Monday, July 19, 2021

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

    LL/R= J+/- I

    is non-vanishing. For c≠ 1, both JL and JR 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 ∂αcIα=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, JK, J and I are sums of M4 and CP2 parts:

    AK= A(M4)+A(CP2),
    JK=JK(M4)+JK(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:

    αIα=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.

    DαIα=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. DαIα≠ 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 Tc by using energy feed providing the energy needed to increase the value of heff. 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 ωD using the already discussed formula ωD = kn cs/a, where kn depends on the effective dimension of the lattice like system and has values kn ∈ {3.14,3.54,2.66} for n=1,2,3. Sound velocity cs 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 ED= .044 eV which is in the range of neuronal membrane potentials.

  3. The role of ℏgr and Bend 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 Bend with its value in Mars? There is evidence that Bend 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 \Lambdagr= rS=2GM\simeq 9 mm to overlap and form a quantum coherent state. For Sun one has \Lambdagr= 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-Tc 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 heff.

    In the TGD framework the first transition temperature leads to a superconductivity but in spatial and time scales (proportional to heff), which are so short that macroscopic super-conductivity is not possible. In the lower transition temperature heff 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 me rather than the effective mass me* 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 me.
  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. heff 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


    with Kähler form Jαβ is absent:

    jβJαβ=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.

Sunday, July 18, 2021

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 $heff> h$ 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 =hgr= 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 λgr= 2GM =rs, 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 λgr 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 heff> 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 heff> h (increase of heff 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 heff> helectrons and holes would liberate the binding energy allowing kicking of something to the magnetic flux with heff> 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 EF< 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. 

Thursday, July 15, 2021

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 heff>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 me instead of its effective mass me* 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. 

Saturday, July 10, 2021

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.

Thursday, July 08, 2021

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.