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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 Arkani-Hamed. One can also find the slides of Trnka . For beginners like me the article of Henriette Envang and Yu-tin 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 one-one 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, space-time, 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 M4 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 CP2 twistorialization and the fact that 4-fermion vertex - rather than 3-boson 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 Arkani-Hamed and Trnka state "everything is positive". This is highly interesting since p-adicization involves canonical identification, which is well defined only for non-negative 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 whole-heartedly. 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 M4×CP2 from twistorial considerations and positivity conditions (whatever they really mean Minkowskian signature!) as a prerequisite for p-adicization will be at the core of the revolution and make TGD a mathematical "must". Mathematicians might be able to generalize the Grassmannian approach to CP2 degrees of freedom without much effort.
Originally the hierarchy of Planck constant was assumed to correspond to a book like structure having as pages the n-fold coverings of the imbedding space for various values of n serving therefore as a page number. The pages are glued together along a 4-D "back" at which the branches of n-furcations 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 n-furcations? This seems to be the case. The conservation of energy in n-furcation in which several sheets are realized simultaneously is consistent with the conservation of classical conserved quantities only if the space-time sheet before n-furcation involves n identical copies of the original space-time sheet or if the Planck constant is heff=nh. This kind of degenerate many-sheetedness is encountered also in the case of branes. The first option means an n-fold covering of imbedding space and heff is indeed effective Planck constant. Second option means a genuine quantization of Planck constant due to the fact the value of Kähler coupling strength αK=gK2/4πhbareff is scaled down by 1/n factor. The scaling of Planck constant consistent with classical field equations since they involve αK as an overall multiplicative factor only.
For details see the chapter Does TGD Predict a Spectrum of Planck Constants?.
4-fermion 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 3-vertices as in SYM, one has 4-fermion 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 n-fermion 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 mk 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 ξi=ξi,1-ξi,2 of the complex CP2 coordinates at the points ξi,1) and ξi,2 of the opposite throats and transforms in a simple manner under scalings of ξi. The simplest expectation is that the correlation function is power r-n, where r2= [ξ1|2+|ξ2|2 defines U(2) invariant coordinate distance squared. The correlation function can be expanded as products of conformal harmonics or ordinary harmonics of CP2 assignable to ξi,1 and ξi,2 and one expects that the values of Y and I3 vanish for the terms in the expansions: this just states that Y and I3 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 mk 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 p-adic length scale and having wormhole contacts at its ends. p-Adic 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 space-time level would be conformal harmonics for induced spinor fields and allowance of all of them in generalized Feynman diagrams.
4. Why twistorialization in CP2 degrees of freedom?
A couple of comments about twistorialization in CP2 degrees of freedom are in order.
N=4 SUSY provides quantitative guidelines concerning the actual construction of the amplitudes.
The interpretation of covariantly constant right-handed neutrinos (briefly νR in what follows) in M4× CP2 has been a continual head-ache. Should they be included to the spectrum or not. If not, then one has no fear/hope about space-time SUSY of any kind and has only conformal SUSY. First some general observations.
If the νR: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 right-handed neutrinos as limiting cases of massless modes
For the first option covariantly constant right-handed neutrinos are obtained as limiting case for the solutions of massless Dirac equation. One obtains 2 complex spinors satisfying Dirac equation nkγku=0 for some momentum direction nk 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 four-fermion vertex is the fundamental vertex and fermions propagate as massless fermions with non-physical helicity in internal lines. This would suggest that if right-handed neutrinos are zero momentum limits, they propagate but give in the residue integral over energy twistor line contribution proportional to pkγk, which is non-vanishing for non-physical helicity in general but vanishes at the limit pk→ 0. Covariantly constant right-handed 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 right-handed neutrinos as limiting cases of massless modes
For the second option covariantly constant neutrinos have vanishing four-momentum 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 νR:s, it seems that the situation reduces to the oscillator algebra for complex M4 spinors since CP2 part of the H-spinor is spinor is fixed. Could twistor approach provide additional insights?
As discussed, M4 and CP2 parts of H-twistors can be treated separately and only M4 part is now interesting. Usually one assigns to massless four-momentum a twistor pair (λa, ξa') such that one has paa'= λaξa' ( ξ denotes for "\hat(\lambda)" which html does not allow to express). Dirac equation gives λa= +/- (ξa')*, where +/- corresponds to positive and negative frequency spinors.
An interesting challenge is to deduce the generalization of conformally invariant part of four-fermion vertices in terms of twistors associated with the four-fermions and also the SUSY extension of this vertex.
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 M4 and CP2 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 space-time metric by lifting twistorialization to the level of the imbedding space containg M4 as a Cartesian factor. Also CP2 allows twistor space identifiable as flag manifold SU(3)/U(1)× U(1) as the self-duality of Weyl tensor indeed suggests. This provides an additional "must" in favor of sub-manifold gravity in M4× CP2. Both octonionic interpretation of M8 and triality possible in dimension 8 play a crucial role in the proposed twistorialization of H=M4× CP2. It also turns out that M4× CP2 allows a natural twistorialization respecting Cartesian product: this is far from obvious since it means that one considers space-like geodesics of H with light-like M4 projection as basic objects. p-Adic mass calculations however require tachyonic ground states and in generalized Feynman diagrams fermions propagate as massless particles in M4 sense. Furthermore, light-like H-geodesics lead to non-compact candidates for the twistor space of H. Hence the twistor space would be 12-dimensional manifold CP3× SU(3)/U(1)× U(1).
Generalisation of 2-D conformal invariance extending to infinite-D variant of Yangian symmetry; light-like 3-surfaces as basic objects of TGD Universe and as generalised light-like geodesics; light-likeness condition for momentum generalized to the infinite-dimensional context via super-conformal 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 M4× CP2 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 Kac-Moody 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.
A question about how non-planar Feynman diagrams could be represented in twistor Grassmannian approach inspired a re-reading of the recent article by recent article by Nima Arkani-Hamed et al.
This inspired the conjecture that non-planar twistor diagrams correspond to non-planar Feynman diagrams and a concrete proposal for realizing the earlier proposal that the contribution of non-planar 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 non-crossing 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 non-planar twistor diagrams at my homepage.
For background see the chapter Generalized Feynman diagrams as generalized braids or the article Still about non-planar twistor diagrams.