What's new inTowards MMatrixNote: Newest contributions are at the top! 
Year 2013 
Scattering amplitudes in positive Grassmannian: TGD perspective
The quite recent but not yet published proposal of Hamed and his former student Trnka has gained a lot of attention. There is a popular article in Quanta Magazine about their work. There is a video talk by Jaroslav Trnka about positive Grassmannian (the topic is actually touched at the end of the talk but it gives an excellent view about the situation) and a video talk by Nima ArkaniHamed. One can also find the slides of Trnka . For beginners like me the article of Henriette Envang and Yutin Huang serves as an enjoyable concretization of the general ideas. The basic claim is that the Grassmannian amplitudes reduce to volumes of positive Grassmannians determined by external particle data and realized as polytopes in Grassmannians such that their facets correspond to logarithmic singularities of a volume form in oneone correspondence with the singularities of the scattering amplitude. Furthermore, t the factorization of the scattering amplitude at singularities corresponds to the singularities at facets. Scattering amplitudes would characterize therefore purely geometric objects. The crucial Yangian symmetry would correspond to diffeomorphisms preserving the positivity property. Unitarity and locality would be implied by the volume interpretation. Nima concludes that unitarity and locality, gauge symmetries, spacetime, and even quantum mechanics emerge. One can however quite well argue that its the positive Grassmannian property and volume interpretation which emerge. In particular, the existence of twistor structure possible in Minkowskian signature only in M^{4} is absolutely crucial for the beautiful outcome, which certainly can mean a revolution as far as calculational techniques are considered and certainly the new view about perturbation theory should be important also in TGD framework. The talks inspired the consideration of the possible Grassmannian formulation in TGD framework in more detail and to ask whether positivity might have some deeper meaning in TGD framework. The vision about what BCFW approach to generalized Feynman diagframs could mean has been fluctuating wildly during last months. The Grassmannian formalism for scattering amplitudes is expected to generalize for generalized Feynman diagrams: the basic modification is due to the possible presence of CP_{2} twistorialization and the fact that 4fermion vertex  rather than 3boson vertex  and its super counterparts define now the fundamental vertices. Both QFT type BFCW and stringy BFCW can be considered. The recent vision is as follows.
As both ArkaniHamed and Trnka state "everything is positive". This is highly interesting since padicization involves canonical identification, which is well defined only for nonnegative reals without further assumptions. This raises the conjecture that positivity is necessary in order to achieve number theoretical universality. Addition: Lubos wrote again about amplituhedrons and managed to write something about which I can agree almost wholeheartedly. Not a single mention of superstrings or Peter Woit! I also agree with Lubos about Scott Aaronson's parody: Scott was entertaining but not deep. The twistors and Grassmannians will revolutionize theoretical physics in many manners: my basic bet is that the uniqueness of M^{4}×CP_{2} from twistorial considerations and positivity conditions (whatever they really mean Minkowskian signature!) as a prerequisite for padicization will be at the core of the revolution and make TGD a mathematical "must". Mathematicians might be able to generalize the Grassmannian approach to CP_{2} degrees of freedom without much effort. For details see the chapter Some fresh ideas about twistorialization of TGD or the article with the same title. 
A little comment about the hierarchy of Planck constantsOriginally the hierarchy of Planck constant was assumed to correspond to a book like structure having as pages the nfold coverings of the imbedding space for various values of n serving therefore as a page number. The pages are glued together along a 4D "back" at which the branches of nfurcations are degenerate. This leads to a very elegant picture about how the particles belonging to the different pages of the book interact. All vertices are local and involve only particles with the same value of Planck constant: this is enough for darkness in the sense of particle physics. The interactions between particles belonging to different pages involve exchange of a particle travelling from page to another through the back of the book and thus experiencing a phase transition changing the value of Planck constant. Is this picture consistent with the picture based on nfurcations? This seems to be the case. The conservation of energy in nfurcation in which several sheets are realized simultaneously is consistent with the conservation of classical conserved quantities only if the spacetime sheet before nfurcation involves n identical copies of the original spacetime sheet or if the Planck constant is h_{eff}=nh. This kind of degenerate manysheetedness is encountered also in the case of branes. The first option means an nfold covering of imbedding space and h_{eff} is indeed effective Planck constant. Second option means a genuine quantization of Planck constant due to the fact the value of Kähler coupling strength α_{K}=g_{K}^{2}/4πhbar_{eff} is scaled down by 1/n factor. The scaling of Planck constant consistent with classical field equations since they involve α_{K} as an overall multiplicative factor only. For details see the chapter Does TGD Predict a Spectrum of Planck Constants?. 
What could 4fermion twistor amplitudes look like?4fermion twistor amplitudes are basic building bricks of twistor amplitudes in TGD framework. What can one conclude about them on basis of N=4 amplitudes? Instead of 3vertices as in SYM, one has 4fermion vertices as fundamental vertices and the challenge is to guess their general form. The basis idea is that N=4 SYM amplitudes could give as special case the nfermion amplitudes and their supersymmetric generalizations 1. Attempt to understand the physical picture One must try to identify the physical picture first.
2. How to identify the bosonic correlation functions inside wormhole contacts? The next challenge is to identify the correlation function for the deformation δ m^{k} inside wormhole contacts. Conformal invariance suggests the identification of the analog of propagator as a correlation function fixed by conformal invariance for a system defined by the wormhole contact. The correlation function should depend on the differences ξ_{i}=ξ_{i,1}ξ_{i,2} of the complex CP_{2} coordinates at the points ξ_{i,1)} and ξ_{i,2} of the opposite throats and transforms in a simple manner under scalings of ξ_{i}. The simplest expectation is that the correlation function is power r^{n}, where r^{2}= [ξ_{1}^{2}+ξ_{2}^{2} defines U(2) invariant coordinate distance squared. The correlation function can be expanded as products of conformal harmonics or ordinary harmonics of CP_{2} assignable to ξ_{i,1} and ξ_{i,2} and one expects that the values of Y and I_{3} vanish for the terms in the expansions: this just states that Y and I_{3} are conserved in the propagation. Second approach relies on the idea about propagator as the inverse of some kind of Laplacian. The approach is not in conflict with the general conformal approach since the Laplacian could occur in the action defining the conformal field theory. One should try to identify a Laplacian defining the propagator for δ m^{k} inside Euclidian regions.
Two general remarks are in order.
3. Do color quantum numbers propagate and are they conserved in vertices? The basic questions are whether one can speak about conservation of color quantum numbers in vertices and their propagation along the internal lines and the closed magnetic flux loops assigned with the elementary particles having size given by padic length scale and having wormhole contacts at its ends. pAdic mass calculations predict that in principle all color partial waves are possible in cm degreees of freedom: this is a description at the level of imbedding space and its natural counterpart at spacetime level would be conformal harmonics for induced spinor fields and allowance of all of them in generalized Feynman diagrams.
4. Why twistorialization in CP_{2} degrees of freedom? A couple of comments about twistorialization in CP_{2} degrees of freedom are in order.
N=4 SUSY provides quantitative guidelines concerning the actual construction of the amplitudes.

About the SUSY generated by covariantly constant righthanded neutrinosThe interpretation of covariantly constant righthanded neutrinos (briefly ν_{R} in what follows) in M^{4}× CP_{2} has been a continual headache. Should they be included to the spectrum or not. If not, then one has no fear/hope about spacetime SUSY of any kind and has only conformal SUSY. First some general observations.
If the ν_{R}:s are included, the pseudoreal analog of N=1 SUSY assumed in the minimal extensions of standard model or the analog of N=2 or even N=4 SUSY is expected so that SUSY type theory might describe the situation. The following is an attempt to understand what might happen. For an earlier attempt see this. 1. Covariantly constant righthanded neutrinos as limiting cases of massless modes For the first option covariantly constant righthanded neutrinos are obtained as limiting case for the solutions of massless Dirac equation. One obtains 2 complex spinors satisfying Dirac equation n^{k}γ_{k}u=0 for some momentum direction n^{k} defining quantization axis for spin. Second helicity is unphysical: one has therefore one helicity for neutrino and one for antineutrino.
Note that in TGD based twistor approach fourfermion vertex is the fundamental vertex and fermions propagate as massless fermions with nonphysical helicity in internal lines. This would suggest that if righthanded neutrinos are zero momentum limits, they propagate but give in the residue integral over energy twistor line contribution proportional to p^{k}γ_{k}, which is nonvanishing for nonphysical helicity in general but vanishes at the limit p^{k}→ 0. Covariantly constant righthanded neutrinos would therefore decouple from the dynamics (natural in continuum approach since they would represent just single point in momentum space). This option is not too attractive. 2. Covariantly constant righthanded neutrinos as limiting cases of massless modes For the second option covariantly constant neutrinos have vanishing fourmomentum and both helicities are allowed so that the number of helicities is 2 for both neutrino and antineutrino.
3. Could twistor approach provide additional insights? Concerning the quantization of ν_{R}:s, it seems that the situation reduces to the oscillator algebra for complex M^{4} spinors since CP_{2} part of the Hspinor is spinor is fixed. Could twistor approach provide additional insights? As discussed, M^{4} and CP_{2} parts of Htwistors can be treated separately and only M^{4} part is now interesting. Usually one assigns to massless fourmomentum a twistor pair (λ^{a}, ξ^{a'}) such that one has p^{aa'}= λ^{a}ξ^{a'} ( ξ denotes for "\hat(\lambda)" which html does not allow to express). Dirac equation gives λ^{a}= +/ (ξ^{a'})^{*}, where +/ corresponds to positive and negative frequency spinors.
An interesting challenge is to deduce the generalization of conformally invariant part of fourfermion vertices in terms of twistors associated with the fourfermions and also the SUSY extension of this vertex. For details see the new chapter Some fresh ideas about twistorialization of TGD or the article with the same title. 
Some fresh ideas about twistorialization of TGD
I found from web an article by Tim Adamo titled "Twistor actions for gauge theory and gravity". The work considers the formulation of N=4 SUSY gauge theory directly in twistor space instead of Minkowski space. The author is able to deduce MHV formalism, tree level amplitudes, and planar loop amplitudes from action in twistor space. Also local operators and null polygonal Wilson loops can be expressed twistorially. This approach is applied also to general relativity: one of the challenges is to deduce MHV amplitudes for Einstein gravity. The reading of the article inspired a fresh look on twistors and a possible answer to several questions (I have written two chapters about twistors and TGD giving a view about development of ideas). Both M^{4} and CP_{2} are highly unique in that they allow twistor structure and in TGD one can overcome the fundamental "googly" problem of the standard twistor program preventing twistorialization in general spacetime metric by lifting twistorialization to the level of the imbedding space containg M^{4} as a Cartesian factor. Also CP_{2} allows twistor space identifiable as flag manifold SU(3)/U(1)× U(1) as the selfduality of Weyl tensor indeed suggests. This provides an additional "must" in favor of submanifold gravity in M^{4}× CP_{2}. Both octonionic interpretation of M^{8} and triality possible in dimension 8 play a crucial role in the proposed twistorialization of H=M^{4}× CP_{2}. It also turns out that M^{4}× CP_{2} allows a natural twistorialization respecting Cartesian product: this is far from obvious since it means that one considers spacelike geodesics of H with lightlike M^{4} projection as basic objects. pAdic mass calculations however require tachyonic ground states and in generalized Feynman diagrams fermions propagate as massless particles in M^{4} sense. Furthermore, lightlike Hgeodesics lead to noncompact candidates for the twistor space of H. Hence the twistor space would be 12dimensional manifold CP_{3}× SU(3)/U(1)× U(1). Generalisation of 2D conformal invariance extending to infiniteD variant of Yangian symmetry; lightlike 3surfaces as basic objects of TGD Universe and as generalised lightlike geodesics; lightlikeness condition for momentum generalized to the infinitedimensional context via superconformal algebras. These are the facts inspiring the question whether also the "world of classical worlds" (WCW) could allow twistorialization. It turns out that center of mass degrees of freedom (imbedding space) allow natural twistorialization: twistor space for M^{4}× CP_{2} serves as moduli space for choice of quantization axes in Super Virasoro conditions. Contrary to the original optimistic expectations it turns out that although the analog of incidence relations holds true for KacMoody algebra, twistorialization in vibrational degrees of freedom does not look like a good idea since incidence relations force an effective reduction of vibrational degrees of freedom to four. The Grassmannian formalism for scattering amplitudes generalizes practically as such for generalized Feynman diagrams. For background and details see the new chapter Some fresh ideas about twistorialization of TGD or the article with the same title. 
Could N=2 or N=4 SUSY have something to do with TGD?
A question about how nonplanar Feynman diagrams could be represented in twistor Grassmannian approach inspired a rereading of the recent article by recent article by Nima ArkaniHamed et al. This inspired the conjecture that nonplanar twistor diagrams correspond to nonplanar Feynman diagrams and a concrete proposal for realizing the earlier proposal that the contribution of nonplanar diagrams could be calculated by transforming them to planar ones by using the procedure applied in knot theories to eliminate crossings by reducing the knot diagram with crossing to a combination of two diagrams for which the crossing is replaced with reconnection. The Wikipedia article about magnetic reconnection explains what reconnection means. More explicitly, the two reconnections for crossing line pair (AB,CD) correspond to the noncrossing line pairs (AD,BC) and (AC,BD). I do not bother to type the 5 pages of text here. Instead I give a link to the article Still about nonplanar twistor diagrams at my homepage. For background see the chapter Generalized Feynman diagrams as generalized braids or the article Still about nonplanar twistor diagrams. 