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The twistor lift of TGD led to the introduction of Kähler form also in M4 factor of imbedding space M4×CP2. The moduli space of causal diamonds (CDs) introduced already early allow to save Poincare invariance at the level of WCW. One of the very nice things is that the self-duality of J(M4) leads to a new mechanism of breaking for P,CP, and T in long scales, where these breakings indeed take place. P corresponds to chirality selection in living matter, CP to matter antimatter asymmetry and T could correspond to preferred arrow of clock time. TGD allows both arrows but T breaking could make other arrow dominant. Also the hierarchy of Planck constant is expected to be important.
Can one say anything quantitative about these various breakings?
M8-H duality maps the preferred extremals in H to those M4× CP2 and vice versa. The tangent spaces of an associative space-time surface in M8 would be quaternionic (Minkowski) spaces.
In M8 one can consider also co-associative space-time surfaces having associative normal space. Could the co-associative normal spaces of associative space-time surfaces in the case of preferred extremals form an integrable distribution therefore defining a space-time surface in M8 mappable to H by M8-H duality? This might be possible but the associative tangent space and the normal space correspond to the same CP2 point so that associative space-time surface in M8 and its possibly existing co-associative companion would be mapped to the same surface of H.
This dead idea however inspires an idea about a duality mapping Minkowskian space-time regions to Euclidian ones. This duality would be analogous to inversion with respect to the surface of sphere, which is conformal symmetry. Maybe this inversion could be seen as the TGD counterpart of finite-D conformal inversion at the level of space-time surfaces. There is also an analogy with the method of images used in some 2-D electrostatic problems used to reflect the charge distribution outside conducting surface to its virtual image inside the surface. The 2-D conformal invariance would generalize to its 4-D quaterionic counterpart. Euclidian/Minkowskian regions would be kind of Leibniz monads, mirror images of each other.
The discovery of dual of the conformal symmetry of gauge theories was crucial for the development of twistor Grassmannian approach. The D=4 conformal generators acting on twistors have a dual representation in which they act on momentum twistors: one has dual conformal symmetry, which becomes manifest in this representation. These two separate symmetries extend to Yangian symmetry providing a powerful constraint on the scattering amplitudes in twistor Grassmannian approach fo N=4 SUSY.
In TGD the conformal Yangian extends to super-symplectic Yangian - actually, all symmetry algebras have a Yangian generalization with multi-locality generalized to multi-locality with respect to partonic 2-surfaces. The generalization of the dual conformal symmetry has however remained obscure. In the following I describe what the generalization of the two conformal symmetries and Yangian symmetry would mean in TGD framework.
One also ends up with a proposal of an information theoretic duality between Euclidian and Minkowskian regions of the space-time surface inspired by number theory: one might say that the dynamics of Euclidian regions is mirror image of the dynamics of Minkowskian regions. A generalization of the conformal reflection on sphere and of the method of image charges in 2-D electrostatics to the level of space-time surfaces allowing a concrete construction reciple for both Euclidian and Minkowskian regions of preferred extremals is in question. One might say that Minkowskian and Euclidian regions are analogous to Leibnizian monads reflecting each other in their internal dynamics.
The first question is what one means with S-matrix in ZEO. I have considered several proposals for the counterparts of S-matrix. In the original U-matrix, M-matrix and S-matrix were introduced but it seems that U-matrix is not needed.
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 magneto-optical 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. Magneto-optic Kerr effect occurs when a circularly polarized plane wave - often with normal incidence - reflects from a sample with planar boundary. In magneto-optic 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 θK 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, magneto-optic 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 super-conductor. Anti-ferromagnetism 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 magneto-optic 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 right-hand 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 M4× CP2 has Kähler structure in generalized sense. M4 has the analog of Kähler form, call it J(M4). J(M4) is assumed to be self-dual and covariantly constant as also CP2 Kähler form, and contributes to the Abelian electroweak U(1) gauge field (electroweak hypercharge) and therefore also to electromagnetic field.
J(M4) implies breaking of Lorentz invariance since it defines decomposition M4= M2× E2 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 non-commutative 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 self-dual to anti-self-dual form and vice versa. If J(M4) is self-dual as also J(CP2), all these 3 discrete symmetries are broken in arbitrarily long length scales. On basis of tensor property of J(M4) one expects P: (J(M2),J(E2)→ (J(M2),-J(E2) and T: (J(M2),J(E2)→ (-J(M2),J(E2). Under C one has (J(M2),J(E2)→ (-J(M2),-J(E2). This gives CPT: (J(M2),J(E2)→ (J(M2),J(E2) as expected.
One can imagine several consequences at the level of fundamental physics.
The generalization of twistor approach from M4 to H=M4× CP2 involves the replacement of twistor space of M4 with that of H. M8-H duality allows also an alternative approach in which one constructs twistor space of octonionic M8. Note that M4,E4, S4, and CP2 are the unique 4-D 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 4-fermion 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.
During 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 6-D 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.
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 M-matrix" or the article with the same title. See also the article About twistor lift of TGD.
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 Λb with quark composition u-d-b 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 2-surface 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 self-dual 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).
I 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 8-D light-likeness allowing to get rid of the problems produced by the mass of particles in 4-D sense, M8-M4× CP2 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 4-fermion scattering amplitude. >
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 2-surfaces defining topological vertices is nice.
Loops are however unavoidable in QFT description and their importance is undeniable. Photon-photon 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 box-like 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 non-point 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 sub-lines 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 3-surfaces 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 space-time surface somehow to reveal that space-time 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 space-time surface via the values of Noether charges. For four-momentum this dependence is obvious. The identification of heff/h=n as order of Galois group would mean that the basic unit for discrete charges depends on the extension characterizing the space-time 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 3-surfaces would be in the intersection of reality and p-adicities 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. p-Adic 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= q1+iq2π. 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 M-matrix".
During last couple years a kind of palace revolution has taken place in the formulation and interpretation of TGD. The notion of twistor lift and 8-D 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 p-adic physics as physics of cognition, heff/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 re-processing 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 re-interpretation for the p-adic evolution of the cosmological constant: cosmological term would correspond to the entire 4-D 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 6-D 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 M-matrix".
The 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 M-matrix".
To 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.
One objection against twistorialization at imbedding space level is that M4-twistorialization requires 4-D 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 8-D sense as also the modes of the imbedding space spinor fields are. M8-H duality indeed provides a solution of the problem. Massless quaternionic momentum in M8 can be for a suitable choice of decomposition M8= M4× E4 be reduce to massless M4 momentum and one can describe the information about 8-momentum using M4 twistor and CP2 twistor.
Second objection is that twistor Grassmann approach uses as twistor space the space T1(M4) =SU(2,2)/SU(2,1)× U(1) whereas the twistor lift of classical TGD uses T(M4)=M4× S2. The formulation of the twistor amplitudes in terms of strong form of holography (SH) using the data assignable to the 2-D surfaces - string world sheets and partonic 2-surfaces perhaps - identified as surfaces in T(M4)× T(CP2) requires the mapping of these twistor spaces to each other - the incidence relations of Penrose indeed realize this map.
It 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 space-time 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 space-time 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 6-D Kähler action. Also one would like to understand strong form of holography (SH). In TGD fields in space-time are are replaced with the imbedding of space-time as 4-surface to H. Twistor lift imbeds the twistor space of the space-time surface as 6-surface into the product of twistor spaces of M4 and CP2. Following Penrose, these imbeddings should be holomorphic in some sense.
Twistor lift T(H) means that M4 and CP2 are replaced with their 6-D twistor spaces.
The 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=M4× CP2 to that of space-time surface realized as its twistor space mean?
In TGD one replaces imbedding space H=M4× CP2 with the product T= T(M4)× T(CP2) of their 6-D twistor spaces, and calls T(H) the twistor space of H. For CP2 the twistor space is the flag manifold T(CP2)=SU(3)/U(1)× U(1) consisting of all possible choices of quantization axis of color isospin and hypercharge.
Symplectic structure for M4, CP breaking, matter-antimatter asymmetry, and electroweak symmetry breaking
The 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 M4 is discussed although it is not directly related to number theoretic aspects of TGD.
In 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 Occam’s razor as a general purpose debunking argument to TGD? tries to emphasize the simplicity of the basic principles of TGD and of the resulting theory.
A. In what aspects TGD extends other theory/theories of physics?
I will replace "extends" with "modifies" since TGD also simplifies in many respects. I shall restrict the considerations to the ontological level which to my view is the really important level.
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).
Many-sheeted space-time however forces to adopt fractal view. Universe would be analogous to Mandelbrot fractal down to CP2 scale. This predicts scaled variants of say hadron physics and electroweak physics. p-Adic length scale hypothesis and hierarchy of phases of matter with heff=n×h interpreted as dark matter gives a quantitative realization of this view.
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.
In the following I will consider some questions related to the twistor lift of TGD and end up to a possible vision about general mechanism of CP breaking and generation of matter antimatter asymmetry.
I have already earlier considered the question whether the analog of Kähler form assignable to M4 could appear in Kähler action. Could one replace the induced Kähler form J(CP2) with the sum J=J(M4)+J(CP2) such that the latter term would give rise to a new component of Kähler form both in space-time interior at the boundaries of string world sheets regarded as point-like particles? This could be done both in the Kähler action for the interior of X4 and also in the topological magnetic flux term ∈t J associated with string world sheet and reducing to a boundary term giving couplings to U(1) gauge potentials Aμ(CP2) and Aμ(M4) associated with J(CP2) and J(M4). The interpretation of this coupling is an interesting challenge.
Consider first the objections against introducing J(M4) to the Kähler action at imbedding space level.
What about the situation at space-time level?
2. About string like objects
String like objects and partonic 2-surfaces carry the information about quantum states and about space-time surfaces as preferred extremals if strong form of holography (SH) holds true. SH has of course some variants. The weakest variant states that fundamental information carrying objects are metrically 2-D. The light-like 3-surfaces separating space-time regions with Minkowskian and Euclidian signature of the induced metric are indeed metrically 2-D, and could thus carry information about quantum state.
An attractive possibility is that this information is basically topological. For instance, the value of Planck constant heff=n× h would tell the number sheets of the singular covering defining this surface such that the sheets co-incide at partonic 2-surfaces at the ends of space-time surface at boundaries of CD. In the following some questions related to string world sheets are considered. The information could be also number theoretical. Galois group for the algebraic extension of rationals defining particular adelic physics would transform to each other the number theoretic discretizations of light-like 3-surfaces and give rise to covering space structure. The action is trivial at partonic 2-surfaces should be trivial if one wants singular covering: this would mean that discretizations of partonic 2-surfaces consist of rational points. heff/h=n could in this case be a factor of the order of Galois group.
The original observation was that string world sheets should carry vanishing W boson fields in order that the em charge for the modes of the induced spinor field is well-defined. This condition can be satisfied in certain situations also for the entire space-time surface. This raises several questions. What is the fundamental condition forcing the restriction of the spinor modes to string world sheets - or more generally, to surface of given dimension? Is this restriction dynamical. Can one have an analog of brane hierarchy in which also higher-D objects can carry modes of induced spinor field Could the analogs of Lagrangian sub-manifolds of X4 ⊂ M4× CP2 satisfying J(M4)+J(CP2)=0 define string world sheets and their variants with varying dimension? The additional condition would be minimal surface property.
2.1 How does the gravitational coupling emerge?
The appearance of G=lP2 has coupling constant remained for a long time actually somewhat of a mystery in TGD. lP defines the radius of the twistor sphere of M4 replaced with its geometric twistor space M4× S2 in twistor lift. G makes itself visible via the coefficients ρvac= 8π Λ/G volume term but not directly and if preferred extremals are minimal surface extremals of Kähler action ρvac makes itself visible only via boundary conditions. How G appears as coupling constant?
Somehow the M4 Kähler form should appear in field equations. 1/G could naturally appear in the string tension for string world sheets as string models suggest. p-Adic mass calculations identify the analog of string tension as something of order of magnitude of 1/R2. This identification comes from the fact that the ground states of super-conformal representations correspond to imbedding space spinor modes, which are solutions of Dirac equation in M4× CP2. This argument is rather convincing and allows to expect that the p-adic mass scale is not determined by string tension and it can be chosen to be of order 1/G just as in string models.
2.2 Non-commutative imbedding space and strong form of holography
The precise formulation of strong form of holography (SH) is one of the technical problems in TGD. A comment in FB page of Gareth Lee Meredith led to the observation that besides the purely number theoretical formulation based on commutativity also a symplectic formulation in the spirit of non-commutativity of imbedding space coordinates can be considered. One can however use only the notion of Lagrangian manifold and avoids making coordinates operators leading to a loss of General Coordinate Invariance (GCI).
Quantum group theorists have studied the idea that space-time coordinates are non-commutative and tried to construct quantum field theories with non-commutative space-time coordinates (see this). My impression is that this approach has not been very successful. In Minkowski space one introduces antisymmetry tensor Jkl and uncertainty relation in linear M4 coordinates mk would look something like [mk, ml] = lP2Jkl, where lP is Planck length. This would be a direct generalization of non-commutativity for momenta and coordinates expressed in terms of symplectic form Jkl.
1+1-D case serves as a simple example. The non-commutativity of p and q forces to use either p or q. Non-commutativity condition reads as [p,q]= hbar Jpq and is quantum counterpart for classical Poisson bracket. Non-commutativity forces the restriction of the wave function to be a function of p or of q but not both. More geometrically: one selects Lagrangian sub-manifold to which the projection of Jpq vanishes: coordinates become commutative in this sub-manifold. This condition can be formulated purely classically: wave function is defined in Lagrangian sub-manifolds to which the projection of J vanishes. Lagrangian manifolds are however not unique and this leads to problems in this kind of quantization. In TGD framework the notion of "World of Classical Worlds" (WCW) allows to circumvent this kind of problems and one can say that quantum theory is purely classical field theory for WCW spinor fields. "Quantization without quantization would have Wheeler stated it.
GCI poses however a problem if one wants to generalize quantum group approach from M4 to general space-time: linear M4 coordinates assignable to Lie-algebra of translations as isometries do not generalize. In TGD space-time is surface in imbedding space H=M4× CP2: this changes the situation since one can use 4 imbedding space coordinates (preferred by isometries of H) also as space-time coordinates. The analog of symplectic structure J for M4 makes sense and number theoretic vision involving octonions and quaternions leads to its introduction. Note that CP2 has naturally symplectic form.
Could it be that the coordinates for space-time surface are in some sense analogous to symplectic coordinates (p1,p2,q1,q2) so that one must use either (p1,p2) or (q1,q2) providing coordinates for a Lagrangian sub-manifold. This would mean selecting a Lagrangian sub-manifold of space-time surface? Could one require that the sum Jμν(M4)+ Jμν(CP2) for the projections of symplectic forms vanishes and forces in the generic case localization to string world sheets and partonic 2-surfaces. In special case also higher-D surfaces - even 4-D surfaces as products of Lagrangian 2-manifolds for M4 and CP2 are possible: they would correspond to homologically trivial cosmic strings X2× Y2⊂ M4× CP2, which are not anymore vacuum extremals but minimal surfaces if the action contains besides Käction also volume term.
But why this kind of restriction? In TGD one has strong form of holography (SH): 2-D string world sheets and partonic 2-surfaces code for data determining classical and quantum evolution. Could this projection of M4 × CP2 symplectic structure to space-time surface allow an elegant mathematical realization of SH and bring in the Planck length lP defining the radius of twistor sphere associated with the twistor space of M4 in twistor lift of TGD? Note that this can be done without introducing imbedding space coordinates as operators so that one avoids the problems with general coordinate invariance. Note also that the non-uniqueness would not be a problem as in quantization since it would correspond to the dynamics of 2-D surfaces.
The analog of brane hierarchy for the localization of spinors - space-time surfaces; string world sheets and partonic 2-surfaces; boundaries of string world sheets - is suggesetive. Could this hierarchy correspond to a hierarchy of Lagrangian sub-manifolds of space-time in the sense that J(M4)+J(CP2)=0 is true at them? Boundaries of string world sheets would be trivially Lagrangian manifolds. String world sheets allowing spinor modes should have J(M4)+J(CP2)=0 at them. The vanishing of induced W boson fields is needed to guarantee well-defined em charge at string world sheets and that also this condition allow also 4-D solutions besides 2-D generic solutions. This condition is physically obvious but mathematically not well-understood: could the condition J(M4)+J(CP2)=0 force the vanishing of induced W boson fields? Lagrangian cosmic string type minimal surfaces X2× Y2 would allow 4-D spinor modes. If the light-like 3-surface defining boundary between Minkowskian and Euclidian space-time regions is Lagrangian surface, the total induced Kähler form Chern-Simons term would vanish. The 4-D canonical momentum currents would however have non-vanishing normal component at these surfaces. I have considered the possibility that TGD counterparts of space-time super-symmetries could be interpreted as addition of higher-D right-handed neutrino modes to the 1-fermion states assigned with the boundaries of string world sheets.
It is relatively easy to construct an infinite family of Lagrangian string world sheets satisfying J(M4) +J(CP2)=0 using generalized symplectic transformations of M4 and CP2 as Hamiltonian flows to generate new ones from a given Lagrangian string world sheets. One must pose minimal surface property as a separate condition. Consider a piece of M2 with coordinates (t,z) and homologically non-trivial geodesic sphere S2 of CP2 with coordinates (u= cos(Θ),Φ). One has J(M4)tz=1 and JuΦ= 1. Identify string world sheet via map (u,Φ)= (kz,ω t) from M2 to S2. The induced CP2 Kahler form is J(CP2)tz= kω. kω=-1 guarantees J(M4) +J(CP2)=0. The strings have necessarily finite length from L=1/k≤ z≤ L. One can perform symplectic transformations of CP2 and symplectic transformations of M4 to obtain new string world sheets. In general these are not minimal surfaces and this condition would select some preferred string world sheets.
An alternative - but of course not necessarily equivalent - attempt to formulate this picture would be in terms of number theoretic vision. Space-time surfaces would be associative or co-associative depending on whether tangent space or normal space in imbedding space is associative - that is quaternionic. These two conditions would reduce space-time dynamics to associativity and commutativity conditions. String world sheets and partonic 2-surfaces would correspond to maximal commutative or co-commutative sub-manifolds of imbedding space. Commutativity (co-commutativity) would mean that tangent space (normal space as a sub-manifold of space-time surface) has complex tangent space at each point and that these tangent spaces integrate to 2-surface. SH would mean that data at these 2-surfaces would be enough to construct quantum states. String world sheet boundaries would in turn correspond to real curves of the complex 2-surfaces intersecting partonic 2-surfaces at points so that the hierarchy of classical number fields would have nice realization at the level of the classical dynamics of quantum TGD.
To sum up, one cannot exclude the possibility that J(M4) is present implying a universal transversal localization of imbedding space spinor harmonics and the modes of spinor fields in the interior of X4: this could perhaps relate to somewhat mysterious de-coherence interaction producing locality and to CP breaking and matter-antimatter asymmetry. The moduli space for M4 Kähler structures proposed by number theoretic considerations would save from the loss of Poincare invariance and the number theoretic vision based on quaternionic and octonionic structure would have rather concrete realization. This moduli space would only extend the notion of "world of classical worlds" (WCW).