How it went?

It took one decade for Einstein to find the final mathematical formulation of General Relativity Theory (GRT). Immediately after having found the final formulation, he predicted gravitional waves, which are easy to discover from the linearized equations. One century later they have been found. Theoretician must be long aged if he wants to enjoy the fruits of his labor.

One should not compare oneself with Gods (the nasty colleagues - you know) but since I am totally crazy (ask colleagues) I talk about both us in the same paragraph. In TGD the process of finding final formulation (see this) took almost four decades and now I dare say that I have finally found the formulation. In the following I summarize what happened during the memorable week during which the final formulation emerged. I have done my best to organize the text to a readable form and I apologize if I have not succeeded completely. An entire flood of ideas emerged and they are still developing. This makes documentation difficult.

Some background

To understand how this is so important I describe briefly the background.

1. Recall that the formulation of classical TGD in terms of Kähler action emerged around 1990 and is therefore quarter century old now. I speak fluently about preferred extremals of Kähler action and I understand reasonably well the dynamics of Kähler action. But about how gravitational constant and cosmological constant emerge from this dynamics I have had only ideas.
   
   
2. This dynamics has one very non-standard feature: huge vacuum degeneracy. All 4-surfaces that have CP₂ projection, which is so called Lagrangian sub-manifold of CP₂ having vanishing induced Kähler form is vacuum extemal. By applying diffeomorphisms of M⁴ and symplectic transformations of CP₂ acting like U(1) gauge transformations one obtains new vacuum extremals. For instance, for the deformations of canonically imbedded empty Minkowski space Kähler action density can be approximated by a fourth order polynomial in CP₂ coordinates and their gradients and perturbation theory fails completely since propagator does not exist.
   
   
3. This spoiled completely the hopes about ordinary quantization of the theory and eventually inspired the idea about "world of classical worlds" (WCW) and led to a beautiful vision about quantum theory as purely classical theory for spinor fields in WCW representing physical states. I could have of course considered the possibility of adding to the action a small volume term to obtain perturbation theory - much like in case of branes - but it looked incredibly ugly. I can only congratulate myself that I refused to consider this possibility although this volume term now emerges from twistorial variant of Kähler action! It had prevented me from discovering WCW and many other deep ideas.
   
   

Only M⁴ and CP₂ allow twistor space with Kähler structure

The situation began to change few years ago as I realized that twistors might be central for understanding of quantum TGD.

1. I am not a jedi master in perturbative QFT and formulas hate me. Furthermore, I do not believe on N=4 SUSY except as a beautiful model able to express some very profound ideas, which have not yet reached our conscious mind. Just the strange beauty of findings of Nima Arkani-Hamed and other pioneers made me convinced that twistors are the key to progress.
   
   
2. Year or two ago came the crucial discovery. I learned that S⁴ (and also conformally compactified version of Minkowski space M⁴) and CP₂ are completely unique 4-D spaces in that only they allow twistor space with Kähler structure. This was discovered by Hitchin (published in 1981). Ironically, this had been discovered roughly year after I ended up with CP₂! Somehow I had failed to learn about this. It was immediately clear that this is something incredibly profound and must mean that TGD a twistorially unique.

"Precisely how?" should have been the immediate question. For some funny reason I did not make quite that question to which the answer would have been totally obvious. I however realized that I should lift space-time surfaces to their 6-D twistor spaces and represent them as 6-D surfaces in the 12-D twistor space of imbedding space inhering their twistor structure from that for the 12-D space. I indeed proposed how to define induction of twistor structure as analog for the induction of metric and spinor structure.

I did not however continue with the obvious question: "How to define dynamics for these 6-D twistor spaces identifiable sphere bundles over space-time surfaces?". This question contains its own answer. Twistor structure involves the identification of an antisymmetric tensor defining preferred quaternionic imaginary unit representing it geometrically. The Kähler form of 12-D twistor space projected to space-time surface should define this preferred imaginary unit. TGD exists only for M⁴× CP₂ and is therefore a completely unique theory. The dynamics is determined by 6-D action variant of Kähler action for 6-D surfaces in twistor space of M⁴× CP₂. I want to repeat: TGD is completely unique if one accepts twistorial formulation.

The existence of twistorial formulation makes TGD unique and leads to unification of gravitation and standard model

After this everything followed within week even though I made all imaginable wrong guesses.

1. 6-D Kähler action has dimension length squared and must be multiplied by a constant with dimensions of 1/length squared. This constant, call it 1/L², is highly analogous to cosmological constant but is coupling constant like parameter since one can replace it with the dimensionless ratio ε²=R²/L², R radius of CP₂. ε is analogous to critical temperature (or perhaps better, critical value of inverse pressure) just like Kähler coupling strength and is expected to have spectrum labelled by p-adic primes p≈ 2^k, k prime- just like Kähler coupling strength. The values of L would be most naturally p-adic length scales and cosmological constant would in first approximation decrease with cosmological time t as 1/t².
   

   The incredibly small cosmological constant (not so in the early Universe) would not be just some nasty trick of Universe making theoreticians crazy but needed to remove the vacuum degeneracy of Kähler action in dimensionally reduced dynamics giving rise to 4-volume as additional term in action and making also perturbation theory around canonically imbedded M⁴ possible. Obviously this action would give rise to the analog of kinetic term for gravitons.
   

   Volume term of course gives just the geometric counterpart of wave equation. In fact, all known extremals of Kähler action are minimal surfaces so that in this sense nothing new has emerged! A coupling between dynamics of volume term and Kähler action is present for more general extremals and Kähler coupling strength and cosmological constant do not completely disappear from the classical dynamics.
   
   

2. Also other fundamental lengths pop up: the radii on S²(M⁴) and S²(CP₂). The radius of S²(CP₂) is essentially CP₂ radius from the definition of twistor space but what about S²(M⁴)?
   Here I had to think thoroughly what the twistor space of M⁴ is. It turned out that the radius is most naturally what Planck length mystic would guess it to be: essentially Planck length l_P. Planck length would emerge from the theory as a purely classical scale! Only two weeks ago I explained that Planck length is purely quantal emergent length scale! However, Planck mass and Newton's constant would emerge as quantal parameters since they depend on Planck constant when expressed in terms of Planck length, hbar and c.
   

   Twistorialization thus brings all the basic scales of gravitation. Earlier I had p-adic length scales emerging from the successful p-adic mass calculations explaining the mass spectra of particles and the CP₂ inspired vision about how gauge coupling strengths and their evolution emerge from quantum criticality. My original belief that G and Λ would emerge from the dynamics of Kähler action alone was therefore wrong.
   
   

3. About how cosmological constant actually emerges I had a slightly wrong view: this was just sloppy thinking. Before continuing it is good to recall how the cosmological constant emerges from TGD framework. The key point is that the 6-D Kähler action contains two terms.
   
   
   1. The first term is essentially the ordinary Kähler action multiplied by the area of S²(X⁴), which is compensated by the length scale, which can be taken to be the area 4π R²(M⁴) of S²(M⁴). This makes sense for winding numbers (w₁,w₂)=(n,0) meaning that S²(CP₂) is effectively absent but S²(M⁴) is present.
      
      
   2. Second term is the analog of Kähler action assignable assignable to the projection of S²(M⁴) Kähler form. The corresponding Kähler coupling strength α_K (M⁴) is huge (really huge, this I did not realize first!) - so huge that one has
      

      α_K (M⁴)4π R²(M⁴)== L² ,
      

      where 1/L² is of the order of cosmological constant and thus of the order of the size of the recent Universe. α_K(M⁴) is also analogous to critical temperature and the earlier hypothesis that the values of L correspond to p-adic length scales implies that the values of come as α_K(M⁴) ∝ p≈ 2^k, p prime, k prime.
      
      

   3. The Kähler form assignable to M⁴ is not assumed to contribute to the action since it does not contribute to spinor connection of M⁴. One can of course ask whether it could be present. For canonically imbedded M⁴ self-duality implies that this contribution vanishes and for vacuum extremals of ordinary Kähler action this contribution is small. Breaking of Lorentz invariance is however a possible problem. If α_K(M⁴) is given by above expression, then this contribution is extremely small.
      
      
   Hence one can consider the possibility that the action is just the sum of full 6-D Kähler actions assignable to T(M⁴) and T(CP₂) but with different values of α_K if one has (w₁,w₂)=(n,0). Also other w₂≠ 0 is possible but corresponds to gigantic cosmological constant.
4. I had also to learn also what twistor space T(M⁴) really is! For CP₂ and S² there are no problems but how to go to the Minkowskian signature - this was the problem. Also here I did all wrong trials. The solution of the problem was simple.
   

   I had actually found the solution for more than one and half decades ago while studying so called massless extremals (MEs) representing radiation in TGD Universe. Also the study of so called M⁸-H duality and the notion of quaternionic structure had led to what I call Hamilton-Jacobi (H-J) structure generalizing Euclidian 4-space E⁴ with complex structure to its Minkowskian variant.
   

   One must first construct M⁴ with H-J structure and then lift it to twistor space. H-J in Minkowski space means the existence of a spatially varying distribution M⁴= M²(x)⊕ E²(x) of decompositions of tangent space of X⁴ to direct orthogonal sum of local 2-D Minkowski space M²(x) and and orthogonal Euclidian 2-space E²(x) . This distribution must be integrable meaning that M²(x) and E²(x) serve as tangent spaces for 2-D surfaces. Euclidian 2-space allows complex structure and complex coordinates (z,z*). M² allows hyper-complex structure and hyper-complex coordinates, which are nothing but light-like coordinates (u=t-z,v=t+z) such that metric of M² is of from ds²=2dudv.
   

   The construction of twistor space looks now rather trivial. Any antisymmetric tensor in the space E³ orthogonal to time like t defines direction that it is point of the sphere defining the twistor space fiber. Metric is induced from the metric for this kind of tensors defined by M⁴ metric. The covariantly constant two-form of E² defines preferred quaternionic imaginary unit. This is also familiar from number theoretic vision demanding its existence. The vision about preferred extremals of Kähler action as quaternionic 4-surfaces of octonionic 8-space relies on this vision. In particular, the twistorial sphere is sphere -not hyperbolic sphere with signature (1,-1) as I believed for 24 hours - and it has metric signature (-1,-1) rather than being time-like!
   
   

5. I had also to learn what the induction of twistor structure means concretely. The preferred quaternionic imaginary unit should be represented as a projection of Kähler form of 12-D twistor space T(H). The preferred imaginary unit defining twistor structure as sum of projections of both T(CP₂) and T(M⁴) Kähler forms would guarantee that vacuum extremals like canonically imbedded M⁴ for which T(CP₂) Kähler form contributes nothing have well-defined twistor structure. T(M⁴) or T(CP₂) are treated completely symmetrically.
   
   
6. For Kähler action M⁴-CP₂ symmetry does not make sense. 4-D Kähler action to which 6-D Kähler action dimensionally reduces can depend on CP₂ Kähler form only. I have also considered the possibility of covariantly constant self-dual M⁴ term in Kähler action but given it up because of problems with Lorentz invariance. One should couple the gauge potential of M⁴ Kähler form to induced spinors. This would mean the existence of vacuum gauge fields coupling to sigma matrices of M⁴ so that the gauge grop would be non-compact SO(3,1) leading to a breakdown of unitarity.
   

   Hence it seems clear that only the projection of T(CP₂) part of Kähler form of T(H) can appear in 6-D Kähler action. This option breaks the symmetry between M⁴ and CP₂ at the level of dynamics but is physically unavoidable and is also mathematically completely acceptable. I cannot but accept the situation.
   
   

7. An important point to notice is that the radius of the sphere associated with the twistor space of X⁴ is dynamical. It however turned that for winding numbres (w₁, w₂(=(n,0) - the only option allowed by the condition that cosmological constant is not gigantic - the radius is essentially that of S²(M⁴) and therefore constant. The cosmological estimate gave same result for all cosmological eras.
   
   

A connection with the hierarchy of Planck constants?

A connection with the hierarchy of Planck constants is highly suggestive. Since also a connection with the p-adic length scale hierarchy suggests itself for the hierarchy of p-adic length scales it seems that both length scale hierarchies might find first principle explanation in terms of twistorial lift of Kähler action.

1. Cosmological considerations encourage to think that R₁≈ l_P and R₂≈ R hold true. One would have in early cosmology (w₁,w₂)=(1,0) and later (w₁,w₂)=(0,1) guaranteeing R_D grows from l_P to R during cosmological evolution. These situations would correspond the solutions (w₁=n,0) and (0,w₂=n) one has A= n 4π R₁² and A=n× 4π R₂² and both Kähler coupling strengths are scaled down to α_K/n. For hbar_eff/h=n exactly the same thing happens!
   

   There are further intriguing similarities. h_eff/h=n is assumed to correspond multi-sheeted (to be distinguished from many-sheeted!) covering space structure for space-time surface. Now one has covering space defined by the lift S²(X⁴)→ S²(M4)× S²(CP₂). These lifts define also lifts of space-time surfaces.
   

   Could the hierarchy of Planck constants correspond to the twistorial surfaces for which S²(M⁴) and S⁽CP₂) are identified in 1-1 manner? The assumption has been that the n-fold multi-sheeted coverings of space-time surface for h_eff/h=n are singular at the ends of space-time surfaces at upper and lower boundaries if causal diamond (CD). Could one consider more precise definition of twistor space in such a manner that CD replaces M⁴ and the covering becomes singular at the light-like boundaries of CD - the branches of space-time surface would collapse to single one. What could this collapse mean geometrically? Or should one give up the assumption about singular nature of the covering used to distinguishes many-sheetedness from multi-sheetedness.
   
   

2. w₁=w₂=w is essentially the first proposal for conditions associated with the lifting of twistor space structure. w₁=w₂=n gives ds²= (R₁²+R₂²)(dθ² +w²dφ²) and A = n × 4π (R₁²+R₂²). Also now Kähler coupling strength is scaled down to α/n. Again a connection with the hierarchy of Planck constants suggests itself.
   
   
3. One can consider also the option R₁=R₂ option giving ds²= R₁²(2dθ² +(w₁²+w₂²)dφ². If the integers w_i define Pythagorean square one has w₁²+w₂²=n² and one has R₁=R₂ option that one has A =n× 4π R². Also now the connection with the hierarchy of Planck constants might make sense.
   
   

Twistorial variant for the imbedding space spinor structure

The induction of the spinor structure of imbedding space is in key role in quantum TGD. The question arises whether one should lift also spinor structure to the level of twistor space. If so one must understand how spinors for T(M⁴) and T(CP₂) are defined and how the induced spinor structure is induced.

1. In the case of CP₂ the definition of spinor structure is rather delicate and one must add to the ordinary spinor connection U(1) part, which corresponds physically to the addition of classical U(1) gauge potential and indeed produces correct electroweak couplings to quarks and leptons. It is assumed that the situation does not change in any essential manner: that is the projections of gauge potentials of spinor connection to the space-time surface give those induced from M⁴× CP₂ spinor connection plus possible other parts coming as a projection from the fiber S²(M²)× S²(CP₂). As a matter of fact, these other parts should vanish if dimensional reduction is what it is meant to be.
   
   
2. The key question is whether the complications due to the fact that the geometries of twistor spaces T(M⁴) and T(CP₂) are not quite Cartesian products (in the sense that metric could be reduced to a direct sum of metrics for the base and fiber) can be neglected so that one can treat the sphere bundles approximately as Cartesian products M⁴× S² and CP₂× S². This will be assumed in the following but should be carefully proven.
   
   
3. Locally the spinors of the twistorspace T(H) are tensor products of imbedding spinors and those for of S²(M⁴)× S²(CP₂) expressible also as tensor products of spinors for S²(M⁴) and S²(CP₂). Obviously, the number of spinor components increases by factor 2× 2 = 4 unless one poses some additional conditions taking care that one has dimensional reduction without the emergence of any new spin like degrees of freedom for which there is no physical evidence. The only possible manner to achieve this is to pose covariant constancy conditions already at the level of twistor spaces T(M⁴) and T(CP₂) leaving only single spin state in these degrees of freedom.
   
   
4. In CP₂ covariant constancy is possible for right-handed neutrino so that CP₂ spinor structure can be taken as a model. In the case of CP₂ spinors covariant constancy is possible for right-handed neutrino and is essentially due to the presence of U(1) part in spinor connection forced by the fact that the spinor structure does not exist otherwise. Ordinary
   S² spinor connection defined by vielbein exists always. One can however add a coupling to a suitable multiple of Kähler potential satisfying the quantization of magnetic charge (the magnetic flux defined by U(1) connection is multiple of 2π so that its imaginary exponential is unity).
   

   S² spinor connections must must have besides ordinary vielbein part determined by S² metric also U(1) part defined by Kähler form coupled with correct coupling so that the curvature form annihilates the second spin state for both S²(M⁴) and S²(CP₂). U(1) part of the spinor curvature is proportional to Kähler form J ∝ sin(theta)dθ dφ so that this is possible. The vielbein and U(1) parts of the spinor curvature ear proportional Pauli spin matrix σ_z= (1,0;0,-1)/2 and unit matrix (1,0;0,1) respectively so that the covariant constancy is possible to satisfy and fixes the spin state uniquely.
   
   

5. The covariant derivative for the induced spinors is defined by the sum of projections of spinor gauge potentials for T(M⁴) and T(CP₂). With above assumptions the contributions gauge potentials from T(M⁴) and T(CP₂) separately annihilate single spinor component. As a consequence there are no constraints on the winding numbers w_i, i=1,2 of the maps S²(X⁴) → S²(M⁴) and S²(X⁴) → S²(CP₂). Winding number w_i corresponds to the imbedding map (Θ_i= θ, Φ_i=w_iφ).
   
   

   
   

6. If the square of the Kähler form in fiber degrees of freedom gives metric to that its square is metric, one obtains just the area of S² from the fiber part of action. This is given by the area A= 4π(2^1/2(w₁²R₁²+w₂²R₂²) since the induced metric is given by ds²= (R₁²+R₂²)dθ² +(w₁²R₁²+w₂²R₂²)dφ² for (Θ₁= θ, Φ=n₁φ, Φ₂= n₂φ).
   
   

To sum up, I strongly feel that the final formulation of TGD has now emerged and it is now clear that TGD is indeed a quantum theory of gravitation allowing to understand standard model symmetries. The existence of twistorial formulation makes possible gravitation and predicts standard model symmetries. This theory is completely unique from extremely general assumptions. This cannot be said about any competitor of TGD.

For background see the article From Principles to Diagrams or the chapter From Principles to Diagrams of "Towards M-matrix".

For a summary of earlier postings see Links to the latest progress in TGD.