Fractons and TGD

In Quanta Magazine there was a highly interesting article about entities known as fractons (see this).

There seems to be two different views about fractons as one learns by going to Wikipedia. Fracton can be regarded as a as self-similar particle-like entity (see this or as "sub-dimensional" particle unable to move in isolation (see this). I do not understand the motivation for "sub-dimensional". It is also unclear whether the two notions are related. The popular article assigns to the fractons both the fractal character and the inability to move in isolation.

The basic idea is however that discrete translational symmetry is replaced with a discrete scaling invariance. The analog of lattice which is invariant under discrete translations is fractal invariant under discrete scalings.

One can also consider the possibility that the time evolution operator would act as scaling rather than translation. This is something totally new from quantum field theory (QFT) point of view. In QFTs energy corresponds to time translational symmetry and Hamiltonian generates infinitesimal translations. In string models the analog of stringy Hamiltonian is the infinitesimal scaling operator, Virasoro generator L₀.

In TGD the extension of physics to adelic physics provides number theoretic and geometric descriptions as dual descriptions of physics (see for instance this, this, and this). This approach also provides insights about fractons as scale invariant entities and.

1. In TGD the analog of time evolution between "small" state function reductions is the exponent of the infinitesimal scaling operator, Virasoro generator L₀. One could imagine fractals as states invariant under discrete scalings defined by the exponential of L₀. They would be counterparts of lattices but realized at the level of space-time surfaces having quite concrete fractal structure.
2. In p-adic mass calculations the p-adic analog of thermodynamics for L₀ proportional to mass squared operator M² replaces energy. This approach is the counterpart of the Higgs mechanism which allows only to reproduce masses but does not predict them. I carried out the calculations already around 1995 and the predictions were amazingly successful and eventually led to what I call adelic physics fusing real and various p-adic physics (see this).
3. Long range coherence and absence of thermal equilibrium are also mentioned as properties of fractons (at least those of the first kind). Long range coherence could be due to the predicted hierarchy of Planck constants h_eff=n×h₀ assigned with dark matter and predicting quantum coherence in arbitrarily long scales and associated with what I called magnetic bodies.

   If translations are replaced by discrete scalings, the analogs of thermodynam equilibria would be possible for L₀ rather than energy. Fractals would be the analogs of thermodynamic equilibria. In p-adic thermodynamic elementary particles are thermodynamic equilibria for L₀ but it is not clear whether the analogy with fractal analog of a plane wave in lattice makes sense.

Fractons are also reported to be able to move only in combinations. This need not relate to the scaling invariance. What this actually means, remained unclear to me from the explanation. What comes to mind is color confinement: free quarks are not possible. Quarks are unable to exist as isolated entities, not only to move as in isolated entities.

In TGD number theoretical vision leads to the notion of Galois confinement analogous to color confinement. The Galois group of a given extension of rationals indeed acts as a symmetry at space-time level. In TGD inspired biology Galois groups would play a fundamental role. For instance, dark analogs of genetic codons, codon pairs, and genes would be singlets (invariant) under an appropriate Galois group and therefore behave as a single quantum coherent dynamical and informational unit. See (see this and this) .

Suppose that one has a system - say a fractal analog of a lattice consisting of Galois singlets. Could fracton be identified as a state which is analogous to quark or gluon and therefore not invariant under the Galois group. The physical states could be formed from these as Galois singlets and are like hadrons.

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

Articles and other material related to TGD.

Hyperon problem of stellar cores in TGD framework

Hyperon problem is a mystery related to the physics of neutron stars (see this). In neutron star neutrons temperature is zero in good approximation and Fermi statistics implies that the all states characterized by momentum and spin are filled up to maximum energy, known as Fermi energy E_F identifiable as a chemical potential determined by the number density of fermions.

The increase of density inside a neutron star increases the total Fermi energy. Above a critical Fermi temperature possible in the core of the neutron star, the transformation of neutrons to hyperons which are baryons with some strange quarks becomes possible. λ hyperon with mass about 10 percent higher than neutron mass becomes possible. In a thermo-dynamical equilibrium the chemical potentials of hyperons and neutrons are identical. Note that chemical potentials are in a good approximation Fermi energies at zero temperature.

If part of neutrons transform to hyperons, the total energy decreases since the Fermi energy scales like 1/mass. One therefore expects the presence of hyperons in the cores of neutron stars, where the density and therefore also Fermi energy is high enough. The problem is that the maximal mass for known stars is above the maximal mass expected if hyperon fraction is present. Hyperon cores seem to be absent.

If further neutrons are added part of them transforms to hyperons and eventually all particles transform to neutrons and one can even think of the doomsday option that all matter transforms to hyperon stars.

Can one imagine any manner to prevent the formation of the hyperon core? Could the Fermi energy in the core remain below the needed critical Fermi energy by some new physics mechanism.

1. Apart from numerical constants, the Fermi energy for effectively n-D system is given by E_F= ℏ² k_F²/2, where k_F is some power of number density (N/V_n)^2/n, where V_n refers to volume, area, or length for n=3, 2, 1. Since zero temperature approximation is good, Fermi energy depends only on the density.
2. Could one think that part of neutrons transforms to dark neutrons in the transformation h_eff→ kh_eff such that neither mass, energy, and Fermi energy are not affected but that wavelength is scaled up as also the volume. For an effectively 3-D system, dark neutrons would occupy a volume which is scaled up by factor k³.
3. The Fermi energies as chemical potentials for both ordinary neutrons and their dark variants could remain the same in thermal equilibrium and remain below the critical value so that the transformation to hyperons would not take place? The condition that Fermi energies are the same implies that the numbers of ordinary and dark neutrons are the same. This would reduce individual Fermi energies by a factor 1/2^2/3 but is only a temporary solution.

   One can however introduce phases with k different values of h_eff and in this case the reduction of Fermi energies is 1/k^2/3.

4. Fermi statistics might however pose a problem. The second quantization of the induced spinor fields at the space-time surface is induced by the second quantization of free spinor fields in the embedding space M⁴×CP₂. Could the CP₂ degrees of freedom give additional degrees of freedom realized as many-sheeted structures allowing to avoid the problems with Fermi statistics?

See the article Solar Metallicity Problem from TGD Perspective.

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

Articles and other material related to TGD. 


Time crystals in TGD framework

Google has reported about a realization of a time crystal as a spin system. A rather hypish layman article at here creates the impression that perpetuum mobile has been discovered. Also the Quanta Magazine article creates this impression. The original research article can be found in arXiv.org.

It is interesting to look at the situation in the TGD framework. From the abstract of the article also from the Wikipedia article about time crystals one learns that the system has periodic energy feed and is therefore not closed so that the finding is not in conflict with the second law and perpetuum mobile is not in question.

1. What is time crystal?

The notion of time crystal (this) is a temporal analog of ordinary crystals in the sense that there is temporal periodicity, was proposed by Frank Wilczeck in 2012. Experimental realization was demonstrated in 2016-2017 but not in the way theorized by Wilczek. Soon also a no-go theorem against the original form of the time crystal emerged and motivated generalizations of Wilzeck's proposal.

The findings reported by Google are however extremely interesting. Very concisely, researchers study a spin system, which has two directions of magnetization and the external laser beam induces the system to oscillate between the two magnetization directions with a period, which is a multiple of the period of the laser beam. It is interesting to consider the system in TGD framework and I have actually discussed time crystals briefly in a recent article.

2. Space-time surfaces as periodic minimal surfaces as counterparts of time crystals

In TGD, classical physics is an exact part of quantum theory and quantum classical correspondence holds true. Hence it is interesting to consider first the situation at the classical space-time level. In TGD time crystals have as classical correlates space-time surfaces which are periodic minimal surfaces.

It is possible to have analogs of time-crystals and also more general structures built as piles of lego like basic pieces in time direction bringing in mind sentences of language and DNA, which is quasi-periodic structure and more general than crystal.

3. What about thermodynamics of time crystals?

Could the time crystal be possible also in thermodynamic sense and even for thermodynamically closed systems? In TGD Negentropy Maximization Principle (NMP) (see this) and zero energy ontology (ZEO) (see this and this) forces to generalize thermodynamics to allow both time arrows. ZEO is forced by TGD inspired theory of consciousness and solves the basic paradox of quantum measurement theory. The arrow of time would change in ordinary ("big") state function reduction (BSFR) and would remain unaffected in "small" SFR (SSFR). Second law holds true at the level of real physics but in the cognitive sector information increases and NMP holds true.

4. Is new quantum theory making possible quantum coherence in long scales needed?

Also new quantum theory might be needed to explain why the period is multiple of the driving period. The first possibly needed new element is hierarchy of effective Planck constants h_eff= n×h_0 having number theoretical interpretation. h_eff measures the scale of quantum coherence and has also interpretation as the order of Galois group for a polynomial defining the space-time surface in M⁸ mapped to M⁴×CP₂ by M⁸-H duality (see this and this).

The replacement of h with h_eff scales the periods by n and keeps energies unchanged. In TGD inspired biology h_eff hierarchy is in a crucial role and its levels behave relative to each other like dark matter.

In the recent case, the magnetic body (MB)of the spin system controlling its behavior would have h_eff=nh. Each period would be initiated by BSFR at the level of MB and change the arrow of time and induce effective change of it also at the level of the ordinary matter.

In ZEO, time crystal-like entities, which live in cycle by extracting back part of the energy that they have dissipated in a time reversed mode, are in principle possible. System "breathes". Various bio-rhythms could correspond to time crystals. The biological analogy is obvious and we know that life requires a metabolic energy feed: in TGD Universe it prevents the decrease of h_eff (see this).

5. Perpetuum mobile?: almost but not quite!

For an thermodynamically open system, part of the dissipated energy leaks into the external world during each half cycle. Same happens in the time reversed mode and would mean that the system apparently receives positive energy also from the external world. Could this energy feed compensate for the energy loss to the external world by dissipation so that no external energy feed would be needed? Perhaps this might be the case in the ideal situation.

One would have almost a perpetuum mobile! Periodic driving feeding energy to the system would be needed to take care that h_eff is not reduced.

See the chapter Quantum Criticality and Dark Matter: part II.

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

Articles and other material related to TGD. 


Huge fluctuations in oxygen concentration during Cambrian Explosion and Expanding Earth model

I encountered two interesting articles related to the Great Oxidation Event that started long before the Cambrian Explosion (CE) and reached its climax during CE (about 541 million years ago) leading to the oxygen based multicellular life in a very rapid time scale.

The standard view is that oceans before CE had very low oxygen content. The emergence of photosynthesizing cyanobacteria producing oxygen as a side product led to the oxygenation of the atmosphere and to mysteriously rapid evolution of life. How this is possible at all is not understood.

The first article (see this) proposes that the slowing down of the spinning of Earth was somehow related to this.

The second article in Quanta Magazine (see this) tells about finding that during the Cambrian Explosion (see this") the oxygen content of the studied shallow ocean show fluctuations with with about 4-5 peaks. The reduction/increase of the oxygen content was even 40 per cent, which is a huge number. The reduction of oxygen content caused extinctions and its increase was accompanied by the emergence of new species. The mystery is how this could happen so fast and which caused the fluctuations.

1. Expanding Earth hypothesis

Expanding Earth theory hypothesis is not originally TGD based but TGD provides its realization. The proposal is that the Cambrian Explosion was caused by a rapid increase of the radius of Earth by factor 2 (see this and this).

This hypothesis also solves one of the basic mysteries of cosmology. Astrophysical objects participate in cosmological expansion by comoving with it but do not expand themselves. Why? The prediction that the expansion of the astrophysical objects did not occur smoothly but as rapid phase transitions and the expansion was very slow in the intermediate states. Cambrian Explosion would correspond to one particular jerk of this kind in which the radius of Earth grew by a factor 2 (p-adic length scale hypothesis). The length of the day increased by factor 4 from conservation of angular momentum. This might relate to the conjecture of the first article.

The rapid expansion led to the breakage of the Earth crust and to the birth of plate tectonics. It also led to the burst of underground oceans to the surface of the Earth. The photosynthesizing multicellular life had developed in these oceans and emerged almost instantaneously and led to a rapid oxygenation of the atmosphere. One can say that life evolved in the womb of Mother Gaia shielded from meteorites and cosmic rays. No superfast evolution was needed. Already Charles Darwin realized that the sudden appearance of trilobites was a heavy objection against the theory of natural selection.

Possible scenarios for the phase transition are discussed here. The thickening of magnetic flux tubes for water blobs at the surface of Earth led to the increase of the volume of water blob and induced the increase of h_eff a factor 2 for valence electrons but not for the inner electrons. Since valence electrons are responsible for chemistry, atoms became effectively dark and the water blobs could leak to the interior of Earth. By their darkness they could have much lower temperature and pressure than the matter around them and the life could evolve.

2. How photosynthesis was possible underground?

What made photosynthesis possible in the underground oceans? One possible explanation is that the photons from the Sun propagated along flux tubes of the "endogenous" part of the Earth's magnetic field as dark photons with h_eff=nh_0>h. Endogenous part would be the part of Earth's magnetic field with a strength about 2/5 of the Earth's magnetic field for which flux tubes carry monopole flux: this is possible in TGD but not in Maxwell's theory.

Since these photons behave like dark matter with respect to the ordinary matter, they were not absorbed considerably and reached the water blobs (or actually their magnetic bodies consisting of flux tubes) in underground oceans having a portion with the same value of h_eff>h. Of course, several values of h_eff were possible since this is the case in quantum critical system (large values of h_eff characterize the quantum scales of long range fluctuations). One can also consider other variants of the model. The ordinary matter in Earth's crust had h_eff =h/2 and photons with h_eff=h propagated to the interior and reached the water blobs with h_eff=h.

3. The sudden emergence of multicellulars and oxygen fluctuations

Before the expansion period was much like the surface of Mars now and contained no oceans, perhaps some ponds allowing primitive monocellular lifeforms. As the ground of Earth broke here and there during the rapid expansion period, lakes and oceans were formed at the surface of Earth. The multicellulars bursted to these oceans and oxygenation of the atmosphere started locally.

Since the oxygen rich water was mixed with the water in the shallow oceans, the local oxygen content of the burst water was reduced and this led to an eventual extinction of many multicellulars in the burst. Burgess Shale fauna contained entire classes, which suffered extinction. In the average sense the oxygen concentration increased and led to the apparent very rapid evolution of multicellulars, which had actually already occurred underground. Of course, also evolution at the surface of Earth took place.

See the article Updated version of Expanding Earth model or the chapter Expanding Earth Model and Pre-Cambrian Evolution of Continents, Climate, and Life .

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

Articles and other material related to TGD. 


Gentral engine of galactic nuclei as a time reversed blackhole-like object

The work to be summarize was inspired by a Quanta Magazine article "Physicists Identify the Engine Powering blackhole Energy Beams" (see this) telling about an empirical support for the model of Blandford and Znajek (BZ model in the sequel) for the central engine providing energy for jets from active galactic nuclei (AGNs). In the BZ model (see this) AGN is identified as a blackhole and the Penrose process would provide the energy of the jets emerging from the blackhole. The energy would come basically from the blackhole mass. The empirical support is found by studying the supermassive blackhole associated with a galaxy known as Messier 87 (M87).

The basic problem is the identification of the central engine of the active galactic nuclei (AGNs) (see this) providing the huge energy feed to the the jets.

1. Typical properties of active galactic nuclei

The power emitted by active galactic nuclei (AGNs) is typically of the order of 10³⁸ W corresponding to a transformation of a mass of 10²² kg per second to energy. The typical radius of the AGN is R ≈ 2 AU for the active region.

One must distinguish between magnetic fields associated with the interior of the central objects, the region near its surface, and the jet region with the scale of visible jets about 10⁵ ly. According to the estimate of (see this), the magnetic field is about 10 Gauss in the jet region. About 10⁶-10⁷ Gauss near horizon. Also near-horizon magnetic fields in the range 10⁸-10¹¹ Tesla have been proposed for some AGNs.

Quasars are examples of AGNs and also M87 central region idenfied as a blackhole is such. In this case the mass is 6.5× 10⁹M_Sun and Scwartschild radius is about 2× 10¹⁰ km = 1.3× 10³ AU. For M87 central object the magnetic field in the jet region is that of refrigerator magnet and about 100 Gauss. For Sagittarius A has a in the center of the Milky Way the radius of the central object .4 AU.

One can pose some general conditions on the central engine serving as the energy source for the jets. The time scale Δt for the luminosity fluctuations in the power should satisfy Δt< R. For M87 one has Δt ≤ 10⁴ s. The gravitational force is assumed to be balanced by the radiation pressure of the outward radiation.

Consider now the observations about the M87 blackhole-like entity (see this).

1. The mass of the M87 black-hole-like entity is about 6.5× 10⁹M_Sun.
2. There are two 5,000 ly long white hot plasma jets travelling in opposite time directions with emitted power of 3× 10³⁶ W. They have blobs at their ends. Synchrotron radiation is emitted at radio wavelengths in the magnetic field, which according to the popular article has the strength of a refrigerator magnet, so that one would have B ≈ 100 Gauss. Both the intensity of B and the size of the emitting region contribute to the intensity of the energy flow.
3. There are two alternatives for the BZ process that have been developed and explored in hundreds of computer simulations in recent decades. They have acronyms MAD and SANE.

   For the SANE option B is weak: charged matter dominates over B. For the MAD option B is strong, has a spiral structure and acts as a "boss" of the matter. The tight spiral structure forms a sleeve around the jet preventing charges from entering the central object. This inspires a critical question: doesn't the object look more like a whitehole.

   The strongly polarized light in the Event Horizon Telescope's new photo suggests strong magnetic fields, and supports the MAD version. B has a strength of about 100 Gauss, that is 200 times the strength of the Earth's magnetic field with the nominal value B_E≈ .5 Gauss. The polarization pattern for the radio waves is found to be stripy and the polarization in a plane locally: this allows us to conclude that the magnetic field is indeed helical and non-random.

2. Central engine as a Penrose process?

In the BZ model, the central object is assumed to be a blackhole and Penrose process would provide the energy feed to the jets of length about 5000 ly. Note that the Milky Way is about 1,0000 ly thick.

1. The blackhole is surrounded by an accretion disk from which the matter ends down to the BH.
2. Kerr solution of Einstein-Maxwell field equations (this) involving magnetic field is the starting point. Matter falling into the Kerr blackhole rotates and the magnetic field lines are twisted to helical shape. By Faraday law, an electric field along field lines is generated by the rotation of the flux lines. Electrons and positrons created in the annihilation photons emitted as the particles fall to the region near the blachole, start to flow along the field lines of the electric field in opposite directions and generate the jets.
3. The model assumes that the electromagnetic field is force free so that it does not dissipate and Lorentz force vanishes. At a single particle level this implies the condition E+qv× B=0. Vanishing dissipation requires v· E=0. This helical structure would be in the direction of the jet.
4. The basic question has been whether it is accretion disk or magnetic field that controls the dynamics. The first option, known as SANE, corresponds to weak and incoherent magnetic fields. The second option, known as MAD, corresponds to strong and coherent magnetic fields.
5. MAD is favoured by the recent observations. Magnetic field would form a sleeve around the jet and the synchrotron radiation pressure would prevent matter from falling into the blackhole. Matter can only occasionally leak to blackhole.

   One can however wonder whether it makes sense to talk about blackhole anymore! Doesn't this look more like white hole as a time reversal of blackhole feeding energy and matter to the environment?

3. TGD inspired view of the central engine

In the TGD framework the model of the central engine as a Penrose process is replaced by the following picture. The key concepts are following:

1. Space-time is identified as a 4-D minimal surface in H=M⁴× CP₂ or as an algebraic surface in complexified M⁸ having octonionic interpretation. These descriptions are related by M⁸-H duality analogous to momentum-position duality, which does not generalize from wave mechanics to quantum field theory (QFT). Therefore the points or M⁸ are 8-momenta.

   The classical dissipation is absent for the generalized Beltrami fields and the proposal is that minimal surfaces (apart form singularities defining dynamically generated frame for space-time surfaces as analog of a soap film) define locally generalized Beltrami fields.

2. Zero energy ontology (ZEO) predicts that time reversal occurs in the TGD counterparts of ordinary state function reductions ("big" SFRs) but not in "small" SFRs (SSFRs).
3. The hierarchy of effective Planck constants predicts a hierarchy of phases of ordinary matter labelled by the values of effective Planck constant h_eff= nh₀. The phases with different values of h_eff behave in many respects like dark matter with respect to each other. The findings of Randell Mills suggest ℏ/ℏ₀=6 but also larger values for this ratio can be considered. I have proposed that the ℏ₀/ℏ is equal to the ratio l_P/R of Planck length l_P to CP₂ radius R.

   As a special case, one obtains gravitational Planck constant satisfying h_eff= h_gr= GMm/β₀, where β₀=v₀/c and β₀<c has dimensions of velocity, as a generalization of Nottale's hypothesis. The gravitational Compton length λ_gr=ℏ_gr/m=GM/β₀ does not depend on m and is equal to Schwartschild radius r_s for β₀= 1/2. Also the cyclotron energy spectrum E_c=n GMqB/β₀ is independent of the mass of the charged particle.

   The hierarchy of Planck constants, the notion of ℏ_gr, and coupling constant evolution are discussed in detail here.

Consider next the key elements of the model.

1. TGD leads to a general model for the formation of galaxies, stars, planets,... in terms of cosmic strings thickening to flux tubes. The energy of the flux tube, which consists of a volume energy and Kähler magnetic energy, is transformed to ordinary matter as the string tension is reduced in a sequence of phase transitions reducing the length scale dependent cosmological constant λ.

   This process is analogous to the decay of an inflaton field to matter. The model (there are actually several basic variants of it) explains the flat velocity spectra associated with the spiral galaxies. For the first option, a long cosmic string normal to the galactic plane causes the gravitational field explaining the flat velocity spectrum of spiral galaxies. For galaxies formed around closed flux loops the velocity spectrum is not flat. There is no dark matter halo although it is possible that the galactic plane contains cosmic strings parallel to the plane.

2. Zero energy ontology (ZEO), which predicts that the TGD counterparts of ordinary state function reductions (SFRs) involve time reversal, is involved in an essential manner. TGD predicts both blackhole-like objects (BH) and whitehole-like objects (WH) as the time reversals of BHs. The seed of the galaxy, active galactic nucleus (AGN), involves WH. Quasars are cases of AGNs as WHs.
3. In the TGD framework, the Kerr blackhole is replaced with a whitehole-like object (WH). Kerr blackhole indeed has an opposite arrow of time reversal as the distant environment. The WH is time reversal of BH and feeds matter and energy to the environment. This serves as an analog of the Penrose process in the TGD based model.
4. The TGD analog for the rotation of spacetime and the twisting of the magnetic field lines near the Kerr blackhole is very concrete. Space-time is a 4-surface and the flux tubes carrying monopole flux are pieces of 3-space as a 3-surface. They quite concretely rotate and get twisted in the process. Analogous process occurs in the Sun with a period of 11 years ending as reconnections untwist the flux tubes.
5. WH would correspond to a tangle of a long cosmic string in the direction of the jet thickened to a flux tube but still carrying an extremely strong magnetic field. The helical magnetic field in the exterior of the jet would not represent return flux of this field as one might first think. There is a current ring associated with the equator of Earth, which carries a parallel magnetic field analogous to the helical magnetic field.

   The magnetic field in the exterior of WH is associated with a space-time surface, which is many-sheeted with respect to CP₂ rather than M⁴ so that either CP₂ or cosmic string world sheet M²× S²⊂ M⁴× CP₂) would serve as the arena of physics rather than M⁴, which is quantum coherent flux tube bundle analogous to BE-condensate. M⁴ coordinates as functions of CP₂ or M²× CP₂ coordinates would be many-valued rather than vice versa. This picture is very natural if one accepts M⁸-H duality.

   Cosmic strings dominate during the primordial cosmology in TGD Universe, and the analog of the inflationary period corresponds to the transition to a phase in which the Einsteinian space-time with M⁴ as the arena of physics is a good approximation. Hence the M²× CP₂ option looks more plausible.

6. The force-free em fields appearing in the BZ model correspond to space-time surfaces as minimal surfaces realizing a 4-D generalization of 3-D Beltrami fields, which do not not dissipate classically. The interpretation of the non-dissipating Kähler currents is as classical correlates for supracurrents. The prediction is that charged particles flow without dissipation that is as supra currents: not only Cooper pairs but also charged fermions. Also the analogs of laser beams of dark photons are expected.
7. The hierarchy of Planck constants is an important piece of the picture emerging from adelic physics. From ℏ_gr=GMm/β₀ realizing Equivalence Principle, the gravitational Compton length λ_gr= r_s/2β₀ is universal and equals to r_S for β₀== β₀=1/2.

   All astrophysical objects are predicted to be quantum coherent in the scale of λ_gr= r_s/2β₀ at least. The quantum coherence would be at the level of magnetic body (MB). WH/BH as a thickened flux tube tangle would not have large h_eff but would be accompanied by a large scale quantum object.

   The astroscopic quantum coherence would be associated with the helical magnetic field surrounding the long cosmic string having BH or WH as a tangle.

8. Also the cyclotron energy spectrum is universal and does not depend on the mass of the charged particles so that all charged particles rather than only electrons are expected to form supracurrents. Dark matter would flow along flux tubes and form the dark core of the jet, perhaps extending over cosmic distances to other galaxies identified as tangles of the one and the same cosmic string.

   Stars and even planets would be parts of this fractal network. Dark cyclotron states have huge energies for h_eff=h_gr serving also as a measure for algebraic complexity and, in the TGD inspired theory of consciousness, also for intelligence and scale of quantum coherence. The analogy with a cosmic nervous system is obvious.

9. The decay of quantum coherent states to ordinary states takes place by the loss of quantum coherence in which h_eff= h_gr is reduced. This would create the visible jets and blobs at their ends. For M87, which is elliptical for which the velocity spectrum is not flat, the flux tubes would be closed in a relatively short scale. Their length scale could be that of the jets in the case of ellipticals. The thickening of the cosmic string at the core leads to the reduction of mass of WH and gives rise to the flow of mass and energy to the environment. One could see this process as a time reversal for the generation of BH and perhaps also as an analogy for the evaporation of BH.
10. M⁸-H duality and adelic physics help to understand the decoherence process geometrically. The reduction of h_eff and thus of the length scale of quantum coherence, allows a number theoretic description at the level of M⁸. An irreducible polynomial, which depends on parameters, reduces to a product of polynomials for some critical values of the parameters. This gives rise to a set of disjoint space-time surfaces, which are not correlated. This means decoherence. This includes as a special case the description of catastrophic changes in catastrophe theory of Thom. The maximal decoherence produces a product of first order polynomials with rational roots.

    At the level of H =M⁴×CP₂ this corresponds to a decay of coherent flux tube bundle to disjoint uncorrelated flux tubes.

See the article TGD view of the engine powering jets from active galactic nuclei or the chapter with the same title.

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

Articles and other material related to TGD.

Evolution of Kähler coupling strength

The evolution of Kähler coupling strength α_K= g_K²/2h_eff gives the evolution of α_K as a function of dimension n of EQ: α_K= g_K²/2nh₀. If g_K² corresponds to electroweak U(1) coupling, it is expected to evolve also with respect to PLS so that the evolutions would factorize.

Note that the original proposal that g_K² is renormalization group invariant was later replaced with a piecewise constancy: α_K has indeed interpretation as piecewise constant critical temperature

1. In the TGD framework, coupling constant as a continuous function of the continuous length scale is replaced with a function of PLS so that coupling constant is a piecewise constant function of the continuous length scale.

   PLSs correspond to p-adic primes p, and a hitherto unanswered question is whether the extension determines p and whether p-adic primes possible for a given extension could correspond to ramified primes of the extension appearing as factors of the moduli square for the differences of the roots defining the space-time surface.

   In the M⁸ picture the moduli squared for differences r_i-r_j of the roots of the real polynomial with rational coefficients associated with the space-time surfaces correspond to energy squared and mass squared. This is the case of p-adic prime corresponds to the size scale of the CD.

   The scaling of the roots by constant factor however leaves the number theoretic properties of the extension unaffected, which suggests that PLS evolution and dark evolution factorize in the sense that PLS reduces to the evolution of a power of a scaling factor multiplying all roots.

2. If the exponent Δ K/log(p) appearing in p^{Δ K/log(p))}=exp(Δ K) is an integer, exp(Δ K) reduces to an integer power of p and exists p-adically. If Δ K corresponds to a deviation from the Kähler function of WCW for a particular path in the tree inside CD, p is fixed and exp(Δ K) is integer. This would provide the long-sought-for identification of the preferred p-adic prime. Note that p must be same for all paths of the tree. p need not be a ramified prime so that the trouble-some correlation between n and ramified prime defining padic prime p is not required.

3. This picture makes it possible to understand also PLS evolution if Δ K is identified as a deviation from the Kähler function. p^{Δ K/log(p))}=exp(Δ K) implies that Δ K is proportional to log(p). Since Δ K as 6-D Kähler action is proportional to 1/α_K, log(p)-proportionality of Δ K could be interpreted as a logarithmic renormalization factor of α_K∝ 1/log(p).

4. The universal CCE for α_K inside CDs would induce other CCEs, perhaps according to the scenario based on M"obius transformations.

Dark and p-adic length scale evolutions of Kähler coupling strength

The original hypothesis for dark CCE was that h_eff=nh is satisfied. Here n would be the dimension of EQ defined by the polynomial defining the space-time surface X⁴subset M⁸_c mapped to H by M⁸-H correspondence. n would also define the order of the Galois group and in general larger than the degree of the irreducible polynomial.

Remark: The number of roots of the extension is in general smaller and equal to n for cyclic extensions only. Therefore the number of sheets of the complexified space-time surface in M⁸_c as the number of roots identifiable as the degree d of the irreducible polynomial would in general be smaller than n. n would be equal to the number of roots only for cyclic extensions (unfortunately, some former articles contain the obviously wrong statement d=n).

Later the findings of Randell Mills, suggesting that h is not a minimal value of h_eff, forced to consider the formula h_eff=nh₀, h₀=h/6, as the simplest formula consistent with the findings of Mills. h₀ could however be a multiple of even smaller value of h_eff, call if h₀ and the formula h₀=h/6 could be replaced by an approximate formula.

The value of h_eff=nh₀ can be understood by noticing that Galois symmetry permutes "fundamental regions" of the space-time surface so that action is n times the action for this kind of region. Effectively this means the replacement of α_K with α_K/n and implies the convergence of the perturbation theory. This was actually one of the basic physical motivations for the hierarchy of Planck constants. In the previous section, it was argued that h₀ is given by the square of the ratio l_P/R of Planck length and CP₂ length scale identified as dark scale and equals to n₀=(7!)².

The basic challenge is to understand p-adic length scale evolutions of the basic gauge couplings. The coupling strengths should have a roughly logarithmic dependence on the p-adic length scale p≈ 2^k/2 and this provides a strong number theoretic constraint in the adelic physics framework.

Since Kähler coupling strength α_K induces the other CCEs it is enough to consider the evolution of α_K.

p-Adic CCE of α from its value at atomic length scale?

If one combines the observation that fine structure constant is rather near to the inverse of the prime p=137 with PLS, one ends up with a number theoretic idea leading to a formula for α_K as a function of p-adic length scale.

1. The fine structure constant in atomic length scale L(k=137) is given α (k)=e²/2h ≈ 1/137. This finding has created a lot of speculative numerology.
2. The PLS L(k)= 2^k/2R(CP₂) assignable to atomic length scale p≈ 2^k corresponds to k=137 and in this scale α is rather near to 1/137. The notion of fine structure constant emerged in atomic physics. Is this just an accident, cosmic joke, or does this tell something very deep about CCE?

   Could the formula

   α(k)= e²(k)/2h= 1/k

   hold true?

There are obvious objections against the proposal.

1. α is length scale dependent and the formula in the electron length scale is only approximate. In the weak boson scale one has α≈ 1/127 rather than α= 1/89.
2. There are also other interactions and one can assign to them coupling constant strengths. Why electromagnetic interactions in electron Compton scale or atomic length scales would be so special?

The idea is however plausible since beta functions satisfy first order differential equation with respect to the scale parameter so that single value of coupling strength determines the entire evolution.

p-Adic CCE from the condition α_K(k=137)= 1/137

In the TGD framework, Kähler coupling strength α_K serves as the fundamental coupling strength. All other coupling strengths are expressible in terms of α_K, and I have proposed that M"obius transformations relate other coupling strengths to α_K. If α_K is identified as electroweak U(1) coupling strength, its value in atomic scale L(k=137) cannot be far from 1/137.

The factorization of dark and p-adic CCEs means that the effective Planck constant h_eff(n,h,p) satisfies

h_eff(n,h,p)=h_eff(n,h) = nh .

and is independent of the p-adic length scale. Here n would be the dimension of the extension of rationals involved. h_eff(1,h,p) corresponding to trivial extension would correspond to the p-adic CCE as the TGD counterpart of the ordinary evolution.

The value of h need not be the minimal one as already the findings of Randel Mills suggest so that one would have h=n₀h₀.

h_eff= nn₀h ,

α_K,0= g_K,max²/2h₀ =n₀ .

This would mean that the ordinary coupling constant would be associated with the non-trivial extension of rationals.

Consider now this picture in more detail.

1. Since dark and p-adic length scale evolutions factorize, one has

   α_K (n)= g_K²(k)/2h_eff ,

   h_eff= nh₀ .

   U(1) coupling indeed evolves with the p-adic length scale, and if one assumes that g_K²(k,n₀) (h=n₀h₀) is inversely proportional to the logarithm of p-adic length scale, one obtains

   g_K²(k,n₀) =g_K²(max)/k ,

   α_K = g_K²(max)/2kh_eff .

2. Since k=137 is prime (here number theoretical physics shows its power!), the condition α_K (k=137,h₀)=1/137 gives

   g_K²(max)/2h₀}= α_K(max) =(7!)² .

   The number theoretical miracle would fix the value of α_K(max) to the ratio of Planck mass and CP₂ mass n₀= M²_P/M²(CP₂)= (7!)² if one takes the argument of the previous section seriously.

   The convergence of perturbation theory could be possible also for h_eff=h₀ if the p-adic length scale L(k) is long enough to make α_K= n₀/k small enough.

3. The outcome is a very simple formula for α_K

   α_K(n,k) = n₀/kn ,

   which is a testable prediction if one assumes that it corresponds to electroweak U(1) coupling strength at QFT limit of TGD. This formula would give a practically vanishing value of α_K for very large values of n associated with h_gr. Here one must have n>n₀.

   For h_eff=nn₀h characterizing extensions of extension with h_eff=h one can write

   α_K(nn₀,k) = 1/kn .

4. The almost vanishing of α_K for the very large values of n associated with ℏ_gr would practically eliminate the gauge interactions of the dark matter at gravitational flux tubes but leave gravitational interactions, whose coupling strength would be beta₀/4pi. The dark matter at gravitational flux tubes would be highly analogous to ordinary dark matter.

See the article Questions about coupling constant evolution or the chapter with the same title.

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

Articles and other material related to TGD.

Minimal value of heff from the ratio of Planck mass and CP2 mass?

Minimal value of h_eff from the ratio of Planck mass and CP₂ mass?

Could one understand and perhaps even predict the minimal value h₀of h_eff? Here number theory and the notion of n-particle Planck constant h_eff(n) suggested by Yangian symmetry could serve as a guidelines.

1. Hitherto I have found no convincing empirical argument fixing the value of r=ℏ/ℏ₀: this is true for both single particle and 2-particle case.

   The value h₀=h/6 as a maximal value of h₀ is suggested by the findings of Randell Mills and by the idea that spin and color must be representable as Galois symmetries so that the Galois group must contain Z₆=Z₂× Z₃. Smaller values of h₀ cannot be however excluded.

2. A possible manner to understand the value r geometrically would be following. It has been assumed that CP₂ radius R defines a fundamental length scale in TGD and Planck length squared l_P²= ℏ G =x^-2 × 10^-6R² defines a secondary length scale. For Planck mass squared one has m_Pl²= m(CP₂,ℏ)²× 10⁶x² , m(CP₂,ℏ)²= ℏ/R². The estimate for x from p-adic mass calculations gives x≈ 4.2. It is assumed that CP₂ length is fundamental and Planck length is a derived quantity.

   But what if one assumes that Planck length identifiable as CP₂ radius is fundamental and CP₂ mass corresponds the minimal value h₀ of h_eff(2)? That the mass formula is quadratic and mass is assignable to wormhole contact connecting two space-time sheets suggests in the Yangian framework that h_eff(2) is the correct Planck constant to consider.

One can indeed imagine an alternative interpretation. CP₂ length scale is deduced indirectly from p-adic mass calculation for electron mass assuming h_eff=h and using Uncertainty Principle. This obviously leaves the possibility that R= l_P apart from a numerical constant near unity, if the value of h_eff to be used in the mass calculations is actually h₀= (l_P/R)²ℏ. This would fix the value of ℏ₀ uniquely.

The earlier interpretation makes sense if R(CP₂) is interpreted as a dark length scale obtained scaling up l_P by ℏ/ℏ₀. Also the ordinary particles would be dark.

h₀ would be very small and α_K(ℏ₀)= (ℏ/ℏ₀)α_K would be very large so that the perturbation theory for it would not converge. This would be the reason for why ℏ and in some cases some smaller values of h_eff such as ℏ/2 and ℏ/4 seem to be realized.

For R=l_P Nottale formula remains unchanged for the identification M²_P= ℏ/R² (note that one could consider also ℏ₀/R² used in p-adic mass calculations).

Various options

Number theoretical arguments allow to deduce precise value for the ratio ℏ/ℏ₀. Accepting the Yangian inspired picture, one can consider two options for what one means with ℏ.

1. ℏ refers to the single particle Planck constant ℏ_eff(1) natural for point-like particles.
2. ℏ refers to h_eff(2). This option is suggested by the proportionality M²∝ ℏ in string models due to the proportionality M²∝ℏ/G in string models. At a deeper level, one has M² ∝ L₀, where L₀ is a scaling generator and its spectrum has scale given by ℏ.

   Since M² is a p-adic thermal expectation of L₀ in the TGD framework, the situation is the same. This also due the fact that one has In TGD framework, the basic building bricks of particles are indeed pairs of wormhole throats.

One can consider two options for what happens in the scaling h_eff→ kh_eff.

Option 1: Masses are scaled by k and Compton lengths are unaffected.

Option 2: Compton lengths are scaled by k and masses are unaffected.

The interpretation of M_P²= (ℏ/ℏ₀) M²(CP₂) assumes Option 1 whereas the new proposal would correspond to Option 2 actually assumed in various applications.

The interpretation of M_P²= (ℏ/ℏ₀) M²(CP₂) assumes Option 1 whereas the new proposal would correspond to Option 2 actually assumed in various applications.

For Option 1 m_Pl²= (ℏ_eff/ℏ) M²(CP₂). The value of M²(CP₂)= ℏ/R² is deduced from the p-adic mass calculation for electron mass. One would have R² ≈ (ℏ_eff/ℏ) l_P² with ℏ_eff/ℏ = 2.54× 10⁷. One could say that the real Planck length corresponds to R.

Quantum-classical correspondence favours Option 2)

In an attempt to select between these two options, one can take space-time picture as a guideline. The study of the imbeddings of the space-time surfaces with spherically symmetric metric carried out for almost 4 decades ago suggested that CP₂ radius R could naturally correspond to Planck length l_P. The argument is described in detail in Appendix and shows that the l_P=R option with h_eff=h used in the classical theory to determine α_K appearing in the mass formula is the most natural.

Deduction of the value of ℏ/ℏ₀

Assuming Option 2), the questions are following.

1. Could l_P=R be true apart from some numerical constant so that CP₂ mass M(CP₂) would be given by M(CP₂)²= ℏ₀/l_P², where ℏ₀≈ 2.4× 10^-7 ℏ (ℏ corresponds to ℏ_eff(2)) is the minimal value of ℏ_eff(2). The value of h₀ would be fixed by the requirement that classical theory is consistent with quantum theory! It will be assumed that ℏ₀ is also the minimal value of ℏ_eff(1) both ℏ_eff(2).
2. Could ℏ(2)/ℏ₀(2)=n₀ correspond to the order of the product of identical Galois groups for two Minkowskian space-time sheets connected by the wormhole contact serving as a building brick of elementary particles and be therefore be given as n₀=m²?

Assume that one has n₀=m².

1. The natural assumption is that Galois symmetry of the ground state is maximal so that m corresponds to the order a maximal Galois group - that is permutation group S_k, where k is the degree of polynomial.

   This condition fixes the value k to k=7 and gives m=k!=7! = 5040 and gives n₀= (k!)²= 25401600=2.5401600 × 10⁷. The value of ℏ₀(2)/ℏ(2)=m^-2 would be rather small as also the value of ℏ₀(1)ℏ(1). p-Adic mass calculations lead to the estimate m_Pl/m(CP₂)= m^1/2 m(CP₂)=4.2× 10^{3, which is not far from m=5040.}

2. The interpretation of the product structure S₇ × S₇ would be as a failure of irreducibility so that the polynomial decomposes into a product of polynomials - most naturally defined for causally isolated Minkowskian space-time sheets connected by a wormhole contact with Euclidian signature of metric representing a basic building brick of elementary particles.

   Each sheet would decompose to 7 sheets. ℏ_gr would be 2-particle Planck constant h_eff(2) to be distinguished from the ordinary Planck constant, which is single particle Planck constant and could be denoted by h_eff(1).

   The normal subgroups of S₇ × S₇ S₇× A₇ and A₇× A₇, S₇, A₇ and trivial group. A₇ is simple group and therefore does not have any normal subgroups expect the trivial one. S₇ and A₇ could be regarded as the Galois group of a single space-time sheet assignable to elementary particles. One can consider the possibility that in the gravitational sector all EQs are extensions of this extension so that ℏ becomes effectively the unit of quantization and m_Pl the fundamental mass unit. Note however that for very small values of α_K in long p-adic length scales also the values of h_eff<h, even h₀, are in principle possible.

   The large value of α_K ∝ 1/ℏ_eff for Galois groups with order not considerably smaller than m=(7!)² suggests that very few values of h_eff(2)<h are realized. Perhaps only S₇ × S₇ S₇× A₇ and A₇× A₇ are allow by perturbation theory. Now however that in the "stringy phase" for which super-conformal invariance holds true, h₀ might be realized as required by p-adic mass calculations. The alternative interpretation is that ordinary particles correspond to dark phase with R identified dark scale.

3. A₇ is the only normal subgroup of S₇ and also a simple group and one has S₇/A₇= Z₂. S₇× S⁷ has S₇× S⁷/A₇× A₇= Z₂× Z₂ with n=n₀/4 and S₇× S⁷/A₇× S₇= Z₂ with n=n₀/2. This would allow the values ℏ/2 and ℏ/4 as exotic values of Planck constant.

   The atomic energy levels scale like 1/ℏ² and would be scaled up by factor 4 or 16 for these two options. It is not clear whether ℏ→ ℏ/2 option can explain all findings of Randel Mills in TGD framework, which effectively scale down the principal quantum number n from n to n/2.

4. The product structure of the Nottale formula suggests

   n=n₁× n₂ = k₁k₂m² .

   Equivalently, n_i would be a multiple of m. One could say that M_Pl=(ℏ/ℏ₀)^1/2M(CP₂) effectively replaces M(CP₂) as a mass unit. At the level of polynomials this would mean that polynomials are composites P○ P₀ where P₀ is ground state polynomial and has a Galois group with degree n₀. Perhaps S₇ could be called the gravitational or ground state Galois group.

See the article Questions about coupling constant evolution or the chapter with the same title.

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

Articles and other material related to TGD. 


TGD view of the engine powering jets from active galactic nuclei

The identification of the energy source (central engine) explaining the energy loss associated with the jets from active galactic nuclei (AGNs) is a long-standing problem of astrophysics. In the model of Blandford and Znajek (BZ model) for the central engine as a blackhole, the Penrose process would provide the energy. The energy would come basically from the blackhole mass.

Empirical support for the BZ model emerges from the study of the supermassive blackhole associated with a galaxy known as Messier 87 (M87). The finding is that the magnetic field associated with the jet structure is tightly wound helical structure and so strong that it would control the dynamics of the matter from falling to blackhole except by occasional leakages. Electron-positron pairs created in the annihilation of photons would accelerate in the force-free helical electromagnetic field having also an electric component.

The TGD based model involves several aspects of the new physics predicted by TDG. TGD leads to a model of galaxies and other astrophysical structures. Inflaton decay is replaced with the thickening of cosmic strings to flux tubes liberating as ordinary matter. Hierarchy of Planck constants h_eff=nh₀, in particular Nottale's hypothesis predicts quantum coherence in the exterior of in scales at least of order Schwartschild radius of the blackhole-like entity. Zero energy ontology (ZEO) predicts that the arrow of time changes in ordinary state function reductions. TGD replaces black-holes with blackhole-like entities (BHs) and white-holes with their time reversals (WHs) allowed in ZEO.

BH (WH) would be a volume filling flux tube but with a relatively small value of h_eff. In the case of WH, it would provide "metabolic energy" for jets and take care that the value of h_eff is preserved (the analogy with living systems is very strong). The jets would be analogous to laser beams/supracurrents with a huge value of h_eff=h_gr. The model would also explain the ultrahigh energy cosmic rays. The force-free fields would be generalized Beltrami fields associated with flux tubes and identifiable as minimal surfaces in the Minkowskian regions of space-time surface. The absence of classical dissipation would be a correlate for the absence of dissipation for supra-currents and dark photon laser beams.

See the article TGD view of the engine powering jets from active galactic nuclei or the chapter with the same title.

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

Articles and other material related to TGD. 


Connection with parity breaking, massivation, and PCAC hypothesis

Conserved vector current hypothesis (CVC) and partially conserved axial current hypothesis (PCAC) are essential elements of old-fashioned hadron physics and hold true also in the standard model.

1. The ansatz, which realizes the Beltrami hypothesis, states that the vectorial Kähler current J equals apart from sign c=+/- 1 to instanton current I, which is axial current:

   J=+/- I .

   The condition states that only the left or right handed current chiral defined as

   L_L/R= J+/- I

   is non-vanishing. For c≠ 1, both J_L and J_R are non-vanishing. Since both right- and left-handed weak currents exist, c≠ 1 seems to be a plausible option.

   By quantum classical correspondence, these currents serve as space-time correlates for the left- and right-handed fermion currents of the standard model. Note however that induced gamma matrices differ from those of M⁴: for instance, they are not covariantly constant but defines a current with divergence which vanishes by field equations.

2. A more general condition would allow c to depend on space-time coordinates. The conservation of J forces conservation of I if the condition ∂_αcI^α=0 is true. This gives a non-trivial condition only in regions with 4-D CP₂ and M⁴ projections.
3. The twistor lift of TGD requires that also M⁴ has Kähler structure. Therefore J and I and corresponding Kähler gauge potential A have both M⁴ part and CP₂ parts and Kähler action K, J_K, J and I are sums of M⁴ and CP₂ parts:

   A_K= A(M⁴)+A(CP₂),
   J_K=J_K(M⁴)+J_K(CP₂) ,
   K = K(M⁴)+K(CP₂) ,
   J =J(M⁴)+J(CP₂) ,
   I= I(M⁴)+I(CP₂) .
   

   Only the divergence for the sum I of M⁴ and CP₂ parts of the instanton currents must vanish:

   ∂_αI^α=0 .

   A possible interpretation is in terms of the 8-D variant of twistorialization by twistor lift requiring masslessness in an 8-D sense.

   PCAC states that the divergence of the axial current is non-vanishing. This is not in conflict with the conservation of the total instanton current I. PCAC corresponds to the non-conservation I(CP₂), whose non-conservation is compensated by that of I(M⁴).

4. For regions with at most 3-D M⁴- and CP₂ projections, the M⁴- and CP₂ instanton currents have identically vanishing divergence. In these regions the conservation of I is not lost if c has both signs. c could be also position dependent and even differ for I(M⁴) and I(CP₂) in these regions.

   D_αI^α=0 is true for the known extremals. For the simplest CP₂ type extremals and for extremals with 2-D CP₂ projection, I itself vanishes. Therefore parity violation is not possible in these regions. This would suggest that these regions correspond to a massless phase.

5. D_αI^α≠ 0 is possible only if both M⁴ and CP₂ projections are 4-D. This phase is interpreted as a chaotic phase and by the non-conservation of electroweak axial currents could correspond to a massive phase.

   CP₂ type extremals have 4-D projection and for them Kähler current and instanton current vanish identically so that also they correspond to massless phase (M⁴ projection is light-like). Could CP₂ type extremals allow deformations with 4-D M⁴ projection (DEs)?

   The wormhole throat between space-time region with Minkowskian signature of the induced metric and CP₂ type extremal (wormhole contact) with Euclidian signature is light-like and the 4-metric is effectively 3-D. It is not clear whether this allows 4-D M⁴ projection in the interior of DE.

6. The geometric model for massivation based on zitterbewegung of DE provides additional insight. M⁸-H duality allows to assign a light-like curve also to DE. For space-time surfaces determined by polynomials (cosmological constant Λ>0), this curve consists of pieces which are light-like geodesics.

   Also real analytic functions (Λ=0) can be considered and they would allow a continuous light-like curve, whose definition boils down to Virasoro conditions. In both cases, the zigzag motion with light-velocity would give rise to velocity v<c in long length scales having interpretation in terms of massivation.

   The interaction with J(M⁴) would be essential for the generation of momentum due to the M⁴ Chern-Simons term assigned with the 3-D light-like partonic orbit. M⁴ Chern-Simons term can be interpreted as a boundary term due to the non-vanishing divergence of I(M⁴) so that a connection with two views about massivation is obtained. Does the Chern-Simons term come from the Euclidean or Minkowskian region?

I have proposed two models for the generation of matter-antimatter asymmetry. In both models, CP breaking by M⁴ Kähler form is essential. Classical electric field induces CP breaking. CP takes self-dual (E,B) to anti-self-dual (-E,B) and self-duality of J(M⁴) does not allow CP as a symmetry.

1. In the first model the electric part of J(M⁴) would induce a small CP breaking inside cosmic strings thickened to flux tubes inducing in turn small matter-antimatter asymmetry outside cosmic strings. After annihilation this would leave only matter outside the cosmic strings.
2. In the simplest variant of TGD only quarks are fundamental particles and leptons are their local composites in CP₂ scale. Both quarks and antiquarks are possible but antiquarks would combine leptons as almost local 3-quark composites and presumably realized CP₂ type extremals with the 3 antiquarks associated with the partonic orbit. I should vanish identically for the DEs representing quarks and leptons but not for antiquarks and antileptons.

   Could the number of DEs with vanishing I be smaller for antiquarks than for quarks by CP breaking and could this induce leptonization of antiquarks and favor baryons instead of antileptons? Could matter-antimatter asymmetry be induced by the interior of DE alone or by its interaction with the Minkowskian space-time region outside DE.

In the standard model also charged weak currents are allowed. Does TGD allow their space-time counterparts? CP₂ allows quaternionic structure in the sense that the conformally invariant Weyl tensor has besides W₃=J(CP₂) also charged components W_+/-, which are however not covariantly constant. One can assign to W_+/- analogs of Kähler currents as covariant divergences and also the analogs of instanton currents. These currents could realize a classical space-time analog of current algebra.

See the article Comparing the Berry phase model of super-conductivity with the TGD based model or the chapter with the same title.

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

Articles and other material related to TGD.

Possible implications of the TGD based model of superconductivity

The universality of the TGD based model of superconductivity provides support for rather far-reaching earlier speculations.

1. The TGD inspired model suggests that SC could be possible also above T_c by using energy feed providing the energy needed to increase the value of h_eff. This would be the basic role of metabolism. This could have far reaching technological consequences and also profound implications concerning the creation of artificial life.

   Furthermore, the TGD based model for "cold fusion" \cite{cfagain,krivit,proposal} led to a reformulation of nuclear physics \cite{darkcore} in which phase transition to dark phase of nuclei has a key role also in the ordinary nuclear reactions as a description of tunnelling phenomenon.

2. In the TGD inspired quantum biology, the cell membrane is identified as a generalized Josephson junction between superconductors assignable to lipid layers of the cell membrane (actually decomposing in a better resolution to membrane proteins acting as Josephson junctions). One can ask what a straightforward application of the basic formulas gives in the case of neuronal membrane.

   One can estimate the gap energy \Delta from the formula \Delta = ℏ ω_D using the already discussed formula ω_D = k_n c_s/a, where k_n depends on the effective dimension of the lattice like system and has values k_n ∈ {3.14,3.54,2.66} for n=1,2,3. Sound velocity c_s can be replaced with the conduction velocity v of nerve pulses varying in the range v/c\in[.1,1]\times 10⁶. The formula would give for n=2 and maximal value v/c=10^-6 E_D= .044 eV which is in the range of neuronal membrane potentials.

3. The role of ℏ_gr and B_end in the model would suggest that the SC observed in laboratories is not a mere local condensed matter phenomenon. What happens to SC on Mars? Is the Earth mass replaced with that of Mars and the monopole part B_end with its value in Mars? There is evidence that B_end is non-vanishing: for instance, Mars has auroras.
4. If the monopole flux tube indeed mediates graviton exchanges, one can wonder whether SC itself is an essentially quantum gravitational phenomenon. Could the attractive interaction between electrons of the Cooper pair be somehow due to gravitation?

   The extremely weak direct gravitational interaction between electrons and nucleons cannot be responsible for the formation of Cooper pairs. One can however argue that Earth takes the role of atomic nuclei in the proposed description. Earth attracts the electrons and causes an effective attraction between them. Could this interaction force the wave functions of the electrons of the Cooper pair with wavelength \Lambda_gr= r_S=2GM\simeq 9 mm to overlap and form a quantum coherent state. For Sun one has \Lambda_gr= 1 Mm, which is slightly below Earth radius. Could Sun's gravitation make the macroscopic quantum phase in the Earth's scale?

   The proposed duality between gauge theories and gravitation, in particular AdS/CFT duality, has a TGD counterpart. The dynamics for the orbits of partonic 2-surfaces and lower-dimensional surface defining a frame for the space-time surface as an analog of soap film \cite{minimal} would be dual to the dynamics in the interior of the space-time surfaces.

   Could the descriptions in terms of cyclotron photon exchanges and graviton exchanges be dual to each other? Note also that at the fundamental level classical TGD are expressible using only 4 classical field-like variables as a selected subset of imbedding space coordinates. This implies extremely strong constraints between fundamental interactions.

See the article Comparing the Berry phase model of super-conductivity with the TGD based model or the chapter with the same title.

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

Articles and other material related to TGD.

The 4 anomalies of BCS model of superconductivity in TGD framework

The article of Koizumi \cite{BerrySC} mentions 4 anomalies of the BCS model of superconductivity (SC) (no generally accepted model of high-Tc SC exists). Besides the absence of the difference of chemical potentials in the condition defining Josephson frequencies, 3 other anomalies are mentioned. These anomalies do not plague the TGD based model. The basic reason is that Cooper pairs reside at the magnetic flux tubes.

1. There is only one transition temperature in the BCS model of SC whereas high-T_c superconductivity involves 2 transition temperatures. Above critial temperature would be that the gap energy is negative above critical temperature so that the energy liberated in the formation of Cooper pairs cannot provide the energy needed to increase h_eff.

   In the TGD framework the first transition temperature leads to a superconductivity but in spatial and time scales (proportional to h_eff), which are so short that macroscopic super-conductivity is not possible. In the lower transition temperature h_eff increases and the flux tubes reconnect in a stable manner to longer flux tubes. The instability of this phase at critical temperature would be due to the geometric instability of the flux tubes.

2. London moment depends on the real electron mass m_e rather than the effective mass m_e^* of the electron. This effect relates to a rotating magnet. There is a supra current in the boundary region creating the magnetic moment. The explanation is that the electrons resulting from the splitting of Cooper pairs at the flux tubes of magnetic field do not interact with the ordinary condensed matter so that the mass is m_e.
3. For SCs of type I, the reversible phase transition from SC to ordinary phase in an external magnetic field does not cause dissipation. One would expect that the splitting of Cooper pairs produces electrons, which continue to flow and dissipate in collisions with the ordinary condensed matter. The reversibility of the phase transition can be understood if the electrons continue to flow at the flux tubes as supracurrents.
4. Magnetic flux tubes also solve the anomaly related to chemical potential: chemical potentials are present but not at the level of magnetic flux tubes so that the erratic calculation gives a correct result in the standard approach.

See the article Comparing the Berry phase model of super-conductivity with the TGD based model or the chapter with the same title.

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

Articles and other material related to TGD.

Beltrami flow as space-time correlate for non-dissipative flow

In the standard model of superconductivity SC is characterized by a complex order parameter for which the Berry phase would serves as an analog in BPM. Berry phase is a consequence of adiabaticity and characterizes collective phase. One can assign to the Berry phase effective U(1) gauge field which reduces to magnetic field in a static situation. What are the TGD counterparts of these notions?

TGD provides the geometrization of classical physics in terms of space-time surfaces carrying gravitational and standard model fields as induced fields so that both the supra current and the phase should have geometric intepretation. This serves as a powerful constraint on the model.

1. Supra current must correspond to a flow. The flow must be integrable in the sense that the coordinate defined along flow lines defines a global coordinate at flux tubes. One can indeed argue that an operational defition of a coordinate system requires that coordinates correspond to coordinates varying along flow lines of some physical flow. The exponential of the coordinate would define the phase factor of the complex order parameter such that its gradient defines the direction of the supracurrent.

   If the motion of particles is random one cannot talk of a hydrodynamic flow but something analogous to the motion of gas particles or Brownian motion. In the TGD framework this situation corresponds to disjoint space-time sheets as a representation of particle orbits. The flow property could however hold true inside the "pieces" of space-time. The coherence scales of flow would become short.

2. One must make it clear that here an approximation is made. Elementary particles have as building bricks wormhole contacts defining light-like partonic orbits to which one can assign light-like curves as M⁴ projections. For a vanishing value \Lambda=0 of cosmological constant (real analytic functions at M⁸ level), these curves are light-like (light-likeness condition reduces to Virasoro conditions) whereas for \Lambda>0 (real polynomials) at M⁸ level the projections consist of pieces which are light-like geodesics somewhat like in the twistor diagrams \cite{minimal}. Smooth curve is replaced with its approximation.

   For massive particles, this orbit would be analogous to zitterbewegung orbit and the motion in the long scales would occur with velocity v<c: this provides a geometric description of particle massiation. The supracurrent would not actually correspond to the flow as such but to CP₂ type extremals along the flow lines.

3. The 4-D generalization of so called Beltrami flow \cite{Beltrami,Beltramia,Beltramib,Beltramic}, which defines an integrable flow in terms of flow lines of magnetic field, could be central in TGD. Superfluid flows and supra currents could be along flux lines of Beltrami flows defined by the Kähler magnetic field \cite{class,prext}.

   If the Beltrami property is universal, one must ask whether even the ordinary hydrodynamics flow could represent Beltrami flow with flow lines interpreted in terms of flow lines Kähler magnetic field appearing as a a part of classical Z⁰ field. Could hydrodynamical flow be stabilized by a superfluid made of neutrino Cooper pairs. h_eff hierarchy of dark matters in turn inspires the question whether weak length scale could be scaled up to say cellular length scales (neutrino mass corresponds to a length scale of a large neuron).

4. The integrability condition

   j∧ dj=0

   of the Beltrami flow states that the flow is of form

   j= Ψ dΦ ,

   where Φ and Ψ are scalar functions, which means that Ψ defines a global coordinate varying along the flow lines.

5. Beltrami property means that the classical dissipation characterized by the contraction of the Kähler current

   j^α=D_βJ^αβ

   with Kähler form J_αβ is absent:

   j^βJ_αβ=0 .

   In absence of Kähler electric field (stationary situation), this condition states the 3-D current is parallel with the magnetic field that it creates.

   In 4-D case, the orthogonality condition guarantees the vanishing of the covariant divergence of the energy momentum tensor associated with the Kähler form. This condition is automatically true for the volume part of the energy momentum tensor but not for the Kähler part, which is essentially energy momentum tensor for Maxwell's field in the induced metric. As far as energetics is considered, the system would be similar to Maxwell's equations.

   The vanishing of the divergence of the energy momentum tensor would support Einstein's equations expected at QFT limit of TGD when many-sheeted space-time is approximated with a slightly curved region of M⁴ and gauge and gravitational fields are defined as the sums of correspond induced fields (experienced by test particles touching all space-time sheets).

6. An interesting question is whether Beltrami condition holds true for all preferred extremals \cite{prext} \cite{minimal}, which have been conjectured to be minimal surfaces analogous to soap films outside the dynamically generated analogs of frames at which the minimal surface property fails but the divergences of isometry currents for volume term and Kähler action have delta function divergences cancelling each other. The Beltrami conditions would be satisfied for the minimal surfaces.

   If the preferred extremals are minimal surfaces and simultaneous extremals of both the volume term and the Kähler action, one expects that they possess a 4-D analog of complex structure \cite{minimal}: the identification of this structure would be as Hamilton-Jacobi structure \cite{prext} to be discussed below.

7. Earlier I have also proposed that preferred extremals involving light-like local direction as direction of the Kähler current and orthogonal local polarization direction. This conforms with the fact that Kähler action is a non-linear generalization of Maxwell action and minimal surface equations generalize massless field equations. Locally the solutions would look like photon like entities.

   This inspires the question whether all preferred extremals except CP₂ type extremals defining basic building bricks of space-time surfaces in H have a 2-D or 3-D CP₂ projection and allow interpretation as thickening of flux tubes? CP₂ type extremals have 4-D CP₂ projection and light-like M⁴ projection and an induced metric with an Euclidean signature.

   See the article Comparing the Berry phase model of superconductivity with the TGD based model or the chapter with the same title.

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

   Articles and other material related to TGD.

Non-dissipative waves in excitonic insulators: a connection with superconductivity?

This comment was inspired by a popular article, which tells that in excitonic insulators, very fast waves with velocity about v∼ .01c, are detected. What caught my attention is that these waves do not dissipate. The theoretical challenge is to explain why this the case. The absence of dissipation means an analogy with superconductors.

I have just worked out the newest version of the TGD based model of superconductivity (see this) with an inspiration coming from the Berry phases model, in particular the anomalies of the BCS model mentioned in the article describing the model.

1. The model suggests a universal framework applying not only to super-conductivity but also to super-fluidity and various phenomena involving absence of dissipative effects.
2. The model predicts that also electrons rather than only Cooper pairs can propagate without dissipation at magnetic flux tubes at which $h_eff> h$ electrons and their Cooper behaving effectively like dark matter are. Also the Berry phase model predicts this.
3. Second prediction is that by external energy feed it is possible to have superconductivity also above Tc: this mechanism (metabolic energy feed) is the basic mechanism of TGD inspired quantum biology making possible high Tc superconductivity.
4. An attractive assumption is that the flux tubes mediated gravitational interaction: in this case one would have h<_eff =h_gr= GMm/v₀, where M is Earth mass, m is the mass of charge carrier, and v₀ is velocity parameter with at Earth surface has value v₀=c/2 giving for the universal gravitational Compton length the value λ_gr= 2GM =r_s, the Scwartshild radius, which is .9 cm for Earth. This would predict universality for various supraphases. Intriguingly, for Sun and inner planets one has v₀= about 2^-11 and λ_gr is very near to the radius of Earth!

Excitonic insulators are described in a second popular article telling about their discovery (see this . They exist in a phase transition region between insulator and conductor as the gap between valence band and conduction band becomes zero. In the TGD framework this means quantum criticality and the presence of h_eff> h phases are associated with the long range correlations and fluctuations at criticality quite generally.

The physical picture looks very similar to that in super-conductivity.

1. Instead of Cooper pairs, one could have bound states of electron and hole bound by Coulomb interaction. The gap energy approaches zero at critical temperature in both cases. For a superconductor the gap energy corresponds to the energy needed to kick out an electron or Cooper pair formed at the level of ordinary matter to the magnetic flux tube with h_eff> h (increase of h_eff increases the energy of the state). The liberated binding energy - gap energy - allows the kicking. The gap energy is negative above Tc and superconductivity is not possible.

   The same would apply also in the case of excitonic insulators. The formation of the bound states of h_eff> helectrons and holes would liberate the binding energy allowing kicking of something to the magnetic flux with h_eff> h.

2. What is this something? The high velocity v ∼ 10^-2c non-dissipating charge neutral waves are observed. v is much higher than sound velocity (or order 10^-4c roughly). The Fermi velocity for electrons for E_F< 10 eV gives a correct order of magnitude so that some kind of charge density waves of this something at flux tubes could be in question. Could this something be Cooper pairs and/or electrons? One would have something resembling superconductivity as a quantum coherent state phase of Cooper pairs.

   The experimentalists believe that the non-dissipating waves are charge neutral - probably because one has an insulator. Is charge neutrality necessary if flux tubes can serve as carriers of dark currents?

For background, see the article Comparing the Berry phase model of super-conductivity with the TGD based model or the chapter with the same title.

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

Articles and other material related to TGD. 


Comparing the Berry phase model of super-conductivity with the TGD based model

Hiroyasu Koizumi (see this) has proposed a new theory of superconductivity (SC) based on the notion of Berry phase related with an effective magnetic field assignable to adiabatically evolving systems. The model shares similarities with the TGD inspired view about SC. The article also mentioned anomalies that were new to me. This motivated a fresh look in the TGD inspired model. The outcome was an integration of two separate ideas about supraphases.

1. Space-time surfaces as preferred extremals with CP₂ projection of dimension D=2 or D=3 would naturally correspond to 4-D generalizations of so called Beltrami flows, which are integrable flows defined by the flow lines of the induced K\"ahler field. The existence of a global coordinate z varying along flow lines requires the integrability of the flow. Classical dissipation is absent so that these surfaces are excellent candidates for the space-time correlates of supra flows. The exponential of z gives a phase factor associated with the complex order parameter of a coherent state of Cooper pairs as a counterpart of the Berry phase. K\"ahler magnetic monopole flux defines the TGD counterpart of "novel" magnetic field.
2. The identification of supra phases as dark matter as h_eff>h phases at magnetic flux quanta (tubes and sheets) implies that Cooper pairs correspond to dark fermions associated with the members of flux tube pair, which actually combine to form a closed flux tube. Also single electrons can define supraflow.
3. The Cooper pairs must be created by bosonic oscillator operators constructed from fermionic oscillator operators by bosonization. This is possible only in 1+1-dimensional situations. Thanks to the Beltrami flow the situation is effectively 1+1-dimensional. Bosonization makes it possible to identify SU(2) Kac-Moody algebra, which has an interpretation in the TGD framework.

The assumption that Cooper pairs reside at the magnetic flux quanta solves the 4 problems of standard framework mentioned by Koizumi: high-Tc SCs have two transition temperatures; electron mass m_e instead of its effective mass m_e^* appears in Thomson moment; the reversible phase transition in an external magnetic field inducing a splitting of Cooper pairs does not involve dissipation; why the erratic calculation of the Josephson frequencies in standard model neglecting the chemical potentials gives a correct result?.

The formation of the Cooper pairs appears as a condition stabilizing the space-time sheets carrying dark matter and all preferred extremals could satisfy the conditions guaranteeing integrable flow and existence of a phase factor varying along flow lines. Could supra phases exist in all scales? Could the breaking of supra phases be only due to the finite size of the space-time sheets? Could even hydrodynamic flow involve super-fluidity of some kind - perhaps based on neutrino Cooper pairs as speculated earlier?

See the article Comparing the Berry phase model of super-conductivity with the TGD based model or the chapter with the same title.

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

Articles and other material related to TGD. 


Galois groups and genetic code

Galois groups are realized as number theoretic symmetry groups realized physically in TGD a symmetries of space-time surfaces. Galois confinement as an analog of color confinement is proposed in TGD inspired quantum biology .

Galois groups, in particular simple Galois groups, play a fundamental role in the TGD view of cognition. The TGD based model of the genetic code involves in an essential manner the groups A₅ (icosahedron), which is the smallest non-abelian simple group, and A₄ (tetrahedron). The identification of these groups as Galois groups leads to a more precise view about genetic code. The question  why the genetic code is a fusion of 3 icosahedral codes and of only a  single tetrahedral code  remained however poorly understood. 

  The identification of the symmetry groups  of  the I, O, and T  as Galois groups  makes it possible  to  answer this question. Icosa-tetrahedral tesselation of 3-D hyperbolic space H³, playing centrl role in TGD, can be replaced with its 3-fold covering replacing  I/O/T with the corresponding symmetry group acting as a Galois group.  T has only  only a single Hamiltonian cycle and  its 3-fold covering  behaves effectively as a  single cycle. Octahedral codons can be regarded as icosahedral and tetrahedral codons so they do not contribute to the code.

See the article Galois groups and genetic code.

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

Articles and other material related to TGD.

About the role of Galois groups in TGD framework

The inverse problem of Galois theory is highly interesting from TGD viewpoint. Galois groups are realized as number theoretic symmetry groups realized physically in TGD a symmetries of space-time surfaces. Galois confinement is as analog of color confinement is proposed in TGD inspired quantum biology .

Two instances of the inverse Galois problem, which are especially interesting in TGD, are following:

Q1: Can a given finite group appear as Galois group over Q? The answer is not known.

Q2: Can a given finite group G appear as a Galois group over some EQ? Answer to Q2 is positive as will be found and the extensions for a given G can be explicitly constructed.

The TGD based formulation based on M⁸-H duality in which space-time surface in complexified M⁸ are coded by polynomials with rational coefficients involves the following open question.

Q: Can one allow only polynomials with coefficients in Q or should one allow also coefficients in EQs?

The idea allowing to answer this question is the requirement that TGD adelic physics is able to represent all finite groups as Galois groups of Q or some EQ acting physical symmetry group.

If the answer to Q1 is positive, it is enough to have polynomials with coefficients in Q. It not, then also EQs are needed as coefficient fields for polynomials to get all Galois groups. The first option would be the more elegant one.

The inverse problem is highly interesting from the perspective of TGD. Galois groups, in particular simple Galois groups, play a fundamental role in the TGD view of cognition. The TGD based model of the genetic code involves in an essential manner the groups A₅ (icosahedron), which is the smallest simple and non-commutative group, and A₄ (tetrahedron). The identification of these groups as Galois groups leads to a more precise view about genetic code and answers to a key open question of the model in its recent form.

About the role of Galois groups in TGD framework or the chapter with the same title.

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

Articles and other material related to TGD.