exotic smooth structures and how they destroy the cosmic censorship hypothesis (CCH of GRT stating that spacetimes of GRT are globally hyperbolic so that there are no time-like loops.

1. Smooth anomaly

No compact smoothable topological 4-manifold is known, which would allow only a single smooth structure. Even worse, the number of exotics is infinite in every known case! In the case of non-compact smoothable manifolds, which are physically of special interest, there is no obstruction against smoothness and they typically carry an uncountable family of exotic smooth structures.

One can argue that this is a catastrophe for classical general relativity since smoothness is an essential prerequisite for tensory analysis and partial differential equations. This also destroys hopes that the path integral formulation of quantum gravitation, involving path integral over all possible space-time geometries, could make sense. The term anomaly is certainly well-deserved.

Note however that for 3-geometries appearing as basic objects in Wheeler's superspace approach, the situation is different since for D≤3 there is only a single smooth structure. If one has holography, meaning that 3-geometry dictates 4-geometry, it might be possible to avoid the catastrophe.

The failure of the CCH is the basic message of Etesi's article. Any exotic R4 fails to be globally hyperbolic and Etesi shows that it is possible to construct exact vacuum solutions representing curved space-times which violate the CCH. In other words, GRT is plagued by causal anomalies.

Etesi constructs a vacuum solution of Einstein's equations with a vanishing cosmological constant which is non-flat and could be interpreted as a pure gravitational radiation. This also represents one particular aspect of the energy problem of GRT: solutions with gravitational radiation should not be vacua.

  1. Etesi takes any exotic R4, which has the topology of S3× R and has an exotic smooth structure, which is not a Cartesian product. Etesi maps maps R4 to CP2, which is obtained from C2 by gluing CP1 to it as a maximal ball B3r for which the radial Eguchi-Hanson coordinate approaches infinity: r → &infty;. The exotic smooth structure is induced by this map. The image of the exotic atlas defines atlas. The metric is that of CP2 but SU(3) does not act as smooth isometries anymore.
  2. After this Etesi performs Wick rotation to Minkowskian signature and obtains a vacuum solution of Einstein's equations for any exotic smooth structure of R4.
The question of exotic smoothness is encountered both at the level of embedding space and associated fixed spaces and at the level of space-time surfaces and their 6-D twistor space analogies.

2. Holography of smoothness

In the TGD framework space-time is 4-surface rather than abstract 4-manifold. 4-D general coordinate invariance, assuming that 3-surfaces as generalization of point-like particles are the basic objects, suggests a fully deterministic holography. A small failure of determinism is however possible and expected, and means that space-time surfaces analogous to Bohr orbits become fundamental objects. Could one avoid the smooth anomaly in this framework?

The 8-D embedding space topology induces 4-D topology. My first naive intuition was that the 4-D smooth structure, which I believed to be somehow inducible from that of H=M4× CP2, cannot be exotic so that in TGD physics the exotics could not be realized. But can one really exclude the possibility that the induced smooth structure could be exotic as a 4-D smooth structure?

What does the induction of a differentiable structure really mean? Here my naive expectations turn out to be wrong.

  1. If a sub-manifold S⊂ H can be regarded as an embedding of smooth manifold N to S⊂ H, the embedding N→ S⊂ H induces a smooth structure in S (see this). The problem is that the smooth structure would not be induced from H but from N and for a given 4-D manifold embedded to H one could also have exotic smooth structures. This induction of smooth structure is of course physically adhoc.

    It is not possible to induce the smooth structure from H to sub-manifold. The atlas defining the smooth structure in H cannot define the charts for a sub-manifold (surface). For standard R4 one has only one atlas.

  2. Could M8-H duality help and holography help? One has holography in M8 and this induces holography in H. The 3-surfaces X3 inducing the holography in M8 are parts of mass shells, which are hyperbolic spaces H3⊂ M4⊂ M8. 3-surfaces X3 could be even hyperbolic 3-manifolds as unit cells of tessellations of H3. These hyperbolic manifolds have unique smooth structures as manifolds with dimension D<4.
  3. One can ask whether the smooth structure at the boundary of a manifold could dictate that of the manifold uniquely. One could speak of holography for smoothness.

    The implication would be that exotic smooth 4-manifold cannot have a boundary. Indeed, R4 does not have a boundary. Could this theorem generalize so that 3-surfaces as sub-manifolds of mass shells H3m determined by the polynomials defining the 4-surface in M8 take the role of the boundaries?

    The regions of X4⊂ M8 connecting two sub-sequent mass shells would have a unique smooth structure induced by the hyperbolic manifolds H3 at the ends. These smooth structures are unique by D<4 and cannot be exotic. Smooth holography would determine the smooth structure from that for the boundary of 4-surface.

  4. However, the holography for smoothness is argued to fail (see this). Assume a 4-manifold W with 2 different smooth structures. Remove a ball B4 belonging to an open set U and construct a smooth structure at its boundary S3. Assume that this smooth structure can be continued to W. If the continuation is unique, the restrictions of the 2 smooth structures in the complement of B4 would be equivalent but it is argued that they are not.

    The first layman objection is that the two smooth structures of W are equivalent in the complement W-B3 of an arbitrary small ball B3⊂ W but not in the entire W. This would be analogous to coordinate singularity. For instance, a single coordinate chart is enough for a sphere in the complement of an arbitrarily small disk. An exotic smooth structure would be like a local defect in condensed matter physics.

    The second layman objection is that smooth structure, unlike topology, cannot be induced from W to W-B3 but only from W-B3 to W. If one a smooth structure at the boundary S3 is chosen, it determines the smooth structure in the interior as standard smooth structure.

  5. In fact, one could argue that the mere fact the 4-surfaces have boundaries as their ends at the light-like boundaries of CD, implies a unique smooth structure by holography. It is however possible that the mass-shells correspond to discontinuities of derivatives so that the smooth holography decomposes to a piece-wise holography. This would mean that M8-H duality is needed.
Amazingly, if the holography of smoothness holds true, the avoidance of the smooth exotics requires holography and both number theoretical vision and general coordinate invariance of geometric vision predict the holography in the TGD framework. For higher space-time dimensions D>4 one cannot avoid the exotics. Also the number theoretic vision fails for them.

3. Can embedding space and related spaces have exotic smooth structure?

One can worry about the exotic smooth structures possibly associated with the M4, CP2, H=M4× CP2, causal diamond CD=cd× CP2, where cd is the intersection of the future and past directed light-cones of M4, and with M8. One can also worry about the twistor spaces CP3 resp. SU(3)/U(1)× U(1) associated with M4 resp. CP2.

The key assumption of TGD is that all these structures have maximal isometry groups so that they relate very closely to Lie groups, whose unique smooth structures are expected to determine their smooth structures.

  1. The first sigh of relief is that all Lie groups have the standard smooth structure. In particular, exotic R4 does not allow translations and Lorentz transformations as isometries. I dare to conclude that also the symmetric spaces like CP2 and hyperbolic spaces such as Hn= SO(1,n)/SO(n) are non-exotic since they provide a representation of a Lie group as isometries and the smoothness of the Lie group is inherited. This would mean that the charts for the coset space G/H would be obtained from the charts for G by an identification of the points of charts related by action of subgroup H.

    Note that the mass shell H3, as any 3-surface, has a unique smooth structure by its dimension.

  2. Second sigh of relief is that twistor spaces CP3 and SU(3)/U(1)× U(1) have by their isometries and their coset space structure a standard smooth structure.

    In accordance with the vision that the dynamics of fields is geometrized to that of surfaces, the space-time surface is replaced by the analog of twistor space represented by a 6-surface with a structure of S2 bundle with space-time surface X4 as a base-space in the 12-D product of twistor spaces of M4 and CP2 and by its dimension D=6 can have only the standard smooth structure unless it somehow decomposes to (S3× R)× R2. Holography of smoothness would prevent this since it has boundaries because X4 as base space has boundaries at the boundaries of CD.

  3. cd is an intersection of future and past directed light-cones of M4. Future/past directed light-cone could be seen as a subset of M4 and implies standard smooth structure is possible. Coordinate atlas of M4 is restricted to cd and one can use Minkowski coordinates also inside the cd. cd could be also seen as a pile of light-cone boundaries S2× R+ and by its dimension S2× R allows only one smooth structure.
  4. M8 is a subspace of complexified octonions and has the structure of 8-D translation group, which implies standard smooth structure.
The conclusion is that continuous symmetries of the geometry dictate standard smoothness at the level of embedding space and related structures.

See the article Intersection form for 4-manifolds, knots and 2-knots, smooth exotics, and TGD or the chapter Does M8 H duality reduce classical TGD to octonionic algebraic geometry?: Part II.

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

Articles related to TGD.

Thursday, November 10, 2022

Intersection form for 4-manifolds, knots and 2-knots, smooth exotics, and TGD

Gary Ehlenberger sent a highly interesting commentary related to smooth structures in R4 discussed in the article of Gompf (see this) and more generally to exotics smoothness discussed from the point of view of mathematical physics in the book of Asselman-Maluga and Brans (see this). I am grateful for these links for Gary. The intersection form of 4-manifold (see this) characterizing partially its 2-homology is a central notion.

The role of intersection forms in TGD

I am not a topologist but I had two good reasons to get interested on intersection forms.

  1. In the TGD framework (see this), the intersection form describes the intersections of string world sheets and partonic 2-surfaces and therefore is of direct physical interest (see this and this).
  2. Knots have an important role in TGD. The 1-homology of the knot complement characterizes the knot. Time evolution defines a knot cobordism as a 2-surface consisting of knotted string world sheets and partonic 2-surfaces. A natural guess is that the 2-homology for the 4-D complement of this cobordism characterizes the knot cobordism. Also 2-knots are possible in 4-D space-time and a natural guess is that knot cobordism defines a 2-knot.

    The intersection form for the complement for cobordism as a way to classify these two-knots is therefore highly interesting in the TGD framework. One can also ask what the counterpart for the opening of a 1-knot by repeatedly modifying the knot diagram could mean in the case of 2-knots and what its physical meaning could be in the TGD Universe. Could this opening or more general knot-cobordism of 2-knot take place in zero energy ontology (ZEO) (see this, this and this) as a sequence of discrete quantum jumps leading from the initial 2-knot to the final one.

Why exotic smooth structures are not possible in TGD?

The article of Gabor Etesi (see this) gives a good idea about the physical significance of the existence of exotic smooth structures (see the book and the article). They mean a mathematical catastrophe for both classical relativity and for the quantization of general relativity based on path integral formulation.

The first naive guess was that the exotic smooth structures are not possible in TGD but it turned out that this is not trivially true. The reason is that the smooth structure of the space-time surface is not induced from that of H unlike topology. One could induce smooth structure by assuming it given for the space-time surface so that exotics would be possible. This would however bring an ad hoc element to TGD. This raises the question of how it is induced.

  1. This led to the idea of a holography of smoothness, which means that the smooth structure at the boundary of the manifold determines the smooth structure in the interior. Suppose that the holography of smoothness holds true. In ZEO, space-time surfaces indeed have 3-D ends with a unique smooth structure at the light-like boundaries of the causal diamond CD= cd× CP2 ⊂ H=M4× CP2, where cd is defined in terms of the intersection of future and past directed light-cones of M4. One could say that the absence of exotics implies that D=4 is the maximal dimension of space-time.
  2. The differentiable structure for X4⊂ M8, obtained by the smooth holography, could be induced to X4⊂ H by M8-H-duality. Second possibility is based on the map of mass shell hyperboloids to light-cone proper time a=constant hyperboloids of H belonging to the space-time surfaces and to a holography applied to these.
  3. There is however an objection against holography of smoothness (see this). In the last section of the article, I develop a counter argument against the objection. It states that the exotic smooth structures reduce to the ordinary one in a complement of a set consisting of arbitrarily small balls so that local defects are the condensed matter analogy for an exotic smooth structure.
See the article Intersection form for 4-manifolds, knots and 2-knots, smooth exotics, and TGD or the chapter Does M8−H duality reduce classical TGD to octonionic algebraic geometry?: Part II.

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

Articles related to TGD.

Saturday, November 05, 2022

Finite Fields and TGD

TGD involves geometric and number theoretic physics as complementary views of physics. Almost all basic number fields: rationals and their algebraic extensions, p-adic number fields and their extensions, reals, complex number fields, quaternions, and octonions play a fundamental role in the number theoretical vision of TGD.

Even a hierarchy of infinite primes and corresponding number fields appears. At the first level of the hierarchy of infinite primes, the integer coefficients of a polynomial Q defining infinite prime have no common prime factors. P=Q hypothesis states that the polynomial P defining space-time surface is identical with a polynomial Q defining infinite prime at the first level of hierarchy.

However, finite fields, which appear naturally as approximations of p-dic number fields, have not yet gained the expected preferred status as atoms of the number theoretic Universe. Also additional constraints on polynomials P are suggested by physical intuition.

Here the notions of prime polynomial and concept of infinite prime come to rescue. Prime polynomial P with prime order n=p and integer coefficients smaller than p can be regarded as a polynomial in a finite field. The proposal is that all physically allowed polynomials are constructible as functional composites of prime polynomials satisfying P=Q condition.

See the article Finite Fields and TGD.

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

Thursday, November 03, 2022

VO2 can remember like a brain

The following comments were inspired by a popular article (see this) with the title "Scientists accidentally discover a material that can 'remember' like a brain". These materials can remember the history of its physical stimuli. The findings are described in the article "Electrical control of glass-like dynamics in vanadium dioxide for data storage and processing" published in Nature (see see this).

The team from the Ecole Polytechnique Federale de Lausanne (EPFL) in Switzerland did this discovery while researching insulator-metal phase transitions of vanadium dioxide (VO2), a compound used in electronics.

  1. PhD student Mohammad Samizadeh Nikoo was trying to figure out how long it takes for VO2 to make a phase transition from insulating to conducting phase under "incubation" by a stimulation by a radio frequency pulse of 10 μs duration and voltage amplitude V= 2.1 V. Note that the Wikipedia article talks about semiconductor-metal transition. The voltage pulse indeed acted like a voltage in a semiconductor.
  2. As the current heated the sample it caused a local phase transition to metallic state in VO2. The induced current moved across the material, following a path until it exited on the other side. A conducting filament connecting the ends of the device was generated by a percolation type process.
  3. Once the current had passed, the material exhibited an insulating state but after incubation time tinc, which was tinc∼ .1 μs for the first pulse, it became conducting. This state lasted at least 10,000 seconds.

    After applying a second electrical current during the experiment, it was observed that tinc appeared to be directly related to its history and was shorter than for the first incubation period .1 μs. The VO2 seemed to remember the first phase transition and anticipate the next. One could say that the system learned from experience.

Before trying to understand the finding in the TGD framework, it is good to list some basic facts about vanadium and vanadium-oxide VO2 or Vanadium(IV) oxide (see this).
  1. Vanadium is a transition metal, which has valence shells d3s2. It is known that the valence electrons of transition metals can mysteriously disappear, for instance in heating (see this). The TGD interpretation (see this) would be that heating provides energy making it possible to transform ordinary valence electrons to dark valence electrons with a higher value of heff and higher energy. In the recent case, the voltage pulses could have the same effect.
  2. VO2 forms a solid lattice of V4+ ions. There are two lattice forms: the monoclinic semiconductor below Tc=340 K and the tetragonal metallic form above Tc. In the monoclinic form, the V4+ ions form pairs along the c axis, leading to alternate short and long V-V distances of 2.65 Angström and 3.12 Angström. In the tetragonal form, the V-V distance is 2.96 Angström. Therefore size of the unit cell for the monoclinic form is 2 times larger than for the tetragonal form. At Tc IMT takes place. The optical band gap of VO2 in the low-temperature monoclinic phase is about 0.7 eV.
  3. Remarkably, the metallic VO2 contradicts the Wiedemann Franz law, which states that the ratio of the electronic contribution of the thermal conductivity (κ) to the electrical conductivity (σ) of a metal is proportional to the temperature. The thermal conductivity that could be attributed to electron movement was 10 per cent of the amount predicted by the Wiedemann Franz law. That the conductivity is 10 times higher than expected, suggests that the mechanism of conductivity is not the usual one.

    Semiconductor property below Tc suggests that a local phase transition modifying the lattice structure from monoclinic to tetragonal takes place at the current path in the incubation.

One can try to understand the chemistry and unconventional conductivity of VO2 in the TGD framework.
  1. Vanadium could give 4 valence electrons to O2: 3 electrons d3:sta and one from s2. In the TGD Universe, the second electron from s2 could become dark and go to the bond between V4+ ions in the VO2 lattice and take the role of conduction electron.
  2. This could explain the non-conventional character of conductivity. In the semiconductor phase, an electric voltage pulse or some other perturbation, such as impurity atoms or heating, can provide the energy needed to increase the value of heff. Electric conductivity could be due to the transformation of electrons to dark electrons possibly forming Cooper pairs at the flux tube pairs connecting V4+ ions or their pairs. The current would run along the flux tubes as a dark current.
  3. In a semi-conducting (insulating) state, the flux tube pairs connecting V4+ ions would be relatively short. The voltage pulse inducing a local metallic state could provide the energy needed to increase heff and thus the quantum coherence scale. This would be accompanied by a reconnection of the short flux tube pairs to longer flux tube pairs serving as bridges along which the dark current could run.

    One can also consider U-shaped closed flux tubes associated with V4+ ions or ion pairs, which reconnect in IMT to longer flux tubes. The mechanism would be very similar to that proposed for the transition to high temperature superconductivity (see this, this, and this).

Experimenters suggest a glass type behavior.
  1. Spin glass corresponds to the existence of a very large number of free energy minima in the energy landscape implying breaking of ergodicity. A system consisting of regions with varying direction of magnetization is the basic example of spin glass. In the recent case, decomposition to metallic and insulating regions could define the spin glass.
  2. TGD predicts the possibility of spin glass type behavior and leads to a model for spin glasses (see this). The quantum counterpart of spin glass behavior would be realized in terms of monopole flux tube structures (magnetic bodies) carrying dark phases of the radinary particles such as electrons serving as current carries in the metallic phase.The length of the flux tube pair would be one critical parameter near Tc. Quantum criticality against the change of heff increasing the length of the flux tube pair by reconnection would make the system very sensitive to perturbations.
  3. These phases are highly sensitive to external perturbations and represent in TGD inspired theory of consciousness higher levels with longer quantum coherence scale and number theoretical complexity measured by the dimension n= heff/h0 of the extension having interpretation as a kind of IQ. These phases would receive sensory information from lower levels of the hierarchy with smaller values of n and control them.

    The large number of free energy minima as a correlate for number theoretical complexity would make possible the representation of "sensory" information as "memories".

See the article TGD and Condensed Matter or the chapter with the same title.

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

Articles related to TGD.

Wednesday, November 02, 2022

More anomalies related to the standard model of galaxy formation

Various anomalies associated with the ΛCDM model assuming that dark matter forms halos and with the general view of galaxy formation have been accumulating rapidly during years. The MOND model assumes no dark mass but modifies Newton's law of gravitation and is inconsistent with the Equivalence Principle. In TGD, the halo is replaced with a cosmic string and the Equivalence Principle and Newtonian gravitation survive. Both MOND and TGD can handle these anomalies because there is no dark mass halo.

In this article, three new anomalies disfavoring ΛCDM but consistent with MOND and TGD are discussed. There are too many thin disk galaxies, dwarf galaxies do not have dark matter halos, and tidal tails associated with star clusters are asymmetric.

See the article More anomalies related to the standard model of galaxy formation

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

Articles related to TGD.<

Too many thin disk galaxies

The title of the article "Breaking Cosmology: Too Many Disk Galaxies – “A Significant Discrepancy Between Prediction and Reality” (see this) describes quite well the situation in the cosmology. The views of the dynamics of galaxies seem to be wrong.

In the current study (see this, Pavel Kroupa’s doctoral student, Moritz Haslbauer, led an international research group to investigate the evolution of the universe using the latest supercomputer simulations. The calculations are based on the Standard Model of Cosmology; they show which galaxies should have formed by today if this theory were correct. The researchers then compared their results with what is currently probably the most accurate observational data of the real Universe visible from Earth.

It is found that the fraction of disk galaxies is much larger than predicted. This suggests that the morphology of disk galaxies is very slowly changing and mergers of galaxies, favoured by dark matter halos, are not so important as though in the dynamics of galaxies. The cold dark matter scenario predicts spherical halos, which does not fit well with the large fraction of disk galaxies. The MOND approach is favored because there are no dark matter halos favoring spherical galaxies and mergers.

In the TGD framework (see this, this, and this), dark matter or rather, dark energy would be associated with what I call cosmic strings so that halos are absent. Cosmic strings are extremely massive and create a 1/ρ transversal gravitational field, which explains the flat velocity spectrum of distant stars automatically.

The orbits of stars are helical since there is free motion in the direction of a long string. This strongly favors the formation of disk galaxies with the plane of the disk orthogonal to the string and correlation between the normals of the disks along the long cosmic string. In accordance with the findings, the concentration of matter at the galactic plane is very natural in the TGD framework.

The intersections of the string-like objects moving at 3-surface are topologically unavoidable and one can ask whether the galaxies are formed as two cosmic string intersect and the resulting perturbation induces their thickening leading to the transformation of the dark energy of the cosmic string to ordinary matter. This would be an analogy for the decay of an inflaton field to ordinary matter.

See the article More anomalies related to the standard model of galaxy formation.

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

Articles related to TGD.<

Monday, October 31, 2022

The asymmetry of tidal tails as a support for the TGD view of dark matter

The most recent puzzling discovery related to the galactic dynamics is that for certain star clusters associated with tidal tails there is an asymmetry with respect to the direction of the motion along the tail (see this). The trailing tail directed to the galactic nucleus is thin and the leading tail is thick and there are many more stars in it. Stars also tend to leak out along the direction of motion along the tail. One would not expect this kind of asymmetry in the Newtonian theory since the contribution of the ordinary galactic matter to the gravitational potential possibly causing the asymmetry is rather small.

MOND theory (see this) is reported to explain the finding satisfactorily.

  1. The tidal tails of the star cluster are directed towards (leading tail) and outwards from it (trailing tail). The standard explanation is that gravitational forces produce them as a purely gravitational effect. These tails can be however often thin and long, which has raised suspicions concerning this explanation.
  2. MOND hypothesis assumes that gravitational acceleration starts to transform above some critical radius from 1/r2 form to 1/r form. This applies to galaxies and star clusters modelled as a point-like object. This idea is realized in terms of a non-linear variant of the Poisson equation by introducing a coefficient μ(a/a0) depending on the ratio a/a0 of the strength of gravitational acceleration a expressible as gradient of the gravitational potential. a0 is the critical acceleration appearing as a fundamental constant in the MOND model. μ approaches unity at large accelerations and a linear function of a/a0 at small accelerations. Note that MOND violates the Equivalence Principle.
  3. For MOND, the effective gravitational potential of the galactic nucleus becomes logarithmic. Therefore the outwards escape velocity in the trailing tail is higher than the inwards escape velocity in the leading tail so that the stars tend to be reflected back from the trailing tail. This would cause tidal asymmetry implying that the tail directed to the galactic nucleus contains more stars than the outwards tail. The MOND model uses the effective gravitational mass of the galaxy to model the situation in a quasi-Newtonian way.
TGD allows us to consider both the variant of the MOND model. The model provides also a possible explanation for the formation of the star cluster itself.
  1. In the TGD framework, cosmic strings are expected to form a network (see this, this, and this). In particular, one can assign to the tidal tails a cosmic string oriented towards the galactic nucleus, call it Lt to distinguish it from the long cosmic string along L along which galaxies are located. The thickening of a long string and the associated formation of a tangle generates ordinary matter as the dark energy of the string transforms to ordinary matter. This is the TGD counterpart for the transformation of the energy of an inflaton field to ordinary matter.

    This process can occur for both the galactic string L and Lt. In the first case it would give rise to galaxies along L and in the case of Lt to the formation of star clusters. Unlike in MOND, the gravitation remains Newtonian and the Equivalence Principle is satisfied in TGD.

  2. The long cosmic string L along which the galaxies are located gives an additive logarithmic contribution to the total gravitational potential of the galaxy. This contribution explains the flat velocity spectrum of distant stars.

    At some critical distance, the contribution of L begins to dominate over the contribution of ordinary matter. The critical acceleration of the MOND model is replaced with the value of acceleration at which this occurs. In contrast to MOND, this acceleration is not a universal constant and depends on the mass of the visible part of the galaxy. TGD predicts a preferred plane for the galaxy and free motion in the direction of the cosmic string orthogonal to it. Also the absence of dark matter halo is predicted.

  3. Concerning the formation of the tidal tails, the simplest TGD based model is very much the same as the MOND model except that one has 2-D logarithmic gravitational potential of string rather than modification of the ordinary 3-D gravitational potential of the galaxy. Therefore TGD allows a very similar model at qualitative level.
One can however challenge the assumption that the mechanism is purely gravitational.
  1. The tidal tails tend to have a linear structure. Could they correspond to linear structures, long strings or tentacles extending towards the galactic nucleus? Could the formation of star clusters itself be a process, which is analogous to the formation of galaxies as a thickening of cosmic string leading to formation of a flux tube tangle?
  2. Why more stars at the rear end rather than the frontal end of the moving star cluster? Could one have a phase transition transforming dark energy to matter proceeding along the cosmic Lt string rather than a star cluster moving? Dark energy would burn to ordinary matter and give rise to the star cluster.
  3. The burning could proceed in both directions or in a single direction only. If the burning proceeds outwards from the galactic nucleus, the star formation is just beginning at the trailing end. In the leading end, the tangle formed by cosmic string has expanded and stretched due to the reduction of string tension. This could explain the asymmetry between trailing and leading ends at least partially.

    If the burning proceeds both outwards and inwards, only the MOND type explanation remains.

  4. Second asymmetry is that the stars tend to leak out along the direction of motion. The gravitational field of the galaxy containing the logarithmic contribution explains this at least partially. Long cosmic string Lt creates a transversal gravitational field and this could strengthen this tendency. The motion along Lt is free so that the stars tend to leak out from the system along the direction of Lt.
See the chapter TGD and Astrophysics.

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

Articles related to TGD.

Evidence for quantum brain

The recent findings suggest quantum coherence in the brain scale. The quantum coherence would make itself visible in the magnetic resonance imaging (MRI). The findings are described in the popular article in Scitechdaily (see this). The research article "Experimental indications of non-classical brain functions" by Christian Matthias Kerskens and David Lopez Perez is published in Journal of Physics Communications (see this).

The system studied is the brain and cyclotron resonance of protons in "brain water" is involved. The goal was to find whether there exists evidence for macroscopic quantum entanglement. The work was based on the proposal that some quantum coherent, non-classical, third party, say quantum gravitation, could mediate quantum entanglement between protons of brain water. NMR methods based on so-called multiple quantum coherence (MQC) act as an entanglement witness.

One source of theoretical inspiration for the work of Kerskens and Perez was the article "Spin Entanglement Witness for Quantum Gravity" of Bose et al (see this).

In the proposal of Bose et al for generating entanglement by quantum gravitational interaction between mesoscopic objects a superposition of two locations for the objects is required. It is assumed that it is possible to correlate the locations with spin values. Entanglement would be generated by different phases, which evolve to different pairs of components of objects and measurement of spin would demonstrate the presence of entanglement.

Mechanisms generating quantum coherence in scales of at least 10 meters and giving rise to a superposition of locations are needed but are difficult to imagine in the standard view of quantum gravitation.

In TGD, the mechanism would be different. Gravitational Planck constant ℏgr= Gm/v0 associated with Earth-test particle interaction could generate quantum coherence in even brain scale and gravitational Compton length Λgr= GM/v0 ∼ .45 meters, where v0∼ c a velocity parameter characterizes the lower bound for the quantum gravitational coherence scale. The analogs of magnetized states assignable to microscopic objects of size scale 10-4 meters take the role of spins and spin-spin interaction generates the entanglement, which is detected by measuring the spin of either object just as in the case of ordinary spins.

Classical interactions, be their gauge or gravitational interactions, cannot generate entanglement whereas their quantum counterparts do so in scales smaller than the scale of quantum coherence.

  1. The first open question is whether quantum gravitation is able to generate quantum coherence in long length scales such as the scale of the brain. The fact that gravitation has infinite range and is unscreened might allow this. This however requires a new view of quantum gravitation.

    A gravitational 2-particle interaction or interaction induced by quantum gravitation is needed to entangle the systems. If spins or possibly magnetizations are in question, the entanglement can be detected by spin measurements as done in the experiment. The interaction must be such that it can be distinguished from ordinary magnetic interactions.

  2. If objects with mass above Planck mass behave like quantum coherent particles with respect to quantum gravitation rather than consisting of small quantum coherent units such as elementary particles, the gravitational fine structure constant αgr=GM1M2/ℏ between objects satisfying M1M2>mPl2 becomes strong and one expects that the situation becomes non-perturbative.

    The condition M1= M2= mPl is satisfied for a water blob of radius ≈ 10-4 meters and corresponds to the size of a large neuron (see this and this). The gravitational interaction energy GM1M2/d for distance d≈ 10-4 m is about 10-2 eV and of the same order of magnitude as thermal energy.

  3. In the interferometer experiment a much larger phase difference could be generated in the TGD framework but the problem is that it is difficult to imagine a mechanism for creating a superposition of 2 locations of mesoscopic or even microscopic objects.
  4. It is also difficult to imagine a mechanism creating 1-1 correlation between location and spin direction (analogous to entanglement associated with spin and angular momentum).
The notion of gravitational Planck constant

The basic problem is what makes the quantum coherence scale so long.

  1. In the TGD framework, the non-perturbative character motivates a generalization of the Nottale's hypothesis stating that the gravitational Planck constant ℏgr= GMm/v0, v0<c a velocity parameter. ℏeff=nh0=ℏgr would be associated with gravitational flux tubes to which interacting masses M and m are attached, and would replace ℏ with the gravitational fine structure constant αgr= GMm/ℏ >1 meaning that Mm>mPl2 is true. One could say that Nature is theoretician friendly and makes perturbation theory possible. This applies also to other interactions.

    The gravitational Compton length Λgr= GM/v0 does not depend on the mass m at all. For the mass of order Planck mass assignable to a large neuron one has Λgr=LPl/v0, which is of order Planck length. Much longer quantum coherence scale is however required.

  2. In the case of the Earth, the basic gravitationally interacting pairs would be Earth mass and particles of various masses. The gravitational Compton length Λgr,E= GME/v0 does not depend on the small mass and is about .45 cm for v0∼ c favored by TGD applications. By the way, this scale corresponds to the size of a snowflake (see this).

    Λgr,E∼ .45 cm defines a minimum value for the gravitational quantum coherence scale but much larger coherence lengths, say of order Earth radius, are possible. The size scale of the brain or even body would define a natural scale of quantum coherence. For objects with a size of order of a large neuron, the gravitational interaction could be quantal in scales of the brain, and actually in the scales of the magnetic bodies assignable to the organism.

  3. Earth-particle interactions can induce quantum coherence in the scale of the brain and the masses could be taken to be of the order of Planck mass so that they would correspond to water blob with size of 10, so that their distance could be larger than d. This raises the hope that the effects of quantum gravitation quantum coherent in cell length scale or even longer scales could be measured although the interaction itself is extremely weak for elementary particles.
  4. For r= 10-4 meters, M=M_E would give E∼ e2/4 × 102 eV ∼ 2.5 eV. For r=5× 10-4 meters this would give E≈ .01 eV, roughly the thermal energy at the physiological temperature.
TGD allows the possibility of detecting gravitational interaction energies for objects of mass of say Planck mass or larger. In fact, the large value of gravitational Planck constant increases the extremely tiny cyclotron energies of ELF photons in EEG range to energies above thermal energy at room temperature (see this, this, this and this).

A possible TGD based mechanism generating spin entanglement

These considerations suggest a TGD based mechanism for the generation of spin entanglement, which is not directly based on quantum gravitational interaction but on microscopic and even macroscopic qravitionally induced quantum coherence making possible a generalization of the spin-spin interaction as a way to generate entanglement.

  1. "Spin" should correspond to an analogy of macroscopic magnetization rather than to individual spin. Spin-spin interaction between "mesoscopic" quantum coherent particles characterized by ℏgr and having mass about Planck mass generates the entanglement, which can be detected by measuring the "spin" of either particle. As a consequence also the "spin" of the other particles is determined and one has a standard situation demonstrating that the particles were entangled before the measurement.

    Large value of the energy due to the large value of ℏgr could mean that one has a dark Bose-Einstein condensate like state with a large number of ordinary particles, say protons at the gravitational flux tube representing the quantal magnet behaving like spin.

    In the TGD framework, Galois confinement provides a universal mechanism for the formation of many-particle bound states from virtual particles with possibly momenta with components in an extension of rationals . The total momentum would have integer components using the unit defined by the size scale of causal diamond (CD).

  2. The dark cyclotron energy Ec=ℏgreB/m= ΛgreB, Λgr= GM/v0 of a mesoscopic particle whose particles are associated with (touching) the dark monopole flux tubes of the Earth's gravitational field, does not depend on its mass and is large.

    The magnetic field created by this kind of particle would correspond in the Maxwellian picture to a field B ∝ ℏgre/mr3. This would give for the magnetic interaction energy of the mesoscopic particles the estimate E≈ μ1 μ2/r3 = e2 Λgr2/r3.

See the article Evidence for Quantum Brain or the chapter Magnetic Sensory Canvas Hypothesis.

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

Articles related to TGD.

Tuesday, October 25, 2022

How also finite fields could define fundamental number fields in Quantum TGD?

How also finite fields could define fundamental number fields in Quantum TGD?

One can represent two objections against the number theoretic vision.

  1. The first problem is related to the physical interpretation of the number theoretic vision. The ramified primes pram dividing the discriminant of the rational polynomial P have a physical interpretation as p-adic primes defining p-adic length- and mass scales.

    The problem is that without further assumptions they do not correlate at all with the degree n of P. However, physical intuition suggests that they should depend on the degree of P so that a small degree n implying a low algebraic complexity should correspond to small ramified primes. This is achieved if the coefficients of P are smaller than n and thus involve only prime factors p<n.

  2. All number fields except finite fields, that is rationals and their extension, p-adic numbers and their extensions, reals, complex numbers, quaternions, and octonsions appear at the fundamental level in TGD. Could there be a manner to make also finite fields a natural part of TGD?
These problems raise the question of whether one could pose additional conditions to the polynomials P of degree n defining 4-surfaces in M8 with roots defining mass shells in M4⊂ M8 (complexification assumed) mapped by M8-H duality to space-time surfaces in H.

1. P=Q condition

One such condition was proposed here. The proposal is that infinite primes forming a hierarchy are central for quantum TGD. It is proposed that the notion of infinite prime generalizes to that of the notion of adele.

  1. Infinite primes at the lowest level of the hierarchy correspond to polynomials of single variable x replaced with the product X=∏p p of all finite primes. The coefficients of the polynomial do not have common prime divisors. At higher levels, one has polynomials of several variables satisfying analogous conditions.
  2. The notion of infinite prime generalizes and one can replace the argument x with Hilbert space,group representation, or algebra and sum and product of ordinary arithmetics with direct sum ⊕ and tensor product ⊗.
  3. The proposal is P=Q: at the lowest level of the hierarchy, the polynomial P(x) defining a space-time surface corresponds to an infinite prime determined by a polynomial Q(X). This would be one realization of quantum classical correspondence. This gives strong constraints to the space-time surface and one might speak of the analog of preferred extremal (PE) at the level of M8 but does not yet give any special role for the finite fields.
  4. The infinite primes at the higher level of the hierarchies correspond to polynomials Q(x1,x2,...,xk) of several variables. How to assign a polynomial of a single argument and thus a 4-surface to Q? One possibility is that one does as in the case of multiple poly-zeta and performs a multiple residue integral around the pole at infinity and obtains a finite result. The remaining polynomial would define the space-time surface.
2. Additional conditions

The speculations related to the p-adicization of ξ inspire the following questions.

  1. Option I: Rational polynomial is apart from scaling a polynomial with integer coefficients having the same roots. Could it make sense to assume that the coefficients of the P(x)= Q(x) of degree n are integers divisible only by primes p<n?
  2. Option II: A stronger condition would be that the integer coefficients of P=Q are smaller than n. This implies that they are divisible by primes p<n, which cannot however appear as common factors of the coefficients. One could say that the corresponding space-time sheet effectively lives in the ring Zn instead of integers. For prime value n=p space-time sheet would effectively "live" the finite field Fp and finite fields would gain a fundamental status in the structure of TGD.

    Should one allow both signs for the coefficients as the interpretation as rationals would suggest? In this case, finite field interpretation would mean the replacement of -1 with p-1.

  3. Option III: A still stronger, perhaps too strong, condition would be that only the prime factors of n appear as factors of the coefficients of P=Q. For integers n with a small number of prime divisors it is easy to find the possible coefficients. For instance, for n= p all coefficients are equal to 1 or 0!

    For n=p1p2, two of the coefficients can be equal to power of p1 or p2 if smaller than n and remaining coefficients equal to 1 or 0. For instance, n=p1p2 for p1=M127=2127-1 and p2=2, one coefficient could be M127, second coefficient power of 2 smaller than 2127 and the remaining coefficients would be equal to 1 or 0.

Option II would solve the two problems whereas Option II is un-necessarily strong.
  1. For n=p, P would make sense in a finite field Fp if the second condition is true. Finite fields, which have been missing from the hierarchy of numbers fields, would find a natural place in TGD if this condition holds true!
  2. The number of polynomial coefficients is n, whereas the number of primes smaller than n behaves as n/log(n). By infinite prime property, the coefficients would not contain common primes p<n. Very few polynomials could define space-time surfaces.
3. How does Option II relate to prime polynomials?

  1. The degree of a composite of polynomials with orders m and n is mn so that a polynomial with prime degree p does not allow expression as a composite of polynomials of lower orders so that any polynomials with prime order is a prime polynomial. Polynomials of order m can in principle be functional composites of prime polynomials with orders, which are prime factors of m.

    Obviously, all prime polynomials cannot satisfy Option II. However, those satisfying Option II could be prime polynomials. Note that the polynomials, which have an interpretation in terms of a finite field Fp have degree p-1.

  2. There are also non-prime polynomials satisfying Option II. P1=xm and P2= xn satisfy Option II as also the composite P= xmn, which is however not a prime polynomial. The composite of P1= x2 and P2= 1+xm gives P= 1+2xm+x2m, which satisfies Option II but is not prime. By the symmetry B(n,k)= B(n,n-k) of binomial coefficients the composite of P1=xm, m>2, and P2= 1+xm does not satisfy the conditions.
  3. Quite generally, polynomials P satisfying Option II and having degree n, which is not prime, can decompose to prime polynomials and probably do so. There the polynomial primeness and Option II do not seem to have a simple relationship.
These observations suggest the tightening of the Option II to the following condition.

All physically allowed polynomials P are functional composites of the prime polynomials of prime degree satisfying Option II. In a rather precise sense, finite fields would serve as basic building blocks of the Universe.

4. Examples of Option II

The following examples illustrate the conditions for Option II.

  1. For instance, for M127=2127-1 assigned with electron by p-adic mass calculations one has n/log(n)∼ M127/log(2)127 ∼ M127/88 so that only about 12 percent of coefficients of P could differ from 0 or 1.
  2. For small values of n it is easy to construct the possible polynomials P.
    1. For n=p=2 one obtains only the coefficients (p0,p1)⊂ {+/- 1,0},{0,+/- 1},{+/- 1,+/- 1} corresponding to P(x)∈{+/- 1,+/- x,+/- 1 +/- x}.
    2. For n=p=3, one of the coefficients is p=2 and the remaining coefficients are equal to 1 or 0. The coefficients are (p0,p1,p2)⊂ {+/- 2,x,y}, {x,+/- 2,y}, {x,y,+/- 2} with x,y ∈{0,1,-1} and (p0,p1,p2) with pi∈ {0,1,-1}.

      A little calculation shows that extensions of rationals containing i, 21/2, i21/2, 31/2, i31/2, 51/2 (from P=x2+x-1 defining Golden Mean), and i71/2 are obtained.

    3. Roots of small primes appear in the Weyl groups, which are reflection groups associated with Dynkin diagrams characterizing Lie groups at Lie algebra level. The finite discrete subgroups of the rotation group SU(2) characterized extensions of hyper-finite factors of type II1 and roots of small primes appear in the matrix elements of these groups. Could the proposed polynomials give in a natural way rise to the extensions of rationals appearing in these two cases?
The above considerations inspire further questions. Could one also allow polynomials P having coefficients in an algebraic extension of rationals? Does this bring in anything new? Could one have coefficients in an extension containing e or even root of e as perhaps the only transcendental extension defining a finite extension of p-adic numbers? The roots would be generalizations of algebraic numbers involving e and could make sense p-adically via Taylor expansion.

See the article New Ideas Related to Langlands Program viz. TGD or the chapter with the same title.

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

Articles related to TGD.

Monday, October 24, 2022

Shocking support for the quantum brain in TGD sense

I learned about findings suggesting quantum coherence in the brain scale. The quantum coherence would make itself visible in the magnetic resonance imaging (MRI). The findings are described in the popular article in Scitechdaily. The research article "Experimental indications of non-classical brain functions" by Christian Matthias Kerskens and David Lopez Perez is published in Journal of Physics Communications, Volume 6, Number 10.

The system studied is the brain and cyclotron resonance of protons in "brain water" is involved. The goal was to find whether there exists evidence for macroscopic quantum entanglement. The work was based on the proposal that some quantum coherent, non-classical, third party, say quantum gravitation, could mediate quantum entanglement between protons of brain water. NMR methods based on so-called multiple quantum coherence (MQC) act as an entanglement witness.

It is far from clear that the ordinary NMR signals can contain quantum correlations of the spectrum in the hot and wet brain environment. Therefore a witness protocol, which eliminated the "classical" background from known sources was used.

To achieve this the "classical" sources of entanglement had to be eliminated. This was achieved by irradiation of the brain region with a radiation inducing cyclotron transitions to higher energy state so that the situation would become saturated and one would have a statistical dynamic equilibrium. In a statistical sense, the temporal patterns associated with the transitions from a higher state to a lower state causing cyclotron radiation patterns visible in MRI would be absent. In this back-ground the presence of "non-classical" sources of cyclotron emission would be visible. This source could correspond to a formation of pure entangled state which would decay by emitting cyclotron radiation.

What was found, was a periodic pattern in MRI with a frequency of heart beat, interpreted in terms of evoked membrane potentials. This pattern is too weak to be visible in the ordinary MRI. What looks surprising is that the frequency was that of heart beat; one would expect some resonance frequency of EEG, say 10 Hz.

The finding fits very nicely with the TGD view of brain and quantum biology, in particular the TGD view of genetic code (see this, this, this, and this).

  1. In the simplest model, sequences of dark protons (ordinary protons with effective Planck constant heff=nh0, which can be very large) at the flux tubes of the magnetic body associated with DNA would realize genetic code as sequences of dark proton triplets. Besides dark nucleotides, also dark codons and dark genes as quantum coherent dark 3N-protons would be possible and characterized by very large value of heff=hgr=GMmv/v0.

    Also dark photon triplets would realize codons and give rise to dark genes as sequences of dark codons: 3N-photons. Communications between dark genes and would occur using dark 3N-photopns by dark 3N-resonance. The 3N-frequency would serve as an address somewhat like in LISP and the modulation of frequency scale would create a sequence of resonances analogous to sequence of nerve pulses.

    EEG would closely relate to the dark photon radiation between the magnetic body and brain. Also generalizations of EEG to other frequency ranges are suggestive.

  2. The dark magnetic flux tubes would be associated with water and its numerous thermodynamic anomalies and exceptional role in biology, could be understood by the presence of a dark phase involving long gravitational flux tubes carrying dark protons with heff=hgr.
  3. The transformation of dark photons (or even dark 3N-photons) from dark DNA to ordinary photons could generate cyclotron photons giving an unexpected contribution to the MRI spectrum. Quite generally, biophotons could result from transitions of this kind.
  4. In MRI (see this) the cyclotron transitions occur in a magnetic field of few Tesla. For protons this corresponds to a cyclotron frequency ∼1.2× 108 Hz in radio frequency range. The associated classical radiation field is detected by a resonance coil.

    The cyclotron frequency ∼1.2× 108 Hz does not correspond to the cyclotron frequency of order 1 Hz assigned with dark DNA, which happens to correspond to the frequency of heart beat. The cyclotron emission at radio frequency would be modulated by the dark photon emission associated with heart beat and dark photon cyclotron frequency should act as a driving frequency for the heart beat.

  5. The quantum correlations due to the large value of hgr, which corresponds to gravitational Compton length Λgr = GM/v0 is for Earth mass and for v0/c∼ 1 about .45 cm. In accordance with the Equivalence Principle, Λgr has no dependence on the "small" mass m , and also cyclotron energy is independent of m. Interestingly, Λgr happens to correspond to the size scale of a snowflake (see this), which has a scaled up variant of the symmetry of the unit cell of 2-D hexagonal layer of ice. This remains a mystery in the standard physics framework.
The striking finding is that the MRI pulse frequency corresponds to that of heart beat, which is between 1-1.67 Hz and .67 Hz for a trained athlete. Why not some EEG resonance frequency such as alpha frequency about 10 Hz?

The TGD view of cyclotron resonances for dark ions can explain this.

  1. The findings of Blackman et al, which led to the TGD view of dark matter, which later was deduced from number theoretic physics, demonstrated that in the case of vertebrates, the radiation of brain at ELF frequencies has completely unexpected quantal effects on both brain physiology and behavior.

    Cyclotron resonance hypothesis (see this and this) states that the irradiation induces a cyclotron resonance. For instance, the effects occur at frequencies which are multiples of Ca++ cyclotron frequency in an "endogenous" magnetic field Bend∼ 2/5 BE, with BE as the strength of the magnetic field of Earth has nominal value of .5 Gauss.

  2. The problem is that the effects should be extremely small since the cyclotron energy is more than 10 orders of magnitude below thermal energy at physiological temperatures. This problem led to the hypothesis that Planck constant as a spectrum and that dark matter could correspond to phases of ordinary matter with effective Planck constant heff = nh0.

    The required values of heff are huge, and this led to a connection with the Nottale hypothesis of gravitational Planck constant ℏgr= GMm/v0, v0≤ c is a velocity parameter. One would have hbareff=hbargr. The value of velocity parameters can be estimated from various applications. It would have a spectrum with the largest value v0/c∼ 1 in the case of Earth with M=ME.

  3. TGD leads also to an identification of Bend. TGD predicts monopole flux tubes (CP2 homology is non-trivial) distinguishing TGD from Maxwellian electrodynamics. Bend=2BE/5 is identified as the monopole flux part of the Earth's magnetic field. The monopole flux tubes would carry dark matter and since they have huge quantum coherence scales, would naturally control ordinary biomatter. The control would involve frequency modulation by the variation of the thickness of the monopole flux tubes which would affect the field strength by the conservation of the monopole flux. The variation of the frequency scale would induce at the end of the receiver sequences of cyclotron resonance analogous to nerve pulse patterns.
  4. Magnetic body of DNA carrying dark DNA is expected to act as controller of the ordinary biomatter using cyclotron resonance mechanism. In particular, important biorhythms could correspond to cyclotron frequencies. Heartbeat defines one such biorhythm.

    DNA nucleotide cyclotron frequencies are about 1 Hz for Bend assigned to the monopole flux tubes. Also for DNA sequences, such as codons and genes, the average cyclotron frequency would be around 1 Hz because the nucleotides carry the same charge and charge to mass ratio Ze/m, so that the cyclotron frequency depends only very weakly on the length of quantum coherent dark DNA segment.

    The variation of the heart beat frequency could be understood in terms of the variation of the monopole flux tube thickness for dark DNA. This variation would be basic motor action of MB making possible control of biomatter using frequency modulation inducing sequences of resonances manifesting as pulses. Nerve pulse patterns could be one manifestation of this mechanism.

See the article Evidence for Quantum Brain or the chapter Magnetic Sensory Canvas Hypothesis.

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

Articles related to TGD.

Friday, October 21, 2022

p-Adic ξ function from Hadamard product

ξ function (see this) is closely related to ζ and is much simpler. In particular, it lacks the trivial zeros forcing to drop from ζ the Euler factor to get ζp. ξ has a very simple representation completely analogous to that for polynomials (see this):

ξ(s)=(1/2)∏k (1-s/sk )(1-s/(sk*) .

Only the non-trivial zeros appear in the product.

  1. For s=O(p), this product is finite but need not converge to a well-defined p-adic number in the infinite extension of p-adic numbers. Also the values of ξ (s) at integer points are known to be transcendental so that the interpretation as a generalization of a rational polynomial fails. Note that the presence of an infinite number terms in the product can cause transcendentality of the coefficients of ξ(s). Algebraic numbers are required. ξ(2n) is proportional to π2 and ξ(2n+1) to ζ(2n+1)/π2. The presence of an infinite number of terms in the expansion of ξ(s) can however cause this.
  2. If ξ is indeed analogous to a rational polynomial, the analog of discriminant

    D=∏k≠ l(sk-sl)(sk-sl*)

    should be proportional to a product of powers of ramified primes for ξ. The simplest option would be that there is complete number theoretic democracy and the ramification is minimal. If so, D would be proportional to the product Y= ∏k pk of all primes, appearing in the definition of infinite primes (see this). This number is infinite in real sense but would have p-adic norms 1/pk.

  3. The Hadamard product can be written in the form

    ξ(s)=(1/2)∏k (1+ s(s-1)/Xk) , Xk= sks*k ,

    in which the s↔ 1-s symmetry is manifest. The power series of ξ(s)== ξ1(u)=∑ an un, u=s(s-1), should converge for all primes p.

    If regard s(s-1) as p-adic number and apply the inverse of I s(s-1) to get real number.

    If the coefficients an of the powers series ∑n an un are algebraic numbers of the assumed extension or ordinary rationals, the power series in s converges for s=O(p) under rather mild conditioons. For instance, the coefficient of the zeroth order term is 1/2. The coefficient of first order term in u is -(1/2)∑k (sks*k) -1= -2∑k (1+4yk2)-1.

    If this sum converges to a well-defined rational if ξ is analogous to a rational polynomial. Same should happen also for the higher coefficients cn. For large values of n the coefficients cn should become integers to guarantee the convergence of the sum for all primes p. A more general condition would be that the sums defining the coefficients give algebraic integers or even transcendentals defining a finite-D extension of rationals with p-adic norm not larger than 1 for the extension defined by zeta.

  4. Algebraic integers are roots of monic polynomials with integer coefficients such that the coefficient of the leading order term is equal to 1. By multiplying ξ by the product Y=∏ sks*k one obtains an expression, which is formally a monic polynomial if one can regard Y as a p-adic integer, which is p-adically finite for every p. Formally this operation does not affect the values of the roots.
One can deduce formal expressions for the Taylor coefficiens of ξ(s).
  1. Taking u=s(s-1) to be the variable, the coefficients of un in ξ(s)= ξ1(u) are given by

    Unk∈ Un Xk-1 , Xk=sks*k .

  2. The calculation of the coefficients cn is simple. In particular, c1 and c2 can written as

    c1= (1/2)∑iXi-1 ,

    c2= (1/2)∑i≠ jXi-1Xj

    =(1/2)∑i,jXi-1Xj -(1/2) ∑iXi-2

    = (1/2)c12 - ∑iXi-2 .

    The calculation reduces to the calculation of sums ∑iXi-k, k=1,2.

  3. Also the higher coefficients cn can be calculated in a similar way recursively by subtracting from the sum ∑i1...inik Xi1-1= c1n without the constraint pi≠ pj≠ ... the sums for which 2,3,...,n primes are identical. One obtains a sum over all partitions of Un. A given partition {i1, ..., ik} contributes to the sum the term

    di1, ... , ikl=1k cil ,

    i=1k ni=n .

    The coefficient di1, ..., ik tells the number of different partitions with same numbers i1, ..., ik of elements, such that the ni elements of the subset correspond to the same prime so that this subset gives cni. Note that the same value of i can appear several times in {i1, ..., ik}.

    The outcome is that the expressions of cn reduce to the calculation of the numbers Ak=∑i Xi-k.

Could one deduce conditions on the coefficients of ξ from number theoretical democracy?

Can one pose additional conditions in the case of ζ or ξ? I have difficulties in avoiding a tendency to bring in some number theoretic mysticism in hope say something interesting of the values of the coefficient Xn in the power series ξ= cnun, u=s(s-1), which can calculated from the Hadamard product representation. Number theoretical democracy between p-adic number fields defines one form of mysticism.

There is however also a real problem involved. There is a highly non-trivial problem involved. One can estimate the real coefficients Xk only as a rational approximation since infinite sums of powers of 1/Xk are involved. The p-adic norm of the approximation is very sensitive to the approximation.

Therefore it seems that one must pose additional conditions and the conditions should be such that the coefficients are mapped to numbers in extension of p-adic numbers by the inverse of I as such so that they should be algebraic numbers or even transcendentals in a finite-D transcendental extension of rationals, if such exists.

  1. One could argue that the coefficients cn must obey a number theoretical democracy, which would mean that they can distinguish p-adically only between the set of primes pk appearing as divisors of n and the remaining primes. One could require that cn is a number in a finite-D extension of rationals involving only rational primes dividing n.
  2. One could pose an even stronger condition: the coefficients cn must belong to an n-D algebraic extension of rationals and thus be determined by a polynomial of degree n. Polynomials P of rational coefficients pn bring in failure of the number theoretic democracy unless one has pn∈ {0,+/- 1}. For p=2 one does not obtain algebraic numbers. For p=3 this would bring in 51/2.
  3. These conditions would guarantee that for a given prime p the coefficients of the expansion would be unaffected by the canonical identification I and at the limit p→ &infty; the Taylor coefficients of p-adic ξp would be identical with those of ξ.
  4. One could allow finite-D transcendental extensions of p-adic numbers. These exist. Since ep is an ordinary p-adic number, there is an infinite number of extensions with a basis given by the powers roots ek/n, k=1,..., np-1 define a finite-D transcendental extension of p-adics for every prime p.

    The strongest hypothesis is that the coefficients ck are expressible solely as polynomials of this kind of extensions with coefficients, which are algebraic numbers of integers in an extension of rationals by a k:th order polynomial Pk, whose coefficients belong to {0,+/- 1}.

This picture suggests a connection with the hyperbolic geometry H2 of the upper half-plane, which is associated with ζ and ξ via Langlands correspondence.
  1. The simplest option is that the roots of Pk correspond to the k:th roots xi of unity satisfying xik=1 so that cos(n2π/k) and sin(n2π/k) would appear as coefficients in the expression of ck. The numbers ek/n would be hyperbolic counterparts for the roots of unity.
  2. The coefficients ck would be of form

    ck= ∑i,jck,ijei/k e(-1) 1/2 2π (j/n) ,

    ck,rs ∈ {0,+/- 1} .

    The coefficients could be seen as Mellin-Fourier transforms of functions defined in a discretized hyperbolic space H2 defined by 2-D mass shell with coordinates (cosh(η), sinh(η)cos(phi), sinh(η)sin(φ)), η = i/k, φ= 2π j/n. η is the hyperbolic angle defining the Lorentz boost to get the momentum from rest momentum and φ defines the direction of space-like part of the momentum. Upper complex plane defines another representation of H2. The values of functions are in the set {0,+/- 1}.

  3. The points of H2 associated with a particular ck would correspond to the orbit of a discrete subgroup of the Lorentz group SO(1,1)× SO(2)⊂ SO(1,2) ⊂ SL(2,R) ( SL(2,R) is the covering of SO(1,2)).

    A good guess is that this discretization could be regarded as a tessellation of H2 and whether other tessellations (there exists an infinite number of them corresponding to discrete subgroups of SL(2,R) could be associated with other L-functions. Riemann zeta is related by Mellin transform to Jacobi theta function (see this) so that SL(2,C), having SL(2,R) as subgroup acting as isometries of H2, is the appropriate group.

  4. The points of H2 associated with a particular ck would correspond to the orbit of a discrete subgroup of SO(1,1)× SO(2)⊂ SO(1,2) ⊂ SL(2,R) (SL(2,R) is the covering of SO(1,2)).

    A good guess is that this discretization could be regarded as a tessellation of H2 and whether other tessellations (there exists an infinite number of them corresponding to discrete subgroups of SL(2,R) could be associated with other L-functions. Mellin transform relates Jacobi theta function (see this), which is a modular form, to 2ξ/s(s-1). Therefore SL(2,C), having SL(2,R) as subgroup acting as isometries of H2, is the appropriate group.

    Note that the modular forms associated with the representations of algebraic subgroups of SL(2,C) defined by finite algebraic extensions of rationals correspond to L-functions analogous to ζ. Now one would have a hyperbolic extension of rationals inducing a finite-D extension of p-adic numbers.

Just for curiosity and to see how the proposal could fail, one can look at what happens for the first coefficient c1 in ξ(s)= ξ1(s(s-1))= ∑ cnsn.
  1. c1 would be exceptional since it cannot depend on any prime. c2 could involve only p=2, and so on.
  2. The only way out of the problem is to allow finite-D transcendental extensions of p-adic numbers. These exist. Since ep is an ordinary p-adic number, there is an infinite number of extensions with a basis given by the powers roots ek/n, k=1,..., np-1 define a finite-D transcendental extension of p-adics for every prime p. For ξ the extension by roots of unity could be infinite-dimensional.

    The roots ek/n, k∈ {1, ..., n} belong to this extension for all primes p and are in this sense universal. One can construct from the powers of ek/n expressions for c1 as c1=∑k ake-k/n, ak ∈{ +/- =0,+/- 1}.

  3. This would allow to get estimates for n using x1=dξ/ds(0)∼ .011547854 =2c1 as an input:

    c1=∑ ake-k/n=x1/2 .

    For instance, the approximation cn= e1- e(n-1)/n would give a rough starting point approximation n ∼ 117. It is of course far from clear whether a reasonably finite value of n can reproduce the approximate value of c1.

See the article Some New Ideas Related to Langlands Program viz. TGD or the chapter with the same title.

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

Articles related to TGD.

Quantum classical correspondence as a feedback loop between the classical space-time level and the quantal WCW level?

Quantum classical correspondence (QCC) has been one of the guidelines in the development of TGD but its precise formulation has been missing. A more precise view of QCC could be that there exists a feedback loop between classical space-time level and quantal "world of classical worlds" (WCW) level. This idea is new and akin to Jack Sarfatti's idea about feedback loop, which he assigned with the conscious experience. The difference between consciousness and cognition at the human resp. elementary particle level could correspond to the difference between L-functions and polynomials.

This vision inspires the question whether the generalization of the number theoretic view of TGD so that besides rational polynomials (subject to some restrictions) also L-functions, which have a nice physical interpretation if RH holds true for them, can be defined via their roots 4-surfaces in M8c and by M8-H duality 4-surfaces in H. Both conformal confinement (in weak and strong form) and Galois confinement (having also weak and strong form) support the view that L-functions are Langlands duals of the partition functions defining quantum states.

If L functions indeed appear as a generalization of polynomials and define space-time surfaces, there must be a very deep reason for this.

  1. The key idea of computationalism is that computers can emulate/mimic each other. Universe should be able to emulate itself. Could WCW level and space-time level mimic each other? If this were the case, it could take place via QCC. If so, it should be possible to assign to a quantum state a space-time surface as its classical space-time correlate and vice versa.
  2. There are several space-time surfaces with a given Galois group but fixing the polynomial P fixes the space-time surface. An interesting possibility is that the observed classical space-time corresponds to superposition of space-time surfaces with the same discretization defined by the extension defined by the polynomial P. If so, the superposition of space-time surfaces would be effectively absent in the measurement resolution used and the quantum world would look classical.
  3. A given polynomial P fixes the mass shells H3 ⊂ M4⊂ M8 but does not fix the space-time surface X4 completely since the polynomial hypothesis says nothing about the intersections of X4 with H3 defining 3-surfaces. The associativity hypothesis for the normal space of X4⊂ M8 (see this and this) implies holography, which fixes X4 to a high degree for a given X3. Holography is not expected to be completely deterministic: this non-determinism is proposed to serve as a correlate for intentionality.

    If space-time has boundaries, the boundaries X2 of X3⊂ H3 could be ends of light-like 3-surfaces X3L (see this). An attractive idea is that they are hyperbolic manifolds or pieces of a tessellation defined by a hyperbolic manifold as the analog of a unit cell (see this). The ends X2 of these 3-surfaces at the boundaries of CD would define partonic 2-surfaces.

    By quantum criticality of the light-like 3-surfaces satisfying det(-g4)=0 (see this), their time evolution is not expected to be completely unique. If the extended conformal invariance of 3-D light-like surfaces is broken to a subgroup with conformal weights, which are multiples of integer n the conformal algebra defines a non-compact group serving as a reductive group allowing extensions of irreps of Galois group to its representations.

    One can also consider space-time surfaces without boundaries. They would define coverings of M4 and there would be several overlapping projections to H3, which would meet along 2-D surfaces as analogies of boundaries of 3-space. Also in this case, the idea that the X3 is a hyperbolic 3-manifold is attractive.

  4. Quantum TGD involves a general mechanism reducing the infinite-D symmetry groups to finite-D groups, which has an interpretation in terms of finite measurement resolution (see this) describable both in terms of inclusions of hyperfinite factors of type II1 and inclusions of extensions of rationals inducing inclusions of cognitive representations. One can also consider an interpretation in terms of symmetry breaking.

    This reduction means that the conformal weights of the generators of the Lie-algebras of these groups have a cutoff so that radial conformal weight associated with the light-like coordinate of δ M4+ is below a maximal value nmax. The generators with conformal weight n>nmax and their commutators with the entire algebra would act like a gauge algebra, whereas for n≤ nmax they generate genuine symmetries. The alternative interpretation is that the gauge symmetry breaks from nmax=0 to nmax>0 by transforming to dynamical symmetry.

    Note that the gauge conditions for the Virasoro algebra and Kac-Moody algebra are assumed to have nmax=0 so that a breaking of conformal invariance would be in question for nmax>0.

  5. The natural expectation is that the representation of the Galois group for these space-time surfaces defines representations in various degrees of freedom in terms of the semi-direct products of the Langlands duals LG0 with the Galois group (here LG0 denotes the connected component of Langlands dual of G). Semi-direct product means that the Galois group acts on the algebraic group G assignable to algebraic extension by affecting the matrix elements of the group element.

    There are several candidates for the group G (see this). G could correspond to a conformal cutoff An of algebra A, which could be the super symplectic algebra SSA of δ M4× CP2, the infinite-D algebra I of isometries of δ M4+, or the algebra Conf extended conformal symmetries of δ M4+. Also the extended conformal algebra and extended Kac-Moody type algebras of H isometries associated with the light-like partonic orbits can be considered.

  6. One could assign to these representations modular forms interpreted as generalized partition functions, kind of complex square roots of thermodynamic partition functions. Quantum TGD can be indeed formally regarded as a complex square root of thermodynamics. This partition function could define a ground state for a space of zero energy state defined in WCW as a superposition over different light-like 3-surfaces.
These considerations boil down to the following questions.
  1. Could the quantum states at WCW level have classical space-time correlates as space-time surfaces, which would be defined by the L-functions associated with the modular forms assignable to finite-D representations of Galois group having a physical interpretation as partition functions?
  2. Could this give rise to a kind of feedback loop representing increasingly higher abstractions as space-time surfaces. This sequence could continue endlessly. This picture brings in mind the hierarchy of infinite primes (see this).

    Many-sheeted space-time would represent a hierarchy of abstractions. The longer the scale of the space-time sheet the higher the level in the hierarchy.

Concerning the concretization of the basic ideas of Langlands program in TGD, the basic principle would be quantum classical correspondence (QCC), which is formulated as a correspondence between the quantum states in WCW characterized by analogs of partition functions as modular forms and classical representations realized as space-time surfaces. L-function as a counter part of the partition function would define as its roots space-time surfaces and these in turn would define via finite-dimensional representations of Galois groups partition functions. Finite-dimensionality in the case of L-functions would have an interpretation as a finite cognitive and measurement resolution. QCC would define a kind of closed loop giving rise to a hierarchy.

If Riemann hypothesis (RH) is true and the roots of L-functions are algebraic numbers, L-functions are in many aspects like rational polynomials and motivate the idea that, besides rationals polynomials, also L-functions could define space-time surfaces as kinds of higher level classical representations of physics.

One concretization of Langlands program would be the extension of the representations of the Galois group to the polynomials P to the representations of reductive groups appearing naturally in the TGD framework. Elementary particle vacuum functionals are defined as modular invariant forms of Teichmüller parameters. Multiple residue integral is proposed as a manner to obtain L-functions defining space-time surfaces.

One challenge is to construct Riemann zeta and the associated ξ function and the Hadamard product leads to a proposal for the Taylor coefficients ck of ξ(s) as a function of s(s-1). One would have ck= ∑i,jck,ijei/ke(-1)1/22πj/n, ck,ij∈ {0,\pm 1}. e1/k is the hyperbolic analogy for a root of unity and defines a finite-D transcendental extension of p-adic numbers and together with n:th roots of unity powers of e1/k define a discrete tessellation of the hyperbolic space H2.

See the article Some New Ideas Related to Langlands Program viz. TGD or the chapter with the same title.

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

Articles related to TGD.