What's new inTowards MMatrixNote: Newest contributions are at the top! 
Year 2016 
Noncommutative imbedding space and strong form of holography
The precise formulation of strong form of holography (SH) is one of the technical problems in TGD. A comment in FB page of Gareth Lee Meredith led to the observation that besides the purely number theoretical formulation based on commutativity also a symplectic formulation in the spirit of noncommutativity of imbedding space coordinates can be considered. One can however use only the notion of Lagrangian manifold and avoids making coordinates operators leading to a loss of General Coordinate Invariance (GCI). Quantum group theorists have studied the idea that spacetime coordinates are noncommutative and tried to construct quantum field theories with noncommutative spacetime coordinates (see this). My impression is that this approach has not been very successful. In Minkowski space one introduces antisymmetry tensor J_{kl} and uncertainty relation in linear M^{4} coordinates m^{k} would look something like [m^{k}, m^{l}] = l_{P}^{2}J^{kl}, where l_{P} is Planck length. This would be a direct generalization of noncommutativity for momenta and coordinates expressed in terms of symplectic form J^{kl}. 1+1D case serves as a simple example. The noncommutativity of p and q forces to use either p or q. Noncommutativity condition reads as [p,q]= hbar J^{pq} and is quantum counterpart for classical Poisson bracket. Noncommutativity forces the restriction of the wave function to be a function of p or of q but not both. More geometrically: one selects Lagrangian submanifold to which the projection of J_{pq} vanishes: coordinates become commutative in this submanifold. This condition can be formulated purely classically: wave function is defined in Lagrangian submanifolds to which the projection of J vanishes. Lagrangian manifolds are however not unique and this leads to problems in this kind of quantization. In TGD framework the notion of "World of Classical Worlds" (WCW) allows to circumvent this kind of problems and one can say that quantum theory is purely classical field theory for WCW spinor fields. "Quantization without quantization would have Wheeler stated it. GCI poses however a problem if one wants to generalize quantum group approach from M^{4} to general spacetime: linear M^{4} coordinates assignable to Liealgebra of translations as isometries do not generalize. In TGD spacetime is surface in imbedding space H=M^{4}× CP_{2}: this changes the situation since one can use 4 imbedding space coordinates (preferred by isometries of H) also as spacetime coordinates. The analog of symplectic structure J for M^{4} makes sense and number theoretic vision involving octonions and quaternions leads to its introduction. Note that CP_{2} has naturally symplectic form. Could it be that the coordinates for spacetime surface are in some sense analogous to symplectic coordinates (p_{1},p_{2},q_{1},q_{2}) so that one must use either (p_{1},p_{2}) or (q_{1},q_{2}) providing coordinates for a Lagrangian submanifold. This would mean selecting a Lagrangian submanifold of spacetime surface? Could one require that the sum J_{μν}(M^{4})+ J_{μν}(CP_{2}) for the projections of symplectic forms vanishes and forces in the generic case localization to string world sheets and partonic 2surfaces. In special case also higherD surfaces  even 4D surfaces as products of Lagrangian 2manifolds for M^{4} and CP_{2} are possible: they would correspond to homologically trivial cosmic strings X^{2}× Y^{2}⊂ M^{4}× CP_{2}, which are not anymore vacuum extremals but minimal surfaces if the action contains besides Käction also volume term. But why this kind of restriction? In TGD one has strong form of holography (SH): 2D string world sheets and partonic 2surfaces code for data determining classical and quantum evolution. Could this projection of M^{4} × CP_{2} symplectic structure to spacetime surface allow an elegant mathematical realization of SH and bring in the Planck length l_{P} defining the radius of twistor sphere associated with the twistor space of M^{4} in twistor lift of TGD? Note that this can be done without introducing imbedding space coordinates as operators so that one avoids the problems with general coordinate invariance. Note also that the nonuniqueness would not be a problem as in quantization since it would correspond to the dynamics of 2D surfaces. The analog of brane hierarchy for the localization of spinors  spacetime surfaces; string world sheets and partonic 2surfaces; boundaries of string world sheets  is suggesetive. Could this hierarchy correspond to a hierarchy of Lagrangian submanifolds of spacetime in the sense that J(M^{4})+J(CP_{2})=0 is true at them? Boundaries of string world sheets would be trivially Lagrangian manifolds. String world sheets allowing spinor modes should have J(M^{4})+J(CP_{2})=0 at them. The vanishing of induced W boson fields is needed to guarantee welldefined em charge at string world sheets and that also this condition allow also 4D solutions besides 2D generic solutions. This condition is physically obvious but mathematically not wellunderstood: could the condition J(M^{4})+J(CP_{2})=0 force the vanishing of induced W boson fields? Lagrangian cosmic string type minimal surfaces X^{2}× Y^{2} would allow 4D spinor modes. If the lightlike 3surface defining boundary between Minkowskian and Euclidian spacetime regions is Lagrangian surface, the total induced Kähler form ChernSimons term would vanish. The 4D canonical momentum currents would however have nonvanishing normal component at these surfaces. I have considered the possibility that TGD counterparts of spacetime supersymmetries could be interpreted as addition of higherD righthanded neutrino modes to the 1fermion states assigned with the boundaries of string world sheets. An alternative  but of course not necessarily equivalent  attempt to formulate this picture would be in terms of number theoretic vision. Spacetime surfaces would be associative or coassociative depending on whether tangent space or normal space in imbedding space is associative  that is quaternionic. These two conditions would reduce spacetime dynamics to associativity and commutativity conditions. String world sheets and partonic 2surfaces would correspond to maximal commutative or cocommutative submanifolds of imbedding space. Commutativity (cocommutativity) would mean that tangent space (normal space as a submanifold of spacetime surface) has complex tangent space at each point and that these tangent spaces integrate to 2surface. SH would mean that data at these 2surfaces would be enough to construct quantum states. String world sheet boundaries would in turn correspond to real curves of the complex 2surfaces intersecting partonic 2surfaces at points so that the hierarchy of classical number fields would have nice realization at the level of the classical dynamics of quantum TGD. For background see the chapter About twistor lift of TGD. 
Minimal surface cosmology
Before the discovery of the twistor lift TGD inspired cosmology has been based on the assumption that vacuum extremals provide a good estimate for the solutions of Einstein's equations at GRT limit of TGD . One can find imbeddings of RobertsonWalker type metrics as vacuum extremals and the general finding is that the cosmological with supercritical and critical mass density have finite duration after which the mass density becomes infinite: this period of course ends before this. The interpretation would be in terms of the emergence of new spacetime sheet at which matter represented by smaller spacetime sheets suffers topological condensation. The only parameter characterizing critical cosmologies is their duration. Critical (overcritical) cosmologies having SO3× E^{3} (SO(4)) as isometry group is the duration and the CP_{2} projection at homologically trivial geodesic sphere S^{2}: the condition that the contribution from S^{2} to g_{rr} component transforms hyperbolic 3metric to that of E^{3} or S^{3} metric fixes these cosmologies almost completely. Subcritical cosmologies have onedimensional CP_{2} projection. Do RobertsonWalker cosmologies have minimal surface representatives? Recall that minimal surface equations read as D_{α}(g^{αβ} ∂_{β}h^{k}g^{1/2})= ∂_{α}[g^{αβ} ∂_{β}h^{k} g^{1/2}] + {_{α}^{k}_{m}} g^{αβ} ∂_{β}h^{m} g^{1/2}=0 , {_{α}^{k}_{m}} ={_{l} ^{k}_{m}} ∂_{α}h^{l} . Subcritical minimal surface cosmologies would correspond to X^{4}⊂ M^{4}× S^{1}. The natural coordinates are RobertsonWalker coordinates, which coincide with lightcone coordinates (a=[(m^{0})^{2}r^{2}_{M}]^{1/2}, r= r_{M}/a,θ, φ) for lightcone M^{4}_{+}. They are related to spherical Minkowski coordinates (m^{0},r_{M},θ,φ) by (m^{0}=a(1+r^{2})^{1/2}, r_{M}= ar). β =r_{M}/m^{0}=r/(1+r^{2})^{1/20},r_{M}). r corresponds to the Lorentz factor r= γ β=β/(1β^{2})^{1/2} The metric of M^{4}_{+} is given by the diagonal form [g_{aa}=1, g_{rr}=a^{2}/(1+r^{2}), g_{θθ}= a^{2}r^{2}, g_{φφ}= a^{2}r^{2}sin^{2}(θ)]. One can use the coordinates of M^{4}_{+} also for X^{4}. The ansatz for the minimal surface reads is Φ= f(a). For f(a)=constant one obtains just the flat M^{4}_{+}. In nontrivial case one has g_{aa}= 1R^{2} (df/da)^{2}. The g^{aa} component of the metric becomes now g^{aa}=1/(1R^{2}(df/da)^{2}). Metric determinant is scaled by g_{aa}^{1/2} =1 → (1R^{2}(df/da)^{2}^{1/2}. Otherwise the field equations are same as for M^{4}_{+}. Little calculation shows that they are not satisfied unless one as g_{aa}=1. Also the minimal surface imbeddings of critical and overcritical cosmologies are impossible. The reason is that the criticality alone fixes these cosmologies almost uniquely and this is too much for allowing minimal surface property. Thus one can have only the trivial cosmology M^{4}_{+} carrying dark energy density as a minimal surface solution! This obviously raises several questions.
See the new chapter Can one apply Occam's razor as a general purpose debunking argument to TGD? or article with the same title. 
LIGO blackhole anomaly and minimal surface model for star
The TGD inspired model of star as a minimal surface with stationary spherically symmetric metric suggests strongly that the analog of blackhole metric as two horizons. The outer horizon is analogous to Scwartschild horizon in the sense that the roles of time coordinate and radial coordinate change. Radial metric component vanishes at Scwartschild horizon rather than divergence. Below the inner horizon the metric has Eucldian signature. Is there any empirical evidence for the existence of two horizons? There is evidence that the formation of the recently found LIGO blackhole (discussed from TGD view point in is not fully consistent with the GRT based model (see this). There are some indications that LIGO blackhole has a boundary layer such that the gravitational radiation is reflected forth and back between the inner and outer boundaries of the layer. In the proposed model the upper boundary would not be totally reflecting so that gravitational radiation leaks out and gave rise to echoes at times .1 sec, .2 sec, and .3 sec. It is perhaps worth of noticied that time scale .1 sec corresponds to the secondary padic time scale of electron (characterized by Mersenne prime M_{127}= 2^{127}1). If the minimal surface solution indeed has two horizons and a layer like structure between them, one might at least see the trouble of killing the idea that it could give rise to repeated reflections of gravitational radiation. The proposed model (see this) assumes that the inner horizon is Schwarstchild horizon. TGD would however suggests that the outer horizon is the TGD counterpart of Schwartschild horizon. It could have different radius since it would not be a singularity of g_{rr} (g_{tt}/g_{rr} would be finite at r_{S} which need not be r_{S}=2GM now). At r_{S} the tangent space of the spacetime surface would become effectively 2dimensional: could this be interpreted in terms of strong holography (SH)? One should understand why it takes rather long time T=.1 seconds for radiation to travel forth and back the distance L= r_{S}r_{E} between the horizons. The maximal signal velocity is reduced for the lightlike geodesics of the spacetime surface but the reduction should be rather large for L∼ 20 km (say). The effective lightvelocity is measured by the coordinate time Δ t= Δ m^{0}+ h(r_{S})h(r_{E}) needed to travel the distance from r_{E} to r_{S}. The Minkowski time Δ m^{0}_{+} would be the from null geodesic property and m^{0}= t+ h(r) Δ m^{0}_{+} =Δ t h(r_{S})+h(r_{E}) , Δ t = ∫_{rE}^{rS}(g_{rr}/g_{tt})^{1/2} dr== ∫_{rE}^{rS} dr/c_{#} . The time needed to travel forth and back does not depend on h and would be given by Δ m^{0} =2Δ t =2∫_{rE}^{rS}dr/c_{#} . This time cannot be shorter than the minimal time (r_{S}r_{E})/c along lightlike geodesic of M^{4} since lightlike geodesics at spacetime surface are in general timelike curves in M^{4}. Since .1 sec corresponds to about 3× 10^{4} km, the average value of c_{#} should be for L= 20 km (just a rough guess) of order c_{#}∼ 2^{11}c in the interval [r_{E},r_{S}]. As noticed, T=.1 sec is also the secondary padic time assignable to electron labelled by the Mersenne prime M_{127}. Since g_{rr} vanishes at r_{E} one has c_{#}→ ∞. c_{#} is finite at r_{S}. There is an intriguing connection with the notion of gravitational Planck constant. The formula for gravitational Planck constant given by h_{gr}= GMm/v_{0} characterizing the magnetic bodies topologically for mass m topologically condensed at gravitational magnetic flux tube emanating from large mass M. The interpretation of the velocity parameter v_{0} has remained open. Could v_{0} correspond to the average value of c_{#}? For inner planets one has v_{0}≈ 2^{11} so that the the order of magnitude is same as for the the estimate for c_{#}. See the new chapter Can one apply Occam's razor as a general purpose debunking argument to TGD? or article with the same title. 
Minimal surface counterpart of ReissnerNordstöm solution
Occarm's razor have been used to debunk TGD. The following arguments provide the information needed by the reader to decide himself. Considerations at three levels. The level of "world of classical worlds" (WCW) defined by the space of 3surfaces endowed with Kähler structure and spinor structure and with the identification of WCW space spinor fields as quantum states of the Universe: this is nothing but Einstein's geometrization program applied to quantum theory. Second level is spacetime level. Spacetime surfaces correspond to preferred extremals of Käction in M^{4}× CP_{2}. The number of field like variables is 4 corresponding to 4 dynamically independent imbedding space coordinates. Classical gauge fields and gravitational field emerge from the dynamics of 4surfaces. Strong form of holography reduces this dynamics to the data given at string world sheets and partonic 2surfaces and preferred extremals are minimal surface extremals of Kähler action so that the classical dynamics in spacetime interior does not depend on coupling constants at all which are visible via boundary conditions only. Continuous coupling constant evolution is replaced with a sequence of phase transitions between phases labelled by critical values of coupling constants: loop corrections vanish in given phase. Induced spinor fields are localized at string world sheets to guarantee welldefinedness of em charge. At imbedding space level the modes of imbedding space spinor fields define ground states of supersymplectic representations and appear in QFTGRT limit. GRT involves postNewtonian approximation involving the notion of gravitational force. In TGD framework the Newtonian force correspond to a genuine force at imbedding space level. For background see the chapter Can one apply Occam's razor as a general purpose debunking argument to TGD?. 
How to build TGD spacetime from legos?
TGD predicts shocking simplicity of both quantal and classical dynamics at spacetime level. Could one imagine a construction of more complex geometric objects from basic building bricks  spacetime legos? Let us list the basic ideas.
What could be the simplest surfaces of this kind  legos?
Geodesic minimal surfaces with vanishing induced gauge fields Consider first static objects with 1D CP_{2} projection having thus vanishing induced gauge fields. These objects are of form M^{1}× X^{3}, X^{3}⊂ E^{3}× CP_{2}. M^{1} corresponds to timelike or possible lightlike geodesic (for CP_{2} type extremals). I will consider mostly Minkowskian spacetime regions in the following.
What about minimal surfaces and geodesic submanifolds carrying nonvanishing gauge fields  in particular em field (Kähler form identifiable as U(1) gauge field for weak hypercharge vanishes and thus also its contribution to em field)? Now one must use 2D geodesic spheres of CP_{2} combined with 1D geodesic lines of E^{2}. Actually both homologically nontrivial resp. trivial geodesic spheres S^{2}_{I} resp. S^{2}_{II} can be used so that also nonvanishing Kähler forms are obtained. The basic legos are now D× S^{2}_{i}, i=I,II and they can be combined with the basic legos constructed above. These legos correspond to two kinds of magnetic flux tubes in the ideal infinitely thin limit. There are good reasons to expected that these infinitely thin flux tubes can be thickened by deforming them in E^{3} directions orthogonal to D. These structures could be used as basic building bricks assignable to the edges of the tensor networks in TGD. Static minimal surfaces, which are not geodesic submanifolds One can consider also more complex static basic building bricks by allowing bricks which are not anymore geodesic submanifolds. The simplest static minimal surfaces are form M^{1}× X^{2}× S^{1}, S^{1} ⊂ CP_{2} a geodesic line and X^{2} minimal surface in E^{3}. Could these structures represent higher level of selforganization emerging in living systems? Could the flexible network formed by living cells correspond to a structure involving more general minimal surfaces  also nonstatic ones  as basic building bricks? The Wikipedia article about minimal surfaces in E^{3} suggests the role of minimal surface for instance in biochemistry (see this). The surfaces with constant positive curvature do not allow imbedding as minimal surfaces in E^{3}. Corals provide an example of surface consisting of pieces of 2D hyperbolic space H^{2} immersed in E^{3} (see this). Minimal surfaces have negative curvature as also H^{2} but minimal surface immersions of H^{2} do not exist. Note that pieces of H^{2} have natural imbedding to E^{3} realized as lightone proper time constant surface but this is not a solution to the problem. Does this mean that the proposal fails?
Dynamical minimal surfaces: how spacetime manages to engineer itself? At even higher level of selforganization emerge dynamical minimal surfaces. Here string world sheets as minimal surfaces represent basic example about a building block of type X^{2}× S^{2}_{i}. As a matter fact, S^{2} can be replaced with complex submanifold of CP_{2}. One can also ask about how to perform this building process. Also massless extremals (MEs) representing TGD view about topologically quantized classical radiation fields are minimal surfaces but now the induced Kähler form is nonvanishing. MEs can be also Lagrangian surfaces and seem to play fundamental role in morphogenesis and morphostasis as a generalization of Chladni mechanism. One might say that they represent the tools to assign material and magnetic flux tube structures at the nodal surfaces of MEs. MEs are the tools of spacetime engineering. Here manysheetedness is essential for having the TGD counterparts of standing waves. For background see the chapter Can one apply Occam's razor as a general purpose debunking argument to TGD?. 
Can one apply Occam's razor as a general purpose debunking argument to TGD?Occarm's razor have been used to debunk TGD. The following arguments provide the information needed by the reader to decide himself. Considerations at three levels. The level of "world of classical worlds" (WCW) defined by the space of 3surfaces endowed with Kählerstructure and spinor structure and with the identification of WCW space spinor fields as quantum states of the Universe: this is nothing but Einstein's geometrization program applied to quantum theory. Second level is spacetime level. Spacetime surfaces correspond to preferred extremals of Kähler action in M^{4}× CP_{2}. The number of field like variables is 4 corresponding to 4 dynamically independent imbedding space coordinates. Classical gauge fields and gravitational field emerge from the dynamics of 4surfaces. Strong form of holography reduces this dynamics to the data given at string world sheets and partonic 2surfaces and preferred extremals are minimal surface extremals ofKähler action so that the classical dynamics in spacetime interior does not depend on coupling constants at all which are visible via boundary conditions only. Continuous coupling constant evolution is replaced with a sequence of phase transitions between phases labelled by critical values of coupling constants: loop corrections vanish in given phase. Induced spinor fields are localized at string world sheets to guarantee welldefinedness of em charge. At imbedding space level the modes of imbedding space spinor fields define ground states of supersymplectic representations and appear in QFTGRT limit. GRT involves postNewtonian approximation involving the notion of gravitational force. In TGD framework the Newtonian force correspond to a genuine force at imbedding space level. For background see the chapter Can one apply Occam's razor as a general purpose debunking argument to TGD?. 
Delicacies of the induced spinor structure and SUSY mysteryThe discussion of induced spinor structure leads to a modification of an earlier idea (one of the many) about how SUSY could be realized in TGD in such a manner that experiments at LHC energies could not discover it and one should perform experiments at the other end of energy spectrum at energies which correspond to the thermal energy about .025 eV at room temperature. I have the feeling that this observation could be of crucial importance for understanding of SUSY. The notion of induced spinor field deserves a more detailed discussion. Consider first induced spinor structures.
From this one ends up to the possibility of identifying the counterpart of SUSY in TGD framework. There are several options to consider.

About minimal surface extremals of Kähler actionIf the spectrum for the critical value of Kähler coupling strength is complex  say given by the complex zeros of zeta  the preferred extremals of Kähler action are minimal surfaces. This means that they satisfy simultaneously the field equations associated with two variational principles. Conservation laws for the minimal surface extremals of Kähler action Consider first the basic conservation laws.
Are minimal surface extremals of Kähler action holomorphic surfaces in some sense? I have considered several ansätze for the general solutions of the field equations for the preferred extremals. One proposal is that preferred extremals as 4surfaces of imbedding space with octonionic tangent space structure have quaternionic tangent space or normal space (so called M^{8}H duality). Second proposal is that preferred extremals can be seen as quaternion analytic surfaces. Third proposal relies on a fusion of complex and hypercomplex structures to what I call HamiltonJacobi structure. In Euclidian regions this would correspond to complex structure. Twistor approach suggests that the condition that the twistor lift of the spacetime surface to a 6D surface in the product of twistor spaces of M^{4} and CP_{2} equals to the twistor space of CP_{2}. This proposal is highly interesting since twistor lift works only for M^{4}× CP_{2}. The intuitive picture is that the field equations are integrable and all these views might be consistent. Preferred extremals of Kähler action as minimal surfaces would be a further proposal. Can one make conclusions about general form of solutions assuming that one has minimal surface extremals of Kähler action? In D=2 case minimal surfaces are holomorphic surfaces or they hypercomplex variants and the imbedding space coordinates can be expressed as complexanalytic functions of complex coordinate or a hypercomplex analog of this. Field equations stating the vanishing of the trace g^{αβ}H^{k}_{αβ} if the second fundamental form H^{k}_{αβ}== D_{α}&partial;_{β}h^{k} are satisfied because the metric is tensor of type (1,1) and second fundamental form of type (2,0) ⊕ (2,0). Field equations reduce to an algebraic identity and functions involved are otherwise arbitrary functions. The constraint comes from the condition that metric is of form (1,1) as holomorphic tensor. This raises the question whether this finding generalizes to the level of 4D spacetime surfaces and perhaps allows to solve the field equations exactly in coordinates generalizing the hypercomplex coordinates for string world sheet and complex coordinates for the partonic 2surface. The known nonvacuum extremals of Kähler action are actually minimal surfaces. The common feature suggested already earlier to be common for all preferred extremals is the existence of generalization of complex structure.

Does cosmological term in twistor action makes Kähler coupling genuine coupling parameter also classically?The addition of the volume term to Kähler action has very nice interpretation as a generalization of equations of motion for a worldline extended to a 4D spacetime surface. The field equations generalize in the same manner for 3D lightlike surfaces at which the signature of the induced metric changes from Minkowskian to Euclidian, for 2D string world sheets, and for their 1D boundaries defining world lines at the lightlike 3surfaces. For 3D lightlike surfaces the volume term is absent. Either lightlike 3surface is freely choosable in which case one would have KacMoody symmetry as gauge symmetry or that the extremal property for ChernSimons term fixes the gauge. The known nonvacuum extremals are minimal surface extremals of Kähler action and it might well be that the preferred extremal property realizing SH quite generally demands this. The addition of the volume term could however make Kähler coupling strength a manifest coupling parameter also classically when the phases of Λ and α_{K} are same. Therefore quantum criticality for Λ and α_{K} would have a precise local meaning also classically in the interior of spacetime surface. The equations of motion for a world line of U(1) charged particle would generalize to field equations for a "world line" of 3D extended particle. This is an attractive idea consistent with standard wisdom but one can invent strong objections against it in TGD framework.
Consider a typical particle physics experiment. There are incoming and outgoing free particles moving along geodesics, these particles interact, and emanate as free particles from the interaction volume. This phenomenological picture does not follow from quantum field theory but is put in by hand, in particular the idea about interaction couplings becoming nonzero is involved. Also the role of the observer remains poorly understood. The motion of incoming and outgoing particles is analogous to free motion along geodesic lines with particles generalized to 3D extended objects. For both options these would correspond to the preferred extremals in the complement of CD within larger CD representing observer or measurement instrument. Decoupling would take place. In the interaction volume interactions are "coupled on" and particles interact inside the volume characterized by causal diamond (CD). What could be the TGD view translation of this picture?
What happens within subCD could be fundamental for the understanding of directed attention and sensorymotor cycle.
The free geodesic line dynamics with vanishing U(1) Kähler force indeed brings in mind the proposed generalization of Chladni mechanism generating nodal surfaces at which charged magnetic flux tubes are driven (see this).
For background see the new chapter About twistor lift of TGD or the article with the same title.

The twistor lift of classical TGD is attractive physically but it is still unclear whether it satisfies all constraints. The basic implication of twistor lift would be the understanding of gravitational and cosmological constants. Cosmological constant removes the infinite vacuum degeneracy of Kähler action but because of the extreme smallness of cosmological constant Λ playing the role of inverse of gauge coupling strength, the situation for nearly vacuum extremals of Kähler action in the recent cosmology is nonperturbative. Cosmological constant and thus twistor lift make sense only in zero energy ontology (ZEO) involving causal diamonds (CDs) in an essential manner.
One motivation for introducing the hierarchy of Planck constants was that the phase transition increasing Planck constant makes possible perturbation theory in strongly interacting system. Nature itself would take care about the converge of the perturbation theory by scaling Kähler coupling strength α_{K} to α_{K}/n, n=h_{eff}/h. This hierarchy might allow to construct gravitational perturbation theory as has been proposed already earlier. This would for gravitation to be quantum coherent in astrophysical and even cosmological scales.
In this chapter this picture is studied in detail. The first interesting finding is that allowing Kähler coupling strength α_{K} to correspond to zeros of zeta implies that for complex zeros the preferred extremals are mimimal surface extremals of Kähler action so that the values of coupling constants do not matter. The dynamics of Kägler action and volume term couple only for real zeros. This leads to an interpretation with profound implications for the views about what happens in particle physics experiment and in quantum measurement, for consciousness theory and for quantum biology. Second observation is that a fundamental length scale of biology  size scale of neuron and axon  would correspond to the padic length scale assignable to vacuum energy density assignable to cosmological constant and be therefore a fundamental physics length scale.
See the new chapter About twistor lift of TGD? or the article with the same title.
Still about induced spinor fields and TGD counterpart for HiggsThe understanding of the modified Dirac equation and of the possible classical counterpart of Higgs field in TGD framework is not completely satisfactory. The emergence of twistor lift of Kähler action inspired a fresh approach to the problem and it turned out that a very nice understanding of the situation emerges. More precise formulation of the Dirac equation for the induced spinor fields is the first challenge. The welldefinedness of em charge has turned out to be very powerful guideline in the understanding of the details of fermionic dynamics. Although induced spinor fields have also a part assignable spacetime interior, the spinor modes at string world sheets determine the fermionic dynamics in accordance with strong form of holography (SH). The welldefinedness of em charged is guaranteed if induced spinors are associated with 2D string world sheets with vanishing classical W boson fields. It turned out that an alternative manner to satisfy the condition is to assume that induced spinors at the boundaries of string world sheets are neutrinolike and that these string world sheets carry only classical W fields. Dirac action contains 4D interior term and 2D term assignable to string world sheets. Strong form of holography (SH) allows to interpret 4D spinor modes as continuations of those assignable to string world sheets so that spinors at 2D string world sheets determine quantum dynamics. Twistor lift combined with this picture allows to formulate the Dirac action in more detail. Welldefinedness of em charge implies that charged particles are associated with string world sheets assignable to the magnetic flux tubes assignable to homologically nontrivial geodesic sphere and neutrinos with those associated with homologically trivial geodesic sphere. This explains why neutrinos are so light and why dark energy density corresponds to neutrino mass scale, and provides also a new insight about color confinement. A further important result is that the formalism works only for imbedding space dimension D=8. This is due the fact that the number of vector components is the same as the number of spinor components of fixed chirality for D=8 and corresponds directly to the octonionic triality. pAdic thermodynamics predicts elementary particle masses in excellent accuracy without Higgs vacuum expectation: the problem is to understand fermionic Higgs couplings. The observation that CP_{2} part of the modified gamma matrices gives rise to a term mixing M^{4} chiralities contain derivative allows to understand the massproportionality of the Higgsfermion couplings at QFT limit. See the chapter Higgs or something else?.

Does GRT really allow gravitational radiation?In Facebook discussion Niklas Grebäck mentioned Weyl tensor and I learned something that I should have noticed long time ago. Wikipedia article lists the basic properties of Weyl tensor as the traceless part of curvature tensor, call it R. Weyl tensor C is vanishing for conformally flat spacetimes. In dimensions D=2,3 Weyl tensor vanishes identically so that they are always conformally flat: this obviously makes the dimension D=3 for space very special. Interestingly, one can have nonflat spacetimes with nonvanishing Weyl tensor but the vanishing Schouten/Ricci/Einstein tensor and thus also with vanishing energy momentum tensor. The rest of curvature tensor R can be expressed in terms of so called KulkarniNomizu product P• g of Schouten tensor P and metric tensor g: R=C+P• g, which can be also transformed to a definition of Weyl tensor using the definition of curvature tensor in terms of Christoffel symbols as the fundamental definition. KulkarniNomizu product • is defined as tensor product of two 2tensors with symmetrization with respect to first and second index pairs plus antisymmetrization with respect to second and fourth indices. Schouten tensor P is expressible as a combination of Ricci tensor Ric defined by the trace of R with respect to the first two indices and metric tensor g multiplied by curvature scalar s (rather than R in order to use index free notation without confusion with the curvature tensor). The expression reads as P= 1/(D2)×[Ric(s/2(D1))×g] . Note that the coefficients of Ric and g differ from those for Einstein tensor. Ricci tensor and Einstein tensor are proportional to energy momentum tensor by Einstein equations relate to the part. Weyl tensor is assigned with gravitational radiation in GRT. What I see as a serious interpretational problem is that by Einstein's equations gravitational radiation would carry no energy and momentum in absence of matter. One could argue that there are no free gravitons in GRT if this interpretation is adopted! This could be seen as a further argument against GRT besides the problems with the notions of energy and momentum: I had not realized this earlier. Interestingly, in TGD framework so called massless extremals (MEs) (see this and this) are foursurfaces, which are extremals of Kähler action, have Weyl tensor equal to curvature tensor and therefore would have interpretation in terms of gravitons. Now these extremals are however nonvacuum extremals.
See the article Does GRT really allow gravitational radiation?. For background see the chapter From Principles to Diagrams. 
What happens to the extremals of Kähler action when volume term is introduced?What happens to the extremals of Kähler action when volume term is introduced?
Consider next vacuum extremals, which have vanishing induced Kähler form and are thus have CP_{2} projection belonging to at most 2D Lagrangian manifold of CP_{2}.

Eigenstates of Yangian coalgebra generators as a manner to generate maximal entanglement?Negentropically entangled objects are key entities in TGD inspired theory of consciousness and in the construction of tensor networks and the challenge is to understand how these could be constructed and what their properties could be. These states are diametrically opposite to unentangled eigenstates of single particle operators, usually elements of Cartan algebra of symmetry group. The entangled states should result as eigenstates of polylocal operators. Yangian algebras involve a hierarchy of polylocal operators, and twistorial considerations inspire the conjecture that Yangian counterparts of supersymplectic and other algebras made polylocal with respect to partonic 2surfaces or endpoints of boundaries of string world sheet at them are symmetries of quantum TGD. Could Yangians allow to understand maximal entanglement in terms of symmetries?
For details see the chapter From Principles to Diagrams or the article with the same title. 
Could N=2 superconformal algebra be relevant for TGD?The concrete realization of the superconformal symmetry (SCS) in TGD framework has remained poorly understood. In particular, the question how SCS relates to superconformal field theories (SCFTs) has remained an open question. The most general superconformal algebra assignable to string world sheets by strong form of holography has N equal to the number of 4+4 =8 spin states of leptonic and quark type fundamental spinors but the spacetime SUSY is badly broken for it. Covariant constancy of the generating spinor modes is replaced with holomorphy  kind of "half covariant constancy". I have considered earlier a proposal that N=4 SCA could be realized in TGD framework but given up this idea. Righthanded neutrino and antineutrino are excellent candidates for generating N=2 SCS with a minimal breaking of the corresponding spacetime SUSY. Covariant constant neutrino is an excellent candidate for the generator of N=2 SCS. The possibility of this SCS in TGD framework will be considered in the sequel. 1. Questions about SCS in TGD framework This work was inspired by questions not related to N=2 SCS, and it is good to consider first these questions. 1. 1 Could the superconformal generators have conformal weights given by poles of fermionic zeta? The conjecture (see this) is that the conformal weights for the generators supersymplectic representation correspond to the negatives of h= ks_{k} of the poles s_{k} fermionic partition function ζ_{F}(ks)=ζ(ks)/ζ(2ks) defining fermionic partition function. Here k is constant, whose value must be fixed from the condition that the spectrum is physical. ζ(ks) defines bosonic partition function for particles whos energies are given by log(p), p prime. These partition functions require complex temperature but is completely sensible in Zero Energy Ontology (ZEO), where thermodynamics is replaced with its complex square root. For nontrivial zeros 2ks=1/2+iy of ζ(2ks) s would correspond pole s= (1/2+iy)/2k of z_{F}(ks). The corresponding conformal weights would be h=(1/2iy)/2k. For trivial zeros 2ks=2n, n=1,2,.. s=n/k would correspond to conformal weights h=n/k>0. Conformal confinement is assumed meaning that the sum of imaginary parts of of generators creating the state vanishes. What can one say about the value of k? The pole of ζ(ks) at s=1/k would correspond to pole and conformal weight h=1/k. For k=1 the trivial conformal weights would be positive integers h=1,2,...: this certainly makes sense. This gives for the real part for nontrivial conformal weights h=1/4. By conformal confinement both pole and its conjugate belong to the state so that this contribution to conformal weight is negative half integers: this is consistent with the facts about superconformal representations. For the ground state of superconformal representation the conformal weight for conformally confined state would be h= K/2. In padic mass calculations one would have K=6 (see this) . The negative ground state conformal weights of particles look strange but padic mass calculations require that the ground state conformal weights of particles are negative: h=3 is required. 1.2 What could be the origin of negative ground state conformal weights? Supersymplectic conformal symmetries are realized at lightcone boundary and various Hamiltonians defined analogs of KacMoody generators are proportional functions f(r_{M})H_{J,m} H_{A}, where H_{J,m} correspond to spherical harmonics at the 2sphere R_{M}=constant and H_{A} is color partial wave in CP_{2}, f(r_{M}) is a partial wave in radial lightlike coordinate which is eigenstate of scaling operator L_{0}=r_{M}d/dR_{M} and has the form (r_{M}/r_{0})^{h}, where h is conformal weight which must be of form h=1/2+iy. To get plane wave normalization for the amplitudes (r_{M}/r_{0})^{ h}=(r_{M}/r_{0})^{1/2}exp(iyx) , x=log(r_{M}/r_{0}) , one must assume h=1/2+iy. Together with the invariant integration measure dr_{M} this gives for the inner product of two conformal plane waves exp(iy_{i}x), x=log(r_{M}/r_{0}) the desired expression ∫ exp[iy_{1}y_{2})x] dx= δ(y_{1}y_{2}), where dx= dr_{M}/r_{M} is scaling invariance integration measure. This is just the usual inner product of plane waves labelled by momenta y_{i}. If r_{M}/r_{0} can be identified as a coordinate along fermionic string (this need not be always the case) one can interpret it as real or imaginary part of a hypercomplex coordinate at string world sheet and continue these wave functions to the entire string world sheets. This would be very elegant realization of conformal invariance. 1.3. How to relate degenerate representations with h>0 to the massless states constructed from tachyonic ground states with negative conformal weight? This realization would however suggest that there must be also an interpretation in which ground states with negative conformal weight h_{vac}=k/2 are replaced with ground states having vanishing conformal weights h_{vac}=0 as in minimal SCAs and what is regarded as massless states have conformal weights h= h_{vac}>0 of the lowest physical state in minimal SCAs. One could indeed start directly from the scaling invariant measure dr_{M}/r_{M} rather than allowing it to emerge from dr_{M}. This would require in the case of padic mass calculations that has representations satisfying Virasoro conditions for weight h=h_{vac}>0. pAdic mass squared would be now shifted downwards and proportional to L_{0}+h_{vac}. There seems to be no fundamental reason preventing this interpretation. One can also modify scaling generator L_{0} by an additive constant term and this does not affect the value of c. This operation corresponds to replacing basis {z^{n}} with basis {z^{n+1/2}}. What makes this interpretation worth of discussing is that the entire machinery of conformal field theories with nonvanishing central charge and nonvanishing but positive ground state conformal weight becomes accessible allowing to determine not only the spectrum for these theories but also to determine the partition functions and even to construct npoint functions in turn serving as basic building bricks of Smatrix elements (see this) . ADE classification of these CFTs in turn suggests at connection with the inclusions of hyperfinite factors and hierarchy of Planck constants. The fractal hierarchy of broken conformal symmetries with subalgebra defining gauge algebra isomorphic to entire algebra would give rise to dynamic symmetries and inclusions for HFFs suggest that ADE groups define KacMoody type symmetry algebras for the nongauge part of the symmetry algebra. 2. Questions about N=2 SCS N=2 SCFTs has some inherent problems. For instance, it has been claimed that they reduce to topological QFTs. Whether N=2 can be applied in TGD framework is questionable: they have critical spacetime dimension D=4 but since the required metric signature of spacetime is wrong. 2.1 Inherent problems of N=2 SCS N=2 SCS has some severe inherent problems.
2.2 Can one really apply N=2 SCFTs to TGD? TGD version of SCA is gigantic as compared to the ordinary SCA. This SCA involves supersymplectic algebra associated with metrically 2dimensional lightcone boundary (lightlike boundaries of causal diamonds) and the corresponding extended conformal algebra (lightlike boundary is metrically sphere S^{2}). Both these algebras have conformal structure with respect to the lightlike radial coordinate r_{M} and conformal algebra also with respect to the complex coordinate of S^{2}. Symplectic algebra replaces finitedimensional Lie algebra as the analog of KacMoody algebra. Also lightlike orbits of partonic 2surfaces possess this SCA but now KacMoody algebra is defined by isometries of imbedding space. String world sheets possess an ordinary SCA assignable to isometries of the imbedding space. An attractive interpretation is that r_{M} at lightcone boundary corresponds to a coordinate along fermionic string extendable to a hypercomplex coordinate at string world sheet. N=8 SCS seems to be the most natural candidate for SCS behind TGD: all fermion spin states would correspond to generators of this symmetry. Since the modes generating the symmetry are however only halfcovariantly constant (holomorphic) this SUSY is badly broken at spacetime level and the minimal breaking occurs for N=2 SCS generated by righthanded neutrino and antineutrino. The key motivation for the application of minimal N=2 SCFTs to TGD is that SCAs for them have a nonvanishing central charge c and vacuum weight h≥ 0 and the degenerate character of ground state allows to deduce differential equations for npoint functions so that these theories are exactly solvable. It would be extremely nice is scattering amplitudes were basically determined by npoint functions for minimal SCFTs. A further motivation comes from the following insight. ADE classification of N=2 SCFTs is extremely powerful result and there is connection with the hierarchy of inclusions of hyperfinite factors of type II_{1}, which is central for quantum TGD. The hierarchy of Planck constants assignable to the hierarchy of isomorphic subalgebras of the supersymplectic and related algebras suggest interpretation in terms of ADE hierarchy a rather detailed view about a hierarchy of conformal field theories and even the identification of primary fields in terms of critical deformations. The application N=2 SCFTs in TGD framework can be however challenged. The problem caused by the negative value of vacuum conformal weight has been already discussed but there are also other problems.
For details see the new chapter Could N=2 SuperConformal Algebra Be Relevant For TGD? or the article with the same title. 
Tensor Networks and SmatricesThe concrete construction of scattering amplitudes has been the toughest challenge of TGD and the slow progress has occurred by identification of general principles with many side tracks. One of the key problems has been unitarity. The intuitive expectation is that unitarity should reduce to a local notion somewhat like classical field equations reduce the time evolution to a local variational principle. The presence of propagators have been however the the obstacle for locally realized unitarity in which each vertex would correspond to unitary map in some sense. TGD suggests two approaches to the construction of Smatrix.
Objections It is certainly clear from the beginning that the possibly existing description of Smatrix in terms of tensor networks cannot correspond to the perturbative QFT description in terms of Feynman diagrams.
The overly optimistic vision With these prerequisites one can follow the optimistic strategy and ask how tensor networks could allow to generalize the notion of unitary Smatrix in TGD framework.
For the details see the new chapter From Principles to Diagrams or the article with the same title. 
Twistor googly problem transforms from a curse to blessing in TGD frameworkThere was a nice story with title "Michael Atiyahâ€™s Imaginative State of Mind" about mathematician Michael Atyiah in Quanta Magazine. The works of Atyiah have contributed a lot to the development of theoretical physics. What was pleasant to hear that Atyiah belongs to those scientists who do not care what others think. As he tells, he can afford this since he has got all possible prices. This is consoling and encouraging even for those who have not cared what others think and for this reason have not earned any prizes. Nor even a single coin from what they have been busily doing their whole lifetime! In the beginning of the story "twistor googly problem" was mentioned. I had to refresh my understanding about googly problem. In twistorial description the modes of massless fields (rather than entire massless fields) in spacetime are lifted to the modes in its 6D twistorspace and dynamics reduces to holomorphy. The analog of this takes place also in string models by conformal invariance and in TGD by its extension. One however encounters googly problem: one can have twistorial description for circular polarizations with welldefined helicity +1/1 but not for general polarization states  say linear polarizations, which are superposition of circular polarizations. This reflects itself in the construction of twistorial amplitudes in twistor Grassmann program for gauge fields but rather implicitly: the amplitudes are constructed only for fixed helicity states of scattered particles. For gravitons the situation gets really bad because of nonlinearity. Mathematically the most elegant solution would be to have only +1 or 1 helicity but not their superpositions implying very strong parity breaking and chirality selection. Parity parity breaking occurs in physics but is very small and linear polarizations are certainly possible! The discusion of Penrose with Atyiah has inspired a possible solution to the problem known as "palatial twistor theory". Unfortunately, the article is behind paywall too high for me so that I cannot say anything about it. What happens to the googly problem in TGD framework? There is twistorialization at spacetime level and imbedding space level.
For background see the chapter From Principles to giagrams or the article From Principles to Diagrams. 
Could M^{4} Kähler form introduce new gravitational physics?The introduction of M^{4} Kähler form strongly suggested by the twistor formulation of TGD could bring in new gravitational physics.
For background see the chapter From Principles to giagrams or the article From Principles to Diagrams. 
Cosmic evolution of the radius of the fiber of the twistor space of spacetime surfaceI 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 6D 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 2D fiber could be hyperbolic sphere H^{2} (t^{2}x^{2}y^{2} =R_{H}^{2}) rather than sphere S^{2} as it is for CP_{2} with Euclidian signature of metric. I however soon realized that the infinite area of H^{2} implies that 6D 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 M^{4}= M^{2}+ E^{2} of local tangent space to longitudinal (defined by lightlike vector) and to transversal directions (polarizations orthogonal to the lightlike vector. The decomposition can depend on point but the distributions of two planes must integrated to 2D surfaces. In E^{2} one has complex structure and in M^{2} its hypercomplex variant. In M^{2} has decomposition of replacing complex numbers by hypercomplex numbers so that complex coordinate x+iy is replaced with w=t+ie, i^{2}=1 and e^{2}=1. It took time to realize I have actually carried out this generalization years ago with quite different motivations and called the resulting structure HamiltonJacobi structure! The twistor fiber is defined by projections of 4D antisymmetric tensors (in particular induced Kähler form) to the orthogonal complement of unique time direction determed by the sum of lightlike vector and its dual in M^{2}. 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 S^{2} 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 timelike coordinates) . The radii of the spheres associated with M^{4} and CP_{2} define two fundamental scales and the scaling of 6D 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 CP_{2} radius. Second option is that the radius of S^{2}(M^{4}) equals to Planck length, which would be therefore a fundamental length scale. The radius R_{D} of the 2D fiber of twistor space assignable to spacetime surfaces is dynamical. In Euclidian spacetime regions the fiber is sphere: a good guess is that its order of magnitude is determined by the winding numbers of the maps from S^{2}(X^{4})→ S^{2}(M^{4}) and S^{2}(X^{4})→ S^{2}(CP_{2}). The winding numbers (1,0) and (0,1) represent the simplest options. The question is whether one could say something nontrivial 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 6D Kähler action contains two terms.
Given the parameter L^{2} 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. 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.
For background see the chapter From Principles to giagrams or the article From Principles to Diagrams. 
Twistorial approach and connection with General RelativityFor year or two ago I ended up with a vision about how twistor approach could generalize to TGD framework. A more explicit realization of twistorialization as lifting of the preferred extremal X^{4} of Kähler action to corresponding 6D twistor space X^{6} identified as surface in the 12D product of twistor spaces of M^{4} and CP_{2} allowing Kähler structure suggests itself: this makes these spaces completely unique twistorially and seems more or less obvious that the Kähler structure must have profound physical meaning. It turned out that it has: the projection of Kähler form defines the representation of preferred quaternionic imaginary unit needed to assign twistor structure to spacetime surface. Almost equally obvious idea is that the lifting of the dynamics for spacetime surface to that for its twistor space in the product of twistor spaces of M^{4} and CP_{2} must be based on 6D Kähler action. Contrary to the original expectations, the twistorial approach is not mere reformulation but leads to a first principle identification of cosmological constant and perhaps also of gravitational constant and to a modification of the dynamics of Kähler action however preserving the known extremals and basic properties of Kähler action and allowing to interpret induced Kähler form in terms of preferred imaginary unit defining twistor structure. There are some new results forcing a profound modification of the recent view about TGD but consistent with the general picture. A more explicit realization of twistorialization as lifting of the preferred extremal X^{4} of Kähler action to corresponding 6D twistor space X^{6} identified as surface in the 12D product of twistor spaces of M^{4} and CP_{2} allowing Kähler structure suggests itself. The action principle in 6D context is also Kähler action, which dimensionally reduces to Kähler action plus cosmological term. This brings in the radii of spheres S^{2} associated with the twistor space of CP_{2} presumably determined by CP_{2} radius and radius of S^{2} associated with M^{4} twistor space for which an attractive identification is as Planck length, which would be now purely classical parameter. The radius of S^{2} associated with spacetime surface is determined by induced metric and is emergent length scale. The normalization of 6D Kähler action by a scale factor with dimension which is inverse length squared brings in a further length scale closely related to cosmological constant which is also dynamical and has correct sign to explain accelerated expansion of the Universe. The dimensionally reduced dynamics is a highly nontrivial modification of the dynamics of Kähler action however preserving the known extremals and basic properties of Kähler action and allowing to interpret induced Kähler form in terms of preferred imaginary unit defining twistor structure. In the sequel I will discuss the recent understanding of twistorizalization, which is considerably improved from that in the earlier formulation. I formulate the dimensional reduction of 6D Kähler action and consider the physical interpretation. After that I proceed to discuss the basic principles behind the recent view about scattering amplitudes as generalized Feynman diagrams. 1. Some mathematical background First I will try to clarify the mathematical details related to the twistor spaces and how they emerge in the recent context. I do not regard myself as a mathematician in technical sense and I can only hope that the representation based on physical intuition does not contain serious mistakes. 1.1. Imbedding space is twistorially unique It took roughly 36 years to learn that M^{4} and CP_{2} are twistorially unique. Spacetimes are surfaces in H=M^{4}× CP_{2}. M^{4} and CP_{2} are unique 4manifolds in the sense that both allow twistor space with Kähler structure: Kähler structure is the crucial concept. Strictly speaking, M^{4} and its Euclidian variant E^{4} allow both twistor space and the twistor space of M^{4} is Minkowskian variant T(M^{4})= SU(2,2)/SU(2,1)× U(1) of 6D twistor space CP_{3}= SU(4)/SU(3)× U(1) of E^{4}. The twistor space of CP_{2} is 6D T(CP_{2})= SU(3)/U(1)× U(1), the space for the choices of quantization axes of color hypercharge and isospin. This leads to a proposal  just a proposal  for the formulation of TGD in which spacetime surfaces X^{4} in H are lifted to twistor spaces X^{6}, which are sphere bundles over X^{4} and such that they are surfaces in 12D product space T(M^{4})× T(CP_{2}) such the twistor structure of X^{4} are in some sense induced from that of T(M^{4})× T(CP_{2}). What is nice in this formulation is that one can use all the machinery of algebraic geometry so powerful in superstring theory (CalabiYau manifolds). 1.2 What does twistor structure in Minkowskian signature really mean? What twistor structure in Minkowskian signature really means geometrically has remained a confusing question for me. The problems associated with the Minkowskian signature of the metric are encountered also in twistor Grassmann approach to scattering amplitudes but are circumvented by performing Wick rotation that is using E^{4} or S^{4} instead of M^{4} and applying algebraic continuation. Also complexification of Minkowksi space for momenta is used. These tricks do not apply now. Let us try to collect thoughts about what is involved.
1.3 What the induction of twistor structure could mean? To proceed one must make explicit the definition of twistor space. The 2D fiber S^{2} consists of antisymmetric tensors of X^{4} which can be taken to be selfdual or antiselfdual by taking any antisymmetric form and by adding to its plus/minus its dual. Each tensor of this kind defines a direction  point of S^{2}. These points can be also regarded as quaternionic imaginary units. One has a natural metric in S^{2} defined by the X^{4} inner product for antisymmetric tensors: this inner product depends on spacetime metric. Kähler action density is example of a norm defined by this inner product in the special case that the antisymmetric tensor is induced Kähler form. Induced Kähler form defines a preferred imaginary unit and is needed to define the imaginary part ω(X,Y)= ig(X,JY) of hermitian form h= h+iω. Consider now what the induction of twistor structure could mean.
2. Surprise: twistorial dynamics does not reduce to a trivial reformulation of the dynamics of Kähler action I have thought that twistorialization classically means only an alternative formulation of TGD. This is definitely not the case as the explicit study demonstrated. Twistor formulation of TGD is in terms of of 6D twistor spaces T(X^{4}) of spacetime surfaces X^{4}⊂ M^{4}× CP_{2} in 12dimensional product T=T(M^{4})× T(CP_{2}) of 6D twistor spaces of T(M^{4}) of M^{4} and T(CP_{2}) of CP_{2}. The induced Kähler form in X^{4} defines the quaternionic imaginary unit defining twistor structure: how stupid that I realized it only now! I experienced during single night many other "How stupid I have been" experiences. Classical dynamics is determined by 6D variant of Kähler action with coefficient 1/L^{2} having dimensions of inverse length squared. Since twistor space is bundle, a dimensional reduction of 6D Kähler action to 4D Kähler action plus a term analogous to cosmological term  spacetime volume  takes place so that dynamics reduces to 4D dynamics also now. Here one must be careful: this happens provided the radius of F associated with X^{4} does not depend on point of X^{4}. The emergence of cosmological term was however completely unexpected: again "How stupid I have been" experience. The scales of the spheres and the condition that the 6D action is dimensionless bring in 3 fundamental length scales! 2.1 New scales emerge The twistorial dynamics gives to several new scales with rather obvious interpretation. The new fundamental constants that emerge are the radius of hyperbolic sphere associated with T(M^{4}) and of sphere associated with T(CP_{2}). The radius of the fiber associated with X^{4} is not a fundamental constant but determined by the induced metric. By above argument the fiber is sphere for Euclidian signature and hyperbolic sphere for Minkowskian signature.
Given the parameter L^{2} as it is defined above, one can deduce an expression for cosmological constant Λ and show that it is positive.
Let us consider now some quantitative estimates. R(X^{4}) depends on homotopy equivalence classes of the maps from S^{2}(X^{4})→ S^{2}(M^{4}) and S^{2}(X^{4})→ S^{2}(CP_{2})  that is winding numbers w_{i} , i=0,2 for these maps. The simplest situations correspond to the winding numbers (w_{1},w_{2})=(1,0) and (w_{1},w_{2})=(0,1) . For (w_{1},w_{2})=(1,0) M^{4} contribution to the metric of S^{2}(X^{4}) dominates and one has R(X^{4})≈ R(M^{4}) . For R(M^{4})=l_{P} so Planck length would define a fundamental length and Planck mass and Newton's constant would be quantal parameters. For (w_{1},w_{2})=(0,1) the radius of sphere would satisfy R_{D}≈ R ( CP_{2} size): now also Planck length would be quantal parameter. Consider next additional scales emerging from TGD picture.
In the case of M^{4} the radius of S^{2} cannot be fixed it remains unclear whether Planck length scale is fundamental constant or whether it emerges. 2.2 What about extremals of the dimensionally reduced 6D Kähler action? It seems that the basic wisdom about extremals of Kähler action remains unaffected and the motivations for WCW are not lost. What is new is that the removal of vacuum degeneracy is forced by twistorial action.
For background see the chapter From Principles to Diagrams or the article From Principles to Diagrams. 
From Principles to DiagramsThe generalization of twistor diagrams to TGD framework has been very inspiring (and also frightening) mission impossible and allowed to gain deep insights about what TGD diagrams could be mathematically. I of course cannot provide explicit formulas but the general structure for the construction of twistorial amplitudes in N=4 SUSY suggests an analogous construction in TGD thanks to huge symmetries of TGD and unique twistorial properties of M^{4}× CP_{2}. I try to summarize the big vision. Several guiding principles are involved and have gradually evolved to a coherent whole. The generalization of twistor diagrams to TGD framework has been very inspiring (and also frightening) mission impossible and allowed to gain deep insights about what TGD diagrams could be mathematically. I of course cannot provide explicit formulas but the general structure for the construction of twistorial amplitudes in N=4 SUSY suggests an analogous construction in TGD thanks to huge symmetries of TGD and unique twistorial properties of M^{4}× CP_{2}. I try to summarize the big vision. Several guiding principles are involved and have gradually evolved to a coherent whole. Imbedding space is twistorially unique It took roughly 36 years to learn that M^{4} and CP_{2} are twistorially unique.
Strong form of holography Strong form of holography (SH) following from general coordinate invariance (GCI) for spacetimes as surfaces states that the data assignable to string world sheets and partonic 2surfaces allows to code for scattering amplitudes. The boundaries of string world sheets at the spacelike 3surfaces defining the ends of spacetime surfaces at boundaries of causal diamonds (CDs) and the fermionic lines along lightlike orbits of partonic 2surfaces representing lines of generalized Feynman diagrams become the basic elements in the generalization of twistor diagrams (I will not use the attribute "Feynman" in precise sense, one could replace it with "twistor" or even drop away). One can assign fermionic lines massless in 8D sense to flux tubes, which can also be braided. One obtains a fractal hierarchy of braids with strands, which are braids themselves. At the lowest level one has braids for which fermionic lines are braided. This fractal hierarchy is unavoidable and means generalization of the ordinary Feynman diagram. I have considered some implications of this hierarchy (see this). The existence of WCW demands maximal symmetries Quantum TGD reduces to the construction of Kähler geometry of infiniteD "world of classical worlds" (WCW), of associated spinor structure, and of modes of WCW spinor fields which are purely classical entities and quantum jump remains the only genuinely quantal element of quantum TGD. Quantization without quantization, would Wheeler say. By its infinitedimensionality, the mere mathematical existence of the Kähler geometry of WCW requires maximal isometries. Physics is completely fixed by the mere condition that its mathematical description exists. Supersymplectic and other symmetries of WCW are in decisive role. These symmetry algebras have conformal structure and generalize and extend the conformal symmetries of string models (KacMoody algebras in particular). These symmetries give also rise to the hierarchy of Planck constants. The supersymplectic symmetries extend to a Yangian algebra, whose generators are polylocal in the sense that they involve products of generators associated with different partonic surfaces. These symmetries leave scattering amplitudes invariant. This is an immensely powerful constraint, which remains to be understood. Quantum criticality Quantum criticality (QC) of TGD Universe is a further principle. QC implies that Kähler coupling strength is mathematically analogous to critical temperature and has a discrete spectrum. Coupling constant evolution is replaced with a discrete evolution as function of padic length scale: sequence of jumps from criticality to a more refined criticality or vice versa (in spin glass energy landscape you at bottom of well containing smaller wells and you go to the bottom of smaller well). This implies that either all radiative corrections (loops) sum up to zero (QFT limit) or that diagrams containing loops correspond to the same scattering amplitude as tree diagrams so that loops can eliminated by transforming them to arbitrary small ones and snipping away moving the end points of internal lines along the lines of diagram (fundamental description). Quantum criticality at the level of superconformal symmetries leads to the hierarchy of Planck constants h_{eff}=n× h labelling a hierarchy of subalgebras of supersymplectic and other conformal algebras isomorphic to the full algebra. Physical interpretation is in terms of dark matter hierarchy. One has conformal symmetry breaking without conformal symmetry breaking as Wheeler would put it. Physics as generalized number theory, number theoretical universality Physics as generalized number theory vision has important implications. Adelic physics is one of them. Adelic physics implied by number theoretic universality (NTU) requires that physics in real and various padic numbers fields and their extensions can be obtained from the physics in their intersection corresponding to an extension of rationals. This is also enormously powerful condition and the success of padic length scale hypothesis and padic mass calculations can be understood in the adelic context. In TGD inspired theory of consciousness various padic physics serve as correlates of cognition and padic spacetime sheets can be seen as cognitive representations, "thought bubbles". NTU is closely related to SH. String world sheets and partonic 2surfaces with parameters (WCW coordinates) characterizing them in the intersection of rationals can be continued to spacetime surfaces by preferred extremal property but not always. In padic context the fact that padic integration constants depend on finite number of pinary digits makes the continuation easy but in real context this need not be possible always. It is always possible to imagine something but not always actualize it! Scattering diagrams as computations Quantum criticality as possibility to eliminate loops has a number theoretic interpretation. Generalized Feynman diagram can be interpreted as a representation of a computation connecting given set X of algebraic objects to second set Y of them (initial and final states in scattering) (trivial example: X={3,4} → 3× 4 = 12 → 2× 6 → {2,6}=Y. The 3vertices (a× b=c) and their timereversals represent algebraic product and coproduct. There is a huge symmetry: all diagrams representing computation connecting given X and Y must produce the same amplitude and there must exist minimal computation. The task of finding this computation is like finding the simplest representation for the formula X=Y and the noble purpose of math teachers is that we should learn to find it during our school days. This generalizes the duality symmetry of old fashioned string models: one can transform any diagram to a tree diagram without loops. This corresponds to quantum criticality in TGD: coupling constants do not evolve. The evolution is actually there but discrete and corresponds to infinite number critical values for Kahler coupling strength analogous to temperature. Reduction of diagrams with loops to braided treediagrams
Scattering amplitudes as generalized braid invariants The last big idea is the reduction of quantum TGD to generalized knot/braid theory (I have talked also about TGD as almost TQFT). The scattering amplitude can be identified as a generalized braid invariant and could be constructed by the generalization of the recursive procedure transforming in a stepbystep manner given braided tree diagram to a nonbraided tree diagram: essentially what Alexander the Great did for Gordian knot but tying the pieces together after cutting. At each step one must express amplitude as superposition of amplitudes associated with the different outcomes of splitting followed by reconnection. This procedure transforms braided tree diagram to a nonbraided tree diagrams and the outcome is the scattering amplitude! For background see the chapter From Principles to Diagrams or the article From Principles to Diagrams. 