About double slit experiments of Dean Radin

Dean Radin and his team have done a very interesting experiment (see this and this) testing the idea that observer induces state function reduction.

Experiment

The experiment is a modified double slit experiment. In double slit experiment a laser beam arrives to the screen via two slits and interference pattern is generated as if photons would behave like waves localized at screen. If one adds detectors at the slits, either detector fires and detects the passing-by by photon, and interference pattern disappears with optimal detection efficiency.

The idea is to add a subject person (S) at distance of two meters. S imagines of measuring that electron passes through either slit. One can say that S intends to add a "detector" to either slit or both of them so that a state function reduction selecting either slit occurs. This experiment differs from experiments in which S tries to affect the ratio of frequencies of 0:s and 1:s in random series of bits: S does not try to force the electrons to pass by either slit. There is a feedback represented as sound/yellow light whose height/intensity coded for the amount of the reduction of the height of the peak. There are two kinds of participants: meditators and those who have no experience in meditation.

The results of the experiment are thoroughly discussed in the Youtube lecture or Radin (see this). To my opinion the results are amazing. In one experiment it was found that the height of the peak of the Fourier transform of the intensity distribution of the diffraction pattern is reduced. In second experiment the depth of bottom of the through of distribution was reduced. As if the intention would induce with some probability to perform the measurement selecting the photon path. The effect was small but appeared systematically for a group consisting of meditators. For persons without experience in meditation the effect averaged out also in this case it was present in the beginning of the experiment when subject person were not bored by the repetitive character of the experiment. The longer attention span of meditators could partially explain this.

Even more amazing finding was that in a variant of the experiment realized in internet the results were also positive although the persons intending to induce the experiment.

Arguments of different skeptic

The standard argument of skeptic is that statistics is poor, that the experiment is even fraud, etc... One can however consider more refined and more imaginative objections. Let us make a digress from the usual behavior of skeptic and assume that the effect was real.

If the meditators could induce the measurement by intention, one expects that also the experimenter could have done it. To how high degree the outcome was due to the experimenters and how much due to the meditators? Experimenter also had the theoretical expectation that meditators are better in inducing the slit detection. Could the wish that the theory is correct have caused subconscious intention about performing the detection in the case of meditators or not doing it in the case of non-meditating subject persons?

In the case of net experiment situation becomes even more problematic. One can imagine that also in this case the intention of experimenter could induce the detection - at least if experimenter is near to the system. Should experimenters have spent the period of experiments in Mars or at least in a distant holiday resort! Experimenters studying remote mental interactions are usually not rich people and presumably they did not do this.

The experimenter effect is well-known in parapsychology. Some experimenters are extremely successful. Could one think that they have strong intentional powers? Ironically, this would demonstrate the reality of paranormal effects of this kind but in a manner that can never convince the skeptics. There is evidence for this kind of effect in the testing of new medicines. Good results are obtained when the testers are enthusiastic and dream of a positive result. When they do same tests after some years, the results are worse.

TGD based model

The challenge is to understand how the S imagining a measurement telling that photon went through either slit could realize this intention. What does the detection mean and what it demands?

1. The measurement should involve a state function reduction selecting between the slits entangled with observer. In principle it is enough to have an interaction of photon in either slit localing the path of the photon to that slit. It is enough that photon interacts with charged particles in either slit with some probability. This measurement is of course not optimal since the interference diagram is only partially changed. Only some fraction of these measurements take place and produce single slit pattern so that the observer pattern is a weighted average of double slit and single slit patterns. In principle one can estimate the probability for single slit pattern from the data.
   
   
2. Quantum classical correspondence requires that in order that the intention to detect could be realized, one must have a physical connection from the S to both slits or at least either of them. Also charged particles assignable to the connection should be involved to make scattering of photon possible. Also entanglement entangling detector fires/does not fire with corresponding states of some other system, say the S would be needed.

How could one realize these connections in TGD?

1. In TGD framework the magnetic flux tubes serve as correlates of entanglement and directed attention. To direct attention to a system means to connected with it by flux tubes. Flux tubes carry dark charged particles essential for TGD view about quantum biology.
   
   
2. Every system has U-shaped flux tubes emanating from it and acting as kind of tentacles scanning the environment. As a U-shaped flux tube from system A encounters another similar flux tube from system B, a reconnection takes place if the quantized fluxes are same. The outcome is a pair of flux tubes connection A and B. The flux tube pair can carry Cooper pairs with members of the pair at the flux tubes. The photons from laser could scatter from the charged particles.
   
   
3. The dark particles the flux tube are dark with h_eff/h=n satisfying an additional condition implying that n is proportional to the mass of the charged particle in turn implying that cyclotron energies E_c= hbar_eff eB/m are universal and assumed to correspond energies in the range of visible and UV.
   

   In order that photon scatters from the charged particles it must have the same value of h_eff as the particles at magnetic flux tubes emanating from the S. Some fraction of laser photons could satisfy this condition. Note that if perturbative quantum theory applies, the classical predictions are same as lowest order quantum predictions so that h_eff makes it visible only in higher orders assuming that perturbation theory works when h_eff/h=n holds true. Unfortunately, it is not possible to estimate the probability that photon enters to the flux tube. Note that the probability depends also on the density of the flux tubes.
   
   

The effect is reported in net experiments for which distances can be long and there is no visual contact. Can one understand this?

1. If there quantum entanglement between A and B already exists one can increase the distance without spoiling the entanglement. But how to achieve the entanglement if n the systems are at large distance from beginning?
   
   
2. The length of the magnetic flux tubes is not a problem. The size scale for the layers of magnetic flux tube corresponding to EEG frequency 7.8 Hz is circumference of Earth. The condition that the size of the flux tube is at least of the order of the cyclotron wavelength λ for cyclotron photons at the flux tube implies that length of the flux tube of of the order of the size scale of Earth for EEG frequencies.
   

   In fact, our MBs could have much larger layers if biological rhythms have cyclotron frequencies as counterparts. The size scales could be of order light-life-time or even longer. This changes totally the view about the role of length scales in biology and consciousness. There is some evidence that galactic day defines the natural rhythm for precognitive phenomena: precognitive phenomena tend to occur at galactic midday. Galactic cyclotron frequencies (the galactic magnetic field is of order nT) could correspond to bio-rhythms up to 12 hours.
   
   

In net experiment the problem is how to generate the connection to a correct target. The same problem is encountered in the attempts to explain the claimed results of remote viewing experiments. Could the density of flux tubes of personal magnetic body (MB) be so high that the connection is generated with high enough probability. S receives data through the web. Could this help to build the desired connection.

1. Skeptic would explain the reported positive result in web experiments by saying that the results were actually induced by the intention of the experimenter who was near to the system. This might of course be the case.
   
   
2. The first possibility is that an entanglement is generated between the camera monitoring the system and slits involving flux tubes. The communication of the image from the camera to computer builds another flux tube bridge. The radiation reflected in satellite to the computer at Earth involves propagation along flux tubes. At the receiver ends similar bridges are build. There is therefore a flux tube connection with the computer of used by S, who generates the last piece of the connection. This kind of flux tube connection would be between all communicating systems. Also the experiments would belong to this entanglement network.
   
   
3. MB has layers with size scale of order Earth size. Could it be able to meet the challenge by using the information coming from web. Could the U-shaped flux tubes be so dense as to be able to build a contact with the experimental arrangement with high enough probability? If they are to represent Maxwellian magnetic field in good approximation, they should be dense. What is important that these flux tubes correspond do different space-time sheets for distinct observers: this is actually the basic distinction between the field concepts of Maxwell and TGD.
   

   Could it be that the feedback from S at her computer via the net to the computer at the other end generates quantum correlated events and this correlation has as correlates magnetic flux tubes connecting the distant systems.
   
   

4. The hyper-imaginative option is that S can delegate the problem with collective consciousness assignable to the magnetosphere of Earth and having all the engineering knowledge that Earth has! Could we be neurons of a gigantic brain of Mother Gaia, which would help S to realize their intention. Can single neuron realize its intention on a distant neuron in brain in the similar manner? Could some kind of resonance mechanism be involved? Could MB detect correlations between distant events and generate flux tube connection and entanglement between these places? Could brain do the same for neutrons?
   

See the article About the double slit experiment of Dean Radin or the chapter TGD inspired view about remote mental interactions and paranormal of "TGD based view about living matter and remote mental interactions".

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

Articles and other material related to TGD.

About unitarity of twistor amplitudes

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.

1. The first question is whether the unitary matrix is between zero energy states or whether it characterizes zero energy states themselves as time-like entanglement coefficients between positive and negative energy parts of zero energy states associated with the ends of CD. One can argue that the first option is not sensible since positive and negative energy parts of zero energy states are strongly correlated rather than forming a tensor product: the S-matrix would in fact characterize this correlation partially.
   

   The latter option is simpler and is natural in the proposed identification of conscious entity - self - as a generalized Zeno effect, that is as a sequence of repeated state function reductions at either boundary of CD shifting also the boundary of CD farther away from the second boundary so that the temporal distance between the tips of CD increases. Each shift of this kind is a step in which superposition of states with different distances of upper boundary from lower boundary results followed by a localization fixing the active boundary and inducing unitary transformation for the states at the original boundary.
   
   

2. The proposal is that the the proper object of study for given CD is M-matrix. M-matrix is a product for a hermitian square root of diagonalized density matrix ρ with positive elements and unitary S-matrix S : M= ρ^1/2S. Density matrix ρ could be interpreted in this approach as a non-trivial Hilbert space metric. Unitarity conditions are replaced with the conditions MM^†= ρ and M^†M=ρ. For the single step in the sequence of reductions at active boundary of CD one has M→ MS (Δ T) so that one has S→ SS(Δ T). S(Δ T) depends on the time interval Δ T measured as the increase in the proper time distance between the tips of CD assignable to the step.
   
   

What does unitarity mean in the twistorial approach?

1. In accordance with the idea that scattering diagrams is a representation for a computation, suppose that the deformations of space-time surfaces defining a given topological diagram as a maximum of the exponent of Kähler function, are the basic objects. They would define different quantum phases of a larger quantum theory regarded as a square root of thermodynamics in ZEO and analogous to those appearing also in QFTs. Unitarity would hold true for each phase separately.
   

   The topological diagrams would not play the role of Feynman diagrams in unitarity conditions although their vertices would be analogous to those appearing in Feynman diagrams. This would reduce the unitarity conditions to those for fermionic states at partonic 2-surfaces at the ends of CDs, actually at the ends of fermionic lines assigned to the boundaries of string world sheets.
   
   

2. The unitarity conditions be interpreted stating the orthonormality of the basis of zero energy states assignable with given topological diagram. Since 3-surfaces as points of WCW appearing as argument of WCW spinor field are pairs consisting of 3-surfaces at the opposite boundaries of CD, unitarity condition would state the orthonormality of modes of WCW spinor field. If might be even that no mathematically well-defined inner product assignable to either boundary of CD exists since it does not conform with the view provided by WCW geometry. Perhaps this approach might help in identifying the correct form of S-matrix.
   
   
3. If only tree diagrams constructed using 4-fermion twistorial vertex are allowed, the unitarity relations would be analogous to those obtained using only tree diagrams. They should express the discontinuity for T in S=1+iT along unitary cut as Disc(T)= TT^†. T and T^† would be T-matrix and its time reversal.
   
   
4. The correlation between the structure of the fermionic scattering diagram and topological scattering diagrams poses very strong restrictions on allowed scattering reactions for given topological scattering diagram. One can of course have many-fermion states at partonic 2-surfaces and this would allow arbitrarily high fermion numbers but physical intuition suggests that for given partonic 2-surface (throat of wormhole contact) the fermion number is only 0, 1, or perhaps 2 in the case of supersymmetry possibly generated by right-handed neutrino.
   

   The number of fundamental fermions both in initial and final states would be finite for this option. In quantum field theory with only masive particles the total energy in the final state poses upper bound on the number of particles in the final state. When massless particles are allowed there is no upper bound. Now the complexity of partonic 2-surface poses an upper bound on fermions.
   

   This would dramatically simplify the unitarity conditions but might also make impossible to satisfy them. The finite number of conditions would be in spirit with the general philosophy behind the notion of hyper-finite factor. The larger the number of fundamental fermions associated with the state, the higher the complexity of the topological diagram. This would conform with the idea about QCC. One can make non-trivial conclusions about the total energy at which the phase transitions changing the topology of space-time surface defined by a topological diagram must take place.
   
   

See the article About twistor lift of TGD.

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

Articles and other material related to TGD.

Kerr effect, breaking of T symmetry, and Kähler form of M4

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 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, 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.

1. One implication is a first principle explanation for the mysterious CP violation and matter antimatter asymmetry not predicted by standard model (see the recent blog post).
   
   
2. A new kind of parity breaking is predicted. This breaking is separate from electroweak parity breaking and perhaps closely related to the chiral selection in living matter.
   
   
3. The breaking of T might in turn relate to Kerr effect if the argument of authors is correct. It could occur in high Tc superconductors in macroscopic scales. Also large h_eff/h=n scaling up quantum scales in high Tc superconductors could be involved as with the breaking of chiral symmetry in living matter. Strontium ruthenate for which Cooper pairs are in S=1 state is is indeed found to exhibit TRSB (for references and explanation see this).
   

   In TGD based model of high Tc superconductivity the members of the Cooper pair are at parallel magnetic flux tubes with the same spin direction of magnetic field. The magnetic fields and thus the direction of spin component in this direction changes under T causing TRSB. The breaking of T for S=1 Cooper pairs is not spontaneous but would occur at the level of physics laws: the time reversed system finds itself experiences in the original self-dual J(M⁴)) rather than in (-J(M²),J(E²)) demanded by T symmetry.
   
   

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

Articles and other material related to TGD.

Key ideas related to the twistor lift of TGD

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.

1. It applies only if the particles are massless. In TGD particles are massless in 8-D sense but the projection of 8-momentum to given M⁴ is in general massive in 4-D sense. This solves the problem. Note that the 4-D M⁴ momenta can be light-like for a suitable choice of M⁴⊂ H. There exist even a choice of M² for which this is the case. For given M² the choices of quaternionic M⁴ are parametrized by CP₂.
   
   
2. The twistor approach has second problem: it works nicely in signature (2,2) rather (1,3) for Minkowski space. For instance, twistor Fourier transform cannot be defined as an ordinary integral. The very nice results by Nima Arkani-Hamed et al about positive Grassmannian follow only in the signature (2,2).
   

   One can always find M²⊂ M⁸ in which the 8-momentum lies and is therefore light-like in 2-D sense. Furthermore, the light-like 8-momenta and thus 2-momenta are prediced already at classical level to be complex. M² as subspace of momentum space M⁸ effectively extends to its complex version with signature (2,2)!
   

   At classical space-time level the presence of preferred M² reflects itself in the properties of massless extremals with M⁴= M²× E² decomposition such that light-like momentum is in M² and polarization in E².
   

   4-D conformal invariance is restricted to its 2-D variant in M². Twistor space of M⁴ reduces to that of M². This is SO(2,2)/SO(2,1)=RP₃. This is 3-D RP₃, the real variant of twistor space CP₃. Complexification of light-like momenta replaces RP₃ with CP₃.
   
   

Light-like M⁸-momenta are in question but they are not arbitrary.

1. They must lie in some quaternionic plane containing fixed M², which corresponds to the plane spanned by real octonion unit and some imaginary unit. . This condition is analogous to the condition that the space-time surfaces as preferred extremals in M⁸ have quaternionic tangent planes.
   
   
2. In particular, the wave functions can be expressed as products of plane waves in M², wave functions in the plane of transverse momenta in E²⊂ M⁴, where M⁴ is quaternionic plane containing M² and wave function in the space for the choices of M⁴, which is CP₂. One obtains exactly the same result in M⁴× CP₂ if delocalization in transversal E² momenta taking place of quarks inside hadrons takes place.
   Transversal wave function can also concentrate on single momentum value.
   

   It should be noticed that quaternionicity forces number theoretical spontaneous compactification. It would be very clumsy to realize the condition that allowed 8-momenta are qiuaternionic. Instead going to M⁴× CP₂, "spontaneously compactifying", description everyting becomes easy.
   
   

3. What is amusing that the geometric twistor space M⁴× S² of M⁴ having bundle projections to M⁴ and ordinary twistor spaces is nothing but the space of choices of causal diamonds with preferred M² and fixed rest frame (time axis connecting the tips). M⁴ point fixes the tip of causal diamond (CD) and S² the spatial direction fixing M² plane. In case of CP₂ the point of twistor space fixes point of CP₂ as analog for tip of CD: the complex CP₂ coordinates have origin at this point. The point of twistor sphere of SU(3)/U(1)× U(1) codes for the selection of quantization axis for hypercharge Y and isospin I₃. The corresponding subgroup U(1)× U(1) affects only the phases of the preferred complex coordinates transforming linearly under SU(2)× U(1).
   

   At the level of momentum space M⁴ twistor codes for the momentum and helicity of particle. For CP₂ it codes for the selection of M⁴⊂ M⁸ and for em charge as analog of helicity. Now one has actually wave function for the selections of CP₂ point labelled by the color numbers of the particle.
   
   

Number theoretical vision inspires the idea that scattering ampitudes define representations for algebraic computations leading from initial set of algebraic objects to to final set of objects. If so, the amplitudes should not depend on how the computation is done and there should exist a minimal computation possibly represented by a tree diagram. There would be no summation over the equivalent diagrams: one can choose any-one of them and the best choice is the simplest one.

To develop this idea one must understand what scattering diagrams are. The scattering diagrams involve two kinds of lines.

1. There are topological "lines" corresponding to light-like orbits of partonic 2-surfaces playing the role of lines of Feynman diagrams. The topological diagram formed by these lines gives boundary conditions for 4-surface: at these light-like partonic orbits Euclidian space-time region changes to Minkowskian one. Vertices correspond to 2-surfaces at which these 3-D lines meet just like line in the case of Feynman diagrams.
   
   
2. There are also fermion lines assignable to fundamental fermions serving as building bricks of elementary particles. They correspond to the boundaries of string world sheets at the orbits of partonic 2-surfaces. Fundamental fermion-fermion scattering takes place via classical interactions at partonic 2-surfaces: there is no 4-vertex in the usual sense (this would lead to non-renormalizable theory).
   

   The conjecture is that he 4-vertex is described by twistor amplitude fixed apart from over all scaling factor. Fermion lines are along parton orbits. Boson lines correspond to pairs of fermion and antifermion at the same parton orbit.
   

   As a matter fact, the situation is more complex for elementary particles since they correspond to pairs of wormhole contacts connected by monopole magnetic tubes and wormhole contacts has two wormhole throats - partonic 2-surfaces.
   
   

For the idea about diagrams as representations of computations to make sense, there should exist moves which allow to glide the 4-fermion vertex and associated flux tubes along the topological line of scattering diagrams in the vicinity of the second end of the loop. Second move should allow to snip away the loop. Is this possible? The possibility to find M² for which momentum is light-like is central in the argument claiming that this is indeed possible.

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.

1. Clearly, one must assume something. If all momenta along at vertices along fermion line are in same M² then they parallel as light-like M²-momenta. Kinematical conditions allow the gliding of two vertices of this kind past each other as is easy to show. The scattering would mean only redistribution of parallel light-like momenta in this particular M².
   

   This kind of scattering would be more general than the scattering in integrable quantum field theories in M²: in this case the scattering would not affect the momenta but would induce phase shifts: particles would spend some time in the vertex before continuing. What is crucial for having non-trivial scatterings, is that in the general frame M²⊂ M⁴ ⊂ M⁸ the momenta would be massive and also different.
   
   

2. The condition would be that all four-fermion vertices along given fermion line correspond to the same preferred M². M²:s can differ only for fermionic sub-diagrams which do not have common vertices.
   

   Note however that tree diagrams for which lines can have different M²s can give rise to non-trivial scattering. One can take tree diagram and assign to the internal lines networks with same M²s as the internal line has. It is quite possible that for general graphs allowing different M²s in internal lines and loops, the reduction to tree graph is not possible.
   

   At least this idea could define precisely what the equivalence of diagrams, if vertices in which M²:s can be different are allowed. One can of course argue, that there is not deep reason for not allowing more general loopy graphs in which the incoming lines can have arbitrary M²:s.
   

One implication is that the BCFW recursion formula allowing to generate loop diagrams from those with lower number of loops must be trivial in TGD - this of course only if one accepts that BCFW formula makes sense in TGD. This requires that the entangled removal appearing as second term in the right hand side of BCFW formula and adding loop gives zero. One can develop and argument for why this must be the case in TGD framework. Also the second term corresponding to removal of BCFW bridge should give zero so that allowed diagrams cannot have BCFW bridges.

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 the article Some questions related to the twistor lift of TGD.

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

Articles and other material related to TGD.

Issues related to the precise formulation of twistor lift of TGD

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.

1. Kähler form has dimension length squared. Kähler form projected to the space-time surface defines Mawell field, which should be however dimensionless. I had assumed that one can just divide Kähler form by CP₂ radius squared to achieve this. The skeptic realizes immediately that this parameter is free coupling parameter albeit CP₂ radius is good guess for it. The correct formulation of the action principle must keep Kähler form dimensional and divides Kähler action by a dimensional parameter with dimension 4: this is new coupling contant type parameter besides α_K. The classical field equations do not depend at all on this scaling parameter. The exponent of action defining vacuum functional however depends on it.
   
   
2. What is so nice that all couplings disappear from classical field equations in the new formulation, and number theoretical universality (NTU) is automatically achieved. In particular, the preferred extremals need not be minimal surface extremals of Kähler action to achieve this as in the original proposal for the twstor lift. It is enough that they are so asymptotically - near the boundaries of CDs, where they behave like free particles. In the interior they couple to Kähler force. This also nicely conforms with the physical idea that they are 4-D generalizations for orbits of particles in induced Kähler field.
   
   
3. I also realized that the exchange of conserved quantities between Euclidian and Minkowskian space-time regions is not possible for the original version of twistor lift. This does not sound physical: quantal interactions should have classical correlates. The reason for the catastrophe is simple. Metric determinant appearing in action integral is identified as g₄^1/2. In Minkowskian regions it is purely imaginary but real in Euclidian regions. Boundary conditions lead to decoupling of Minkowskian and Euclidian regions.
   

   This forced to return to an old nagging question whether one should use a) g₄^1/2 (imaginary in Minkowskian regions) or b)|g₄^1/2| in the action. For real α_K the option a) is unavoidable and the need to have exponent of imaginary action in Minkowskian regions indeed motivated option a).
   

   For complex α_K forced by other considerations the situation however changes - something that I had not noticed. Complex α_K allows |g₄^1/2|. The study of so called CP₂ extremals assuming that 1/α_K= s, s=1/2+iy zero of Riemann zeta shows that NTU is realized in the sense that the exponent of action exists in some extension of rationals, provided that the imaginary part of zero of zeta satisfies y= qπ, q rational, implying that the exponent of y is root of unity. This possibility has been considered already earlier. This is highly non-trivial hypothesis about zeros of zeta.
   

4. Option b) allows transfer of conserved quantities between Minkowskian and Euclidian regions as required. Option a) also predicts separate conservation of Noether charges for Kähler action and volume term. This can make sense only asymptotically. Therefore only Option b) remains under serious consideration. In the new picture the interaction region in particle physics experiences corresponds to the region, where there is coupling between volume and Kähler terms: extrenal particles correpond to minimal surface extremals of Kähler action and all known extremals indeed are such.
   

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.

1. Quantum Classical Correspondence (QCC) breaks the invariance with respect to the scalings via fermionic anti-commutation relations and NTU can fix the spectrum of values of the over-all scaling parameter of the action. Fermionic anticommutation relations introduce the constraint removing the projective invariance.
   
   
2. One ends up to a condition guaranteeing NTU of the action exponentiale xp(S). One must have S= q₁+iq₂π , q_i rational. This guarantees that exp(S) is in some extension
   of rationals and therefore number theoretically universal. S itself is however not number theoretically universal.
   The overall scaling parameter for action contrained by fermionic anticommutations must have a value allowing to satisfy the condition.
   

3. The vision about scattering amplitude as a representation of computation however suggests the action exponential disappears from twistorial scattering amplitudes altogether as it does in quantum field theories. This would require that one defines scattering amplitude - actually zero energy state - by allowing functional integral
   only around single maximum of action. Whether this makes sense is not obvious but ZEO might allow it. I have not yet discussed seriously the constraints from unitary - or its generalization to ZEO, and these constraints might force sum
   over several maxima.
   

   This looks at first a catastrophe but the scattering amplitudes depend on the preferred extremal in implicit manner. For instance, the h_eff/h= n depends on extremal. Also quantum classical correspondence (QCC) realized as boundary conditions stating that the classical Noether charges are equal to the eigenvalues of fermionic charges in Cartan algebra bring in the dependence of scattering amplitudes on preferred extremal. Furhermore, the maxima of Kähler function could correspond to the points of WCW for which WCW coordinates are in the extension of rationals: if the exponent of action is such a coordinate this could be the case.
   

   One could see the situation in two manners. The standard view in which preferred extremals are maxima of Kähler function, whose exponentials however disappear from the scattering amplitudes, and the number theoretic view in which maxima correspond to WCW points in the intersection of real and various p-adic WCWs defining cognitive representation at the level of WCW similar to that provided by the discretization at the level of space-time surface. Maybe there is a maximization of cognitive information (classical correlate for NMP): say in the sense that the number of points in the intersection of real and p-adic space-time surfaces is maximal for the preferred extremals.
   

   This kind of connection would mean deep connection between cognition and sensory perception, p-adic physics and real physics, and geometric and number theoretic views about physics.
   

Trouble with cosmological constant

Also an unpleasant observation about cosmological constant forces to challenge the original view about twistor lift.

1. The original vision for the p-adic evolution of cosmological constant assumed that α_K(M⁴) and α_K(CP₂) are different for the twistor lift. This is definitely somewhat ad hoc choice but in principle possible. If one assumes that the Kähler form has also M⁴ part J(M⁴) this option becomes very artificial. In fact, the assumption
   that the twistor space M⁴× S² associated with M⁴ allows Kähler structure, J(M⁴) must be non-vanishing and is completely fixed. It is now clear that
   J(M⁴) allows to understand both CP breaking and parity breaking (in particular chiral selection in living matter). The introduction of moduli space for CDs means also introduction of moduli space for the choices of J(M⁴), which is nothing but the twistor space T(M⁴)!
   
   
2. One indeed finds in the more geometric formulation of 6-D Kähler action that single value of α_K is the only natural choice. The nice outcome guaranteeing NTU is that the preferred extremals do not depend on the coupling parameters at all. In the original version one had to assume that extremals of Kähler action are also minimal surfaces to guarantee this.
   
   
3. One however loses the original proposal for the p-adic length scale evolution of cosmological constant explaining why it is so small in cosmological scale. The solution to the problem would be that the entire 6-D action decomposing to 4-D Kähler action and volume term is identified in terms of cosmological constant. The cancellation of Kähler electric contribution and remaining contributions would explain why the cosmological constant is so small in cosmological scales and also allows to understand p-adic coupling constant evolution of cosmological constant.
   

   One important implication is that there are two kind string like objects. Those for which string tension is very large and which are analogous to the strings of super-string theories and those for which string tension is small due to the cancellation of Kähler action and volume term. These strings appear in all scales and they also mediate gravitational interaction. Also hadronic strings are this kind of strings as also elementary particles as string like objects. In this framework one additional reason for the superstring tragedy becomes manifest: they predict only the strings giving rise to a gigantic cosmological constant.
   
   

To sum up, it is fair to say that the twistor lift of TGD has now achieved rather stable form. There are also a lot of details to be polished but this requires only hard work and a lot of counter argumentation. What is so fascinating is that the formalism produces now rather precise predictions and new detailed fresh insights to the basic problems of standard model. The problems of cosmological constant and CP breaking represent only two examples in this respect. There is also an explicit proposal for twistor four-fermion amplitudes and one can understand how the QFT picture with central role played by the loops emerges although there are no loops at the fundamental level: when particles are approximated by point like objects, some tree diagrams are contracted to loop diagrams. Consider only exchange between two particle lines replaced with single line in pointlike this approximation.

See the article About twistor lift of TGD.

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

Articles and other material related to 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 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).

What says TGD? I have considered this here and in the earlier blog posting (see this).

1. M⁴ and CP₂ are unique in allowing twistor space with Kähler structure (in generalized sense for M⁴). If the twistor space T(M⁴)= M⁴× S² having bundle projections to both M⁴ and to the conventional twistor space CP₃, or rather its non-compact version) allows Kähler structure then also M⁴ allow the generalized Kähler structure and the analog symplectic structure.
   

   This boils down to the existence of self-dual and covariantly constant U(1) gauge field J(M⁴) for which electric and magnetic fields E and B are equal and constant and have the same direction. This field is not dynamical like gauge fields but would characterize the geometry of M⁴. J(M⁴) implies violation Lorentz invariance. TGD however leads to a moduli space for causal diamonds (CDs) effectively labelled by different choices of direction for these self-dual Maxwell fields. The common direction of E and B could correspond to that for spin quantization axis. J(M⁴) has nothing to do with instanton field.
   It should be noticed that also the quantum group inspired attempts to build quantum field theories for which space-time geometry is non-commutative introduce the analog of Kähler form in M⁴, and are indeed plagued by the breaking of Lorentz invariance. Here there is no moduli space saving the situation (see this) .
   
   

2. The choice of quantization axis would therefore have a correlate at the level of "world of classical worlds" (WCW). Different choices would correspond to different sectors of WCW. The moduli space for the choices of preferred point of CP₂ and color quantization axis corresponds to the twistor space T(CP₂)= SU(3)/U(1)× U(1) of WCW. One could interpret also the twistor space T(M⁴)= M⁴× S² as the space with given point representing the position of the tip of CD and the direction of the quantization axis of angular momentum. This choice requires a characterization of a unique rest system and the directions of quantization axis and time axes defines plane M² playing a key role in TGD approach to twstorialization(see this) .
   
   
3. The prediction would be CP violation for a given choice of J(M⁴). Usually this violation would be averaged out in the average over the moduli space for the choices of M² but in some situation this would not happen. Why the CP violation does not average out when there is CKM mixing of quarks? Why the parity violation due to the preferred direction is not compensated by C violation meaning that the directions of E and B fields would be exactly opposite for quarks and antiquarks. Could the fact that quarks are not free but inside hadron induce CP violation? Could a more abstract formulation say that the wave function in the moduli space for J(M⁴) (wave function for the choices of spin quantization axis!) is not CP symmetric and this is reflected in the CKM matrix.
   
   
4. An important delicacy is that J(M⁴) can be both self-dual and anti-self-dual depending on whether the magnetic and electric field have same or opposite directions. It will be found that reflection P and CP transform self-dual J(M⁴) to anti-self-dual one. If only self-dual J(M⁴) is allowed, one has both parity breaking and CP violations at the level of WCW.
   
   

Can one understand the emergence of CP violation in TGD framework?

1. Zero energy state is pair of two positive and negative energy parts. Let us assume that positive energy part is fixed - one can call corresponding boundary of CD passive. This state corresponds to the outcome of state function reduction fixing the direction of quantization axes and producing eigenstates of measured observables, for instance spin. Single system at passive boundary is by definition unentangled with the other systems. It can consists of entangled subsystems hadrons are basic example of systems having entanglement in spin degrees of freedom of quarks: only the total spin of hadron is precisely defined.
   

   The states at the active boundary of CD evolve by repeated unitary steps by the action of the analog of S-matrix and are not anymore eigenstates of single particle observables but entangled. There is a sequence of trivial state function reductions at passive boundary inducing sequence of unitary time evolutions to the state at the active boundary of CD and shifting it. This gives rise to self as a generalized Zeno effect.
   

   Classically the time evolution of hadron corresponds to a superposition of space-time surfaces inside CD. The passive ends of the space-time surface or rather, the quantum superposition of them - is fixed. At the active end one has a superposition of 3-surfaces defining classical correlates for quantum states at the active end: this superposition changes in each unitary step during repeated measurements not affecting the passive end. Also time flows, which means that the distance between the tips of CD defining clock-time increases as the active boundary of CD shifts farther away.
   
   

2. The classical field equations for space-time surface follow from an action, which at space-time level is sum of Kähler action and volume term. If Kähler form at space-time surface is induced (projected to space-time surface) from J=J(M⁴)+J(CP₂), the classical time evolution is CP violating. CKM mixing is induced by different topological mixings for U and D type quarks (recall that 3 particle generations correspond to different genera for partonic 2-surfaces: sphere, torus, and sphere with two handles). J(M⁴)+J(CP₂) defines the electroweak U(1) component of electric field so that J(M⁴) contributes to U(1) part of em field and is thus physically observable.
   
   
3. Topological mixing of quarks corresponds to a superposition of time evolutions for the partonic 2-surfaces, which can also change the genus of partonic 2-surface defined as the number of handles attached to 2-sphere. For instance, sphere can transform to torus or torus to a sphere with two handles. This induces mixing of quantum states. For instance, one can say that a spherical partonic 2-surface containing quark would develop to quantum superposition of sphere, torus, and sphere with two handles. The sequence of state function reductions leaving the passive boundary of CD unaffected (generalized Zeno effect) by shifting the active boundary from its position after the first state function reduction to the passive boundary could but need not give rise to a further evolution of CKM matrix.
   

If the topological mixings are different for U and D type quarks, one obtains CKM mixing. How could the classical time evolution for quarks and for antiquarks as their CP transforms differ? To answer the question one must look how J(M⁴) transforms under C, P, T and CP.

1. J(M⁴)=(J_0z, J_xy= ε J_0z), ε=+/- 1, characterizes hadronic space-time sheet (all space-time sheets in fact). Since J(M⁴) is tensor, P changes only the sign of J_0z giving J(M⁴)→ (-J_0z, J_xy). Since C changes the signs of charges and therefore the signs of fields created by them, one expects J(M⁴)→ -J(M4) under C. CP would give J(M⁴)→ (J_0z, -J_xy) transforming selfdual J(M⁴) to anti-selfdual J(M⁴). If WCW has no anti-self-dual sector, CP is violated at the level of WCW.
   
   
2. If CPT leaves J(M⁴) invariant, one must have J(M⁴) → (J_0z, -J_xy) under T rather than J(M⁴)→ (-J_0z, J_xy). The anti-unitary character of T could correspond for additional change of sign under T. Otherwise CPT should act as J(M⁴)→ -J(M⁴) and only (CPT)² would correspond to unity.
   
   
3. Same considerations apply to J(CP₂) but the difference would be that induced J(M⁴) for space-time surfaces, which are small deformations of M⁴ covariantly constant in good approximation. Also for string world sheets corresponding to small cosmological constant J(M⁴)× J(M⁴)-2≈ 0 holds true in good approximation and induced J(M⁴) at string world sheet is in good approximation covariantly constant. If the string world sheet is just M² characterizing J(M⁴) the condition is exact and was has Kähler electric field induced by J(M⁴) but no corresponding magnetic field. This would make the CP breaking effect large.
   
   

If CP is not violated, particles and their CP transforms correspond to different sectors of WCW with self dual and anti-self dual J(M⁴). If only self-dual sector of WCW is present then CP is violated. Also P is violated at the level of WCW and this parity breaking is different from that associated with weak interactions and could relate to the geometric parity breaking manifesting itself via chiral selection in living matter. Classical time evolutions induce different CKM mixings for quarks and antiquarks reflecting itself in the small imaginary part of the determinant of CKM matrix. CP breaking at the level of WCW could explain also matter-antimatter asymmetry. For instance, antimatter could be dark with different value of h_eff/h=n.

See the articles About twistor lift of TGD and Questions related to the twistor lift of TGD.

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

Articles and other material related to TGD.

Criticizing the TGD based construction of twistor amplitudes

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.

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 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.

1. The fundamental fermions are assigned to the boundaries of string world sheets at the light-like orbits of partonic 2-surfaces: both fermions and bosons are built from them. The classical scatterings of fundamental fermions at the 2-D partonic 2-surface defining the vertices of topological scattering diagrams give rise to scattering amplitudes at the level of fundamental fermions and twistor lift with 8-D light-likeness suggests essentially unique expressions for the 4-fermion vertex.
   
   
2. Elementary particle is modelled as a pair of wormhole contacts (Euclidian signature of metric) connecting two space-time sheets with throats at the two sheets connected by monopole flux tubes. All elementary particles are hadronlike systems but at recent energies the substructure is not visible. The fundamental fermions at the wormhole throats at given space-time sheet are connected by strings. There are altogether 4 wormhole throats per elementary particle in the simplest model.
   

   Elementary boson corresponds to fundamental fermion and antifermion at opposite wormhole throats with very small size (CP₂ size). Elementary fermion has only single fundamental fermion at either throat. There is ν_Lνbar_R pair or its CP conjugate at the other end of the flux tube to neutralize the weak isospin. The flux tube has length of order Compton length (or elementary particle or of weak boson) gigantic as compared to the size of the wormhole contact.
   
   

3. The vertices of topological diagram involve joining of the stringy diagrams associated with elementary particles at their ends defined by wormhole contacts. Wormhole contacts defining the ends of partonic orbits of say 3 interacting particles meet at the vertex - like lines in Feynman diagram - and fundamental fermion scattering redistributes fundamental fermions between the outgoing partonic orbits.
   
   
4. The important point is that there are 2× 2=4 manners for the wormhole contacts at the ends of two elementary particle flux tubes to join together. This makes a possible a diagrams in which particle described by a string like object is emitted at either end and glued back at the other end of string like object. This is basically tree diagram at the level of wormhole contacts but if one looks it at a resolution reducing string to a point, it becomes a loop diagram.
   
   
5. Improvement of the resolution reveals particles inside particles, which can scatter by tree diagrams. This allows to "unloop" the QFT loops. By increasing resolution new space-time sheets with smaller size emerge and one obtains "unlooped" loops in shorter scales. The space-time sheets are characterized by p-adic length scale and primes near powers of 2 are favored. p-Adic coupling constant evolution corresponds to the gradual "unlooping" by going to shorter and shorter p-adic length scales revealing smaller and smaller space-time sheets.
   
   

The loop diagrams of QFTs could thus be seen as a direct evidence of the fractal many-sheeted space-time and quantum criticality and number theoretical universality (NTU) of TGD Universe. Quantum critical dynamics makes the dynamics universal and this explains the unreasonable success of QFT models as far as length scale dependence of couplings constants is considered. The weak point of QFT models is that they are not able to describe bound states: this indeed requires that the extended structure of particles as 3-surfaces is taken into account.

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 h_eff/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.

See the article Questions related to twistor lift TGD and for background the article About twistor lift of TGD.

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

Articles and other material related to TGD.

Questions related to the twistor lift of Kähler action

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, 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 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" or the article with the same title.

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

Articles and other material related to TGD.

Questions related to the quantum aspects of twistorialization

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.

1. There are several notions of twistor. Twistor space for M⁴ is T(M⁴) =M⁴× S² (see this) having projections to both M⁴ and to the standard twistor space T₁(M⁴) often identified as CP₃. T(M⁴)=M⁴× S² is necessary for the twistor lift of space-time dynamics. CP₂ gives the factor T(CP₂)= SU(3)/U(1)× U(1) to the classical twistor space T(H). The quantal twistor space T(M⁸)= T₁(M⁴)× T(CP₂) assignable to momenta. The possible way out is M⁸-H duality relating the momentum space M⁸ (isomorphic to the tangent space H) and H by mapping space-time associative and co-associative surfaces in M⁸ to the surfaces which correspond to the base spaces of in H: they construction would reduce to holomorphy in complete analogy with the original idea of Penrose in the case of massless fields.
   
   
2. The standard twistor approach has problems. Twistor Fourier transform reduces to ordinary Fourier transform only in signature (2,2) for Minkowski space: in this case twistor space is real RP₃ but can be complexified to CP₃. Otherwise the transform requires residue integral to define the transform (in fact, p-adically multiple residue calculus could provide a nice manner to define integrals and could make sense even at space-time level making possible to define action).
   

   Also the positive Grassmannian requires (2,2) signature. In M⁸-H relies on the existence of the decomposition M²⊂ M²= M²× E²⊂ M⁸. M² could even depend on position but M²(x) should define an integrable distribution. There always exists a preferred M², call it M²₀, where 8-momentum reduces to light-like M² momentum. Hence one can apply 2-D variant of twistor approach. Now the signature is (1,1) and spinor basis can be chosen to be real! Twistor space is RP₃ allowing complexification to CP₃ if light-like complex momenta are allowed as classical TGD suggests!
   
   

3. A further problem of the standard twistor approach is that in M⁴ twistor approach does not work for massive particles. In TGD all particles are massless in 8-D sense. In M⁸ M⁴-mass squared corresponds to transversal momentum squared coming from E⁴⊂ M⁴× E⁴ (from CP₂ in H). In particular, Dirac action cannot contain anyo mass term since it would break chiral invariance.
   

   Furthermore, the ordinary twistor amplitudes are holomorphic functions of the helicity spinors λ_i and have no dependence on &lambda tile;_i: no information about particle masses! Only the momentum conserving delta function gives the dependence on masses. These amplitudes would define as such the M⁴ parts of twistor amplitudes for particles massive in TGD sense. The simplest 4-fermion amplitude is unique.
   
   

Twistor approach gives excellent hopes about the construction of the scattering amplitudes in ZEO. The construction would split into two pieces corresponding to the orbital degrees of freedom in "world of classical worlds" (WCW) and to spin degrees of freedom in WCW: that is spinors, which correspond to second quantized induced spinor fields at space-time surface (actually string world sheets- either at fundamental level or for effective action implied by strong form of holography (SH)).

1. At WCW level there is a perturbative functional integral over small deformations of the 3-surface to which space-time surface is associated. The strongest assumption is that this 3-surface corresponds to maximum for the real part of action and to a stationary phase for its imaginary part: minimal surface extremal of Kähler action would be in question. A more general but number theoretically problematic option is that an extremal for the sum of Kähler action and volume term is in question.
   

   By Kähler geometry of WCW the functional integral reduces to a sum over contributions from preferred extremals with the fermionic scattering amplitude multiplied by the ration X_i/X, where X=∑_i X_i is the sum of the action exponentials for the maxima. The ratios of exponents are however number theoretically problematic.
   

   Number theoretical universality is satisfied if one assigns to each maximum independent zero energy states: with this assumption ∑ X_i reduces to single X_i and the dependence on action exponentials becomes trivial! ZEO allow this. The dependence on coupling parameters of the action essential for the discretized coupling constant evolution is only via boundary conditions at the ends of the space-time surface at the boundaries of CD.
   

   Quantum criticality of TGD demands that the sum over loops associated with the functional integral over WCW vanishes and strong form of holography (SH) suggests that the integral over 4-surfaces reduces to that over string world sheets and partonic 2-surfaces corresponding to preferred extremals for which the WCW coordinates parametrizing them belong to the extension of rationals defining the adele. Also the intersections of the real and various p-adic space-time surfaces belong to this extension.
   
   

2. Second piece corresponds to the construction of twistor amplitude from fundamental 4-fermion amplitudes. The diagrams consists of networks of light-like orbits of partonic two surfaces, whose union with the 3-surfaces at the ends of CD is connected and defines a boundary condition for preferred extremals and at the same time the topological scattering diagram.
   

   Fermionic lines correspond to boundaries of string world sheets. Fermion scattering at partonic 2-surfaces at which 3 partonic orbits meet are analogs of 3-vertices in the sense of Feynman and fermions scatter classically. There is no local 4-vertex. This scattering is assumed to be described by simplest 4-fermion twistor diagram. These can be fused to form more complex diagrams. Fermionic lines runs along the partonic orbits defining the topological diagram.
   
   

3. Number theoretic universality suggests that scattering amplitudes have interpretation as representations for computations. All space-time surfaces giving rise to the same computation wold be equivalent and tree diagrams corresponds to the simplest computation. If the action exponentials do not appear in the amplitudes as weights this could make sense but would require huge symmetry based on two moves. One could glide the 4-vertex at the end of internal fermion line along the fermion line so that one would eventually get the analog of self energy loop, which should allow snipping away. An argument is developed stating that this symmetry is possible if the preferred M²₀ for which 8-D momentum reduces to light-like M²-momentum having unique direction is same along entire fermion line, which can wander along the topological graph.
   

   The vanishing of topological loops would correspond to the closedness of the diagrams in what might be called BCFW homology. Boundary operation involves removal of BCFW bridge and entangled removal of fermion pair. The latter operation forces loops. There would be no BCFW bridges and entangled removal should give zero. Indeed, applied to the proposed four fermion vertex entangled removal forces it to correspond to forward scattering for which the proposed twistor amplitude vanishes.
   
   

To sum up, the twistorial approach leads to a proposal for an explicit construction of scattering amplitudes for the fundamental fermions. Bosons and fermions as elementary particles are bound states of fundamental fermions assignable to pairs of wormhole contacts carrying fundamental fermions at the throats. Clearly, this description is analogous to a quark level description of hadron. Yangian symmetry with multilocal generators is expected to crucial for the construction of the many-fermion states giving rise to elementary particles. The problems of the standard twistor approach find a nice solution in terms of M⁸-H duality, 8-D masslessness, and holomorphy of twistor amplitudes in λ_i and their indepence on &lambda tilde;_i.

See the new chapter Some Questions Related to the Twistor Lift of TGD of "Towards M-matrix".

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

Articles and other material related to TGD.

A new view about color, color confinement, and twistors

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 (notice that this posting is a corrected version of earlier).

1. As Witten shows, the twistor transform is problematic in signature (1,3) for Minkowski space since the the bi-spinor μ playing the role of momentum is complex. Instead of defining the twistor transform as ordinary Fourier integral, one must define it as a residue integral. In signature (2,2) for space-time the problem disappears since the spinors μ can be taken to be real.
   
   
2. The twistor Grassmannian approach works also nicely for (2,2) signature, and one ends up with the notion of positive Grassmannians, which are real Grassmannian manifolds. Could it be that something is wrong with the ordinary view about twistorialization rather than only my understanding of it?
   
   
3. For M⁴ the twistor space should be non-compact SU(2,2)/SU(2,1)× U(1) rather than CP₃= SU(4)/SU(3)× U(1), which is taken to be. I do not know whether this is only about short-hand notation or a signal about a deeper problem.
   
   
4. Twistorilizations does not force SUSY but strongly suggests it. The super-space formalism allows to treat all helicities at the same time and this is very elegant. This however forces Majorana spinors in M⁴ and breaks fermion number conservation in D=4. LHC does not support N=1 SUSY. Could the interpretation of SUSY be somehow wrong? TGD seems to allow broken SUSY but with separate conservation of baryon and lepton numbers.
   
   

In number theoretic vision something rather unexpected emerges and I will propose that this unexpected might allow to solve the above problems and even more, to understand color and even color confinement number theoretically. First of all, a new view about color degrees of freedom emerges at the level of M⁸.

1. One can always find a decomposition M⁸=M²₀× E⁶ so that the complex light-like quaternionic 8-momentum restricts to M²₀. The preferred octonionic imaginary unit represent the direction of imaginary part of quaternionic 8-momentum. The action of G₂ to this momentum is trivial. Number theoretic color disappears with this choice. For instance, this could take place for hadron but not for partons which have transversal momenta.
   
   
2. One can consider also the situation in which one has localized the 8-momenta only to M⁴ =M²₀× E². The distribution for the choices of E² ⊂ M²₀× E²=M⁴ is a wave function in CP₂. Octonionic SU(3) partial waves in the space CP₂ for the choices for M²₀× E² would correspond ot color partial waves in H. The same interpretation is also behind M⁸-H correspondence.
   
   
3. The transversal quaternionic light-like momenta in E²⊂ M²₀× E² give rise to a wave function in transversal momenta. Intriguingly, the partons in the quark model of hadrons have only precisely defined longitudinal momenta and only the size scale of transversal momenta can be specified.
   

   The introduction of twistor sphere of T(CP₂) allows to describe electroweak charges and brings in CP₂ helicity identifiable as em charge giving to the mass squared a contribution proportional to Q_em² so that one could understand electromagnetic mass splitting geometrically.
   

   The physically motivated assumption is that string world sheets at which the data determining the modes of induced spinor fields carry vanishing W fields and also vanishing generalized Kähler form J(M⁴) +J(CP₂). Em charge is the only remaining electroweak degree of freedom. The identification as the helicity assignable to T(CP₂) twistor sphere is natural.
   
   

4. In general case the M² component of momentum would be massive and mass would be equal to the mass assignable to the E⁶ degrees of freedom. One can however always find M²₀× E⁶ decomposition in which M² momentum is light-like. The naive expectation is that the twistorialization in terms of M² works only if M² momentum is light-like, possibly in complex sense. This however allows only forward scattering: this is true for complex M² momenta and even in M⁴ case.
   

   The twistorial 4-fermion scattering amplitude is however holomorphic in the helicity spinors λ_i and has no dependence on λtilde;_i. Therefore carries no information about M² mass! Could M² momenta be allowed to be massive? If so, twistorialization might make sense for massive fermions!
   

M²₀ momentum deserves a separate discussion.

1. A sharp localization of 8-momentum to M²₀ means vanishing E² momentum so that the action of U(2) would becomes trivial: electroweak degree of freedom would simply disappear, which is not the same thing as having vanishing em charge (wave function in T(CP₂) twistorial sphere S² would be constant). Neither M²₀ localization nor localization to single M⁴ (localization in CP₂) looks plausible physically - consider only the size scale of CP₂. For the generic CP₂ spinors this is impossible but covariantly constant right-handed neutrino spinor mode has no electro-weak quantum numbers: this would most naturally mean constant wave function in CP₂ twistorial sphere.
   

   For the preferred extremals of twistor lift of TGD either M⁴ or CP₂ twistor sphere can effectively collapse to a point. This would mean disappearence of the degrees of freedom associated with M⁴ helicity or electroweak quantum numbers.
   
   

2. The localization to M⁴⊃ M²₀ is possible for the tangent space of quaternionic space-time surface in M⁸. This could correlate with the fact that neither leptonic nor quark-like induced spinors carry color as a spin like quantum number. Color would emerge only at the level of H and M⁸ as color partial waves in WCW and would require de-localization in the CP₂ cm coordinate for partonic 2-surface. Note that also the integrable local decompositions M⁴= M²(x)× E²(x) suggested by the general solution ansätze for field equations are possible.
   
   
3. Could it be possible to perform a measurement localization the state precisely in fixed M²₀ always so that the complex momentum is light-like but color degrees of freedom disappear? This does not mean that the state corresponds to color singlet wave function! Can one say that the measurement eliminating color degrees of freedom corresponds to color confinement. Note that the subsystems of the system need not be color singlets since their momenta need not be complex massless momenta in M²₀. Classically this makes sense in many-sheeted space-time. Colored states would be always partons in color singlet state.
   
   
4. At the level of H also leptons carry color partial waves neutralized by Kac-Moody generators, and I have proposed that the pion like bound states of color octet excitations of leptons explain so called lepto-hadrons. Only right-handed covariantly constant neutrino is an exception as the only color singlet fermionic state carrying vanishing 4-momentum and living in all possible M²₀:s, and might have a special role as a generator of supersymmetry acting on states in all quaternionic subs-spaces M⁴.
   
   
5. Actually, already p-adic mass calculations performed for more than two decades ago forced to seriously consider the possibility that particle momenta correspond to their projections o M²₀⊂ M⁴. This choice does not break Poincare invariance if one introduces moduli space for the choices of M²₀⊂ M⁴ and the selection of M²₀ could define quantization axis of energy and spin. If the tips of CD are fixed, they define a preferred time direction assignable to preferred octonionic real unit and the moduli space is just S². The analog of twistor space at space-time level could be understood as T(M⁴)=M⁴× S² and this one must assume since otherwise the induction of metric does not make sense.
   
   

What happens to the twistorialization at the level of M⁸ if one accepts that only M²₀ momentum is sharply defined?

1. What happens to the conformal group SO(4,2) and its covering SU(2,2) when M⁴ is replaced with M²₀⊂ M⁸? Translations and special conformational transformation span both 2 dimensions, boosts and scalings define 1-D groups SO(1,1) and R respectively. Clearly, the group is 6-D group SO(2,2) as one might have guessed. Is this the conformal group acting at the level of M⁸ so that conformal symmetry would be broken? One can of course ask whether the 2-D conformal symmetry extends to conformal symmetries characterized by hyper-complex Virasoro algebra.
   
   
2. Sigma matrices are by 2-dimensionality real (σ₀ and σ₃ - essentially representations of real and imaginary octonionic units) so that spinors can be chosen to be real. Reality is also crucial in signature (2,2), where standard twistor approach works nicely and leads to 3-D real twistor space.
   

   Now the twistor space is replaced with the real variant of SU(2,2)/SU(2,1)× U(1) equal to SO(2,2)/SO(2,1), which is 3-D projective space RP³ - the real variant of twistor space CP₃, which leads to the notion of positive Grassmannian: whether the complex Grassmannian really allows the analog of positivity is not clear to me. For complex momenta predicted by TGD one can consider the complexification of this space to CP₃ rather than SU(2,2)/SU(2,1)× U(1). For some reason the possible problems associated with the signature of SU(2,2)/SU(2,1)× U(1) are not discussed in literature and people talk always about CP₃. Is there a real problem or is this indeed something totally trivial?
   

   
   

3. SUSY is strongly suggested by the twistorial approach. The problem is that this requires Majorana spinors leading to a loss of fermion number conservation. If one has D=2 only effectively, the situation changes. Since spinors in M² can be chosen to be real, one can have SUSY in this sense without loss of fermion number conservation! As proposed earlier, covariantly constant right-handed neutrino modes could generate the SUSY but it could be also possible to have SUSY generated by all fermionic helicity states. This SUSY would be however broken.
   
   
4. The selection of M²₀ could correspond at space-time level to a localization of spinor modes to string world sheets. Could the condition that the modes of induced spinors at string world sheets are expressible using real spinor basis imply the localization? Whether this localization takes place at fundamental level or only for effective action being due to SH, is a question to be settled. The latter options looks more plausible.
   
   

To sum up, these observation suggest a profound re-evalution of the beliefs related to color degrees of freedom, to color confinement, and to what twistors really are.

For details see the new chapter Some Questions Related to the Twistor Lift of TGD of "Towards M-matrix" of "Towards M-matrix" or the article Some questions related to the twistor lift of TGD.

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

Articles and other material related to TGD.

How does the twistorialization at imbedding space level emerge?

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.

For details see the new chapter Some Questions Related to the Twistor Lift of TGD of "Towards M-matrix" or the article Some questions related to the twistor lift of TGD.

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

Articles and other material related to TGD.

Twistor lift and the reduction of field equations by SH to holomorphy

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.

1. If S² for M⁴ has 2 time-like dimensions one has 3+3 dimensions, and one can speak about hyper-complex variants of holomorphic functions with time-like and space-like coordinate paired for all three hypercomplex coordinates. For the Minkowskian regions of the space-time surface X⁴ the situation is the same.
   
   
2. For T(CP₂) Euclidian signature of twistor sphere guarantees this and one has 3 complex coordinates corresponding to those of S² and CP₂. One can also now also pair two real coordinates of S² with two coordinates of CP₂ to get two complex coordinates. For the Euclidian regions of the space-time surface the situation is the same.
   

Consider now what the general solution could look like. Let us continue to use the shorthand notations S²₁= S²(X⁴); S²₂= S²(CP₂);S²₃= S²(M⁴).

1. Consider first solution of type (1,0) so that coordinates of S²₂ are constant. One has holomorphy in hypercomplex sense (light-like coordinate t-z and t+z correspond to hypercomplex coordinates).
   
   
   1. The general map T(X⁴) to T(M⁴) should be holomorphic in hyper-complex sense. S²₁ is in turn identified with S²₃ by isometry realized in real coordinates. This could be also seen as holomorphy but with different imaginary unit. One has analytical continuation of the map S²₁→ S²₃ to a holomorphic map. Holomorphy might allows to achieve this rather uniquely. The continued coordinates of S²₁ correspond to the coordinates assignable with the integrable surface defined by E²(x) for local M²(x)× E²(x) decomposition of the local tangent space of X⁴. Similar condition holds true for T(M⁴). This leaves only M²(x) as dynamical degrees of freedom. Therefore one has only one holomorphic function defined by 1-D data at the surface determined by the integrable distribution of M²(x) remains. The 1-D data could correspond to the boundary of the string world sheet.
      
      
   2. The general map T(X⁴) to T(CP₂) cannot satisfy holomorphy in hyper-complex sense. One can however provide the integrable distribution of E²(x) with complex structure and map it holomorphically to CP₂. The map is defined by 1-D data.
      
      
   3. Altogether, 2-D data determine the map determining space-time surface. These two 1-D data correspond to 2-D data given at string world sheet: one would have SH.
      
      
2. What about solutions of type (0,1) making sense in Euclidian region of space-time? One has ordinary holomorphy in CP₂ sector.
   
   
   1. The simplest picture is a direct translation of that for Minkowskian regions. The map S²₁→ S²₂ is an isometry regarded as an identification of real coordinates but could be also regarded as holomorphy with different imaginary unit. The real coordinates can be analytically continued to complex coordinates on both sides, and their imaginary parts define coordinates for a distribution of transversal Euclidian spaces E²₂(x) on X⁴ side and E²(x) on M⁴ side. This leaves 1-D data.
      
      
   2. What about the map to T(M⁴)? It is possible to map the integrable distribution E²₂(x) to the corresponding distribution for T(M⁴) holomorphically in the ordinary sense of the word. One has 1-D data. Altogether one has 2-D data and SH and partonic 2-surfaces could carry these data. One has SH again.
      
      
3. The above construction works also for the solutions of type (1,1), which might make sense in Euclidian regions of space-time. It is however essential that the spheres S²₂ and S²₃ have real coordinates.
   
   

SH thus would thus emerge automatically from the twistor lift and holomorphy in the proposed sense.

1. Two possible complex units appear in the process. This suggests a connection with quaternion analytic functions suggested as an alternative manner to solve the field equations. Space-time surface as associative (quaterionic) or co-associate (co-quaternionic) surface is a further solution ansatz.
   

   Also the integrable decompositions M²(x)× E²(x) resp. E²₁(x)× E²₂(x) for Minkowskian resp. Euclidian space-time regions are highly suggestive and would correspond to a foliation by string wold sheets and partonic 2-surfaces. This expectation conforms with the number theoretically motivated conjectures.
   
   

2. The foliation gives good hopes that the action indeed reduces to an effective action consisting of an area term plus topological magnetic flux term for a suitably chosen stringy 2-surfaces and partonic 2-surfaces. One should understand whether one must choose the string world sheets to be Lagrangian surfaces for the Kähler form including also M⁴ term. Minimal surface condition could select the Lagrangian string world sheet, which should also carry vanishing classical W fields in order that spinors modes can be eigenstates of em charge.
   

   The points representing intersections of string world sheets with partonic 2-surfaces defining punctures would represent positions of fermions at partonic 2-surfaces at the boundaries of CD and these positions should be able to vary. Should one allow also non-Lagrangian string world sheets or does the space-time surface depend on the choice of the punctures carrying fermion number (quantum classical correspondence)?
   
   

3. The alternative option is that any choice produces of the preferred 2-surfaces produces the same scattering amplitudes. Does this mean that the string world sheet area is a constant for the foliation - perhaps too strong a condition - or could the topological flux term compensate for the change of the area?
   

   The selection of string world sheets and partonic 2-surfaces could indeed be also only a gauge choice. I have considered this option earlier and proposed that it reduces to a symmetry identifiable as U(1) gauge symmetry for Kähler function of WCW allowing addition to it of a real part of complex function of WCW complex coordinates to Kähler action. The additional term in the Kähler action would compensate for the change if string world sheet action in SH. For complex Kähler action it could mean the addition of the entire complex function.
   
   

For details see the new chapter Some Questions Related to the Twistor Lift of TGD of "Towards M-matrix" or the article Some questions related to the twistor lift of TGD.

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

Articles and other material related to TGD.