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Towards M-Matrix

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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 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.

  1. Fermions of internal lines are massless in real sense and have unphysical helicity. Wormhole contacts carrying fermion and antifermion at their opposite throats correspond to basic building bricks of bosons. For fermions second throat is empty. The residue integral over the momenta of internal lines replaces fermionic propagator with its inverse and replaces 4-D momentum integration with integration over light-cone using the standard Lorentz invariant integration measure.
  2. 4-fermion vertex defines the fundamental vertex and contains constant proportional to length squared. This is definitely a problem. If all 4-fermion vertices contain one bosonic wormhole contact, one can replace regard the verties effectively as BFF or BBB vertices. The four-fermion coupling constant L2 having dimensions length squared can be replaced with 1/p2 for the bosonic line: this is the new ingredient allowing to overcome the difficulties assignable to four-fermion vertex.
  3. For QFT type BFCW BFF and BBB vertices would be an outcome of bosonic emergence (bosons as wormhole contacts) and 4-fermion vertex is proportional to factor with dimensions of inverse mass squared and naturally identifiable as proportional to the factor 1/p2 assignable to each boson line. This predicts a correct form for the bosonic propagators for which mass squared is in general non-vanishing unlike for fermion lines. The usual BFCW construction would emerge naturally in this picture. There is however a problem: the emergent bosonic propagator diverges or vanishes depending on whether one assumes SUSY at the level of single wormhole throat or not. By the special properties of the analog of N=4 SUSY SUSY generated by right handed neutrino the SUSY cannot be applied to single wormhole throat but only to a pair of wormhole throats.
  4. This as also the fact that physical particles are necessarily pairs of wormhole contacts connected by fermionic strings forces stringy variant of BFCW avoiding the problems caused by non-planar diagrams. Now boson line BFCW cuts are replaced with stringy cuts and loops with stringy loops. By generalizing the earlier QFT twistor Grassmannian rules one ends up with their stringy variants in which super Virasoro generators G, G and L bringing in CP2 scale appear in propagator lines: most importantly, the fact that G and G carry fermion number in TGD framework ceases to be a problem since a string world sheet carrying fermion number has 1/G and 1/G at its ends. The general rules is simple: each line emerging from 4-fermion vertex carries 1/G and 1/G as vertex factor. Twistorialization applies because all fermion lines are light-like.
  5. A more detailed analysis of the properties of right-handed neutrino demonstrates that modified gamma matrices in the modified Dirac action mix right and left handed neutrinos but that this happens markedly only in very short length scales comparable to CP2 scale. This makes neutrino massive and also strongly suggests that SUSY generated by right-handed neutrino emerges as a symmetry at very short length scales so that spartners would be very massive and effectively absent at low energies. Accepting CP2 scale as cutoff in order to avoid divergent gauge boson propagators QFT type BFCW makes sense. The outcome is consistent with conservative expectations about how QFT emerges from string model type description.
  6. The generalization to gravitational sector is not a problem in sub-manifold gravity since M4 - the only space-time geometry with Minkowski signature allowing twistor structure - appears as a Cartesian factor of the imbedding space. A further finding is that CP2 and S4 are the only Euclidian 4-manifolds allowing twistor space with Kähler structure. Since S4 does not allow Kähler structure, CP2 is completely unique just like M4. Stringy picture indeed treats gravitons and other elementary particles completely democratically.
  7. The analog of twistorial construction in CP2 degrees of freedom based on the notion of flag manifold and geometric quantization is proposed. Light-likeness in real sense poses a powerful constraint analogous to constraints posed by moves in the case of SYMs and if volume of a convex polytope dictated by the external momenta and helicities provides a representation of the scattering amplitude, the tree diagrams would give directly the full volume.
Perhaps it is not exaggeration to say that the architecture of generalized Feynman diagrams and their connection to twistor approach is now reasonably well-understood. There are of course several problems to be solved. On must feed in p-adic thermodynamics for external particles (here zero energy ontology might be highly relevant). Also the description of elementary particle families in terms of elementary particle functionals in the space of conformal equivalence classes of partonic 2-surface must be achieved.

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.

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 constants

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?.

What could 4-fermion twistor amplitudes look like?

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.

  1. Elementary particles consist of pairs of wormhole contacts connecting two space-time sheets. The throats are connected by magnetic fluxes running in opposite directions so that a closed monopole flux loop is in question. One can assign to the ordinary fermions open string world sheets whose boundary belong to the light-like 3-surfaces assignable to these two wormhole contacts. The question is whether one can restrict the consideration to single wormhole contact or should one describe the situation as dynamics of the open string world sheets so that basic unit would involve two wormhole contacts possibly both carrying fermion number at their throats.

    Elementary particles are bound states of massless fermions assignable to wormhole throats. Virtual fermions are massless on mass shell particles with unphysical helicity. Propagator for wormhole contact as bound state - or rather entire elementary particle would be from p-adic thermodynamics expressible in terms of Virasoro scaling generator as 1/L0 in the case of boson. Super-symmetrization suggests that one should replace L0 by G0 in the wormhole contact but this leads to problems if G0 carries fermion number. This might be a good enough motivation for the twistorial description of the dynamics reducing it to fermion propagator along the light-like orbit of wormhole throat. Super Virasoro algebra would emerged only for the bound states of massless fermions.

  2. Suppose that the construction of four-fermion vertices reduces to the level of single wormhole contact. 4-fermion vertex involves wormhole contact giving rise to something analogous to a boson exchange along wormhole contact. This kind of exchange might allow interpretation in terms of Euclidian correlation function assigned to a deformation of CP2 type vacuum extremal with Euclidian signature.

    A good guess for the interaction terms between fermions at opposite wormhole contacts is as current-current interaction jα (x) jα(y), where x and y parametrize points of opposite throats. The current is defined in terms of induced gamma matrices as ‾ΨΓαΨ and one functionally integrates over the deformations of the wormhole contact assumed to correspond in vacuum configuration to CP2 type vacuum extremal metrically equivalent with CP2 itself. One can expand the induced gamma matrix as a sum of CP2 gamma matrix and contribution from M4 deformation Γα = ΓαCP2 + ∂α mkγk. The transversal part of M4 coordinates orthogonal to M2⊂ M4 defines the dynamical part of mk so that one obtains strong analogy with string models and gauge theories.

The deformation Δ mk can be expanded in terms of CP2 complex coordinates so that the modes have well defined color hyper-charge and isospin. There are two options to be considered.
  1. One could use CP2 spherical harmonics defined as eigenstates of CP2 scalar Laplacian D2. The scale of eigenvalues would be 1/R2, where R is CP2 radius of order 104 Planck lengths. The spherical harmonics are in general not holomorphic in CP2 complex coordinates ξi, i=1,2. The use of CP2 spherical harmonics is however not necessary since wormhole throats mean that wormhole contact involves only a part of CP2 is involved.
  2. Conformal invariance suggests the use of holomorphic functions ξ1mξ2n as analogs of zn in the expansion. This would also be the Euclidian analog for the appearance of massless spinors in internal lines. Holomorphic functions are annihilated by the ordinary scalar Laplacian. For conformal Laplacian they correspond to the same eigenvalue given by the constant curvature scalar R of CP2. This might have interpretation as a spontaneous breaking of conformal invariance.

    The holomorphic basis zn reduces to phase factors exp(inφ) at unit circle and can be orthogonalized. Holomorphic harmonics reduce to phase factors exp(imφ1)exp(inφ2) and torus defined by putting the moduli of ξi constant and can thus be orthogonalized. Inner product for the harmonics is however defined at partonic 2-surface. Since partonic 2-surfaces represent Kähler magnetic monopoles they have 2-dimensional CP2 projection. The phases exp(imφi) could be functionally independent and a reduction of inner product to integral over circle and reduction of phase factors to powers exp(inφ) could take place and give rise to the analog of ordinary conformal invariance at partonic 2-surface. This does not mean that separate conservation of I3 and Y is broken for propagator.

  3. Holomorphic harmonics are very attractive but the problem is that they are annihilated by the ordinary Laplacian. Besides ordinary Laplacian one can however consider (Conformal Laplacian) defined as

    Dc2= -6D2+R

    and relating the curvatures of two conformally scaled metrics R denotes now curvature scalar). The overall scale factor and also its sign is just a convention. This Laplacian has the same eigenvalue for all conformal harmonics. The interpretation would be in terms of a breaking of conformal invariance due to CP2 geometry: this could also relate closely to the necessity to assume tachyonic ground state in the p-adic mass calculations.

    The breaking of conformal invariance is necessary in order to avoid infrared divergences. The replacement of M4 massless propagators with massive CP2 bosonic propagators in 4-fermion vertices brings in the needed breaking of conformal invariance. Conformal invariance is however retained at the level of M4 fermion propagators and external lines identified as bound states of massless states.

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 ξii,1i,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.

  1. The propagator defined by the ordinary Laplacian D2 has infinite value for all conformal harmonics appearing in the correlation function. This cannot be the case.
  2. If the propagator is defined by the conformal Laplacian Dc2 of CP2 multiplied by some numerical factor it gives fro a given model besides color quantum numbers conserving delta function a constant factor nR2 playing the same role as weak coupling strength in the four-fermion theory of weak interactions. Propagator in CP2 degrees of freedom would give a constant contribution if the total color quantum numbers for vanish for wormhole throat so that one would have four-fermion vertex.
  3. One can consider also a third - perhaps artificial option - motivated for Dirac spinors by the need to generalize Dirac operator to contain only I3 and Y. Holomorphic partial waves are also eigenstates of a modified Laplacian D2C defined in terms of Cartan algebra as

    D2C== [aY2+bI32]/R2 ,

    where a and b suitable numerical constants and R denotes the CP2 radius defined in terms of the length 2π R of CP2 geodesic circle. The value of a/b is fixed from the condition Tr(Y2)=Tr(I32) and spectra of Y and I3 given by (2/3,-1/3,-1/3) and (0,1/2,-1/2) for triplet representation. This gives a/b= 9/20 so that one has

    D2C= ((9/20) Y2+ I32]× a/R2 .

    In the fermionic case this kind of representation is well motivated since fermionic Dirac operator would be Yk eAkγA+I3k eAkγA, where the vierbein projections YkeAk YkeAk and I3keAk of Killing vectors represent the conserved quantities along geodesic circles and by semiclassical quantization argument should correspond to the quantized values of Y and I3 as vectors in Lie algebra of SU(3) and thus tangent vectors in the tangent space of CP2 at the point of geodesic circle along which these quantities are conserved. In the case of S2 one would have Killing vector field Lz at equator.

Two general remarks are in order.

  1. That a theory containing only fermions as fundamental elementary particles would have four-fermion vertex with dimensional coupling as a basic vertex at twistor level, would not be surprising. As a matter of fact, Heisenberg suggested for long time ago a unified theory based on use of only spinors and this kind of interaction vertex. A little book about this theory actually inspired me to consider seriously the fascinating challenge of unification.
  2. A common problem of all these options seems to be that the 4-fermion coupling strength is of order R2 - about 108 times gravitational coupling strength and quite too weak if one wants to understand gauge interactions. It turns out however that color partial waves for the deformations of space-time surface propagating in loops can increase R2 to the square Lp2= pR2 of p-adic length scale. For D2C assumed to serve as an propagator in an effective action of a conformal field theory one can argue that large renormalization effects from loops increase R2 to something of order pR2.

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.

  1. The analog of massless propagation in Euclidian degrees of freedom would correspond naturally to the conservation of Y and I3 along propagator line and conservation of Y and I3 at vertices. The sum of fermionic and bosonic color quantum numbers assignable to the color partial waves woul be conserved. For external fermions the color quantum numbers are fixed but fermions in internal lines could move also in color excited states.
  2. One can argue that the correlation function for the M4 coordinates for points at the ends of fermionic line do not correlate as functions of CP2 coordinates since the distance between partonic 2-surface is much longer than CP2 scale but do so as functions of the string world sheet coordinates as stringy description strongly suggests and that stringy correlation function satisfying conformal invariance gives this correlation. One can however couner argue that for hadrons the color correlations are different in hadronic length scale. This in turn suggests that the correlations are non-trivial for both the wormhole magnetic flux tubes assignable to elementary particles and perhaps also for the internal fermion lines.
  3. I3 and Y assignable to the exchanged boson should have interpretation as an exchange of quantum numbers between the fermions at upper and lower throat or change of color quantum numbers in the scattering of fermion. The problem is that induced spinors have constant anomalous Y and I3 in given coordinate patch of CP2 so that the exchange of these quantum numbers would vanish if upper and lower coordinate patches are identical. Should one expand also the induced spinor fields in Euclidian regions using the harmonics or their holomorphic variants as suggested by conformal invariance?

    The color of the induced spinor fields as analog of orbital angular momentum would realized as color of the holomorphic function basis in Euclidian regions. If the fermions in the internal lines cannot carry anomalous color, the sum over exchanges trivializes to include only a constant conformal harmonic. The allowance of color partial waves would conform with the idea that all color partial waves are allowed for quarks and leptons at imbedding space level but define very massive bound states of massless fermions.

  4. The 4-fermion vertex would involve a sum over the exchanges defined by spherical harmonics or - more probably - by their holomorphic analogs. For both the spherical and conformal harmonic option the 4-fermion coupling strength would be of order R2, where R is CP2 length. The coupling would be extremely weak - about 108 times the gravitational coupling strength G if the coupling is of order one. This is definitely a severe problem: one would want something like Lp2, where p is p-adic prime assignable to the elementary particle involved.

    This problem provides a motivation for why a non-trivial color should propagate in internal lines. This could amplify the coupling strength of order R2 to something of order Lp2=pR2. In terms of Feynman diagrams the simplest color loops are associated with the closed magnetic flux tubes connecting two elementary wormhole contacts of elementary particle and having length scale given by p-adic length scale Lp. Recall that νL R)c pair or its conjugate neutralizes the weak isospin of the elementary fermion. The loop diagrams representing exchange of neutrino and the fermion associated with the two wormhole contacts and thus consisting of two fermion lines assignable to "long" strings and two boson lines assignable to "short strings" at wormhole contacts represent the first radiative correction to 4-fermion diagram. They would give sum over color exchanges consistent with the conservation of color quantum numbers at vertices. This sum, which in 4-D QFT gives rise to divergence, could increase the value of four-fermion coupling to something of order Lp2= kpR2 and induce a large scaling factor of $D^2_C$.

  5. Why known elementary fermions correspond to color singlets and triplets? p-Adic mass calculations provide one explanation for this: colored excitations are simply too massive. There is however evidence that leptons possess color octet excitations giving rise to light mesonlike states. Could the explanation relate to the observation that color singlet and triplet partial waves are special in the sense that they are apart from the factor 1/(1+r2)1/2 , r2=∑ |ξi|2 for color triplet holomorphic functions?

4. Why twistorialization in CP2 degrees of freedom?

A couple of comments about twistorialization in CP2 degrees of freedom are in order.

  1. Both M4 and CP2 twistors could be present for the holomorphic option. M4 twistors would characterize fermionic momenta and CP2 twistors to the quantum numbers assignable to deformations of CP2 type vacuum extremals. CP2 twistors would be discretized since I3 and Y have discrete spectrum and it is not at all clear whether twistorialization is useful now. There is excellent motivation for the integration over the flag-manifold defining the choices of color quantization axes. The point is that the choice of conformal basis with well-defined Y and I3 breaks overall color symmetry SU(3) to U(2) and an integration over all possible choices restores it.
  2. Four-fermion vertex has a singularity corresponding to the situation in which p1, p2 and p1+p2 assignable to emitted virtual wormhole throat are collinear and thus all light-like. The amplitude must develop a pole as p3+p3= p1+p2 becomes massless. These wormhole contacts would behave like virtual boson consisting of almost collinear pair of fermion and anti-fermion at wormhole throats.
5. Reduction of scattering amplitudes to subset of N =4 scattering amplitudes

N=4 SUSY provides quantitative guidelines concerning the actual construction of the amplitudes.

  1. For single wormhole contact carrying one fermion, one obtains two N=2 SUSY multiplets from fermions by adding to ordinary one-fermion state right-handed neutrino, its conjugate with opposite spin, or their pair. The net spin projections would be 0, 1/2 ,1 with degeneracies (1,2,1) for fermion helicity 1/2 and (0,-1/2, -1) with same degeneracies for fermion helicity -1/2. These N=2 multiplets can be imbedded to the N=4 multiplet containing 24 states with spins (1,1/2,0,-1/2,-1) and degeneracies given by (1, 4, 6, 4, 1). The amplitudes in N=2 case could be special cases of N=4 amplitudes in the same manner as they amplitudes of gauge theories are special cases of those of super-gauge theories. The only difference would be that propagator factors 1/p2 appearing in twistorial construction would be replaced by propagators in CP2 degrees of freedom.
  2. In twistor Grassmannian approach to planar SYM one obtains general formulas for n-particle scattering amplitudes with k positive (or negative helicities) in terms of residue integrals in Grassmann manifold G(n,k). 4-particle scattering amplitudes of TGD, that is 4-fermion scattering amplitudes and their super counterparts would be obtained by restricting to N=2 sub-multiplets of full N=4 SYM. The only non-vanishing amplitudes correspond for n=4 to k=2=n-2 so that they can be regarded as either holomorphic or anti-holomorphic in twistor variables, an apparent paradox understandable in terms of additional symmetry as explained and noticed by Witten. Four-particle scattering amplitude would be obtained by replacing in Feynman graph description the four-momentum in propagator with CP2 momentum defined by I3 and Y for the particle like entity exchanged between fermions at opposite wormhole throats. Analogous replacement should work for twistorial diagrams.
  3. In fact, single fermion per wormhole throat implying 4-fermion amplitudes as building blocks of more general amplitudes is only a special case although it is expected to provide excellent approximation in the case of ordinary elementary particles. Twistorial approach could allow the treatment of also n>4-fermion case using subset of twistorial n-particle amplitudes with Euclidian propagator. One cannot assign right-handed neutrino to each fermion separately but only to the elementary particle 3-surface so that the degeneration of states due to SUSY is reduced dramatically. This means strong restrictions on allowed combinations of vertices.
Some words of critism is in order.
  1. Should one use CP2 twistors everywhere in the 3-vertices so that only fermionic propagators would remain as remnants of M4? This does not look plausible. Should one use include to 3-vertices both M4 and CP2 type twistorial terms? Do CP2 twistorial terms trivialize as a consequence of quantization of Y and I3?
  2. Nothing has been said about modified Dirac operator. The assumption has been that it disappears in the functional integration and the outcome is twistor formalism. The above argument however implies functional integration over the deformations of CP2 type vacuum extremals.
For details see the new chapter Some fresh ideas about twistorialization of TGD or the article with the same title.

About the SUSY generated by covariantly constant right-handed neutrinos

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.

  1. In TGD framework right-handed neutrinos differ from other electroweak charge states of fermions in that the solutions of the modified Dirac equation for them are delocalized at entire 4-D space-time sheets whereas for other electroweak charge states the spinors are localized at string world sheets (see this).
  2. Since right-handed neutrinos are in question so that right-handed neutrino are in 1-1 correspondence with complex 2-component Weyl spinors, which are eigenstates of γ5 with eigenvalue say +1 (I never remember whether +1 corresponds to right or left handed spinors in standard conventions).
  3. The basic question is whether the fermion number associated with covariantly constant right-handed neutrinos is conserved or conserved only modulo 2. The fact that the right-handed neutrino spinors and their conjugates belong to unitarily equivalent pseudoreal representations of SO(1,3) (by definition unitarily equivalent with its complex conjugate) suggests that generalized Majorana property is true in the sense that the fermion number is conserved only modulo 2. Since νR decouples from other fermion states, it seems that lepton number is conserved.
  4. The conservation of the number of right-handed neutrinos in vertices could cause some rather obvious mathematical troubles if the right-handed neutrino oscillator algebras assignable to different incoming fermions are identified at the vertex. This is also suggested by the fact that right-handed neutrinos are delocalized.
  5. Since the νR:s are covariantly constant complex conjugation should not affect physics. Therefore the corresponding oscillator operators would not be only hermitian conjugates but hermitian apart from unitary transformation (pseudo-reality). This would imply generalized Majorana property.
  6. A further problem would be to understand how these SUSY candidates are broken. Different p-adic mass scale for particles and super-partners is the obvious and rather elegant solution to the problem but why the addition of right-handed neutrino should increase the p-adic mass scale beyond TeV range?

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.

  1. If the oscillator operators for νR and its conjugate are hermitian conjugates, which anticommute to zero (limit of anticommutations for massless modes) one obtains the analog of N=2 SUSY.

  2. If the oscillator operators are hermitian or pseudohermitian, one has pseudoreal analog of N=1 SUSY. Since νR decouples from other fermion states, lepton number and baryon number are conserved.

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.

  1. The analog of N=4 SUSY is obtained if oscillator operators are not hermitian apart from unitary transformation (pseudo reality) since there are 2+2 oscillator operators.
  2. If hermiticity is assumed as pseudoreality suggests, N=2 SUSY with right-handed neutrino conserved only modulo two in vertices obtained.
  3. In this case covariantly constant right-handed neutrinos would not propagate and would naturally generate SUSY multiplets.

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.

  1. The first - presumably non-physical - option would correspond to limiting case and the twistors λ and ξ would both approach zero at the pk→ 0 limit, which again would suggest that covariantly constant right-handed neutrinos decouple completely from dynamics.
  2. For the second option one can assume that either λ or ξa' vanishes. In this manner one obtains 2 spinors λi, i=1,2 and their complex conjugates ξa'i as representatives for the super-generators and could assign the oscillator algebra to these. Obviously twistors would give something genuinely new in this case. The maximal option would give 4 anti-commuting creation operators and their hermitian conjugates and the non-vanishing anti-commutators would be proportional to δa,bλaib)j* and δa,bξa'ia'j)*. If the oscillator operators are hermitian conjugates of each other and (pseudo-)hermitian, the anticommutators vanish.

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.

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 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.

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 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.

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