Sunday, January 15, 2017

Anomalies of water as evidence for dark matter in TGD sense

The motivation for this brief comment came from a popular article telling that a new phase of water has been discovered in the temperature range 50-60 oC (see this ). Also Gerald Pollack (see this ) has introduced what he calls the fourth phase of water. For instance, in this phase water consists of hexagonal layers with effective H1.5O stoichiometry and the phase has high negative charge. This phase plays a key role in TGD based quantum biology. These two fourth phases of water could relate to each other if there exist a deeper mechanism explaining both these phases and various anomalies of water.

Martin Chaplin (see this ) has an extensive web page about various properties of water. The physics of water is full of anomalous features and therefore the page is a treasure trove for anyone ready to give up the reductionistic dogma. The site discusses the structure, thermodynamics, and chemistry of water. Even academically dangerous topics such as water memory and homeopathy are discussed.

One learns from this site that the physics of water involves numerous anomalies (see this ). The structural, dynamic and thermodynamic anomalies form a nested in density-temperature plane. For liquid water at atmospheric pressure of 1 bar the anomalies appear in the temperature interval 0-100 oC.

Hydrogen bonding creating a cohesion between water molecules distinguishes water from other substances. Hydrogen bonds induce the clustering of water molecules in liquid water. Hydrogen bonding is also highly relevant for the phase diagram of H2O coding for various thermodynamical properties of water (see this ). In biochemistry hydrogen bonding is involved with hydration. Bio-molecules - say amino-acids - are classified to hydrophobic, hydrophilic, and amphiphilic ones and this characterization determines to a high extent the behavior of the molecule in liquid water environment. Protein folding represents one example of this.

Anomalies are often thought to reduce to hydrogen bonding. Whether this is the case, is not obvious to me and this is why I find water so fascinating substance.

TGD indeed suggests that water decomposes into ordinary water and dark water consisting of phases with effective Planck constant heff=n× h residing at magnetic flux tubes. Hydrogen bonds would be associated with short and rigid flux tubes but for larger values of n the flux tubes would be longer by factor n and have string tension behaving as 1/n so that they would softer and could be loopy. The portional of water molecules connected by flux tubes carrying dark matter could be identified as dark water and the rest would be ordinary water. This model allows to understand various anomalies. The anomalies are largest at the physiological temperature 37 C, which conforms with the vision about the role of dark matter and dark water in living matter since the fraction of dark water would be highest at this temperature. The anomalies discussed are density anomalies, anomalies of specific heat and compressibility, and Mpemba effect. I have discussed these anomalies already for decade ago. The recent view about dark matter allows however much more detailed modelling.

For details see the chapter Dark Nuclear Physics and Condensed Matter of "Hyper-finite factors, p-adic length scale hypothesis, and dark matter hierarchy" or the article
The anomalies of water as evidence for the existence of dark matter in TGD sense
.

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

Articles and other material related to TGD.

Thursday, January 05, 2017

What does Negentropy Maximization Principle really say?

There is something in NMP that I still do not understand: every time I begin to explain what NMP is I have this unpleasant gut feeling. I have the habit of making a fresh start everytime rather than pretending that everything is crystal clear. I have indeed considered very many variants of NMP. In the following I will consider two variants of NMP. Second variant reduces to a pure number theory in adelic framework inspired by number theoretic vision. It is certainly the simplest one since it says nothing explicit about negentropy. Second variant says essentially the as "strong form of NMP", when the reduction occurs to an eigen-space of density matrix.

I will not consider zero energy ontology (ZEO) related aspects and the aspects related to the hierarchy of subsystems and selves since I dare regard these as "engineering" aspects.

What NMP should say?

What NMP should state?

  1. NMP takes in some sense the role of God and the basic question is whether we live in the best possible world or not. Theologists asks why God allows sin. I ask whether NMP demand increase of negentropy always or does it allow also reduction of negentropy? Why? Could NMP lead to increase of negentropy only in statistical sense - evolution? Could it only give potential for gaining a larger negentropy?

    These questions have turned to be highly non-trivial. My personal experience is that we do not live in the best possible world and this experience plus simplicity motivates the proposal to be discussed.

  2. Is NMP a separate principle or could NMP be reduced to mere number theory? For the latter option state function would occur to an eigenstate/eigenspace of density matrix only if the corresponding eigenvalue and eigenstate/eigenspace are expressible using numberes in the extension of rationals defining the adele considered. A phase transition to an extension of an extension containing these coefficients would be required to make possible reduction. A step in number theoretic evolution would occur. Also an entanglement of measured state pairs with those of measuring system in containing the extension of extension would make possible the reduction. Negentropy would be reduced but higher-D extension would provide potential for more negentropic entanglement. I will consider this option in the following.

  3. If one has higher-D eigenspace of density matrix, p-adic negentropy is largest for the entire subspace and the sum of real and p-adic negentropies vanishes for all of them. For negentropy identified as total p-adic negentropy strong from of NMP would select the entire sub-space and NMP would indeed say something explicit about negentropy.

The notion of entanglement negentropy

  1. Number theoretic universality demands that density matrix and entanglement coefficients are numbers in an algebraic extension of rationals extended by adding root of e. The induced p-adic extensions are finite-D and one obtains adele assigned to the extension of rationals. Real physics is replaced by adelic physics.

  2. The same entanglement in coefficients in extension of rationals can be seen as numbers is both real and various p-adic sectors. In real sector one can define real entropy and in various p-adic sectors p-adic negentropies (real valued).

  3. Question: should one define total entanglement negentropy as

    1. sum of p-adic negentropies or

    2. as difference for the sum of p-adic negentropies and real etropy. For rational entanglement probabilities real entropy equals to the sum of p-adic negentropies and total negentropy would vanish. For extensions this negentropy would be positive under natural additional conditions as shown earlier.
    Both options can be considered.

State function reduction as universal measurement interaction between any two systems


  1. The basic vision is that state function reductions occur all the for all kinds of matter and involves a measurement of density matrix ρ characterizing entanglement of the system with environment leading to a sub-space for which states have same eigenvalue of density matrix. What this measurement really is is not at all clear.

  2. The measurement of the density matrix means diagonalization of the density matrix and selection of an eigenstate or eigenspace. Diagonalization is possible without going outside the extension only if the entanglement probabilities and the coefficients of states belong to the original extension defining the adele. This need not be the case!

    More precisely, the eigenvalues of the density matrix as roots of N:th order polynomial with coefficients in extension in general belong to N-D extension of extension. Same about the coefficients of eigenstates in the original basis. Consider as example the eigen values and eigenstates of rational valued N× N entanglement matrix, which are roots of a polynomial of degree N and in general algebraic number.

    Question: Is state function reduction number theoretically forbidden in the generic case? Could entanglement be stable purely number theoretically? Could NMP reduce to just this number theoretic principle saying nothing explicit about negentropy? Could phase transition increasing the dimension of extension but keeping the entanglement coefficients unaffected make reduction possible. Could entanglement with an external system in higher-D extension -intelligent observer - make reduction possible?

  3. There is a further delicacy involved. The eigen-space of density matrix can be N-dimensional if the density matrix has N-fold degenerate eigenvalue with all N entanglement probabilities identical. For unitary entanglement matrix the density matrix is indeed N×N unit matrix. This kind of NE is stable also algebraically if the coefficients of eigenstates do not belong to the extension. If they do not belong to it then the question is whether NMP allows a reduction to subspace of and eigen space or whether only entire subspace is allowed.

    For total negentropy identified as the sum of real and p-adic negentropies for any eigenspace would vanish and would not distinguish between sub-spaces. Identification of negentropy as as p-adic negentropy would distinguish between sub-spaces and´NMP in strong form would not allow reduction to sub-spaces. Number theoretic NMP would thus also say something about negentropy.

    I have also consider the possibility of weak NMP. Any subspace could be selected and negentropy would be reduced. The worst thing to do in this case would be a selection of 1-D subspace: entanglement would be totally lost and system would be totally isolated from the rest of the world. I have proposed that this possibility corresponds to the fact that we do not seem to live in the best possible world.

NMP as a purely number theoretic constraint?

Let us consider the possibility that NMP reduces to the number theoretic condition tending to stabilize generic entanglement.

  1. Density matrix characterizing entanglement with the environment is a universal observable. Reduction can occur to an eigenspace of the density matrix. For rational entanglement probabilities the total negentropy would vanish so that NMP formulated in terms of negentropy cannot say anything about the situation. This suggests that NMP quite generally does not directly refer to negentropy.

  2. The condition that eigenstates and eigenvalues are in the extension of rationals defining the adelic physics poses a restriction. The reduction could occur only if these numbers are in the original extension. Also rational entanglement would be stable in the generic case and a phase transition to higher algebraic extension is required for state function reduction to occur. Standard quantum measurement theory would be obtained when the coefficients of eigenstates and entanglement probabilities are in the original extension.

  3. If this is not the case, a phase transition to an extension of extension containing the N-D extension of it could save the situation. This would be a step in number theoretic evolution. Reduction would lead to a reduction of negentropy but would give potential for gaining a larger entanglement negentropy. Evolution would proceed through catastrophes giving potential for more negentropic entanglement! This seems to be the case!

    Alternatively, the state pairs of the system + complement could be entangled with observer in an extension of rationals containg the needed N-D extension of extension and state function possible for observer would induce reduction in the original system. This would mean fusion with a self at higher level of evolutionary hierarchy - kind of enlightment. This would give an active role to the intelligent observer (intelligence characterized by the dimension of extension of rationals). Intelligent observer would reduce the negentropy and thus NMP would not hold true universally.

    Since higher-D extension allows higher negentropy and in the generic case NE is stable, one might hope that NMP holds true statistically (for rationals total negentropy as sum or real and total p-adic negentropies vanishes).

    The Universe would evolve rather than being a paradize: the number theoretic NMP would allow temporary reduction of negentropy but provide a potential for larger negentropy and the increase of negentropy in statistical sense is highly suggestive. To me this option looks like simplest and most realistic one.

  4. If negentropy is identified as total p-adic negentropy rather than sum of real and p-adic negentropies, strong form of NMP says something explicit about negentropy: the reduction would take place to the entire subspace having the largest p-adic negentropy.

See the article About number theoretical aspects of NMP.

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

Articles and other material related to TGD.

Answer to a question about general aspects of TGD

In FB I received a question about general aspects of TGD. It was impossible to answer the question with few lines and I decided to write a blog posting. I am sorry for typos in the hastily written text. A more detailed article Can one apply Occam’s razor as a general purpose debunking argument to TGD? tries to emphasize the simplicity of the basic principles of TGD and of the resulting theory.


A. In what aspects TGD extends other theory/theories of physics?

I will replace "extends" with "modifies" since TGD also simplifies in many respects. I shall restrict the considerations to the ontological level which to my view is the really important level.

  1. Space-time level is where TGD started from. Space-time as an abstract 4-geometry is replaced as space-time as 4-surface in M4× CP2. In GRT space-time is small deformation of Minkowski space.

    In TGD both Relativity Principle (RP) of Special Relativity (SRT) and General Coordinate Invariance (GCI) and Equivalence Principle (EP) of General Relativity hold true. In GRT RP is given up and leads to the loss of conservation laws since Noether theorem cannot be applied anymore: this is what led to the idea about space-time as surface in H. Strong form of holography (SH) is a further principle reducing to strong form of GCI (SGCI).

  2. TGD as a physical theory extends to a theory of consciousness and cognition. Observer as something external to the Universe becomes part of physical system - the notion of self - and quantum measurement theory which is the black sheet of quantum theory extends to a theory of consciousness and also of cognition relying of p-adic physics as correlate for cognition. Also quantum biology becomes part of fundamental physics and consciousness and life are seen as basic elements of physical existence rather than something limited to brain.

    One important aspect is a new view about time: experienced time and geometric time are not one and same thing anymore although closely related. ZEO explains how the experienced flow and its direction emerges. The prediction is that both arrows of time are possible and that this plays central role in living matter.

  3. p-Adic physics is a new element and an excellent candidate for a correlate of cognition. For instance, imagination could be understood in terms of non-determinism of p-adic partial differential equations for p-adic variants of space-time surfaces. p-Adic physics and fusion of real and various p-adic physics to adelic physics provides fusion of physics of matter with that of cognition in TGD inspired theory of cognition. This means a dramatic extension of ordinary physics. Number Theoretical Universality states that in certain sense various p-adic physics and real physics can be seen as extensions of physics based on algebraic extensions of rationals (and also those generated by roots of e inducing finite-D extensions of p-adics).

  4. Zero energy ontology (ZEO) in which so called causal diamonds (CDs, analogs Penrose diagrams) can be seen as being forced by very simple condition: the volume action forced by twistorial lift of TGD must be finite. CD would represent the perceptive field defined by finite volume of imbedding space H=M4× CP2.

    ZEO implies that conservation laws formulated only in the scale of given CD do not anymore fix select just single solution of field equations as in classical theory. Theories are strictly speaking impossible to test in the old classical ontology. In ZEO testing is possible be sequence of state function reductions giving information about zero energy states.

    In principle transition between any two zero energy states - analogous to events specified by the initial and final states of event - is in principle possible but Negentropy Maximization Principle (NMP) as basic variational principle of state function reduction and of consciousness restricts the possibilities by forcing generation of negentropy: the notion of negentropy requires p-adic physics.

    Zero energy states are quantum superpositions of classical time evolutions for 3-surfaces and classical physics becomes exact part of quantum physics: in QFTs this is only the outcome of stationary phase approximation. Path integral is replaced with well-defined functional integral- not over all possible space-time surface but pairs of 3-surfaces at the ends of space-time at opposite boundaries of CD.

    ZEO leads to a theory of consciousness as quantum measurement theory in which observer ceases to be outsider to the physical world. One also gets rid of the basic problem caused by the conflict of the non-determinism of state function reduction with the determinism of the unitary evolution. This is obviously an extension of ordinary physics.

  5. Hierarchy of Planck constants represents also an extension of quantum mechanics at QFT limi. At fundamental level one actually has the standard value of h but at QFT limit one has effective Planck constant heff =n× h, n=1,2,... this generalizes quantum theory. This scaling of h has a simple topological interpretation: space-time surface becomes n-fold covering of itself and the action becomes n-multiple of the original which can be interpreted as heff=n×h.

    The most important applications are to biology, where quantum coherence could be understood in terms of a large value of heff/h. The large n phases resembles the large N limit of gauge theories with gauge couplings behaving as α ∝ 1/N used as a kind of mathematical trick. Also gravitation is involved: heff is associated with the flux tubes mediating various interactions (being analogs to wormholes in ER-EPR correspondence). In particular, one can speak about hgr, which Nottale introduced originally and heff= hgr plays key role in quantum biology according to TGD.

B. In what sense TGD is simplification/extension of existing theory?

  1. Classical level: Space-time as 4-surface of H means a huge reduction in degrees of freedom. There are only 4 field like variables - suitably chosen 4 coordinates of H=M4× CP2. All classical gauge fields and gravitational field are fixed by the surface dynamics. There are no primary gauge fields or gravitational fields nor any other fields in TGD Universe and they appear only at the QFT limit.

    GRT limit would mean that many-sheeted space-time is replaced by single slightly curved region of M4. The test particle - small particle like 3-surface - touching the sheets simultaneously experience sum of gravitational forces and gauge forces. It is natural to assume that this superposition corresponds at QFT limit to the sum for the deviations of induced metrics of space-time sheets from flat metric and sum of induce gauge potentials. These would define the fields in standard model + GRT. At fundamental level effects rather than fields would superpose. This is absolutely essential for the possibility of reducing huge number field like degrees of freedom. One can obviously speak of emergence of various fields.

    A further simplification is that only preferred extremals for which data coding for them are reduced by SH to 2-D string like world sheets and partonic 2-surfaces are allowed. TGD is almost like string model but space-time surfaces are necessary for understanding the fact that experiments must be analyzed using classical 4-D physics. Things are extremely simple at the level of single space-time sheet.

    Complexity emerges from many-sheetedness. From these simple basic building bricks - minimal surface extremals of Kähler action (not the extremal property with respect to Kähler action and volume term strongly suggested by the number theoretical vision plus analogs of Super Virasoro conditions in initial data) - one can engineer space-time surfaces with arbitrarily complex topology - in all length scales. An extension of existing space-time concept emerges. Extremely simple locally, extremely complex globally with topological information added to the Maxwellian notion of fields (topological field quantization allowing to talk about field identify of system/field body/magnetic body.

    Another new element is the possibility of space-time regions with Euclidian signature of the induced metric. These regions correspond to 4-D "lines" of general scattering diagrams. Scattering diagrams has interpretation in terms of space-time geometry and topology.

  2. The construction of quantum TGD using canonical quantization or path integral formalism failed completely for Kähler action by its huge vacuum degeneracy. The presence of volume term still suffers from complete failure of perturbation theory and extreme non-linearity. This led to the notion of world of classical worlds (WCW) - roughly the space of 3-surfaces. Essentially pairs of 3-surfaces at the boundaries of given CD connected by preferred extremals of action realizing SH and SGCI.

    The key principle is geometrization of the entire quantum theory, not only of classical fields geometrized by space-time as surface vision. This requires geometrization of hermitian conjugation and representation of imaginary unit geometrically. Kähler geometry for WCW makes this possible and is fixed once Kähler function defining Kähler metric is known. Kähler action for a preferred extremal of Kähler action defining space-time surface as an analog of Bohr orbit was the first guess but twistor lift forced to add volume term having interpretation in terms of cosmological constant.

    Already the geometrization of loop spaces demonstrated that the geometry - if it exists - must have maximal symmetries (isometries). There are excellent reasons to expect that this is true also in D=3. Physics would be unique from its mathematical existence!


  3. WCW has also spinor structure. Spinors correspond to fermionic Fock states using oscillator operators assignable to the induced spinor fields - free spinor fiels. WCW gamma matrices are linear combinations of these oscillator operators and Fermi statistics reduces to spinor geometry.


  4. There is no quantization in TGD framework at the level of WCW. The construction of quantum states and S-matrix reduces to group theory by the huge symmetries of WCW. Therefore zero energy states of Universe (or CD) correspond formally to classical WCW spinor fields satisfying WCW Dirac equation analogous to Super Virasoro conditions and defining representations for the Yangian generalization of the isometries of WCW (so called super-symplectic group). In ZEO stated are analogous to pairs of initial and final states and the entanglement coefficients between positive and negative energy parts of zero energy states expected to be fixed by Yangian symmetry define scattering matrix and have purely group theoretic interpretation. If this is true, entire dynamics would reduce to group theory in ZEO.

C. What is the hypothetical applicability of the extension - in energies, sizes, masses etc?

TGD is a unified theory and is meant to apply in all scales. Usually the unifications rely on reductionistic philosophy and try to reduce physics to Planck scale. Also super string models tried this and failed: what happens at long length scales was completely unpredictable (landscape catastrophe).

Many-sheeted space-time however forces to adopt fractal view. Universe would be analogous to Mandelbrot fractal down to CP2 scale. This predicts scaled variants of say hadron physics and electroweak physics. p-Adic length scale hypothesis and hierarchy of phases of matter with heff=n×h interpreted as dark matter gives a quantitative realization of this view.

  1. p-Adic physics shows itself also at the level of real physics. One ends up to the vision that particle mass squared has thermal origin: the p-adic variant of particle mass square is given as thermal mass squared given by p-adic thermodynamics mappable to real mass squared by what I call canonical identification. p-Adic length scale hypothesis states that preferred p-adic primes characterizing elementary particles correspond to primes near to power of 2: p=about 2k. p-Adic length scale is proportional to p1/2.

    This hypothesis is testable and it turns out that one can predict particle mass rather accurately. This is highly non-trivial since the sensitivity to the integer k is exponential. So called Mersenne primes turn out to be especially favoured. This part of theory was originally inspired by the regularities of particle mass spectrum. I have developed arguments for why the crucial p-adic length scale hypothesis - actually its generalization - should hold true. A possible interpretation is that particles provide cognitive representations of themselves by p-adic thermodynamics.

  2. p-Adic length scale hypothesis leads also to consider the idea that particles could appear as different p-adically scaled up variants. For instance, ordinary hadrons to which one can assign Mersenne prime M107=2107-1 could have fractally scaled variants. M89 and MG,107 (Gaussian prime) would be two examples and there are indications at LHC for these scaled up variants of hadron physics. These fractal copies of hadron physics and also of electroweak physics would correspond to extension of standard model.

  3. Dark matter hierarchy predicts zoomed up copies of various particles. The simplest assumption is that masses are not changed in the zooming up. One can however consider that binding energy scale scales non-trivially. The dark phases would emerge are quantum criticality and give rise to the associated long range correlations (quantum lengths are typically scaled up by heff/h=n).

D. What is the leading correction/contribution to physical effects due to TGD onto particles, interactions, gravitation, cosmology?

  1. Concerning particles I already mentioned the key predictions.

    1. The existence of scaled variants of various particles and entire branches of physics. The fundamental quantum numbers are just standard model quantum numbers code by CP2 geometry.


    2. Particle families have topological description meaning that space-time topology would be an essential element of particle physics. The genus of partonic 2-surfaces (number of handles attached to sphere) is g=0,1,2,... and would give rise to family replication. g<2 partonic 2-surfaces have always global conformal symmetry Z2 and this suggests that they give rise to elementary particles identifiable as bound states of g handles. For g>2 this symmetry is absent in the generic case which suggests that they can be regarded as many-handle states with mass continuum rather than elementary particles. 2-D anyonic systems could represent an example of this.

    3. A hierarchy of dynamical symmetries as remnants of super-symplectic symmetry however suggests itself. The super-symplectic algebra possess infinite hierarchy of isomorphic sub-algebras with conformal weights being n-multiples of for those for the full algebra (fractal structure again possess also by ordinary conformal algebras). The hypothesis is that sub-algebra specified by n and its commutator with full algebra annihilate physical states and that corresponding classical Noether charges vanish. This would imply that super-symplectic algebra reduces to finite-D Kac-Moody algebra acting as dynamical symmetries. The connection with ADE hierarchy of Kac-Moody algebras suggests itself. This would predict new physics. Condensed matter physics comes in mind.

    4. Number theoretic vision suggests that Galois groups for the algebraic extensions of rationals act as dynamical symmetry groups. They would act on algebraic discretizations of 3-surfaces and space-time surfaces necessary to realize number theoretical universality. This would be completely new physics.

  2. Interactions would be mediated at QFT limit by standard model gauge fields and gravitons. QFT limit however loses all information about many-sheetedness and there would be anomalies reflecting this information loss. In many-sheeted space-time light can propagate along several paths and the time taken to travel along light-like geodesic from A to B depends on space-time sheet since the sheet is curved and warped. Neutrinos and gamma rays from SN1987A arriving at different times would represent a possible example of this. It is quite possible that the outer boundaries of even macroscopic objects correspond to boundaries between Euclidian and Minkowskian regions at the space-time sheet of the object.

    The failure of QFTs to describe bound states of say hydrogen atom could be second example: many-sheetedness and identification of bound states as single connected surface formed by proton and electron would be essential and taken into account in wave mechanical description but not in QFT description.

  3. Concerning gravitation the basic outcome is that by number theoretical vision all preferred extremals are extremals of both Kähler action and volume term. This is true for all known extremals what happens if one introduces the analog of Kähler form in M4 is an open question).

    Minimal surfaces carrying no K&aum;lher field would be the basic model for gravitating system. Minimal surface equation are non-linear generalization of d'Alembert equation with gravitational self-coupling to induce gravitational metric. In static case one has analog for the Laplace equation of Newtonian gravity. One obtains analog of gravitational radiation as "massless extremals" and also the analog of spherically symmetric stationary metric.

    Blackholes would be modified. Besides Schwartschild horizon which would differ from its GRT version there would be horizon where signature changes. This would give rise to a layer structure at the surface of blackhole.

  4. Concerning cosmology the hypothesis has been that RW cosmologies at QFT limit can be modelled as vacuum extremals of Kä hler action. This is admittedly ad hoc assumption inspired by the idea that one has infinitely long p-adic length scale so that cosmological constant behaving like 1/p as function of p-adic length scale assignable with volume term in action vanishes and leaves only Kähler action. This would predict that cosmology with critical is specified by a single parameter - its duration as also over-critical cosmology. Only sub-critical cosmologies have infinite duration.

    One can look at the situation also at the fundamental level. The addition of volume term implies that the only RW cosmology realizable as minimal surface is future light-cone of M4. Empty cosmology which predicts non-trivial slightly too small redshift just due to the fact that linear Minkowski time is replaced with lightcone proper time constant for the hyperboloids of M4+. Locally these space-time surfaces are however deformed by the addition of topologically condensed 3-surfaces representing matter. This gives rise to additional gravitational redshift and the net cosmological redshift. This also explains why astrophysical objects do not participate in cosmic expansion but only comove. They would have finite size and almost Minkowski metric.

    The gravitational redshift would be basically a kinematical effect. The energy and momentum of photons arriving from source would be conserved but the tangent space of observer would be Lorentz-boosted with respect to source and this would course redshift.

    The very early cosmology could be seen as gas of arbitrarily long cosmic strings in H (or M4) with 2-D M4 projection. Horizon would be infinite and TGD suggests strongly that large values of heff makes possible long range quantum correlations. The phase transition leading to generation of space-time sheets with 4-D M4 projection would generate many-sheeted space-time giving rise to GRT space-time at QFT limit. This phase transition would be the counterpart of the inflationary period and radiation would be generated in the decay of cosmic string energy to particles.

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

Articles and other material related to TGD.

Wednesday, January 04, 2017

Bullet cluster and TGD based model of dark matter

Sabine Hossenfelder wrote about Bullet Cluster. Usually Bullet Cluster is seen to favor dark matter and disfavor MOND theory introducing a modification of Newtonian gravity. Sabine Hossenfelder saw it differently.

Cold dark matter model (ΛCDM) and MOND are two competing mainstream models explaining the constant velocity spectrum of stars in galaxies.

  1. ΛCDM assumes that dark matter forms a spherical halo around galaxy and that its density profile is such that it gives the observed velocity spectrum of distant stars. The problem of the model is that dark matter distribution can have many shapes and it is not easy to understand why approximately constant velocity spectrum is obtained. Also the attempts to find dark matter particles identified as some exoticons have failed one after another. The recent finding that the velocity spectrum of distant stars around galaxies correlates strongly with the density of baryonic matter also challenges this model: it is difficult to believe that the halo would have so universal baryonic mass density.

  2. MOND does not assume dark matter but makes an ad hoc modification of gravitational force for small accelerations. The problem of MOND is that it is indeed an ad hoc modification and it is not easy to see how to make it consistent with general relativity: it is difficult to do cosmology using MOND. For small accelerations (small space-time curvatures) one would expect Newtonian theory to be an excellent approximation.

Consider now how Bullet Cluster relates to these two options. Bullet cluster is a pair of galaxy clusters which has emerged from collision (see the figure). There exists data at optical wavelenghts about stars. Stars experience only a small gravitational slowing down and are expected to go through the collision region rather fast. Data from X-ray measurements give information about the intergalactic gas associated with clusters. This gas interacts electromagnetically and is slowed down much more and remains in the collision region for a longer time. The red regions in the figure correspond to the gas. Gravitational lensing in turn gives information about space-time curvature and these two regions are farthest away from the collision center. These regions are blue and would naturally correspond to dark matter in ΛCDM model. Both models have severe problems.
  1. In cold dark matter model the event would require too high relative velocity for colliding clusters - about c/100. The probability for this kind of collision in cold dark matter model is predicted to be very low - about 6.4×10-6. Something seems to be wrong with ΛCDM model.

  2. In MOND the relative collision velocities are argued to be much more frequent. Bee however forgot to mention that in MOND the lensing is expected to be associated with X-ray region (hot gas in the center of figure) rather than with the blue regions disjoint from it. This observation is a very severe blow against MOND model.

The logical conclusion is that there indeed seems to be dark matter there but it is something different from the cold dark matter. What it could be?

What could be the interpretation in TGD?

  1. In TGD galaxies are associated with cosmic string or more general string like objects like pearls with necklace: that this is the case is known for decades but for some mysterious reason to me has not been used as guideline in dark matter models. Maybe it is very difficult to see things from bigger perspective than galaxies.

    The flux tubes carry Kähler magnetic energy, dark energy, and dark matter in TGD sense having heff=n×h. The galactic matter experiences transversal 1/ρ gravitational force predicting constant velocity spectrum for distant stars when baryonic matter is neglected. Note that one avoids a model for the profile of the halo altogether. The motion of the galaxy along the flux tube is free apart from the forces caused by galaxy. The presence of baryonic matter implies that the velocity increases slowly with distance up to some critical radius. By recent findings correlating observed velocity spectrum with density of baryonic matter one can deduce the density of baryonic matter (see this). A possible interpretation is as remnants of cosmic string like object produced in its decay to ordinary matter completely analogous to the decay of the vacuum energy of inflaton field to matter in inflation theory.

    The order of magnitude for velocity vgal for distant stars in galaxies is about vgal∼ c/1000. In absence of baryonic matter it is predicted to be constant and proportional satisfy v∝ (TG)1/2, T string tension and G Newton's constant (c=1). T in turn is proportional to 1/R2, where R is CP2 radius. Maximal velocity is obtained for cosmic strings. For magnetic flux tubes resulting when cosmic strings develop 4-D M4 projection string tension T and thus vgal is reduced. One obtains larger velocities if there are several parallel flux tubes forming a gravitational bound state so that tensions add.

  2. By fractality also galaxy clusters are expected to form similar linear structures. Concerning the interpretiong of the Bullet Cluster one can imagine two options.

    1. The two colliding clusters could belong to the same string like object and move in opposite directions along it. In this case gravitational lensing would be most naturally associated with the flux tube and there would be single linear blue region instead of the two blue spots of the figure.

    2. The clusters could also belong to different flux tubes, which pass by each other and induce the collision of clusters and the gas associated with them. If the flux tubes are more or less parallel and orthogonal to the plane of the figure, the gravitational lensing would be from the two string like objects and two disjoint blue spots would appear in the figure. This option conforms with the figure.



  3. The collision velocity would correspond to the relative velocity of flux tubes. Can one say anything about the needed collision velocities? The naive first guess of dimensional analyst is that the rotation velocity vgal ∝ (TG)1/2 determining galactic rotation spectrum determines also the typical relative velocity between galaxies. Here T would be the string tension of flux tubes containing galaxy clusters along it. T would gradually decrease during the cosmic evolution as flux tubes gets thicker and magnetic energy density is reduced. The velocity v∼ c/100 suggested by ΛCDM model is 10 times larger than c/1000 for distant stars in galaxies.
    By fractality similar view would apply to galaxy clusters assigned to flux tubes. Cluster flux tubes containing clusters along them could correspond to bound states of parallel galactic flux tubes containg galaxies along them.

  4. The simplest model for collision of flux tubes treats them as parallel rigid strings so that dimensional reduction to D=2 occurs. The gravitational potential is logarithmic potential: V= Klog(ρ). One can use conservation laws of angular momentum and energy to solve the equations of motion just as in 3-D central force problem. The initial and final angular momentum per mass equals to J= v0a, where a is the impact parameter and v0 the initial velocity. The initial energy per unit mass equals to e= v02/2 and is same in the final state. Conservation law for e gives e= v2/2+Klog(ρ) = v02/2.
    Conservation law for angular momentum reads j= v ρsin(φ)=v0a and gives v =j/ρsin(φ). Velocity is given from v2=(dρ/dt)2+ ρ2(dφ/dt)2 and leads together with conservation laws a first order differential equation for &drho;/dt.

    Since the potential is logarithmic, there is rather small variation of energy in the collision so that the clusters interact rather weakly. This could produce the same effect as larger relative collision velocity in ΛCDM model with kinetic energy dominating over gravitational potential.

See the chapter TGD and astrophysics.

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

Articles and other material related to TGD.

Tuesday, January 03, 2017

TGD and hydrogen atom as anomaly of QED

Certain crackpot hunter made in FB an attempt to debunk TGD by claiming that TGD cannot describe hydrogen atom, which he regards as kindergarten physics. This couldn't be farther from truth: hydrogen atom (and more generally, bound states) are anomaly of QED (QDT) and the many-sheeted space-time of TGD can cure it! See the following arguments. I hope they might help also this anti-crackpot to learn the basics.

  1. Hydrogen atom is problem in QED and therefore also in standard model. The relativistic generalization of hydrogen atom based on Bethe-Salpeter equation does not work properly but predicts a lot of non-existing states. This can be found from the text book or Iztykson-Zuber. Schroedinger equation and Dirac equation give excellent predictions. What goes wrong with QED?

  2. In superstring theories one cannot say anything about hydrogen atom unless one just assumes that spontaneous compactification yields M4 as base space. It is still unclear whether one really obtains standard model gauge group at low energy limit. Actually one cannot even predict space-time dimension. I do not know whether 3-brane identification can yield the desired QFT limit. Probably not.

  3. In TGD framework standard model symmetries and fields are coded in CP2 geometry. Standard model and general relativity are obtained at QFT limit when one replaces sheets with single slightly curved region of Minkowski space. Gravitational field (deviation of metric from M4 metric) and gauge potentials are obtained as sums of those for sheets by simple arguments. Effects on test particle touching the sheet sum up at fundamental level and corresponding fields sum up at QFT limit.

  4. QFT limit of TGD would have same problems as hydrogen atom in QED. That wave mechanics works so well and QED fails must be due to approximation replacing many-sheeted space-time with a region of empty Minkowski space M4. Indeed, in QFT approach one treats proton and electron as disjoint surfaces approximated as point like particles. For bound states they however form single 3-surface obtained by connecting the 3-surfaces by flux tubes. The outcome is too many degrees of freedom and non-physical states as Bethe-Salpeter indeed predicts. In wave mechanics one makes the needed approximation by considering time evolution of wave functions in 3-space instead of 4-D correlation functions and gets rid of spurious states.

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

Articles and other material related to TGD.

Monday, January 02, 2017

Generalized Kähler structure for Minkowski space and CP breaking and matter antimatter asymmetry

In previous postings I have considered the possibility that the analog of Kähler form as self-dual U(1) gauge field representing constant magnetic and electric fields with same sign, magnitude and direction could be present as symmetry with CP2 would suggest. Poincare invariance would be still achieved by allowing the moduli space for CDs and extension of WCW by this moduli space necessitated also by quantum measurement theory and theory of consciousness (flow and arrow of time).

Why J(M4) would be needed? First of all, it provides the breaking of Poincare invariance involve with quantum measurement. The time direction defining rest system and quantization axis of spin indeed imply symmetry breaking. Secondly, it might allow to understand CP breaking in kaon-anti-kaon system and other similar systems from first principles since M4-electric dipole momentum would contribute to mass splitting between particle. This small asymmetry could lead to matter-antimatter asymmetry. Matter and antimatter would have different values of heff and dark relative to each other. The following arguments already given in previous posting (the reposting is motivated by the enormous importance of these mysteries for modern physics) support this view quantitatively.

CP breaking in hadronic systems is one of the poorly understood aspects of fundamental physics and relates closely to the mysterious matter-antimatter asymmetry. The constant electric part of self dual J(M4) implies CP breaking. I have earlier considered the possibility that Kähler electric fields could cause this breaking but this breaking would be local. Second possibility is that matter and antimatter correspond to different values of heff and are dark relative to each other.

Could J(M4) explain the observed CP breaking as appearing already at the level of imbedding space M4× CP2 and could this breaking explain hadronic CP breaking and matter anti-matter asymmetry? Could M4 part of Kähler electric field induce different heff/h=n for particles and antiparticles?

To answer these questions one can study Dirac equation at imbedding space level coupled to the gauge potential A(M4) for J(M4).

  1. The coupling of Kähler form to leptons is 3 times larger than to to quarks as in the case of A(CP2). This would give coupling k=1 for quarks an k=3 for leptons. k corresponds to fermion number which is opposite for fermions and antifermions having therefore opposite values of k at the respective space-time sheets.

  2. The potential satisfies ∂μAμ(M4)=0. Let the non-vanishing components of the Kähler gauge potential be (A0,Az)=ε (x,+/- y). The sign fact ε+/- 1 corresponds to self dual and antiself-dual options, let us assume self-duality as in the case of CP2 Kähler form. Scalar d'Alembertian reads as (∂μμ+ AμAμ)Ψ= -m2 Ψ.

  3. Assuming momentum eigenstate in time and z-direction (plane M2), one obtains by separation of variables (H1+H2)Ψ= (E-m2-kz2)Ψ. Hx= -∂x2+k2x2 and Hy= -∂y2+k2y2) are oscillator Hamiltonians. The spectrum is of Hx+Hy is given by kT2= (n1+n2+1)21/2|k| and one obtains E2=m2 +kz2 +kT2. This contribution is CP invariant and same for fermions and anti-fermions. The special feature is the presence of zero point transversal momentum. It is not possible to have a particle, which would be completely at rest. One can also say that m2 is increased 21/2|k| hbar2/L2, L= 1 m if standard convention for metric is used. For other conventions the numerical value of CP2 radius is scale by L/Lnew. L must correspond to some physical scale assignable to particle: secondary p-adic length scale is the natural identification.

  4. Spinor d'Alembertian contains also dipole moment term kX=JmuνΣμν giving a contribution, which depends on the sign of k: E2=m2 +kz2 +kT2+ kX. The term is sum of magnetic and electric dipole moment terms. The coupling k changes sign in CP operation and be of opposite sign for fermions and anti-fermions. One has a breaking of CP for given spin state. The dependence of X on spin state gives a test for the theory and also for the predicted CP breaking.

  5. Scaling covariance allows in principle all values L. To estimate the size of the effect one must fix the length scale L. CP2 size has only different value using L as unit and in flat background it does not matter. L should correspond to the size scale of the CD associated with particle. The secondary p-adic length scale of fermion defining also the size scale of its magnetic body is a natural guess so that Δ E2/E2≈ 2Δ E/E≈ Δ m/m ∼ 2/p1/2, p≈ 2k would hold true. This mass splitting is very small. For weak bosons having k=89 the mass splitting would be of order 3× 10-4 eV. For small values of p at ultrahigh energies the scale of CP breaking is larger, which conforms with the idea that matter-antimatter-asymmetry has emerged in very early cosmology.

    The recent experiment found that the mass difference Δ m/m for proton and antiproton satisfies Δ m <69× 10-12m ≈ 6.9× 10-2 eV (see this) so that this gives no constraints. Kaon-antikaon mass difference is estimated to be about 3.5× 10-6 eV (see this). This would correspond to a p-adic length scale k=96. Top quark is mainly responsible for the mixing of neutral kaon and its antiparticle in the model of based on loops involving decay to virtual quark pairs. The estimate from p-adic mass calculations for top quark mass scale is k=94 so that the order of magnitude estimate has correct of order of magnitude (being by factor 4 too large). This is an encouraging sign.

    How the mass splitting of neutral kaons would result? In quark model kaon and antikaon can be regarded as sdbbar and dsbar pairs. The net spins vanishes but the mass splitting due to electric moment dipole moment term X is non-vanishing due to the different sign of coupling k. The sign of the mass splitting is also opposite for kaon and antikaon.

  6. One can also consider the modified Dirac equation for canonically imbedded M4 which is simplest preferred extremal. The coupling to J(M4) to modified Dirac equation in space-time interior with gamma matrices replaced with modified gamma matrices are obtained as contractions of canonical momentum currents with M4 gamma matrices. Completely analogous phenomenon happens for CP2 type extremals. Tαβ=0 so that the modified gamma comes from Jαβ Jklβmlγk. These give just ordinary gamma matrices so that the two Dirac equations are identical.

For background see the article Some questions related to the twistor lift of TGD or the chapter How the hierarchy of Planck constants might relate to the almost vacuum degeneracy for twistor lift of TGD? of "Towards M-matrix".

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

Articles and other material related to TGD.

Sunday, January 01, 2017

Some questions related to the twistor lift of TGD

In the following I will consider some questions related to the twistor lift of TGD and end up to a possible vision about general mechanism of CP breaking and generation of matter antimatter asymmetry.

  1. Can the analog of Kähler form J(M4) assignable to M4 suggested by the symmetry between M4 and CP2 and by number theoretical vision appear in the theory? What would be the physical implications? The basic objection is the loss of Poincare invariance is lost. This can be however avoided by introducing the moduli space for Kähler forms. This moduli space is actually the moduli space of causal diamonds (CDs) forced in any case by zero energy ontology (ZEO) and playing central role in the generalization of quantum measurement theory to a theory of consciousness and in the explanation of the relationship between geometric and subjective time.

    Why J(M4) would be needed? J(M4) corresponds to parallel constant electric and magnetic fields in given direction. Constant E and B=E fix directions of quantization axes for energy (rest system) and spin. One implication is transversal localization of imbedding space spinor modes: imbedding space spinor modes are products of harmonic oscillator Gaussians in transversal degrees of freedom very much like quarks inside hadrons.

    Also CP breaking is implied by the electric field and the question is whether this could explain the observed CP breaking as appearing already at the level of imbedding space M4× CP2. The estimate for the CP mass splitting of neutral kaon and anti-kaon is of correct order of magnitude. Whether stationary spherically symmetric metric as minimal surface allows a sensible physical generalization is a killer test for the hypothesis.

  2. How does gravitational coupling emerge at fundamental level? The answer is obvious: string area action is scaled by 1/G as in string models. The objection is that p-adic mass calculations suggest that string tension is determined by CP2 size R: the analog of string tension appearing in mass formula given by p-adic mass calculations would be by a factor about 10-8 smaller than that estimated from string tension. The discrepancy evaporates by noticing that p-adic mass calculations rely on p-adic thermodynamics at imbedding space level whereas string world sheets appear at space-time level.

  3. Could one regard the localization of spinor modes to string world sheets as a localization to Lagrangian sub-manifolds of space-time surface having by definition vanishing induced Kähler form: J(M4)+J(CP2)=0. Lagrangian sub-manifolds would be commutative in the sense of Poisson bracket. Could string world sheets be minimal surfaces satisfying J(M4)+J(CP2)=0. The Lagrangian condition allows also more general solutions - even 4-D space-time surfaces and one obtains analog of brane hierarchy. Could one allow spinor modes also at these analogs of branes. Is Lagrangian condition equivalent with the original condition that induced W boson fields making the em charge of induced spinor modes ill-defined vanish and allowing also solution with other dimensions. How Lagrangian property relates to the idea that string world sheets correspond to complex (commutative) surfaces of quaternionic space-time surface in octonionic imbedding space.

1. Can the Kähler form of M4 appear in Kähler action?

I have already earlier considered the question whether the analog of Kähler form assignable to M4 could appear in Kähler action. Could one replace the induced Kähler form J(CP2) with the sum J=J(M4)+J(CP2) such that the latter term would give rise to a new component of Kähler form both in space-time interior at the boundaries of string world sheets regarded as point-like particles? This could be done both in the Kähler action for the interior of X4 and also in the topological magnetic flux term ∈t J associated with string world sheet and reducing to a boundary term giving couplings to U(1) gauge potentials Aμ(CP2) and Aμ(M4) associated with J(CP2) and J(M4). The interpretation of this coupling is an interesting challenge.

Consider first the objections against introducing J(M4) to the Kähler action at imbedding space level.

  1. J(M4) would would break translational and Lorentz symmetries at the level of imbedding space since J(M4) cannot be Lorentz invariant. For imbedding space spinor modes this term would bring in coupling to the self-dual Kähler form in M4. The simplest choice is A=(At=z, Az=0,Ax=y,Ay=0) defining decomposition M4 =M2× E2. For Dirac equation in M4 one would have free motion in preferred time-like (t,z)-plane plane M2 in whereas in x- and y-directions (E2 plane) would one have harmonic oscillator potentials due to the gauge potentials of electric and magnetic fields. One would have something very similar to quark model of hadron: quark momenta would have conserved longitudinal part and non-conserved transversal part. The solution spectrum has scaling invariance Ψ(mk)→ Ψ(λ mk) so that there is no preferred scale and the transversal scales scale as 1/E and 1/kx.

  2. Since J(M4) is not Lorentz invariant Lorentz boosts would produce new M2× E2 decomposition. If one assumes above kind of linear gauge as gauge invariance suggests, the choices with fixed second tip of causal diamond (CD) define finite-dimensional moduli space SO(3,1)/SO(1,1)× SO(2) having in number theoretic vision an interpretation as a choice of preferred hypercomplex plane and its orthogonal complement. This is the moduli space for hypercomplex structures in M4 with the choices of origins parameterized by M4. The introduction of the moduli space would allow to preserve Poincare invariance.

  3. If one generalizes the condition for Kähler metric to J2(M4)=-g(M4) fixing the scaling of J, the coupling to A(M4) is also large and suggests problems with the large breaking of Poincare symmetry for the spinor modes of the imbedding space for given moduli. The transversal localization by the self-dual magnetic and electric fields for J(M4) would produce wave packets in transversal degrees of freedom: is this physical?

    This moduli space is actually the moduli space introduced for causal diamonds (CDs) in zero energy ontology (ZEO) forced by the finite value of volume action: fixing of the line connecting the tips of CD the Lorentz boost fixing the position for the second tip of CD parametrizes this moduli space apart from division with the group of transformations leaving the planes M2 and E2 having interpretation a plane defined by light-like momentum and polarization plane associated with a given CD invariant.

  4. Why this kind of symmetry breaking for Poincare invariance? A possible explanation proposed already earlier is that quantum measurement involves a selection of quantization axis. This choice necessarily breaks the symmetries and J(M4) would be an imbedding space correlate for the selection of rest frame and quantization axis of spin. This conforms with the fact that CD is interpreted as the perceptive field of conscious entity at imbedding space level: the contents of consciousness would be determined by the superposition of space-time surfaces inside CD. The choice of J(M4) for CD would select preferred rest system (quantization axis for energy as a line connecting tips of CD) via electric part of J(M4) and quantization axis of spin (via magnetic part of J(M4). The moduli space for CDs would be the space for choices of these particular quantization axis and in each state function reduction would mean a localization in this moduli space. Clearly, this reduction would be higher level reduction and correspond to a decision of experimenter.

To summarize, for J(M4)=0 Poincare symmetries are realized at the level of imbedding space but obviously broken slightly by the geometry of CD. The allowance of J(M4)≠ 0 implies that both translational and rotational symmetries are reduced for a given CD: the interpretation would be in terms of a choice of quantization axis in state function reduction. They are however lifted to the level of moduli space of CDs and exact in this more abstract sense. This is nothing new: already the introduction of ZEO and CDs force by volume term in action forced by twistor lift of TGD implies the same. Also the view about state function reduction requires wave functions in the moduli space of CDs. This is also essential for understanding how the arrow of geometric time is inherited from that of subjective time in TGD inspired theory of consciousness.

What about the situation at space-time level?

  1. The introduction of J(M4) part to Kähler action has nice number theoretic aspects. In particular, J selects the preferred complex and quaternionic sub-space of octonionic space of imbedding space. The simplest possibility is that the Kähler action is defined by the Kähler form J(M4)+J(CP2).

    Since M4 and CP2 Kähler geometries decouple it should be possible to take the counterpart of Kähler coupling strength in M4 to be much larger than in CP2 degrees of freedom so that M4 Kähler action is a small perturbation and slowly varying as a functional of preferred extremal. This option is however not in accordance with the idea that entire Kähler form is induced.

  2. Whether the proposed ansätze for general solutions make still sense is not clear. In particular, can one still assume that preferred extremals are minimal surfaces? Number theoretical vision strongly suggests - one could even say demands - the effective decoupling of Kähler action and volume term. This would imply the universality of quantum critical dynamics. The solutions would not depend at all on the coupling parameters except through the dependence on boundary conditions. The coupling between the dynamics of Kähler action and volume term would come also from the conservation conditions at light-like 3-surfaces at which the signature of the induced metric changes.

  3. At space-time level the field equations get more complex if the M4 projection has dimension D(M4)>2 and also for D(M4)=2 if it carries non-vanishing induced J(M4). One would obtain cosmic strings of form X2× Y2 as minimal surface extremals of ordinary Kähler action or X2 Lagrangian manifold of M4 as also CP2 type vacuum extremals and their deformations with M4 projection Lagrangian manifold. Thus the differences would not be seen for elementary particle and string like objects. Simplest string worlds sheet for which J(M4) vanishes would correspond to a piece of plane M2.

    M4 is the simplest minimal surface extremal of Kähler action necessarily involving also J(M4). The action in this case vanishes identically by self-duality (in Euclidian signature self-duality does not imply this). For perturbations of M4 such as spherically symmetric stationary metric the contribution of M4 Kähler term to the action is expected to be small and the come mainly from cross term mostly and be proportional to the deviation from flat metric. The interpretation in terms of gravitational contribution from M4 degrees of freedom could make sense.

  4. What about massless extremals (MEs)? How the induced metric affects the situation and what properties second fundamental form has? Is it possible to obtain a situation in which the energy momentum tensor Tαβ and second fundamental form Hkαβ have in common components which are proportional to light-like vector so that the contraction TαβHkαβ vanishes?

    Minimal surface property would help to satisfy the conditions. By conformal invariance one would expect that the total Kähler action vanishes and that one has JαγJγβ = a× gαβ+b × kαkβ.
    These conditions together with light-likeness of Kähler current guarantee that field equations are satisfied.

    In fact, one ends up to consider a generalization of MEs by starting from a generalization of holomorphy. Complex CP2 coordinates ξi would be functions of light-like M2 coordinate u+=k• m, k light-like vector, and of complex coordinate w for E2 orthogonal to M2. Therefore the CP2 projection would 3-D rather than 2-D now.

    The second fundamental form has only components of form Hku+w, Hku+w* and Hkww, Hkw*w*. The CP2 contribution to the induced metric has only components of form Δ gu+w, Δ g+w*, and gw*w. There is also contribution gu+u-=1, where v is the light-like dual of u in plane M2. Contravariant metric can be expanded as a power series for in the deviation (Δ gu+w, Δ gu+w*) of the metric from (gu+u-, gww*). Only components of form gu+,ui and gww* are obtained and their contractions with the second fundamental form vanish identically since there are no common index pairs with simultaneously non-vanishing components. Hence it seems that MEs generalize!

    I have asked earlier whether this construction might generalize for ordinary MEs. One can introduce what I have called Hamilton-Jacobi structure for M4 consisting of locally orthogonal slicings by integrable 2-surfaces having tangent space having local decomposition M2x× E2x with light-like direction depending on point x. An objection is that the direction of light-like momentum depends on position: this need not be inconsistent with momentum conservation but would imply that the total four-momentum is not light-like anymore. Topological condensation for MEs and at MEs could imply this kind modification.

  5. There is also a topological magnetic flux type term for string world sheet. Topological term can be transformed to a boundary term coupling classical particles at the boundary of string world sheet to CP2 Kähler gauge potential (added to the equation for a light-like geodesic line). Now also the coupling to M4 gauge potential would be obtained. The condition J(M4)+ J(CP2)=0 at string world sheets is very attractive manner to identify string world sheets as analogs of Lagrangian manifolds but does not imply the vanishing of the net U(1) couplings at boundary since the induce gauge potentials are in general different.

    Also topological term including also M4 Kähler magnetic flux for string world sheet contributes also to the modified Dirac equation since the gamma matrices are modified gamma matrices required by super-conformal symmetries and defined as contractions of canonical momentum densities with imbedding space gamma matrices. This is true both in space-time interior, at string world sheets and at their boundaries. CP2 (M4) term gives a contribution proportional to CP2 (M4) gamma matrices.

    At imbedding space level transversal localization would be the outcome and a good guess is that the same happens also now. This is indeed the case for M4 defining the simplest extremal. The general interpretation of M4 Kähler form could be as a quantum tool for transversal dynamical localization of wave packets in Kähler magnetic and electric fields of M4. Analog for decoherence occurring in transversal degrees of freedom would be in question. Hadron physics could be one application.

How to test this idea?
  1. It might be possible to kill the idea by showing that one does not obtain spherically symmetric Schwartschild type metric as a minimal surface extremal of generalized Kähler action: these extremals are possible for ordinary Kähler action. For the canonical imbedding of M4 field equations are satisfied since energy momentum tensor vanishes identically. For the small deformations the presence of J(M4) would reduce rotational symmetry to cylindrical symmetry.

  2. J(M4) could make its presence manifest in the physics of right-handed neutrino having no direct couplings to electroweak gauge fields. Mixing with left handed neutrino is however induced by mixing of M4 and CP2 gamma matrices. The transversal localization of right-handed neutrino in a background, which is a small deformation of M4 could serve as an experimental signature.

  3. CP breaking in hadronic systems is one of the poorly understood aspects of fundamental physics and relates closely to the mysterious matter-antimatter asymmetry. The constant electric part of self dual J(M4) implies CP breaking. I have earlier considered the possibility that Kähler electric fields could cause this breaking but this breaking would be local. Second possibility is that matter and antimatter correspond to different values of heff and are dark relative to each other.

    Could J(M4) explain the observed CP breaking as appearing already at the level of imbedding space M4× CP2 and could this breaking explain hadronic CP breaking and matter anti-matter asymmetry? Could M4 part of Kähler electric field induce different heff/h=n for particles and antiparticles?

To answer these questions one can study Dirac equation at imbedding space level coupled to the gauge potential A(M4) for J(M4).
  1. The coupling of Kähler form to leptons is 3 times larger than to to quarks as in the case of A(CP2). This would give coupling k=1 for quarks an k=3 for leptons. k corresponds to fermion number which is opposite for fermions and antifermions having therefore opposite values of k at the respective space-time sheets.

  2. The potential satisfies ∂μAμ(M4)=0. Let the non-vanishing components of the Kähler gauge potential be (A0,Az)=ε (x,+/- y). The sign fact ε+/- 1 corresponds to self dual and antiself-dual options, let us assume self-duality as in the case of CP2 Kähler form. Scalar d'Alembertian reads as (∂μμ+ AμAμ)Ψ= -m2 Ψ.

  3. Assuming momentum eigenstate in time and z-direction (plane M2), one obtains by separation of variables (H1+H2)Ψ= (E-m2-kz2)Ψ. Hx= -∂x2+k2x2 and Hy= -∂y2+k2y2) are oscillator Hamiltonians. The spectrum is of Hx+Hy is given by kT2= (n1+n2+1)21/2|k| and one obtains E2=m2 +kz2 +kT2. This contribution is CP invariant and same for fermions and anti-fermions. The special feature is the presence of zero point transversal momentum. It is not possible to have a particle, which would be completely at rest. One can also say that m2 is increased 21/2|k| hbar2/L2, L= 1 m if standard convention for metric is used. For other conventions the numerical value of CP2 radius is scale by L/Lnew. L must correspond to some physical scale assignable to particle: secondary p-adic length scale is the natural identification.

  4. Spinor d'Alembertian contains also dipole moment term kX=JmuνΣμν giving a contribution, which depends on the sign of k: E2=m2 +kz2 +kT2+ kX. The term is sum of magnetic and electric dipole moment terms. The coupling k changes sign in CP operation and be of opposite sign for fermions and anti-fermions. One has a breaking of CP for given spin state. The dependence of X on spin state gives a test for the theory and also for the predicted CP breaking.

  5. Scaling covariance allows in principle all values L. To estimate the size of the effect one must fix the length scale L. CP2 size has only different value using L as unit and in flat background it does not matter. L should correspond to the size scale of the CD associated with particle. The secondary p-adic length scale of fermion defining also the size scale of its magnetic body is a natural guess so that Δ E2/E2≈ 2Δ E/E≈ Δ m/m ∼ 2/p1/2, p≈ 2k would hold true. This mass splitting is very small. For weak bosons having k=89 the mass splitting would be of order 3× 10-4 eV. For small values of p at ultrahigh energies the scale of CP breaking is larger, which conforms with the idea that matter-antimatter-asymmetry has emerged in very early cosmology.

    The recent experiment found that the mass difference Δ m/m for proton and antiproton satisfies Δ m <69× 10-12m ≈ 6.9× 10-2 eV (see this) so that this gives no constraints. Kaon-antikaon mass difference is estimated to be about 3.5× 10-6 eV (see this). This would correspond to a p-adic length scale k=96. Top quark is mainly responsible for the mixing of neutral kaon and its antiparticle in the model of based on loops involving decay to virtual quark pairs. The estimate from p-adic mass calculations for top quark mass scale is k=94 so that the order of magnitude estimate has correct of order of magnitude (being by factor 4 too large). This is an encouraging sign.

    How the mass splitting of neutral kaons would result? In quark model kaon and antikaon can be regarded as sdbbar and dsbar pairs. The net spins vanishes but the mass splitting due to electric moment dipole moment term X is non-vanishing due to the different sign of coupling k. The sign of the mass splitting is also opposite for kaon and antikaon.

  6. One can also consider the modified Dirac equation for canonically imbedded M4 which is simplest preferred extremal. The coupling to J(M4) to modified Dirac equation in space-time interior with gamma matrices replaced with modified gamma matrices are obtained as contractions of canonical momentum currents with M4 gamma matrices. Completely analogous phenomenon happens for CP2 type extremals. Tαβ=0 so that the modified gamma comes from Jαβ Jk~lβmlγk. These give just ordinary gamma matrices so that the two Dirac equations are identical.

2. About string like objects

String like objects and partonic 2-surfaces carry the information about quantum states and about space-time surfaces as preferred extremals if strong form of holography (SH) holds true. SH has of course some variants. The weakest variant states that fundamental information carrying objects are metrically 2-D. The light-like 3-surfaces separating space-time regions with Minkowskian and Euclidian signature of the induced metric are indeed metrically 2-D, and could thus carry information about quantum state.

An attractive possibility is that this information is basically topological. For instance, the value of Planck constant heff=n× h would tell the number sheets of the singular covering defining this surface such that the sheets co-incide at partonic 2-surfaces at the ends of space-time surface at boundaries of CD. In the following some questions related to string world sheets are considered. The information could be also number theoretical. Galois group for the algebraic extension of rationals defining particular adelic physics would transform to each other the number theoretic discretizations of light-like 3-surfaces and give rise to covering space structure. The action is trivial at partonic 2-surfaces should be trivial if one wants singular covering: this would mean that discretizations of partonic 2-surfaces consist of rational points. heff/h=n could in this case be a factor of the order of Galois group.

The original observation was that string world sheets should carry vanishing W boson fields in order that the em charge for the modes of the induced spinor field is well-defined. This condition can be satisfied in certain situations also for the entire space-time surface. This raises several questions. What is the fundamental condition forcing the restriction of the spinor modes to string world sheets - or more generally, to surface of given dimension? Is this restriction dynamical. Can one have an analog of brane hierarchy in which also higher-D objects can carry modes of induced spinor field Could the analogs of Lagrangian sub-manifolds of X4 ⊂ M4× CP2 satisfying J(M4)+J(CP2)=0 define string world sheets and their variants with varying dimension? The additional condition would be minimal surface property.

2.1 How does the gravitational coupling emerge?

The appearance of G=lP2 has coupling constant remained for a long time actually somewhat of a mystery in TGD. lP defines the radius of the twistor sphere of M4 replaced with its geometric twistor space M4× S2 in twistor lift. G makes itself visible via the coefficients ρvac= 8π Λ/G volume term but not directly and if preferred extremals are minimal surface extremals of Kähler action ρvac makes itself visible only via boundary conditions. How G appears as coupling constant?

Somehow the M4 Kähler form should appear in field equations. 1/G could naturally appear in the string tension for string world sheets as string models suggest. p-Adic mass calculations identify the analog of string tension as something of order of magnitude of 1/R2. This identification comes from the fact that the ground states of super-conformal representations correspond to imbedding space spinor modes, which are solutions of Dirac equation in M4× CP2. This argument is rather convincing and allows to expect that the p-adic mass scale is not determined by string tension and it can be chosen to be of order 1/G just as in string models.

2.2 Non-commutative imbedding space and strong form of holography

The precise formulation of strong form of holography (SH) is one of the technical problems in TGD. A comment in FB page of Gareth Lee Meredith led to the observation that besides the purely number theoretical formulation based on commutativity also a symplectic formulation in the spirit of non-commutativity of imbedding space coordinates can be considered. One can however use only the notion of Lagrangian manifold and avoids making coordinates operators leading to a loss of General Coordinate Invariance (GCI).

Quantum group theorists have studied the idea that space-time coordinates are non-commutative and tried to construct quantum field theories with non-commutative space-time coordinates (see this). My impression is that this approach has not been very successful. In Minkowski space one introduces antisymmetry tensor Jkl and uncertainty relation in linear M4 coordinates mk would look something like [mk, ml] = lP2Jkl, where lP is Planck length. This would be a direct generalization of non-commutativity for momenta and coordinates expressed in terms of symplectic form Jkl.

1+1-D case serves as a simple example. The non-commutativity of p and q forces to use either p or q. Non-commutativity condition reads as [p,q]= hbar Jpq and is quantum counterpart for classical Poisson bracket. Non-commutativity forces the restriction of the wave function to be a function of p or of q but not both. More geometrically: one selects Lagrangian sub-manifold to which the projection of Jpq vanishes: coordinates become commutative in this sub-manifold. This condition can be formulated purely classically: wave function is defined in Lagrangian sub-manifolds to which the projection of J vanishes. Lagrangian manifolds are however not unique and this leads to problems in this kind of quantization. In TGD framework the notion of "World of Classical Worlds" (WCW) allows to circumvent this kind of problems and one can say that quantum theory is purely classical field theory for WCW spinor fields. "Quantization without quantization would have Wheeler stated it.

GCI poses however a problem if one wants to generalize quantum group approach from M4 to general space-time: linear M4 coordinates assignable to Lie-algebra of translations as isometries do not generalize. In TGD space-time is surface in imbedding space H=M4× CP2: this changes the situation since one can use 4 imbedding space coordinates (preferred by isometries of H) also as space-time coordinates. The analog of symplectic structure J for M4 makes sense and number theoretic vision involving octonions and quaternions leads to its introduction. Note that CP2 has naturally symplectic form.

Could it be that the coordinates for space-time surface are in some sense analogous to symplectic coordinates (p1,p2,q1,q2) so that one must use either (p1,p2) or (q1,q2) providing coordinates for a Lagrangian sub-manifold. This would mean selecting a Lagrangian sub-manifold of space-time surface? Could one require that the sum Jμν(M4)+ Jμν(CP2) for the projections of symplectic forms vanishes and forces in the generic case localization to string world sheets and partonic 2-surfaces. In special case also higher-D surfaces - even 4-D surfaces as products of Lagrangian 2-manifolds for M4 and CP2 are possible: they would correspond to homologically trivial cosmic strings X2× Y2⊂ M4× CP2, which are not anymore vacuum extremals but minimal surfaces if the action contains besides Käction also volume term.

But why this kind of restriction? In TGD one has strong form of holography (SH): 2-D string world sheets and partonic 2-surfaces code for data determining classical and quantum evolution. Could this projection of M4 × CP2 symplectic structure to space-time surface allow an elegant mathematical realization of SH and bring in the Planck length lP defining the radius of twistor sphere associated with the twistor space of M4 in twistor lift of TGD? Note that this can be done without introducing imbedding space coordinates as operators so that one avoids the problems with general coordinate invariance. Note also that the non-uniqueness would not be a problem as in quantization since it would correspond to the dynamics of 2-D surfaces.

The analog of brane hierarchy for the localization of spinors - space-time surfaces; string world sheets and partonic 2-surfaces; boundaries of string world sheets - is suggesetive. Could this hierarchy correspond to a hierarchy of Lagrangian sub-manifolds of space-time in the sense that J(M4)+J(CP2)=0 is true at them? Boundaries of string world sheets would be trivially Lagrangian manifolds. String world sheets allowing spinor modes should have J(M4)+J(CP2)=0 at them. The vanishing of induced W boson fields is needed to guarantee well-defined em charge at string world sheets and that also this condition allow also 4-D solutions besides 2-D generic solutions. This condition is physically obvious but mathematically not well-understood: could the condition J(M4)+J(CP2)=0 force the vanishing of induced W boson fields? Lagrangian cosmic string type minimal surfaces X2× Y2 would allow 4-D spinor modes. If the light-like 3-surface defining boundary between Minkowskian and Euclidian space-time regions is Lagrangian surface, the total induced Kähler form Chern-Simons term would vanish. The 4-D canonical momentum currents would however have non-vanishing normal component at these surfaces. I have considered the possibility that TGD counterparts of space-time super-symmetries could be interpreted as addition of higher-D right-handed neutrino modes to the 1-fermion states assigned with the boundaries of string world sheets.

It is relatively easy to construct an infinite family of Lagrangian string world sheets satisfying J(M4) +J(CP2)=0 using generalized symplectic transformations of M4 and CP2 as Hamiltonian flows to generate new ones from a given Lagrangian string world sheets. One must pose minimal surface property as a separate condition. Consider a piece of M2 with coordinates (t,z) and homologically non-trivial geodesic sphere S2 of CP2 with coordinates (u= cos(Θ),Φ). One has J(M4)tz=1 and J= 1. Identify string world sheet via map (u,Φ)= (kz,ω t) from M2 to S2. The induced CP2 Kahler form is J(CP2)tz= kω. kω=-1 guarantees J(M4) +J(CP2)=0. The strings have necessarily finite length from L=1/k≤ z≤ L. One can perform symplectic transformations of CP2 and symplectic transformations of M4 to obtain new string world sheets. In general these are not minimal surfaces and this condition would select some preferred string world sheets.

An alternative - but of course not necessarily equivalent - attempt to formulate this picture would be in terms of number theoretic vision. Space-time surfaces would be associative or co-associative depending on whether tangent space or normal space in imbedding space is associative - that is quaternionic. These two conditions would reduce space-time dynamics to associativity and commutativity conditions. String world sheets and partonic 2-surfaces would correspond to maximal commutative or co-commutative sub-manifolds of imbedding space. Commutativity (co-commutativity) would mean that tangent space (normal space as a sub-manifold of space-time surface) has complex tangent space at each point and that these tangent spaces integrate to 2-surface. SH would mean that data at these 2-surfaces would be enough to construct quantum states. String world sheet boundaries would in turn correspond to real curves of the complex 2-surfaces intersecting partonic 2-surfaces at points so that the hierarchy of classical number fields would have nice realization at the level of the classical dynamics of quantum TGD.

To sum up, one cannot exclude the possibility that J(M4) is present implying a universal transversal localization of imbedding space spinor harmonics and the modes of spinor fields in the interior of X4: this could perhaps relate to somewhat mysterious de-coherence interaction producing locality and to CP breaking and matter-antimatter asymmetry. The moduli space for M4 Kähler structures proposed by number theoretic considerations would save from the loss of Poincare invariance and the number theoretic vision based on quaternionic and octonionic structure would have rather concrete realization. This moduli space would only extend the notion of "world of classical worlds" (WCW).

For background see the article Some questions related to the twistor lift of TGD or the chapter How the hierarchy of Planck constants might relate to the almost vacuum degeneracy for twistor lift of TGD? of "Towards M-matrix".

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

Articles and other material related to TGD.

Thursday, December 29, 2016

Non-commutative space and strong form of holography

The precise formulation of strong form of holography (SH) is one of the technical problems in TGD. A comment in FB page of Gareth Lee Meredith led to the observation that besides the purely number theoretical formulation based on commutativity also a symplectic formulation in the spirit of non-commutativity of imbedding space coordinates can be considered. One can however use only the notion of Lagrangian manifold and avoids making coordinates operators leading to a loss of General Coordinate Invariance (GCI).

Quantum group theorists have studied the idea that space-time coordinates are non-commutative and tried to construct quantum field theories with non-commutative space-time coordinates (see this). My impression is that this approach has not been very successful. In Minkowski space one introduces antisymmetry tensor Jkl and uncertainty relation in linear M4 coordinates mk would look something like [mk, ml] = lP2Jkl, where lP is Planck length. This would be a direct generalization of non-commutativity for momenta and coordinates expressed in terms of symplectic form Jkl.

1+1-D case serves as a simple example. The non-commutativity of p and q forces to use either p or q. Non-commutativity condition reads as [p,q]= hbar Jpq and is quantum counterpart for classical Poisson bracket. Non-commutativity forces the restriction of the wave function to be a function of p or of q but not both. More geometrically: one selects Lagrangian sub-manifold to which the projection of Jpq vanishes: coordinates become commutative in this sub-manifold. This condition can be formulated purely classically: wave function is defined in Lagrangian sub-manifolds to which the projection of J vanishes. Lagrangian manifolds are however not unique and this leads to problems in this kind of quantization. In TGD framework the notion of "World of Classical Worlds" (WCW) allows to circumvent this kind of problems and one can say that quantum theory is purely classical field theory for WCW spinor fields. "Quantization without quantization would have Wheeler stated it.

GCI poses however a problem if one wants to generalize quantum group approach from M4 to general space-time: linear M4 coordinates assignable to Lie-algebra of translations as isometries do not generalize. In TGD space-time is surface in imbedding space H=M4× CP2: this changes the situation since one can use 4 imbedding space coordinates (preferred by isometries of H) also as space-time coordinates. The analog of symplectic structure J for M4 makes sense and number theoretic vision involving octonions and quaternions leads to its introduction. Note that CP2 has naturally symplectic form.

Could it be that the coordinates for space-time surface are in some sense analogous to symplectic coordinates (p1,p2,q1,q2) so that one must use either (p1,p2) or (q1,q2) providing coordinates for a Lagrangian sub-manifold. This would mean selecting a Lagrangian sub-manifold of space-time surface? Could one require that the sum Jμν(M4)+ Jμν(CP2) for the projections of symplectic forms vanishes and forces in the generic case localization to string world sheets and partonic 2-surfaces. In special case also higher-D surfaces - even 4-D surfaces as products of Lagrangian 2-manifolds for M4 and CP2 are possible: they would correspond to homologically trivial cosmic strings X2× Y2⊂ M4× CP2, which are not anymore vacuum extremals but minimal surfaces if the action contains besides Käction also volume term.

But why this kind of restriction? In TGD one has strong form of holography (SH): 2-D string world sheets and partonic 2-surfaces code for data determining classical and quantum evolution. Could this projection of M4 × CP2 symplectic structure to space-time surface allow an elegant mathematical realization of SH and bring in the Planck length lP defining the radius of twistor sphere associated with the twistor space of M4 in twistor lift of TGD? Note that this can be done without introducing imbedding space coordinates as operators so that one avoids the problems with general coordinate invariance. Note also that the non-uniqueness would not be a problem as in quantization since it would correspond to the dynamics of 2-D surfaces.

The analog of brane hierarchy for the localization of spinors - space-time surfaces; string world sheets and partonic 2-surfaces; boundaries of string world sheets - is suggesetive. Could this hierarchy correspond to a hierarchy of Lagrangian sub-manifolds of space-time in the sense that J(M4)+J(CP2)=0 is true at them? Boundaries of string world sheets would be trivially Lagrangian manifolds. String world sheets allowing spinor modes should have J(M4)+J(CP2)=0 at them. The vanishing of induced W boson fields is needed to guarantee well-defined em charge at string world sheets and that also this condition allow also 4-D solutions besides 2-D generic solutions. This condition is physically obvious but mathematically not well-understood: could the condition J(M4)+J(CP2)=0 force the vanishing of induced W boson fields? Lagrangian cosmic string type minimal surfaces X2× Y2 would allow 4-D spinor modes. If the light-like 3-surface defining boundary between Minkowskian and Euclidian space-time regions is Lagrangian surface, the total induced Kähler form Chern-Simons term would vanish. The 4-D canonical momentum currents would however have non-vanishing normal component at these surfaces. I have considered the possibility that TGD counterparts of space-time super-symmetries could be interpreted as addition of higher-D right-handed neutrino modes to the 1-fermion states assigned with the boundaries of string world sheets.

An alternative - but of course not necessarily equivalent - attempt to formulate this picture would be in terms of number theoretic vision. Space-time surfaces would be associative or co-associative depending on whether tangent space or normal space in imbedding space is associative - that is quaternionic. These two conditions would reduce space-time dynamics to associativity and commutativity conditions. String world sheets and partonic 2-surfaces would correspond to maximal commutative or co-commutative sub-manifolds of imbedding space. Commutativity (co-commutativity) would mean that tangent space (normal space as a sub-manifold of space-time surface) has complex tangent space at each point and that these tangent spaces integrate to 2-surface. SH would mean that data at these 2-surfaces would be enough to construct quantum states. String world sheet boundaries would in turn correspond to real curves of the complex 2-surfaces intersecting partonic 2-surfaces at points so that the hierarchy of classical number fields would have nice realization at the level of the classical dynamics of quantum TGD.

For background see the chapter How the hierarchy of Planck constants might relate to the almost vacuum degeneracy for twistor lift of TGD?.

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

Articles and other material related to TGD.