Still one attempt understand generalized Feynman diagrams

The only manner to develop the understanding about generalized Feynman diagrams is to articulate the basic questions again and again in the hope that something new might emerge. There are many questions to be answered.

What Grasmannian twistorialization means when imbedding space spinor fields are the fundamental objects. How does ZEO make twistorialization possible? How twistorialization emerges from the functional integral in WCW from the proposed stringy construction of spinor modes.

One must also understand in detail the realization of super-conformal symmetries and how n-point functions of conformal field theory are associated with scattering amplitudes, and how cm degrees of freedom described using imbedding space spinor harmonics are treated in the scattering amplitudes. Also the braiding and knotting should be understood. The challenge is to find a universal form for the vertices and to identify the propagators. Also the modular degrees of freedom of partonic 2-surfaces explaining family replication phenomenon should be taken into account.

Zero energy ontology, twistors, and Grassmannian description?

In ZEO also virtual wormhole throats are massless particles and four-momentum conservation at vertices identifiable as partonic 2-surfaces at which wormhole throats meet expressed in terms of twistors leads to Grassmannian formulation automatically. This feature is thus not specific to N=4 SYM.

Momentum conservation and massless on mass-shell conditions at vertices defined as partonic 2-surfaces at which the orbits of wormhole contacts meet, are extremely restrictive, and one has good hopes that huge reduction in the number of twistorial diagrams takes place and could even lead to finite number of diagrams (number theoretic arguments favor this).

Realization of super-conformal algebra

Thanks to the advances in the construction of preferred extremals and solutions of the modified Dirac equation there has been considerable progress in the understanding of super-conformal invariance and its 4-D generalization (see this).

1. In ordinary SYM ground states correspond to both maximal helicites or only second maximal helicity of super multiplet ( N=4 case). Now these ground states are replaced by the modes of imbedding space spinor fields assignable to center of mass degrees of freedom for partonic 2-surfaces. The light-like four-momenta of these modes can be expressed in terms of twistor variables. Spin-statistics connection seems to require that the total number of fermions and antifermions associated with given wormhole throat is always odd.
   
   
2. Super-algebra consists of oscillator operators with non-vanishing quark or lepton number. By conformal invariance fermionic oscillator operators obey 1-D anti-commutation relations. The integral over CD boundary defines a bi-linear form analogous to inner product. If a reduction to single particle level takes place, the vertex is expressible as a matrix element between two fermion-anti-fermion states: the first one assignable to the incoming and outgoing wormhole throats one and second to the virtual boson identified as wormhole contact on one hand. The exchange boson entangled fermion-anti-fermion state represented by a bi-local generalization of the gauge current. This picture applying to gauge boson exchanges generalizes in rather obvious manner.
   
   
3. Unitary demands correlation between fermionic oscillator operators and spinor harmonics of imbedding space
   as following argument suggets. The bilocal generalization of gauge current defines a "norm" for spinor modes as generalization of what in QFT regarded as charge. On basis of experience with Dirac spinors one expect that this norm is not positive definite. This "norm" must be consistent with the unitarity of the scattering amplitude and the experience with QFT suggests a correlation between creation/annihilation operator character of fermionic oscillator operators and the sign of the "norm" in imbedding space degrees of freedom.
   
   
4. The modes with negative norm should correspond to negative energy fermions and annihilation operators and modes with positive norm to positive energy fermions and creation operators. Therefore the anti-commutators of fermionic oscillator operators must be linear in four-momentum or its longitudinal projection and thus proportional to p^kγ_k or p^k_Lγ_k.
   

   On the other hand, the primary anti-commutators for the induced spinor fields are proportional to the modified gamma matrix in a direction normal to the 1-D quantization curve at the boundary of string world sheet or at the partonic 2-surface. These two anti-commutators should be consistent.
   
   

   1. Does the functional integral somehow lead from the primordial anti-commutators to the anti-commutators involving longitudinal momentum and perhaps 1-D delta function in the intersection of M² with CD boundary (light-like line)?
      
      
   2. Or does the connection between the two quantizations emerge as boundary conditions stating that the normal component of modified gamma matrix at the boundary and along string world sheet equals to p_L^kγ_k? This would also realize quantum classical correspondence in the sense that the longitudinal momentum is reflected in the geometry of the space-time sheet. Quaternionic space-time surfaces indeed contain integrable distribution of M²(x) subset M⁴ at their tangent spaces. The restriction to braid strands would mean that the condition indeed makes sense. Note that braid strands should correspond to same M²(x).
      

How conformal time evolution corresponds to physical time evolution?

The only internally consistent option is conformally invariant meaning that induced spinor fields anti-commute only along as set of 1-D curves belonging to partonic 2-surfaces. This means that one can speak about conformal time evolution at partonic 2-surface.

This suggests a huge simplification of the conformal dynamics.

1. Conformal time evolution can be translated to time evolution along light-like orbit of wormhole throat by projecting the intersections of this surface with shifted light-cone boundary to the upper or lower light-like boundary of CD: whether it is upper or lower boundary of CD depends on the arrow of imbedding space time associated with the zero energy state. All partonic 2-surfaces would be mapped to same light-cone boundary. The orbits of braid strands at wormhole throat project to orbits at light-cone boundary in question and can be further projected to the sphere r_M=constant at light-boundary. 3-D dynamics would project to simplest possible stringy 2-D dynamics (spherical string orbit) and dictated by conformal invariance.
   
   
2. The conformal field theory in question is for conformal fermionic fields realized in terms of fermionic oscillator operators and n-point functions correspond to fermionic n-point functions. The non-triviality of dynamics in these degrees of freedom follows from the non-triviality of the conformal field theory. The entire collection of partonic 2-surfaces at the ends of CD would reduce to its projection to S².
   
   
3. One can try to build a geometric view about the situation using as a guideline conformal Hamiltonian quantum evolution. Time=constant slices would correspond to 1-D curve or collection of them. At these slices fermionic oscillator operators would satisfy the conformal anti-commutation relations. This kind of slice would be associated with both ends of CD. Braid strands would connect these 1-D slices as kind of hairs. One can however ask whether there is any need to restrict the end points of braid strands to line on a curve at which fermionic oscillator operators satisfy stringy anti-commutation relations.
   

What happens in 3-vertices?

The vision is that only 3-vertices are needed. Idealize particles as wormhole contacts (in reality pair of wormhole contacts connected by a flux tube would describe elementary particles). A very convenient visualization of wormhole contact is as a very short string like object with throats at its ends so that stringy diagrammatics allows to identify the vertices as the analogs of open string vertices. One can even consider the possibility that string theory amplitudes define the vertices. This would conform with the p-adic mass calculations applying conformal invariance in CP₂ scale. Note also that partonic 2-surfaces are effectively replaced by braids so that very stringy picture results.

1. Consider a three vertex representing the emission of boson by incoming fermion (FFB) or by incoming boson (BBB) described as wormhole contact such that throats carry fermion and anti-fermion number in the bosonic case. In the fermionic the first throat carries fermion and second one represents vacuum state. The exchanged boson can be regarded as fermion anti-fermion pair such that second fermion travels to future and second one to the past in the vertex. 3-vertex would reduce to two 2-vertices representing the transformation of fermion line from incoming line to exchanged line or from latter to outgoing line.
   
   
2. The minimal option is that the same vertex describes the situation if both cases. Essentially a combination of incoming free fermions to boson like state is in question and corresponds in string picture an exchange of open string between open strings. If so, second wormhole throat is passive and suffers forward scattering in the vertex. The fermion and anti-fermion of the exchanged virtual boson (the light-like momenta of wormhole throats need not be collinear for virtual bosons and also the sign of energy can be different form them) would suffer scattering before the transformation to fermions belonging to incoming and outgoing wormhole contact.
   

One expects the vertices to factorize into products of two kinds of factors: the inner products of fermionic Fock states defined by conformal n-point functions at sphere of light-cone boundary, and the bi-linear forms for the modes of imbedding space spinor fields involving integral over cm degrees of freedom and allowing twistorialization by previous arguments. Let us continue with the simple example in which wormhole throats carry fermion number 0 or 1.

1. If second wormhole throat is passive, it is enough to construct only FFB vertex, with B identified as a wormhole contact carrying fermion and anti-fermion. One has 4 fermions altogether, and one expects that in cm degrees of freedom incoming and outgoing fermion are un-correlated whereas the fermions of the boson exchange are correlated and the correlation is expressible as the analog of gauge current.
   
   
2. This suggests a sum over bi-local counterparts of electro-weak and color gauge currents at opposite ends of the exchanged line. Bi-local gauge currents would contain a spinor mode from both wormhole throats, and the strict locality of M⁴ gauge currents would be replaced with a bi-locality in CP₂ scale.
   
   
3. The current assignable to a particular boson exchange must involve the matrix element of corresponding charge matrix between spinor modes besides the quantity. Is it possible to find a general expression for the sum over current - current interaction terms? If this is the case, there would be no need to perform the summation over bosonic exchanges explicitly. One would have the analog for the ∑_n |n ><| in propagator line but summation allowing the momenta of fermion and antifermion to be arbitrary massless momenta rather than summing up to the on mass shell momentum of boson. The counterpart of gauge coupling should be universal and naturally given by Kähler coupling.
   
   
4. The TGD counterparts of scalar and pseudo-scalar bosons would be vector bosons with polarization in CP₂ direction and they could be also seen both as Higgs like states and Euclidian pions assignable to wormhole contacts. Genuine H-scalars are excluded implied by 8-D chiral symmetry implying also separate conservation of B and L.
   

In the general case the wormhole throats carry arbitrary odd fermion number but for fermion numbers n>1 at any wormhole throat exotic super-partner with propagator decaying faster than 1/p² is in question. Furthermore, wormhole contact is accompanied by second wormhole contact since the flux lines of monopole flux must closed. Therefore one has a pair of "long" string like flux tubes connected by short flux tubes at their ends. Its length is given by weak length scale quite generally or possibly by Compton length. The other end of the long flux tube can also contain fermions at both flux tubes.

The identification of propagators

A natural guess is that the propagator for single fermion state is just the longitudinal Dirac propagator D_pL for a massless fermion in M⁴⊃ M². For states, which by statistics constraint always contain an odd number M=2N+1 of fermions and antifermions, the propagator would be M:th power of fermionic longitudinal propagator so that it would reduce to p_L^-2ND_pL meaning that only the single fermion states would be behave like ordinary elementary particles. States with higher fermion number would represent radiative corrections reflecting the non-point-like nature of partons. Longitudinal mass squared would be equal to the sum of the contribution from CP₂ degrees of freedom and the integer valued conformal contribution from spinor harmonics. The M⁴ momenta associated with wormhole throats would be light-like. In the prescription using fermionic longitudinal propagators assigned to the braid strands, braid strands are analogous to the edges of polygons appearing in twistor Grassmannian approach.

Some open questions

A long list of open questions remains without a final answer. Consider first twistor Grassmannian approach.

1. Does this prescription follow from quantum criticality? Recall that quantum criticality formulated in terms of preferred extremals and modified Dirac equation leads to a stringy perturbation theory involving fermionic propagator defined by the modified Dirac operator and functional integral over WCW for the deformations of space-time surface preserving the preferred extremal property (see this). This propagator could be called space-time propagator to distinguish it from the imbedding space propagator associated with the longitudinal momentum.
   
   
2. One expects that one still has topological Feynman diagrammatic expansion (besides that defined by functional integral over small deformations of space-time surface with given topology) involving in principle an arbitrary number of vertices defined by the intermediate partonic 2-surfaces. Momentum conservation and massless on mass-shell conditions however pose powerful restrictions on the allowed diagrams, and one might hope that the simplicity of the outcome is comparable to Grassmannian twistor approach for N=4 SYM. One can even hope that the number of contributing diagrams is finite. The important point would be that Grassmannian diagrams would give the outcome of the functional integral over 3-surfaces. Twistorial Grassmann representation is the first guess hitherto for the explicit outcome of the functional integral over WCW.
   
   
3. The lines of Feynman graph are replaced with braids. A new element is that braid strands are braided as curves inside light-like 3-surfaces defined by the orbit of the wormhole throat. Twistorial construction applies only to the planar amplitudes of N=4 SYM. Can one imagine TGD counterparts for non-planar amplitudes in TGD framework or does the stringy picture imply that they are completely absent?
   

   A possible answer to the question is based on the M² projection of the lines of braid strands (or on the projection to the 2-surface defined by an integrable distribution of tangent planes M²(x)). For non-planar diagrams the projections intersect and the intersection cannot be eliminated by a small deformation. It does not make sense to say that line goes over or below the second line. One can speak only about crossings. In the theory or algebraic knots (see this) algebraic knots with crossings are possible. Could algebraic knot theory allow to reduce non-planar diagrams to sums of planar diagrams?
   
   

4. Does one obtain Yangian symmetry using longitudinal propagators and by integrating over the moduli labeling among other things the choices of the preferred plane M²⊂ M⁴ or integrable distribution of preferred planes M²(x)⊂ M⁴? The integral over the choices M²⊂ M⁴ gives formally a Lorentz invariant outcome. Does it also give rise to physically acceptable scattering amplitudes? Are the gauge conditions for the incoming gauge boson states formulated in terms of longitudinal momentum and thus allowing also the third polarization physical? Can one apply this gauge condition also to the virtual boson like exchanges?
   
   
5. It is still somewhat unclear whether one should assume single global choice of M² or an integrable distribution of M²(x).
   
   1. The choice of M²(x) must be same for all braid strands of given partonic 2-surface and remain constant along braid strand and therefore be same also at second end of the strand. Otherwise the fermionic propagator would vary along braid strand. A possible additional condition on braids is that braid strands correspond to the same choice of M²(x). In quantum measurement theory this corresponds to the choice of same spin quantization axes for all fermions inside parton and is physically extremely natural condition. The implication is that one can indeed assign a fixed M² with CD and choice of braid strands via boundary conditions. The simplest boundary conditions would require M²(x) to be constant at light-like 3-surfaces and at the ends of space-time surface at boundaries of CD. This is in spirit with holography stating that quantum measurements can be carried out only at these 3-surfaces (or at least those at the ends of CD).
      
      
   2. One cannot exclude the possibility that M²(x) does not depend on x for a particular space-time sheet and even entire CD although this looks rather strong a restriction. On the other hand, one can ask whether the preferred M² assigned with CD should be generalized to an integrable distribution M²(x) assigned with CD such that M²(x) is contained in the tangent space of preferred Minkowskian extremal.
      
      
   3. Is the functional integral over integrable distributions M²(x) needed? It would be analogous to a functional integral over string world sheets. It is enough to integrate over Lorentz transforms of a given intebrable distribution M²(x) to achieve Lorentz invariance. This because the choice of the integrable distribution reduces effectively to the choice of M² for the disconnected pieces of generalized Feynman diagram. Physical intuition suggests that a particular choice of M²(x) corresponds to fixing of zero modes of WCW and is essentially fixing of classical variables needed to fix quantization axes. The fixing of value distributions of induced Kähler fields n 4-D sense at partonic 2-surfaces would be similar fixation of zero modes.
      
      
   4. If only M² momentum makes it visible in anti-commutators, how the other components of four-momentum can make themselves visible in dynamics? This is possible via momentum conservation at vertices making possible twistor Grassmannian approach. The dynamics in transversal momenta would be dictated completely by the conservation laws.
      

There are also other challenges.

1. Family replication phenomenon has TGD based explanation in terms of the conformal moduli of partonic 2-surfaces. How conformal moduli should be taken into account in the Feynman diagrammatics? Phenomena like topological mixing inducing in turn the mixing of partonic 2-topologies responsible for CKM mixing in TGD Universe should be understood in this description.
   
   
2. Number theoretical universality requires that also the p-adic variants of the amplitudes should make sense. One could even require that the amplitudes decompose to products of parts belonging to different number fields (see this). If one were able to formulate this vision precisely, it would provide powerful constraints on the amplitudes. For instance, a reduction of the amplitudes to a sum over finite number of generalized Feynman diagrams is plausible since this would guarantee that individual contributions which must give rise to algebraic numbers for algebraic 4-momenta, would sum up to an algebraic number.