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.

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

One can represent two objections against the number theoretic vision.

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

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

2. All number fields except finite fields, that is rationals and their extension, p-adic numbers and their extensions, reals, complex numbers, quaternions, and octonsions appear at the fundamental level in TGD. Could there be a manner to make also finite fields a natural part of TGD?

These problems raise the question of whether one could pose additional conditions to the polynomials P of degree n defining 4-surfaces in M⁸ with roots defining mass shells in M⁴⊂ M⁸ (complexification assumed) mapped by M⁸-H duality to space-time surfaces in H.

1. P=Q condition

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

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

2. Additional conditions

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

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

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

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

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

Option II would solve the two problems whereas Option II is un-necessarily strong.

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

3. How does Option II relate to prime polynomials?

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

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

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

These observations suggest the tightening of the Option II to the following condition.

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

4. Examples of Option II

The following examples illustrate the conditions for Option II.

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

      A little calculation shows that extensions of rationals containing i, 2^1/2, i2^1/2, 3^1/2, i3^1/2, 5^1/2 (from P=x²+x-1 defining Golden Mean), and i7^1/2 are obtained.

   3. Roots of small primes appear in the Weyl groups, which are reflection groups associated with Dynkin diagrams characterizing Lie groups at Lie algebra level. The finite discrete subgroups of the rotation group SU(2) characterized extensions of hyper-finite factors of type II₁ and roots of small primes appear in the matrix elements of these groups. Could the proposed polynomials give in a natural way rise to the extensions of rationals appearing in these two cases?

The above considerations inspire further questions. Could one also allow polynomials P having coefficients in an algebraic extension of rationals? Does this bring in anything new? Could one have coefficients in an extension containing e or even root of e as perhaps the only transcendental extension defining a finite extension of p-adic numbers? The roots would be generalizations of algebraic numbers involving e and could make sense p-adically via Taylor expansion.

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

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

Articles related to TGD.

Shocking support for the quantum brain in TGD sense

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

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

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

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

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

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

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

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

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

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

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

5. The quantum correlations due to the large value of h_gr, which corresponds to gravitational Compton length Λ_gr = GM/v₀ is for Earth mass and for v₀/c∼ 1 about .45 cm. In accordance with the Equivalence Principle, Λ_gr has no dependence on the "small" mass m , and also cyclotron energy is independent of m. Interestingly, Λ_gr happens to correspond to the size scale of a snowflake (see this), which has a scaled up variant of the symmetry of the unit cell of 2-D hexagonal layer of ice. This remains a mystery in the standard physics framework.

The striking finding is that the MRI pulse frequency corresponds to that of heart beat, which is between 1-1.67 Hz and .67 Hz for a trained athlete. Why not some EEG resonance frequency such as alpha frequency about 10 Hz?

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

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

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

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

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

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

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

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

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

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

Articles related to TGD.

p-Adic ξ function from Hadamard product

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

ξ(s)=(1/2)∏_k (1-s/s_k )(1-s/(s_k^*) .

Only the non-trivial zeros appear in the product.

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

   D=∏_{k≠ l}(s_k-s_l)(s_k-s_l^*)

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

3. The Hadamard product can be written in the form

   ξ(s)=(1/2)∏_k (1+ s(s-1)/X_k) , X_k= s_ks^*_k ,

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

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

   If the coefficients a_n of the powers series ∑_n a_n uⁿ are algebraic numbers of the assumed extension or ordinary rationals, the power series in s converges for s=O(p) under rather mild conditioons. For instance, the coefficient of the zeroth order term is 1/2. The coefficient of first order term in u is -(1/2)∑_k (s_ks^*_k) ^-1= -2∑_k (1+4y_k²)^-1.

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

4. Algebraic integers are roots of monic polynomials with integer coefficients such that the coefficient of the leading order term is equal to 1. By multiplying ξ by the product Y=∏ s_ks^*_k one obtains an expression, which is formally a monic polynomial if one can regard Y as a p-adic integer, which is p-adically finite for every p. Formally this operation does not affect the values of the roots.

One can deduce formal expressions for the Taylor coefficiens of ξ(s).

1. Taking u=s(s-1) to be the variable, the coefficients of uⁿ in ξ(s)= ξ₁(u) are given by

   ∑_Un ∏_{k∈ Un} X_k^-1 , X_k=s_ks^*_k .

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

   c₁= (1/2)∑_iX_i^-1 ,

   c₂= (1/2)∑_{i≠ j}X_i^-1X_j

   =(1/2)∑_i,jX_i^-1X_j -(1/2) ∑_iX_i^-2

   = (1/2)c₁² - ∑_iX_i^-2 .

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

3. Also the higher coefficients c_n can be calculated in a similar way recursively by subtracting from the sum ∑_i1...in ∏_ik X_i1^-1= c₁ⁿ without the constraint p_i≠ p_j≠ ... the sums for which 2,3,...,n primes are identical. One obtains a sum over all partitions of U_n. A given partition {i₁, ..., i_k} contributes to the sum the term

   d_{i1, ... , ik}∏_l=1^k c_il ,

   ∑_i=1^k n_i=n .

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

   The outcome is that the expressions of c_n reduce to the calculation of the numbers A_k=∑_i X_i^-k.

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

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

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

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

1. One could argue that the coefficients c_n must obey a number theoretical democracy, which would mean that they can distinguish p-adically only between the set of primes p_k appearing as divisors of n and the remaining primes. One could require that c_n is a number in a finite-D extension of rationals involving only rational primes dividing n.
2. One could pose an even stronger condition: the coefficients c_n must belong to an n-D algebraic extension of rationals and thus be determined by a polynomial of degree n. Polynomials P of rational coefficients p_n bring in failure of the number theoretic democracy unless one has p_n∈ {0,+/- 1}. For p=2 one does not obtain algebraic numbers. For p=3 this would bring in 5^1/2.
3. These conditions would guarantee that for a given prime p the coefficients of the expansion would be unaffected by the canonical identification I and at the limit p→ &infty; the Taylor coefficients of p-adic ξ_p would be identical with those of ξ.
4. One could allow finite-D transcendental extensions of p-adic numbers. These exist. Since e^p is an ordinary p-adic number, there is an infinite number of extensions with a basis given by the powers roots e^k/n, k=1,..., np-1 define a finite-D transcendental extension of p-adics for every prime p.

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

This picture suggests a connection with the hyperbolic geometry H² of the upper half-plane, which is associated with ζ and ξ via Langlands correspondence.

1. The simplest option is that the roots of P_k correspond to the k:th roots x_i of unity satisfying x_i^k=1 so that cos(n2π/k) and sin(n2π/k) would appear as coefficients in the expression of c_k. The numbers e^k/n would be hyperbolic counterparts for the roots of unity.
2. The coefficients c_k would be of form

   c_k= ∑_i,jc_k,ije^i/k e^{(-1) 1/2 2π (j/n)} ,

   c_k,rs ∈ {0,+/- 1} .

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

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

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

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

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

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

Just for curiosity and to see how the proposal could fail, one can look at what happens for the first coefficient c₁ in ξ(s)= ξ₁(s(s-1))= ∑ c_nsⁿ.

1. c₁ would be exceptional since it cannot depend on any prime. c₂ could involve only p=2, and so on.
2. The only way out of the problem is to allow finite-D transcendental extensions of p-adic numbers. These exist. Since e^p is an ordinary p-adic number, there is an infinite number of extensions with a basis given by the powers roots e^k/n, k=1,..., np-1 define a finite-D transcendental extension of p-adics for every prime p. For ξ the extension by roots of unity could be infinite-dimensional.

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

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

   c₁=∑ a_ke^-k/n=x₁/2 .

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

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

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

Articles related to TGD.

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

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

This vision inspires the question whether the generalization of the number theoretic view of TGD so that besides rational polynomials (subject to some restrictions) also L-functions, which have a nice physical interpretation if RH holds true for them, can be defined via their roots 4-surfaces in M⁸_c and by M⁸-H duality 4-surfaces in H. Both conformal confinement (in weak and strong form) and Galois confinement (having also weak and strong form) support the view that L-functions are Langlands duals of the partition functions defining quantum states.

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

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

   If space-time has boundaries, the boundaries X² of X³⊂ H³ could be ends of light-like 3-surfaces X³_L (see this). An attractive idea is that they are hyperbolic manifolds or pieces of a tessellation defined by a hyperbolic manifold as the analog of a unit cell (see this). The ends X² of these 3-surfaces at the boundaries of CD would define partonic 2-surfaces.

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

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

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

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

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

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

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

6. One could assign to these representations modular forms interpreted as generalized partition functions, kind of complex square roots of thermodynamic partition functions. Quantum TGD can be indeed formally regarded as a complex square root of thermodynamics. This partition function could define a ground state for a space of zero energy state defined in WCW as a superposition over different light-like 3-surfaces.

These considerations boil down to the following questions.

1. Could the quantum states at WCW level have classical space-time correlates as space-time surfaces, which would be defined by the L-functions associated with the modular forms assignable to finite-D representations of Galois group having a physical interpretation as partition functions?
2. Could this give rise to a kind of feedback loop representing increasingly higher abstractions as space-time surfaces. This sequence could continue endlessly. This picture brings in mind the hierarchy of infinite primes (see this).

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

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

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

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

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

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

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

Articles related to TGD.

Dwarf galaxies do not have dark matter halo

Evidence for the failure of dark halo model has been steadily accumulating during years. The popular article New Discovery Indicates an Alternative Gravity Theory published in SciTechDaily tells of the most recent discovery challenging the halo model. The dwarf galaxies of one of Earth s closest galaxy clusters do not behave as the halo model predicts.

Elena Asencio, a Ph.D. student at the University of Bonn was the lead author of the article "The distribution and morphologies of Fornax Cluster dwarf galaxies suggest they lack dark matter by Elena Asencio, Indranil Banik, Steffen Mieske, Aku Venhola, Pavel Kroupa and Hongsheng Zhao published in 25 June 2022, Monthly Notices of the Royal Astronomical Society (see this.

The presence of dark matter halo surrounding dwarf galaxy shields the dwarf galaxy from tidal forces by acting as a kind of matress. What was found was that the tidal forces are too large to be consistent with the presence of dark matter halo. In the TGD framework, dark matter halo is replaced with a long cosmic string producing automatically the flat velocity spectrum. There is no shielding.

See the article Cosmic string model for the formation of galaxies and stars or the chapter with the same title.

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

Articles related to TGD.

Why do meteors always land in craters?

Why do meteors always land in craters? I encountered this innocent layman question on Face Book and from the TGD perspective it looked brilliant. I did a web search and found this question at some pages accompanied by strong emotional responses in style "craters are of course made by meteorites, you idiot!".

This is of course true and one must formulate the question more precisely. One must characterize meteor crater by size. Suppose that smaller craters assigned to meteors indeed have a tendency to appear inside larger craters. One would have a fractal like structure.

What is known about the size destribition of meteor craters and its correlation with the distribution of their locations? Is the fractal structure only an illusion: is it easier to spot the crateres if they are inside craters? Or is this tendency real? I do not know for certain but I can make what-if... questions.

I have proposed a model for the craters created in meteor collisions based on the TGD view of the magnetic body of a planet, say Earth (see this).

1. The model was inspired by an anomaly: the meteor craters seem to favor meteor orbits orthogonal to the surface of the planet so that the craters tend to look like circular disks rather than ellipsoids.
2. The craters assigned with meteor collisions could be created by matter, which arrives along magnetic flux tubes roughly perpendicular to the surface of the planet. Part of the material of the meteor could end up as dark, possibly charged, matter at the magnetic flux tubes or bundle of tubes. Since the friction and electric resistance of the dark matter inside the flux tube are much smaller than for ordinary matter, dark particles could achieve very high energies before collision with Earth. This would also explain the gamma rays and ultrahigh energy electrons associated with lightning.
3. If the magnetic flux tube bundles form rather stable structures with fractal flux tubes inside flux tubes inside ... inside flux tubes, which emanate from larger craters, the meteors or the material created in their decay could tend to land in craters. This hypothesis should be testable. For instance, could lightnings have tendency to be associated with craters?

One exotic application of the idea relates to the observation that the craters in the Moon are accompanied by glass spheres (see this). Also the crop circles, which any real academic physicist of course regards as pseudoscience, involve glass spheres suggesting very high temperatures created somehow (see this).

See the article Comparing Electric Universe hypothesis and TGD or the chapter with the same title.

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

Articles related to TGD.

Some comments of the physical interpretation of Riemann zeta in TGD

The Riemann zeta function ζ and its generalizations are very interesting from the point of view of the TGD inspired physics. M⁸-H duality assumes that rational polynomials define cognitive representations as unique discretizations of space-time regions interpreted in terms of a finite measurement resolution. One implication is that virtual momenta for fermions are algebraic integers in an extension of rationals defined by a rational polynomial P and by Galois confinement integers for the physical states.

In principle, also real analytic functions, with possibly rational coefficients, make sense. The notion of conformal confinement with zeros of ζ interpreted as mass squared values and conformal weights, makes ζ and L-functions as its generalizations physically unique real analytic functions.

If the conjecture stating that the roots of ζ are algebraic numbers is true, the virtual momenta of fermions could be algebraic integers for virtual fermions and integers for the physical states also for ζ. This makes sense if the notions of Galois group and Galois confinement are sensible notions for ζ.

In this article, the properties of ζ and its symmetric variant ξ and their multi-valued inverses are studied. In particular, the question whether ξ might have no finite critical points is raised.

See the article Some comments of the physical interpretation of Riemann zeta in TGD.

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

Articles related to TGD.