Why would primes near powers of two (or small primes) be important?

The earlier posting What could be the role of complexity theory in TGD? was an abstract of an article about how complexity theory based thinking might help in attempts to understand the emergence of complexity in TGD. The key idea is that evolution corresponds to an increasing complexity for Galois group for the extension of rationals inducing also the extension used at space-time and Hilbert space level. This leads to rather concrete vision about what happens and the basic notions of complexity theory helps to articulate this vision more concretely.

Also new insights about how preferred p-adic primes identified as ramified primes of extension emerge. The picture suggests strong resemblance with the evolution of genetic code with conserved genes having ramified primes as their analogs. Category theoretic thinking in turn suggests that the positions of fermions at partonic 2-surfaces correspond to singularities of the Galois covering so that the number of sheets of covering is not maximal and that the singularities has as their analogs what happens for ramified primes.

p-Adic length scale hypothesis states that physically preferred p-adic primes come as primes near prime powers of two and possibly also other small primes. Does this have some analog to complexity theory, period doubling, and with the super-stability associated with period doublings?

Also ramified primes characterize the extension of rationals and would define naturally preferred primes for a given extension.

1. Any rational prime p can be decomposes to a product of powers P^k_i of primes P_i of extension given by p= ∏_i P_i^k_i, ∑ k_i=n. If one has k_i≠ 1 for some i, one has ramified prime. Prime p is Galois invariant but ramified prime decomposes to lower-dimensional orbits of Galois group formed by a subset of P_i^k_i with the same index k_i . One might say that ramified primes are more structured and informative than un-ramified ones. This could mean also representative capacity.
   
   
2. Ramification has as its analog criticality leading to the degenerate roots of a polynomial or the lowering of the rank of the matrix defined by the second derivatives of potential function depending on parameters. The graph of potential function in the space defined by its arguments and parameters if n-sheeted singular covering of this space since the potential has several extrema for given parameters. At boundaries of the n-sheeted structure some sheets degenerate and the dimension is reduced locally . Cusp catastrophe with 3-sheets in catastrophe region is standard example about this.
   

   Ramification also brings in mind super-stability of n-cycle for the iteration of functions meaning that the derivative of n:th iterate f(f(...)(x)== fⁿ⁾(x) vanishes. Superstability occurs for the iteration of function f= ax+bx² for a=0.
   
   

3. I have considered the possibility that that the n-sheeted coverings of the space-time surface are singular in that the sheet co-incide at the ends of space-time surface or at some partonic 2-surfaces. One can also consider the possibility that only some sheets or partonic 2-surfaces co-incide.
   

   The extreme option is that the singularities occur only at the points representing fermions at partonic 2-surfaces. Fermions could in this case correspond to different ramified primes. The graph of w=z^1/2 defining 2-fold covering of complex plane with singularity at origin gives an idea about what would be involved. This option looks the most attractive one and conforms with the idea that singularities of the coverings in general correspond to isolated points. It also conforms with the hypothesis that fermions are labelled by p-adic primes and the connection between ramifications and Galois singularities could justify this hypothesis.
   
   

4. Category theorists love structural similarities and might ask whether there might be a morphism mapping these singularities of the space-time surfaces as Galois coverings to the ramified primes so that sheets would correspond to primes of extension appearing in the decomposition of prime to primes of extension.
   

   Could the singularities of the covering correspond to the ramification of primes of extension? Could this degeneracy for given extension be coded by a ramified prime? Could quantum criticality of TGD favour ramified primes and singularities at the locations of fermions at partonic 2-surfaces?
   

   Could the fundamental fermions at the partonic surfaces be quite generally localize at the singularities of the covering space serving as markings for them? This also conforms with the assumption that fermions with standard value of Planck constants corresponds to 2-sheeted coverings.
   
   

5. What could the ramification for a point of cognitive representation mean algebraically? The covering orbit of point is obtained as orbit of Galois group. For maximal singularity the Galois orbit reduces to single point so that the point is rational. Maximally ramified fermions would be located at rational points of extension. For non-maximal ramifications the number of sheets would be reduced but there would be several of them and one can ask whether only maximally ramified primes are realized. Could this relate at the deeper level to the fact that only rational numbers can be represented in computers exactly.
   
   
6. Can one imagine a physical correlate for the singular points of the space-time sheets at the ends of the space-time surface? Quantum criticality as analogy of criticality associated with super-stable cycles in chaos theory could be in question. Could the fusion of the space-time sheets correspond to a phenomenon analogous to Bose-Einstein condensation? Most naturally the condensate would correspond to a fractionization of fermion number allowing to put n fermions to point with same M⁴ projection? The largest condensate would correspond to a maximal ramification p= P_iⁿ.
   
   

Why ramified primes would tend to be primes near powers of two or of small prime? The attempt to answer this question forces to ask what it means to be a survivor in number theoretical evolution. One can imagine two kinds of explanations.

1. Some extensions are winners in the number theoretic evolution, and also the ramified primes assignable to them.
   These extensions would be especially stable against further evolution representing analogs of evolutionary fossils.
   As proposed earlier, they could also allow exceptionally large cognitive representations that is large number of points of real space-time surface in extension.
   
   
2. Certain primes as ramified primes are winners in the sense the further extensions conserve the property of being ramified.
   
   
   1. The first possibility is that further evolution could preserve these ramified primes and only add new ramified primes. The preferred primes would be like genes, which are conserved during biological evolution. What kind of extensions of existing extension preserve the already existing ramified primes. One could naively think that extension of an extension replaces P_i in the extension for P_i= Q_ik^{k_i so that the ramified primes would remain ramified primes.}
      
      
   2. Surviving ramified primes could be associated with a exceptionally large number of extensions and thus with their Galois groups. In other words, some primes would have strong tendency to ramify. They would be at criticality with respect to ramification. They would be critical in the sense that multiple roots appear.
      

      Can one find any support for this purely TGD inspired conjecture from literature? I am not a number theorist so that I can only go to web and search and try to understand what I found. Web search led to a thesis (see this) studying Galois group with prescribed ramified primes.
      

      The thesis contained the statement that not every finite group can appear as Galois group with prescribed ramification. The second statement was that as the number and size of ramified primes increases more Galois groups are possible for given pre-determined ramified primes. This would conform with the conjecture. The number and size of ramified primes would be a measure for complexity of the system, and both would increase with the size of the system.
      
      

   3. Of course, both mechanisms could be involved.
      
      

Why ramified primes near powers of 2 would be winners? Do they correspond to ramified primes associated with especially many extension and are they conserved in evolution by subsequent extensions of Galois group. But why? This brings in mind the fact that n=2^k-cycles becomes super-stable and thus critical at certain critical value of the control parameter. Note also that ramified primes are analogous to prime cycles in iteration. Analogy with the evolution of genome is also strongly suggestive.

For details see the chapter Unified Number Theoretic Vision or the article What could be the role of complexity theory in TGD?.

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

Articles and other material related to TGD.

heff/h=n hypothesis and Galois group

The previous posting What could be the role of complexity theory in TGD? was an abstract of an article about how complexity theory based thinking might help in attempts to understand the emergence of complexity in TGD. The key idea is that evolution corresponds to an increasing complexity for Galois group for the extension of rationals inducing also the extension used at space-time and Hilbert space level. This leads to rather concrete vision about what happens and the basic notions of complexity theory helps to articulate this vision more concretely.

I ended up to rather interesting information theoretic interpretation about the understanding of effective Planck constant assigned to flux tubes mediating as gravitational/electromagnetic/etc... interactions. The real surprise was that this leads to a proposal how mono-cellulars and multicellulars differ! The emergence of multicellulars would have meant emergence of systems with mass larger than critical mass making possible gravitational quantum coherence. Penrose's vision about the role of gravitation would be correct although Orch-OR as such has little to do with reality!

The natural hypothesis is that h_eff/h=n equals to the order of Galois group in the case that it gives the number of sheets of the covering assignable to the space-time surfaces. The stronger hypothesis is that h_eff/h=n is associated with flux tubes and is proportional to the quantum numbers associated with the ends.

1. The basic idea is that Mother Nature is theoretician friendly. As perturbation theory breaks down, the interaction strength expressible as a product of appropriate charges divided by Planck constant, is reduced in the phase transition hbar→ hbar_eff.
   
   
2. In the case of gravitation GMm→ = GMm (h/h_eff). Equivalence Principle is satisfied if one has hbar_eff=hbar_gr = GMm/v₀, where v₀ is parameter with dimensions of velocity and of the order of some rotation velocity associated with the system. If the masses move with relativistic velocities the interaction strength is proportional to the inner product of four-momenta and therefore to Lorentz boost factors for energies in the rest system of the entire system. In this case one must assume quantization of energies to satisfy the constraint or a compensating reduction of v₀. Interactions strength becomes equal to β₀= v₀/c having no dependence on the masses: this brings in mind the universality associated with quantum criticality.
   
   
3. The hypothesis applies to all interactions. For electromagnetism one would have the replacements Z₁Z₂α→ Z₁Z₂α (h/ h_em) and hbar_em=Z₁Z₂α/&beta₀ giving Universal interaction strength. In the case of color interactions the phase transition would lead to the emergence of hadron and it could be that inside hadrons the valence quark have h_eff/h=n>1. In this case one could consider a generalization in which the product of masses is replaced with the inner product of four-momenta. In this case quantization of energy at either or both ends is required. For astrophysical energies one would have effective energy continuum.
   
   

This hypothesis suggests the interpretation of h_eff/h=n as either the dimension of the extension or the order of its Galois group. If the extensions have dimensions n₁ and n₂, then the composite system would be n₂-dimensional extension of n₁-dimensional extension and have dimension n₁× n₂. This could be also true for the orders of Galois groups. This would be the case if Galois group of the entire system is free group generated by the G₁ and G₂. One just takes all products of elements of G₁ and G₂ and assumes that they commute to get G₁× G₂. Consider gravitation as example.

1. The order of Galois group should coincide with hbar_eff/hbar=n= hbar_gr/hbar= GMm/v₀hbar. The transition occurs only if the value of hbar_gr/hbar is larger than one. One can say that the order of Galois group is proportional the product of masses using as unit Planck mass. Rather large extensions are involved and the number of sheets in the Galois covering is huge.
   

   Note that it is difficult to say how larger Planck constants are actually involved since by gravitational binding the classical gravitational forces are additive and by Equivalence principle same potential is obtained as sum of potentials for splitting of masses into pieces. Also the gravitational Compton length λ_gr= GM/v₀ for m does not depend on m at all so that all particles have same λ_gr= GM/v₀ irrespective of mass (note that v₀ is expressed using units with c=1).
   

   The maximally incoherent situation would correspond to ordinary Planck constant and the usual view about gravitational interaction between particles. The extreme quantum coherence would mean that both M and m behave as single quantum unit. In many-sheeted space-time this could be understood in terms of a picture based on flux tubes. The interpretation for the degree of coherence is in terms of flux tube connections mediating gravitational flux.
   
   

2. h_gr/h would be order of Galois group, and there is a temptation to associated with the product of masses the product n=n₁n₂ of the orders n_i of Galois groups associated masses M and m. The order of Galois group for both masses would have as unit m_P/β₀^1/2, β₀=v₀/c, rather than Planck mass m_P. For instance, the reduction of the Galois group of entire system to a product of Galois groups of parts would occur if Galois groups for M and m are cyclic groups with orders with have no common prime factors but not generally.
   

   The problem is that the order of the Galois group associated with m would be smaller than 1 for masses m<m_P/β₀^1/2. Planck mass is about 1.3 × 10¹⁹ proton masses and corresponds to a blob of water with size scale 10^-4 meters - size scale of a large neuron so that only above these scale gravitational quantum coherence would be possible. For v₀<1 it would seem that even in the case of large neurons one must have more than one neurons. Maybe pyramidal neurons could satisfy the mass constraint and would represent higher level of conscious as compared to other neurons and cells. The giant neurons discovered by the group led by Christof Koch in the brain of of mouse having axonal connections distributed over the entire brain might fulfil the constraint (see this).
   
   

3. It is difficult to avoid the idea that macroscopic quantum gravitational coherence for multicellular objects with mass at least that for the largest neurons could be involved with biology. Multicellular systems can have mass above this threshold for some critical cell number. This might explain the dramatic evolutionary step distinguishing between prokaryotes (mono-cellulars consisting of Archaea and bacteria including also cellular organelles and cells with sub-critical size) and eukaryotes (multi-cellulars).
   
   
4. I have proposed an explanation of the fountain effect appearing in super-fluidity and apparently defying the law of gravity. In this case m was assumed to be the mass of ⁴He atom in case of super-fluidity to explain fountain effect. The above arguments however allow to ask whether anything changes if one allows the blobs of superfluid to have masses coming as a multiple of m_P/β₀^1/2. One could check whether fountain effect is possible for super-fluid volumes with mass below m_P/β₀^1/2.
   
   

What about h_em? In the case of super-conductivity the interpretation of h_em/h as product of orders of Galois groups would allow to estimate the number N= Q/2e of Cooper pairs of a minimal blob of super-conducting matter from the condition that the order of its Galois group is larger than integer. The number N=Q/2e is such that one has 2N(α/β₀)^1/2=n. The condition is satisfied if one has α/β₀=q², with q=k/2l such that N is divisible by l. The number of Cooper pairs would be quantized as multiples of l. What is clear that em interaction would correspond to a lower level of cognitive consciousness and that the step to gravitation dominated cognition would be huge if the dark gravitational interaction with size of astrophysical systems is involved \citeallbhgrprebio. Many-sheeted space-time allows this in principle.

These arguments support the view that quantum information theory indeed closely relates not only to gravitation but also other interactions. Speculations revolving around blackhole, entropy, and holography, and emergence of space would be replaced with the number theoretic vision about cognition providing information theoretic interpretation of basic interactions in terms of entangled tensor networks (see this). Negentropic entanglement would have magnetic flux tubes (and fermionic strings at them) as topological correlates. The increase of the complexity of quantum states could occur by the "fusion" of Galois groups associated with various nodes of this network as macroscopic quantum states are formed. Galois groups and their representations would define the basic information theoretic concepts. The emergence of gravitational quantum coherence identified as the emergence of multi-cellulars would mean a major step in biological evolution.

For details see the chapter Unified Number Theoretic Vision or the article What could be the role of complexity theory in TGD?.

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

Articles and other material related to TGD.

What could be the role of complexity theory in TGD?

Chaotic (or actually extremely complex and only apparently chaotic) systems seem to be the diametrical opposite of completely integrable systems about which TGD is a possible example. There is however also something common: in completely integrable classical systems all orbits are cyclic and in chaotic systems they form a dense set in the space of orbits. Furthermore, in chaotic systems the approach to chaos occurs via steps as a control parameter is changed. Same would take place in adelic TGD fusing the descriptions of matter and cognition.

In TGD Universe the hierarchy of extensions of rationals inducing finite-dimensional extension of p-adic number fields defines a hierarchy of adelic physics and provides a natural correlate for evolution. Galois groups and ramified primes appear as characterizers of the extensions. The sequences of Galois groups could characterize an evolution by phase transitions increasing the dimension of the extension associated with the coordinates of "world of classical worlds" (WCW) in turn inducing the extension used at space-time and Hilbert space level. WCW decomposes to sectors characterized by Galois groups G₃ of extensions associated with the 3-surfaces at the ends of space-time surface at boundaries of causal diamond (CD) and G₄ characterizing the space-time surface itself. G₃ (G₄) acts on the discretization and induces a covering structure of the 3-surface (space-time surface). If the state function reduction to the opposite boundary of CD involves localization into a sector with fixed G₃, evolution is indeed mapped to a sequence of G₃s.

Also the cognitive representation defined by the intersection of real and p-adic surfaces with coordinates of points in an extension of rationals evolve. The number of points in this representation becomes increasingly complex during evolution. Fermions at partonic 2-surfaces connected by fermionic strings define a tensor network, which also evolves since the number of fermions can change.

The points of space-time surface invariant under non-trivial subgroup of Galois group define singularities of the covering, and the positions of fermions at partonic surfaces could correspond to these singularities - maybe even the maximal ones, in which case the singular points would be rational. There is a temptation to interpret the p-adic prime characterizing elementary particle as a ramified prime of extension having a decomposition similar to that of singularity so that category theoretic view suggests itself.

One also ends up to ask how the number theoretic evolution could select preferred p-adic primes satisfying the p-adic length scale hypothesis as a survivors in number theoretic evolution, and ends up to a vision bringing strongly in mind the notion of conserved genes as analogy for conservation of ramified primes in extensions of extension. h_eff/h=n has natural interpretation as the order of Galois group of extension. The generalization of hbar_gr= GMm/v₀=hbar_eff hypothesis to other interactions is discussed in terms of number theoretic evolution as increase of G₃, and one ends up to surprisingly concrete vision for what might happen in the transition from prokaryotes to eukaryotes.

For details see the chapter Unified Number Theoretic Vision or the article What could be the role of complexity theory in TGD?.

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

Articles and other material related to TGD.

Missing dark matter

One problem of ΛCDM scenario is missing of matter and dark matter in some places (see this). There missing dark matter in the scale of R=.2 Gy and also in the vicinity of solar system in the scale 1.5-4 kpc.

In the work titled "Missing Dark Matter in the Local Universe", Igor D. Karachentsev studied a sample of 11,000 galaxies in the local Universe around the MW (see this). He summed up the masses of individual galaxies and galaxy-groups and used this to test a very fundamental prediction of ΛCDM.

1. Standard cosmology predicts the average fraction of matter density to be Ω_m,glob=28+/- 3 per cent of critical mass density (83 percent of this would be dark and 17 per cent visible matter). 72 per cent would be dark energy, 28 per cent dark matter, and 4.8 per cent visible matter.
   
   To test this one can simply sum up all the galactic masses in some volume Karachentsev chose the volume to be a sphere of radius R= .2 Gy surrounding Milky Way and containing 11,000 galaxies. In this scale the density is expected to fluctuate only 10 per cent. Note that horizon radius is estimated to be about R_H=14 Gly giving R_H= 70 R.
   
   
2. The visible galactic mass in certain large enough volume of space was estimated as also the sum of galactic dark masses estimated as so called virial mass (see this). The sum of these masses gave the estimate for the total mass.
   
   
3. The estimate for the total mass (dark matter plus visible matter assuming halo model) in a volume of radius .2 Gy gives Ω_m,glob=8+/- 3 per cent, which is only 28 per cent of the predicted fraction. The predicted fraction of visible matter is 4.8 per cent and marginally consistent with 8+/- 3 per cent but it seems plausible that also dark matter is present although its amount is much smaller than expected. The total contribution to the dark matter could be at most of the same size as that of visible matter.
   
   
4. One explanation is that all matter has not been included. Second not very plausible explanation is that the measurement region corresponds to a region with abnormally low density.
   
   

Can on understand the finding in TGD framework?

1. In TGD based model part of dark energy/matter would reside at the long flux tubes with which galaxies form bound states. Constraints come from accelerated expansion and galactic velocity curves allowing to determine string tension for given galaxy. Let us assume that the GRT limit of TGD and its predictions hold true.
   

   The estimate for the virial mass assumes that galaxy's dark mass forms a halo. The basic observation is that in TGD flux tubes give the dark energy and mass and virial mass would underestimate the the dark mass of the galaxy.
   
   

2. How long length of the flux tube effectively corresponds to the dark and visible mass of disk galaxy? This length should be roughly the length containing the dark mass and energy estimated from cosmology: L=M_dark/T. If GRT limit of TGD makes sense, one has M_dark =xM_vis/T, where M_dark is the amount of dark energy + matter associated with the flux tube, M_vis is visible mass, x≈ ρ_dark/ρ_vis≈ 83/17 , and T is string tension deduced from the asymptotic rotation velocity.
   

   If these segments do not cover the entire flux tubes containing the galaxies along it, the amount of dark matter and energy will be underestimated. By the above argument elliptic galaxies would not have considerable amount of dark matter and energy so that only disk galaxies should contribute unless there are flux tubes in shorter scales inside elliptic galaxies.
   

   Also larger and smaller scale flux tube structures contribute to the dark energy + dark matter. Fractality suggests the presence of both larger and smaller flux tube structures than those associated with spiral galaxies (even stars couldbe associated with flux tubes).
   

   One should have estimates for the lengths of various flux tubes involved. Unfortunately this kind of estimate is not available.
   
   

3. If GRT limit makes sense then the total dark mass then the dark energy and matter obtained in this manner should give 95 per cent of critical mass density. The fraction of dark matter included would be at most a fraction 5/28≈ 18 per cent of the total dark matter. 82 per cent of dark matter and energy would be missed in the estimate. This could allow to get some idea about the lengths of flux tubes and density of galaxies along flux tubes.
   
   

The amount of dark matter in the solar neighborhood was investigated in the work "Kinematical and chemical vertical structure of the Galactic thick disk II. A lack of dark matter in the solar neighborhood" by Christian Moni Bidin and collaborators (see this). Moni Bidin et al have studied as sample of 400 red giants in the vicinity of solar system at vertical distances 1.5 to 4 kpc and deduce 3-D kinematics for these start. From this data they estimate the surface mass density of the Milky Way within this range of heights from the disk. This surface density should be sum of both visual and dark mass.

According to their analysis, the visible mass is enough to explain the data. No additional mass is needed. Only highly flattened dark matter halo would be consistent with the findings. This conforms with the TGD prediction that dark mass/energy are associated with magnetic flux tubes.

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

Articles and other material related to TGD.

New pseudoscalar meson at LHC?

This posting is an good example of blunders that one cannot avoid when targeted by huge information torrents! The article telling about the bump was one year old. Thanks for "Mitchell"! I however want to leave posting here since it I have a strong suspicion that the M₈₉ physics is indeed there. It might serve as a reminder!

Extremely interesting finding at LHC. Not 5 sigma finding but might be something real. There is evidence for the existence of a meson with mass 750 GeV decaying to gamma pair. The only reasonable candidate is pseudo-scalar or scalar meson.

What says TGD? M₈₉ hadron physics is the basic "almost prediction" of TGD. Mass scale is scaled up from that of ordinary hadron physics characterized by M₁₀₇ by a factor of 512.

About two handfuls of bumps with masses identifiable in TGD framework as scaled up masses for the mesons of ordinary hadron physics have been reported. See the article. The postings of Lubos trying to interpret the bumps as Higgses predicted by SUSY have been extremely helpful. No-one in the hegemony has of course taken this proposal seriously and the bumps have been forgotten since people have been trying to find SUSY and dark matter particles, certainly not TGD!

What about this new bump? It has mass about 750 GeV. The scaling down by 1/512 gives mass about 1.465 GeV
for the corresponding meson of the ordinary hadron physics. It should be flavorless and have spin 0 and would be most naturally pseudoscalar.

By going to Particle Data Tables and clicking "Mesons" and looking for "Light Unflavored Mesons" one finds that there are several unflavored mesons with mass near 1.465 GeV. Vector mesons do not decay to gamma pair and also most pseudoscalar mesons decay mostly via strong interactions. There is however only one decaying also to gamma pair: η(1475)! The error for the predicted mass is 1.3 per cent.

There are of many other ordinary mesons decaying also to gamma pairs and LHC might make history of science by trying to find them at masses scaled up by 512.

See the article M₈₉ Hadron Physics and Quantum Criticality or the chapter New Particle Physics Predicted by TGD: Part I of p-Adic Physics.

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

Articles and other material related to TGD.

Zwicky paradox and models of galactic dark matter

The anomalies of the halo model of dark matter have begun to accumulate rapidly. The problems of the halo model are discussed in detail in the blog "Dark matter crisis" of Prof. Pavel Kroupa and Marcel S. Pawlowski (see this). MOND is the most well-known competitor of the halo model for dark matter but has its own problems. TGD is less known alternative for the halo model. In the following brief comments about Zwicky paradox (see this) implying that neither cold nor warm dark matter particles in the usual sense (different from that in TGD based model) can play a significant role in cosmology.

The standard/concordance model of dark matter relies on two hypothesis formulated originally by Zwicky assuming that a) GRT is correct in all scales and b) all matter is created during Big Bang. Zwicky formulated two hypothesis (for references see the article) leading to the halo model of dark matter and also to Zwicky paradox.

1. Zwicky noticed (1937) that galaxies must about 500 heavier in the Coma galaxy cluster than judged from their light emission: cold or hot dark matter halo must exist. Note that this does not actually require that the dark matter consists of some exotic particles or that the dark matter forms halos. To get historical perspective note that Vera Rubin published 1976 an article about the constancy of velocity curves for distant stars for Andromeda which is spiral galaxy.
   
   
2. Zwicky noticed (1956) that when galaxies collide, the expelled matter can condense in new regions and form new smaller dwarf galaxies. These so called tidal galaxies are thus formed from the collisional debris of other galaxies.
   
   

From these observations one ends up with a computer model allowing to simulate the formation of galaxies (for a detailed discussion see this). The basic elements of the model are collisions of galaxies possibly leading to a fusion and formation of tidal galaxies. The model assumes a statistical distribution of dark matter lumps defining the halos of the dwarf galaxies formed in the process.

The model predicts a lot of dark matter dominated dwarf galaxies formed around the dark matter lumps: velocity spectrum should approach constant. There are also tidal dwarf galaxies formed from collision debris of other galaxies. Unless also now condensation around a dark matter lump is involved, these should not contain dark matter and velocity spectrum for tidal dwarfs should be declining. It turns out that tidal dwarfs alone are able to explain the observed dwarf galaxies, which are typically elliptic. Furthermore, there is no empirical manner to distinguish between tidal dwarfs and other dwarfs.

Do the elliptic galaxies contain dark matter? What does one know about the rotation curves of elliptic galaxies? There is an article "The rotation curves of elliptic galaxies" of J. Binney published around 1979 about the determination of the rotation curves of elliptic galaxies giving also some applications (see this). The velocity curves are declining as if no dark matter were present. Therefore dark matter would not be present in dwarf galaxies so that the prediction of the halo model would be wrong.

Could this finding be also a problem for MOND? Assuming that the laws governing gravitation are modified for small accelerations, shouldn't elliptic and spiral galaxies have similar velocity curves?

What about TGD?

1. In TGD Universe dark energy and matter reside at flux tubes along which disk galaxies condense like pearls in string.
   
   
2. The observation about velocity curves suggests a TGD based explanation for the difference between elliptic and spiral galaxies. Elliptic galaxies - in particular tidal dwarfs - are not associated with a flux tube containing dark matter. Spiral galaxy can form as elliptic galaxy if it becomes bound with flux tube as the recent finding about declining velocity curves for galaxies with age about 10 Gy suggest. Dark matter would not be present in dwarf galaxies so that the prediction of the halo model is wrong. This also conforms with the fact that the stars in elliptic galaxies are much older than in spiral galaxies (see this).
   
   
3. Dwarf galaxies produced from the collision debris contain only ordinary matter. Elliptic galaxies can later condense around magnetic flux tubes so that velocity spectrum approaches constant at large distances. The breaking of spherical symmetry to cylindrical symmetry might allow to understand why the oblate spheroidal shape is flattened to that of disk.
   
   

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

Articles and other material related to TGD.

Is conscious experience without definite causal order possible?

The exciting question is what the superposition of causal orders could mean from the point of view of conscious experience. What seems obvious is that in the superposition of selves with opposite arrows of clock-time there should be no experience about the flow of time in definite direction. Dissipation is associated with the thermodynamical arrow of time. Therefore also the sensory experience about dissipation expected to have unpleasant emotional color should be absent. This brings in mind the reports of meditators about experiences of timelessness. These states are also characterized by words like "bliss" and "enlightenment".

Why I find this aspects so interesting is due to my personal experience for about 32 years ago. I of course know that this kind of personal reminiscences in an article intended to be scientific, is like writing one's own academic death sentence. But I also know I long ago done this so that I have nothing to lose! The priests of the materialistic church will never bother to take seriously anything that I have written so that it does not really matter! This experience - I dared to talk about enlightenment experience - changed my personal life profoundly, and led to the decision to continue work with TGD instead of doing full-day job to make money and keeping TGD as a kind of hobby. The experience also forced to realize that our normal conscious experience is only a dim shadow of what it can be and stimulated the passion to understand consciousness.

In this experience my body went to a kind of light flowing state: liquid is what comes in mind. All unpleasant sensations in body characterizing the everyday life (at least mine!) suddenly disappeared as this phase transition propagated through my body. As a physicist I characterized this as absence of dissipation, and I talked to myself about a state of whole-body consciousness.

There was also the experience about moving in space in cosmic scales and the experience about the presence of realities very different the familiar one. Somehow I saw these different worlds from above, in bird's eye of view. I also experienced what I would call time travel and re-incarnation in some other world.

Decades later I would ask whether my sensory consciousness could have been replaced with that only about my magnetic body only. In the beginning of the experience there was indeed a concrete feeling that my body size had increased with some factor. I even had the feeling the factor was about 137 (inverse of the fine structure constant) but this interpretation was probably forced by my attempt to associate the experience with something familiar to physicist! Although I did all the time my best to understand what I was experiencing, I did not direct my attention to my time experience, and cannot say whether I experienced the presence or absence of time or time flow.

Towards the end of the experience I was clinically unconscious for about day or so. I was however conscious. For instance, I experienced quite concretely how the arrow of time flow started to fluctuate forth and back. I somehow knew that permanent change would mean death and I was fighting to preserve the usual arrow of time. My childhood friend, who certainly did not know much about physics, told about about alternation of the arrow of time during a state that was classified by psychiatrists as an acute psychosis.

See the chapter Topological Quantum Computation in TGD Universe and the article Quantum computations without definite causal structure: TGD view

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

Articles and other material related to TGD.

About some unclear technical issues of TGD

TGD has been in the middle of palace revolution during last two years and it is almost impossible to keep the chapters of the books updated. Adelic vision and twistor lift of TGD are the newest developments and there are still many details to be understood and errors to be corrected. The description of fermions in TGD framework has contained some unclear issues. Hence the motivation for the following brief comments.

Adelic vision and symmetries

In the adelic TGD SH is weakened: also the points of the space-time surface having imbedding space coordinates in an extension of rationals (cognitive representation) are needed so that data are not precisely 2-D. I have believed hitherto that one must use preferred coordinates for the imbedding space H - a subset of these coordinates would define space-time coordinates. These coordinates are determined apart from isometries. Does the number theoretic discretization imply loss of general coordinate invariance and also other symmetries?

The reduction of symmetry groups to their subgroups (not only algebraic since powers of e define finite-dimensional extension of p-adic numbers since e^p is ordinary p-adic number) is genuine loss of symmetry and reflects finite cognitive resolution. The physics itself has the symmetries of real physics.

The assumption about preferred imbedding space coordinates is actually not necessary. Different choices of H-coordinates means only different and non-equivalent cognitive representations. Spherical and linear coordinates in finite accuracy do not provide equivalent representations.

Quantum-classical correspondence for fermions

Quantum-classical correspondence (QCC) for fermions is rather well-understood but deserves to be mentioned also here.

QCC for fermions means that the space-time surface as preferred extremal should depend on fermionic quantum numbers. This is indeed the case if one requires QCC in the sense that the fermionic representations of Noether charges in the Cartan algebras of symmetry algebras are equal to those to the classical Noether charges for preferred extremals.

Second aspect of QCC becomes visible in the representation of fermionic states as point like particles moving along the light-like curves at the light-like orbits of the partonic 2-surfaces (curve at the orbit can be locally only light-like or space-like). The number of fermions and antifermions dictates the number of string world sheets carrying the data
needed to fix the preferred extremal by SH. The complexity of the space-time surface increases as the number of fermions increases.

Strong form of holography for fermions

It seems that scattering amplitudes can be formulated by assigning fermions with the boundaries of strings defining the lines of twistor diagrams. This information theoretic dimensional reduction from D=4 to D=2 for the scattering amplitudes can be partially understood in terms of strong form of holography (SH): one can construct the theory by using the data at string worlds sheets and/or partonic 2-surfaces at the ends of the space-time surface at the opposite boundaries of causal diamond (CD).

4-D modified Dirac action would appear at fundamental level as supersymmetry demands but would be reduced for preferred extremals to its 2-D stringy variant serving as effective action. Also the value of the 4-D action determining the space-time dynamics would reduce to effective stringy action containing area term, 2-D Kähler action, and topological Kähler magnetic flux term. This reduction would be due to the huge gauge symmetries of preferred extremals. Sub-algebra of super-symplectic algebra with conformal weigths coming as n-multiples of those for the entire algebra and the commutators of this algebra with the entire algebra would annihilate the physical states, and thecorresponding classical Noether charges would vanish.

One still has the question why not the data at the entire string world sheets is not needed to construct scattering amplitudes. Scattering amplitudes of course need not code for the entire physics. QCC is indeed motivated by the fact that quantum experiments are always interpreted in terms of classical physics, which in TGD framework reduces to that for space-time surface.

The relationship between spinors in space-time interior and at boundaries between Euclidian and Minkoskian regions

Space-time surface decomposes to interiors of Minkowskian and Euclidian regions. At light-like 3-surfaces at which the four-metric changes, the 4-metric is degenerate. These metrically singular 3-surfaces - partonic orbits- carry the boundaries of string world sheets identified as carriers of fermionic quantum numbers. The boundaries define fermion lines in the twistor lift of TGD. The relationship between fermions at the partonic orbits and interior of the space-time surface has however remained somewhat enigmatic.

So: What is the precise relationship between induced spinors Ψ_B at light-like partonic 3-surfaces and Ψ_I in the interior of Minkowskian and Euclidian regions? Same question can be made for the spinors Ψ_B at the boundaries of string world sheets and Ψ_I in interior of the string world sheets. There are two options to consider:

• Option I: Ψ_B is the limiting value of Ψ_I .
  
  
• Option II: Ψ_B serves as a source of Ψ_I .
  
  

For Opion I it is difficult to understand the preferred role of Ψ_B.

I have considered Option II already years ago but have not been able to decide.

1. That scattering amplitudes could be formulated only in terms of sources only, would fit nicely with SH, twistorial amplitude construction, and also with the idea that scattering amplitudes in gauge theories can be formulated in terms of sources of boson fields assignable to vertices and propagators. Now the sources would become fermionic.
   
   
2. One can take gauge theory as a guideline. One adds to free Dirac equation source term γ^kkA_kΨ. Therefore the natural boundary term in the action would be of the form (forgetting overall scale factor)
   

   S_B=Ψ_IΓ^α(C-S)A_αΨ_B+ c.c .
   

   Here the modified gamma matrix is Γ^α(C-S) (contravariant form is natural for light-like 3-surfaces) is most naturally defined by the boundary part of the action - naturally Chern-Simons term for Kähler action. A denotes the Kähler gauge potential.
   
   

3. The variation with respect to Ψ_B gives
   

   G^α(C-S)A_αΨ_I=0
   

   at the boundary so that the C-S term and interaction term vanish. This does not however imply vanishing of the source term! This condition can be seen as a boundary condition.
   
   

The same argument applies also to string world sheets.

About second quantization of the induced spinor fields

The anti-commutation relations for the induced spinors have been a long-standing issue and during years I have considered several options. The solution of the problem looks however stupifuingly simple. The conserved fermion currents are accompanied by super-currents obtained by replacing Ψ with a mode of the induced spinor field to get u_nΓ^αΨ or ΨΓ^αu_n with the conjugate of the mode. One obtains infinite number of conserved super currents. One can also replace both Ψ and Ψ in this manner to get purely bosonic conserved currents Ψ_mΓ^αu_n to which one can assign a conserved bosonic charges Q_mn.

I noticed this years ago but did not realize that these bosonic charges define naturally anti-commutators of fermionic creation and annihilation operators! The ordinary anti-commutators of quantum field theory follow as a special case! By a suitable unitary transformation of the spinor basis one can diagonalize the hermitian matrix defined by Q_mn and by performing suitable scalings one can transform anti-commutation relations to the standard form. An interesting question is whether the diagonalization is needed, and whether the deviation of the diagonal elements from unity could have some meaning and possibly relate to the hierarchy h_eff=n× h of Planck constants - probably not.

Is statistical entanglement "real" entanglement?

The question about the "reality" of statistical entanglement has bothered me for years. This entanglement is maximal and it cannot be reduced by measurement so that one can argue that it is not "real". Quite recently I learned that there has been a longstanding debate about the statistical entanglement and that the issue still remains unresolved.

The idea that all electrons of the Universe are maximally entangled looks crazy. TGD provides several variants for solutions of this problem. It could be that only the fermionic oscillator operators at partonic 2-surfaces associated with the space-time surface (or its connected component) inside given CD anti-commute and the fermions are thus indistinguishable. The extremist option is that the fermionic oscillator operators belonging to a network of partonic 2-surfaces connected by string world sheets anti-commute: only the oscillator operators assignable to the same scattering diagram would anti-commute.

What about QCC in the case of entanglement. ER-EPR correspondence introduced by Maldacena and Susskind for 4 years ago proposes that blackholes (maybe even elementary particles) are connected by wormholes. In TGD the analogous statement emerged for more than decade ago - magnetic flux tubes take the role of wormholes in TGD. Magnetic flux tubes were assumed to be accompanied by string world sheets. I did not consider the question whether string world sheets are always accompanied by flux tubes.

What could be the criterion for entanglement to be real? "Reality" of entanglement demands some space-time correlate. Could the presence of the flux tubes make the entanglement "real"? If statistical entanglement is accompanied by string connections without magnetic flux tubes, it would not be "real": only the presence of flux tubes would make it "real". Or is the presence of strings enough to make the statistical entanglement "real". In both cases the fermions associated with disjoint space-time surfaces or with disjoint CDs would not be indistinguishable. This looks rather sensible.

The space-time correlate for the reduction of entanglement would be the splitting of a flux tube and fermionic strings inside it. The fermionic strings associated with flux tubes carrying monopole flux are closed and the return flux comes back along parallel space-time sheet. Also fermionic string has similar structure. Reconnection of this flux tube with shape of very long flattened square splitting it to two pieces would be the correlate for the state function reduction reducing the entanglement with other fermions and would indeed decouple the fermion from the network.

See the article Comments about some unclear issues of TGD .

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

Articles and other material related to TGD.

Could the two photon emissions of dark particles reveal the value of heff?

Again a new anomaly! Photon pairs have been created by a new mechanism. Photons emerge at different points! See this.

Could this give support for the TGD based general model for elementary particle as a string like object (flux tube) with first end (wormhole contact) carrying the quantum numbers - in the case of gauge boson fermion and antifermion at opposite throats of the contact. Second end would carry neutrino-right-handed neutrino pair neutralizing the possible weak isospin. This would give only local decays. Also emissions of photons from charged particle would be local.

Could the bosonic particle be a mixture of two states. For the first state flux tube would have fermion and antifermion at the same end of the fluxtube: only local decays. For the second state fermion and antifermion would reside at the ends of the flux tubes residing at throats associated with different wormhole contacts. This state in state would give rise to non-local two-photon emissions. Mesons of hadron physics would correspond to this kind of states and in old-fashioned hadron physics one speaks about photon-vector meson mixing in the description of the photon-hadron interactions.

If the Planck constant h_eff/h=n of the emitting particle is large, the distance between photon emissions would be long. The non-local days could make the visible both exotic decay and allow to deduce the value of n! This would how require the transformation of emitted dark photon to ordinary (same would happen when dark photons transform to biophotons).

Can one say anything about the length of fux tube? Magnetic flux tube contains fermionic string. The length of this string is of order Compton length and of the order of p-adic length scale.

What about photon itself - could it have non-local fermion-antifermion decays based on the same mechanism? What the length of photonic string is is not clear. Photon is massless, no scales! One identification of length would be as wavelength defining also the p-adic length scale.

To sum up: the nonlocal decays and emissions could lend strong support for both flux tube identification of particles and for hierarchy of Planck constants. It might be possible to even measure the value of n associated with quantum critical state by detecting decays of this kind.

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

Articles and other material related to TGD.

Quantum computations without definite causal structure and TGD

I encountered a link to a interesting popular article "Causal Witness" Provides First Experimental Evidence Of Indefinite Causal Order (see this). The article tells about an article Experimental verification of an indefinite causal order by Rubio et al (see this). In the following are my first impressions. Probably I mest update impressions. The findings are however so fascinating that I cannot resists the temptation to
post them in blog.

1. Indefinite causal order

The abstract of the article Rubio et al might give some idea about what is involved.

Investigating the role of causal order in quantum mechanics has recently revealed that the causal relations of events may not be a priori well-defined in quantum theory. Although this has triggered a growing interest on the theoretical side, creating processes without a causal order is an experimental task. We report the first decisive demonstration of a process with an indefinite causal order. To do this, we quantify how incompatible our setup is with a definite causal order by measuring a "causal witness">. This mathematical object incorporates a series of measurements that are designed to yield a certain outcome only if the process under examination is not consistent with any well-defined causal order. In our experiment, we perform a measurement in a superposition of causal orders—without destroying the coherence—to acquire information both inside and outside of a causally non-ordered process. Using this information, we experimentally determine a causal witness, demonstrating by almost 7 SDs that the experimentally implemented process does not have a definite causal order.

Unfortunately, I do not have prerequisites to say anything interesting about the delicacies of the experiment itself. Since causal order is fixed by that associated with space-time in standard physics, the implications of the experiment could be world view changing.

The key quantum information theoretic notions are causal order, causal separability, quantum wittness, quantum process called SWITCH changing causal order, and superposition of causal orders.

1. The notion of causal order is discussed in the article "Quantum correlations with no causal order" by Oreshkov et al (see this). One has two events A and B. If they are causally separable, one can tell which causes which. In Minkowski space causally separable events would connected by a time-like curve. If not, one cannot speak about causal order. One can tell whether A precedes B or vice versa. For light-like distances, the situation is not so clear.
   

   Relaxing the standard assumption about fixed arrow of time one can at quantum level consider also a situation in which one has quantum superposition of different causal orders. One has causal non-separability.
   
   

2. The notion of causal wittness (see this) provides a method allowing to deduce experimentally whether the process is causally separable or not. The notion is similar to that of entanglement wittness (see this) allowing to deduce whether the two systems are entangled. Essentially one has observable whose expectation is negative for states with indefinite causal order and positive for those with definite causal order. Causal wittness is not universal but must be constructed for each causally indefinite state separately. The construction of causal wittness expectation value of operator is far from trivial and requires deeper understanding of operator theory. The abstract definition goes as follows:
   

   Causal wittness represents a set of quantum operations, such as unitaries, channels, state preparations, and measurements, whose expectation value is non-negative as long as all the operations are performed in a definite causal order, i.e., as long as only causally separable resources are used. The observation of a negative expectation value is thus sufficient to conclude that the operations were not performed in a definite order.
   

   Causal witness can be constructed efficiently and the construction is discussed here).
   
   

3. SWITCH is a further basic notion. One has two events A and B, which can be connected by a time-like curve. One can tell whether A precedes B or vice versa. SWITCH is a quantum operation switching the causal order. The obvious manner to do this would permute A and B and would require "time travel" not allowed in standard physics. Obviously, SWITCH cannot be realized as operation respecting fixed causal order.
   
   
4. If superpositions of causal orders are possible, one can have a situation in which causal order is indefinite. Also this is something which does not conform the ordinary view about physics in fixed space-time but is allowed by postulates for quantum computation and SWITCH represents an example of quantum computation impossible with a fixed causal order.
   
   

Needless to say, the notions of causal order and superposition of causal orders are revolutionary ideas and the article claims that they have been experimentally verified. Therefore there are excellent motivations to find whether they could be be understood in TGD framework.

In TGD Zero Energy Ontology (ZEO) replaces ordinary ontology and the arrow of time is not fixed, and it is interesting to see whether superposition of different causal orders related by time inversion T for causal diamond (CD) and SWITCH could be realized in ZEO. The twistor lift of TGD leads to the proposal that CD is accompanied by a Minkowskian generalization of self-dual Kähler form J(CD). Although the moduli space of CDs allows to avoid breaking of Poincare invariance, self-duality of J(CD) leads to violation of T implying that different causal orders correspond to disjoint sectors of "world of classical worlds". This makes possible also superposition of different causal orders and SWITCH would map these sectors to each other.

2. ZEO and discrete symmetries for twistor lift of TGD

Some background about TGD is necessary in order to to proceed.

1. Zero Energy Ontology (ZEO) is the cornerstone of TGD and TGD inspired theory of consciousness. Zero energy states appear as two variants and correspond to different WCW spinor fields (WCW for "world of classical worlds"). I have proposed that they correspond also to WCW spinor fields localized to different sectors of WCW but this might be un-necessarily strong assumption.
   

   Zero energy states for given basis have been subject to a state function reduction at either boundary of CD - passive boundary. Neither the passive boundary nor the members of state pairs at it appearing in the superposition of state pairs are affected in repeated state function reductions. One has what I have called generalized Zeno effect identified as conscious entity - self.
   

   At the opposite boundary the states evolve: every state function reduction at active boundary is preceded by a unitary time evolution ending to a localization of the active boundary of CD, which can be also seen as a state function reduction (see this). The temporal distance between the tips of CD increases in this process and gives rise to clock time and experienced flow of time.
   

   Eventually the first reduction to the opposite boundary occurs and the roles of active and passive boundary of CD are changed. One can say that time reversed zero energy state is obtained and begins to evolve. The first reduction to the opposite boundary would mean death of the conscious entity defined by the sequence of state function reductions at the same boundary and generation of time reversed re-incarnation of self.
   
   

2. Violation of T - to be discussed below - would also imply asymmetry between selves and their time reversals. For instance, the average duration for the state reduction sequences keeping boundary fixed could be different and second causal order could dominate giving rise to a dominating arrow of time. Since these reduction sequences are identified as correlates for conscious selves, the time reversed re-incarnations would live much shorter time. Biological systems might be an exception: in TGD inspired theory of consciousness sensory perception and motor action are time reversals of each other.
   
   
3. Fermionic oscillator operators associated with induced spinor fields allow to represent WCW gamma matrices as their linear combinations: Fermi statistics is geometrized. Fermionic oscillator operators define also quantum Boolean algebra in the sense that fermion numbers 1/0 correspond to the two Boolean values. One could say that quantum logic is square root of WCW Kähler geometry. This allows to interpretation the S-matrix for fermions as quantum Boolean map between quatum Boolean algebras at opposite boundaries. This is obviously important when one talks about quantum computation.
   

   In the approach to twistor amplitudes (see this, this, and this) fermions are localized at the boundaries of string world sheets defining light-like curves at the 3-D light-like orbits of partonic 2-surfaces, at which the signature of the induced metric changes from Minkowskian to Euclidian and has a vanishing determinant so that tangent space is effectively 3-D. The interpretation is in terms of strong form of holography (SH) stating that the data determining both space-time surface as preferred extremal and modes of the induced spinor field in the interior of space-time surface. SH predicts that both the bosonic and fermionic 4-D actions reduce to 2-D effective actions for string world sheets.
   

   The implication is that fermion states at the boundaries of CD are localized at discrete points of partonic 2-surfaces. One has of course amplitude over different locations of fermions at partonic 2-surfaces. The presence of fermionic string world sheets correlates the fermions at different partonic 2-surfaces and serves as correlate for entanglement in fermionic degrees of freedom.
   
   

4. The twistor lift of TGD (see this) has led to a rather detailed understanding of discrete symmetries CP, P,T. If M⁴ factor of imbedding space - or more precisely CD - is endowed with a generalized self-dual Kähler form J(M⁴) (analogs of magnetic and electric fields of same magnitude and direction), new violations of CP, P and T occurring in long scales and having no counterpart in standard model emerge. The reason is that CP, P and T do not respect the self-duality of M⁴ Kähler form. The violations of Poincare invariance are avoided if one assumes moduli space for CDs containing the Lorentz boosts and translations of CD.
   

   The first guess is that T leaving the center point of CD invariant applied to the CD maps the 3-surfaces at the boundaries of CD to each other. The violation of T however implies that the image of the pair need not allow preferred extremal (space-time surface) connecting its members.
   

   One can however define the temporal mirror image of pair by mapping only the 3-surface at the passive boundary to the opposite boundary: preferred extremal property would determine the 3-surface at the passive boundary. This could imply that the sub-WCWs formed by pairs and the time reversals are disjoint and form different sectors of WCW (see this). This realization of T allows also the possibility that the dynamics of preferred extremals is not strictly deterministic (true at least for p-adic space-time sheets).
   

   In absence of T violation T operation would also permute the values fermionic states partonic 2-surfaces at the boundaries of CD but if T is violated, can map only the state at the passive boundary to the opposite boundary and determine the state at original boundary from the hermitian conjugate of S-matrix in opposite time direction. The fermionic state at the opposite boundary would be superposition of states having only same total quantum numbers as the state at the passive boundary. The quantum numbers of individual fermions would not be sharp for non-trivial S-matrix if the zero energy states and their T images correspond to same sector of WCW.
   

   If T is not violated, zero energy states and their T-images would not correspond to the same sector of WCW. Obviously they would correspond to opposite causal orders, see below) This would force to give up the assumption that states at passive boundary are state function reduced. If T is violated globally, the zero energy states and their T-reversals would correspond to disjoint sectors of WCW, and the sectors would only correspond to different arrows of time. This option gives hopes about WCW localization the outcome of the measurement of causal order.
   

   The union of sub-WCWs with opposite arrow of time is a space with fixed causal order plus additional binary digit characterizing the causal order. The state function for this binary digit could fix the causal order and quantum computation generating superposition of causal orders should generate entanglement with this bit.
   
   

5. What about the situation for a union of CDs? Different CDs should be able to have their own arrow of time. For instance, there are reasons to think that in living matter one can have subsystems with non-standard arrow of time. Also phase conjugate laser ray could be also example of this. This requires that WCW spinor fields associated with a union of CDs form a tensor product. Causal order need not be same for all CDs but characterizes the sub-WCW associated with CD forming a Cartesian factor of WCW so that WCW spinors for the CDs form tensor product.
   
   

3. Two views about SWITCH and superposition of causal orders

One can imagine two approaches to the identification of SWITCH operation and superposition of causal orders.

3.1 Option I: Unitary SWITCH as time reversal

The first option corresponds to corresponds to unitary "time travel" option.

1. As already proposed, SWITCH as a unitary operation could correspond to T. Time reversal operation T applied to the 3-surfaces at the boundaries of CD would naturally change the causal order for the zero energy state. If T is violated the state and its T-image belong to separate sectors of WCW. One could also have a superposition of zero energy states related by T and having different causal orders and localizable to the two sectors of WCW.
   
   
2. Is it possible to perform SWITCH as a unitary (as a matter fact, antiunitary) operation? If T maps the disjoint sectors to each other and maps fermionic time evolutions to their time reversals, SWITCH maps the two sectors of WCW to each other.
   

   Can one realize SWITCH mathematically as a unitary operation between fermionic state spaces. This seems possible: the tensor product of S⊗ S^† on tensor product of fermionic Fock spaces would realize this map. Whether SWITCH can be realized physically is of course another question.
   
   

3.2 Option II: Non-unitary SWITCH as the first state function reduction to the opposite boundary of CD

Could non-unitary SWITCH be realized as the first state function reduction to the opposite boundary - death of self followed by a re-incarnation as time-reversed self? In this case SWITCH is neither unitary nor deterministic. This SWITCH corresponds to non-unitary "time travel" option in the sense that self identified as passive boundary makes a time travel to the opposite boundary of CD by re-incarnating in non-deterministic manner.

What about the superposition of self and time reversed self as a superposition of causal orders? Schrödinger cat would be more than a catchy metaphor: it would indeed be a superposition of cat and re-incarnated cat! Should one take this seriously?

If the CDs with different arrow of time correspond to different sectors of WCW, different causal orders correspond to states localized in these sectors. A superposition of causal orders would correspond to WCW spinor field having component in both these sectors. If the state function reduction to opposite boundary of CD takes place at the level of entire WCW - or more realistically, for a Cartesian factor of WCW, it must be accompanied by a localization to either sector of WCW in order to avoid paradoxes.

I have already earlier work with TGD inspired ideas related to quantum computation. For more than decade ago I developed rather speculative model topological quantum computation in TGD framework (see this). One speculation about super-effective quantum computations is inspired by the analogy of selves with quantum computations halting, when the self dies and re-incarnates as time reversed self with clock time running in opposite direction at opposite boundary of CD (see this). This would mean using the non-unitary SWITCH realized as state function reduction in quantum computation.

This allows to imagine a series of quantum computations at opposite boundaries proceeding as a sequence of re-incarnations so that the size of CD and thus clock time would grow in opposite directions during subsequent incarnations (see this). Although the re-incarnation as time reversed self could have long life-time, it would not be seen by an observer near the former re-incarnation and the re-re-incarnation would appear at clock-time, which need not be much later than the time of previous death. One could imagine that the time-reversed selves could have very long life-time or that the self could die and re-incarnate very many times without any-one noticing it! The large amount of time spent in time reversed mode could explain the miraculous cognitive feats of mathematicians like Ramajunan and also the magic computational abilities of idiot savants able to factorize large integers without any idea about the notion of prime.

4. Higher level quantum computations and ZEO

From the article of Rubio et al one ends up to an article Quantum computations without definite causal structure by Chiribella et al (see this). The article considers a rather far reaching generalization of quantum computation. Ordinary quantum computation is a time evolution of quantum states followed by a state function reduction. Since the outcome of state function reduction halting the quantum computer program is non-deterministic, the extraction of the result involves statistical averaging over a large enough number of quantum computations to get the outcome, say prime factorization or a period of periodic function.

The notion of classical computation is generalized by Church. The computation need not be a function but can assign function to a function. One can continue this abstraction hierarchy indefinitely and it is realized formally in terms of so called Λ calculus (see this). Could this hierarchy be extended to quantum computations? The quantum computation in question would be kind of super-computation assigning to quantum computation a quantum computation and entire hierarchy of quantum computations.

In TGD this kind of hierarchies emerge naturally. At space-time level there is hierarchy of space-time sheets: space-time sheet is (topologically) condensed to a larger space-time sheet and contains smaller space-time sheets condensed at it. The hierarchy of infinite primes corresponds to an infinite hierarchy of second quantizations and could relate to this hierarchy (see this). In each scale space-time sheets would be particles consisting of smaller particles consisting of ... Even galaxy could be seen as elementary particle in some scale characterizing the galactic space-time sheet.

The analog of quantum computational hierarchy emerges quite concretely in ZEO. The simplest zero energy states have positive and negative energy states with opposite total quantum numbers at the opposite boundaries of CD (intersection of future and past directed light-cones). One can have CDs within CDs. Furthermore, the positive/negative energy states assignable to the boundaries of CD could be also zero energy states associated with smaller CDs near the boundaries of CD. The simplest zero energy states correspond to quantum evolution representing ordinary quantum computation. Higher level zero energy stats would represent time evolution assigning to a quantum computation represented as zero energy state at the boundary of CD a second quantum computation at opposite boundary of CD.

The fermionic representation of quantum Boolean algebra makes this hierarchy quite concrete. At lowest level unitary evolution connects positive and negative energy fermionic states at opposite boundaries of CD and unitary S-matrix characterizes the computation. Higher level computations connect zero energy states assignable to sub-CDs near the boundaries of CD.

See the chapter Topological Quantum Computation in TGD Universe and the article Quantum computations without definite causal structure: TGD view

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

Articles and other material related to TGD.

Early galactic collision gives support for TGD based model of galactic dark matter

The discoveries related to galaxies and dark matter emerge with an accelerating pace, and from TGD point of view it seems that puzzle of galactic dark matter is now solved.

The newest finding is described in popylar article This Gigantic Ring of Galaxies Could Bring Einstein's Gravity Into Question. What has been found that in a local group of 54 galaxies having Milky Way and Andromeda near its center the other dwarf galaxies recede outwarts as a ring. The local group is in good approximation in plane and the situation is said to look like having a spinning umbrella from which the water droplets fly radially outwards.

The authors of the article Anisotropic Distribution of High Velocity Galaxies in the Local Group argue that the finding can be understood aif Milky Way and Andromeda had nearly head-on collision about 10 billion light-years ago. The Milky Way and Andromeda would have lost the radially moving dwarf galaxies in this collision during the rapid acceleration turning the direction of motion of both. Coulomb collision is good analog.

There are however problems. The velocities of the dwards are quite too high and the colliding Milky Way and Andromeda would have fused together by the friction caused by dark matter halo.

What says TGD? In TGD galactic dark matter (actually also energy) is at cosmic strings thickened to magnetic flux tubes like pearls along necklace. The finding could be perhaps explained if the galaxies in same plane make a near hit and generate in the collision the dwarf galaxies by the spinning umbrella mechanism.

In TGD Universe dark matter is at cosmic strings and this automatically predicts constant velocity distribution. The friction created by dark matter is absent and the scattering in the proposed manner could be possible. The scattering event could be basically a scattering of approximately parallel cosmic strings with Milky Way and Andromeda forming one pearl in their respective cosmic necklaces.

But were Milky Way and Andromeda already associated with cosmic strings at that time? The time would be about 10 billion years. One annot exclude this possibility. Note however that the binding to strings might have helped to avoid the fusion. The recent finding about effective absence of dark matter about 10 billion light years ago - velocity distributions decline at large distances - suggests that galaxies formed bound states with cosmic strings only later. This would be like formation of neutral atoms from ions as energies are not too high! How fast the things develope becomes clear from the fact that I posted TGD explanation to my blog yesterday and replaced with it with a corrected version this morning!.

See the chapter TGD and Astrophysics of "Physics in Many-Sheeted Space-time" or the article TGD interpretation for the new discovery about galactic dark matter.

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

Articles and other material related to TGD.

Velocity curves of galaxies flatten for large redshifts

Sabine Hossenfelder gave a link to a popular article "Declining Rotation Curves at High Redshift" (see this) telling about a new strange finding about galactic dark matter. The rotation curves are declining in the early Universe meaning distances about 10 billion light years (see this). In other words, the rotation velocity of distant stars decreases with radius rather than approaching constant - as if dark matter would be absent and galaxies were baryon dominated. This challenges the halo model of dark matter. For the illustrations of the rotation curves see the article. Of course, the conclusions of the article are uncertain.

Some time ago also a finding about correlation of baryonic mass density with density of dark matter emerged: the ScienceDaily article "In rotating galaxies, distribution of normal matter precisely determines gravitational acceleration" can be found here. The original article can be found in arXiv.org (see this). TGD explanation (see this) involves only the string tension of cosmic strings and predicts the behavior of baryonic matter on distance from the center of the galaxy.

In standard cosmology based on single-sheeted GRT space-time large redshifts mean very early cosmology at the counterpart of single space-time sheet, and the findings are very difficult to understand. What about the interpretation of the results in TGD framework? Let us first summarize the basic assumptions behind TGD inspired cosmology and view about galactic dark matter.

1. The basic difference between TGD based and standard cosmology is that many-sheeted space-time brings in fractality and length scale dependence. In zero energy ontology (ZEO) one must specify in what length scale the measurements are carried out. This means specifying causal diamond (CD) parameterized by moduli including the its size. The larger the size of CD, the longer the scale of the physics involved. This is of course not new for quantum field theorists. It is however a news for cosmologists. The twistorial lift of TGD allows to formulate the vision quantitatively.
   
   
2. TGD view resolves the paradox due to the huge value of cosmological constant in very small scales. Kähler action and volume energy cancel each other so that the effective cosmological constant decreases like inverse of the p-adic length scale squared because these terms compensate each other. The effective cosmological constant suffers huge reduction in cosmic scales and solves the greatest (the "most gigantic" would be a better attribute) quantitative discrepancy that physics has ever encountered. The smaller value of Hubble constant in long length scales finds also an explanation (see this). The acceleration of cosmic expansion due to the effective cosmological constant decreases in long scales.
   
   
3. In TGD Universe galaxies are located along cosmic strings like pearls in necklace, which have thickened to magnetic flux tubes. The string tension of cosmic strings is proportional to the effective cosmological constant. There is no dark matter hallo: dark matter and energy are at the magnetic flux tubes and automatically give rise to constant velocity spectrum for distant stars of galaxies determined solely by the string tension. The model allows also to understand the above mentioned finding about correlation of baryonic and dark matter densities (see this) .
   
   

What could be the explanation for the new findings about galactic dark matter?

1. The idea of the first day is that the string tension of cosmic strings depends on the scale of observation and this means that the asymptotic velocity of stars decreases in long length scales. The asymptotic velocity would be constant but smaller than for galaxies in smaller scales. The velocity graphs show that in the velocity range considered the velocity decreases. One cannot of course exclude the possibility that velocity is asymptotically constant.
   

   The grave objection is that the scale is galactic scale and same for all galaxies irrespective of distance. The scale characterizes the object rather than its distance for observer. Fractality suggests a hierarchy of string like structures such that string tension in long scales decreases and asymptotic velocity associated with them decreases with the scale.
   
   

2. The idea of the next day is that the galaxies at very early times have not yet formed bound states with cosmic strings so that the velocities of starts are determined solely by the baryonic matter and approach to zero at large distances. Only later the galaxies condense around cosmic strings - somewhat like water droplets around blade of grass. The formation of these gravitationally bound states would be analogous to the formation of bound states of ions and electrons below ionization temperature or formation of hadrons from quarks but taking place in much longer scale. The early galaxies are indeed baryon dominated and decline of the rotation velocities would be real.
   
   

See the chapter TGD and Astrophysics of "Physics in Many-Sheeted Space-time" or the article TGD interpretation for the new discovery about galactic dark matter.

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

Articles and other material related to TGD.

Getting quantitative about violations of CP, T, and P

The twistor lift of TGD led to the introduction of Kähler form also in M⁴ factor of imbedding space M⁴×CP₂. The moduli space of causal diamonds (CDs) introduced already early allow to save Poincare invariance at the level of WCW. One of the very nice things is that the self-duality of J(M⁴) leads to a new mechanism of breaking for P,CP, and T in long scales, where these breakings indeed take place. P corresponds to chirality selection in living matter, CP to matter antimatter asymmetry and T could correspond to preferred arrow of clock time. TGD allows both arrows but T breaking could make other arrow dominant. Also the hierarchy of Planck constant is expected to be important.

Can one say anything quantitative about these various breakings?

1. J(M⁴) is proportional to Newton's constant G in the natural scale of Minkowski coordinates defined by twistor sphere of T(M⁴). Therefore CP breaking is expected to be proportional to l_P²/R² or to its square root l_P/R. The estimate for l_P/R is X== l_P/R≈ 2^-12≈ 2.5× 10^-4.
   

   The determinant of CKM matrix is equal to phase factor by unitarity (UU^†=1) and its imaginary part characterizes CP breaking. The imaginary part of the determinant should be proportional to the Jarlskog invariant J= +/- Im(V_usV_cbV*_ub V*_cs) characterizing CP breaking of CKM matrix (see this).
   

   The recent experimental estimate is J≈ 3.0× 10^-5. J/X≈ 0 .1 so that there is and order of magnitude deviation. Earlier experimental estimate used in p-adic mass calculations was by almost order of magnitude larger consistent with the value of X. For B mesons CP breading is about 50 times larger than for kaons and it is clear that Jarlskog invariant does nto distinguish between different meson so that it is better to talk about orders of magnitude only.
   

   The parameter used to characterize matter antimatter asymmetry (see this) is the ratio R=[n(B-n(B*)]/n(γ)) ≈ 9× 10^-11 of the difference of baryon and antibaryon densities to photon density in cosmological scales. One has X³ ≈ 1.4 × 10^-11, which is order of magnitude smaller than R.
   
   

2. What is interesting that P is badly broken in long length scales as also CP. The same could be true for T. Could this relate to the thermodynamical arrow of time? In ZEO state function reductions to the opposite boundary change the direction of clock time. Most physicist believe that the arrow of thermodynamical time and thus also clock time is always the same. There is evidence that in living matter both arrows are possible. For instance, Fantappie has introduced the notion of syntropy as time reversed entropy. This suggests that thermodynamical arrow of time could correspond to the dominance of the second arrow of time and be due to self-duality of J(M⁴) leading to breaking of T. For instance, the clock time spend in time reversed phase could be considerably shorter than in the dominant phase. A quantitative estimate for the ratio of these times might be given some power of the the ratio X =l_P/R.
   

For background see chapter Some questions related to the twistor lift of TGD of "Towards M-matrix" or the article with the same title.

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

Articles and other material related to TGD.

Is there a duality between associative and co-associative space-time surfaces?

A more appropriate title of this posting would be "A new duality or an old duality seen from a number theoretic perspective?". The original proposal turned out to be partially wrong and I can only blame myself for breaking the rule "Wait for a week before posting!".

M⁸-H duality maps the preferred extremals in H to those M⁴× CP₂ and vice versa. The tangent spaces of an associative space-time surface in M⁸ would be quaternionic (Minkowski) spaces.

In M⁸ one can consider also co-associative space-time surfaces having associative normal space. Could the co-associative normal spaces of associative space-time surfaces in the case of preferred extremals form an integrable distribution therefore defining a space-time surface in M⁸ mappable to H by M⁸-H duality? This might be possible but the associative tangent space and the normal space correspond to the same CP₂ point so that associative space-time surface in M⁸ and its possibly existing co-associative companion would be mapped to the same surface of H.

This dead idea however inspires an idea about a duality mapping Minkowskian space-time regions to Euclidian ones. This duality would be analogous to inversion with respect to the surface of sphere, which is conformal symmetry. Maybe this inversion could be seen as the TGD counterpart of finite-D conformal inversion at the level of space-time surfaces. There is also an analogy with the method of images used in some 2-D electrostatic problems used to reflect the charge distribution outside conducting surface to its virtual image inside the surface. The 2-D conformal invariance would generalize to its 4-D quaterionic counterpart. Euclidian/Minkowskian regions would be kind of Leibniz monads, mirror images of each other.

1. If strong form of holography (SH) holds true, it would be enough to have this duality at the informational level relating only 2-D surfaces carrying the holographic information. For instance, Minkowskian string world sheets would have duals at the level of space-time surfaces in the sense that their 2-D normal spaces in X⁴ form an integrable distribution defining tangent spaces of a 2-D surface. This 2-D surface would have induced metric with Euclidian signature.
   

   The duality could relate either a) Minkowskian and Euclidian string world sheets or b) Minkowskian/Euclidian string world sheets and partonic 2-surfaces common to Minkowskian and Euclidian space-time regions. a) and b) is apparently the most powerful option information theoretically but is actually implied by b) due to the reflexivity of the equivalence relation. Minkowskian string world sheets are dual with partonic 2-surfaces which in turn are dual with Euclidian string world sheets.
   
   

   1. Option a): The dual of Minkowskian string world sheet would be Euclidian string world sheet in an Euclidian region of space-time surface, most naturally in the Euclidian "wall neighbour" of the Minkowskian region. At parton orbits defining the light-like boundaries between the Minkowskian and Euclidian regions the signature of 4-metric is (0,-1,-1,-1) and the induced 3-metric has signature (0,-1,-1) allowing light-like curves. Minkowskian and Euclidian string world sheets would naturally share these light-like curves aas common parts of boundary.
      
      
   2. Option b): Minkowskian/Euclidian string world sheets would have partonic 2-surfaces as duals. The normal space of the partonic 2-surface at the intersection of string world sheet and partonic 2-surface would be the tangent space of string world sheets so that this duality could make sense locally. The different topologies for string world sheets and partonic 2-surfaces force to challenge this option as global option but it might hold in some finite region near the partonic 2-surface. The weak form of electric-magnetic duality could closely relate to this duality.
      
   In the case of elementary particles regarded as pairs of wormhole contacts connected by flux tubes and associated strings this would give a rather concrete space-time view about stringy structure of elementary particle. One would have a pair of relatively long (Compton length) Minkowskian string sheets at parallel space-time sheets completed to a parallelepiped by adding Euclidian string world sheets connecting the two space-time sheets at two extremely short (CP₂ size scale) Euclidian wormhole contacts. These parallelepipeds would define lines of scattering diagrams analogous to the lines of Feynman diagrams.

This duality looks like new but as already noticed is actually just the old electric-magnetic duality seen from number-theoretic perspective.

For background see chapter Some questions related to the twistor lift of TGD of "Towards M-matrix" or the article with the same title.

About the generalization of dual conformal symmetry and Yangian in TGD

The discovery of dual of the conformal symmetry of gauge theories was crucial for the development of twistor Grassmannian approach. The D=4 conformal generators acting on twistors have a dual representation in which they act on momentum twistors: one has dual conformal symmetry, which becomes manifest in this representation. These two separate symmetries extend to Yangian symmetry providing a powerful constraint on the scattering amplitudes in twistor Grassmannian approach fo N=4 SUSY.

In TGD the conformal Yangian extends to super-symplectic Yangian - actually, all symmetry algebras have a Yangian generalization with multi-locality generalized to multi-locality with respect to partonic 2-surfaces. The generalization of the dual conformal symmetry has however remained obscure. In the following I describe what the generalization of the two conformal symmetries and Yangian symmetry would mean in TGD framework.

One also ends up with a proposal of an information theoretic duality between Euclidian and Minkowskian regions of the space-time surface inspired by number theory: one might say that the dynamics of Euclidian regions is mirror image of the dynamics of Minkowskian regions. A generalization of the conformal reflection on sphere and of the method of image charges in 2-D electrostatics to the level of space-time surfaces allowing a concrete construction reciple for both Euclidian and Minkowskian regions of preferred extremals is in question. One might say that Minkowskian and Euclidian regions are analogous to Leibnizian monads reflecting each other in their internal dynamics.

See the chapter Some Questions Related to the Twistor Lift of TGD of "Towards M-matrix" or the article with the same title.

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

Articles and other material related to TGD.