Three good reasons for the localization of fundamental fermions to string world sheets

There are now three good reasons for the modes of the induced spinor fields to be localized to 2-D string world sheets and partonic 2-surfaces - inf fact, to the boundaries of string world sheets at them defining fermionic world lines. I list these three good reasons in the same order as I became aware of them.

1. The first good reason is that this condition allows spinor modes to have well-defined electromagnetic charges - the induced classical W boson fields and perhaps also Z field vanish at string world sheets so that only em field and possibly Z field remain and one can have eigenstates of em charge.
   
   
2. Second good reason actually a set of closely related good reasons. First, strong form of holography implied by the strong form of general coordinate invariance demands the ocalization: string world sheets and partonic 2-surfaces are "space-time genes". Also twistorial picture follows naturally if the locus for the restriction of spinor modes at the light-like orbits of partonic 2-surfaces at which the signature of the induced metric changes from Minkowskian to Euclidian is 1-D fermion world line. Thanks to holography fermions behave like point like particles, which are massless in 8-D sense. Thirdly, conformal invariance in the fermionic sector demands the localization.
   
   
3. The third good reason emerges from the mathematical problem of field theories involving fermions: also in the models of condensed matter systems this problem is also encountered - in particular, in the models of high T_c superconductivity. For instance, AdS/CFT correspondence involving 10-D blackholes has been proposed as a solution - the reader can decide whether to take this seriously.
   

   Fermionic path integral is the source of problems. It can be formally reduced to the analog of partition function but the Boltzman weights (analogous to probabilities) are not necessary positive in the general case and this spoils the stability of the numerical computation. One gets rid of the sign problem if one can diagonalize the Hamiltonian, but this problem is believed to be NP-hard in the generic case. A further reason to worry in QFT context is that one must perform Wick rotation to transform action to Hamiltonian and this is a trick. It seems that the problem is much more than a numerical problem: QFT approach is somehow sick.
   

   The crucial observation giving the third good reason is that this problem is encountered only in dimensions D≥3 - not in dimensions D=1,2! No sign problem in TGD where second quantized fundamental fermions are at string world sheets!
   
   

A couple of comments are in order.

1. Although the assumption about localization 2-D surfaces might have looked first a desperate attempt to save em charge, it now seems that it is something very profound. In TGD approach standard model and GRT emerge as an approximate description obtained by lumping the sheets of the many-sheeted space-time together to form a slightly curved region of Minkowski space and by identifying gauge potentials and gravitational field identified as sums of those associated with the sheets lumped together. The more fundamental description would not be plagued by the mathematical problem of QFT approach .
   
   
2. Although fundamental fermions as second quantized induced spinor fields are 2-D character, it is the modes of the classical imbedding space spinor fields - eigenstates of four-momentum and standard model quantum numbers - that define the ground states of the super-conformal representations. It is these modes that correspond to the 4-D spinor modes of QFT limit. What goes wrong in QFT is that one assigns fermionic oscillator operators to these modes although second quantization should be carried out at deeper level and for the 2-D modes of the induced spinor fields: 2-D conformal symmetry actually makes the construction of these modes trivial.
   
   

To conclude, the condition that the theory is computable would pose a powerful condition on the theory. As a matter fact, this is not a new finding. The mathematical existence of Kähler geometry of "world of classical worlds" fixes its geometry more or less uniquely and therefore also the physics: one obtains a union of symmetric spaces labelled by zero modes of the metric and for symmetric space all points (now 3-surfaces) are geometrically equivalent meaning a gigantic simplification allowing to handle the infinite-dimensional case. Even for loop spaces the Kähler geometry is unique and has infinite-dimensional isometry group (Kac-Moody symmetries).

For a summary of earlier postings see Links to the latest progress in TGD.

Could TGD provide a unified description of high Tc superconductivity and other exotic conductivities?

During years I have been developing a model for high T_c superconductivity ). The recent view is already rather detailed but the fact that I am not a condensed matter physicist implies that professional might regard the model as rather lopsided. Quite recently I read several popular articles related to superconductivity and various types of other exotic conductivities: one can say that condensed matter physics has experienced an inflation of poorly understood conductivities. This of course is an fascinating challenge for TGD. In fact, super string theorist Subir Sachdev has taken the same challenge.

In particular, the article about superconductivity provides a rather general sketch about the phase diagram for a typical high T_c super conductor and discusses experimental support for the idea quantum criticality in standard sense and thus defined only at zero temperature could be crucial for the understanding of high T_c super conductivity.

The cuprates doped with holes by adding atoms binding some fraction of conduction electrons are very rich structured. The transition from antiferromagnetic insulator to ordinary metal involves several steps described by a 2-D phase diagram in the plane defined by temperature and doping fraction. Besides high T_c super conducting region the phases include pseudogap region, a region allowing charge oscillations, strange metal region, and metal region.

The attempt to find a unifed TGD based view about these conductivities led to an article I consider the general vision based on magnetic flux tubes carrying the dark h_eff/h=n variants of electrons as Cooper pairs or as free electrons allowing to understand not only high T_c super-conductivity and various accompanying phases but also exotic variants of conductivity associated with strange and bad metals, charge density waves and spin density waves. One should also understand the anomalous conductivity of SmB₆, and the fact that electron currents in graphene behave more like viscous liquid current than ohmic current (see this).

The TGD inspired model for the anomalous conductivity of SmB₆ as flux tube conductivity developed during last year forms an essential element of the mode. This model implies that Fermi energy controlled by the doping fraction would serve as a control variable whose value determines whether electrons can be transferred to magnetic flux tubes to form cyclotron orbits at the surface of the tube.

Also the metals (such as graphene) for which current behaves more like a viscous flow rather than Ohmic current can be understood in this framework: the liquid flow character comes from magnetic field which is mathematically like incompressible liquid flow. The electric field at flux tube space-time sheets is parallel to the flux tube and can reverse its direction since flux tubes resemble liquid flow and only the average of the flux tube electric field is is parallel to the applied electric field.

The outcome is a unified view about various conductivities relying on universal concepts and mechanisms. For a detailed representation see the article About high T_c superconductivity and other exotic conductivities.

For a summary of earlier postings see Links to the latest progress in TGD.

Does M4 Kähler form imply new physics?

The introduction of M⁴ Kähler form strongly suggested by the twistor formulation of TGD could bring in new gravitational physics.

1. As found, the twistorial formulation of TGD assigns to M⁴ a self dual Kähler form whose square gives Minkowski metric. It can (but need not if M⁴ twistor space is trivial as bundle) contribute to the 6-D twistor counterpart of Kähler action inducing M⁴ term to 4-D Kähler action vanishing for canonically imbedded M⁴.
   
   
2. Self-dual Kähler form in empty Minkowski space satisfies automatically Maxwell equations and has by Minkowskian signature and self-duality a vanishing action density. Energy momentum tensor is proportional to the metric so that Einstein Maxwell equations are satisfied for a non-vanishing cosmological constant! M⁴ indeed allows a large number of self dual Kähler fields (I have christened them as Hamilton-Jacobi structures). These are probably the simplest solutions of Einstein-Maxwell equations that one can imagine!
   
   
3. There however exist quite a many Hamilton-Jacobi structures. However, if this structure is to be assigned with a causal diamond (CD) it must satisfy additional conditions, say SO(3) symmetry and invariance under time translations assignable to CD. Alternatively, covariant constancy and SO(2)⊂ SO(3) symmetry might be required.
   
   
   1. In the case of causal diamond (CD) a spherically symmetric self-dual monopole Kähler form with non-vanishing components J^tr= ε^trθφJ_θφ, J_θφ=cos(θ) carrying radial electric and magnetic fields with identical gravitational charges looks rather natural option. The time-like line connecting the tips of CD would carry a genuine self-dual monopole so that Dirac monopole would not be in question. The potential associated with J could be chosen to be A_μ ↔ (1/r,0,0,sin(θ). I have considered this kind of possibility earlier in context of TGD
      inspired model of anyons but gave up the idea.
      

      The moduli space for CDs with second tip fixed would be hyperbolic space H³=SO(3,1)/SO(3) or a space obtained by identifying points at the orbits of some discrete subgroup of SO(3,1) as suggested by number theoretic considerations. This induced Kähler field could make the blackholes with center at this line to behave like M⁴ magnetic monopoles if the M⁴ part of Kähler form is induced into the 6-D lift of Kähler action with extremely small coefficients of order of magnitude of cosmological constant. Cosmological constant and the possibility of CD monopoles would thus relate to each other.
      
      

   2. Covariant constancy is an alternative option. This would leave only the fields J_tz =J_xy=1 unique apart from Lorentz transformation: it would be attractive to assign this Kähler with given CD to define a preferred plane M² required also by the number theoretic vision. Now however rotational invariance is broken to SO(1,1)× SO(2). SO(3,1)/SO(1,1)× SO(2) would define moduli for CDs. Magnetic and electric parts of Kähler form would be in z-direction and flux tubes would tend to be in this direction. One would have clearly a preferred direction and it is difficult to imagine how the gravitational field of blackhole could correlate with these fluxes unless one assigns to each flux tube its own CD.
      
      
4. A further interpretational problem is that the classical coupling of M⁴ Kähler gauge potential to induced spinors is not small. Can one really tolerate this kind of coupling equivalent to a coupling to a self dual monopole field carrying electric and magnetic charges? One could of course consider the condition that the string world sheets carrying spinor modes are such that the induced M⁴ Kähler form vanishes and gauge potential become pure gauge. M⁴ projection would be 2-D Lagrange manifold whereas CP₂ projection would carry vanishing induce W and possibly also Z⁰ field in order that em charge is well defined for the modes. These conditions would fix the string world sheets to a very high degree in terms of maps between this kind of 2-D sub-manifolds of M⁴ and CP₂. Spinor dynamics would be determined by the avoidance of interaction!
   

   It must be emphasized that the imbedding space spinor modes characterizing the ground states of super-symplectic representations would not couple to the monopole field so that at this level Poincare invariance is not broken. The coupling would be only at the space-time level and force spinor modes to Lagrangian sub-manifolds.
   
   

5. At the static limit of GRT and for g_ij≈ δ_ij implying SO(3) symmetry there is very close analogy with Maxwell's equations and one can speak of gravi-electricity and gravi-magnetism with 4-D vector potential given by the components of g_tα. The genuine U(1) gauge potential does not however relate to the gravimagnetism in GRT sense. Situation would be analogous to that for CP₂, where one must add to the spinor connection U(1) term to obtain respectable spinor structure. Now the U(1) term would be added to trivial spinor connection of flat M⁴: its presence would be justified by twistor space Kähler structure. If the induced M⁴ Kähler form is present as a classical physical field it means genuinely new contribution to gravitational interaction and assignable to cosmological constant.
   

   I have talked much about gravitational flux or gravitons are carried along Kähler magnetic monopole flux tubes. This is quite respectable hypothesis. One can however ask whether the gravitational interaction could be mediated along flux tubes of M⁴ Kähler magnetic field carrying monopole flux. For the proposed SO(3) symmetric option the flux tubes would emanate radially from the origin and one could assign to each gravitating object CD. It is of course quite possible that Kähler magnetic flux tubes and gravitational flux tubes are one and same thing in astrophysical systems. Note however that Kähler magnetic monopole fluxes do not involve genuine monopole like M⁴ Kähler fluxes in SO(3) symmetric case.
   
   

For background see the chapter From Principles to giagrams of "Towards M-matrix" or the article From Principles to Diagrams.

For a summary of earlier postings see Links to the latest progress in TGD.

Reactor antineutrino anomaly as indication for new nuclear physics predicted by TGD

A highly interesting new neutrino anomaly has emerged recently. The anomaly appears in two experiments and is referred to as reactor antineutrino anomaly. There is a popular article in Symmetry Magazine about the discovery of the anomalyf in Daya Bay experiment. Bee mentioned in Backreaction blog Reno experiment exhibiting the same anomaly. What happens that more antineutrinos with energies around 5 MeV are produced as should: the anomaly seems to extened to antineutrino energy about 6.3 MeV.

What makes me happy is that this anomaly might provide a new evidence for TGD based model of atomic nuclei.

1. In nuclear string model nucleons are assumed to be bonded to nuclear strings by color magnetic flux tubes with quarks at ends. These nuclear quarks are different from hadronic quarks and can have different p-adic mass scales. Nuclear d quark is expected to be heavier than nuclear d quark and can decay to nuclear u quark by emission of a virtual W boson decaying to electron antineutrino pair. These decays are anomalous from the point of view of standard nuclear physics.
   
   
2. The virtual W boson decaying to electron antineutrino pair in the anomalous region around 5 MeV should have energy which is two times neutrino energy since electron is relativistic. Since the upper boundary of anomalous region corresponds to about 6.3 MeV antineutrino energy, W energy should be below d-u mass difference, which must be therefore around 12.6 MeV . This is a highly valuable bit of information.
   
   

To proceed one can use p-adic mass calculations.

1. The topological mixing of quark generations (characterized by handle number for partonic two surfaces) must make u and d quark masses almost but quite not identical in the lowest p-adic order. In the model for CKM mixing of hadronic quarks they would be identical in this order.
   
   
2. p-Adic mass squared can be expressed as m²(q)/m_e²= 2^(k-127)/2(s(q)+X(q))/(s(e)+X(e)), where s is positive integer and and X <1 a parameter characterizing the poorly known second order contribution in p-adic mass calculations. For topologically unmixed u and d quarks one has s(d)=8 and s(u)=5 =s(e). p=≈ 2^k characterizes the p-adic scale of quark (for p-adic mass calculations see this).
   
   

Assume first that there is no breaking of isospin symmetry so that the p-adic mass scales of u and d type nuclear quarks are same.

1. By using the information about the mass difference m(d)-m(u) < about 12.3 MeV and the above p-adic mass squared formula one can estimate the common p-adic mass scale of the nuclear quarks to be k=113. This is nothing but the p-adic mass scale assigned with nuclei and corresponds to Gaussian Mersenne M_G,113=(1+i)¹¹³-1. Very natural!
   
   
2. The maximal value 6.3 MeV for mass difference would be obtained for s(d)=8 and s(u)= 7 and X(e)=X(u)=X(d)=0 one obtains mass m(d)-m(u)= 5.49 MeV. Interestingly, figure 2 of the Reno article) shows a sharp downwards shoulder at 5.5 MeV.
   

   m(d)-m(u) =6.3 MeV can be reproduced accurately for X(d)/8^1/2- X(u)/7^1/2≈ .01. There are several manners to reproduce the estimate for d-u mass difference by varying second order contributions. Mixing with higher quark generations would occur for both u quark. The mass of nuclear u (d) quark would be (s(q)/5)^1/2× 64 MeV, s(u)=2 (s(d)=8) for m(d)-m(u)=5.5 MeV. This mass is assumed to include the color magnetic energy of the color magnetic body of quark and would correspond to constituent quark mass rather than current quark mass, which is rather small.
   

   What is interesting that the sum of the u and d quark masses m(d)+m(u)= 144.95 MeV in absence of topological mixing is about 4 per cent larger than the charged pion mass m(π⁺)= 139.57 MeV. In any case, it is difficult to see how this large additional mass could be compensated.
   
   

In an alternative scenario, which is in accordance with the original picture, the isospin symmetry would be broken in the sense that p-adic mass scales of u and d would be different so that the mass difference would corresponds to the mass scale of (say) d quark and could be much smaller.

1. For k(d)=119, s(d)= 10 (small topological mixing) and s(u)=5 (no topological mixing), k(u)=127 (say) one would have m(d)-m(u)=10.8 MeV so that neutrino energy would be below 5.4 MeV, which is near to the steep shoulder of the figure. One would have m(d)=11.3 MeV and m(u)=.5 MeV (electron mass) in absence of topological mixing. Now k(d) is however not prime as the strongest form of p-adic length scale hypothesis demands. k(u)=127 is only the first guess. Also k(u)=137 corresponding to atomic length scale can be considered.
   
   
2. The accepted values for hadronic current quark masses deduced from lattice calculations are about m(u)=2 MeV for m(d)=5 MeV and smaller than the values deduced above suggesting the interpretation of the masses estimate above as nuclear constituent quark masses.
   
   
3. Beta stable configurations would correspond to u-ubar bonds with total energy about 2m(e)=1 MeV, which is consistent with the general view about nuclear binding energy scale. Also exotic nuclear excitations containing charged color bonds with quark or antiquark or both transformed to d type state are predicted. The first guess for the excitation energy of charged color bond is m(d)-m(u) ≈ 10.8 MeV. Each charged color bond increases the nuclear charge by one unit but proton and neutron numbers remain the same as for the original nucleus: I have called these states exotic nuclei (see this).
   
   
4. The so called leptohadron hypothesis postulates color excitations of leptons having as bound states leptopions with mass equal to 2m(e) in good approximation. An alternative option would replace colored leptons with quarks and assumes that unmixed u quark has electron mass and their production in heavy ion collisions would be natural if they appear as color bonds between nucleons. This would fix s(u) to s(u)=5 (no topological mixing).
   
   
5. X rays from Sun have anomalous effects on the observed nuclear decay rate with a periodicity of year and with magnitude varying like inverse of the distance from the Sun with which also solar X ray intensity varies: this is known as GSI anomaly. I have proposed earlier that the energy scale of the excitations of nuclear color bonds is 1-10 keV on basis of these findings (see this). Nuclei could be in exited states with excitation energies in 1-10 keV range and the X ray radiation would affect the fraction of excited states thus changing also the average decay rates.
   

   One can try to understand the keV energy scale to the 1 MeV energy scale of beta stable color bonds in terms of fractal scaling. Above it was found that for k=113 charged color bond would have energy m(d)+m(u)= 144.95 MeV if quarks are free. Since the actual charged pion mass is m(π⁺)= 139.57 MeV, the pionic binding energy would be 5.38 MeV which makes about 3.7 per cent of the total mass. If one applies same fractal logic to the k=127 color bond with 2m(u)= 1 MeV, one obtains 37 keV, which has somewhat too high value. The Coulombic interaction is attractive between u and ubar in k=127 pion with broken isospin symmetry. The naive perturbative estimate is as α/m_e≈ 3.6 keV reducing the estimate to 34.4 keV. The fact that π⁺ has positive Coulombic interaction energy reduces the estimate further but this need not be enough.
   

   For k(u)=137 (atomic length scale) one would obtain binding energy scale, which is by factor 1/32 lower and about 1.2 keV. The simplest model for color bond would be as harmonic oscillator predicting multiples of 1.2 keV as excitation energies. This would conform with the earlier suggestion that color magnetic flux tubes are loops with size of even atom. This cold also explain the finding that the charge radius of proton is not quite what it is expected to be.
   
   

For the background see the chapter Nuclear string model of "p-Adic length scale hypothesis and dark matter hierarchy".

For a summary of earlier postings see Links to the latest progress in TGD.

What could the gamma ray pulse detected .4 seconds after the LIGO merger mean?

This posting begins with a part from earlier posting to which I added more material, which turns out to relate also to the no-hair theorem and to Hawking's recent work (discussed from TGD perspective here) in an interesting manner. It might be that the blackhole formed in the merger breaks non-hair theorem by having magnetic moment.

The Fermi Gamma-ray Burst Monitor detected 0.4 seconds after the merger a pulse of gamma rays with red shifted energies about 50 keV (see the posting of Lubos and the article from Fermi Gamma Ray Burst Monitor). At the peak of gravitational pulse the gamma ray power would have been about one millionth of the gravitational radiation. Since the gamma ray bursts do not occur too often, it is rather plausible that the pulse comes from the same source as the gravitational radiation. The simplest model for blackholes does not suggest this but it is not difficult to develop more complex models involving magnetic fields.

Could this observation be seen as evidence for the assumption that dark gravitons are associated with magnetic flux tubes?

1. The radiation would be dark cyclotron gravitation generated at the magnetic flux tubes carrying the dark gravitational radiation at cyclotron frequency f_c= qB/m and its harmonics (q denotes the charge of charge carrier and B the intensity of the magnetic field and its harmonics and with energy E=h_effeB/m .
   
   
2. If h_eff= h_gr= GMm/v₀ holds true, one has E= GMB/v₀ so that all particles with same charge respond at the same the same frequency irrespective of their mass: this could be seen as a magnetic analog of Equivalence Principle. The energy 50 keV corresponds to frequency f∼ 5× 10¹⁸ Hz. For scaling purposes it is good to remember that the cyclotron frequency of electron in magnetic field B_end=.2 Gauss (value of endogenous dark magnetic field in TGD inspired quantum biology) is f_c=.6 Mhz.
   

   From this the magnetic field needed to give 50 keV energy as cyclotron energy would be B_ord= (f/f_c)B_end=.4 GT corresponds to electrons with ordinary value of Planck constant the strength of magnetic field. If one takes the redshift of order v/c∼ .1 for cosmic recession velocity at distance of Gly one would obtain magnetic field of order 4 GT. Magnetic fields of with strength of this order of magnitude have been assigned with neutron stars.
   
   

3. On the other hand, if this energy corresponds to h_gr= GMm_e c/v₀ one has B= (h/h_gr)B_ord = (v₀m_P²/Mm_e)× B_ord∼ (v₀/c)× 10^-11 T (c=1). This magnetic field is rather weak (fT is the bound for detectability) and can correspond only to a magnetic field at flux tube near Earth. Interstellar magnetic fields between arms of Milky way are of the order of 5× 10^-10 T and are presumably weaker in the intergalactic space.
   
   
4. Note that the energy of gamma rays is by order or magnitude or two lower than that for dark gravitons. This suggests that the annihilation of dark gamma rays could not have produced dark gravitons by gravitational coupling bilinear in collinear photons.
   

One can of course forget the chains of mundane realism and ask whether the cyclotron radiation coming from distant sources has its high energy due to large value of h_gr rather than due to the large value of magnetic field at source. The presence of magnetic fields would reflects itself also via classical dynamics (that is frequency). In the recent case the cyclotron period would be of order (.03/v₀) Gy, which is of the same order of magnitude as the time scale defined by the distance to the merger.

In the case of Sun the prediction for energy of cyclotron photons would be E=[v₀(Sun)/v₀] × [M(Sun)/M(BH)] × 50 keV ∼ [v₀(Sun)/v₀] keV. From v₀(Sun)/c≈ 2^-11 one obtains E=(c/v₀)× .5 eV> .5 eV. Dark photons in living matter are proposed to correspond to h_gr=h_eff and are proposed to transform to bio-photons with energies in visible and UV range (see this).

Good dialectic would ask next whether both views about the gamma rays are actually correct. The "visible" cyclotron radiation with standard value of Planck constant at gamma ray energies would be created in the ultra strong magnetic field of blackhole, would be transformed to dark gamma rays with the same energy, and travel to Earth along the flux tubes. In TGD Universe the transformation ordinary photons to dark photons would occur in living matter routinely. One can of course ask whether this transformation takes place only at quantum criticality and whether the quantum critical period corresponds to the merger of blackholes.

The time lag was .4 second and the merger event lasted .2 seconds. Many-sheeted space-time provides one possible explanation. If the gamma rays were ordinary photons so that dark gravitons would have travelled along different flux tubes, one can ask whether the propagation velocities differed by Δ c/c∼ 10^-17. In the case of SN1987A neutrino and gamma ray pulses arrived at different times and neutrinos arrived as two different pulses (see this so that this kind of effect is not excluded. Since the light-like geodesics of the space-time surface are in general not light-like geodesics of the imbedding space signals moving with light velocity along space-time sheet do not move with maximal signal velocity in imbedding space and the time taken to travel from A to B depends on space-time sheet. Could the later arrival time reflect slightly different signal velocities for photons and gravitons?

Could one imagine a function for the gamma ray pulse possibly explaining also why it came considerably later than gravitons (0.4 seconds after the merger which lasted 2. seconds)? This function might relate to the transfer of surplus angular momentum from the system.

1. The merging blackholes were reported to have opposite spins. Opposite directions of spins would make the merger easier since local velocities at the point of contact are in same direction. The opposite directions spins suggest an analogy with two vortices generated from water and this suggests that their predecessors were born inside same star. There is also relative orbital angular momentum forming part of the spin of the final state blackhole, which was modelled as a Kerr blackhole. Since the spins of blackholes were opposite, the main challenge is to understand the transition to the situation in all matter has same direction of spin. The local spin directions must have changed by some mechanism taking away spin.
   
   
2. Magnetic analogs blackholes seem to be needed. They would be analogs of magnetars, which are pulsars with very strong magnetic fields. Magnetic fields are needed to carry out angular momentum from the matter as blackhole is formed. Same should apply now. Outgoing matter spirals along the helical jets (and carries away the spin which is liberated as the rotating matter in two spinning blackholes slows down to rest and the orbital angular momentum becomes the total spin.
   
   
3. If cyclotron adiation left .4 later, it would be naturally assignable to the liberation of temporarily stored surplus angular momentum which blackhole could not carry stably. This cyclotron radiation could have carried out the surplus angular momentum. Amusingly, it could be also seen as a dark analog of Hawking radiation.
   
   

Here one must be ready to update the beliefs about what black hole like objects are. About their interiors empirical data tell of course nothing.

1. The exteriors could contain magnetic fields and must do so in TGD Universe. No exact rotating and magnetic blackhole solutions of Einstein-Maxwell theory are known - otherwise we would not have "blackhole has no hair theorem" stating that blackhole is completely characterized by conserved charges associated with long range interactions: mass, angular momentum and electric charge. In this framework one cannot speak about the magnetic dipole moment of blackhole.
   
   
2. No hair theorem has been challenged quite recently by Hawking (for TGD inspired commentary see this). This suggest the possibility that higher multiple moments characterize blackhole like entities. An extension of U(1) gauge symmetries allowing gauge transformations, which become constant in radial direction at large distances but depend on angle degrees of freedom, is in question. In TGD framework the situation is analogous but much more general and super-symplectic and other symmetries with conformal structure extend the various conformal symmetries and allow to understand also the hierarchy of Planck constants in terms of a fractal hierarchy of symmetry breakins to sub-algebra isomorphic with the full algebra of symmetries in question (see this).
   
   
3. There exist also experimental data challenging the no-hair theorem. The supermassive blackhole like entity near the galactic center is known to have a magnetic field (see this) and thus magnetic moment if the magnetic field is assignable to the blackhole itself rather than matter surrounding it.
   
   

Be as it may, any model should explain why the cyclotron radiation pulse came .4 seconds later than gravitaton pulse rather than at the same time. Compared to .2 seconds for blackhole formation this is quite a long time.

1. Suppose that blackhole like objects have - as any gravitating astrophysical object in TGD Universe must have - a magnetic body making possible the transfer of gravitons and carrying classical gravitational fields. Suppose that radial monopole flux tubes carrying gravitons can carry also BE condensates for which charged particles have varying mass m. h_gr= GMm/v₀=h_eff=n× h implies that particles with different masses reside at their own flux tubes like books in book shelves - something very important in TGD inspired quantum biology (see this).
   

   One might argue that hbar_gr serves as a very large spin unit and makes the storage very effective but here one must be very cautious: spin fractionization suggested by the covering property of space-time sheets could scale down the spin unit to hbar/n. I do not really understand this issue well enough. In any case, already the spontaneously magnetized BE condensate with relative angular momentum of Cooper makes at pairs of helical flux tubes possible effective angular momentum storage.
   
   

2. The spontaneously magnetized dark Bose-Einstein condensate would consist of charged bosons - say charged fermion pairs with members located at parallel flux tubes as in the TGD inspired model of hight T_c superconductor with spin S=1 Cooper pairs. This BE condensate would be ideal for the temporary storage of surplus spin and relative angular momentum of members of pairs at parallel helical flux tubes. This angular momentum would have been radiated away as gamma ray pulse in a quantum phase transition to a state without dark spontaneous magnetization.
   
   

To sum up, LIGO could mean also a new era in the theory of gravitation. The basic problem of GRT description of blackholes relates to the classical conservation laws and it becomes especially acute in the non-stationary situation represented by a merger. Post-Newtonian approximation is more than a calculational tool since it brings in conservation laws from Newtonian mechanics and fixes the coordinate system used to that assignable to empty Minkowski space. Further observations about blackhole mergers might force to ask whether Post-Newtonian approximation actually feeds in the idea that space-time is surface in imbedding space. If the mergers are accompanied by gamma ray bursts as a rule, one is forced to challenge the notion of blackhole and GRT itself.

For details see the chapter Quantum Astrophysics of "Physics in Many-Sheeted Space-time" or the article LIGO and TGD.

For a summary of earlier postings see Links to the latest progress in TGD.

LIGO and TGD

The recent detection of gravitational radiation by LIGO (see the posting of Lubos at and the article) can be seen as birth of gravito-astronomy. The existence of gravitational waves is however an old theoretical idea: already Poincare proposed their existence at the time when Einstein was starting the decade lasting work to develop GRT (see this).

Gravitational radiation has not been observed hitherto. This could be also seen as indicating that gravitational radiation is not quite what it is believed to be and its detection fails for this reason. This has been my motivation for considering the TGD inspired possibility that part or even all of gravitational radiation could consist of dark gravitons (see this). Their detection would be different from that for ordinary gravitons and this might explain why they have not been detected although they are present (Hulse-Taylor binary).

In this respect the LIGO experiment provided extremely valuable information: the classical detection of gravitational waves - as opposed to quantum detection of gravitons - does not seem to differ from that predicted by GRT. On the other hand, TGD suggests that the gravitational radiation between massive objects is mediated along flux tubes characterized by dark gravitational Planck constant h_gr =GMm/v₀ identifiable as h_eff=n× h (see this). This allows to develop in more detail TGD view about the classical detection of dark gravitons.

A further finding was that there was an emission of gamma rays .4 seconds after the merger (see the posting of Lubos and the article from Fermi Gamma Ray Burst Monitor). The proposal that dark gravitons arrive along dark magnetic flux tubes inspires the question whether these gamma rays were actually dark cyclotron radiation in extremely weak magnetic field associated with these flux tubes. There was also something anomalous involved. The mass scale of the merging blackholes deduced from the time evolution for so called chirp mass was 30 solar masses and roughly twice too large as compared to the upper bound from GRT based models (see this).

Development of theory of gravitational radiation

A brief summary about the development of theory of gravitational radiation is useful.

1. After having found the final formulation of GRT around 1916 after ten years hard work Einstein found solutions representing gravitational radiation by linearizing the field equations. The solutions are very similar in form to the radiation solutions of Maxwell's equations. The interpretation as gravitational radiation looks completely obvious in the light of after wisdom but the existence of gravitational radiation was regarded even by theoreticians far from obvious until 1957. Einstein himself wrote a paper claiming that gravitons might not exist after all: fortunately the peer review rejected it (see this)!
   
   
2. During 1916 Schwartschild published an exact solution of field equations representing a non-rotating black hole. At 1960 Kerr published an exact solution representing rotating blackhole. This gives an idea about how difficult the mathematics involved is.
   
   
3. After 1970 the notion of quasinormal mode was developed. Quasinormal modes are like normal modes and characterized by frequencies. Dissipation is however taken into account and this makes the frequencies complex. In the picture representing the gravitational radiation detected by LIGO, the damping is clearly visible after the maximum intensity is reached. These modes represent radiation, which can be thought of as incoming radiation totally reflected at horizon. These modes are needed to describe gravitational radiation after the blackhole is formed.
   
   
4. After 1990 post-Newtonian methods and numerical relativity developed and extensive calculations became possible allowing also precise treatment of the merger of two blackholes to single one.
   
   

I do not have experience in numerics nor in findings solutions to field equations of GRT. General Coordinate Invariance is extremely powerful symmetry but it also makes difficult the physical interpretation of solutions and finding of them. One must guess the coordinates in which everything is simple and here symmetries are of crucial importance. This is why I have been so enthusiastic about sub-manifold gravity: M⁴ factor of imbedding space provides preferred coordinates and physical interpretation becomes straightforward. In TGD framework the construction of extremals - mostly during the period 1980-1990 - was surprisingly easy thanks to the existence of the preferred coordinates. In TGD framework also conservations laws are exact and geodesic motion can be interpreted in terms of analog of Newton's equations at imbedding level: at this level gravitation is a genuine force and post-Newtonian approximation can be justified in TGD framework.

Evolution of the experimental side

1. The first indirect proof for gravitational radiation was Hulse-Taylor binary pulsar (see this. The observed increase of the rotation period could be understood as resulting from the loss of rotational energy by gravitational radiation.
   
   
2. Around 1960 Weber suggests a detector based on mass resonance with resonance frequency 1960 Hz. Weber claimed of detecting gravitational radiation on daily basis but his observations could not be reproduced and were probably due to an error in computer program used in the data analysis.
   
   
3. At the same time interferometers as detectors were proposed. Interferometer has two arms and light travels along both arms arms, is reflected from mirror at the end, and returns back. The light signals from the two arms interfere at crossing. Gravitational radiation induces the oscillation of the distance between the ends of interferometer arm and this in turn induces an oscillating phase shift. Since the shifts associated with the two arms are in general different, a dynamical interference pattern is generated. Later laser interferometers emerged.
   

   One can also allow the laser light to move forth and back several times so that the phase shifts add and interference pattern becomes more pronounced. This requires that the time spent in moving forth and back is considerably shorter than the period of gravitational radiation. Even more importantly, this trick also allows to use arms much shorter than the wavelength of gravitational radiation: for 35 Hz defining the lower bound for frequency in LIGO experiment the wavelength is of the order of Earth radius!
   
   

4. One can also use several detectors positioned around the globe. If all detectors see the signal, there are good reasons to take it seriously. It becomes also possible to identity precisely the direction of the source. A global network of detectors can be constructed.
   
   
5. The fusion of two massive blackholes sufficiently near to Earth (now they were located at distance of about Gly!) is optimal for the detection since the total amount of radiation emitted is huge.
   

What was observed?

LIGO detected an event that lasted for about .2 seconds. The interpretation was as gravitational radiation and numerical simulations are consistent with this interpretation. During the event the frequency of gravitational radiation increased from 35 Hz to 250 Hz. Maximum intensity was reached at 150 Hz and correspond to the moment when the blackholes fuse together. The data about the evolution of frequency allows to deduce information about the source if post-Newtonian approximation is accepted and the final state is identified as Kerr blackhole.

1. The merging objects could be also neutron stars but the data combined with the numerical simulations force the interpretation as blackholes. The blackholes begin to spiral inwards and since energy is conserved (in post-Newtonian approximation), the kinetic energy increases because potential energy decreases. The relative rotational velocity for the fictive object having reduced mass increases. Since gravitational radiation is emitted at the rotational frequency and its harmonics, its frequency increases and the time development of frequency codes for the time development of the rotational velocity. This rising frequency is in audible range and known as chirp.
   

   In the recent situation the rotational frequency increases from 35 Hz to maximum of 150 Hz at which blackholes fuse together. After that a spherically symmetric blackhole is formed very rapidly and exponentially damped gravitational radiation is generated (quasinormal modes) as frequency increases to 250 Hz. A ball bouncing forth and back in gravitational field of Earth and losing energy might serve as a metaphor.
   
   

2. The time evolution of the frequency of radiation coded to the time evolution of interference pattern provides the data allowing to code the masses of the initial objects and of final state object using numerical relativity. So called chirp mass can be expressed in two manners: using the masses of fusing initial objects and the rotation frequency and its time derivative. This allows to estimate the masses of the fusing objects. They are 36 and 29 solar masses respectively. The sizes of these blackholes are obtained by scaling from the blackhole radius 3 km of Sun. The objects must be blackholes. For neutron stars the radii would be much larger and the fusion would occur at much lower rotation frequency.
   
   
3. Assuming that the rotating final state blackhole can be described as Kerr's blackhole, one can model the situation in post-Newtonian approximation and predict the mass of the final state blackhole. The mass of the final state blackhole would be 62 solar masses so that 3 solar masses would transform to gravitational radiation! The intensity of the gravitational radiation at peak was more than the entire radiation by stars int the observed Universe. The second law of blackhole thermodynamics holds true: the sum of mass squared for the initial state is smaller than the mass squared for the final state (32²+29²< 62²).
   
   

Are observations consistent with TGD predictions

The general findings about masses of blackholes and their correlations with the frequency and about the net intensity of radiation are also predictions of TGD. The possibility of dark gravitons as large h_eff quanta however brings in possible new effects and might affect the detection. The consistency of the experimental findings with GRT based theory of detection process raises critical question: are dark gravitons there?

About the relationship between GRT and TGD

The proposal is that GRT plus standard model defines the QFT limit of TGD replacing many-sheeted space-time with slightly curved region of Minkowski space carrying gauge potentials defined as sums of the components of the induced spinor connection and the deviation of metric from flat metric as sum of similar deviations for space-time sheets (see this). This picture follows from the assumption that the test particle touching the space-time sheets experience the sum of the classical fields associated with the sheets.

The open problems of GRT limit of TGD have been the origin of Newton's constant - CP₂ size is almost four orders of magnitude longer than Planck length.Amusingly, a dramatic progress occurred in this respect just during the week when LIGO results were published.

The belief has been that Planck length is genuine quantal scale not present in classical TGD. The progress in twistorial approach to classical TGD however demonstrated that this belief was wrong. The idea is to lift the dynamics of 6-D space-time surface to the dynamics of their 6-D twistor spaces obeying the analog of the variational principle defined by Kähler action. I had thought that this would be a passive reformulation but I was completely wrong (see this).

1. The 6-D twistor space of the space-time surface is a fiber bundle having space-time as base space and sphere as fiber and assumed to be representable as a 6-surface in 12-D twistor space T(M⁴)× T(CP₂). The lift of Kähler action to Kähler action requires that the twistor spaces T(M⁴) T(CP₂) have Kähler structure. These structures exist only for S⁴, E⁴ and its Minkowskian analog M⁴ and CP₂ so that TGD is completely unique if one requires the existence of twistorial formulation. In the case of M⁴ one has a hybrid of complex and hyper-complex structure.
   
   
2. The radii of the two spheres bring in new length scales. The radius in the case of CP₂ is essentially CP₂ radius R. In the case of M⁴ the radius is very naturally Planck length so that the origin of Planck length is understood and it is purely classical notion whereas Planck mass and Newton's constant would be quantal notions.
   
   
3. The 6-D Kähler action must be made dimensionless by dividing with a constant with dimensions of length squared. The scale in question is actually the area of S²(M⁴), not the inverse of cosmological constant as the first guess was. The reason is that this would predict extremely large Kähler coupling strength for the CP₂ part of Kähler action.
   

   There are however two contributions to Kähler action corresponding to T(CP₂) and T(M⁴) and the corresponding Kähler coupling strengths - the already familiar α_K and the new α_K(M⁴) - are independent. The value of α_K(M⁴)× 4π R(S²(M⁴) corresponds essentially to the inverse of cosmological constant and to a length scale which is of the order of the size of Universe in the recent cosmology. Both Kähler coupling strengths are analogous to critical temperature and are predicted to have a spectrum of values. According to the earlier proposal, α_K(M⁴) would be proportional to p-adic prime p≈ 2^k, k prime, so that in very early times cosmological constant indeed becomes extremely large. This has been the problem of GRT based view about gravitation. The prediction is that besides the volume term coming from S⁽M⁴) there is also the analog of Kähler action associated with M⁴ but is extremely small except in very early cosmology.
   
   

4. A further new element is that TGD predicts the possibility of large h_eff=n× h gravitons. One has h_eff=h_gr= GMm/v₀, where v₀ has dimensions of velocity and satisfies v₀/c<1: the value of v₀/c is of order .5 × 10^-3 for the inner planets. h_gr seems to be absolutely essential for understanding how perturbative quantum gravitation emerges.
   

   What is nice is that the twistor lift of Kähler action suggests also a concrete explanation for h_eff/h=n. It would correspond to winding number for the map S²(X⁴)→ S²(M⁴) and one would indeed have covering of space-time surface induced by the winding as assumed earlier. This covering would have the special property that the base base for each branch of covering would reduce to same 3-surface at the ends of the space-time surface at the light-like boundaries of causal diamond (CD)
   defining fundamental notion in zero energy ontology (ZEO).
   
   

Twistor approach thus shows that TGD is completely unique in twistor formulation, explains Planck length geometrically, predicts cosmological constant and assigns p-adic length scale hypothesis to the cosmic evolution of cosmological constant, and also suggests an improved understanding of the hierarchy of Planck constants.

Can one understand the detection of gravitational waves if gravitons are dark?

The problem of quantum gravity is that if the parameter GMm/h=Mm/m_P² associated with two masses characterizes the interaction strength and is larger than unity, perturbation theory fails to converge. If one can assume that there is no quantum coherence, the interactions can be reduced to those between elementary particles for which this parameter is below unity so that the problem would disappear. In TGD framework however fermionic strings mediate connecting partonic 2-surface mediate the interaction even between astrophysical objects and quantum coherence in astrophysical scales is unavoidable.

The proposal is that Nature has been theoretician friendly and arranged so that a phase transition transforming gravitons to dark gravitons takes place so that Planck constant is replaced with h_gr=GMm/v₀. This implies that v₀/c<1 becomes the expansion parameter and perturbation theory converges. Note that the notion of h_gr makes sense only of one has Mm/m_P²>1. The notion generalizes also to other interactions and their perturbative description when the interaction strength is large. Plasmas are excellent candidates in this respect.

1. The notion of h_gr was proposed first by Nottale from quite different premises was that planetary orbits are analogous to Bohr orbits and that the situation is characterized by gravitational Planck constant h_gr= GMm/v₀. This replaces the parameter GMm/h with v₀ as perturbative parameter and perturbation theory converges. h_gr would characterize the magnetic flux tubes connecting masses M and m along which gravitons mediating the interaction propagate.
   

   According to the model of Nottale > for planetary orbits as Bohr orbits the entire mass of star behaves as dark mass from the point of view particles forming the planet. h_gr=GMm/v₀ appears as in the quantization of angular momentum and if dark mass M_D<M is assumed, the integer characterizing the angular momentum must be scaled up by M/M_D. In some sense all astrophysical objects would behave like quantum coherent systems and many-sheeted space-time suggests that the magnetic body of the system along which gravitons propagate is responsible for this kind of behavior.
   
   

2. The crucial observation is that h_gr depends on the product of interacting masses so that h_gr characterizes a pair of systems satisfying Mm/m_P²>1 rather than either mass. If so, the gravitons at magnetic flux tubes mediating gravitational interaction between masses M and m are always dark and have h_gr=h_eff. One cannot say that the systems themselves are characterized by h_gr. Rather, only the magnetic bodies or parts of them can be characterized by h_gr. The magnetic bodies can be associated with mass pairs and also with self interactions of single massive object (as analog of dipole field).
   
   
3. The general vision is that ordinary particles and large h_eff particles can transform to each other at quantum criticality (see this). Above temperatures corresponding to critical temperature particle would be ordinary, in a finite temperature range both kind of particles would be present, and below the lower critical temperature the particles would be dark. High T_c super-conductivity would provide a school example about this.
   
   

One would expect that for pairs of quantum coherent objects satisfying GMm/h>1, the graviton exchange is by dark gravitons. This could affect the model for the detection of gravitons.

1. Since Planck constant does not appear in classical physics, one might argue that the classical detection does not distinguish between dark and ordinary gravitons. Gravitons corresponds classically to radiation with same frequency but amplitude scaled up by n^1/2. One would obtain for h_gr/h>1 a sequence of pulses with large amplitude length oscillations rather than continuous oscillation as in GRT. The average intensity would be same as for classical gravitational radiation.
   

   Interferometers detect gravitational radiation classically as distance oscillations and the finding of LIGO suggests that all of the radiation is detected. Irrespective of the value of h_eff all gravitons couple to the geometry of the measuring space-time sheets. This looks very sensible in the geometric picture for this coupling. A more quantitative statement would be that dark and ordinary gravitons do not differ for detection times longer than the oscillation period. This would be the case now.
   

   The detection is based on laser light which goes forth and back along arm. The total phase shift between beams associated with the two arms matters and is a sum over the shifts associated with pulses. The quantization to bunches should be smoothed out by this summation process and the outcome is same as in GRT since average intensity must be same irrespective of the value of h_gr. Since all detection methods use interferometers there would be no difference in the detection of gravitons from other sources.
   
   

2. The quantum detection h_eff gravitons - as opposed to classical detection - is expected to differ from that of ordinary gravitons. Dark gravitons can be regarded as bunches of n ordinary gravitons and thus is n times higher energy. Genuine quantum measurement would correspond to an absorption of this kind of giant graviton. Since the signal must be "visible dark gravitons must transform to ordinary gravitons with same energy in the detection. For 35 Hz graviton the energy would have been GMm/v₀h times the energy or ordinary graviton with the same frequency. This would give energy of 19 (c/v₀) MeV: one would have gravitational gamma rays. The detection system should be quantum critical. The transformation of dark gravitons with frequency scale done by 1/n and energy increased correspondingly would serve as a signature for darkness.
   

   Living systems in TGD Universe are quantum critical and bio-photons are interpreted as dark photons with energies in visible and UV range but frequencies in EEG range and even below (see this). It can happen that only part of dark graviton radiation is detected and it can remain completely undetected if the detecting system is not critical. One can also consider the possibility that dark gravitons first decay to a bunch of n ordinary gravitons. Now however the detection of individual gravitons is impossible in practice.
   
   

A gamma ray pulse was detected .4 seconds after the merger

The Fermi Gamma-ray Burst Monitor detected 0.4 seconds after the merger a pulse of gamma rays with red shifted energies about 50 keV (see the posting of Lubos and the article from Fermi Gamma Ray Burst Monitor). At the peak of gravitational pulse the gamma ray power would have been about one millionth of the gravitational radiation. Since the gamma ray bursts do not occur too often, it is rather plausible that the pulse comes from the same source as the gravitational radiation. The simplest model for blackholes does not suggest this but it is not difficult to develop more complex models involving magnetic fields.

Could this observation be seen as evidence for the assumption that dark gravitons are associated with magnetic flux tubes?

1. The radiation would be dark cyclotron gravitation generated at the magnetic flux tubes carrying the dark gravitational radiation at cyclotron frequency f_c= qB/m and its harmonics (q denotes the charge of charge carrier and B the intensity of the magnetic field and its harmonics and with energy E=h_effeB/m .
   
   
2. If h_eff= h_gr= GMm/v₀ holds true, one has E= GMB/v₀ so that all particles with same charge respond at the same the same frequency irrespective of their mass: this could be seen as a magnetic analog of Equivalence Principle. The energy 50 keV corresponds to frequency f∼ 5× 10¹⁸ Hz. For scaling purposes it is good to remember that the cyclotron frequency of electron in magnetic field B_end=.2 Gauss (value of endogenous dark magnetic field in TGD inspired quantum biology) is f_c=.6 Mhz.
   

   From this the magnetic field needed to give 50 keV energy as cyclotron energy would be B_ord= (f/f_c)B_end=.4 GT corresponds to electrons with ordinary value of Planck constant the strength of magnetic field. If one takes the redshift of order v/c∼ .1 for cosmic recession velocity at distance of Gly one would obtain magnetic field of order 4 GT. Magnetic fields of with strength of this order of magnitude have been assigned with neutron stars.
   
   

3. On the other hand, if this energy corresponds to h_gr= GMm_e c/v₀ one has B= (h/h_gr)B_ord = (v₀m_P²/Mm_e)× B_ord∼ (v₀/c)× 10^-11 T (c=1). This magnetic field is rather weak (fT is the bound for detectability) and can correspond only to a magnetic field at flux tube near Earth. Interstellar magnetic fields between arms of Milky way are of the order of 5× 10^-10 T and are presumably weaker in the intergalactic space.
   
   
4. Note that the energy of gamma rays is by order or magnitude or two lower than that for dark gravitons. This suggests that the annihilation of dark gamma rays could not have produced dark gravitons by gravitational coupling bilinear in collinear photons.
   

One can of course forget the chains of mundane realism and ask whether the cyclotron radiation coming from distant sources has its high energy due to large value of h_gr rather than due to the large value of magnetic field at source. The presence of magnetic fields would reflects itself also via classical dynamics (that is frequency). In the recent case the cyclotron period would be of order (.03/v₀) Gy, which is of the same order of magnitude as the time scale defined by the distance to the merger.

In the case of Sun the prediction for energy of cyclotron photons would be E=[v₀(Sun)/v₀] × [M(Sun)/M(BH)] × 50 keV ∼ [v₀(Sun)/v₀] keV. From v₀(Sun)/c≈ 2^-11 one obtains E=(c/v₀)× .5 eV> .5 eV. Dark photons in living matter are proposed to correspond to h_gr=h_eff and are proposed to transform to bio-photons with energies in visible and UV range (see this).

Good dialectic would ask next whether both views about the gamma rays are actually correct. The "visible" cyclotron radiation with standard value of Planck constant at gamma ray energies would be created in the ultra strong magnetic field of blackhole, would be transformed to dark gamma rays with the same energy, and travel to Earth along the flux tubes. In TGD Universe the transformation ordinary photons to dark photons would occur in living matter routinely. One can of course ask whether this transformation takes place only at quantum criticality and whether the quantum critical period corresponds to the merger of blackholes.

The time lag was .4 second and the merger event lasted .2 seconds. If the gamma rays were ordinary photons so that dark gravitons would have travelled along different flux tubes, one can ask whether the propagation velocities differed by &Delta: c/c∼ 10^-17. Since the geodesics of the space-time surface are in general not geodesics of the imbedding space signals moving with light velocity along space-time sheet do not move with maximal signal velocity in imbedding space and the time taken to travel from A to B depends on space-time sheet. Could the later arrival time reflect slightly different signal velocities for photons and gravitons?

For details see the chapter Quantum Astrophysics of "Physics in Many-Sheeted Space-time" or the article LIGO and TGD.

For a summary of earlier postings see Links to the latest progress in TGD.

How it went?

It took one decade for Einstein to find the final mathematical formulation of General Relativity Theory (GRT). Immediately after having found the final formulation, he predicted gravitional waves, which are easy to discover from the linearized equations. One century later they have been found. Theoretician must be long aged if he wants to enjoy the fruits of his labor.

One should not compare oneself with Gods (the nasty colleagues - you know) but since I am totally crazy (ask colleagues) I talk about both us in the same paragraph. In TGD the process of finding final formulation (see this) took almost four decades and now I dare say that I have finally found the formulation. In the following I summarize what happened during the memorable week during which the final formulation emerged. I have done my best to organize the text to a readable form and I apologize if I have not succeeded completely. An entire flood of ideas emerged and they are still developing. This makes documentation difficult.

Some background

To understand how this is so important I describe briefly the background.

1. Recall that the formulation of classical TGD in terms of Kähler action emerged around 1990 and is therefore quarter century old now. I speak fluently about preferred extremals of Kähler action and I understand reasonably well the dynamics of Kähler action. But about how gravitational constant and cosmological constant emerge from this dynamics I have had only ideas.
   
   
2. This dynamics has one very non-standard feature: huge vacuum degeneracy. All 4-surfaces that have CP₂ projection, which is so called Lagrangian sub-manifold of CP₂ having vanishing induced Kähler form is vacuum extemal. By applying diffeomorphisms of M⁴ and symplectic transformations of CP₂ acting like U(1) gauge transformations one obtains new vacuum extremals. For instance, for the deformations of canonically imbedded empty Minkowski space Kähler action density can be approximated by a fourth order polynomial in CP₂ coordinates and their gradients and perturbation theory fails completely since propagator does not exist.
   
   
3. This spoiled completely the hopes about ordinary quantization of the theory and eventually inspired the idea about "world of classical worlds" (WCW) and led to a beautiful vision about quantum theory as purely classical theory for spinor fields in WCW representing physical states. I could have of course considered the possibility of adding to the action a small volume term to obtain perturbation theory - much like in case of branes - but it looked incredibly ugly. I can only congratulate myself that I refused to consider this possibility although this volume term now emerges from twistorial variant of Kähler action! It had prevented me from discovering WCW and many other deep ideas.
   
   

Only M⁴ and CP₂ allow twistor space with Kähler structure

The situation began to change few years ago as I realized that twistors might be central for understanding of quantum TGD.

1. I am not a jedi master in perturbative QFT and formulas hate me. Furthermore, I do not believe on N=4 SUSY except as a beautiful model able to express some very profound ideas, which have not yet reached our conscious mind. Just the strange beauty of findings of Nima Arkani-Hamed and other pioneers made me convinced that twistors are the key to progress.
   
   
2. Year or two ago came the crucial discovery. I learned that S⁴ (and also conformally compactified version of Minkowski space M⁴) and CP₂ are completely unique 4-D spaces in that only they allow twistor space with Kähler structure. This was discovered by Hitchin (published in 1981). Ironically, this had been discovered roughly year after I ended up with CP₂! Somehow I had failed to learn about this. It was immediately clear that this is something incredibly profound and must mean that TGD a twistorially unique.

"Precisely how?" should have been the immediate question. For some funny reason I did not make quite that question to which the answer would have been totally obvious. I however realized that I should lift space-time surfaces to their 6-D twistor spaces and represent them as 6-D surfaces in the 12-D twistor space of imbedding space inhering their twistor structure from that for the 12-D space. I indeed proposed how to define induction of twistor structure as analog for the induction of metric and spinor structure.

I did not however continue with the obvious question: "How to define dynamics for these 6-D twistor spaces identifiable sphere bundles over space-time surfaces?". This question contains its own answer. Twistor structure involves the identification of an antisymmetric tensor defining preferred quaternionic imaginary unit representing it geometrically. The Kähler form of 12-D twistor space projected to space-time surface should define this preferred imaginary unit. TGD exists only for M⁴× CP₂ and is therefore a completely unique theory. The dynamics is determined by 6-D action variant of Kähler action for 6-D surfaces in twistor space of M⁴× CP₂. I want to repeat: TGD is completely unique if one accepts twistorial formulation.

The existence of twistorial formulation makes TGD unique and leads to unification of gravitation and standard model

After this everything followed within week even though I made all imaginable wrong guesses.

1. 6-D Kähler action has dimension length squared and must be multiplied by a constant with dimensions of 1/length squared. This constant, call it 1/L², is highly analogous to cosmological constant but is coupling constant like parameter since one can replace it with the dimensionless ratio ε²=R²/L², R radius of CP₂. ε is analogous to critical temperature (or perhaps better, critical value of inverse pressure) just like Kähler coupling strength and is expected to have spectrum labelled by p-adic primes p≈ 2^k, k prime- just like Kähler coupling strength. The values of L would be most naturally p-adic length scales and cosmological constant would in first approximation decrease with cosmological time t as 1/t².
   

   The incredibly small cosmological constant (not so in the early Universe) would not be just some nasty trick of Universe making theoreticians crazy but needed to remove the vacuum degeneracy of Kähler action in dimensionally reduced dynamics giving rise to 4-volume as additional term in action and making also perturbation theory around canonically imbedded M⁴ possible. Obviously this action would give rise to the analog of kinetic term for gravitons.
   

   Volume term of course gives just the geometric counterpart of wave equation. In fact, all known extremals of Kähler action are minimal surfaces so that in this sense nothing new has emerged! A coupling between dynamics of volume term and Kähler action is present for more general extremals and Kähler coupling strength and cosmological constant do not completely disappear from the classical dynamics.
   
   

2. Also other fundamental lengths pop up: the radii on S²(M⁴) and S²(CP₂). The radius of S²(CP₂) is essentially CP₂ radius from the definition of twistor space but what about S²(M⁴)?
   Here I had to think thoroughly what the twistor space of M⁴ is. It turned out that the radius is most naturally what Planck length mystic would guess it to be: essentially Planck length l_P. Planck length would emerge from the theory as a purely classical scale! Only two weeks ago I explained that Planck length is purely quantal emergent length scale! However, Planck mass and Newton's constant would emerge as quantal parameters since they depend on Planck constant when expressed in terms of Planck length, hbar and c.
   

   Twistorialization thus brings all the basic scales of gravitation. Earlier I had p-adic length scales emerging from the successful p-adic mass calculations explaining the mass spectra of particles and the CP₂ inspired vision about how gauge coupling strengths and their evolution emerge from quantum criticality. My original belief that G and Λ would emerge from the dynamics of Kähler action alone was therefore wrong.
   
   

3. About how cosmological constant actually emerges I had a slightly wrong view: this was just sloppy thinking. Before continuing it is good to recall how the cosmological constant emerges from TGD framework. The key point is that the 6-D Kähler action contains two terms.
   
   
   1. The first term is essentially the ordinary Kähler action multiplied by the area of S²(X⁴), which is compensated by the length scale, which can be taken to be the area 4π R²(M⁴) of S²(M⁴). This makes sense for winding numbers (w₁,w₂)=(n,0) meaning that S²(CP₂) is effectively absent but S²(M⁴) is present.
      
      
   2. Second term is the analog of Kähler action assignable assignable to the projection of S²(M⁴) Kähler form. The corresponding Kähler coupling strength α_K (M⁴) is huge (really huge, this I did not realize first!) - so huge that one has
      

      α_K (M⁴)4π R²(M⁴)== L² ,
      

      where 1/L² is of the order of cosmological constant and thus of the order of the size of the recent Universe. α_K(M⁴) is also analogous to critical temperature and the earlier hypothesis that the values of L correspond to p-adic length scales implies that the values of come as α_K(M⁴) ∝ p≈ 2^k, p prime, k prime.
      
      

   3. The Kähler form assignable to M⁴ is not assumed to contribute to the action since it does not contribute to spinor connection of M⁴. One can of course ask whether it could be present. For canonically imbedded M⁴ self-duality implies that this contribution vanishes and for vacuum extremals of ordinary Kähler action this contribution is small. Breaking of Lorentz invariance is however a possible problem. If α_K(M⁴) is given by above expression, then this contribution is extremely small.
      
      
   Hence one can consider the possibility that the action is just the sum of full 6-D Kähler actions assignable to T(M⁴) and T(CP₂) but with different values of α_K if one has (w₁,w₂)=(n,0). Also other w₂≠ 0 is possible but corresponds to gigantic cosmological constant.
4. I had also to learn also what twistor space T(M⁴) really is! For CP₂ and S² there are no problems but how to go to the Minkowskian signature - this was the problem. Also here I did all wrong trials. The solution of the problem was simple.
   

   I had actually found the solution for more than one and half decades ago while studying so called massless extremals (MEs) representing radiation in TGD Universe. Also the study of so called M⁸-H duality and the notion of quaternionic structure had led to what I call Hamilton-Jacobi (H-J) structure generalizing Euclidian 4-space E⁴ with complex structure to its Minkowskian variant.
   

   One must first construct M⁴ with H-J structure and then lift it to twistor space. H-J in Minkowski space means the existence of a spatially varying distribution M⁴= M²(x)⊕ E²(x) of decompositions of tangent space of X⁴ to direct orthogonal sum of local 2-D Minkowski space M²(x) and and orthogonal Euclidian 2-space E²(x) . This distribution must be integrable meaning that M²(x) and E²(x) serve as tangent spaces for 2-D surfaces. Euclidian 2-space allows complex structure and complex coordinates (z,z*). M² allows hyper-complex structure and hyper-complex coordinates, which are nothing but light-like coordinates (u=t-z,v=t+z) such that metric of M² is of from ds²=2dudv.
   

   The construction of twistor space looks now rather trivial. Any antisymmetric tensor in the space E³ orthogonal to time like t defines direction that it is point of the sphere defining the twistor space fiber. Metric is induced from the metric for this kind of tensors defined by M⁴ metric. The covariantly constant two-form of E² defines preferred quaternionic imaginary unit. This is also familiar from number theoretic vision demanding its existence. The vision about preferred extremals of Kähler action as quaternionic 4-surfaces of octonionic 8-space relies on this vision. In particular, the twistorial sphere is sphere -not hyperbolic sphere with signature (1,-1) as I believed for 24 hours - and it has metric signature (-1,-1) rather than being time-like!
   
   

5. I had also to learn what the induction of twistor structure means concretely. The preferred quaternionic imaginary unit should be represented as a projection of Kähler form of 12-D twistor space T(H). The preferred imaginary unit defining twistor structure as sum of projections of both T(CP₂) and T(M⁴) Kähler forms would guarantee that vacuum extremals like canonically imbedded M⁴ for which T(CP₂) Kähler form contributes nothing have well-defined twistor structure. T(M⁴) or T(CP₂) are treated completely symmetrically.
   
   
6. For Kähler action M⁴-CP₂ symmetry does not make sense. 4-D Kähler action to which 6-D Kähler action dimensionally reduces can depend on CP₂ Kähler form only. I have also considered the possibility of covariantly constant self-dual M⁴ term in Kähler action but given it up because of problems with Lorentz invariance. One should couple the gauge potential of M⁴ Kähler form to induced spinors. This would mean the existence of vacuum gauge fields coupling to sigma matrices of M⁴ so that the gauge grop would be non-compact SO(3,1) leading to a breakdown of unitarity.
   

   Hence it seems clear that only the projection of T(CP₂) part of Kähler form of T(H) can appear in 6-D Kähler action. This option breaks the symmetry between M⁴ and CP₂ at the level of dynamics but is physically unavoidable and is also mathematically completely acceptable. I cannot but accept the situation.
   
   

7. An important point to notice is that the radius of the sphere associated with the twistor space of X⁴ is dynamical. It however turned that for winding numbres (w₁, w₂(=(n,0) - the only option allowed by the condition that cosmological constant is not gigantic - the radius is essentially that of S²(M⁴) and therefore constant. The cosmological estimate gave same result for all cosmological eras.
   
   

A connection with the hierarchy of Planck constants?

A connection with the hierarchy of Planck constants is highly suggestive. Since also a connection with the p-adic length scale hierarchy suggests itself for the hierarchy of p-adic length scales it seems that both length scale hierarchies might find first principle explanation in terms of twistorial lift of Kähler action.

1. Cosmological considerations encourage to think that R₁≈ l_P and R₂≈ R hold true. One would have in early cosmology (w₁,w₂)=(1,0) and later (w₁,w₂)=(0,1) guaranteeing R_D grows from l_P to R during cosmological evolution. These situations would correspond the solutions (w₁=n,0) and (0,w₂=n) one has A= n 4π R₁² and A=n× 4π R₂² and both Kähler coupling strengths are scaled down to α_K/n. For hbar_eff/h=n exactly the same thing happens!
   

   There are further intriguing similarities. h_eff/h=n is assumed to correspond multi-sheeted (to be distinguished from many-sheeted!) covering space structure for space-time surface. Now one has covering space defined by the lift S²(X⁴)→ S²(M4)× S²(CP₂). These lifts define also lifts of space-time surfaces.
   

   Could the hierarchy of Planck constants correspond to the twistorial surfaces for which S²(M⁴) and S⁽CP₂) are identified in 1-1 manner? The assumption has been that the n-fold multi-sheeted coverings of space-time surface for h_eff/h=n are singular at the ends of space-time surfaces at upper and lower boundaries if causal diamond (CD). Could one consider more precise definition of twistor space in such a manner that CD replaces M⁴ and the covering becomes singular at the light-like boundaries of CD - the branches of space-time surface would collapse to single one. What could this collapse mean geometrically? Or should one give up the assumption about singular nature of the covering used to distinguishes many-sheetedness from multi-sheetedness.
   
   

2. w₁=w₂=w is essentially the first proposal for conditions associated with the lifting of twistor space structure. w₁=w₂=n gives ds²= (R₁²+R₂²)(dθ² +w²dφ²) and A = n × 4π (R₁²+R₂²). Also now Kähler coupling strength is scaled down to α/n. Again a connection with the hierarchy of Planck constants suggests itself.
   
   
3. One can consider also the option R₁=R₂ option giving ds²= R₁²(2dθ² +(w₁²+w₂²)dφ². If the integers w_i define Pythagorean square one has w₁²+w₂²=n² and one has R₁=R₂ option that one has A =n× 4π R². Also now the connection with the hierarchy of Planck constants might make sense.
   
   

Twistorial variant for the imbedding space spinor structure

The induction of the spinor structure of imbedding space is in key role in quantum TGD. The question arises whether one should lift also spinor structure to the level of twistor space. If so one must understand how spinors for T(M⁴) and T(CP₂) are defined and how the induced spinor structure is induced.

1. In the case of CP₂ the definition of spinor structure is rather delicate and one must add to the ordinary spinor connection U(1) part, which corresponds physically to the addition of classical U(1) gauge potential and indeed produces correct electroweak couplings to quarks and leptons. It is assumed that the situation does not change in any essential manner: that is the projections of gauge potentials of spinor connection to the space-time surface give those induced from M⁴× CP₂ spinor connection plus possible other parts coming as a projection from the fiber S²(M²)× S²(CP₂). As a matter of fact, these other parts should vanish if dimensional reduction is what it is meant to be.
   
   
2. The key question is whether the complications due to the fact that the geometries of twistor spaces T(M⁴) and T(CP₂) are not quite Cartesian products (in the sense that metric could be reduced to a direct sum of metrics for the base and fiber) can be neglected so that one can treat the sphere bundles approximately as Cartesian products M⁴× S² and CP₂× S². This will be assumed in the following but should be carefully proven.
   
   
3. Locally the spinors of the twistorspace T(H) are tensor products of imbedding spinors and those for of S²(M⁴)× S²(CP₂) expressible also as tensor products of spinors for S²(M⁴) and S²(CP₂). Obviously, the number of spinor components increases by factor 2× 2 = 4 unless one poses some additional conditions taking care that one has dimensional reduction without the emergence of any new spin like degrees of freedom for which there is no physical evidence. The only possible manner to achieve this is to pose covariant constancy conditions already at the level of twistor spaces T(M⁴) and T(CP₂) leaving only single spin state in these degrees of freedom.
   
   
4. In CP₂ covariant constancy is possible for right-handed neutrino so that CP₂ spinor structure can be taken as a model. In the case of CP₂ spinors covariant constancy is possible for right-handed neutrino and is essentially due to the presence of U(1) part in spinor connection forced by the fact that the spinor structure does not exist otherwise. Ordinary
   S² spinor connection defined by vielbein exists always. One can however add a coupling to a suitable multiple of Kähler potential satisfying the quantization of magnetic charge (the magnetic flux defined by U(1) connection is multiple of 2π so that its imaginary exponential is unity).
   

   S² spinor connections must must have besides ordinary vielbein part determined by S² metric also U(1) part defined by Kähler form coupled with correct coupling so that the curvature form annihilates the second spin state for both S²(M⁴) and S²(CP₂). U(1) part of the spinor curvature is proportional to Kähler form J ∝ sin(theta)dθ dφ so that this is possible. The vielbein and U(1) parts of the spinor curvature ear proportional Pauli spin matrix σ_z= (1,0;0,-1)/2 and unit matrix (1,0;0,1) respectively so that the covariant constancy is possible to satisfy and fixes the spin state uniquely.
   
   

5. The covariant derivative for the induced spinors is defined by the sum of projections of spinor gauge potentials for T(M⁴) and T(CP₂). With above assumptions the contributions gauge potentials from T(M⁴) and T(CP₂) separately annihilate single spinor component. As a consequence there are no constraints on the winding numbers w_i, i=1,2 of the maps S²(X⁴) → S²(M⁴) and S²(X⁴) → S²(CP₂). Winding number w_i corresponds to the imbedding map (Θ_i= θ, Φ_i=w_iφ).
   
   

   
   

6. If the square of the Kähler form in fiber degrees of freedom gives metric to that its square is metric, one obtains just the area of S² from the fiber part of action. This is given by the area A= 4π(2^1/2(w₁²R₁²+w₂²R₂²) since the induced metric is given by ds²= (R₁²+R₂²)dθ² +(w₁²R₁²+w₂²R₂²)dφ² for (Θ₁= θ, Φ=n₁φ, Φ₂= n₂φ).
   
   

To sum up, I strongly feel that the final formulation of TGD has now emerged and it is now clear that TGD is indeed a quantum theory of gravitation allowing to understand standard model symmetries. The existence of twistorial formulation makes possible gravitation and predicts standard model symmetries. This theory is completely unique from extremely general assumptions. This cannot be said about any competitor of TGD.

For background see the article From Principles to Diagrams or the chapter From Principles to Diagrams of "Towards M-matrix".

For a summary of earlier postings see Links to the latest progress in TGD.

Cosmic evolution of the radius of the fiber of the twistor space of space-time surface

I have continued the little calculations inspired by the surprising finding that twistorial lift of Kähler action based dynamics immediately leads to the identification of cosmological length scales as fundamental classical length scales appearing in 6-D Kähler action, whose dimensional reduction gives Kähler action plus small cosmological term with correct sign to explain together with magnetic flux tube tension accelerating cosmic expansion. Whether Planck length emerges classically from from quantum theory remains still an open question.

For a fleeting moment I thought that for the twistor space of Minkowski space the 2-D fiber could be hyperbolic sphere H² (t²-x²-y² =-R_H²) rather than sphere S² as it is for CP₂ with Euclidian signature of metric. I however soon realized that the infinite area of H² implies that 6-D Kähler action is infinite and that there are many other difficulties.

The correct manner to define Minkowskian variant of twistor space is by starting from the generalization of complex and Kähler structures for M⁴= M²+ E² of local tangent space to longitudinal (defined by light-like vector) and to transversal directions (polarizations orthogonal to the light-like vector. The decomposition can depend on point but the distributions of two planes must integrated to 2-D surfaces. In E² one has complex structure and in M² its hyper-complex variant. In M² has decomposition of replacing complex numbers by hyper-complex numbers so that complex coordinate x+iy is replaced with w=t+ie, i²=-1 and e²=-1.

It took time to realize I have actually carried out this generalization years ago with quite different motivations and called the resulting structure Hamilton-Jacobi structure! The twistor fiber is defined by projections of 4-D antisymmetric tensors (in particular induced Kähler form) to the orthogonal complement of unique time direction determed by the sum of light-like vector and its dual in M². This part of tensor could be called magnetic. Th magnetic part of the tensor defines a direction and one has natural metric making the space of directions sphere S² with metric having signature (-1,-1). This requires that twistor space has metric signature (-1,-1,1,-1,-1,-1) (I also considered seriously the signature (1,1,1,-1,-1,-1) so that there are three time-like coordinates) .

The radii of the spheres associated with M⁴ and CP₂ define two fundamental scales and the scaling of 6-D Käler action brings in third fundamental length scale. On possibility is that the radii of the two spheres are actually identical and essentially equal to CP₂ radius. Second option is that the radius of S²(M⁴) equals to Planck length, which would be therefore a fundamental length scale.

The radius R_D of the 2-D fiber of twistor space assignable to space-time surfaces is dynamical. In Euclidian space-time regions the fiber is sphere: a good guess is that its order of magnitude is determined by the winding numbers of the maps from S²(X⁴)→ S²(M⁴) and S²(X⁴) → S²(CP₂). The winding numbers (1,0) and (0,1) represent the simplest options. The question is whether one could say something non-trivial about cosmic evolution of R_D as function of cosmic time. This seems to be the case.

Before continuing it is good to recall how the cosmological constant emerges from TGD framework. The key point is that the 6-D Kähler action contains two terms.

1. The first term is essentially the ordinary Kähler action multiplied by the area of S²(X⁴), which is compensated by the length scale, which can be taken to be the area 4π R²(M⁴) of S²(M⁴). This makes sense for winding numbers (w₁,w₂)=(n,0) meaning that S²(CP₂) is effectively absent but S²(M⁴) is present.
   
   
2. Second term is the analog of Kähler action assignable assignable to the projection of S²(M⁴) Kähler form. The corresponding Kähler coupling strength α_K (M⁴) is huge - so huge that one has
   

   α_K (M⁴)4π R²(M⁴)== L² ,
   

   where 1/L² is of the order of cosmological constant and thus of the order of the size of the recent Universe. α_K(M⁴) is also analogous to critical temperature and the earlier hypothesis that the values of L correspond to p-adic length scales implies that the values of come as α_K(M⁴) ∝ p≈ 2^k, p prime, k prime.
   
   

3. The Kähler form assignable to M⁴ is not assumed to contribute to the action since it does not contribute to spinor connection of M⁴. One can of course ask whether it could be present. For canonically imbedded M⁴ self-duality implies that this contribution vanishes and for vacuum extremals of ordinary Kähler action this contribution is small.Breaking of Lorentz invariance is however a possible problem. If α_K(M⁴) is given by above expression, then this contribution is extremely small.
   
   

Hence one can consider the possibility that the action is just the sum of full 6-D Kähler actions assignable to T(M⁴) and T(CP₂) but with different values of α_K if one has (w₁,w₂)=(n,0). Also other w₂≠ 0 is possible but corresponds to gigantic cosmological constant.

Given the parameter L² as it is defined above, one can deduce an expression for cosmological constant Λ and show that it is positive. One can actually get estimate for the evolution of R_D as function of cosmic time if one accepts Friedman cosmology as an approximation of TGD cosmology.

1. Assume critical mass density so that one has
   

   ρ_cr= 3H²/8π G .
   

   
   

2. Assume that the contribution of cosmological constant term to the mass mass density dominates. This gives ρ≈ ρ_vac=Λ/8π G. From ρ_cr=ρ_vac one obtains
   

   Λ= 3H² .
   
   

3. From Friedman equations one has H²= ((da/dt)/a)², where a corresponds to light-cone proper time and t to cosmic time defined as proper time along geodesic lines of space-time surface approximated as Friedmanncosmology. One has
   

   Λ= 3/g_aaa²
   

   in Robertson-Walker cosmology with ds²= g_aada²-a² dσ₃².
   
   

4. Combining this equations with the TGD based equation
   

   Λ= 8π²G/L²R_D²
   

   one obtains
   

   8π²G/L²R_D²= 3/g_aaa².
   
   

5. Assume that quantum criticality applies so that L has spectrum given by p-adic length scale hypothesis so that one discrete p-adic length scale evolution for the values of L. There are two options to consider depending on whether p-adic length scales are assigned with light-cone proper time a or with cosmic time t
   

   T= a (Option I) ,

   T=t (Option II).

   Both options give the same general formula for the p-adic evolution of L(k) but with different interpretation of T(k).
   

   L(k)/Lnow= T(k)/Tnow ,

   T(k)= L(k) = 2(k-151)/2× L(151) ,

   L(151)≈ 10 nm .

   Here T(k) is assumed to correspond to primary p-adic length scale. An alternative - less plausible - option is that T(k) corresponds to secondary p-adic length scale L₂(k)=2^k/2L(k) so that T(k) would correspond to the size scale of causal diamond. In any case one has L ∝ L(k). One has a discretized version of smooth evolution
   

   L(a) = L_now × (T/T_now) .
   
   

Consider now the predictions.

1. Feeding into the formula following from two expressions for Λ one obtains an expression for R_D(a)
   

   R_D/l_P= (8/3)^1/2π× (a/L(a)× g_aa^1/2
   

   This equation tells that R_D is indeed dynamical, and becomes very small at very early times since g_aa becomes very small. As a matter of fact, in very early cosmic string dominated cosmology g_aa would be extremely small constant (see this). In late cosmology g_aa→ 1 holds true and one obtains at this limit
   

   R_D(now)= (8/3)^1/2π× (a_now/L_now) × l_P ≈ 4.4 ×(a_now/L_now) × l_P .
   
   

2. For T= t option R_D/l_P remains constant during both matter dominated cosmology, radiation dominated cosmology, and string dominated cosmology since one has a∝ tⁿ with n= 1/2 during radiation dominated era, n= 2/3 during matter dominated era, and n=1 during string dominated era (see this). This gives
   

   R_D/l_P=(8/3)^1/2π× at (g_aa^1/2(t(end)/L(end)) = (8/3)^1/2π×(1/n)(t(end)/L(end)) .
   

   Here "end"> refers the end of the string or radiation dominated period or to the recent time in the case of matter dominated era. The value of n would have evolved as R_D/l_P∝ (1/n (t_end/L_end), n∈ [1,3/2,2}. During radiation dominated cosmology R_D ∝ a^1/2 holds true. The value of R_D would be very nearly equal to R(M⁴) and R(M⁴) would be of the same order of magnitude as Planck length. In matter dominated cosmology would would have R_D ≈ 2.2 (t(now)/L(now)) × l_P .
   
   

3. For R_D(now)=l_P one would have
   

   L_now/a_now =(8/3)^1/2π≈ 4.4 .
   

   In matter dominated cosmology g_aa=1 gives t_now=(2/3)× a_now so that predictions differ only by this factor for options I and II. The winding number for the map S²(X⁴)→ S²(CP₂) must clearly vanish since otherwise the radius would be of order R.
   
   

4. For R_D(now)= R one would obtain
   

   a_now/L_now =(8/3)^1/2π× (R/l_P)≈ 2.1× 10⁴ .
   

   One has L_now=10⁶ ly: this is roughly the average distance scale between galaxies. The size of Milky Way is in the range 1-1.8 × 10⁵ ly and of an order of magnitude smaller.
   
   

5. An interesting possibility is that R_D(a) evolves from R_D ∼ R(M⁴) ∼ l_P to R_D ∼ R. This could happen if the winding number pair (w₁,w₂)=(1,0) transforms to (w₁,w₂)=(0,1) during transition to from radiation (string) dominance to matter (radiation) dominance. R_D/l_P radiation dominated cosmology would be related by a factor
   

   R_D(rad)/R_D(mat)= (3/4)(t(rad,end)/L(rad,end))×(L(now)/t(now))
   

   to that in matter dominated cosmology. Similar factor would relate the values of R_D/l_P in string dominated and radiation dominated cosmologies. The condition R_D(rad)/R_D(mat) =l_P/R expressing the transformation of winding numbers would give
   

   L(now)/L(rad,end) =(4/3) (l_P/R) (t(now)/t(rad,end)) .
   

   One has t(now)/t(rad,end)≈ .5× 10⁶ and l_P/R =2.5× 10^-4 giving L(now)/L(rad,end)≈ 125, which happens to be near fine structure constant.
   
   

6. For the twistorial lifts of space-time surfaces for which cosmological constant has a reasonable value , the winding numbers are equal to (w₁,w₂)=(n,0) so that R_D=n^1/2 R(S²(M⁴)) holds true in good approximation. This conforms with the observed constancy of R_D during various cosmological eras, and would suggest that the ratio t(end)/L(end) characterizing these periods is same for all periods. This determines the evolution for the values of α_K(M⁴).
   
   

R(M⁴)∼ l_P seems rather plausible option so that Planck length would be fundamental classical length scale emerging naturally in twistor approach. Cosmological constant would be coupling constant like parameter with a spectrum of critical values given by p-adic length scales.

For background see the article From Principles to Diagrams or the chapter From Principles to Diagrams of "Towards M-matrix".

For a summary of earlier postings see Links to the latest progress in TGD.