### Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds"

The construction of Kähler geometry of WCW ("world of classical worlds") is fundamental to TGD program. I ended up with the idea about physics as WCW geometry around 1985 and made a breakthrough around 1990, when I realized that Kähler function for WCW could correspond to Kähler action for its preferred extremals defining the analogs of Bohr orbits so that classical theory with Bohr rules would become an exact part of quantum theory and path integral would be replaced with genuine integral over WCW. The motivating construction was that for loop spaces leading to a unique Kähler geometry. The geometry for the space of 3-D objects is even more complex than that for loops and the vision still is that the geometry of WCW is unique from the mere existence of Riemann connection.

The basic idea is that WCW is union of symmetric spaces G/H labelled by zero modes which do not contribute to the WCW metric. There have been many open questions and it seems the details of the earlier approach must be modified at the level of detailed identifications and interpretations. What is satisfying that the overall coherence of the picture has increased dramatically and connections with string model and applications of TGD as WCW geometry to particle physics are now very concrete.

- A longstanding question has been whether one could assign Equivalence Principle (EP) with the coset representation formed by the super-Virasoro representation assigned to G and H in such a manner that the four- momenta associated with the representations and identified as inertial and gravitational four-momenta would be identical. This does not seem to be the case. The recent view will be that EP reduces to the view that the classical four- momentum associated with Kähler action is equivalent with that assignable to modified Dirac action supersymmetrically related to Kähler action: quantum classical correspondence (QCC) would be in question. Also strong form of general coordinate invariance implying strong form of holography in turn implying that the super-symplectic representations assignable to space-like and light-like 3-surfaces are equivalent could imply EP with gravitational and inertial four-momenta assigned to these two representations.

- The detailed identification of groups G and H and corresponding algebras has been a longstanding problem. Symplectic algebra associated with δM
^{4}_{+/-}× CP_{2}(δM^{4}_{+/-}is light-cone boundary - or more precisely, with the boundary of causal diamond (CD) defined as Cartesian product of CP_{2}with intersection of future and past direct light cones of M^{4}has Kac-Moody type structure with light-like radial coordinate replacing complex coordinate z. Virasoro algebra would correspond to radial diffeomorphisms.

I have also introduced Kac-Moody algebra assigned to the isometries and localized with respect to internal coordinates of the light-like 3-surfaces at which the signature of the induced metric changes from Minkowskian to Euclidian and which serve as natural correlates for elementary particles (in very general sense!). This kind of localization by force could be however argued to be rather ad hoc as opposed to the inherent localization of the symplectic algebra containing the symplectic algebra of isometries as sub-algebra. It turns out that one obtains direct sum of representations of symplectic algebra and Kac-Moody algebra of isometries naturally as required by the success of p-adic mass calculations.

- The dynamics of Kähler action is not visible in the earlier construction. The construction also expressed WCW Hamiltonians as 2-D integrals over partonic 2-surfaces. Although strong form of general coordinate invariance (GCI) implies strong form of holography meaning that partonic 2-surfaces and their 4-D tangent space data should code for quantum physics, this kind of outcome seems too strong. The progress in the understanding of the solutions of modified Dirac equation led however to the conclusion that spinor modes other than right-handed neutrino are localized at string world sheets with strings connecting different partonic 2-surfaces.

This leads to a modification of earlier construction in which WCW super-Hamiltonians were essentially 2-D flux integrals. Now they are 2-D flux integrals with super-Hamiltonian replaced Noether super charged for the deformations in G and obtained by integrating over string at each point of partonic 2-surface. Each spinor mode gives rise to super current and the modes of right-handed neutrino and other fermions differ in an essential manner. Right-handed neutrino would correspond to symplectic algebra and other modes to the Kac-Moody algebra and one obtains the crucial 5 tensor factors of Super Virasoro required by p-adic mass calculations.

The matrix elements of WCW metric between Killing vectors are expressible as anticommutators of super-Hamiltonians identifiable as contractions of WCW gamma matrices with these vectors and give Poisson brackets of corresponding Hamiltonians. The anti-commutation relates of induced spinor fields are dictated by this condition. Everything is 3-dimensional although one expects that symplectic transformations localized within interior of X

^{3}act as gauge symmetries so that in this sense effective 2-dimensionality is achieved. The components of WCW metric are labelled by standard model quantum numbers so that the connection with physics is extremely intimate.

- An open question in the earlier visions was whether finite measurement resolution is realized as discretization at the level of fundamental dynamics. This would mean that only certain string world sheets from the slicing by string world sheets and partonic 2-surfaces are possible. The requirement that anti-commutations are consistent suggests that string world sheets correspond to surfaces for which Kähler magnetic field is constant along string in well defined sense (J
_{μν}ε^{μν}g^{1/2}remains constant along string). It however turns that by a suitable choice of coordinates of 3-surface one can guarantee that this quantity is constant so that no additional constraint results.

See the new chapter Recent View about
Kähler Geometry and Spin Structure of "World of Classical Worlds of "Quantum TGD as
Infinite-dimensional Spinor Geometry" or the article
with the same title.

## 4 Comments:

Dear Matti,

Thanks for the last answer and the new postings.

see is there any misunderstanding?

In classical TGD, one can assign to a macroscopic object(for example a system of gas) both classical kahler field and classical spinor field at the same time. information of distributions of particles speeds in the gas is encoded to classical spinor field.

and information of geometrical structures of the space-time sheet of the system of gas is encoded to induced kahler field and induced metric.

one can ask about action principle for both of the fields(kahler field and spinor field). determining one of them is enough to determine another one.

in classical TGD, particles glued to the larger space time but in quantum TGD this is not the correct. in quantum TGD particles are points of larger space time sheet. hence in quantum TGD nothing is glued to another.

suppose that one say also in classical world, particles are points of larger space time and nothing glued to the larger space time.

does the picture make sense?

Dear Hamed,

I would say that geometry - space-time as surface - is the fundamental. When you know

this surface you can deduce various classical fields - induced K\"abler form (Kahler field), electroweak gauge fields, color gauge field, gravitational field as induced metric.

Second quantised induced spinor field whose components other than right-handed neutrino are localised at 2-D string world sheets and they bring in fermions and representation of particles

as field quanta. Gauge bosons and gravitons emerge from fermion antifermion pairs at

wormhole throats. WCW spinor field represents fermion Fock states as functional of 3-surface.

Modified Dirac action and Kahler action are closely related and consistency for the modified Dirac

action based on Kahler action requires extremals of Kahler action.

One can of course wonder whether one could reduce the dynamics to that for modified Dirac action. For years ago I proposed that Dirac determinant could indeed give rise to Kahler action. I ended up with Riemann Zeta.

Naively one expects that the determinant reduces to a product of sub-determinants associated with string world sheets. Their number is in principle infinite: one for each point of partonic 2-surface unless finite measurement resolution somehow makes it finite.

How to define this product? Does one take only product over sub-determinants associated with finite number of braid strands serving as ends of allow string world sheets. What principle would fix which string world sheets are allow? Or does one try to define the determinant as product over sub-determinants for a continuum of points of partonic 2-surface? This kind of situation is of course encountered also in QFT so that one might be able to consider logarithm of determinant defined as continuous integral over contributions from different sheets.

As I see it, there is no difference between classical and quantum views about particle in TGD framework. Particle has as geometric correlate space-time sheet or to be precise- double structure forming at least double covering of M^4 locally since boundaries are not allowed. Wormhole throats connecting the sheets are building bricks of particle. This double sheet can be glued by wormhole contacts (not carrying monopole flux through them) to large space-time sheet. Quantum description introduces second quantized spinor modes and quantum superpositions of space-time sheets but it does not make particle point.

One can of course idealise that situation and this is done at QFT limit where twistorial amplitudes

functionally integrate over all the details and effectively only cm degrees of freedom are left in consideration in case of light (originally massless particles).

Dear Matti,

Thanks. as i understand, classical spinor field are the limit of Second quantised induced spinor field when 3-surfaces replaced with points.

also induced kahler field are the limit of kahler field of WCW when the 3-surfaces replaced with points.

induced spinor connection of CP2 to macroscopic objects have large Planck constants that we can not observe any electroweak field at this level?

This is more complicated.

One must distinguish between fundamental spinor fields and those used to described various particles. Fundamental spinor fields appear in modified Dirac action: all modes expect right handed neutrino are localised at 2-D string world sheets.

The spinor fields that we assign with particles correspond to *imbedding space* spinor fields assignable to cm degrees of freedom of 3-surfaces. This means effective replacement of particle with its cm. This gives QFT description

which I believe relies on twistor amplitudes.

In string models the situation is exactly the same: the microscopic string level and observable target space level. Also in string models the spinors assigned to physical particles are distinct from those associated with string world sheets. What we observe is not directly the spinors at space-time sheets but something which involves a lot of integration over degrees of freedom which we cannot observe.

The large Planck constant can be assigned with any particle: I would not assign it to induced spinor field but to topology of space-time sheet. h_eff=nh, n=n1_n_2 corresponds to n_1- resp. n_2-sheeted coverings of M^4 and CP_2 by space-time sheet. The topology of space-time sheets manifests itself in this manner although otherwise 3-surface representing particle is replaced with its cm. And do not forget that we have always quantum superpositions of surfaces!

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