Pollack Effect and Some Anomalies of Water

In the  Pollack effect (PE) negatively charged exclusion zones (EZs) are induced at the boundary between the  gel phase and water  by an energy feed such as IR radiation.  Pollack has  introduced the notion of fourth phase of water, which obeys effective stoichiometry H_1.5O and consists of hexagonal layers  having therefore an  ice-like structure. EZs e are able to clean up inpurities from their interior, which seems to be in conflict with the second law of thermodynamics. I have collected in the article Pollack Effect and Some Anomalies of Water examples of hydrodynamic anomalies, which might have an explanation in terms of the Pollack effect.  

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Some facts about birational geometry

Birational geometry has as its morphisms birational maps: both the map and its inverse are expressible in terms of rational functions. The coefficients of polynomials appearing in rational functions are in the TGD framework rational. They map rationals to rationals and also numbers of given extension E of rationals to themselves (one can assign to each space-time region an extension defined by a polynomial).

Therefore birational maps map cognitive representations, defined as discretizations of the space-time surface such that the points have physically/number theoretically preferred coordinates in E, to cognitive representations. They therefore respect cognitive representations and are morphisms of cognition. They are also number-theoretically universal, making sense for all p-adic number fields and their extensions induced by E. This makes birational maps extremely interesting from the TGD point of view.

The following lists basic facts about birational geometry as I have understood them on the basis of Wikipedia articles about birational geometry and Enriques-Kodaira classification. I have added physics inspired associations with TGD.

Birational geometries are one central approach to algebraic geometry.

1. They provide classification of complex varieties to equivalence classes related by birational maps. The classification complex curves (real dimension 2) is best understood and reduces to the classification of projective curves of projective space CP_n determined as zeros of a homogeneous polynomial. I had good luck since complex surfaces (real dimension 4) are of obvious interest in TGD: now however the notion of complex structure is generalized and one has Hamilton-Jacobi structure and Minkowski signature is allowed.
2. In TGD, a generalization of complex surfaces of complex dimension 2 in the embedding space H=M⁴× CP₂ of complex dimension 4 is considered. What is new is the presence of the Minkowski signature requiring a combination of hypercomplex and complex structures to the Hamilton-Jacobi structure. Note however the space-time surfaces also have counterparts in the Euclidean signature E⁴× CP₂: whether this has a physical interpretation, remains an open question. Second representation is provided as 4-surfaces in the space M⁸_c of complexified octonions and an attractive idea is that M⁸-H duality corresponds to a birational mapping of cognitive representations to cognitive representations.
3. Every algebraic variety is birationally equivalent with a sub-variety of CP_n so that their classification reduces to the classification of projective varieties of CP_n defined in terms of homogeneous polynomials. n=2 (4 real dimensions) is of special relevance from the TGD point of view. A variety is said to be rational if it is birationally equivalent to some projective variety: for instance CP₂ is rational.
4. A concrete example of birational equivalence is provided by stereographic projections of quadric hypersurfaces in n+1-D linear space. Circle in plane is the simplest example. Let p be a point of quadric. The stereographic projection sends a point q of the quadric to the line going through p and q, that is a point of CP_n in the complex case. One can select one point on the line as its representative. Another exammple is provided by Möbius transformations representing Lorentz group as transformations of complex plane.

The notion of a minimal model is important.

1. The basic observation is that it is possible to eliminate or add singularities by using birational maps of the space in which the surface is defined to some other spaces, which can have a higher dimension. Peaks and self-intersections are examples of singularities. The zeros of a birational map can be used to eliminate singularities of the algebraic surface of dimension n by blowups replacing the singularity with CP_n. Poles in turn create singularities.

   The idea is to apply birational maps to find a birationally equivalent surface representation, which has no singularities. There is a very counter-intuitive formal description for this. For instance, complex curves of CP₂ have intersections since their sum of their real dimensions is 4. The same applies to 4-surfaces in H. My understanding is as follows: the blowup for CP₂ makes it possible to get rid of an intersection with intersection number 1. One can formally say that the blow up by gluing a CP₁ defines a curve which has negative intersection number -1.

2. In the TGD framework, wormhole contacts are Euclidian regions of space-time surface, which have the same metric and Kähler structure as CP₂ and light-like M⁴ projection (or even H projection). They appear as blowups of singularities of 4-surfaces along a light-like curve of M⁸. The union of the quaternionic/associative normal spaces along the curve is not a line of CP₂ but CP₂ itself with two holes corresponding to the ends of the light-like curve. The 3-D normal spaces at the points of the light-like curve are not unique and form a local slicing of CP₂ by 3-D surfaces. This is a Minkowskian analog of a blow-up for a point and also an analog of cut of analytic function.
3. The Italian school of algebraic geometry has developed a rather detailed classification of these surfaces. The main result is that every complex surface X is birational either to a product CP¹× C for some curve C or to a minimal surface Y. Preferred extremals are indeed minimal surfaces so that space-time surfaces might define minimal models. The absence of singularities (typically peaks or self-intersections) characterizing minimal models is indeed very natural since physically the peaks do not look acceptable.

Mathematicians use invariants to characterize mathematical structures. In TGD birational invariants would be cognitive invariants. They would be extremely interesting physically if the 4-D generalization of holomorphy really to a fusion of complex and hypercomplex structrures make sense (see this and this).

There are several birationals invariants listed in the Wikipedia article. Many of them are rather technical in nature. The canonical bundle K_X for a variety of complex dimension n corresponds to n:th exterior power of complex cotangent bundle that is holomorphic n-forms. For space-time surfaces one would have n=2 and holomorphic 2-forms.

1. Plurigenera corresponds to the dimensions for the vector space of global sections H₀(X,K_X^d) for smooth projective varieties and are birational invariants. The global sections define global coordinates, which define birational maps to a projective space of this dimension.
2. Kodaira dimension measures the complexity of the variety and characterizes how fast the plurigenera increase. It has values -∞,0,1,..n and has 4 values for space-time surfaces. The value -∞ corresponds to the simplest situation and for n=2 characterizes CP₂, which is rational and has vanishing plurigenera.
3. The dimensions for the spaces of global sections of the tensor powers of complex cotangent bundle (holomorphic 1-forms) define birational invariants. In particular, holomorphic forms of type (p,0) are birational invariants unlike the more general forms having type (p,q). Betti numbers are not in general birational invariants.
4. Fundamental group is birational invariant as is obvious from the blowup construction. Other homotopy groups are not birational invariants.
5. Gromow-Witten invariants are birational invariants. They are defined for pseudo-holomorphic curves (real dimension 2) in a symplectic manifold X. These invariants give the number of curves with a fixed genus and 2-homology class going through n marked points. Gromow-Witten invariants have also an interpretation as symplectic invariants characterizing the symplectic manifold X.

   In TGD, the application would be to partonic 2-surfaces of given genus g and homology charge (Kähler magnetic charge) representatable as holomorphic surfaces in X=CP₂ containing n marked points of CP₂ identifiable as the loci of fermions at the partonic 2-surface. This number would be of genuine interest in the calculation of scattering amplitudes.

What birational classification could mean in the TGD framework?

1. Holomorphic ansatz gives the space-time surfaces as Bohr orbits. Birational maps give new solutions from a given solution. It would be natural to organize the Bohr orbits to birational equivalence classes, which might be called cognitive equivalence classes. This should induce similar organization at the level of M⁸_c.
2. An interesting possibility is that for certain space-time surfaces CP₂ coordinates can be expressed in terms of preferred M⁴ coordinates using birational functions and vice versa. Cognitive representation in M⁴ coordinates would be mapped to a cognitive representation in CP₂ coordinates.
3. The interpretation of M⁸-H duality as a generalization of momentum position duality suggests information theoretic interpretation and the possibility that it could be seen as a cognitive/birational correspondence. This is indeed the case M⁴ when one considers linear M⁴ coordinates at both sides.
4. An intriguing question is whether the pair of hypercomplex and complex coordinates associated with the Hamilton-Jacobi structure could be regarded as cognitively acceptable coordinates. If Hamilton-Jacobi coordinates are cognitively acceptable, they should relate to linear M⁴ coordinates by a birational correspondence so that M⁸-H duality in its basic form could be replaced with its composition with a coordinate transformation from the linear M⁴ coordinates to particular Hamilton-Jacobi coordinates. The color rotations in CP₂ in turn define birational correspondences between different choices of Eguchi-Hanson coordinates.

   If this picture makes sense, one could say that the entire holomorphic space-time surfaces, rather than only their intersections with mass shells H³ and partonic orbits, correspond to cognitive explosions. This interpretation might make sense since holomorphy has a huge potential for generating information: it would make TGD exactly solvable.

See the article Birational maps as morphisms of cognitive structures or the chapter New findings related to the number theoretical view of TGD.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Cymatics, ringing bells, water memory, homeopathy, Pollack effect, turbulence

Warning: This post contains many words, which induce deep aggression in academic colleagues receiving a monthly salary: cymatics, the ringing bells of Buddhist monks, water memory, and homeopathy(!!). Pollack effect is perhaps not so aggression inducing and turbulence is quite neutral. All these words are linked: this is message that I try to communicate in the following.

Cymatics (see this) is a very interesting phenomenon. Thanks to Jukka Sarno for a post inspiring this comment. I lost the original link: Facebook has started to suddenly change the page content completely and this makes it very difficult to respond to the posts. Maybe some kind of virus is in question.

I came across a related phenomenon recently. The ringing of Buddhist monks' bells by running the bell along its edge has strange effects. The water started to boil so that a strong transfer of energy had to happen to the water by sound. Energy was supplied to the system by the ringer of the bells. This energy could play a role of metabolic energy and help in the problems resulting from its local deficiency in the patient's body.

Something analogous to turbulence also arises in cymatics. Turbulence and its generation are very interesting phenomena and poorly understood. Standard hydrodynamics, which was developed centuries ago, can't really cope with the challenges of the modern world: if only someone could tell this to the theoreticians working on it!

I myself have built a model for turbulence and related phenomena (see this and this). A core element of the model is the anomalous phenomenon observed by Pollack related to water. When water is irradiated in the presence of a gel phase with, for example, infrared light, negatively charged gel-like volumes are created in the water: Pollack talks about the fourth phase of water. Living matter is full of them: for instance cell interior is negatively charged as also DNA.

Some of the water's protons disappear somewhere: in the TGD world they would go to the magnetic body of the water and form dark matter there precisely because we cannot detect them with standard methods. This dark matter would be a phase of ordinary matter with a nonstandard, and often very large value of effective Planck constant. This would make it quantum coherent in much longer scales than ordinary matter.

Pollack's fourth phase resembles ice and very recently it has been discovered that there is a thin ice-like layer at the interface between water and air (see this and this). Could it be Pollack's fourth phase? The energy input is essential. In cymatics and in the case of bells the energy feeder would be sound rather than light. In homeopathy (one of the most hated phenomena of physics besides water memory; I have never understood why it generates so deep a hatred), the shaking of the homeopathic preparation would supply the energy. A fourth phase of water would be created and the water would become "living" as its magnetic body would "wake up" and start to control ordinary matter.

Homeopathy is one of the most hated phenomena of physics besides water memory; I have never understood why it generates so deep hatred), the shaking of the homeopathic preparation would supply the energy. A fourth phase of water would be created and the water would become "living" as its magnetic body would "wake up" and start to control ordinary matter.

In homeopathy, shaking would provide the energy making it possible to create magnetic organisms consisting of flux tubes associated with water molecule clusters connected by hydrogen bonds. Their cyclotron frequency spectrum would mimic the corresponding spectrum of the molecules dissolved in water. Water would magnetically mimic the intruder molecule and from the perspective of biology this would be enough for water memory explaining homeopathic effects. This should be trivial for scientists living in the computer age but some kind of primitive regression makes it impossible for colleagues to stay calm and rational when they hear the word "homeopathy".

For a summary of earlier postings see Latest progress in TGD. For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Birational maps as morphisms of cognitive structures

Birational maps and their inverses are defined in terms of rational functions. They are very special in the sense that they map algebraic numbers in a given extension E of rationals to E itself. In the TGD framework, E defines a unique discretization of the space-time surface if the preferred coordinates of the allowed points belong to E. I refer to this discretization as cognitive representation. Birational maps map points in E to points in E so that they define what might be called cognitive morphism.

M⁸-H duality duality (H=M⁴× CP₂) relates the number vision of TGD to the geometric vision. M⁸-H duality maps the 4-surfaces in M⁸_c to space-time surfaces in H: a natural condition is that in some sense it maps E to E and cognitive representations to cognitive representations. There are special surfaces in M⁸_c that allow cognitive explosion in the number-theotically preferred coordinates. M⁴ and hyperbolic spaces H³ (mass shells), which contain 3-surfaces defining holographic data, are examples of these surfaces. Also the 3-D light-like partonic orbits defining holographic data. Possibly also string world sheets define holographic data. Does cognitive explosion happen also in these cases?

In M⁸_c octonionic structure allows to identify natural preferred coordinates. In H, in particular M⁴, the preferred coordinates are not so unique but should be related by birational mappings. So called Hamilton-Jacobi structures define candidates for preferred coordinates: could different Hamilton-Jacobi structures relate to the each other by birational maps? In this article these questions are discussed.

See the article Birational maps as morphisms of cognitive structures or the chapter New findings related to the number theoretical view of TGD.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Boundary conditions at partonic orbits and holography

TGD reduces coupling constant evolution to a number theoretical evolution of the coupling parameters of the action identified as Kähler function for WCW. An interesting question is how the 3-D holographic data at the partonic orbits relates to the corresponding 3-D data at the ends of space-time surfaces at the boundary of CD, and how it relates to coupling constant evolution.

1. The twistor lift of TGD strongly favours 6-D Kähler action, which dimensionally reduces to Kähler action plus volume term plus topological ∫ J∧ J term reducing to Chern Simons-Kähler  action. The coefficients of these terms are proposed to be expressible in terms of number theoretical invariants characterizing the algebraic extensions of rationals and polynomials determining the space-time surfaces by M⁸-H duality.

   Number theoretical coupling constant evolution would be discrete. Each extension of rationals would give rise to its own coupling parameters involving also the ramified primes characterizing the polynomials involved and identified as p-adic length scales.

2. The time evolution of the partonic orbit would be non-deterministic but subject to the light-likeness constraint and boundary conditions guaranteeing conservation laws. The natural expectation is that the boundary/interface conditions for a given action cannot be satisfied for all partonic orbits (and other singularities). The deformation of the partonic orbit requiring that boundary conditions are satisfied,  does not affect X³  but   the time derivatives ∂_t h^k at X³  are affected since the form of the holomorphic functions defining the space-time surface would change.   The interpretation would be in terms of duality of the holographic data associated with the partonic orbits resp. X³.  

    There can of course exist deformations, which require the change of the coupling parameters of the action to satisfy the boundary conditions. One can consider an analog of  renormalization group equations in which the deformation corresponds to a modification of the  coupling parameters of the action, most plausibly determined by the  twistor lift. Coupling parameters would label different regions of WCW and  the space-time surfaces possible for two different sets of coupling parameters would define interfaces between these regions.

 In order to build a more detailed view one must fix the details related to the action whose value defines the WCW  Kähler function.  

1. If Kähler action is identified as Kähler action, the identification is unique. There is however the possibility that the imaginary exponent of the instanton term or the contribution from the Euclidean region is not included in the definition of Kähler function. For instance instanton term could be  interpreted as a phase of quantum state and would not contribute.
2. Both Minkowskian and Euclidean regions are involved and the Euclidean signature poses problems. The definition of the  determinant as (-g₄)^1/2 is natural in Minkowskian regions but gives an imaginary contribution in Euclidean regions. (|g₄|)^1/2 is real in both regions. i(g₄)^1/2 is real in Minkowskian regions but imaginary in the Euclidean regions.

   There is also a problem related to the instanton term, which does not depend on the  metric determinant at all.  In QFT context the instanton term is imaginary and this is important for instance in QCD in  the definition of CP breaking vacuum functional. Should one include only the 4-D  or possibly only Minkowskian contribution to the Kähler function  imaginary coefficient for the instanton/Euclidian term would be possible?

3.   Boundary conditions guaranteeing the conservation laws at the partonic orbits must be satisfied. Consider the  |g₄| case.  Charge transfer between Euclidean and Minkowskian  regions. If the C-S-K term is real, also the  charge transfer between partonic orbit and 4-D  regions is possible.  The boundary conditions at the partonic orbit fix it to a high degree and also affect the time derivatives ∂_th^k at X³. This option looks physically rather attractive because classical conserved charges would be real.

   If the C-S-K term is imaginary it behaves like a free particle since charge exchange  with Minkowskian and Euclidean regions is not possible. A possible interpretation of the  possible M⁴ contribution to momentum could be in terms of decay width.  The symplectic charges do not however involve momentum. The imaginary contribution to momentum could therefore come only from the Euclidean region.

4. If the Euclidean contribution is imaginary, it seems that it cannot be included in the Kähler function. Since in M⁸ picture the momenta of virtual fermions are in general complex, one could consider the possibility that  Euclidean contribution  to the momentum is imaginary and allows an  interpretation as a decay width.

See the article Symmetries and Geometry of the "World of Classical Worlds" or the chapter Recent View about K\"ahler Geometry and Spin Structure of "World of Classical Worlds".

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Why the water flowing out of bathtub rotates always in the same direction?

In FB Wes Johnson wondered whether Coriolis force could explain why the water flowing out of bathtub forms a vortex with direction which is opposite at Northern and Southern hemispheres.

Coriolis effect is a coordinate force proportional to ω× v, where ω is the angular velocity of Earth directed to Noth and v is the velocity of the object. For bathtub v would be downwards, that is in the direction of Earth radius. At the  equator Coriolis force  is  along the equator  and non-vanishing. On the other hand, the force causing rotation of water in the bathtub is of opposite sign below and above equator and therefore vanishes at equator. Therefore Coriolis force is excluded as an explanation.

My own view is that this is a hydrodynamical effect and new physics might be involved.   Turbulence is involved and   vortex is generated.  The direction  of the  rotation of the vortex should be understood. The selection of a specific direction violates parity symmetry and this gives in the TGD framework strong guidelines.  

1.   The   vortex is in the direction of the  Earth's gravitational force. In the TGD framework,  gravitational interaction is mediated by monopole flux tubes in the direction of the gravitational field. Quantum gravitation is involved and it is quite possible that the gravitational magnetic body (MB) induces the effect since quite generally MB plays a control role, in particular in living matter.
2.    The induced Kähler field contributes to both electromagnetic and classical (weak) Z⁰ field:  since the matter is em neutral but not Z⁰ neutral, it  seems that   Z⁰ field  must be in question. Could the gravitational MB  of Earth consist  of Z⁰ monopole flux tubes?

   If this is the case, a macroscopic quantum effect involving a very large value ℏ_gr=GMm/β₀ of gravitational Planck constant of the pair formed by Earth mass and particle must be in question since ordinary Z⁰ has extremely short range. The gravitational Compton length Λ_gr = ℏ_gr/m= GM/β₀= r_S/2β₀ does not depend on particle mass and Z⁰ is about .5 cm, one half of the Schwartschild radius of the Earth, for the favored β₀=v₀/c=1.

3.   In the classical Z⁰ field,  particles with Z⁰ charges rotate around the axis of the field and since magnetic flux is approximately dipole field, the flux lines  are radial but  are upwards/downwards above/below the equator. This would explain why the rotation directions of the vortex are  opposite and Northern and Southern hemispheres. The presence of the classical Z⁰ field, which violates parity symmetry, would also conform with the parity breaking and would be essential for the understanding of the mystery of chiral selection in biomatter.

For a summary of earlier postings see Latest progress in TGD. For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Mysterious lift of drill in downwards water flow

I learned of a very interesting and paradoxical looking phenomenon. Thanks for Shamoon Ahmed for the link. A drill with a helical geometry raises in a downwards fluid flow (see this) This is in conflict with the naive expectations.

1. Suppose first that momentum is conserved. By momentum conservation water must get downwards directed momentum if the drill obtains upwards directed momentum. If there is no slipping, just the opposite should happen. Therefore the situation could be like in a turbulent flow: the water and the drill do not directly touch each other. There is indeed turbulence as one can see.

   But what makes possible the slipping? It has been quite recently learned that the surface of water in air has thin ice-like layer for which TGD suggests and explanation (see this). The surface between drill and water would be covered by a very thin ice layer so that slipping would take place naturally. Drill is like a skater. Also the boundary layer in the water (liquid) flow past a body could be a thin ice-sheet. Second analogy is as a screw penetrating upstream.

2. But is the momentum really conserved? Water is accelerated in the gravitational field: this gives it momentum. Water is rotating already before the addition of the drill. The downwards kinematic pressure, which increases downwards, pushes the drill having a helical geometry. If there is no friction fixing the drill to water flow, the drill has no other option than raise. The constraint due to helicality forces the drill to rotate.

   Water in the vortex and drill would rotate in opposite directions and helicality constraint would transform the rotational motion of the drill to a translational motion and force the rotation of drill to gain upwards directed momentum.

3. This raises some questions.
   1. Could there be a connection with the fact that in the Northern/Southern hemisphere water flowing in a water tub rotates in a unique direction (kind of parity breaking)?
   2. What is the role of the handedness of the drill? One would expect that the drill with an opposite handedness rotate in an opposite direction? What if the handedness of the drill does not favor the natural rotation direction for the vortex? Do these effects tend to cancel.

There might be a connection with the "ordinary" hydrodynamics. The drill raising in the fluid flow is analogous to a propeller. Could also ordinary propeller involve the same basic mechanism and act like a skater and in this way minimize dissipative energy losses? It is known that propellers induce cavitation as evaporation of water and there is anecdotal evidence from power plants that more energy is liberated in the process than one would expect. Recently it was found that the mere irradiation of water by light leads to its evaporation as a generation of droplets, which would have ice-like surface layer consisting of the fourth phase of water (this requires energy): Pollack effect again! Could dark photons with a non-standard value of Planck constant provide the energy needed for the cavitation creating a vapour phase with a larger total area of fourth phase of water?

Runcel D. Arcaya informed me of the work of a brilliant experimentalist and inventor Victor Schauberger related to the strange properties of flowing water. This work relates in an interesting manner to the effect discussed. I have written about Schauberger's findings about to the ability of fishes too swim "too" easily upstream. Gravitation is involved also now. Could the bodily posture of the fish generate the counterpart of the helical geometry? Could the fish as a living organism help to generate the fourth phase of water in the water bounding their skin by Pollack effect, which requires the presence of a gel phase besides energy source (IR radiation for instance) to transform part of protons of water molecules to dark photons with a higher energy.

Schauberger also invented a method of water purification using vortex flow: the reason for why the method works remained unclear. In Pollack effect, the negatively charged exclusion zones (EZs) spontaneously purify themselves. This conflicts with the thermodynamical intuitions. The TGD explanation is in terms of reversed arrow of time which explains the purification process as normal diffusion leading to the decay of gradients but taking place with an opposite arrow of time. Could the purification of in vortex flow be caused by the Pollack effect creating the surface layers consisting of the fourth phase of water (EZs)?

Schauberger developed the notion of living water and believed that spring water is somehow very special in this respect. In TGD water is regarded as a multiphase system involving magnetic body with layers labelled by the values of effective Planck constant h_eff. The larger the value of the h_eff, the higher the (basically algebraic complexity) and "IQ" of the system. Gravitational magnetic body has the largest value of effective Planck constant. Spring water is pure and could be this kind of highly complex system. Also systems involving turbulence and vortices are very complex.

See the article TGD Inspired Model for Freezing in Nano Scales and the chapter TGD and Quantum Hydrodynamics.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

About the universality of the holomorphic solution ansatz

The explicit solution of field equations in terms of the generalized holomorphy is now known. Also the emergence of supersymplectic symmetry is understood: it emerges as symmetries of Chern-Simons-Kähler action at the 3-D partonic orbits defining part of 3-D holographic data.

The solution ansatz is independent of action as long it is general coordinate invariance depending only on the induced geometric structures. Space-time surfaces would be minimal surfaces apart from lower-dimensional singular surfaces at which the field equations involve the entire action. Only the singularities, classical charges and positions of topological interaction vertices depend on the choice of the action (see this). Kähler action plus volume term is the choice of action forced by twistor lift making the choice of H unique.

The universality has a very intriguing implication. One can assign to any action of this kind conserved Noether currents and their fermionic counterparts (also super counterparts). One would have a huge algebra of conserved currents characterizing the space-time geometry. The corresponding charges need not be conserved since the conservation conditions at the partonic orbits and other singularities depend on the action. The discussion of the symplectic symmetries leads to the conclusion that they give rise to conserved charges at the partonic 3-surfaces obeying Chern-Simons-Kähler dynamics, which is non-deterministic.

Partonic 3-surfaces could be in the same role as space-like 3-surfaces as initial data: the time coordinate for this time evolution would be dual to the light-like coordinate of the partonic orbit. Could one say that the measurement localizing the partonic orbit leads to a phase characterized by a particular action? The classical conserved quantities are determined by the action. The WCW K\"ahler function should correspond to this action and different actions would correspond to different regions of WCW. Could phase transition between these regions take place when the 4-surface determined by the partonic orbit belongs to regions corresponding to two different effective actions. The twistor lift suggests that action must be the sum of the Kähler action and volume term so that only Kähler couplings strength, the coefficient of instanton term and the dynamically determined cosmological constant would vary.

See the article Symmetries and Geometry of the "World of Classical Worlds" or the chapter Recent View about K\"ahler Geometry and Spin Structure of "World of Classical Worlds".

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Some comments about the identification of leptons and matter-antimatter asymmetry

The mathematical formulation of TGD has now reached a stage in which one can seriously consider fixing the details of the physical interpretation of the theory. The discussion of the detailed physical interpretation in articles (see this), inspired by what I called Platonization, led to a proposal for a unification of hadron, nuclear, atomic, and molecular physics in terms of the notion of Hamiltonian cycles defined by monopole flux tubes at Platonic solids. This generated quite unexpected insights and killer predictions.

In (see this) a construction of strong, electroweak and gravitational interaction vertices, reducing them to partly topological 2-vertex describing a creation of fermion-antifermion pair in a classical induced electroweak gauge potentials, led to very concrete predictions relating also the topological explanation of family replication phenomenon and its correlation with homology charge of the partonic 2-surface.

In the articles (see this and this) the identification of the isometries of WCW were considered and explicit realizations of symplectic and holomorphic isometry generators demonstrated that the original intuitive view is almost correct. The new aspects were related to the holography suggesting also a duality between symplectic and holomorphic isometry charges and supercharges. Also the relationship of holography, apparently in conflict with path integral approach, was understood. The dynamics of light-like partonic orbits is almost topological and not completely deterministic, which implies that the finite sum over partonic orbits can be approximated with path integral at QFT limit.

The emergence of all these constraints raises the hope that one could fix the interpretation of the theory at the level of details. In this article, the identification of leptons and matter-antimatter asymmetry are reconsidered in light of the new understanding.

About two competing identifications for leptons

In the TGD Universe, one can imagine two competing identifications of leptons.

1. Leptons and quarks correspond to different chiralities of spinors of H=M⁴× CP₂.
2. Only quarks are fundamental fermions and leptons are anti-baryon like objects.

Option A: Are both leptons and quarks fundamental fermions?

The first option means that both lepton and quark chiralities appear in the modified Dirac action fixed by hermicity once the action fixing space-time surfaces or their lower-dimensional submanifolds such as string world sheets and partonic orbits is known. In accordance with the experimental facts, lepton and quark numbers are conserved separately for this option. This is the original proposal and seems to be the most realistic option although the geometry of "world of classical worlds" (WCW) in terms of anticommutators of WCW gamma matrices, expressible as super generators of symmetries of H inducing isometries of WCW, seems to require only a single fermionic chirality.

The challenge is to explain why both chiralities are needed. It seems now clear that all elementary particles should be assignable to 2-sheeted monopole flux tubes with the fermion lines at wormhole throats (partonic 2-surfaces) of wormhole contacts identifiable as boundaries of string world sheets. The fermion numbers could be also delocalized inside string world sheets inside the flux tubes or inside flux tubes. Also in condensed matter physics states localized to geometric objects of various dimensions are accepted as a basic notion (for TGD view of condensed matter see (see this).

One can identify several dichotomies, which are analogous to the lepton-quark dichotomy. There is holomorphic-symplectic dichotomy, the dichotomy between light-like partonic orbits and 3-surfaces at the boundaries of Δ M⁴₊× CP₂, the dichotomy between Euclidean and Minkowskian space-time regions and the dichotomy between the 3-D holographic boundary data and interiors of the space-time surface. Could one unify all these dichotomies?

1. Consider first the Minkowskian-Euclidean dichotomy. Leptons could reside inside (string world sheets of) the Minkowskian regions of space-time surface and quarks inside (the string world sheets of) the Euclidean wormhole contacts. Euclidean regions with a fixed Minkowskian region or vice versa could be regarded as two sub-WCWs.

   The nice feature of this option is that it allows us to understand both quark/color confinement without any quark propagation and propagation of quarks in QCD. We would not see free quarks because they live inside the Euclidean regions of the space-time surface and do not propagate inside the Minkowskian regions of the space-time surface. Embedding space spinor spinor fields would however propagate in accordance in H which gives rise to quark propagators in the scattering amplitudes and conforms with the QCD picture. Notice that both quarks and leptons can appear at the partonic orbits forming the interfaces between Euclidean and Minkowskian space-time regions.

2. The holomorphic-symplectic dichotomy for the isometries of WCW is now well-established and one has explicit expression for the corresponding conserved charges and their fermionic counters defining gamma matrices as fermionic super charges which in anticommute to WCW metric.

   The symplectic representations of the fermionic isometry generators associated with the light-like partonic orbits at boundaries of CD defining 3-D holomorphic data could correspond to quarks. In accordance with color confinement, quarks would not appear at Euclidean 3-surfaces at light-cone boundaries Δ M⁴₊× CP₂. Classical gluon fields would define the simplest Hamiltonian fluxes and conserved quantities in the 3-D dynamics would be determined by the Chern-Simons-Kähler action with time defined by the light-like time coordinate. The Hamiltonians of S²× CP₂,organized to representations of color group and of rotation group restricted to partonic 2-surface, would define the Hamiltonian fluxes.

   The 4-D holomorphic representations in the interior of space-time surfaces and also assignable to the 3-surfaces at the boundaries of space-time surface at Δ M⁴₊× CP₂ would correspond to leptons. Also the anticommutators of the lepton-like gamma matrices would give contributions to the metric of WCW.

3. Holography as a dichotomy would suggest at quantum level that in an information theoretic sense the 4-D holomorphic dynamics of leptons represents the 3-D symplectic dynamics of quarks. The possibility of a kind of holography-like relation was already discussed in (see this), where it was found that the states of nuclei could be in rather precise correspondence with the states of atomic electrons. A generalization of this holography would correspond to a kind of quark-lepton holography.

One can argue that this general view could lead to a conflict with the possible holography-like relation between ordinary and dark quarks considered in (see this) as a way to guarantee that perturbation theory converges.

1. According to an intuitive argument, strong coupling strength is proportional to 1/h_eff so that the increase of effective Planck constant h_eff could guarantee the convergence of QFT type description expected at QFT limit of TGD. Ordinary quarks could transform in the h→ h_eff transition to states, which consist of pair of quark and dark antiquark with a vanishing total color, electroweak quantum numbers and spin whereas the second dark quark with a larger value of h_eff would have quantum numbers of the ordinary quark.

   The proposal was that dark quark and antiquark reside at the Minkowskian string world sheet. This does not conform with the above proposal, which requires that all quarks are at the partonic orbits.

2. This problem can be solved. Many-sheeted space-time however makes it possible to imagine that the dark quark and antiquark reside at the wormhole contacts of a larger space-time sheet and form a dark meson-like object. For instance, the ordinary quark would be associated with a wormhole contact connecting the other large space-time sheet to a third smaller space-time sheet, itself part of the monopole flux tube defining the ordinary quark. This would conform with the hierarchy formed by flux tubes topologically condensed to larger flux tubes.

Option B: Are only quarks fundamental fermions?

The second option stating that only quarks are fundamental particles, was motivated by the fact that only single fermion chirality seemed to be needed to construct WCW geometry. Leptons would be antibaryon-like states such that the 3 antiquarks are associated with single wormhole contact (see this). Lepton itself would be a closed monopole flux tube with geometric size defined by the Compton scale.

1. The first critical question is whether it makes sense to put 3 quarks to the same wormhole contact defining 2-D surface in CP₂ when the color degrees of freedom correspond to the "rotational" degrees of freedom in CP₂ but realized as spinor modes. One would have at least 2 antiquarks at the same wormhole contact. If the partonic 2-surface is homologically non-trivial geodesic sphere, the reduction of symmetry from SU(3) to U(2) subgroup with the same Cartan algebra occurs and the rotational degrees of freedom reduce to those at the partonic 2-surface. Wave functions for quarks would be wave functions for the end of the string at a partonic 2-surface having well-defined U(2) quantum numbers.
2. If multi-quark states at partonic 2-surfaces make sense, one can ask how to avoid the counterparts of Δ baryons with spin 3/2. Statistics constraint does not help to achieve this. Oscillator operators for color partial waves of quarks are anticommuting and there seems to be no reason excluding these states. In this sense color quantum numbers are like spin-like quantum numbers. One could of course hope that Δ-like states have a very high mass scale.
3. The third critical question concerns the origin of the CP breaking which would allow baryons as stable 3-quark states and only leptons as stable bound states of 3 antiquarks. Matter antimatter asymmetry would correspond to the stable condensation of antiquarks to leptons and quarks to baryons. This mechanism looks really elegant.

How matter antimatter asymmetry could be generated?

Both options A and B for the identification of leptons must be able to explain the generation of matter-antimatter asymmetry.

1. CP breaking involving M⁴ and/or CP₂ Kähler forms could explain the matter antimatter asymmetry along the same lines as in the standard picture. For Option A a small asymmetry between the densities of fermions and antifermions should be generated in the early cosmology and annihilation would lead to the antisymmetry. There would be space-time regions with opposite sign of asymmetries. For Option B the densities of leptons and antileptons and baryons and antibaryons would be slightly different before the annihilation and there would be no actual asymmetry.
2. For both A and B option, many-sheeted space-time and the hierarchy of magnetic bodies makes it possible to imagine many different realizations for the separation of fermion and antifermion numbers. For instance, cosmic strings could contain antimatter. The ordinary matter would be generated in the decay of the energy of cosmic strings to ordinary matter. This process is the TGD counterpart of inflation and highly analogous to black hole evaporation.

   In the simplest model, this energy would be associated with classical M⁴ type Kähler electric fields and CP₂ type Kähler magnetic fields inside the cosmic string. The decay of the volume energy and the energy of the classical electroweak fields would take place by a generation of fermion-antifermion pairs via fermion 2-vertex and classical electroweak gauge potentials would appear in the vertex.

3. If the CP breaking induced by the classical Kähler fields makes it more probable for antifermions to remain inside the monopole flux tubes, antimatter-matter asymmetry is generated. Also the CP breaking observed in meson decays could relate to the asymmetry caused by the induced Kähler field of the meson-like monopole flux tube.
4. In QCD, the topological instanton term gives rise to strong CP breaking as a CP violation of the vacuum state which is not invariant under CP (so called theta parameter describes the situation, (see this).

   In the TGD framework, the "instanton density" ∫ J∧ J for the induced Kähler field is non-vanishing and analogous to the theta therm. As a matter fact, instanton density is equal to the gluonic istanto action for the classical gluon field g_A= H_AJ. Instanton density can be transformed to the Chern-Simons-Kähler action as a boundary term and their contribution to the action is analogous to instanton number in QCD although it need not be integer valued. The Chern-Simons-Kähler action gives a contribution to the modified Dirac action at partonic orbits giving rise to fermionic vertices.

   The strong Kähler magnetic fields at the monopole flux tubes give rise to the analog of strong CP violation and provide a possible quantitative description for the generation of the matter-antimatter asymmetry in the decay of the energy of cosmic strings to fermion-antifermion pairs and bosons. For cosmic strings the Kähler magnetic field is extremely strong.

See the article Some comments about the identification of leptons and matter-antimatter asymmetry or the chapter About the TGD based views of family replication phenomenon, color confinement, identification of leptons, and matter-antimatter asymmetry For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Objection against the idea about theoretician friendly Mother Nature

One of the key ideas behind the TGD view of dark matter is that Nature is theoretician friendly (see this). When the coupling strength proportional to ℏ_eff becomes so large that perturbation series ceases to converge, a phase transition increasing the value of h_eff takes place so that the perturbation series converges.

One can however argue that this argument is quantum field-theoretic and does not apply in TGD since holography changes the very concept of perturbation theory. There is no path integral to worry about. Path integral is indeed such a fundamental concept that one expects it to have some approximate counterpart also in the TGD Universe. Bohr orbits are not completely deterministic: could the sum over the Bohr orbits however translate to an approximate description as a path integral at the QFT limit? The dynamics of light-like partonic orbits is indeed non-deterministic and could give rise to an analog of path integral as a finite sum.

1. The dynamics implied by Chern-Simons-Kähler action assignable to the partonic 3-surface with light-one coordinate in the role of time, is very topological in that the partonic orbits is light-like 3-surface and has 2-D CP₂ and M⁴ projections unless the induced M⁴ and CP₂ Kähler forms sum up to zero. The light-likeness of the projection is a very loose condition and and the sum over partonic orbits as possible representation of holographic data analogous to initial values (light-likeness!) is therefore analogous to the sum over all paths appearing as a representation of Schrödinger equation in wave mechanics.

   One would have an analog of 1-D QFT. This means that the infinities of quantum field theories are absent but for a large enough coupling strength g²/4πℏ the perturbation series fails to converge. The increase of h_eff would resolve the problem. For instance,   Dirac equation in atomic physics makes unphysical predictions when the value of nucler charge is larger than Z≈ 137.

2. I have also considered a discretized variant of this picture. The light-like orbits would consist of pieces of light-like geodesics. The points at which the direction of segment changes would correspond to points at which energy and momentum transfer between the partonic orbit and environment takes place. This kind of quantum number transfer might occur at least for the fermionic lines as boundaries of string world sheets. They could be described quantum mechanically as interactions with classical fields in the same way as the creation of fermion pairs as a fundamental vertex (see this). The same universal 2-vertex would be in question.
3. What is intriguing, that the light-likeness of the projection of the CP₂ type extremals in M⁴ leads to Virasoro conditions assignable to M⁴ coordinates and this eventually led to the idea of conformal symmetries as isometries as WCW. In the case of the partonic orbits, the light-like curve would be in M⁴× CP₂ but it would not be surprising if the generalization of the Virasoro conditions would emerge also now.

   One can write M⁴ and CP₂ coordinates for the light-like curve as Fourier expansion in powers of exp(it), where t is the light-like coordinate. This gives h^k= ∑ h^k_n exp(int). If the CP₂ projection of the orbits of the partonic 2-surface is geodesic circle, CP₂ metric s_kl is constant, the light-likeness condition h_kl∂_th^k∂l_th^l=0 gives Re(h_kl∑_m h^k_n-mh^l_m=0). This does not give Virasoro conditions.

   The condition d/dt(h_kl∂_th^k∂_th^l=0)=0 however gives the standard Virasoro conditions stating that the normal ordered operators L_n= Re(h_kl∑_m (n-m) h^k_n-mh^l_m) annihilate the physical states. What is interesting is that the latter condition also allows time-like (and even space-like) geodesics.

4. Could massivation mean a failure of light-likeness? For piecewise light-like geodesics the light-likeness condition would be true only inside the segments. By taking Fourier transform one expects to obtain Virasoro conditions with a cutoff analogous to the momentum cutoff in condensed matter physics for crystals. For piecewise light-like geodesics the condition would be trivially true inside the segments and therefore discretized. By taking Fourier transform one expects to obtain Virasoro conditions with a cutoff analogous to the momentum cutoff in condensed matter physics for crystals.
5. In TGD the Virasoro, Kac-Moody algebras and symplectic algebras are replaced by half-algebras and the gauge conditions are satisfied for conformal weights which are n-multiples of fundamentals with with n larger than some minimal value. This would dramatically reduce the effects of the non-determinism and could make the sum over all paths allowed by the light-likeness manifestly finite and reduce it to a sum with a finite number of terms. This cutoff in degrees of freedom would correspond to a genuinely physical cutoff due to the finite measurement resolution coded to the number theoretical anatomy of the space-time surfaces. This cutoff is analogous to momentum cutoff and could at the space-time picture correspond to finite minimum length for the light-like segments of the orbit of the partoic 2-surface.

See the articles About the Relationships Between Weak and Strong Interactions and Quantum Gravity in the TGD Universe, Holography and Hamilton-Jacobi Structure as 4-D generalization of 2-D complex structure, and Symmetries and Geometry of the "World of Classical Worlds".

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

The Great Narrative of theoretical physics

I have used a lot of time in pondering the Great Narrative of theoretical physics. It has two components. The stubborn belief on the idea of a continuous progress that arose during the Enlightenment: it has proven impossible to admit that huge mistakes have been made at the level of basic postulates so that progress has become illusory.

The second component comes from colonialism: the development of theoretical physics is seen as a series of wars of conquest. Reductionism encapsulates how conquest-war progresses. Each conquered area corresponds to a new field of physics. The community is still unable to see that the triumphal march of reductionism is a very similar illusion as colonialism was.

Wars of conquest have progressed in the direction of both short and long length scales. We have progressed from the planetary system to astrophysical and cosmic scales. The narrative of cosmology has started to crack more and more: there is a crisis related to the understanding of dark matter and energy and inflationary theory has been in crisis from the beginning. As a matter of fact, a state of stagnation here might be a better word than crisis since crisis means criticality and a promise for something new.

The observations already made earlier, and especially James Webb, have now once and for all destroyed the grand cosmological narrative. The Big Bang remains, but Webb's observations call into question the concept of time at the base of cosmology (galaxies older than the universe), the assumption that coherence is only possible on short scales (correlations on cosmic scales), the assumptions about how signals propagate (or rather do not propagate) on cosmological scales, and also the existing view of the formation of astro-physical objects.

On the other hand, progress has been made in both directions by starting from atomic physics which was a real triumph, but molecular physics is already just phenomenology without any real theory (for example, the concept of a chemical bond is not understood on a basic level at all). In biochemistry, biocatalysis remains a complete mystery.

The troops marched also in the direction of nuclear physics, but it was necessary to decide that it is a completely separate area from atomic physics, even though correlations were noticed very early on. The march proceeded to electroweak interactions: this was a real success and also to hadron physics and QCD. It was agreed that hadrons have been understood even though color confinement remained a complete mystery. Standard model emerged and all that was left was the jump to the Planck length scales. The GUTs were a leap into the void producing nothing, but were accepted as a part of the great narrative, partly for reasons related to funding.

Finally, the super string model was built as a theory that was supposed to unify the standard model and quantum gravity. The trial was based on two theories, both of which have a huge gap. The gaps were already noticed a hundred years ago.

In Einstein's theory, conservation laws of the Special Relativity are lost, but perhaps because the discoverer of the gap was Emmy Noether, a woman and a Jew, this discovery was not allowed to mess with the unfolding Grand Narrative.

The basic paradox of quantum measurement theory was the big gap of quantum mechanics. In the spirit of pragmatism, even that was not allowed to interfere with the development of the Great Narrative so that an endless variety of interpretations were invented. So it's no wonder that the superstring theory built above these two great gaps eventually collapsed.

When one thinks about reductionist wars of conquest, one can't avoid comparisons to Alexander the Great's victories and the rapid collapse of the empire that followed. The Colonial Wars is another point of comparison. In between all the areas of the physics landscape agreed to be conquered, there are white areas on the map, about which nothing is actually known. The last hundred years of theoretical physics will probably be seen as the greatest intellectual self-deception in human intellectual history.

I remember the novel, was it the core of Darkness, which told about a similar illusion related to colonialism. It told about a commander of a British base in Africa, a drunkard who desperately tried to maintain the illusion that the situation was under control after all. In the same way the community of theoretical physicists tries to preserve its Grand Illusions. Internalized censorship takes care that new ideas challenging the basic dogmas are neither published in "prestigious" journals nor funded.

Can one find reasons for the recent situation? Sloppy thinking is certainly one basic reason. Colleagues must respect the rules of logic when they write computer code but when it comes to the consistency between fundamental assumptions and mathematics and empirical facts, the basic rules of logic are given up: the justification for this deadly sin of theoretician comes from "pragmatism".

It has been said that great narratives are dead. I do not agree with this. Great Narrative of theoretical physics is possible but it can be developed only by a continual challenging of the basic assumptions of the existing narrative. This has not been done for a century.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Kundalini phenomenon from TGD point of view

There was a very interesting post by Hunter Glenn in Qualia Computing Network about the Kundalini phenomenon. The proposal of Glenn was that the charge separations relate somehow to the Kundalini phenomenon involving both the bliss and the dark night of the soul suggesting that (quantum) criticality is involved.

In bioelectromagnetism it has been known for decades that the sign of electric gradient along the longitudinal axis of the body correlates with the state of consciousness. The sign of the electric gradient changes when one falls asleep. This sign matters also at the level of brain hemispheres in horizontal direction. At the neuronal level the membrane potential changes temporarily sign during the nerve pulses. At the axonal microtubular level the sign of gradient matters and the tubule is in (quantum?) critical state in the sense that it is decaying and re-assembling all the time.

This suggests that the electric gradient along the spine correlates with the contents of consciousness and has a lot to do with the kundalini phenomenon. The appearance of chills in the spine could reflect the generation of electrical gradients. In my own Great Experience around 1985I experienced these chills and the subsequent "whole body consciousness" completely free from the usual "thermal noise". The attempt to understand this experience led to the development of TGD inspired theory of consciousness.

Charges are needed to create these gradients and the natural question is where these charges reside. Between what kind of systems the charge separations are generated?

Electric gradients and charge separations seem to be fundamental. In the TGD inspired quantum model of the nerve pulse, the cell membrane is regarded as a Josephson junction. Standard physics does not of course allow this: according to the Hodgkin-Huxley model the currents are ohmic currents. There is very intriguing experimental evidence in conflict with the assumption of Ohmic currents as cause of nerve pulse: they could be of course caused by it. This evidence justifies TGD inspired model of nerve pulse discussed here. The model involves the Pollack effect as a way to generate charge separations. In presence of suitable energy feed and gel phase, water develops negatively charged regions with very high charge. 1/4:th of protons of water molecules to somewhere, "outside" the system in some sense. This generates electric gradients and all electric gradients in living matter could be created in this way by metabolic energy feed.

Where could the protons go? In the TGD Universe they would go to the magnetic body (MB), the TGD geometric counterpart for Maxwellian magnetic fields, and form a dark phase there. This would mean that they have non-standard and very large values of effective Planck constant so that they form a large-scale quantum coherent phase at MB: this would induce the coherence of ordinary biomatter as forced coherence. The MB in question would be a gravitational magnetic body and the value of gravitational Planck constant was proposed already by Nottale. Charge separations would reduce to those between the biological body (cell membrane, etc) and corresponding (gravitational) MB and this would allow us to understand how the electric gradients develop. Also nerve pulse would be based on the Pollack effect.

The spiritual aspect would come to play via the magnetic body representing higher level consciousness (an entire hierarchy of them is predicted). In Kundalini a connection to some magnetic body would be created (or lost) and could also give rise to the experience of becoming God.

See the article Some new aspects of the TGD inspired model of the nerve pulse or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see No comments:

Symmetries and Geometry of the "World of Classical Worlds"

Still about the symmetries of WCW

I have been analyzing the basic visions of TGD trying to identify weak points. WCW geometry exists only if it has maximal isometries. I have proposed that WCW could be regarded as a union of generalized symmetric spaces labelled by zero modes which do not contribute to the metric. The induced Kähler field is invariant under symplectic transformations of CP₂ and would therefore define zero mode degrees of freedom if one assumes that WCW metric has symplectic transformations as isometries. In particular, Kähler magnetic fluxes would define zero modes and are quantized closed 2-surfaces. The induced metric appearing in Kähler action is however not zero mode degree of freedom. If the action contains volume term, the assumption about union of symmetric spaces is not well-motivated.

Symplectic transformations are not the only candidates for the isometries of WCW. The basic picture about what these maximal isometries could be, is partially inspired by string models.

1. A weaker proposal is that the symplectomorphisms of H define only symplectomorphisms of WCW. Extended conformal symmetries define also a candidate for isometry group. Remarkably, light-like boundary has an infinite-dimensional group of isometries which are in 1-1 correspondence with conformal symmetries of S²⊂ S²× R₊= δ M⁴₊.
2. Extended Kac Moody symmetries induced by isometries of δ M⁴₊ are also natural candidates for isometries. The motivation for the proposal comes from physical intuition deriving from string models. Note they do not include Poincare symmetries, which act naturally as isometries in the moduli space of causal diamonds (CDs) forming the "spine" of WCW.
3. The light-like orbits of partonic 2-surfaces might allow separate symmetry algebras. One must however notice that there is exchange of charges between interior degrees of freedom and partonic 2-surfaces. The essential point is that one can assign to these surface conserved charges when the dual light-like coordinate defines time coordinate. This picture also assumes a slicing of space-time surface by by the partonic orbits for which partonic orbits associated with wormrhole throats and boundaries of the space-time surface would be special. This slicing would correspond to Hamilton-Jacobi structure.
4. Fractal hierarchy of symmetry algebras with conformal weights, which are non-negative integer multiples of fundamental conformal weights, is essential and distinguishes TGD from string models. Gauge conditions are true only the isomorphic subalgebra and its commutator with the entire algebra and the maximal gauge symmetry to a dynamical symmetry with generators having conformal weights below maximal value. This view also conforms with p-adic mass calculations.
5. The realization of the symmetries for 3-surfaces at the boundaries of CD and for light-like orbits of partonic 2-surfaces is known. The problem is how to extend the symmetries to the interior of the space-time surface. It is natural to expect that the symmetries at partonic orbits and light-cone boundary extend to the same symmetries.

Could generalized holomorphy allow to sharpen the existing views?

This picture is rather speculative, allows several variants, and is not proven. There is now however a rather convincing ansatz for the general form of preferred extremals. This proposal relies on the realization of holography as generalized 4-D holomorphy. Could it help to make the picture more precise?

1. Explicit solution of field equations in terms of the generalized holomorphy is now known. The solution ansatz is independent of action as long it is general coordinate invariance depending only on the induced geometric structures. Space-time surfaces would be minimal surfaces apart from lower-dimensional singular surfaces at which the field equations involve the entire action. Only the singularities, classical charges and positions of topological interaction vertices depend on the choice of the action (see this). Kähler action plus volume term is the choice of action forced by twistor lift making the choice of H unique.
2. Hamilton-Jacobi structures emerge naturally as generalized conformal structures of space-time surfaces and M⁴ (see this). This inspires a proposal for a generalization of modular invariance and of moduli spaces as subspaces of Teichmüller spaces.
3. One can assign to holomorphy conserved Noether charges. The conservation reduces to the algebraic conditions satisfied for the same reason as field equations, i.e. the conservation conditions involving contractions of complex tensors of type (1,1) with tensors of type (2,0) and (0,2). The charges have the same form as Noether charges but it is not completely clear whether the action remains invariant under these transformations. This point is non-trivial since Noether theorem says that invariance of the action implies the existence of conserved charges but not vice versa. Could TGD represent a situation in which the equivalence between symmetries of action and conservation laws fails?

   Also string models have conformal symmetries but in this case 2-D area form suffers conformal scaling. Also the fact that holomorphic ansatz is satisfied for such a large class of actions apart from singularities suggests that the action is not invariant.

4. The action should define Kähler function for WCW identified as the space of Bohr orbits. WCW Kähler metric is defined in terms of the second derivatives of the Kähler action of type (1,1) with respect to complex coordinates of WCW. Does the invariance of the action under holomorphies imply a trivial Kähler metric and constant Kähler function?

   Here one must be very cautious since by holography the variations of the space-time surface are induced by those of 3-surface defining holographic data so that the entire space-time surface is modified and the action can change. The presence of singularities, analogous to poles and cuts of an analytic function and representing particles, suggests that the action represents the interactions of particles and must change. Therefore the action might not be invariant under holomorphies. The parameters characterizing the singularities should affect the value of the action just as the positions of these singularities in 2-D electrostatistics affect the Coulomb energy.

   Generalized conformal charges and supercharges define a generalization of Super Virasoro algebra of string models. Also Kac-Moody algebras assignable to the isometries of δ M⁴₊× CP₂ and light-like 3 surfaces generalize trivially.

5. An absolutely essential point is that generalized holomorphisms are not symmetries of Kähler function since otherwise Kähler metric involving second derivatives of type (1,1) with respect to complex coordinates of WCW is non-trivial if defined by these symmetry generators as differential operators. If Kähler function is equal to Kähler action, as it seems, Kähler action cannot be invariant under generalized holomorphies.

   Noether's theorem states that the invariance of the action under a symmetry implies the conservation of corresponding charge but does not claim that the existence of conserved Noether currents implies invariance of the action. Since Noether currents are conserved now, one would have a concrete example about the situation in which the inverse of Noether's theorem does not hold true. In a string model based on area action, conformal transformations of complex string coordinates give rise to conserved Noether currents as one easily checks. The area element defined by the induced metric suffers a conformal scaling so that the action is not invariant in this case.

Challenging the existing picture of WCW geometry

These findings make it possible to challenge and perhaps sharpen the existing speculations concerning the metric and isometries of WCW.

I have considered the possibility that also the symplectomorphisms of δ M⁴+× CP₂ could define WCW isometries. This actually the original proposal. One can imagine two options.

1. The continuation of symplectic transformations to transformations of the space-time surface from the boundary of light-cone or from the orbits partonic 2-surfaces should give rise to conserved Noether currents but it is not at all obvious whether this is the case.
2. One can assign conserved charges to the time evolution of the 3-D boundary data defining the holographic data: the time coordinate for the evolution would correspond to the light-like coordinate of light-cone boundary or partonic orbit. This option I have not considered hitherto. It turns out that this option works!

The conclusion would be that generalized holomorphies give rise to conserved charges for 4-D time evolution and symplectic transformations give rise to conserved charged for 3-D time evolution associated with the holographic data.

About extremals of Chern-Simons-Kähler action

Let us look first the general nature of the solutions to the extremization of Chern-Simons-Kähler action.

1.  The light-likeness of the partonic orbits requires Chern-Simons action, which is equivalent to the topological action J∧ J, which is total divergence and   is a symplectic in variant.  The field equations at the boundary cannot involve  induced metric so that only induced symplectic structure remains. The 3-D holographic data   at partonic orbits would extremize Cherns-Simons-Kähler action. Note that at the ends of the space-time surface about boundaries of CD one cannot pose any dynamics.
2. If the induced Kähler form has only the CP₂ part, the variation of Chern-Simons-Kähler form would give equations  satisfied if the CP₂ projection is at most 2-dimensional and Chern-Simons action would vanish and imply that instanton number vanishes.
3. If the action is the sum of M⁴ and CP₂  parts, the field equations in M⁴ and CP₂ degrees of freedom would give the same result. If the induced Kähler form is  identified as the sum of the M⁴ and CP₂ parts, the equations also allow solutions for which the induced M⁴ and CP₂ Kähler forms sum up to zero.  This phase would involve a map identifying M⁴ and CP₂ projections and force induce Kähler forms to be identical. This would force magnetic charge in M⁴ and the question is whether the line connecting the tips of the CD makes non-trivial homology possible.  The homology charges and the 2-D ends of the partonic orbit cancel each other so that partonic surfaces can have monopole charge.

   The conditions at the partonic orbits do not pose conditions on the interior and should allow generalized holomorphy. The following considerations show that besides homology charges as Kähler magnetic fluxes also Hamiltonian fluxes are conserved in Chern-Simons-Kähler dynamics.

Can one assign conserved charges with symplectic transformations or partonic orbits and 3-surfaces at light-cone boundary?

The geometric picture is that symplectic symmetries are Hamiltonian flows along the light-like partonic orbits generated by the projection A_t of the Kähler gauge potential in the direction of the light-like time coordinate. The physical picture is that the partonic 2-surface is a Kähler charged particle that couples to the Hamilton H=A_t. The Hamiltonians H_A are conserved in this time evolution and give rise to conserved Noether currents. The corresponding conserved charge is integral over the 2-surface defined by the area form defined by the induced Kähler form.

Let's examine the change of the Chern-Simons-Kähler action in a deformation that corresponds, for example, to the CP₂ symplectic transformation generated by Hamilton H_A. M⁴ symplectic transformations can be treated in the same way:here however M⁴ Kähler form would be involved, assumed to accompany Hamilton-Jacobi structure as a dynamically generated structure.

1. Instanton density for the induced Kähler form reduces to a total divergence and gives Chern-Simons-Kähler action, which is TGD analog of topological action. This action should change in infinitesimal symplectic transformations by a total divergence, which should vanish for extremals and give rise to a conserved current. The integral of the divergence gives a vanishing charge difference between the ends of the partonic orbit. If the symplectic transformations define symmetries, it should be possible to assign to each Hamiltonian H_A a conserved charge. The corresponding quantal charge would be associated with the modified Dirac action.

2. The conserved charge would be an integral over X². The surface element is not given by the metric but by the symplectic structure, so that it is preserved in symplectic transformations. The 2-surface of the time evolution should correspond to the Hamiltonian time transformation generated by the projection A_α=A_k ∂_αs^k of the Kähler gauge potential A_k to the direction of light-like time coordinate xα== t.

3. The effect of the generator j_A^k= J^kl∂_lH_A on the Kähler potential A_l is given by j^k_A∂_kA_l. This can be written as ∂_kA_l=J_kl + ∂_lA_k. The first term gives the desired total divergence ∂_α (ε^αβγJ_βγ H_A).

   The second term is proportional to the term ∂_αH_A- {A_α,H}. Suppose that the induced Kähler form is transversal to the light-like time coordinate t, i.e. the induced Kähler form does not have components of form J_tμ. In this kind of situation the only possible choice for α corresponds to the time coordinate t. In this situation one can perform the replacement ∂_αH_A-{A_α,H}→ dH_A/dt-{A_t,H}. This corresponds to a Hamiltonian time evolution generated by the projection A_t acting as a Hamiltonian. If this is really a Hamiltonian time evolution, one has dH_A/dt-{A,H}=0. Because the Poisson bracket represents a commutator, the Hamiltonian time evolution equation is analogous to the vanishing of a covariant derivative of H_A along light-like curves: ∂_tH_A +[A,H_A]= 0. The physical interpretation is that the partonic surface develops like a particle with a Kähler charge. As a consequence the change of the action reduces to a total divergence.

   An explicit expression for the conserved current J_A^α=H_A ε^αβγJ_βγ can be derived from the vanishing of the total divergence. Symplectic transformations on X² generate an infinite-dimensional symplectic algebra. The charge is given by the Hamiltonian flux Q_A =∫ H_A J_αβdx^α∧dx^β.

4. If the projection of the partonic path CP₂ or M⁴ is 2-D, then the light-like geodesic line corresponds to the path of the parton surface. If A_l can be chosen parallel to the surface, its projection in the direction of time disappears and one has A_t=0. In the more general case, X² could, for example, rotate in CP₂. In this case A_t is nonvanishing. If J is transversal (no Kähler electric field), charge conservation is obtained.

Do the above observations apply at the boundary of the light-cone?

1. Now the 3-surface is space-like and Chern-Simons-Kähler action makes sense. It is not necessary but emerges from the "instanton density" for the Kähler form. The symplectic transformations of δ M⁴₊× CP₂ are the symmetries. The most time evolution associated with the radial light-like coordinate would be from the tip of the light-cone boundary to the boundary of CD. Conserved charges as homological invariants defining symplectic algebra would be associated with the 2-D slices of 3-surfaces. For closed 3-surfaces the total charges from the sheets of 3-space as covering of δ M⁴₊ must sum up to zero.
2. Interestingly, the original proposal for the isometries of WCW was that the Hamiltonian fluxes assignable to M⁴ and CP₂ degrees of freedom at light-like boundary act define the charges associated with the WCW isometries as symplectic transformations so that a strong form of holography would have been be realized and space-time surface would have been effectively 2-dimensional. The recent view is that these symmetries pose conditions only on the 3-D holographic data. The holographic charges would correspond to additional isometries of WCW and would be well-defined for the 3-surfaces at the light-cone boundary.

To sum up, one can imagine many options but the following picture is perhaps the simplest one and is supported by physical intuition and mathematical facts. The isometry algebra ofδ M⁴₊× CP₂ consists of generalized conformal and KM algebras at 3-surfaces in δ M⁴₊× CP₂ and symplectic algebras at the light cone boundary and 3-D light-like partonic orbits. The latter symmetries give constraints on the 3-D holographic data. It is still unclear whether one can assign generalized conformal and Kac-Moody charges to Chern-Simons-K\"ahler action. The isomorphic subalgebras labelled by a positive integer and their commutators with the entire algebra would annihilate the physical states.

The TGD counterparts of the gauge conditions of string models

The string model picture forces to ask whether the symplectic algebras and the generalized conformal and Kac-Moody algebras could act as gauge symmetries.

1. In string model picture conformal invariance would suggest that the generators of the generalized conformal and KM symmetries act as gauge transformations annihilate the physical states. In the TGD framework, this does not however make sense physically. This also suggests that the components of the metric defined by supergenerators of generalized conformal and Kac Moody transformations vanish. If so, the symplectomorphisms δ M⁴₊× CP₂ localized with respect to the light-like radial coordinate acting as isometries would be needed. The half-algebras of both symplectic and conformal generators are labelled by a non-negative integer defining an analog of conformal weight so there is a fractal hierarchy of isomorphic subalgebras in both cases.
2. TGD forces to ask whether only subalgebras of both conformal and Kac-Moody half algebras, isomorphic to the full algebras, act as gauge algebras. This applies also to the symplectic case. Here it is essential that only the half algebra with non-negative multiples of the fundamental conformal weights is allowed. For the subalgebra annihilating the states the conformal weights would be fixed integer multiples of those for the full algebra. The gauge property would be true for all algebras involved. The remaining symmetries would be genuine dynamical symmetries of the reduced WCW and this would reflect the number theoretically realized finite measurement resolution. The reduction of degrees of freedom would also be analogous to the basic property of hyperfinite factors assumed to play a key role in thee definition of finite measurement resolution.
3. For strong holography, the orbits of partonic 2-surfaces and boundaries of the spacetime surface at δ M⁴₊ would be dual in the information theoretic sense. Either would be enough to determine the space-time surface.

See the articles About the Relationships Between Weak and Strong Interactions and Quantum Gravity in the TGD Universe, Holography and Hamilton-Jacobi Structure as 4-D generalization of 2-D complex structure, Symmetries and Geometry of the "World of Classical Worlds" and the chapter Recent View about K\"ahler Geometry and Spin Structure of "World of Classical Worlds".

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Still about the symmetries of WCW

I have been analyzing the basic visions of TGD trying to identify weak points. WCW geometry exists only if it has maximal isometries. I have proposed that WCW could be regarded as a union of generalized symmetric spaces labelled by zero modes which do not contribute to the metric. The induced Kähler field is invariant under symplectic transformations of CP₂ and would therefore define zero mode degrees of freedom if one assumes that WCW metric has symplectic transformations as isometries. In particular, Kähler magnetic fluxes would define zero modes and are quantized closed 2-surfaces. The induced metric appearing in Kähler action is however not zero mode degree of freedom. If the action contains volume term, the assumption about union of symmetric spaces is not well-motivated.

Symplectic transformations are not the only candidates for the isometries of WCW. The basic picture about what these maximal isometries could be, is partially inspired by string models.

1. A weaker proposal is that the symplectomorphisms of H define only symplectomorphisms of WCW. Extended conformal symmetries define also a candidate for isometry group. Remarkably, light-like boundary has an infinite-dimensional group of isometries which are in 1-1 correspondence with conformal symmetries of S²⊂ S²× R₊= δ M⁴₊.
2. Extended Kac Moody symmetries induced by isometries of δ M⁴₊ are also natural candidates for isometries. The motivation for the proposal comes from physical intuition deriving from string models. Note they do not include Poincare symmetries, which act naturally as isometries in the moduli space of causal diamonds (CDs) forming the "spine" of WCW.
3. The light-like orbits of partonic 2-surfaces might allow separate symmetry algebras. One must however notice that there is exchange of charges between interior degrees of freedom and partonic 2-surfaces. The essential point is that one can assign to these surface conserved charges when the dual light-like coordinate defines time coordinate. This picture also assumes a slicing of space-time surface by by the partonic orbits for which partonic orbits associated with wormrhole throats and boundaries of the space-time surface would be special. This slicing would correspond to Hamilton-Jacobi structure.
4. Fractal hierarchy of symmetry algebras with conformal weights, which are non-negative integer multiples of fundamental conformal weights, is essential and distinguishes TGD from string models. Gauge conditions are true only the isomorphic subalgebra and its commutator with the entire algebra and the maximal gauge symmetry to a dynamical symmetry with generators having conformal weights below maximal value. This view also conforms with p-adic mass calculations.
5. The realization of the symmetries for 3-surfaces at the boundaries of CD and for light-like orbits of partonic 2-surfaces is known. The problem is how to extend the symmetries to the interior of the space-time surface. It is natural to expect that the symmetries at partonic orbits and light-cone boundary extend to the same symmetries.

Could generalized holomorphy allow to sharpen the existing views?

This picture is rather speculative, allows several variants, and is not proven. There is now however a rather convincing ansatz for the general form of preferred extremals. This proposal relies on the realization of holography as generalized 4-D holomorphy. Could it help to make the picture more precise?

1. Explicit solution of field equations in terms of the generalized holomorphy is now known. The solution ansatz is independent of action as long it is general coordinate invariance depending only on the induced geometric structures. Space-time surfaces would be minimal surfaces apart from lower-dimensional singular surfaces at which the field equations involve the entire action. Only the singularities, classical charges and positions of topological interaction vertices depend on the choice of the action (see this). Kähler action plus volume term is the choice of action forced by twistor lift making the choice of H unique.
2. Hamilton-Jacobi structures emerge naturally as generalized conformal structures of space-time surfaces and M⁴ (see this). This inspires a proposal for a generalization of modular invariance and of moduli spaces as subspaces of Teichmüller spaces.
3. One can assign to holomorphy conserved Noether charges. The conservation reduces to the algebraic conditions satisfied for the same reason as field equations, i.e. the conservation conditions involving contractions of complex tensors of type (1,1) with tensors of type (2,0) and (0,2). The charges have the same form as Noether charges but it is not completely clear whether the action remains invariant under these transformations. This point is non-trivial since Noether theorem says that invariance of the action implies the existence of conserved charges but not vice versa. Could TGD represent a situation in which the equivalence between symmetries of action and conservation laws fails?

   Also string models have conformal symmetries but in this case 2-D area form suffers conformal scaling. Also the fact that holomorphic ansatz is satisfied for such a large class of actions apart from singularities suggests that the action is not invariant.

4. The action should define Kähler function for WCW identified as the space of Bohr orbits. WCW Kähler metric is defined in terms of the second derivatives of the Kähler action of type (1,1) with respect to complex coordinates of WCW. Does the invariance of the action under holomorphies imply a trivial Kähler metric and constant Kähler function?

   Here one must be very cautious since by holography the variations of the space-time surface are induced by those of 3-surface defining holographic data so that the entire space-time surface is modified and the action can change. The presence of singularities, analogous to poles and cuts of an analytic function and representing particles, suggests that the action represents the interactions of particles and must change. Therefore the action might not be invariant under holomorphies. The parameters characterizing the singularities should affect the value of the action just as the positions of these singularities in 2-D electrostatistics affect the Coulomb energy.

   Generalized conformal charges and supercharges define a generalization of Super Virasoro algebra of string models. Also Kac-Moody algebras assignable to the isometries of δ M⁴₊× CP₂ and light-like 3 surfaces generalize trivially.

5. An absolutely essential point is that generalized holomorphisms are not symmetries of Kähler function since otherwise Kähler metric involving second derivatives of type (1,1) with respect to complex coordinates of WCW is non-trivial if defined by these symmetry generators as differential operators. If Kähler function is equal to Kähler action, as it seems, Kähler action cannot be invariant under generalized holomorphies.

   Noether's theorem states that the invariance of the action under a symmetry implies the conservation of corresponding charge but does not claim that the existence of conserved Noether currents implies invariance of the action. Since Noether currents are conserved now, one would have a concrete example about the situation in which the inverse of Noether's theorem does not hold true. In a string model based on area action, conformal transformations of complex string coordinates give rise to conserved Noether currents as one easily checks. The area element defined by the induced metric suffers a conformal scaling so that the action is not invariant in this case.

Challenging the existing picture of WCW geometry

These findings make it possible to challenge and perhaps sharpen the existing speculations concerning the metric and isometries of WCW.

I have considered the possibility that also the symplectomorphisms of δ M⁴+× CP₂ could define WCW isometries. This actually the original proposal. One can imagine two options.

1. The continuation of symplectic transformations to transformations of the space-time surface from the boundary of light-cone or from the orbits partonic 2-surfaces should give rise to conserved Noether currents but it is not at all obvious whether this is the case.
2. One can assign conserved charges to the time evolution of the 3-D boundary data defining the holographic data: the time coordinate for the evolution would correspond to the light-like coordinate of light-cone boundary or partonic orbit. This option I have not considered hitherto. It turns out that this option works!

The conclusion would be that generalized holomorphies give rise to conserved charges for 4-D time evolution and symplectic transformations give rise to conserved charged for 3-D time evolution associated with the holographic data.

About extremals of Chern-Simons-Kähler action

Let us look first the general nature of the solutions to the extremization of Chern-Simons-Kähler action.

1.  The light-likeness of the partonic orbits requires Chern-Simons action, which is equivalent to the topological action J∧ J, which is total divergence and   is a symplectic in variant.  The field equations at the boundary cannot involve  induced metric so that only induced symplectic structure remains. The 3-D holographic data   at partonic orbits would extremize Cherns-Simons-Kähler action. Note that at the ends of the space-time surface about boundaries of CD one cannot pose any dynamics.
2. If the induced Kähler form has only the CP₂ part, the variation of Chern-Simons-Kähler form would give equations  satisfied if the CP₂ projection is at most 2-dimensional and Chern-Simons action would vanish and imply that instanton number vanishes.
3. If the action is the sum of M⁴ and CP₂  parts, the field equations in M⁴ and CP₂ degrees of freedom would give the same result. If the induced Kähler form is  identified as the sum of the M⁴ and CP₂ parts, the equations also allow solutions for which the induced M⁴ and CP₂ Kähler forms sum up to zero.  This phase would involve a map identifying M⁴ and CP₂ projections and force induce Kähler forms to be identical. This would force magnetic charge in M⁴ and the question is whether the line connecting the tips of the CD makes non-trivial homology possible.  The homology charges and the 2-D ends of the partonic orbit cancel each other so that partonic surfaces can have monopole charge.

   The conditions at the partonic orbits do not pose conditions on the interior and should allow generalized holomorphy. The following considerations show that besides homology charges as Kähler magnetic fluxes also Hamiltonian fluxes are conserved in Chern-Simons-Kähler dynamics.

Can one assign conserved charges with symplectic transformations or partonic orbits and 3-surfaces at light-cone boundary?

The geometric picture is that symplectic symmetries are Hamiltonian flows along the light-like partonic orbits generated by the projection A_t of the Kähler gauge potential in the direction of the light-like time coordinate. The physical picture is that the partonic 2-surface is a Kähler charged particle that couples to the Hamilton H=A_t. The Hamiltonians H_A are conserved in this time evolution and give rise to conserved Noether currents. The corresponding conserved charge is integral over the 2-surface defined by the area form defined by the induced Kähler form.

Let's examine the change of the Chern-Simons-Kähler action in a deformation that corresponds, for example, to the CP₂ symplectic transformation generated by Hamilton H_A. M⁴ symplectic transformations can be treated in the same way:here however M⁴ Kähler form would be involved, assumed to accompany Hamilton-Jacobi structure as a dynamically generated structure.

1. Instanton density for the induced Kähler form reduces to a total divergence and gives Chern-Simons-Kähler action, which is TGD analog of topological action. This action should change in infinitesimal symplectic transformations by a total divergence, which should vanish for extremals and give rise to a conserved current. The integral of the divergence gives a vanishing charge difference between the ends of the partonic orbit. If the symplectic transformations define symmetries, it should be possible to assign to each Hamiltonian H_A a conserved charge. The corresponding quantal charge would be associated with the modified Dirac action.

2. The conserved charge would be an integral over X². The surface element is not given by the metric but by the symplectic structure, so that it is preserved in symplectic transformations. The 2-surface of the time evolution should correspond to the Hamiltonian time transformation generated by the projection A_α=A_k ∂_αs^k of the Kähler gauge potential A_k to the direction of light-like time coordinate xα== t.

3. The effect of the generator j_A^k= J^kl∂_lH_A on the Kähler potential A_l is given by j^k_A∂_kA_l. This can be written as ∂_kA_l=J_kl + ∂_lA_k. The first term gives the desired total divergence ∂_α (ε^αβγJ_βγ H_A).

   The second term is proportional to the term ∂_αH_A- {A_α,H}. Suppose that the induced Kähler form is transversal to the light-like time coordinate t, i.e. the induced Kähler form does not have components of form J_tμ. In this kind of situation the only possible choice for α corresponds to the time coordinate t. In this situation one can perform the replacement ∂_αH_A-{A_α,H}→ dH_A/dt-{A_t,H}. This corresponds to a Hamiltonian time evolution generated by the projection A_t acting as a Hamiltonian. If this is really a Hamiltonian time evolution, one has dH_A/dt-{A,H}=0. Because the Poisson bracket represents a commutator, the Hamiltonian time evolution equation is analogous to the vanishing of a covariant derivative of H_A along light-like curves: ∂_tH_A +[A,H_A]= 0. The physical interpretation is that the partonic surface develops like a particle with a Kähler charge. As a consequence the change of the action reduces to a total divergence.

   An explicit expression for the conserved current J_A^α=H_A ε^αβγJ_βγ can be derived from the vanishing of the total divergence. Symplectic transformations on X² generate an infinite-dimensional symplectic algebra. The charge is given by the Hamiltonian flux Q_A =∫ H_A J_αβdx^α∧dx^β.

4. If the projection of the partonic path CP₂ or M⁴ is 2-D, then the light-like geodesic line corresponds to the path of the parton surface. If A_l can be chosen parallel to the surface, its projection in the direction of time disappears and one has A_t=0. In the more general case, X² could, for example, rotate in CP₂. In this case A_t is nonvanishing. If J is transversal (no Kähler electric field), charge conservation is obtained.

Do the above observations apply at the boundary of the light-cone?

1. Now the 3-surface is space-like and Chern-Simons-Kähler action makes sense. It is not necessary but emerges from the "instanton density" for the Kähler form. The symplectic transformations of δ M⁴₊× CP₂ are the symmetries. The most time evolution associated with the radial light-like coordinate would be from the tip of the light-cone boundary to the boundary of CD. Conserved charges as homological invariants defining symplectic algebra would be associated with the 2-D slices of 3-surfaces. For closed 3-surfaces the total charges from the sheets of 3-space as covering of δ M⁴₊ must sum up to zero.
2. Interestingly, the original proposal for the isometries of WCW was that the Hamiltonian fluxes assignable to M⁴ and CP₂ degrees of freedom at light-like boundary act define the charges associated with the WCW isometries as symplectic transformations so that a strong form of holography would have been be realized and space-time surface would have been effectively 2-dimensional. The recent view is that these symmetries pose conditions only on the 3-D holographic data. The holographic charges would correspond to additional isometries of WCW and would be well-defined for the 3-surfaces at the light-cone boundary.

To sum up, one can imagine many options but the following picture is perhaps the simplest one and is supported by physical intuition and mathematical facts. The isometry algebra ofδ M⁴₊× CP₂ consists of generalized conformal and KM algebras at 3-surfaces in δ M⁴₊× CP₂ and symplectic algebras at the light cone boundary and 3-D light-like partonic orbits. The latter symmetries give constraints on the 3-D holographic data. It is still unclear whether one can assign generalized conformal and Kac-Moody charges to Chern-Simons-K\"ahler action. The isomorphic subalgebras labelled by a positive integer and their commutators with the entire algebra would annihilate the physical states.

The TGD counterparts of the gauge conditions of string models

The string model picture forces to ask whether the symplectic algebras and the generalized conformal and Kac-Moody algebras could act as gauge symmetries.

1. In string model picture conformal invariance would suggest that the generators of the generalized conformal and KM symmetries act as gauge transformations annihilate the physical states. In the TGD framework, this does not however make sense physically. This also suggests that the components of the metric defined by supergenerators of generalized conformal and Kac Moody transformations vanish. If so, the symplectomorphisms δ M⁴₊× CP₂ localized with respect to the light-like radial coordinate acting as isometries would be needed. The half-algebras of both symplectic and conformal generators are labelled by a non-negative integer defining an analog of conformal weight so there is a fractal hierarchy of isomorphic subalgebras in both cases.
2. TGD forces to ask whether only subalgebras of both conformal and Kac-Moody half algebras, isomorphic to the full algebras, act as gauge algebras. This applies also to the symplectic case. Here it is essential that only the half algebra with non-negative multiples of the fundamental conformal weights is allowed. For the subalgebra annihilating the states the conformal weights would be fixed integer multiples of those for the full algebra. The gauge property would be true for all algebras involved. The remaining symmetries would be genuine dynamical symmetries of the reduced WCW and this would reflect the number theoretically realized finite measurement resolution. The reduction of degrees of freedom would also be analogous to the basic property of hyperfinite factors assumed to play a key role in thee definition of finite measurement resolution.
3. For strong holography, the orbits of partonic 2-surfaces and boundaries of the spacetime surface at δ M⁴₊ would be dual in the information theoretic sense. Either would be enough to determine the space-time surface.

See the articles About the Relationships Between Weak and Strong Interactions and Quantum Gravity in the TGD Universe and Holography and Hamilton-Jacobi Structure as 4-D generalization of 2-D complex structure)

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.