The classical part of the twistor story

Twistors Grassmannian formalism has made a breakthrough in N=4 supersymmetric gauge theories and the Yangian symmetry suggests that much more than mere technical breakthrough is in question. Twistors seem to be tailor made for TGD but it seems that the generalization of twistor structure to that for 8-D imbedding space H=M⁴× CP₂ is necessary. M⁴ (and S⁴ as its Euclidian counterpart) and CP₂ are indeed unique in the sense that they are the only 4-D spaces allowing twistor space with Kähler structure. These twistor structures define define twistor structure for the imbedding space H and one can ask whether this generalized twistor structure could allow to understand both quantum TGD and classical TGD defined by the extremals of Kähler action. In the following I summarize the background and develop a proposal for how to construct extremals of Kähler action in terms of the generalized twistor structure.

Summary about background

Consider first some background.

1. The twistors originally introduced by Penrose (1967) have made breakthrough during last decade. First came the twistor string theory of Edward Witten proposed twistor string theory and the work of Nima-Arkani Hamed and collaborators led to a revolution in the understanding of the scattering amplitudes of scattering amplitudes of gauge theories. Twistors do not only provide an extremely effective calculational method giving even hopes about explicit formulas for the scattering amplitudes of N=4 supersymmetric gauge theories but also lead to an identification of a new symmetry: Yangian symmetry which can be seen as multilocal generalization of local symmetries.
   

   This approach, if suitably generalized, is tailor-made also for the needs of TGD. This is why I got seriously interested on whether and how the twistor approach in empty Minkowski space M⁴ could generalize to the case of H=M⁴× CP₂. In particular, is the twistor space associated with H should be just the cartesian product of those associated with its Cartesian factors. Can one assign a twistor space with CP₂?
   
   

2. First a general result: any oriented manifold X with Riemann metric allows 6-dimensional twistor space Z as an almost complex space. If this structure is integrable, Z becomes a complex manifold, whose geometry describes the conformal geometry of X. In general relativity framework the problem is that field equations do not imply conformal geometry and twistor Grassmann approach certainly requires the complex manifold structure.
   
   
3. One can also consider also a stronger condition: what if twistor space allows also Kähler structure? The twistor space of empty Minkowski space M⁴ is 3-D complex projective space P³ and indeed allows Kähler structure. Rather remarkably, there are no other space-times with Minkowski signature allowing twistor space with Kähler structure. Does this mean that the empty Minkowski space of special relativity is much more than a limit at which space-time is empty?
   

   This also means a problem for GRT. Twistor space with Kähler structure seems to be needed but general relativity does not allow it. Besides twistor problem GRT also has energy problem: matter makes space-time curved and the conservation laws and even the definition of energy and momentum are lost since the underlying symmetries giving rise to the conservation laws through Noether's theorem are lost. GRT has therefore two bad mathematical problems which might explain why the quantization of GRT fails. This would not be surprising since quantum theory is to high extent representation theory for symmetries and symmetries are lost. Twistors would extend these symmetries to Yangian symmetry but GRT does not allow them.
   
   

4. What about twistor structure in CP₂? CP₂ allows complex structure (Weyl tensor is self-dual), Kähler structure plus accompanying symplectic structure, and also quaternion structure. One of the really big personal surprises of the last years has been that CP₂ twistor space indeed allows Kähler structure meaning the existence of antisymmetric tensor representing imaginary unit whose tensor square is the negative of metric in turn representing real unit.
   

   The article by Nigel Hitchin, a famous mathematical physicist describes a detailed argument identifying S⁴ and CP₂ as the only compact Riemann manifolds allowing Kählerian twistor space. Hitchin sent his discovery for publication 1979. An amusing co-incidence is that I discovered CP₂ just this year after having worked with S² and found that it does not really allow to understand standard model quantum numbers and gauge fields. It is difficult to avoid thinking that maybe synchrony indeed a real phenomenon as TGD inspired theory of consciousness predicts to be possible but its creator cannot quite believe. Brains at different side of globe discover simultaneously something closely related to what some conscious self at the higher level of hierarchy using us as instruments of thinking just as we use nerve cells is intensely pondering.
   

   Although 4-sphere S⁴ allows twistor space with Kähler structure, it does not allow Kähler structure and cannot serve as candidate for S in H=M⁴× S. As a matter of fact, S⁴ can be seen as a Wick rotation of M⁴ and indeed its twistor space is P³.
   

   In TGD framework a slightly different interpretation suggests itself. The Cartesian products of the intersections of future and past light-cones - causal diamonds (CDs)- with CP₂ - play a key role in zero energy ontology (ZEO). Sectors of "world of classical worlds" (WCW) correspond to 4-surfaces inside CD× CP₂ defining a the region about which conscious observer can gain conscious information: state function reductions - quantum measurements - take place at its light-like boundaries in accordance with holography. To be more precise, wave functions in the moduli space of CDs are involved and in state
   function reductions come as sequences taking place at a given fixed boundary. These sequences define the notion of self and give rise to the experience about flow of time. When one replaces Minkowski metric with Euclidian metric, the light-like boundaries of CD are contracted to a point and one obtains topology of 4-sphere S⁴.
   
   

5. The really big surprise was that there are no other compact 4-manifolds with Euclidian signature of metric allowing twistor space with Kähler structure! The imbedding space H=M⁴× CP₂ is not only physically unique since it predicts the quantum number spectrum and classical gauge potentials consistent with standard model but also mathematically unique!
   

   After this I dared to predict that TGD will be the theory next to GRT since TGD generalizes string model by bringing in 4-D space-time. The reasons are manyfold: TGD is the only known solution to the two big problems of GRT: energy problem and twistor problem. TGD is consistent with standard model physics and leads to a revolution concerning the identification of space-time at microscopic level: at macroscopic level it leads to GRT with some anomalies for which there is empirical evidence. TGD avoids the landscape problem of M-theory and anthropic non-sense. I could continue the list but I think that this is enough.
   
   

6. The twistor space of CP₂ is 3-complex dimensional flag manifold F₃= SU(3)/U(1)× U(1) having interpretation as the space for the choices of quantization axes for the color hypercharge and isospin. This choice is made in quantum measurement of these quantum numbers and a means localization to single point in F₃. The localization in F₃ could be higher level measurement leading to the choice of quantizations for the measurement of color quantum numbers.
   

   Analogous interpretation could make sense for M⁴ twistors represented as points of P³. Twistor corresponds to a light-like line going through some point of M⁴ being labelled by 4 position coordinates and 2 direction angles: what higher level quantum measurement could involve a choice of ligh-like line going through a point of M⁴? Could the associated spatial direction specify spin quantization axes? Could the associated time direction specify preferred rest frame? Does the choice of position mean localization in the measurement of position? Do momentum twistors relate to the localization in momentum space? These questions remain fascinating open questions and I hope that they will lead to a considerable progress in the understanding of quantum TGD.
   
   

7. It must be added that the twistor space of CP₂ popped up much earlier in a rather unexpected context: I did not of course realize that it was twistor space. Topologist Barbara Shipman has proposed a model for the honeybee dances leading to the emerge of F₃. The model led her to propose that quarks and gluons might have something to do with biology. Because of her position and specialization the proposal was forgiven and forgotten by community. TGD however suggests both dark matter hierarchies and p-adic hierarchies of physics. For dark hierarchies the masses of particles would be the standard ones but the Compton scales would be scaled up by h_eff/h=n. Below the Compton scale one would have effectively massless gauge boson: this could mean free quarks and massless gluons even in cell length scales. For p-adic hierarchy mass scales would be scaled up or down from their standard values depending on the value of the p-adic prime.
   
   

Why twistor spaces with Kähler structure?

I have not yet even tried to answer an obvious question. Why the fact that M⁴ and CP₂ have twistor spaces with Kähler structure could be so important that it would fix the entire physics? Let us consider a less general question. Why they would be so important for the classical TGD - exact part of quantum TGD - defined by the extremals of Kähler action?

1. Properly generalized conformal symmetries are crucial for the mathematical structure of TGD. Twistor spaces have almost complex structure and in these two special cases also complex, Kähler, and symplectic structures (note that the integrability of the almost complex structure to complex structure requires the self-duality of the Weyl tensor of the 4-D manifold).
   

   The Cartesian product CP₃× F₃ of the two twistor spaces with Kähler structure is expected to be fundamental for TGD. The obvious wishful thought is that this space makes possible the construction of the extremals of Kähler action in terms of holomorphic surfaces defining 6-D twistor sub-spaces of CP₃× F₃ allowing to circumvent the technical problems due to the signature of M⁴ encountered at the level of M⁴× CP₂. For years ago I considered the possibility that complex 3-manifolds of CP₃×CP₃ could have the structure of S² fiber space but did not realize that CP₂ allows twistor space with Kähler structure so that CP₃× F₃ is a more plausible choice.
   
   

2. It is possible to construct so called complex symplectic manifolds by Kähler manifolds using as complexified symplectic form ω₁+Iω₂. Could the twistor space CP₃× F₃ be seen as complex symplectic sub- manifold of real dimension 6?
   

   The safest option is to identify the imaginary unit I as same imaginary unit as associated with the complex coordinates of CP₃ and F₃. At space-time level however complexified quaternions and octonions could allow alternative formulation. I have indeed proposed that space-time surfaces have associative of co-associative meaning that the tangent space or normal space at a given point belongs to quaternionic subspace of complexified octonions.
   
   

3. Recall that every 4-D orientable Riemann manifold allows a twistor space as 6-D bundle with CP₁ as fiber and possessing almost complex structure. Metric and various gauge potentials are obtained by inducing the corresponding bundle structures. Hence the natural guess is that the twistor structure of space-time surface defined by the induced metric is obtained by induction from that for CP₃× F₃ by restricting its twistor structure to a 6-D (in real sense) surface of CP₃× F₃ with a structure of twistor space having at least almost complex structure with CP₁ as a fiber. If so then one can indeed identify the base space as 4-D space-time surface in M⁴× SCP₂ using bundle projections in the factors CP₃ and F₃.

About the identification of 6-D twistor spaces as sub-manifolds of CP₃× F₃

How to identify the 6-D sub-manifolds with the structure of twistor space? Is this property all that is needed? Can one find a simple solution to this condition? In the following intuitive considerations of a simple minded physicist. Mathematician could probably make much more interesting comments.

Consider the conditions that must be satisfied using local trivializations of the twistor spaces. Before continuing let us introduce complex coordinates z_i=x_i+iy_i resp. w_i=u_i+iv_i for CP₃ resp. F₃.

1. 6 conditions are required and they must give rise by bundle projection to 4 conditions relating the coordinates in the Cartesian product of the base spaces of the two bundles involved and thus defining 4-D surface in the Cartesian product of compactified M⁴ and CP₂.
   
   
2. One has Cartesian product of two fiber spaces with fiber CP₁ giving fiber space with fiber CP₁¹× CP₁². For the 6-D surface the fiber must be CP₁. It seems that one must identify the two spheres CP₁ⁱ. Since holomorphy is essential, holomorphic identification w₁=f(z₁) or z₁=f(w₁) is the first guess. A stronger condition is that the function f is meromorphic having thus only finite numbers of poles and zeros of finite order so that a given point of CP₁ⁱ is covered by CP₁ⁱ⁺¹. Even stronger and very natural condition is that the identification is bijection so that only Möbius transformations parametrized by SL(2,C) are possible.
   
   
3. Could the Möbius transformation f: CP₁¹ → CP₁² depend parametrically on the coordinates z₂,z₃ so that one would have w₁= f₁(z₁, z₂,z₃), where the complex parameters a,b,c,d (ad-bc=1) of Möbius transformation depend on z₂ and z₃ holomorphically?
   

   What conditions can one pose on the dependence of the parameters a,b,c,d of the Möbius transformation on (z₂,z₃)? The spheres CP₁ defined by the conditions w₁= f(z₁, z₂,z₃) and z₁= g(w₁, w₂,w₃) must be identical. Inverting the first condition one obtains z₁= f^-1(w₁, z₂,z₃) and this must allow an expression as z₁= g(w₁,w₂,w₃). This is true if z₂ and z₃ can be expressed as holomorphic functions of (w₂,w₃): z_i= f_i(w_k), i=2,3, k=2,3. Non-holomorphic correspondence cannot be excluded.
   
   

4. Further conditions are obtained by demanding that the known extremals - at least non-vacuum extremals - are allowed. The known extremals can be classified into CP₂ type vacuum extremals with 1-D light-like curve as M⁴ projection, to vacuum extremals with CP₂ projection, which is Lagrangian sub-manifold and thus at most 2-dimensional, to string like objects with string world sheet as M⁴ projection (minimal surface) and 2-D complex sub-manifold of CP₂ as CP₂ projection, to massless extremals with 2-D CP₂ projection such that CP₂ coordinates depend on arbitrary manner on light-like coordinate defining local propagation direction and space-like coordinate defining a local polarization direction. There are certainly also other extremals such as magnetic flux tubes resulting as deformations of string like objects. Number theoretic vision relying on classical number fields suggest a very general construction based on the notion of associativity of tangent space or co-tangent space.
   
   
5. The conditions coming from these extremals reduce to 4 conditions expressible in the holomorphic case in terms of the base space coordinates (z₂,z₃) and (w₂,w₃) and in the more general case in terms of the corresponding real coordinates. It seems that holomorphic ansatz is not consistent with the existence of vacuum extremals, which however give vanishing contribution to transition amplitudes since WCW ("world of classical worlds") metric is completely degenerate for them.
   

   The mere condition that one has CP₁ fiber bundle structure does not force field equations since it leaves the dependence between real coordinates of the base spaces free. On the other hand, CP₁ bundle structure alone need not of course guarantee twistor space structure. One can ask whether non vacuum extremals could correspond to holomorphic constraints between (z₂,z₃) and (w₂,w₃).
   
   

6. Pessimist could of course argue that field equations are additional conditions completely independent of the conditions realizing the bundle structure! One cannot exclude this possibility. Mathematician could easily answer the question about whether the proposed CP₁ bundle structure is enough to produce twistor space or not and whether field equations could be the additional condition and realized using the holomorphic ansatz.
   
   

To sum up, the construction of space-times as surfaces of H lifted to that of (almost) complex sub-manifolds in CP₃× F₃ with induced twistor structure shares the spirit of the vision that induction procedure is the key element of classical and quantum TGD.

For background see the new chapter Classical part of twistor story of "Towards M-matrix". See also the article Classical part of twistor story.