Ultradiffuse galaxies as a problem of cold dark matter and MOND scenarios

The existence of ultra-diffuse galaxies for which the velocity of distant rotating stars is extremely low (see this), means difficulties for the cold dark matter scenario since in some cases there seems to be no dark matter at all, and in some cases there seems to be only dark matter. These objects have been proposed as a support for MOND, but also MOND has grave difficulties with them.

In the TGD framework (see this, this, and this), the rotation velocity is proportional to the square root of the product GT, where T is the string tension of a long magnetic flux tube formed from a cosmic string carrying dark energy and possibly also matter. No dark matter halo is needed. In the ordinary situation, the flux tube would be considerably thickened only in a tangle associated with the galaxy as part of volume- and magnetic energies would have decayed to ordinary matter, in analogy with the decay of the inflaton field.

Whereas the dark halo model predicts that any rotation plane for the distant stars is possible, TGD allows only the galactic plane orthogonal to the long cosmic string, and distant stars move in the general case along helical orbits along the cosmic string.

The TGD view also allows us to understand why MOND works in some cases: in the TGD framework the critical acceleration a_cr, suggested originally to be constant of Nature, is replaced with string tension.

If the flux tube itself has a very long thickened portion such that ordinary matter has left this region by free helical motion along the string or by gravitational attraction of some other object, the string tension T is small and very small rotation velocity is possible. Ordinary bound states of matter are not necessary since the gravitational force of the flux tubes binds the stars to the flux tube. This might explain why the galaxy can be ultradiffuse.

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

Articles related to TGD.

Is pair creation really understood in the twistorial picture?

Twistorialization leads to a beautiful picture about scattering amplitudes at the level of M⁸ (see this). In the simplest picture, scattering would be just a re-organization of Galois singlets to new Galois singlets. Fundamental fermions would move as free particles.

The components of the 4-momentum of virtual fundamental fermion with mass m would be algebraic integers and therefore complex. The real projection of 4-momentum would be mapped by M⁸-H duality to a geodesic of M⁴ starting from either vertex of the causal diamond (CD) . Uncertainty Principle at classical level requires inversion so that one has a= ℏ_eff/m, where ab denotes light-cone proper time assignable to either half-cone of CD and m is the mass assignable to the point of the mass shell H³⊂ M⁴⊂ M⁸.

The geodesic would intersect the a=ℏ_eff/m 3-surface and also other mass shells and the opposite light-cone boundaries of CDs involved. The mass shells and CDs containing them would have a common center but Uncertainty Principle at quantum level requires that for each CD and its contents there is an analog of plane wave in CD cm degrees of freedom.

One can however criticize this framework. Does it really allow us to understand pair creation at the level of the space-time surfaces X⁴⊂ H?

1. All elementary particles consist of fundamental fermions in the proposed picture. Conservation of fermion number allows pair creation occurring for instance in the emission of a boson as fermion-antifermion pair in f→ f+b vertex.
2. The problem is that if only non-space-like geodesics of H are allowed, both fermion and antifermion numbers are conserved separately so that pair creation does not look possible. Pair creation is both the central idea and source of divergence problems in QFTs.
3. One can identify also a second problem: what are the anticommutation relations for the fermionic oscillator operators labelled by tachyonic and complex valued momenta? Is it possible to analytically continue the anticommutators to complexified M⁴⊂ H and M⁴⊂ M⁸? Only the first problem will be considered in the following.

Is it possible to understand pair creation in the proposed picture based on twistor scattering amplitudes or should one somehow bring the bff 3-vertex or actually ff fbar fbar vertex to the theory at the level of quark lines? This vertex leads to a non-renormalizable theory and is out of consideration.

One can first try to identify the key ingredients of the possible solution of the problem.

1. Crossing symmetry is fundamental in QFTs and also in TGD. For non-trivial scattering amplitudes, crossing moves particles between initial and final states. How should one define the crossing at the space-time level in the TGD framework? What could the transfer of the end of a geodesic line at the boundary of CDs to the opposite boundary mean geometrically?
2. At the level of H, particles have CP₂ type extremals - wormhole contacts - as building bricks. They have an Euclidean signature (of the induced metric) and connect two space-time sheets with a Minkowskian signature.

   The opposite throats of the wormhole contacts correspond to the boundaries between Euclidean and Minkowskian regions and their orbits are light-like. Their light-like boundaries, orbits of partonic 2-surfaces, are assumed to carry fundamental fermions. Partonic orbits allow light-like geodesics as possible representation of massless fundamental fermions.

   Elementary particles consist of at least two wormhole contacts. This is necessary because the wormhole contacts behave like Kähler magnetic charges and one must have closed magnetic field lines. At both space-time sheets, the particle could look like a monopole pair.

3. The generalization of the particle concept allows a geometric realization of vertices. At a given space-time sheet a diagram involving a topological 3-vertex would correspond to 3 light-like partonic orbits meeting at the partonic 2-surface located in the interior of X⁴. Could the topological 3-vertex be enough to avoid the introduction of the 4-fermion vertex?

Could one modify the definition of the particle line as a geodesic of H to a geodesic of the space-time surface X⁴ so that the classical interactions at the space-time surface would make it possible to describe pair creation without introducing a 4-fermion vertex? Could the creation of a fermion pair mean that a virtual fundamental fermion moving along a space-like geodesics of a wormhole throat turns backwards in time at the partonic 3-vertex. If this is the case, it would correspond to a tachyon. Indeed, in M⁸ picture tachyons are building bricks of physical particles identified as Galois singlets.

1. If fundamental fermion lines are geodesics at the light-like partonic orbits, they can be light-like but are space-like if there is motion in transversal degrees of freedom.
2. Consider a geodesic carrying a fundamental fermion, starting from a partonic 2-surface at either light-like boundary of CD. As a free fermion, it would propagate to the opposite boundary of CD along the wormhole throat.

   What happens if the fermion goes through a topological 3-vertex? Could it turn backwards in time at the vertex by transforming first to a space-like geodesic inside the wormhole contact leading to the opposite throat and return back to the original boundary of CD? It could return along the opposite throat or the throat of a second wormhole contact emerging from the 3-vertex. Could this kind of process be regarded as a bifurcation so that it would correspond to a classical non-determinism as a correlate of quantum non-determinism?

3. It is not clear whether one can assign a conserved space-like M⁴ momentum to the geodesics at the partonic orbits. It is not possible to assign to the partonic 2-orbit a 3-momentum, which would be well-defined in the Noether sense but the component of momentum in the light-like direction would be well-defined and non-vanishing.

   By Lorentz invariance, the definition of conserved mass squared as an eigenvalue of d'Alembertian could be possible. For light-like 3-surfaces the d'Alembertian reduces to the d'Alembertian for the Euclidean partonic 2-surface having only non-positive eigenvalues. If this process is possible and conserves M⁴ mass squared, the geodesic must be space-like and therefore tachyonic.

   The non-conservation of M⁴ momentum at single particle level (but not classically at n-particle level) would be due to the interaction with the classical fields.

4. In the M⁸ picture, tachyons are unavoidable since there is no reason why the roots of the polynomials with integer coefficients could not correspond to negative and even complex mass squared values. Could the tachyonic real parts of mass squared values at M⁸ level, correspond to tachyonic geodesics along wormhole throats possibly returning backwards along the another wormhole throat?

How does this picture relate to p-adic thermodynamics (see this) as a description of particle massivation?

1. The description of massivation in terms of p-adic thermodynamics suggests that at the fundamental level massive particles involve non-observable tachyonic contribution to the mass squared assignable to the wormhole contact, which cancels the non-tachyonic contribution.

   All articles, and for the most general option all quantum states could be massless in this sense, and the massivation would be due the restriction of the consideration to the non-tachyonic part of the mass squared assignable to the Minkowskian regions of X⁴.

2. p-Adic thermodynamics would describe the tachyonic part of the state as "environment" in terms of the density matrix dictated to a high degree by conformal invariance, which this description would break. A generalization of the blackhole entropy applying to any system emerges and the interpretation for the fact that blackhole entropy is proportional to mass squared. Also gauge bosons and Higgs as fermion-antifermion pairs would involve the tachyonic contribution and would be massless in the fundamental description.
3. This could solve a possible and old problem related to massless spin 1 bosons. If they consist of a collinear fermion and antifermion, which are massless, they have a vanishing helicity and would be scalars, because the fermion and antifermion with parallel momenta have opposite helicities. If the fermion and antifermion are antiparallel, the boson has correct helicity but is massive.

   Massivation could solve the problem and p-adic thermodynamics indeed predicts that even photons have a very small thermal mass. Massless gauge bosons (and particles in general) would be possible in the sense that the positive mass squared is compensated by equally small tachyonic contribution.

4. It should be noted however that the roots of the polynomials in M⁸ can also correspond to energies of massless states. This phase would be analogous to the Higgs=0 phase. In this phase, Galois symmetries would not be broken: for the massive phase Galois group permutes different mass shells (and different a=constant hyperboloids) and one must restrict Galois symmetries to the isotropy group of a given root. In the massless phase, Galois symmetries permute different massless momenta and no symmetry breaking takes place.

See the article About TGD counterparts of twistor amplitudes: part II or the chapter About TGD counterparts of twistor amplitudes.

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

Articles related to TGD.

Universe as a dodecahedron?: two decades later

I encountered a link to a popular article in Physics World with the title "Is the Universe a dodecahedron" (see this) telling about the proposal of Luminet et al that the Universe has a geometry of dodecahedron. I have commented on this finding almost 20 years ago (see this). A lot has happened during these two decades and it is interesting to take a fresh TGD inspired view.

In the TGD framework, one can imagine two starting points concerning the explanation of the findings.

1. Could there be a connection with the redshift quantization along some lines ("God's fingers"") proposed by Halton Arp (see this) and Fang-Sato. I have considered several explanations for the quantization. In TGD cosmic=time constant surface corresponds to hyperbolic 3-space H³ of Minkowski space in TGD. H³ allows an infinite number of tessellations (lattice-like structures).

   I have proposed an explanation for the redshift quantization in terms of tessellations of H³. The magnetic bodies (MBs) of astrophysical objects and even objects themselves could tend to locate at the unit cells of the tessellation.

2. Icosa-tetrahedral tessellation (lattice-like structure in hyperbolic space H³) plays a key role in the TGD model of genetic code (see this) suggested to be universal. Lattice-like structures make possible diffraction if the incoming light has a wavelength, which is of the same order as the size of the unit cell.

In the sequel I will consider only the latter option.

1. In X ray diffraction, the diffraction pattern reflects the structure of the dual lattice: the same should be true now. Only the symmetries of the unit cell are reflected in diffraction. If CMB is diffracted in the tessellation, the diffraction pattern reflects the symmetries of the dual of the tessellation and does not depend on the value of the effective Planck constant h_eff. Large values of Planck constant make possible large crystal-like structures realized as part of the magnetic body having large enough size, now realized at the magnetic body (MB).
2. Icosatetrahedral tessellation plays a key role in the TGD inspired model of the genetic code. Dodecahedron is the dual of icosahedron and tetrahedron is self-dual! [Note however that also the octahedron is involved with the unit cell although "icosa-tetrahedral" does not reflect its presence. Cube is the dual of the octahedron.]

   So: could the gravitational diffraction of CMB on a local crystal having the structure of icosa-tetrahedral tessellation create the illusion that the Universe is a dodecahedron?

Could the possible dark part of the CMB radiation diffract in local tessellations assigned with the local MBs?

1. In diffraction, the wavelength of diffracted radiation must correspond to the size of the unit cell of the lattice-like structure involved. The maximum wavelength of CMB intensity as function of wavelength corresponds to a wavelength of about .5 cm. Can one imagine a tessellation with the unit cell of size about .5 cm?
2. The gravitational Planck constant ℏ_gr =GMm/β₀, where M is large mass and m a small mass, say proton mass (see this, this, this, this and this). Both masses are assignable to the monopole flux tubes mediating gravitational interaction. β₀=v₀/c is velocity parameter and near to unity in the case of Earth.
3. The size scale of the unit cell of the dark gravitational crystal would be naturally given by Λ_gr = ℏ_gr/m= GM/β₀ and would be depend on M only and would be rather large and depend on the local large mass M, say that of Earth. Λ_gr does not depend on m (Equivalence Principle).
4. For Earth, the size scale of the unit cell would be of the order of Λ_gr= GM_E/β₀ ≈ .45 cm, where β₀= 0=v₀/c ≈ 1 is near unity from the experimental inputs emerging from quantum hydrodynamics (see this) and quantum model of EEG (see this) and quantum gravitational model for metabolism (see this and this). Λ_gr could define the size of the unit cell of the icosa-tetrahedral tessellation. Note that Earth's Schwartschild radius r_S=2GM≈ .9 cm.

   Encouragingly, the wavelength of CMB intensity as a function of wavelength around .5 cm to be compared with Λ_gr ≈ .45 cm! Quantum gravitational diffraction might take place for dark CMB and give rise to the diffraction peaks!

5. Diffraction pattern would reflect astroscopic quantum coherence, and the findings of Luminet et al could have an explanation in terms of the geometry of local gravitational MB rather than the geometry of the Universe! Diffraction could also explain the strange deviations of CMB correlation functions from predictions for large values of the angular distance. It might be also possible to understand the finding that CMB seems to depend on the features of the local environment of Earth, which is in a sharp conflict with the cosmological principle. According to Wikipedia article (see this), even in the COBE map, it was observed that the quadrupole (l=2, spherical harmonic) has a low amplitude compared to the predictions of the Big Bang. In particular, the quadrupole and octupole (l=3) modes appear to have an unexplained alignment with each other and with both the ecliptic plane and equinoxes.
6. Could the CMB photon transform to a gravitationally dark photon in the diffraction? This would be a reversal for the transformation of dark photons to ordinary photons interpreted as biophotons. Also in quantum biology the transformation of ordinary photons to dark ones takes place. If so the wave length for a given CMB photon would be scaled up by the factor ℏ_gr/ℏ =(GM_Em/β₀)/ℏ ≈ 3.5× 10¹² for proton. This gives Λ=1.75 × 10⁷ km, to be compared with the radius of Earth about 6.4× 10⁶ km.

See the chapter TGD and cosmology.

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

Articles related to TGD

Antipodal duality and TGD

The so called antipodal duality has received considerable attention. The calculations of Dixon et al based on the earlier calculations of Goncharow et al suggests a new kind of duality relating color and electroweak interactions. The calculations lead to an explicit formula for the loop contributions to the 6-gluon scattering amplitude in N=4 SUSY. The new duality and relates 6-gluon amplitude for the forward scattering to a 3-gluon form factor of stress tensor analogous to a quantum field describing a scalar particle. This amplitude can be identified as a contribution to the scattering amplitude h+g→ g+g at the soft limit when the stress tensor particle scatters in forward direction. The result is somewhat mysterious since in the standard model strong and electroweak interactions are completely separate.

In TGD, there are indeed quite a number of pieces of evidence for this kind of duality but the possibility that only electroweak or color interactions could provide a full description of scattering amplitudes. The number-theoretical view of TGD could however come into rescue.

See the article Antipodal duality and TGD or the chapter About TGD counterparts of twistor amplitudes.

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

Articles related to TGD

Quantization of cosmic redshifts in the TGD framework

In a FB discussion Jivan Coquat asked for my opinion about Halton Arp (see this).

Halton Arp brings to my mind redshift quantization along lines, which has been considered also by Fang-Sato, who talked of "God's fingers". I have considered several explanations for the quantization.

1. To my opinion, the most convincing explanation is in terms of lattice-like structures, tessellations, in hyperbolic space H³, which corresponds to cosmic time a= constant hyperboloid of future light-one (see this).
2. H³ allows an infinite number of tessellations, which correspond to discrete subgroups of Lorentz group SO(1,3) having as covering SL(2,C) (spinors). In E³ only 17 lattices are possible. The so called icosa-tetrahedral tessellation is in a key role in TGD model of genetic code realized at a deeper level in terms of dark (h_eff=nh₀>h) proton triplets and flux tubes of magnetic body (see this).
3. For lattices in the Euclidean space E³, the radial distance from origin is quantized. For H³ redshift proportional to H³ distance replaces Euclidian radial distance and the tessellation gives redshift quantization. Astrophysical objects would tend to be associated with the unit cells of the tessellation.
4. The tessellation itself could be associated with the magnetic (/field) body carrying dark matter in the TGD sense as h_eff=nh₀>h phases: this is a prediction of number theoretic vision about physics as dual of geometric vision. Very large values of h_eff =h_gr= GMm/v₀ (Nottale hypothesis for gravitational Planck constant, see this) assignable to gravitational flux tubes are possible, and this makes these tessellations possible as gravitationally quantum coherent structures even in cosmological scales. This is a diametric opposite to what superstring models where quantum gravitation appears only in Planck length scale, suggests.

See the chapter (see Quantum Theory of Self-Organization.

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

Articles related to TGD

The mysterious precession of the Earth's spin axis from the TGD point of view

These comments were inspired by two interesting Youtube videos by Sören Backman (see this and this) with a provocative title "Gravity's biggest failure - precession, what is it hiding". The precession of the Earth's spin axis cannot be explained as an effect caused by other planets and Sun and even the nearest stars are too far in order to explain the precession as an effect caused by them. Precession is therefore a real problem for the standard view of gravitation.

The proposal for the explanation of the precession of Earth discussed in the videos is inspired by the notion of Electric Universe, and has several similarities with the TGD inspired model. I could expect this from my earlier discussions with the proponents of Electric Universe. About 3 years ago, I wrote a chapter inspired by these discussions (see this).

My view is that the extremist view that gravitation reduces to electromagnetism is wrong but that electromagnetism, in particular magnetic fields, have an important role even in cosmological scales. In standard physics, magnetic fields in long scales would require coherent currents, which tend to be random and dissipate. Even the understanding of the stability of the magnetic field of Earth is a challenge, to say nothing of the magnetic fields able to survive in cosmological scales. In TGD, monopole flux tubes define magnetic fields which need no currents as sources.

Consider first monopole flux tubes, which are present in all length scales in the TGD Universe and distinguish TGD from both Maxwell's electrodynamics and general relativity.

1. Flux tubes can carry monopole flux, in which case they are highly stable. The cross section is not a disk but a closed 2-surface so that no current is needed to create the magnetic flux. The flux tubes with vanishing flux are not stable against splitting.
2. Flux tubes relate to the model for the emergence of galaxies (see this and this) and explain galactic jets propagating along flux tubes (see this). Dark energy and possible matter assignable to the cosmic strings predicts correctly the flat velocity spectrum of stars around galaxies.
3. In the MOND model it is assumed that the gravitational force transforms for certain critical acceleration from 1/r² to 1/r force. In TGD this would mean that the 1/ρ forces caused by the cosmic string would begin to dominate over the 1/rho² force. The predictions of MOND TGD are different since in TGD the motion takes place in the plane orthogonal to the cosmic string.
4. The flux tubes can appear as torus-like circular loops. Also flux tube pairs carrying opposite fluxes, resembling a DNA double strand, are possible and might be favoured by stability. Flux tubes are possible in all scales and connect astrophysical structures to a fractal quantum network. The flux tubes could connect to each other nodes, which are deformations of membrane-like entities having 3-D M⁴ projections and 2-D E³ projections (time= constant) (also an example of "non-Einsteinian" space-time surface).
5. Pairs of monopole flux tubes with opposite direction of fluxes can connect two objects: this could serve as a prerequisite of entanglement. The splitting of a flux tube pair to a pair of U-shaped flux tubes by a reconnection in a state function reduction destroying the entanglement. Reconnection would play an essential role in bio-catalysis.
6. Flux tube pairs can form helical structures and stability probably requires helical structure. Cosmic analog of DNA could be in question: fractality and gravitational quantum coherence in arbitrarily long scales are a basic prediction of TGD so that monopole flux tubes should appear in all scales. Also flux tubes inside flux tubes inside and hierarchical coilings as for DNA are possible.

A possible TGD inspired solution of the precession problem relies on the TGD view about the formation of galaxies and stellar systems.

1. Just like galaxies, also the stellar systems would have been formed as local tangles of a long monopole flux tube (thickened cosmic string), which itself could be part of or have been reconnected from a tangle of flux tube giving rise to the galaxy. The thickening liberates dark energy of cosmic strings and gives rise to the ordinary matter and is the TGD counterpart of inflation involving no inflaton fields.
2. In the same way as galaxies, stellar systems would be like pearls along string. This predicts correlations in the dynamics and positions of distant stars and galaxies and there is evidence for these correlations.
3. The flux tubes could connect the solar system to some distant stellar system. A good candidate for this kind of system is Pleiades, a star cluster located at a distance of 444 light years (the nearest star has a distance of 4 light years). There would also be an analog of solar wind along this flux tube giving for the solar magnetosphere a bullet-like shape.
4. The transversal gravitational force of the flux tube would cause the precession of the solar system around the flux tube. The entire solar system, as a tangle of the flux tube, would precess like a bullet-like top around the direction of this flux tube. The details of this picture are discussed in this and this .
5. The TGD analogs of Birkeland currents and the analogy of solar wind would flow along the monopole flux tubes, perhaps as dark particles in the TGD sense, that is having effective Planck constant h_eff=nh₀ which can be much larger than h, even so large that gravitational quantum coherence is possible in astrophysical and even cosmological scales.

See the the chapter TGD and astrophysics.

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

Articles related to TGD.

MOND and TGD view of dark matter

The TGD based model explains the MOND (Modified Newton Dynamics) model of Milgrom for the dark matter. Instead of dark matter, the model assumes a modification of Newton's laws. The model is based on the observation that the transition to a constant velocity spectrum in the galactic halos seems to occur at a constant value of the stellar acceleration equal to a_cr =about 10^-11g, where g is the gravitational acceleration at the Earth. MOND theory assumes that Newtonian laws are modified below a_cr.

1. In TGD, dark energy plus magnetic energy would be associated with cosmic strings, which are "non-Einsteinian" 4-surfaces of M⁴× CP₂ with 2-D M⁴ projection. Cosmic strings are unstable agains thickening of the M⁴ projection so that one obtains Einsteinian monopole flux tube.

   In accordance with the observations of Zeldovich, galaxies would correspond to tangles along a long cosmic string at which the string has thickened and liberated its energy as ordinary matter (TGD counterpart for the decay of the inflaton field). The flux tubes create 1/ρ type gravitational field orthogonal to string and this gives rise to the observed flat velocity spectrum (see this, this, and this).

2. In MOND theory, it is assumed that gravitation starts to behave differently when it becomes very weak and predicts the critical acceleration. In the TGD framework, the critical acceleration would be of the same order of magnitude as the acceleration created by the gravitational field of the cosmic string and would also define a critical distance depending only on the string tension.
3. If 1/r² changes to 1/r in MOND, model one obtains the same predictions as in TGD for the planar orbits orthogonal to the long string along which galaxies correspond to a flux tube tangled. The models are not equivalent. In TGD, general orbit corresponds to a helical motion of the star along the cosmic string so that the concentration on a preferred plane is predicted. This has been recently reported as an anomaly of dark matter models (see this).
4. The critical acceleration predicted would correspond to acceleration of the same order of magnitude as the acceleration caused by cosmic string. From M²/R_cr= GM/R²_cr= TG/R_cr (assuming that dark matter dominates) one obtains the estimate R_cr=M/T and a_cr =GT²/M, where M is the visible mass of the object - for instance the ordinary matter of a galaxy. If critical acceleration is always the same, one would have T=(a_crM/G)^1/2 so that the visible mass would scale like M∝ T² if a_cr is constant of Nature.

See the chapter TGD and astrophysics.

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

Articles related to TGD

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Wheels and quantum arithmetics

Gary Ehlenberg gave a link to a Wikipedia article telling of Wheel theory (this) and said that he now has a name for what he has been working with. I am sure that this kind of adventure is a wonderful mathematical experience.

I looked at the link and realized that it might be very relevant to the TGD inspired idea about quantum arithmetics (see this).

I understood that Wheel structure is special in the sense that division by zero is well defined and multiplication by zero gives a non-vanishing result. The wheel of fractions, discussed in the Wikipedia article as an example of wheel structure, brings into mind a generalization of arithmetics and perhaps even of number theory to its quantum counterpart obtained by replacing + and - with direct sum ⊕ and tensor product &otimes: for irreps of finite groups with trivial representation as multiplicative unit: Galois group is the natural group in TGD framework.

One could also define polynomial equations for the extension of integers (multiples of identity representation) by irreps and solve their roots.

This might allow us to understand the mysterious McKay correspondence. McKay graph codes for the tensor product structure for irreps of a finite group, now Galois group. For subgroups of SL(n,C), the graphs are extended Dynkin diagrams for affine ADE groups.

Could wheel structure provide a more rigorous generalization of the notion of the additive and multiplicative inverses of the representation somehow to build quantum counterparts of rationals, algebraic numbers and p-adics and their extensions?

1. One way to achieve this is to restrict consideration to the quantum analogs of finite fields G(p,n): + and x would be replaced with ⊕ and ⊗ obtained as extensions by the irreps of the Galois group in TGD picture. There would be quantum-classical correspondence between roots of quantum polynomials and ordinary monic polynomials.
2. The notion of rational as a pair of integers (now representations) would provide at least a formal solution of the problem, and one could define non-negative rationals.

   p-Adically one can also define quite concretely the inverse for a representation of form R=1 ⊕ O(p), where O(p) is proportional to p (p-fold direct sum) of a representation, as a geometric series.

3. Negative integers and rationals pose a problem for ordinary integers and rationals: it is difficult to imagine what direct sum of -n irreps could mean.

   The definition of the negative of representation could work in the case of p-adic integers: -1 = (p-1)⊗ (1 ⊕ p*1 ⊕ p^2*1 ⊕...) would be generalized by replacing 1 with trivial representation. Infinite direct sum would be obtained but it would converge rapidly in p-adic topology.

4. Could 1/pⁿ make sense in the Wheel structure so that one would obtain the analog of a p-adic number field? The definition of rationals as pairs might allow this since only non-negative powers of p need to be considered. p would represent zero but multiplication by p would give a non-vanishing result.

See the article McKay Correspondence from Quantum Arithmetics Replacing Sum and Product with Direct Sum and Tensor Product? or the chapter with the same title.

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

Articles related to TGD

The first findings of the James Webb telescope might revolutionize the views of the formation of galaxies

The first preliminary findings of the James Webb telescope, the successor of the Hubble telescope, are in conflict with the standard view of the formation of galaxies. The YouTube video (see this) "James Webb Found Galaxies That Sort of Break Modern Theories" gives a good summary of these findings. The findings are also summarized in an article in Nature (see this) with the title "Four revelations from the Webb telescope about distant galaxies".

The official story of the formation of galaxies goes roughly as follows.

1. Around 3 minutes of cosmic time, the cosmic microwave background emerged as the first atoms formed and radiation decoupled from matter.
2. When the age of the Universe was more than about .1 billion years, the first stars were formed. They lived their life and exploded as supernovas and yielded interstellar hydrogen gas. Galaxies started to form. One can see this process as a gravitational condensation. What is essential is that this process went from long to short scales, just as the formation of stars in the earlier phase.
3. The model gives a stringent upper bound for the age of the galaxies. They should be younger than the oldest observed stars. This limit gives an upper bound for the distance of the galaxy, that is for its redshift.

The first, preliminary, observations of the James Webb telescope were galaxies with redshifts up to 16. Even redsshift extending to 20 have been speculated in arXiv papers. Redshift 16 would correspond to the age of 250 million years and redshift of 20 to the age of 200 million years. They are too far to fit into the official picture. To get some perspective, note that the estimate for the age of the Universe is 13.8 billion years.

The ages of these galaxies were few hundred million years and of the same order as the estimated ages of about 100 million years of the hypothetical population III stars (see this), which are thought to be the oldest stars but have not not (yet?) detected. The criterion for the age of the star is its metal content: the first stars should have contained only hydrogen and Helium and "metal" here means anything heavier than Helium. The suggestive conclusion is that there was a significant population of star forming galaxies in the early universe. This challenges the standard view stating that stars came first and led to the formation of galaxies.

TGD proposes an unofficial view of the formation of galaxies (see this, this and this).

1. In the very beginning the Universe was dominated by cosmic strings, which were space-time surfaces in H=M⁴× CP₂ having 2-dimensional M⁴ projection. They were not "Einsteinian" space-time surfaces with 4-D M⁴ projection and have no counterpart in general relativity.
2. Cosmic strings were unstable against thickening of the M⁴ projection to 4-D one. Phase transitions thickening the cosmic strings occurred and increased their thickness and reduced string tension so that part of their energy transformed to ordinary matter. This is the TGD counterpart for inflation.

   This process led to radiation dominated Universe and the local description of the Universe as an Einsteinian 4-surface became a good approximation and is used in standard cosmology based on the standard model as a QFT limit of TGD.

   At this moment the thickness of the thickened strings would be around 100 micrometers, which corresponds to a length scale around large neuron size. Water blob with this size has mass of order Planck mass. The connection with biology is suggestive \cite{btart/penrose,watermorpho,waterbridge}.

3. The liberated dark energy (and possible dark matter, dark in the TGD sense) assignable to cosmic strings produced quasars, which in the TGD framework are identified as time reversals of the ordinary galactic blackholes. They did not extract matter from the environment but feeded darl energy as matter to the environment as jets. Jets are observed and explained in terms of the magnetic field due to the rotation of the galaxy.

   The jets are somewhat problematic in the GRT based cosmology since the simplest, non-rotating Schwarzschild blackholes do not allow them. The rotating blackholes identifiable as Kerr-Newman blackholes accompanied by magnetic fields, also have some interpretational problems. For instance, the arrow of time can be said to be different in the nearby and faraway regions and closed time-like geodesics are possible. In TGD, this could have an interpretation in terms of zero energy ontology (ZEO). The matter from the jets would have eventually led to the formation of atoms, stars, and galaxies.

4. What is essential is that the formation of galaxies proceeds from short to long scales rather than vice versa as in the standard cosmology. A second essential point is that the dark energy (and possible dark matter) concentrated at cosmic strings was added to the ordinary matter predicted by the standard model to be present in the radiation dominated cosmology. This led to the formation of galaxies. Therefore this picture is consistent with the standard story as far as the formation of atoms and emergence of CMB is considered.

The possibility considered in the TGD framework (see this, this and this) is that quasars are time reversed black-holes (this property can be formulated precisely in zero energy ontology (ZEO), which forms the basis of TGD based quantum measurement theory) (see this, this and this). Note that the time reversal property would hold true in long time scales at the magnetic body (MB) defined by the monopole flux tubes produced by the thickening of the cosmic strings. For ordinary matter, the scale for the time spent with a given arrow of time is very short but MB with a large gravitational Planck constant can force ordinary matter to effectively behave like its time reversed version.

There is indeed quite recent support for the proposal that quasars are time reversals of blackhole-like objects identified in the TGD framework as monopole flux tube tangles. The Hubble telescope detected a dwarf galaxy at a distance of 30 million light years for which the number of stars is about 10 per cent for that in the Milky Way. Its center contains a blackhole-like object (see this), which did not extract matter from the environment but did just the opposite by jets, which gave rise to a formation of stars.

The observations challenging the basic dogma of blackhole physics are not new and during the writing of an article about galactic jets I got the impression that one of the basic challenges is to explain why some blackholes do just the opposite of what they should do.

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 related to TGD

Dark 3N-resonances and quantum teleportation

Dark 3N -resonances and quantum teleportation Could the communication by dark 3N-resonances (see for instance this), which is central for the TGD view about genetic code (see this and this), relate to quantum teleportation? This is possible but requires modifying the previous assumption that the states of dark proton sequences are fixed and correspond to those of ordinary genes with which they are in energy resonance when communicating. One must loosen this assumption. Give up the assumption that cyclotron states of the dark 3N-proton are always the same and correspond to a gene. Assume that in some time scale, perhaps of order cyclotron time, dark proton sequences representing genes decay to the ground state configuration defining an analog of ferromagnet. Assume that some excited dark 3N-photon states, dark geme states, can be in energy resonance with ordinary genes, most naturally the nearest one if dark DNA strands are parallel to an ordinary DNA strand. Even this assumption might be unnecessarily strong. Dark 3N-proton would interact with its ordinary counterpart by energy resonance only when it corresponds to the dark variant of the gene. Same applies to dark genes in general. Only identical dark genes can have resonance interaction. This applies also to the level of other fundamental biomolecules RNA,tRNA and amino acids. What is this interaction in its simplest form? Suppose dark 3N- proton is in an excited state and thus defines a dark gene. Suppose that it decays by SFR to the ground state (magnetization) by emitting dark 3N-photon. If this 3N photon is absorbed in SFR by a dark proton sequence originally in ferromagnetic state, it excites by resonance the same gene. The transfer of entanglement takes place. This is nothing but quantum teleportation but without Alice, doing Bell measurements and sending the resulting bit sequences to Bob , performing the reversals of Bell measurements to rebuild the entanglement. This suggests a modification of the earlier picture of the relation between dark and chemical genetic code and the function of dark genetic code. Dark DNA (DDNA) strand is dynamical and has the ordinary DNA strand associated with it and dark gene state can be in resonant interaction with ordinary gene only when it corresponds to the ordinary gene. This applies also to DRNA, DtRNA and DAA (AA is for amino acids). This would allow DDNA, DRNA, DtRNA and DAA to perform all kinds of information processing such as TQC by applying dark-dark resonance in quantum communications. The control of fundamental biomolecules by their dark counterparts by energy resonance would be only one particular function. One can also allow superpositions of the dark genes representing 6-qubit units. A generalization of quantum computation so that it would use 6-qubits units instead of a single qubit as a unit, is highly suggestive. Genetic code code could be interpreted as an error code in which dark proteins correspond to logical 6-qubits and the DNA codons coding for the protein correspond to the physical qubits associated with the logical qubit. The teleportation mechanism could make possible remote replication and remote transcription of DNA by sending the information about ordinary DNA strand to corresponding dark DNA strand by energy resonance. After that, the information would be teleported to a DNA strand in a ferromagnetic ground state at the receiver. After this, ordinary replication or transcription, which would also use the resonance mechanism, would take place. Could there be a connection with bioharmony as a model of harmony providing also a model of genetic code (see this and this)? In the icosa-tetrahedral model, the orbit of the face of icosahedron under the group Z6,Z4, Z2,rot or Z2,refl would correspond to single physical 6-qubit represented as dark protein. This representation of the logical qubit would be geometric: orbit rather than sub-space of a state space. One could however assign to this kind of orbit a state space as wave functions defined at the orbit. This representation of Z6, Z4, Z2,rot or Z2,refl would correspond to a set of 6-qubits, which replaces a single 6-qubit. The TGD proposal for TQC \cite{btart/TQCTGD,QCCC} is that the irreps of Galois groups could replace qubits as analogs of anyons. Could these orbits correspond to irreps of Galois groups or their subgroups, say isotropy groups of roots? Another option is the finite subgroups G of quaternionic automorphisms, whose MacKay diagrams, characterizing the tensor products of irreps of G with the canonical 2-D irrep, give rise to extended Dynkin diagrams (see this). What puts bells ringing is that Z6,Z4, Z2,rot or Z2,refl are subgroups of the icosahedral group, which corresponds to the Dynkin diagram of E8. These alternatives need not be mutually exclusive. I have proposed (see this) that Galois groups could act as the Weyl groups of extended ADE Dynkin diagrams given by McKay graphs of finite subgroups of SU(2) interpreted as the covering group for the automorphism group of quaternions. The Galois group and its subgroup would define a cognitive representation for the subgroup of the covering group of quaternion automorphisms. The communications by the modulation of frequency scale 3N-Josephson frequency scale are still possible. The 3N-resonance occurs when the receiver 3N-proton is in ferromagnetic ground state and the 3N-Josephson frequency corresponds to 3N-cyclotron frequency. If the time scale for the return to the ferromagnetic state is considerably shorter than the time scale of modulations, a sequence of resonance pulses results and codes for the frequency modulation as an analog of nerve pulse pattern. This communication can lead to communication if the ordinary gene accompanying the excited dark gene is in energy resonance with it. It must be noticed that the communications by dark 3N-resonances are not possible in standard physics and are made possible only by. Galois confinement and heff hierarchy. In standard physics only single photon fermion interactions would be present and would be relatively weak. In quantum computation, this suggests the possibility of quantum coherent manipulation of N-qubit states by dark N-photons instead of qubit-wise manipulations prone to errors and destroying the coherence. There is evidence for N-photon states with these properties (see this and this). For the TGD inspired comments see this. See the chapter TGD View about Water Memory and the Notion of Morphogenetic Field or the article Quantum biological teleportation using multiple 6-qubits.. 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For last 37 years Topological Geometrodynamics has been both the passion and mission of my life. TGD is a noble attempt to construct a theory of everything, not forgetting consciousness. I have four children, who have brought a lot of happiness to my life. I live in Hanko, a small seaside town in southern Finland. I love almost all kinds of music but if I had to give just one name I would have difficulties in deciding between Chopin and Beethoven. View my complete profile Simple theme. Powered by Blogger.