Summary about breakthrough in the construction of U-matrix and understanding of twistorialization

During last few days a dramatic development in the understanding of the notions of U-matrix, M-matrix and S-matrix- a trinity of matrices replacing in zero energy ontology the notion of S-matrix of positive energy ontology and allowing fusion of thermodynamics with quantum theory. Also twistorialization reduces to pure group theory-albeit infinite-dimensional: zero energy states define a Yangian algebra generalizing that of Grassmannian twistor approach. For this reason I feel it as appropriate to add a further posting trying to give bird's eye of view about the ideas since previous posting is rather long and detailed and reader easily loses the thread.

1. The realization that the hermitian square roots of density matrices form infinite-D unitary algebra and that their commutativity with universal S-matrix implies that zero energy states define the generalization of Kac-Moody algebra became only after I had realized the possibility to construct U-matrix. The Hermitian sub-Lie-algebra commuting with S is large: for SU(N) it would correspond to SU(N-1)× U(1). It is this observation which reduces the construction of U-matrix (or matrices if they form algebra) to that for S is expected to correspond directly to the ordinary S-matrix. A possible interpretation of the algebra of U-matrices is in terms of scales of CDs coming as positive integer powers of two. Another possibility more in line with the usual interpretation of S-matrix as time evolution operator is that scales of CDs come as integers and these integers correspond to powers of S.

   What is so fascinating is that zero energy states themselves define the symmetry algebra of the theory and that this algebra can be interpreted as a generalization of Yangian responsible for the successes of Grassmannian twistor approach by replacing finite-dimensional conformal group of Minkowski space with infinite-dimensional super-conformal algebras associated with partonic 2-surfaces in accordance with the replacement of point-like particles with surfaces. The basic characteristic of Yangian algebra is the multilocality of its generators and zero energy states are indeed multilocal since they involve partonic surfaces at both light-like boundaries of CD. Quantum TGD reduces to pure group theory! Note only states but also dynamics is coded completely by symmetries since M-matrices code for quantum dynamics! This powerful aspect of zero energy ontology I have not realized before.

2. In ordinary QFT Feynman diagrams are purely algebraic objects. In TGD framework they reduce to space-time topology and geometry with Euclidian regions of space-time surfaces having interpretation as generalized Feynman diagrams. At the vertices of generalized Feynman diagrams in coming partonic 2-surfaces meet just like in ordinary Feynman diagrams which means deep difference from string theory. A more general assumption is that entire 4-D lines of generalized Feynman diagram meet at vertices. This could apply to the Euclidian regions only.

   There is also a second kind of branching involved with the hierarchy of Planck constants. In Minkowskian regions similar meeting would take place for the branches of space-time sheets with same values of canonical momentum densities of Kähler action at the ends of CDs and have interpretation in terms of fractionization and hierarchy of Planck constants. The value of Planck constant for single branch would be effectively and integer multiple of the ordinary one. For the entire multi-sheeted structure describable naturally in terms of singular covering space of M⁴× CP₂ it would be just the ordinary value.

3. Zero energy ontology with massless external wormhole wormholes implies as such twistorialization of the theory although external wormhole momenta must be assumed to be massive bounds states of massless throats (see this). This also guarantees exact Yangian symmetry and the absence of IR divergences. If also virtual wormhole throats are massless, twistorialization takes place in strong sense. This is possible only in zero energy ontology and accepting the identification of wormhole throats as basic building blocks of particles.

4. The notion of a bosonic emergence means that bosonic propagators emerge as radiative loops for wormhole contacts. The emergence generalizes to all states associated with wormhole contacts and also to flux tubes having wormhole contacts at their ends. What is nice that coupling constants emerge as normalization factors of propagators. Note that for single wormhole throat as opposed to wormhole contact having two throats bosonic propagator would result as a product of two collinear fermionic propagators and have the standard form. For states with higher total number of fermions and anti-fermions the propagator of wormhole throat behaves as pⁿ, n>2. Here however p is replaced with what I call pseudo-momentum.

5. Number theoretical universality suggest that at given level (CD) only finite sum of diagrams appears: otherwise there is a danger that one obtains sum of rational functions which is not rational anymore. This gives strong constraints on generalized Feynman diagrams at the lowest level of the hierarchy. This follows naturally if twistor diagrams are identified as sums of Feynman diagrams which are irreducible in the sense that they do not represent two subsequence scatterings. Only these diagrams contribute to twistor diagram and the number of these diagrams is finite if all particles have small mass (even photon which would eat the remaining Higgs component).

6. Category theoretical approach to TGD based on planar operad proposed for few years ago fits nicely with the twistorial construction of amplitudes interpreting radiative corrections in terms of CDs within CDs picture. The generalized Feynman diagrams with radiative corrections define the analog of planar operad with disk containing within itself disks containing.... replaced with causal diamond containing causal diamonds containing....

To make this more concrete it deserves to describe TGD counterparts for the recursion formula and duality between descriptions using twistors and momentum twistors.

1. The great victory of twistor approach is the recursion formula for the amplitudes (see also the representation in TGD framework) applying to all planar diagrams of N=4 SYM becoming an exact formula at the large N limit for gauge group SU(N). In the recent case the infinite-dimensional character of the Yangian symmetry algebra of S-matrix could be correlate for large N limit so that the planar limit should make sense. Also the fact that string worlds sheets are an essential aspect of TGD approach suggests that stringy picture deduced by t'Hooft for gauge theories at this limit implies planarity. What is relevant in the recent case is the general structure of the reduction formula, not the details which as such are of course extremely interesting also in TGD framework since Grassmannian amplitudes are claimed to provide a universal representation of Yangian invariants.

   The recursive formula expresses scattering amplitude with n external particles with k negative helicities up to l loops is expressible as a sum of two terms. The first term-referred to as classical contribution- involves a fusion of twistor amplitudes with smaller number of particles and with the number of loops not larger than l by a procedure used already for tree diagrams. Second term - called quantum contribution- involves l loops and is irreducible in the sense that it is not expressible as a fusion of lower amplitudes and is obtained from n+2 particle by a process eliminating two particles. The identification of the TGD counterparts of these terms is obvious. The "classical" term corresponds to the proposed fusion of the lower level amplitudes associated with polygons for sub-CDs. The "quantum" term corresponds to the contribution appearing at the level of CD itself and involves genuine loops in Feynman sense but only a finite number of them.

   Since zero energy states correspond to generators of Yangian algebra or rather- its Kac-Moody variant with integer power of phase factor identified as integer power of S, the recursion formula might allow an interpretation as a direct counterpart for the recursive definition of Yangian algebra in terms of relations allowing the construction of generators labeled by non-negative integers.

2. One of the fascinating findings of twistor Grassmannian approach is that conformal invariance and its dual correspond in twistor approach to descriptions in terms of twistors in ordinary Minkowski space by starting from Feynman diagrams and in terms of momentum twistors in its dual by starting from Wilson loops. Also this duality has counterpart in TGD.

   String world sheets are an essential part of quantum TGD and the translation of Witten's work with knots to TGD context led to a precise identification of string world sheets and a deep connection between TGD and the theory of knots, braids, braid cobordisms, and 2-knots emerges (see this).

   Amusingly, the basic idea of this connection emerged from the model of DNA as topological quantum computer developed for few years ago. The braiding defining the quantum computation is time-like and can be illustrated using dance metaphor: the world lines of dancers define the running topological computation program. If you connect the feet of dancers to a wall with threads (dancers are lipids at cell membrane forming 2-D liquid, wall is represented by DNA nucleotide sequence, and threads are magnetic flux tubes), the threads entangle during dance and give rise to a space-like braiding and code the computer program to memory: a fundamental mechanism of memory. These braidings are clearly dual and this duality relates closely to the duality just to the duality between Feynman graphs and Wilson loops! The time evolution of this space-like braiding defines braid cobordism and also a 2-knot.

   The natural implication of strong form of holography made possible by preferred extremal (Bohr orbit in generalized sense) property of space-time surfaces is that the descriptions in terms of string world sheets and partonic 2-surfaces are dual. The twistorial representation of this duality is as the duality of descriptions in terms of Feynman diagrammatics in ordinary space-time and Wilson sheets-rather than loops- in the dual space-time assigned with region momenta.

To conclude, it seems that all the basic visions about quantum TGD combine to single coherent whole! Maybe the long march is in some sense over!

For background see the chapter Construction of Quantum Theory: M-matrix of "Towards M-Matrix". The recent results are summarized at the end of the chapter. I added to my homepage also an article summarizing the recent progress.

By the way, there is excellent video of a lecture by Nima Arkani-Hamed to mathematicians with title Grassmannians, Polytopes and Quantum Field Theory. If you identify yourself as a theoretician do not miss it! Just the opportunity to experience Nima's enthusiasm, joy, and energy convinces that theoretical physics can also be an exploration and adventure.