What's new inTowards MMatrixNote: Newest contributions are at the top! 
Year 2017 
Could McKay correspondence generalize in TGD framework?McKay correspondence states that the McKay graphs for the irreducible representations (irreps) of finite subgroups of G⊂ SU(2) characterizing their fusion algebra is given by extended Dynkin diagram of ADE type Lie group. Minimal conformal models with SU(2) KacMoody algebra (KMA) allow a classification by the same diagrams as fusion algebras of primary fields. The resolution of the singularities of complex algebraic surfaces in C^{3} by blowing implies the emergence of complex lines CP_{1}. The intersection matrix for the CP_{1}s is Dynkin diagram of ADE type Lie group. These results are highly inspiring concerning adelic TGD.
See the new chapter Are higher structures needed in the categorification of TGD? or the article Could McKay correspondence generalize in TGD framework?. 
About McKay and Langlands correspondences in TGD frameworkIn adelic TGD Galois groups for extensions of rationals become discrete symmetry groups acting on dark matter, identified as h_{eff}/h=n phases of ordinary matter. n gives the number of sheet of covering assignable to spacetime surface. Since Galois group acts on the cognitive representation defined by a discrete set of points of spacetime surface with coordinates having values in extension of rationals, the action of Galois group defines nsheeted covering, where n is the order of Galois group thus identifiable in terms of Planck constant. Adelic TGD inspires the question whether the representations of Galois groups could correspond to representations of Lie groups defining the ground states of KacMoody representations emerging in TGD in two manners: as representations of KacMoody algebra assignable the Poincare, color and electroweak symmetries on one hand and with dynamical generated from supersymplectic symmetry assignable with the boundaries of causal diamond (CD) and extended KacMoody symmetres assignable to the lightlike orbits of partonic 2surfaces defining boundaries between spacetime regions with Minkowskian and Euclidian signatures of the induced metric. McKay correspondence states that the finite discrete subgroups of SU(2) can be characterized by McKay graphs characterizing the fusion rules for the tensor products for the representations of these groups. These graphs correspond to the Dynkin diagrams for KacMoody algebras of ADE type group (all roots have same unit length in Dynkin diagram). This inspires the conjecture that finite subgroups of SU(2) indeed correspond to KacMoody algebras. Could the representations of discrete subgroups appearing in the McKay graph define also representations for the ground states of corresponding ADE type KacMoodyt algebra? More generally, could the McKay graps of the Galois groups? Number theoretic Langlands correspondence in turn states roughly that the representations of Galois group for extensions of rationals correspond to the so called automorphic representations of algebraic variants of reductive Lie groups. This is not totally surprising since the matrices defining algebraic matrix group has matrix elements in the extension of rationals. This raises the question how closely the number theoretic Langlands correspondence corresponds to the basic physical picture of TGD. 1. Could normal subgroups of symplectic group and of Galois groups correspond to each other? Measurement resolution realized in terms of various inclusion is the key principle of quantum TGD. There is an analogy between the hierarchies of Galois groups, of fractal subalgebras of supersymplectic algebra (SSA), and of inclusions of hyperfinite factors of type II_{1} (HFFs). The inclusion hierarchies of isomorphic subalgebras of SSA and of Galois groups for sequences of extensions of extensions should define hierarchies for measurement resolution. Also the inclusion hierarchies of HFFs are proposed to define hierarcies of measurement resolutions. How closely are these hierarchies related and could the notion of measurement resolution allow to gain new insights about these hierarchies and even about the mathematics needed to realize them?
1.1 Some basic facts about Galois groups and finite groups Some basic facts about Galois groups mus be listed before continuing. Any finite group can appear as a Galois group for an extension of some number field. It is known whether this is true for rationals (see this). Simple groups appear as building bricks of finite groups and are rather well understood. One can even speak about periodic table for simple finite groups (see this). Finite groups can be regarded as a subgroup of permutation group S_{n} for some n. They can be classified to cyclic, alternating , and Lie type groups. Note that alternating group A_{n} is the subgroup of permutation group S_{n} that consists of even permutations. There are also 26 sporadic groups and Tits group. Most simple finite groups are groups of Lie type that is rational subgroups of Lie groups. Rational means ordinary rational numbers or their extension. The groups of Lie type (see this) can be characterized by the analogs of Dynkin diagrams characterizing Lie algebras. For finite groups of Lie type the McKay correspondence could generalize. 1.2 Representations of Lie groups defining KacMoody ground states as irreps of Galois group? The goal is to generalize the McKay correspondence. Consider extension of rationals with Galois group Gal. The ground staes of KMA representations are irreps of the Lie group G defining KMA. Could the allow ground states for given Gal be irreps of also Gal? This constraint would determine which group representations are possible as ground states of SKMA representations for a given Gal. The better the resolution the larger the dimensions of the allowed representations would be for given G. This would apply both to the representations of the SKMA associated with dynamical symmetries and maybe also those associated with the standard model symmetries. The idea would be quantum classical correspondence (QCC) spacetime sheets as coverings would realize the ground states of SKMA representations assignable to the various SKMAs. This option could also generalize the McKay correspondence since one can assign to finite groups of Lie type an analog of Dynkin diagram (see this). For Galois groups, which are discrete finite groups of SU(2) the hypothesis would state that the KacMoody algebra has same Dynkin diagram as the finite group in question. To get some perspective one can ask what kind of algebraic extensions one can assign to ADE groups appearing in the McKay correspondence? One can get some idea about this by studying the geometry of Platonic solids (see this). Also the geometry of Dynkin diagrams telling about the geometry of root system gives some idea about the extension involved.
2. A possible connection with number theoretic Langlands correspondence I have discussed number theoretic version of Langlands correspondence in \citeallb/Langland,Langlandsnew trying to understand it using physical intuition provided by TGD (the only possible approach in my case). Concerning my unashamed intrusion to the territory of real mathematicians I have only one excuse: the number theoretic vision forces me to do this. Number theoretic Langlands correspondence relates finitedimensional representations of Galois groups and so called automorphic representations of reductive algebraic groups defined also for adeles, which are analogous to representations of Poincare group by fields. This is kind of relationship can exist follows from the fact that Galois group has natural action in algebraic reductive group defined by the extension in question. The "Resiprocity conjecture" of Langlands states that so called Artin Lfunctions assignable to finitedimensional representations of Galois group Gal are equal to Lfunctions arising from so called automorphic cuspidal representations of the algebraic reductive group G. One would have correspondence between finite number of representations of Galois group and finite number of cuspidal representations of G. This is not far from what I am naively conjecturing on physical grounds: finiteD representations of Galois group are reductions of certain representations of G or of its subgroup defining the analog of spin for the automorphic forms in G (analogous to classical fields in Minkowski space). These representations could be seen as induced representations familiar for particle physicists dealing with Poincare invariance. McKay correspondence encourages the conjecture that the allowed spin representations are irreducible also with respect to Gal. For a childishly naive physicist knowing nothing about the complexities of the real mathematics this looks like an attractive starting point hypothesis. In TGD framework Galois group could provide a geometric representation of "spin" (maybe even spin 1/2 property) as transformations permuting the sheets of the spacetime surface identifiable as Galois covering. This geometrization of number theory in terms of cognitive representations analogous to the use of algebraic groups in Galois correspondence might provide a totally new geometric insights to Langlands correpondence. One could also think that Galois group represented in this manner could combine with the dynamical KacMoody group emerging from SSA to form its Langlands dual. Skeptic physicist taking mathematics as high school arithmetics might argue that algebraic counterparts of reductive Lie groups are rather academic entities. In adelic physics the situation however changes completely. Evolution corresponds to a hierarchy of extensions of rationals reflected directly in the physics of dark matter in TGD sense: that is as phases of ordinary matter with h_{eff}/h=n identifiable as order of Galois group for extension of rationals. Algebraic groups and their representations get physical meaning and also the huge generalization of their representation to adelic representations makes sense if TGD view about consciousness and cognition is accepted. In attempts to understand what Langlands conjecture says one should understand first the rough meaning of many concepts. Consider first the Artin Lfunctions appearing at the number theoretic side. Consider first the Artin Lfunctions appearing at the number theoretic side.

Are higher structures needed in the categorification of TGD?The notion of higher structures promoted by John Baez looks very promising notion in the attempts to understand various structures like quantum algebras and Yangians in TGD framework. The stimulus for this article came from the nice explanations of the notion of higher structure by Urs Screiber. The basic idea is simple: replace "=" as a blackbox with an operational definition with a proof for $A=B$. This proof is called homotopy generalizing homotopy in topological sense. nstructure emerges when one realizes that also the homotopy is defined only up to homotopy in turn defined only up... In TGD framework the notion of measurement resolution defines in a natural manner various kinds of "="s and this gives rise to resolution hierarchies. Hierarchical structures are characteristic for TGD: hierarchy of spacetime sheet, hierarchy of padic length scales, hierarchy of Planck constants and dark matters, hierarchy of inclusions of hyperfinite factors, hierarchy of extensions of rationals defining adeles in adelic TGD and corresponding hierarchy of Galois groups represented geometrically, hierarchy of infinite primes, self hierarchy, etc... In this article the idea of nstructure is studied in more detail. A rather radical idea is a formulation of quantum TGD using only cognitive representations consisting of points of spacetime surface with imbedding space coordinates in extension of rationals defining the level of adelic hierarchy. One would use only these discrete points sets and Galois groups. Everything would reduce to number theoretic discretization at spacetime level perhaps reducing to that at partonic 2surfaces with points of cognitive representation carrying fermion quantum numbers. Even the"{world of classical worlds" (WCW) would discretize: cognitive representation would define the coordinates of WCW point. One would obtain cognitive representations of scattering amplitudes using a fusion category assignable to the representations of Galois groups: something diametrically opposite to the immense complexity of the WCW but perhaps consistent with it. Also a generalization of McKay's correspondence suggests itself: only those irreps of the Lie group associated with KacMoody algebra that remain irreps when reduced to a subgroup defined by a Galois group of Lie type are allowed as ground states. See the new chapter Are higher structures needed in the categorification of TGD? or the article with the same title. 
Could categories, tensor networks, and Yangians provide the tools for handling the complexity of TGD?TGD Universe is extremely simple but the presence of various hierarchies make it to look extremely complex globally. Category theory and quantum groups, in particular Yangian or its TGD generalization are most promising tools to handle this complexity. The arguments developed in the sequel suggest the following overall view.
One should assign Yangian to various KacMoody algebras (SKMAs) involved and even with superconformal algebra (SSA), which however reduces effectively to SKMA for finitedimensional Lie group if the proposed gauge conditions meaning vanishing of Noether charges for some subalgebra H of SSA isomorphic to it and for its commutator [SSA,H] with the entire SSA. Strong form of holography (SH) implying almost 2dimensionality motivates these gauge conditions. Each SKMA would define a direct summand with its own parameter defining coupling constant for the interaction in question. See the new chapter Could categories, tensor networks, and Yangians provide the tools for handling the complexity of TGD? of "Towards Mmatrix" or the article with the same title. 
Getting even more quantitative about CP violationThe twistor lift of TGD forces to introduce the analog of Kähler form for M^{4}, call it J. J is covariantly constant selfdual 2form, whose square is the negative of the metric. There is a moduli space for these Kähler forms parametrized by the direction of the constant and parallel magnetic and electric fields defined by J. J partially characterizes the causal diamond (CD): hence the notation J(CD) and can be interpreted as a geometric correlate for fixing quantization axis of energy (rest system) and spin. Kähler form defines classical U(1) gauge field and there are excellent reasons to expect that it gives rise to U(1) quanta coupling to the difference of BL of baryon and lepton numbers. There is coupling strength α_{1} associated with this interaction. The first guess that it could be just Kähler coupling strength leads to unphysical predictions: α_{1} must be much smaller. Here I do not yet completely understand the situation. One can however check whether the simplest guess is consistent with the empirical inputs from CP breaking of mesons and antimatter asymmetry. This turns out to be the case. One must specify the value of α_{1} and the scaling factor transforming J(CD) having dimension length squared as tensor square root of metric to dimensionless U(1) gauge field F= J(CD)/S. This leads to a series of questions. How to fix the scaling parameter S?
How can one fix the value of U(1) coupling strength α_{1}? As a guideline one can use CP breaking in K and B meson systems and the parameter characterizing matterantimatter symmetry.
For ε=2^{30} the value of l_{P}^{2}/R^{2}(CP_{2}) is l_{P}^{2}/R^{2}(CP_{2})=(2π)^{2} × R^{2}(S^{2})/R^{2}(CP_{2}) ≈ 3.7× 10^{8}. l_{P}/R(S^{2}) would be a transcendental number but since it would not be a fundamental constant but appear only at the QFTGRT limit of TGD, this would not be a problem. One can make order of magnitude estimates for the Jarlskog parameter J and the fraction r= n(B)/n(γ). Here it is not however clear whether one should use ε or α_{1} as the basis of the estimate
Quantal U(1) force must be also consistent with atomic physics. The value of the parameter α_{1} consistent with the size of CP breaking of K mesons and with matter antimatter asymmetry is α_{1}= εα_{K} = 2^{30}α_{K}.

About some unclear issues of TGDTGD has been in the middle of palace revolution during last two years and it is almost impossible to keep the chapters of the books updated. Adelic vision and twistor lift of TGD are the newest developments and there are still many details to be understood and errors to be corrected. The description of fermions in TGD framework has contained some unclear issues. Hence the motivation for the following brief comments. Adelic vision and symmetries In the adelic TGD SH is weakened: also the points of the spacetime surface having imbedding space coordinates in an extension of rationals (cognitive representation) are needed so that data are not precisely 2D. I have believed hitherto that one must use preferred coordinates for the imbedding space H  a subset of these coordinates would define spacetime coordinates. These coordinates are determined apart from isometries. Does the number theoretic discretization imply loss of general coordinate invariance and also other symmetries? The reduction of symmetry groups to their subgroups (not only algebraic since powers of e define finitedimensional extension of padic numbers since e^{p} is ordinary padic number) is genuine loss of symmetry and reflects finite cognitive resolution. The physics itself has the symmetries of real physics. The assumption about preferred imbedding space coordinates is actually not necessary. Different choices of Hcoordinates means only different and nonequivalent cognitive representations. Spherical and linear coordinates in finite accuracy do not provide equivalent representations. Quantumclassical correspondence for fermions Quantumclassical correspondence (QCC) for fermions is rather wellunderstood but deserves to be mentioned also here. QCC for fermions means that the spacetime surface as preferred extremal should depend on fermionic quantum numbers. This is indeed the case if one requires QCC in the sense that the fermionic representations of Noether charges in the Cartan algebras of symmetry algebras are equal to those to the classical Noether charges for preferred extremals. Second aspect of QCC becomes visible in the representation of fermionic states as point like particles moving along the lightlike curves at the lightlike orbits of the partonic 2surfaces (curve at the orbit can be locally only lightlike or spacelike). The number of fermions and antifermions dictates the number of string world sheets carrying the data needed to fix the preferred extremal by SH. The complexity of the spacetime surface increases as the number of fermions increases. Strong form of holography for fermions It seems that scattering amplitudes can be formulated by assigning fermions with the boundaries of strings defining the lines of twistor diagrams. This information theoretic dimensional reduction from D=4 to D=2 for the scattering amplitudes can be partially understood in terms of strong form of holography (SH): one can construct the theory by using the data at string worlds sheets and/or partonic 2surfaces at the ends of the spacetime surface at the opposite boundaries of causal diamond (CD). 4D modified Dirac action would appear at fundamental level as supersymmetry demands but would be reduced for preferred extremals to its 2D stringy variant serving as effective action. Also the value of the 4D action determining the spacetime dynamics would reduce to effective stringy action containing area term, 2D Kähler action, and topological Kähler magnetic flux term. This reduction would be due to the huge gauge symmetries of preferred extremals. Subalgebra of supersymplectic algebra with conformal weigths coming as nmultiples of those for the entire algebra and the commutators of this algebra with the entire algebra would annihilate the physical states, and thecorresponding classical Noether charges would vanish. One still has the question why not the data at the entire string world sheets is not needed to construct scattering amplitudes. Scattering amplitudes of course need not code for the entire physics. QCC is indeed motivated by the fact that quantum experiments are always interpreted in terms of classical physics, which in TGD framework reduces to that for spacetime surface. The relationship between spinors in spacetime interior and at boundaries between Euclidian and Minkoskian regions Spacetime surface decomposes to interiors of Minkowskian and Euclidian regions. At lightlike 3surfaces at which the fourmetric changes, the 4metric is degenerate. These metrically singular 3surfaces  partonic orbits carry the boundaries of string world sheets identified as carriers of fermionic quantum numbers. The boundaries define fermion lines in the twistor lift of TGD. The relationship between fermions at the partonic orbits and interior of the spacetime surface has however remained somewhat enigmatic. So: What is the precise relationship between induced spinors Ψ_{B} at lightlike partonic 3surfaces and Ψ_{I} in the interior of Minkowskian and Euclidian regions? Same question can be made for the spinors Ψ_{B} at the boundaries of string world sheets and Ψ_{I} in interior of the string world sheets. There are two options to consider:
I have considered Option II already years ago but have not been able to decide.
About second quantization of the induced spinor fields The anticommutation relations for the induced spinors have been a longstanding issue and during years I have considered several options. The solution of the problem looks however stupifuingly simple. The conserved fermion currents are accompanied by supercurrents obtained by replacing Ψ with a mode of the induced spinor field to get u_{n}Γ^{α}Ψ or ΨΓ^{α}u_{n} with the conjugate of the mode. One obtains infinite number of conserved super currents. One can also replace both Ψ and Ψ in this manner to get purely bosonic conserved currents Ψ_{m}Γ^{α}u_{n} to which one can assign a conserved bosonic charges Q_{mn}. I noticed this years ago but did not realize that these bosonic charges define naturally anticommutators of fermionic creation and annihilation operators! The ordinary anticommutators of quantum field theory follow as a special case! By a suitable unitary transformation of the spinor basis one can diagonalize the hermitian matrix defined by Q_{mn} and by performing suitable scalings one can transform anticommutation relations to the standard form. An interesting question is whether the diagonalization is needed, and whether the deviation of the diagonal elements from unity could have some meaning and possibly relate to the hierarchy h_{eff}=n× h of Planck constants  probably not. Is statistical entanglement "real" entanglement? The question about the "reality" of statistical entanglement has bothered me for years. This entanglement is maximal and it cannot be reduced by measurement so that one can argue that it is not "real". Quite recently I learned that there has been a longstanding debate about the statistical entanglement and that the issue still remains unresolved. The idea that all electrons of the Universe are maximally entangled looks crazy. TGD provides several variants for solutions of this problem. It could be that only the fermionic oscillator operators at partonic 2surfaces associated with the spacetime surface (or its connected component) inside given CD anticommute and the fermions are thus indistinguishable. The extremist option is that the fermionic oscillator operators belonging to a network of partonic 2surfaces connected by string world sheets anticommute: only the oscillator operators assignable to the same scattering diagram would anticommute. What about QCC in the case of entanglement. EREPR correspondence introduced by Maldacena and Susskind for 4 years ago proposes that blackholes (maybe even elementary particles) are connected by wormholes. In TGD the analogous statement emerged for more than decade ago  magnetic flux tubes take the role of wormholes in TGD. Magnetic flux tubes were assumed to be accompanied by string world sheets. I did not consider the question whether string world sheets are always accompanied by flux tubes. What could be the criterion for entanglement to be "real"? "Reality" of entanglement demands some spacetime correlate. Could the presence of the flux tubes make the entanglement "real"? If statistical entanglement is accompanied by string connections without magnetic flux tubes, it would not be "real": only the presence of flux tubes would make it "real". Or is the presence of strings enough to make the statistical entanglement "real". In both cases the fermions associated with disjoint spacetime surfaces or with disjoint CDs would not be indistinguishable. This looks rather sensible. The spacetime correlate for the reduction of entanglement would be the splitting of a flux tube and fermionic strings inside it. The fermionic strings associated with flux tubes carrying monopole flux are closed and the return flux comes back along parallel spacetime sheet. Also fermionic string has similar structure. Reconnection of this flux tube with shape of very long flattened square splitting it to two pieces would be the correlate for the state function reduction reducing the entanglement with other fermions and would indeed decouple the fermion from the network. See the chapter Number Theoretical Vision.

Getting quantitative about breaking of CP, P, and TThe twistor lift of TGD led to the introduction of Kähler form also in M^{4} factor of imbedding space M^{4}×CP_{2}. The moduli space of causal diamonds (CDs) introduced already early allow to save Poincare invariance at the level of WCW. One of the very nice things is that the selfduality of J(M^{4}) leads to a new mechanism of breaking for P,CP, and T in long scales, where these breakings indeed take place. P corresponds to chirality selection in living matter, CP to matter antimatter asymmetry and T could correspond to preferred arrow of clock time. TGD allows both arrows but T breaking could make other arrow dominant. Also the hierarchy of Planck constant is expected to be important. Can one say anything quantitative about these various breakings?

A new kind of duality of old duality from a new perspective?M^{8}H duality maps the preferred extremals in H to those M^{4}× CP_{2} and vice versa. The tangent spaces of an associative spacetime surface in M^{8} would be quaternionic (Minkowski) spaces. In M^{8} one can consider also coassociative spacetime surfaces having associative normal space. Could the coassociative normal spaces of associative spacetime surfaces in the case of preferred extremals form an integrable distribution therefore defining a spacetime surface in M^{8} mappable to H by M^{8}H duality? This might be possible but the associative tangent space and the normal space correspond to the same CP_{2} point so that associative spacetime surface in M^{8} and its possibly existing coassociative companion would be mapped to the same surface of H. This dead idea however inspires an idea about a duality mapping Minkowskian spacetime regions to Euclidian ones. This duality would be analogous to inversion with respect to the surface of sphere, which is conformal symmetry. Maybe this inversion could be seen as the TGD counterpart of finiteD conformal inversion at the level of spacetime surfaces. There is also an analogy with the method of images used in some 2D electrostatic problems used to reflect the charge distribution outside conducting surface to its virtual image inside the surface. The 2D conformal invariance would generalize to its 4D quaterionic counterpart. Euclidian/Minkowskian regions would be kind of Leibniz monads, mirror images of each other.
For background see chapter Some questions related to the twistor lift of TGD or the article with the same title.

About the generalization of dual conformal symmetry and Yangian in TGDThe discovery of dual of the conformal symmetry of gauge theories was crucial for the development of twistor Grassmannian approach. The D=4 conformal generators acting on twistors have a dual representation in which they act on momentum twistors: one has dual conformal symmetry, which becomes manifest in this representation. These two separate symmetries extend to Yangian symmetry providing a powerful constraint on the scattering amplitudes in twistor Grassmannian approach fo N=4 SUSY. In TGD the conformal Yangian extends to supersymplectic Yangian  actually, all symmetry algebras have a Yangian generalization with multilocality generalized to multilocality with respect to partonic 2surfaces. The generalization of the dual conformal symmetry has however remained obscure. In the following I describe what the generalization of the two conformal symmetries and Yangian symmetry would mean in TGD framework. One also ends up with a proposal of an information theoretic duality between Euclidian and Minkowskian regions of the spacetime surface inspired by number theory: one might say that the dynamics of Euclidian regions is mirror image of the dynamics of Minkowskian regions. A generalization of the conformal reflection on sphere and of the method of image charges in 2D electrostatics to the level of spacetime surfaces allowing a concrete construction reciple for both Euclidian and Minkowskian regions of preferred extremals is in question. One might say that Minkowskian and Euclidian regions are analogous to Leibnizian monads reflecting each other in their internal dynamics. See the chapter Some Questions Related to the Twistor Lift of TGD or the article with the same title. 
About unitarity for scattering amplitudesThe first question is what one means with Smatrix in ZEO. I have considered several proposals for the counterparts of Smatrix. In the original Umatrix, Mmatrix and Smatrix were introduced but it seems that Umatrix is not needed.
For background see chapter Some questions related to the twistor lift of TGD or the article with the same title. 
Kerr effect, breaking of T symmetry, and Kähler form of M^{4}I encountered in Facebook (thanks to Ulla) a link to a very interesting article Here is the abstract. We prove an instance of the Reciprocity Theorem that demonstrates that Kerr rotation, also known as the magnetooptical Kerr effect, may only arise in materials that break microscopic time reversal symmetry. This argument applies in the linear response regime, and only fails for nonlinear effects. Recent measurements with a modified Sagnac Interferometer have found finite Kerr rotation in a variety of superconductors. The Sagnac Interferometer is a probe for nonreciprocity, so it must be that time reversal symmetry is broken in these materials. I had to learn some basic condensed matter physics. Magnetooptic Kerr effect occurs when a circularly polarized plane wave  often with normal incidence  reflects from a sample with planar boundary. In magnetooptic Kerr effect there are many options depending on the relative directions of the reflection plane (incidence is not normal in the general case so that one can talk about reflection plane) and magnetization. Also the incoming polarization can be linear or circular. Reflected circular polarized beams suffers a phase change in the reflection: as if they would spend some time at the surface before reflecting. Linearly polarized light reflects as elliptically polarized light. Kerr angle θ_{K} is defined as 1/2 of the difference of the phase angle increments caused by reflection for oppositely circularly polarized plane wave beams. As the name tells, magnetooptic Kerr effect is often associated with magnetic materials. Kerr effect has been however observed also for high Tc superconductors and this has raised controversy. As a layman in these issues I can naively wonder whether the controversy is created by the expectation that there are no magnetic fields inside the superconductor. Antiferromagnetism is however important for high Tc superconductivity. In TGD based model for high Tc superconductors the supracurrents would flow along pairs of flux tubes with the members of S=0 (S=1) Cooper pairs at parallel flux tubes carrying magnetic fields with opposite (parallel) magnetic fluxes. Therefore magnetooptic Kerr effect could be in question after all. The author claims to have proven that Kerr effect in general requires breaking of microscopic time reversal symmetry. Time reversal symmetry breaking (TRSB) caused by the presence of magnetic field and in the case of unconventional superconductors is explained nicely here. Magnetic field is required. Magnetic field is generated by a rotating current and by righthand rule time reversal changes the direction of the current and also of magnetic field. For spin 1 Cooper pairs the analog of magnetization is generated, and this leads to T breaking. This result is very interesting from the point of TGD. The reason is that twistorial lift of TGD requires that imbedding space M^{4}× CP_{2} has Kähler structure in generalized sense. M^{4} has the analog of Kähler form, call it J(M^{4}). J(M^{4}) is assumed to be selfdual and covariantly constant as also CP_{2} Kähler form, and contributes to the Abelian electroweak U(1) gauge field (electroweak hypercharge) and therefore also to electromagnetic field. J(M^{4}) implies breaking of Lorentz invariance since it defines decomposition M^{4}= M^{2}× E^{2} Implying preferred rest frame and preferred spatial direction identifiable as direction of spin quantization axis. In zero energy ontology (ZEO) one has moduli space of causal diamonds (CDs) and therefore also moduli space of Kähler forms and the breaking of Lorentz invariance cancels. Note that a similar Kähler form is conjectured in quantum group inspired noncommutative quantum field theories and the problem is the breaking of Lorentz invariance. What is interesting that the action of P,CP, and T on Kähler form transforms it from selfdual to antiselfdual form and vice versa. If J(M^{4}) is selfdual as also J(CP_{2}), all these 3 discrete symmetries are broken in arbitrarily long length scales. On basis of tensor property of J(M^{4}) one expects P: (J(M^{2}),J(E^{2})→ (J(M^{2}),J(E^{2}) and T: (J(M^{2}),J(E^{2})→ (J(M^{2}),J(E^{2}). Under C one has (J(M^{2}),J(E^{2})→ (J(M^{2}),J(E^{2}). This gives CPT: (J(M^{2}),J(E^{2})→ (J(M^{2}),J(E^{2}) as expected. One can imagine several consequences at the level of fundamental physics.
For background see chapter Some questions related to the twistor lift of TGD or the article with the same title. 
Key ideas related to the twistor lift of TGDThe generalization of twistor approach from M^{4} to H=M^{4}× CP_{2} involves the replacement of twistor space of M^{4} with that of H. M^{8}H duality allows also an alternative approach in which one constructs twistor space of octonionic M^{8}. Note that M^{4},E^{4}, S^{4}, and CP_{2} are the unique 4D spaces allowing twistor space with Kähler structure. This makes TGD essentially unique. Ordinary twistor approach has two problems.
To develop this idea one must understand what scattering diagrams are. The scattering diagrams involve two kinds of lines.
The basic problem is that the kinematics for 4fermion vertices need not be consistent with the gliding of vertex past another one so that this move is not possible.
In TGD Universe allowed diagrams would represent closed objects in what one might call BCFW homology. The operation appearing at the right hand side of BCFW recursion formula is indeed boundary operation, whose square by definition gives zero. For background see chapter Some questions related to the twistor lift of TGD or the article with the same title. 
Issues related to the precise formulation of twistor lift of TGDDuring last two weeks I have worked hardly to deduce the implications of some observations relating to the twistor lift of Kähler action. Some of these observations were very encouraging but some observations were a cold shower forcing a thorough criticism of the first view about the details of the twistor lift of TGD. New formulation of Kähler action The first observation was that the correct formulation of 6D Kähler action in the framework of adelic physics implies that the classical physics of TGD does not depend on the overall scaling of Kähler action.
Realizing NTU The independence of the classical physics on the scale of the action in the new formulation inspires a detailed discussion of the number theoretic vision.
Trouble with cosmological constant Also an unpleasant observation about cosmological constant forces to challenge the original view about twistor lift.
See the articles About twistor lift of TGD and and See the new chapter Some Questions Related to the Twistor Lift of TGD of "Towards Mmatrix" or the article with the same title. See also the article About twistor lift of TGD. 
What causes CP violation?CP violation and matter antimatter asymmetry involving it represent white regions in the map provided by recent day physics. Standard model does not predict CP violation necessarily accompanied by the violation of time reflection symmetry T by CPT symmetry assumed to be exact. The violation of T must be distinguished from the emergence of time arrow implies by the randomness associated with state function reduction. CP violation was originally observed for mesons via the mixing of neutral kaon and antikaon having quark content nsbar and nbars. The lifetimes of kaon and antikaon are different and they transform to each other. CP violation has been also observed for neutral mesons of type nbbar. Now it has been observed also for baryons Λ_{b} with quark composition udb and its antiparticle (see this). Standard model gives the Feynman graphs describing the mixing in standard model in terms of CKM matrix (see this). The CKM mixing matrix associated with weak interactions codes for the CP violation. More precisely, the small imaginary part for the determinant of CKM matrix defines the invariant coding for the CP violation. The standard model description of CP violation involves box diagrams in which the coupling to heavy quarks takes place. b quark gives rise to anomalously large CP violation effect also for mesons and this is not quite understood. Possible new heavy fermions in the loops could explain the anomaly. Quite generally, the origin of CP violation has remained a mystery as also CKM mixing. In TGD framework CKM mixing has topological explanation in terms of genus of partonic 2surface assignable to quark (sphere, torus or sphere with two handles). Topological mixings of U and D type quarks are different and the difference is not same for quarks and antiquarks. But this explains only CKM mixing, not CP violation. Classical electric field  not necessary electromagnetic  prevailing inside hadrons could cause CP violation. So called instantons are basic prediction of gauge field theories and could cause strong CP violation since selfdual gauge field is involved with electric and magnetic fields having same strength and direction. That this strong CP violation is not observed is a problem of QCD. There are however proposals that instantons in vacuum could explain the CP violation of hadron physics (see this). What says TGD? I have considered this here and in the earlier blog posting (see this).
See the new chapter Some Questions Related to the Twistor Lift of TGD or the article with the same title. See also the article About twistor lift of TGD. 
Criticizing the TGD based twistorial construction of scattering amplitudesI have developed a rather detailed vision about twistorial construction of scattering amplitudes of fundamental fermions in TGD framework. These amplitudes serve as building bricks of scattering amplitudes of elementary particles. The construction allows to solve the basic problems of ordinary twistor approach. Some of the key notions are 8D lightlikeness allowing to get rid of the problems produced by the mass of particles in 4D sense, M^{8}M^{4}× CP_{2} duality having nice interpretation in twistor space of $H$, quantum criticality demanding the vanishing of loops associated with functional integral and together with Kähler property implying that functional integral reduces to mere action exponential around given maximum of K\"ahler function, and number theoretical universality (NTU) suggesting that scattering diagrams could be seen as representations of computations reducible to minimal computation represented by tree diagram. One ends up with an explicit representations for the fundamentl 4fermion scattering amplitude. The vision is discussed in Questions related to twistor lift TGD. For the necessary background see About twistor lift of TGD. One can however criticize the proposed vision. What about loops of QFT? The idea about cancellation of loop corrections in functional integral and moves allowing to transform scattering diagrams represented as networks of partonic orbits meeting at partonic 2surfaces defining topological vertices is nice. Loops are however unavoidable in QFT description and their importance is undeniable. Photonphoton cattering is described by a loop diagram in which fermions appear in box like loop. Magnetic moment of muon) involves a triangle loop. A further interesting case is CP violation for mesons involving boxlike loop diagrams. Apart from divergence problems and problems with bound states, QFT works magically well and loops are important. How can one understand QFT loops if there are no fundamental loops? How could QFT emerge from TGD as an approximate description assuming lengths scale cutoff? The key observation is that QFT basically replaces extended particles by point like particles. Maybe loop diagrams can be "unlooped" by introducing a better resolution revealing the nonpoint like character of the particles. What looks like loop for a particle line becomes in an improved resolution a tree diagram describing exchange of particle between sublines of line of the original diagram. In the optimal resolution one would have the scattering diagrams for fundamental fermions serving as building bricks of elementary particles. To see the concrete meaning of the "unlooping" in TGD framework, it is necessary to recall the qualitative view about what elementary particles are in TGD framework.
Can action exponentials really disappear? The disappearance of the action exponentials from the scattering amplitudes can be criticized. In standard approach the action exponentials associated with extremals determine which configurations are important. In the recent case they should be the 3surfaces for which Kähler action is maximum and has stationary phase. But what would select them if the action exponentials disappear in scattering amplitudes? The first thing to notice is that one has functional integral around a maximum of vacuum functional and the disappearance of loops is assumed to follow from quantum criticality. This would produce exponential since Gaussian and metric determinants cancel, and exponentials would cancel for the proposal inspired by the interpretation of diagrams as computations. One could in fact define the functional integral in this manner so that a discretization making possible NTU would result. Fermionic scattering amplitudes should depend on spacetime surface somehow to reveal that spacetime dynamics matters. In fact, QCC stating that classical Noether charges for bosonic action are equal to the eigenvalues of quantal charges for fermionic action in Cartan algebra would bring in the dependence of scattering amplitudes on spacetime surface via the values of Noether charges. For fourmomentum this dependence is obvious. The identification of h_{eff}/h=n as order of Galois group would mean that the basic unit for discrete charges depends on the extension characterizing the spacetime surface. Also the cognitive representations defined by the set of points for which preferred imbedding space coordinates are in this extension. Could the cognitive representations carry maximum amount of information for maxima? For instance, the number of the points in extension be maximal. Could the maximum configurations correspond to just those points of WCW, which have preferred coordinates in the extension of rationals defining the adele? These 3surfaces would be in the intersection of reality and padicities and would define cognitive representation. These ideas suggest that the usual quantitative criterion for the importance of configurations could be equivalent with a purely number theoretical criterion. pAdic physics describing cognition and real physics describing matter would lead to the same result. Maximization for action would correspond to maximization for information. Irrespective of these arguments, the intuitive feeling is that the exponent of the bosonic action must have physical meaning. It is number theoretically universal if action satisfies S= q_{1}+iq_{2}π. This condition could actually be used to fix the dependence of the coupling parameters on the extension of rationals (see this). By allowing sum over several maxima of vacuum functional these exponentials become important. Therefore the above ideas are interesting speculations but should be taken with a big grain of salt. For details see the new chapter Some Questions Related to the Twistor Lift of TGD of "Towards Mmatrix". 
Two observations about twistor lift of Kähler actionDuring last couple years a kind of palace revolution has taken place in the formulation and interpretation of TGD. The notion of twistor lift and 8D generalization of twistorialization have dramatically simplified and also modified the view about what classical TGD and quantum TGD are. The notion of adelic physics suggests the interpretation of scattering diagrams as representations of algebraic computations with diagrams producing the same output from given input are equivalent. The simplest possible manner to perform the computation corresponds to a tree diagram. As will be found, it is now possible to even propose explicit twistorial formulas for scattering formulas since the horrible problems related to the integration over WCW might be circumvented altogether. From the interpretation of padic physics as physics of cognition, h_{eff}/h=n could be interpreted as the order of Galois group. Discrete coupling constant evolution would correspond to phase transitions changing the extension of rationals and its Galois group. TGD inspired theory of consciousness is an essential part of TGD and the crucial Negentropy Maximization Principle in statistical sense follows from number theoretic evolution as increase of the order of Galois group for extension of rationals defining adeles. During the reprocessing of the details related to twistor lift, it became clear that the earlier variant for the twistor lift can be criticized and allows an alternative. This option led to a simpler view about twistor lift, to the conclusion that minimal surface extremals of Kähler action represent only asymptotic situation near boundaries of CD (external particles in scattering), and also to a reinterpretation for the padic evolution of the cosmological constant: cosmological term would correspond to the entire 4D action and the cancellation of Kähler action and cosmological term would lead to the small value of the effective cosmological constant. The pleasant observation was that the correct formulation of 6D Kähler action in the framework of adelic physics implies that the classical physics of TGD does not depend on the overall scaling of Kähler action but that quantum classical correspondence implies this dependence. It is however too early to select between the two options. For details see the new chapter Some Questions Related to the Twistor Lift of TGD of "Towards Mmatrix". 
Questions related to the quantum aspects of twistorializationThe progress in the understanding of the classical aspects of twistor lift of TGD makes possible to consider in detail the quantum aspects of twistorialization of TGD and for the first time an explicit proposal for the part of scattering diagrams assignable to fundamental fermions emerges.
See the new chapter Some Questions Related to the Twistor Lift of TGD of "Towards Mmatrix". 
A new view about color, color confinement, and twistorsTo my humble opinion twistor approach to the scattering amplitudes is plagued by some mathematical problems. Whether this is only my personal problem is not clear.
For details see the new chapter Some Questions Related to the Twistor Lift of TGD or the article Some questions related to the twistor lift of TGD. 
How does the twistorialization at imbedding space level emerge?One objection against twistorialization at imbedding space level is that M^{4}twistorialization requires 4D conformal invariance and massless fields. In TGD one has towers of particle with massless particles as the lightest states. The intuitive expectation is that the resolution of the problem is that particles are massless in 8D sense as also the modes of the imbedding space spinor fields are. M^{8}H duality indeed provides a solution of the problem. Massless quaternionic momentum in M^{8} can be for a suitable choice of decomposition M^{8}= M^{4}× E^{4} be reduce to massless M^{4} momentum and one can describe the information about 8momentum using M^{4} twistor and CP_{2} twistor. Second objection is that twistor Grassmann approach uses as twistor space the space T_{1}(M^{4}) =SU(2,2)/SU(2,1)× U(1) whereas the twistor lift of classical TGD uses T(M^{4})=M^{4}× S^{2}. The formulation of the twistor amplitudes in terms of strong form of holography (SH) using the data assignable to the 2D surfaces  string world sheets and partonic 2surfaces perhaps  identified as surfaces in T(M^{4})× T(CP_{2}) requires the mapping of these twistor spaces to each other  the incidence relations of Penrose indeed realize this map. For details see the new chapter Some Questions Related to the Twistor Lift of TGD or the article Some questions related to the twistor lift of TGD.

Twistor lift and the reduction of field equations and SH to holomorphyIt has become clear that twistorialization has very nice physical consequences. But what is the deep mathematical reason for twistorialization? Understanding this might allow to gain new insights about construction of scattering amplitudes with spacetime surface serving as analogs of twistor diatrams. Penrose's original motivation for twistorilization was to reduce field equations for massless fields to holomorphy conditions for their lifts to the twistor bundle. Very roughly, one can say that the value of massless field in spacetime is determined by the values of the twistor lift of the field over the twistor sphere and helicity of the massless modes reduces to cohomology and the values of conformal weights of the field mode so that the description applies to all spins. I want to find the general solution of field equations associated with the Kähler action lifted to 6D Kähler action. Also one would like to understand strong form of holography (SH). In TGD fields in spacetime are are replaced with the imbedding of spacetime as 4surface to H. Twistor lift imbeds the twistor space of the spacetime surface as 6surface into the product of twistor spaces of M^{4} and CP_{2}. Following Penrose, these imbeddings should be holomorphic in some sense. Twistor lift T(H) means that M^{4} and CP_{2} are replaced with their 6D twistor spaces.

More details about the induction of twistor structureThe notion of twistor lift of TGD (see this and this) has turned out to have powerful implications concerning the understanding of the relationship of TGD to general relativity. The meaning of the twistor lift really has remained somewhat obscure. There are several questions to be answered. What does one mean with twistor space? What does the induction of twistor structure of H=M^{4}× CP_{2} to that of spacetime surface realized as its twistor space mean? In TGD one replaces imbedding space H=M^{4}× CP_{2} with the product T= T(M^{4})× T(CP_{2}) of their 6D twistor spaces, and calls T(H) the twistor space of H. For CP_{2} the twistor space is the flag manifold T(CP_{2})=SU(3)/U(1)× U(1) consisting of all possible choices of quantization axis of color isospin and hypercharge.
For details see the new chapter Some Questions Related to the Twistor Lift of TGD or the article Some questions related to the twistor lift of TGD. 
Symplectic structure for M^{4}, CP breaking, matterantimatter asymmetry, and electroweak symmetry breakingThe preparation of an article about number theoretic aspects of TGD forced to go through various related ideas and led to a considerable integration of the ideas. In this note idea about the symplectic structure of M^{4} is discussed although it is not directly related to number theoretic aspects of TGD.
See the chapter About twistor lift of TGD or the article with the same title. 
Questions about TGDIn FB I was made a question about general aspects of TGD. It was impossible to answer the question with few lines and I decided to write a blog posting. I am sorry for typos in the hastily written text. A more detailed article Can one apply Occamâ€™s razor as a general purpose debunking argument to TGD? tries to emphasize the simplicity of the basic principles of TGD and of the resulting theory. A. In what aspects TGD extends other theory/theories of physics? I will replace "extends" with "modifies" since TGD also simplifies in many respects. I shall restrict the considerations to the ontological level which to my view is the really important level.
B. In what sense TGD is simplification/extension of existing theory?
C. What is the hypothetical applicability of the extension  in energies, sizes, masses etc? TGD is a unified theory and is meant to apply in all scales. Usually the unifications rely on reductionistic philosophy and try to reduce physics to Planck scale. Also super string models tried this and failed: what happens at long length scales was completely unpredictable (landscape catastrophe). Manysheeted spacetime however forces to adopt fractal view. Universe would be analogous to Mandelbrot fractal down to CP_{2} scale. This predicts scaled variants of say hadron physics and electroweak physics. pAdic length scale hypothesis and hierarchy of phases of matter with h_{eff}=n×h interpreted as dark matter gives a quantitative realization of this view.
D. What is the leading correction/contribution to physical effects due to TGD onto particles, interactions, gravitation, cosmology?
See the new chapter Can one apply Occam's razor as a general purpose debunking argument to TGD? or article with the same title. 
Generalized Kähler structure for M^{4} and CP breaking and matter antimatter asymmetryIn the following I will consider some questions related to the twistor lift of TGD and end up to a possible vision about general mechanism of CP breaking and generation of matter antimatter asymmetry.
I have already earlier considered the question whether the analog of Kähler form assignable to M^{4} could appear in Kähler action. Could one replace the induced Kähler form J(CP_{2}) with the sum J=J(M^{4})+J(CP_{2}) such that the latter term would give rise to a new component of Kähler form both in spacetime interior at the boundaries of string world sheets regarded as pointlike particles? This could be done both in the Kähler action for the interior of X^{4} and also in the topological magnetic flux term ∈t J associated with string world sheet and reducing to a boundary term giving couplings to U(1) gauge potentials A_{μ}(CP_{2}) and A_{μ}(M^{4}) associated with J(CP_{2}) and J(M^{4}). The interpretation of this coupling is an interesting challenge. Consider first the objections against introducing J(M^{4}) to the Kähler action at imbedding space level.
What about the situation at spacetime level?
2. About string like objects String like objects and partonic 2surfaces carry the information about quantum states and about spacetime surfaces as preferred extremals if strong form of holography (SH) holds true. SH has of course some variants. The weakest variant states that fundamental information carrying objects are metrically 2D. The lightlike 3surfaces separating spacetime regions with Minkowskian and Euclidian signature of the induced metric are indeed metrically 2D, and could thus carry information about quantum state. An attractive possibility is that this information is basically topological. For instance, the value of Planck constant h_{eff}=n× h would tell the number sheets of the singular covering defining this surface such that the sheets coincide at partonic 2surfaces at the ends of spacetime surface at boundaries of CD. In the following some questions related to string world sheets are considered. The information could be also number theoretical. Galois group for the algebraic extension of rationals defining particular adelic physics would transform to each other the number theoretic discretizations of lightlike 3surfaces and give rise to covering space structure. The action is trivial at partonic 2surfaces should be trivial if one wants singular covering: this would mean that discretizations of partonic 2surfaces consist of rational points. h_{eff}/h=n could in this case be a factor of the order of Galois group. The original observation was that string world sheets should carry vanishing W boson fields in order that the em charge for the modes of the induced spinor field is welldefined. This condition can be satisfied in certain situations also for the entire spacetime surface. This raises several questions. What is the fundamental condition forcing the restriction of the spinor modes to string world sheets  or more generally, to surface of given dimension? Is this restriction dynamical. Can one have an analog of brane hierarchy in which also higherD objects can carry modes of induced spinor field Could the analogs of Lagrangian submanifolds of X^{4} ⊂ M^{4}× CP_{2} satisfying J(M^{4})+J(CP_{2})=0 define string world sheets and their variants with varying dimension? The additional condition would be minimal surface property.
2.1 How does the gravitational coupling emerge? The appearance of G=l_{P}^{2} has coupling constant remained for a long time actually somewhat of a mystery in TGD. l_{P} defines the radius of the twistor sphere of M^{4} replaced with its geometric twistor space M^{4}× S^{2} in twistor lift. G makes itself visible via the coefficients ρ_{vac}= 8π Λ/G volume term but not directly and if preferred extremals are minimal surface extremals of Kähler action ρ_{vac} makes itself visible only via boundary conditions. How G appears as coupling constant? Somehow the M^{4} Kähler form should appear in field equations. 1/G could naturally appear in the string tension for string world sheets as string models suggest. pAdic mass calculations identify the analog of string tension as something of order of magnitude of 1/R^{2}. This identification comes from the fact that the ground states of superconformal representations correspond to imbedding space spinor modes, which are solutions of Dirac equation in M^{4}× CP_{2}. This argument is rather convincing and allows to expect that the padic mass scale is not determined by string tension and it can be chosen to be of order 1/G just as in string models. 2.2 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. It is relatively easy to construct an infinite family of Lagrangian string world sheets satisfying J(M^{4}) +J(CP_{2})=0 using generalized symplectic transformations of M^{4} and CP_{2} as Hamiltonian flows to generate new ones from a given Lagrangian string world sheets. One must pose minimal surface property as a separate condition. Consider a piece of M^{2} with coordinates (t,z) and homologically nontrivial geodesic sphere S^{2} of CP_{2} with coordinates (u= cos(Θ),Φ). One has J(M^{4})_{tz}=1 and J_{uΦ}= 1. Identify string world sheet via map (u,Φ)= (kz,ω t) from M^{2} to S^{2}. The induced CP_{2} Kahler form is J(CP_{2})_{tz}= kω. kω=1 guarantees J(M^{4}) +J(CP_{2})=0. The strings have necessarily finite length from L=1/k≤ z≤ L. One can perform symplectic transformations of CP_{2} and symplectic transformations of M^{4} to obtain new string world sheets. In general these are not minimal surfaces and this condition would select some preferred string world sheets. An alternative  but of course not necessarily equivalent  attempt to formulate this picture would be in terms of number theoretic vision. 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. To sum up, one cannot exclude the possibility that J(M^{4}) is present implying a universal transversal localization of imbedding space spinor harmonics and the modes of spinor fields in the interior of X^{4}: this could perhaps relate to somewhat mysterious decoherence interaction producing locality and to CP breaking and matterantimatter asymmetry. The moduli space for M^{4} Kähler structures proposed by number theoretic considerations would save from the loss of Poincare invariance and the number theoretic vision based on quaternionic and octonionic structure would have rather concrete realization. This moduli space would only extend the notion of "world of classical worlds" (WCW). For background see the chapter Questions related to the twistor lift of TGD or the article with the same title.
