What's new inTowards MMatrixNote: Newest contributions are at the top! 
Year 2014 
Twistor spaces as TGD counterparts of CalabiYau manifolds of super string modelsThe understanding of twistor structure of imbedding space and its relationship to the consctruction of extremals of Kähler action is certainly greatest breakthroughs in the mathematical understanding of TGD for years. One can say that the good physics provided by TGD can be now combined with the marvelous mathematics produced by string theorists by replacing CalabiYau manifolds with twistor spaces assignable to spacetime surfaces and representable as submanifolds of the twistor space CP_{3}× F_{3} of imbedding space M^{4}× CP_{2} (strictly speaking M^{4} twistor space is the noncompact space SU(2,2)/SU(2,1) ×U(1)). What it so wonderful is that the enormous collective knowhow involved with algebraic geometry becomes avaiblable in TGD and now this mathematics makes sense physically. My sincere hope is that also colleagues would finally realize that TGD is the only way out from the recent dead alley in fundamental physics. Below is the introduction of the article article Classical part of twistor story. I will later add some key pieces of the article.
Twistor 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 8D imbedding space H=M^{4}× CP_{2} is necessary. M^{4} (and S^{4} as its Euclidian counterpart) and CP_{2} are indeed unique in the sense that they are the only 4D spaces allowing twistor space with Kähler structure. The Cartesian product of twistor spaces CP_{3} and F_{3} defines twistor space 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. One ends up with a scenario in which spacetime surfaces are lifted to twistor spaces by adding CP_{1} fiber so that the twistor spaces give an alternative representation for generalized Feynman diagrams having as lines spacetime surfaces with Euclidian signature of induced metric and having wormhole contacts as basic building bricks. There is also a very close analogy with superstring models. Twistor spaces replace CalabiYau manifolds and the modification recipe for CalabiYau manifolds by removal of singularities can be applied to remove selfintersections of twistor spaces and mirror symmetry emerges naturally. The overall important implication is that the methods of algebraic geometry used in superstring theories should apply in TGD framework. The basic problem of TGD has been indeed the lack of existing mathematical methods to realize quantitatively the view about spacetime as 4surface. The physical interpretation is totally different in TGD. Twistor space has spacetime as basespace rather than forming with it Cartesian factors of a 10D spacetime. The CalabiYau landscape is replaced with the space of twistor spaces of spacetime surfaces having interpretation as generalized Feynman diagrams and twistor spaces as submanifolds of CP_{3}× F_{3} replace Witten's twistor strings. The space of twistor spaces is the lift of the "world of classical worlds" (WCW) by adding the CP_{1} fiber to the spacetime surfaces so that the analog of landscape has beautiful geometrization. For background see the new chapter Classical part of twistor story or the article Classical part of twistor story. 
Classical TGD and imbedding space twistorsThe understanding of twistor structure of imbedding space and its relationship to the consctruction of extremals of Kähler action is certainly greatest breakthroughs in the mathematical understanding of TGD for years. One can say that the good physics provided by TGD can be now combined with the marvelous mathematics produced by string theorists by replacing CalabiYau manifolds with twistor spaces assignable to spacetime surfaces and representable as submanifolds of the twistor space CP_{3}× F_{3} of imbedding space M^{4}× CP_{2} (strictly speaking M^{4} twistor space is the noncompact space SU(2,2)/SU(2,1) ×U(1)). What it so wonderful is that the enormous collective knowhow involved with algebraic geometry becomes avaiblable in TGD and now this mathematics makes sense physically. My sincere hope is that also colleagues would finally realize that TGD is the only way out from the recent dead alley in fundamental physics. Below is the introduction of the article article Classical part of twistor story. I will later add some key pieces of the article.
Twistor 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 8D imbedding space H=M^{4}× CP_{2} is necessary. M^{4} (and S^{4} as its Euclidian counterpart) and CP_{2} are indeed unique in the sense that they are the only 4D spaces allowing twistor space with Kähler structure. The Cartesian product of twistor spaces CP_{3} and F_{3} defines twistor space 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. One ends up with a scenario in which spacetime surfaces are lifted to twistor spaces by adding CP_{1} fiber so that the twistor spaces give an alternative representation for generalized Feynman diagrams having as lines spacetime surfaces with Euclidian signature of induced metric and having wormhole contacts as basic building bricks. There is also a very close analogy with superstring models. Twistor spaces replace CalabiYau manifolds and the modification recipe for CalabiYau manifolds by removal of singularities can be applied to remove selfintersections of twistor spaces and mirror symmetry emerges naturally. The overall important implication is that the methods of algebraic geometry used in superstring theories should apply in TGD framework. The basic problem of TGD has been indeed the lack of existing mathematical methods to realize quantitatively the view about spacetime as 4surface. The physical interpretation is totally different in TGD. Twistor space has spacetime as basespace rather than forming with it Cartesian factors of a 10D spacetime. The CalabiYau landscape is replaced with the space of twistor spaces of spacetime surfaces having interpretation as generalized Feynman diagrams and twistor spaces as submanifolds of CP_{3}× F_{3} replace Witten's twistor strings. The space of twistor spaces is the lift of the "world of classical worlds" (WCW) by adding the CP_{1} fiber to the spacetime surfaces so that the analog of landscape has beautiful geometrization. For background see the new chapter Classical part of twistor story or the article Classical part of twistor story. 
Classical part of the twistor storyTwistors 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 8D imbedding space H=M^{4}× CP_{2} is necessary. M^{4} (and S^{4} as its Euclidian counterpart) and CP_{2} are indeed unique in the sense that they are the only 4D 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.
Why twistor spaces with Kähler structure? I have not yet even tried to answer an obvious question. Why the fact that M^{4} and CP_{2} 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?
About the identification of 6D twistor spaces as submanifolds of CP_{3}× F_{3} How to identify the 6D submanifolds 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 CP_{3} resp. F_{3}.
For background see the new chapter Classical part of twistor story or the article Classical part of twistor story. 
Class field theory and TGD: does TGD reduce to number theory?The intriguing general result of class field theory) something extremely abstract for physicist's brain  is that the the maximal Abelian extension for rationals is homomorphic with the multiplicative group of ideles. This correspondence plays a key role in Langlands correspondence (see this,this, this, and this). Does this mean that it is not absolutely necessary to introduce padic numbers? This is actually not so. The Galois group of the maximal abelian extension is rather complex objects (absolute Galois group, AGG, defines as the Galois group of algebraic numbers is even more complex!). The ring Z of adeles defining the group of ideles as its invertible elements homeomorphic to the Galois group of maximal Abelian extension is profinite group. This means that it is totally disconnected space as also padic integers and numbers are. What is intriguing that pdic integers are however a continuous structure in the sense that differential calculus is possible. A concrete example is provided by 2adic units consisting of bit sequences which can have literally infinite nonvanishing bits. This space is formally discrete but one can construct differential calculus since the situation is not democratic. The higher the pinary digit in the expansion is, the less significant it is, and padic norm approaching to zero expresses the reduction of the insignificance. 1. Could TGD based physics reduce to a representation theory for the Galois groups of quaternions and octonions? Number theoretical vision about TGD raises questions about whether adeles and ideles could be helpful in the formulation of TGD. I have already earlier considered the idea that quantum TGD could reduce to a representation theory of appropriate Galois groups. I proceed to make questions.
2. Adelic variant of spacetime dynamics and spinorial dynamics? As an innocent novice I can continue to pose stupid questions. Now about adelic variant of the spacetime dynamics based on the generalization of Kähler action discussed already earlier but without mentioning adeles (see this).
3. Objections The basic idea is that appropriately defined invertible quaternionic/octonionic adeles can be regarded as elements of Galois group assignable to quaternions/octonions. The best manner to proceed is to invent objections against this idea.
For background see the chapter Various General Ideas Related to Quantum TGD. 
Class field theory and TGD: does TGD reduce to number theory?The intriguing general result of class field theory) something extremely abstract for physicist's brain  is that the the maximal Abelian extension for rationals is homomorphic with the multiplicative group of ideles. This correspondence plays a key role in Langlands correspondence (see this,this, this, and this). Does this mean that it is not absolutely necessary to introduce padic numbers? This is actually not so. The Galois group of the maximal abelian extension is rather complex objects (absolute Galois group, AGG, defines as the Galois group of algebraic numbers is even more complex!). The ring Z of adeles defining the group of ideles as its invertible elements homeomorphic to the Galois group of maximal Abelian extension is profinite group. This means that it is totally disconnected space as also padic integers and numbers are. What is intriguing that pdic integers are however a continuous structure in the sense that differential calculus is possible. A concrete example is provided by 2adic units consisting of bit sequences which can have literally infinite nonvanishing bits. This space is formally discrete but one can construct differential calculus since the situation is not democratic. The higher the pinary digit in the expansion is, the less significant it is, and padic norm approaching to zero expresses the reduction of the insignificance. 1. Could TGD based physics reduce to a representation theory for the Galois groups of quaternions and octonions? Number theoretical vision about TGD raises questions about whether adeles and ideles could be helpful in the formulation of TGD. I have already earlier considered the idea that quantum TGD could reduce to a representation theory of appropriate Galois groups. I proceed to make questions.
2. Adelic variant of spacetime dynamics and spinorial dynamics? As an innocent novice I can continue to pose stupid questions. Now about adelic variant of the spacetime dynamics based on the generalization of Kähler action discussed already earlier but without mentioning adeles (see this).
3. Objections The basic idea is that appropriately defined invertible quaternionic/octonionic adeles can be regarded as elements of Galois group assignable to quaternions/octonions. The best manner to proceed is to invent objections against this idea.
See the new chapter Unified Number Theoretical Vision or the article with the same title. 
General ideas about octonions, quaternions, and twistorsOctonions, quaternions, quaternionic spacetime surfaces, octonionic spinors and twistors and twistor spaces are highly relevant for quantum TGD. In the following some general observations distilled during years are summarized. There is a beautiful pattern present suggesting that H=M^{4}× CP_{2} is completely unique on number theoretical grounds. Consider only the following facts. M^{4} and CP_{2} are the unique 4D spaces allowing twistor space with Kähler structure. M^{8}H duality allows to deduce M^{4}× CP_{2} via number theoretical compactification. Octonionic projective space OP_{2} appears as octonionic twistor space (there are no higherdimensional octonionic projective spaces). Octotwistors generalise the twistorial construction from M^{4} to M^{8} and octonionic gamma matrices make sense also for H with quaternionicity condition reducing OP_{2 to to the twistor space of H. } A further fascinating structure related to octotwistors is the nonassociated analog of Lie group defined by automorphisms by octonionic imaginary units: this group is topologically sixsphere. Also the analogy of quaternionicity of preferred extremals in TGD with the Majorana condition central in super string models is very thought provoking. All this suggests that associativity indeed could define basic dynamical principle of TGD. See the new chapter Unified Number Theoretical Vision or the article. 
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 3D 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.
