Summary about background
Consider first some background.
- 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?
- 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.
- 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.
- 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.
- 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.
- 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.
- 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 heff/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?
- 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.
- 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.
- 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 zi=xi+iyi resp. wi=ui+ivi for CP3 resp. F3.
- 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.
- 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.
- 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): zi= fi(wk), i=2,3, k=2,3. Non-holomorphic correspondence cannot be excluded.
- 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.
- 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).
- 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.