### Updated view about the Kähler geometry of "world of classical worlds"

TGD differs in several respects from quantum field theories and string models. The basic mathematical difference is that the mathematically poorly defined notion of path integral is replaced with the mathematically well-defined notion of functional integral defined by the Kähler function defining Kähler metric for WCW ("world of classical worlds"). Apart from quantum jump, quantum TGD is essentially theory of classical WCW spinor fields with WCW spinors represented as fermionic Fock states. One can say that Einstein's geometrization of physics program is generalized to the level of quantum theory.

It has been clear from the beginning that the gigantic super-conformal symmetries generalizing ordinary super-conformal symmetries are crucial for the existence of WCW Kähler metric. The detailed identification of Kähler function and WCW Kähler metric has however turned out to be a difficult problem. It is now clear that WCW geometry can be understood in terms of the analog of AdS/CFT duality between fermionic and space-time degrees of freedom (or between Minkowskian and Euclidian space-time regions) allowing to express Kähler metric either in terms of Kähler function or in terms of anti-commutators of WCW gamma matrices identifiable as super-conformal Noether super-charges for the symplectic algebra assignable to δ M^{4}_{+/-}× CP_{2}. The string model description of gravitation emerges and also the TGD based view about dark matter becomes more precise.

**Kähler function, Kähler action, and connection with string models**

The definition of Kähler function in terms of Kähler action is possible because space-time regions can have also Euclidian signature of induced metric. Euclidian regions with 4-D CP_{2} projection - wormhole contacts - are identified as lines of generalized Feynman diagrams - space-time correlates for basic building bricks of elementary particles. Kähler action from Minkowskian regions is imaginary and gives to the functional integrand a phase factor crucial for quantum field theoretic interpretation. The basic challenges are the precise specification of Kähler function of "world of classical worlds" (WCW) and Kähler metric.

There are two approaches concerning the definition of Kähler metric: the conjecture analogous to AdS/CFT duality is that these approaches are mathematically equivalent.

- The Kähler function defining Kähler metric can be identified as Kähler action for space-time regions with Euclidian signature for a preferred extremal containing 3-surface as the ends of the space-time surfaces inside causal diamond (CD). Minkowskian space-time regions give to Kähler action an imaginary contribution interpreted as the counterpart of quantum field theoretic action. The exponent of Kähler function defines functional integral in WCW. WCW metric is dictated by the Euclidian regions of space-time with 4-D CP
_{2}projection.

The basic question concerns the attribute "preferred". Physically the preferred extremal is analogous to Bohr orbit. What is the mathematical meaning of preferred extremal of Kähler action? The latest step of progress is the realization that the vanishing of generalized conformal charges for the ends of the space-time surface fixes the preferred extremals to high extent and is nothing but classical counterpart for generalized Virasoro and Kac-Moody conditions.

- Fermions are also needed. The well-definedness of electromagnetic charge led to the hypothesis that spinors are restricted to string world sheets. It has become also clear that string world sheets are most naturally minimal surfaces with 1-D CP
_{2}projection (this brings in gravitational constant) and that Kähler action in Minkowskian regions involves also the string area (, which does not contribute to Kähler function) giving the entire action in the case of M^{4}type vacuum extremals with vanishing Kähler form. Hence vacuum extremals might serve as an excellent approximation for the sheets of the many-sheeted space-time in Minkowskian space-time regions.

- Second manner to define Kähler metric is as anticommutators of WCW gamma matrices identified as super-symplectic Noether charges for the Dirac action for induced spinors with string tension proportional to the inverse of Newton's constant. These charges are associated with the 1-D space-like ends of string world sheets connecting the wormhole throats. WCW metric contains contributions from the spinor modes associated with various string world sheets connecting the partonic 2-surfaces associated with the 3-surface.

It is clear that the information carried by WCW metric about 3-surface is rather limited and that the larger the number of string world sheets, the larger the information. This conforms with strong form of holography and the notion of measurement resolution as a property of quantums state. Clearly. Duality means that Kähler function is determined either by space-time dynamics inside Euclidian wormhole contacts or by the dynamics of fermionic strings in Minkowskian regions outside wormhole contacts. This duality brings strongly in mind AdS/CFT duality. One could also speak about fermionic emergence since Kähler function is dictated by the Kähler metric part from a real part of gradient of holomorphic function: a possible identification of the exponent of Kähler function is as Dirac determinant.

**Realization of super-conformal symmetries**

The detailed realization of various super-conformal symmetries has been also a long standing problem but recent progress leads to very beautiful overall view.

- Super-conformal symmetry requires that Dirac action for string world sheets is accompanied by string world sheet area as part of bosonic action. String world sheets are implied and can be present only in Minkowskian regions if one demands that octonionic and ordinary representations of induced spinor structure are equivalent (this requires vanishing of induced spinor curvature to achieve associativity in turn implying that CP
_{2}projection is 1-D). Note that 1-dimensionality of CP_{2}projection is symplectically invariant property. Neither string world sheet area nor Kähler action is invariant under symplectic transformations. This is necessary for having non-trivial Kähler metric. Whether WCW really possesses super-symplectic isometries remains an open problem.

- Super-conformal symmetry also demands that Kähler action is accompanied by what I call Kähler-Dirac action with gamma matrices defined by the contractions of the canonical momentum currents with imbedding space-gamma matrices. Hence also induced spinor fields in the space-time interior must be present. Indeed, inside wormhole contacts Kähler-Dirac equation reducing to CP
_{2}Dirac equation for CP_{2}vacuum extremals dictates the fermionic dynamics.

Strong form of holography implied by strong form of general coordinate invariance strongly suggests that super-conformal invariance in the interior of the space-time surface is a broken gauge invariance in the sense that the super-conformal charges for a sub-algebra with conformal weights vanishing modulo some integer n vanish. The proposal is that n corresponds to the effective Planck constant as h

_{eff}/h=n. For string world sheets super-conformal symmetries are not gauge symmetries and strings dominate in good approximation the fermionic dynamics.

**Interior dynamics for fermions, the role of vacuum extremals, dark matter, and SUSY**

The key role of CP_{2}-type and M^{4}-type vacuum extremals has been rather obvious from the beginning but the detailed understanding has been lacking. Both kinds of extremals are invariant under symplectic transformations of δ M^{4}× CP_{2}, which inspires the idea that they give rise to isometries of WCW. The deformations CP_{2}-type extremals correspond to lines of generalized Feynman diagrams. M^{4} type vacuum extremals in turn are excellent candidates for the building bricks of many-sheeted space-time giving rise to GRT space-time as approximation. For M^{4} type vacuum extremals CP_{2} projection is (at most 2-D) Lagrangian manifold so that the induced Kähler form vanishes and the action is fourth-order in small deformations. This implies the breakdown of the path integral approach and of canonical quantization, which led to the notion of WCW.

If the action in Minkowskian regions contains also string area, the situation changes dramatically since strings dominate the dynamics in excellent approximation and string theory should give an excellent description of the situation: this of course conforms with the dominance of gravitation.

String tension would be proportional to 1/hbar G and this raises a grave classical counter argument. In string model massless particles are regarded as strings, which have contracted to a point in excellent approximation and cannot have length longer than Planck length. How this can be consistent with the formation of gravitationally bound states is however not understood since the required non-perturbative formulation of string model required by the large valued of the coupling parameter GMm is not known.

In TGD framework strings would connect even objects with macroscopic distance and would obviously serve as correlates for the formation of bound states in quantum level description. The classical energy of string connecting say the two wormhole contacts defining elementary particle is gigantic for the ordinary value of hbar so that something goes wrong.

I have however proposed that gravitons - at least those mediating interaction between dark matter have large value of Planck constant. I talk about gravitational Planck constant and one has h_{eff}= h_{gr}=GMm/v_{0}, where v_{0}/c<1 (v_{0} has dimensions of velocity). This makes possible perturbative approach to quantum gravity in the case of bound states having mass larger than Planck mass so that the parameter GMm analogous to coupling constant is very large. The velocity parameter v_{0}/c becomes the dimensionless coupling parameter. This reduces the string tension so that for string world sheets connecting macroscopic objects one would have T ∝ v_{0}/G^{2}Mm. For v_{0}= GMm/hbar, which remains below unity for Mm/m_{Pl}^{2} one would have h_{gr}/h=1. Hence the action remains small and its imaginary exponent does not fluctuate wildly to make the bound state forming part of gravitational interaction short ranged. This is expected to hold true for ordinary matter in elementary particle scales. The objects with size scale of large neutron (100 μm in the density of water) - probably not an accident - would have mass above Planck mass so that dark gravitons and also life would emerge as massive enough gravitational bound states are formed. h_{gr}=h_{eff} hypothesis is indeed central in TGD based view about living matter.
In this framework superstring theory with single value of Planck constant would not give rise to macroscopic
gravitationally bound matter and would be thus simply wrong.

If one assumes that for non-standard values of Planck constant only n-multiples of super-conformal algebra in interior annihilate the physical states, interior conformal gauge degrees of freedom become partly dynamical. The identification of dark matter as macroscopic quantum phases labeled by h_{eff}/h=n conforms with this.

The emergence of dark matter corresponds to the emergence of interior dynamics via breaking of super-conformal symmetry. The induced spinor fields in the interior of flux tubes obeying Kähler Dirac action should be highly relevant for the understanding of dark matter. The assumption that dark particles have essentially same masses as ordinary particles suggests that dark fermions correspond to induced spinor fields at both string world sheets and in the space-time interior: the spinor fields in the interior would be responsible for the long range correlations characterizing h_{eff}/h=n. Magnetic flux tubes carrying dark matter are key entities in TGD inspired quantum biology. Massless extremals represent second class of M^{4} type non-vacuum extremals.

This view forces once again to ask whether space-time SUSY is present in TGD and how it is realized. With a motivation coming from the observation that the mass scales of particles and sparticles most naturally have the same p-adic mass scale as particles in TGD Universe I have proposed that sparticles might be dark in TGD sense. The above argument leads to ask whether the dark variants of particles correspond to states in which one has ordinary fermion at string world sheet and 4-D fermion in the space-time interior so that dark matter in TGD sense would almost by definition correspond to sparticles!

See the chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds" of "Towards M-matrix" .

## 31 Comments:

Dear Matti,

Could you also describe your Kähler geometry of "world of classical worlds" in a more classical ontological way?

I am not quite sure what you mean by "classical".

The very definition of WCW Kahler metric must assign to given 3-surface 4-D space-time surface in order that 4-D general coordinate invariance can be realised. This translates to holography.

Classical world translates to 3-surface and by holography space-time surface identified as preferred extremal of Kahler action analogous to Bohr orbit.

GRT space-time and gauge theory description can be seen as "classical". GRT space-time is obtained by replacing many-sheeted space-time with region of Minkowski space with the deviation of metric from flat one obtains as sum of the deviations for the sheets. Gauge potentials in QFT limit correspond to sums for gauge potentials at sheets.

Euclidian space-time regions and the boundaries between Euclidian and Minkowskian regions are definitely something "non-classical" but absolutely essential for the reduction of Feynman diagrams to space-time topology and geometry. Point like particles of QFT emerge when one makes the 4- "lines" of generalised Feynman diagrams infinitely thin.

You asked about classical description of Kaehler

geometry: above I talked about this description for WCW.

Kaehler geometry is defined usually=classically in terms of Kahler function, call it K (say in the case of CP_2). One generalises this approach.

This is the first approach and gives K as Kahler action for a preferred extremal in Euclidian regions (lines of generalised Feynman diagrams having purely geometric and topological meaning, not so classical;-)).

Not so classical manner to define K is in terms of anticommutators of WCW gamma matrices identifiable as symplectic Noether super-charges, which can be calculated immediately and one obtains explicit expressions for the matrix elements of WCW metric.

This is the analog of AdS/CFT duality: I have talked also about strong form of holography and quantum classical correspondence. WCW geometry can be described in terms of fermions at strings in Minkowskian regions or geometric degrees of freedom of space-time surface in Euclidian regions.

Above I have cheated a little bit to create impression of classicality;-) : 3-surface in ZEO is union of space-like 3-surfaces at the ends of space-time surface located a opposite boundaries of causal diamond CD plus the light-like orbits of partonic surfaces at which the signature of the induced metric changes. In standard ontology it would be space-like 3-surface.

Thanks Matti, I have to soak that in before I come back.

Matti, about Kahler structures, Vector space and configuration space, could you imagine even the sort of vector space which is Chiral? or better have only spiral ( left or righthanded spiral curved vectors?

see perhaps as an example:

https://www.flickr.com/photos/93308747@N05/14210603469/

Chirality makes sense geometrically. Helical magnetic flux tubes would be chiral surface with well-defined handedness in geometric sense.Biomolecules would correspond to this kind of space-time surfaces.

Don't you think that a chiral vacuum could be the origin of our MATERIAL universe in contrast with an Anti-Material universe?

https://medium.com/the-physics-arxiv-blog/the-origin-of-life-and-the-hidden-role-of-quantum-criticality-ca4707924552

This comment has been removed by the author.

Stuart Kauffman by the way works in Finland. Quantum criticality is one of the basic predictions of TGD. I wrote just some time ago a series of postings about quantum criticality and hierarchy of Planck constants as crucial for life. Both dark matter and life are critical phenomena It is amazing that all these TGD notions are discovered but no one mentions TGD;-). Academic outsiders do not exist academically!

The picture proposed still introduction of high Tc superconductivity, magnetic flux tubes, macroscopic quantum coherence, hierarchy of Planck constants-dark matter connection, and many other things. This is a problem since all this is against the existing belief systems.

TGD is there just waiting but how to discover all these fantastic things without mentioning TGD? This defines perhaps the greatest challenge that recent day academic science is facing;-).

To Leo Vyuk:

Living matter is good example of chiral vacuum. Chirality selection of biomolecules is a good example about chiral ground state. In TGD molecule of given handedness represents its own space-time sheet defining mini-sub-Universe containing smaller space-time sheets as sub-Univserses and belonging to a bigger one.

Chirality selection is basically due to the presence of weak interactions in the Compton scale of weak bosons which is now scaled up by h_eff/h =n to a value which is of the order of molecule or cell or even bigger. This makes the system also macroscopic quantum system in this scaled. Dark weak interactions select molecules with preferred handedness as minimum energy states. Geometric chirality in turn induces polarisation of light for instance.

A righthanded chiral vacuum vector lattice should have a direct relation with the right handed double helkix structure of the DNA molecule?

Sorry, it must be: "the right handed double helix structure of the DNA molecule?" (not helkix)

http://arxiv.org/pdf/1103.1833.pdf

To Ulla:

What article proposes are entire critical phase transitions, which should play key role in biology. 4-dimensionality instead of 3-dimensionality. These authors have "discovered" zero energy ontology without saying anything about TGD. Very clever!

TGD ideas are now being discovered with constant rate. In ResearchGate there where more than 120 loadings of TGD related material so that the ideas will now find their way to the articles.

It would be of course nice if the discoverers would mention TGD but this is too much to hope in recent the day science, which tries to cope without ethics and moral.

To Leo:

I am not sure whether one needs lattice at all. Just the 3-surface which has geometric chirality is enough. Surface geomery of course induces chirality to possible lattice like structure - say sequence of DNA nucleotides. Essential is that one has sub-manifold geometry: one sees the structure from outside.

I do not know how to define chiral lattices in abstract manifold geometry where "seeing from outside" is not possible.

http://en.wikipedia.org/wiki/Leggett%E2%80%93Garg_inequality

compared to Bells inequality

Would this give the 4D in biology? Jenny Nielsen told she is doing her thesis on this. I asked if she is interested in quantum biology. Her answer: hmmmm

She does not yet know...

4-D brain, even 4-D society were the catch words that I introduced long time ago. 3-D state is replaced with 4-D behaviour pattern in zero energy ontology. This justifies notions like function, behavioural pattern, etc… used in bio-sciences but having no counterpart in fundamental physics.

Pleas let me become a bit more specific about my view on the multiverse:

I think that Stephen Hawking did not calculate with the possibility of a chiral oscillating Higgs field vacuum lattice combined with propeller shaped Fermions. Then, due to Vacuum chirality, Electron- and Positron propellers could both pushed away from the BH horizon after spin flip polarization at different distances, forming two charged separated spheres. With quark ( plasma) formation in between.

Based on such a simple object ( propeller and process) oriented ontology, Black Holes could be imagined as charge splitters violating the 2e law af thermodynamics, combined with a continuous microscopic big bang plasma creation process!

The result I try to describe :

1: Black holes are the same as Dark Matter, they all consume photons, even gravitons and the Higgs field, but REPEL Fermions due to their propeller shape. They produce electric charged plasma.

2: Dark Energy is the oscillating ( Casimir) energy of the Higgs Field equipped with a tetrahedron lattice structure with variable Planck length..

3: Quantum Gravity = Dual Push gravity= Attraction (Higgs-Casimir opposing Graviton push).

4: The Big Bang is a Splitting dark matter Big Bang Black Hole (BBBH), splitting into smaller primordial BBBH Splinters forming the Fractalic Lyman Alpha forest and evaporating partly into a zero mass energetic oscillating Higgs particle based Higgs field.

5: Dual PBBSs hotspots, produce central plasma concentration in electric Herbig Haro systems as a base for star formation in open star clusters as a start for Spiral Galaxies.

6: Spiral Galaxies will keep both Primordial Dark Matter Black Holes as Galaxy Anchor Black Holes (GABHs) at long distance.

7: After Galaxy Merging, these GABHs are the origin of

Galaxy- and Magnetic field complexity and distant dwarf galaxies .

8: Black Holes produce Plasma direct out of the Higgs field because two Higgs particles are convertible into symmetric electron and positron (or even dual quark-) propellers (by BH horizon fluctuations).

9: The chirality of the (spiralling) vacuum lattice is the origin our material universe. (propeller shaped positrons merge preferentially first with gluons to form (u) Quarks to form Hydrogen.

10: The first Supernovas produce medium sized Black Holes as the base for secondary Herbig Haro systems and open star clusters.

11: ALL Dark Matter Black Holes are supposed to be CHARGE SEPARATORS with internal positive charge and an external globular shell of negative charged Quark electron plasma.

12: The lightspeed is related to gravity fields like the earth with long extinction distances to adapt with the solar gravity field.

See also: vixra.org/author/leo_vuyk

Quantum FFF Theory as I propose: states that the raspberry shaped multiverse is symmetric and instant entangled down to the smallest quantum level. Also down to living and dying CATS in BOXES.

If our material universes has a chiral oscillating Higgs field, then our material Right Handed DNA helix molecule could be explained.

However it also suggests that in our opposing ANTI-MATERIAL multiverse neighbour universe the DNA helix should have a LEFT HANDED spiral.

Interestingly, according to Max Tegmark: in a multiverse we may ask: is there COPY PERSON over there, who is reading the same lines as I do?

If this COPY person is indeed living over there, then even our consciousness should be shared in a sort of DEMOCRATIC form,

Then we are not alone with our thoughts and doubts,see:

Democratic Free Will in the instant Entangled Multiverse.

http://vixra.org/pdf/1401.0071v2.pdf

I believe in multiverse in the restricted sense that there is an interacting hierarchy of space-time sheets collapsed to single sheet in GRT approximation leading to analogous such as several maximal signal speeds. I believe also in a hierarchy of conscious entities, which in certain sense means multiverse too. Subselves represent mental images of self. Self is identified as a sequences of state function reductions at same boundary of causal diamond.

I cannot take seriously the multiverse in the sense of inflationary theories: in TGD the sheets of many-sheeted space-time obey essentially the same standard model physics at the level of symmetries and there is no need to introduce inflaton fields. Conscious entities can mimic each other and by looking around it becomes clear that they might love to do so but the idea about additional copies of colleagues looks extremely unattractive to me;-).

As a whole, multiverse in standard sense has ended up to playing with mathematical and conceptual pathologies such as Boltzmann brains. This could have been avoided with some critical philosophical thought and contact with experiment: not only cosmology.

Superstring theory has also suffered the same fate. When the connection with experimental world splits, theoreticians begin to hallucination begin just as under sensory deprivation induced by think tanks. Places like Harward are indeed think tanks carefully isolated from external world;-).

I think it could much more simple, see:

“The Navel Cord Multiverse with Raspberry Shape, a Super Symmetric Entangled 12 Fold Bubble Universe.”

http://vixra.org/pdf/1312.0143v2.pdf

http://www.dailymail.co.uk/sciencetech/article-2975606/Did-Homer-Simpson-discover-HIGGS-BOSON-Maths-1998-episode-predicts-particle-s-mass-14-years-CERN.html

mmmmm donuts

Lubos wrote about Homer's discovery recently. As a matter fact, the formula giving the prediction was of order Planck mass as one learns from the picture so that it was by a factor of order 10^19 too large!

Now this is interesting...http://m.phys.org/news/2015-03-quantum-scheme-states-transmitting-physical.html still no one else has thought of a hierarchy of plank constants? Could it also be thought of as an array ? Hierarchies more resemble graphs than sequences

Looks interesting. It seems that that it is difficult to distinguish quantum information scientists from parapsychologists who have talked about information transfer without physical signal!;-).

*How to generate entanglement without sending particles between the systems to be entangled? *

This seems to be the basic question: prior entanglement is necessary for quantum commutation schemes. Have never thought about this kind of bottleneck question.

Authors claim that it is possible to generate entanglement without interaction that is sending physical particles between them. *Interaction free measurement* is mentioned: this is central in TGD inspired theory of consciousness: kind of telepathic effect too!

The notion of *chained Zeno effect* is introduced and claimed to make possible entanglement between two distant object. This in turn would drake possible information transfer without transfer of particles. I hope I understood.

Quantum version of obstructing object is a further notion: object obstructs if it it absorbs the incoming radiation. Is *quantum obstructing object* in superposition of obstructing and non-obstructing states?

How could this relate to TGD.

a) Interaction free measurement provides a manner

to transformed negentropic entanglement to conscious information without destroying it.

b) Zeno effect in TGD means repeated reductions on same boundary of CD and gives rise to self as conscious entity. During this period no decoherence occurs (interpreted as first reduction to opposite boundary of CD).

c) In TGD based quantum biology communication involving cyclotron resonance of dark photons is essential element. Magnetic body can become absorber if it changes flux tube thickness controlling magnetic field strength controlling cyclotron frequency.

Could the proposed idea have some relevance for consciousness theory and quantum biology: kind of telepathic information transfer would be in question!

Information moves without movement of informational particles also in nerves, according to the new model, if I remember right. This was the new thing for me.

To Ulla:

The ideas in quantum computation are being rapidly transferred to biology. I hope I will find time to understand what the chain Zeno effect means. Basic prerequisites for it seems to be met in TGD model.

Some kind of holography must be involved, as instance in form of imprints in light?

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