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.

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 B³_r 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 H³_m 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.

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.

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.

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 t_inc, which was t_inc∼ .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 t_inc 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 h_eff 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 T_c=340 K and the tetragonal metallic form above T_c. 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 T_c 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 T_c 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 h_eff. Electric conductivity could be due to the transformation of electrons to dark electrons possibly forming Cooper pairs at the flux tube pairs connecting V^{4+ 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 h_eff 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 T_c. Quantum criticality against the change of h_eff 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= h_eff/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.

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.<

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 L_t 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 L_t. In the first case it would give rise to galaxies along L and in the case of L_t 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 L_t 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 L_t creates a transversal gravitational field and this could strengthen this tendency. The motion along L_t is free so that the stars tend to leak out from the system along the direction of L_t.

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 ℏ_gr= Gm/v₀ associated with Earth-test particle interaction could generate quantum coherence in even brain scale and gravitational Compton length Λ_gr= 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 α_gr=GM₁M₂/ℏ between objects satisfying M₁M₂>m_Pl² becomes strong and one expects that the situation becomes non-perturbative.

   The condition M₁= M₂= m_Pl 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 ℏ_gr= GMm/v₀, v₀<c a velocity parameter. ℏ_eff=nh₀=ℏ_gr 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 α_gr= GMm/ℏ >1 meaning that Mm>m_Pl² 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 Λ_gr= 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 Λ_gr=L_Pl/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 Λ_gr,E= GM_E/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).

   Λ_gr,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 ℏ_gr 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 ℏ_gr 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 E_c=ℏ_greB/m= Λ_greB, Λ_gr= 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 ∝ ℏ_gre/mr³. This would give for the magnetic interaction energy of the mesoscopic particles the estimate E≈ μ₁ μ₂/r³ = e² Λ_gr²/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.