The twistor Grassmann approach has led to a phenomenal progress in the understanding of the scattering amplitudes of gauge theories, in particular the N=4 SUSY.
As a non-specialist I have been frustrated about the lack of concrete picture, which would help to see how twistorial amplitudes might generalize to TGD framework. A pleasant surprise in this respect was the proposal of a particle interpretation for the twistor amplitudes by Nima Arkani Hamed et al in the article "Unification of Residues and Grassmannian Dualities" .
In this interpretation incoming particles correspond to spheres CP1 so that n-particle state corresponds to (CP1)n/Gl(2) (the modding by Gl(2) might be seen as a kind of formal generalization of particle identity by replacing permutation group S2 with Gl(2) of 2× 2 matrices). If the number of "wrong" helicities in twistor diagram is k, this space is imbedded to CPk-1n/Gl(k) as a surface having degree k-1 using Veronese map to achieve the imbedding. The imbedding space can be identified as Grassmannian G(k,n) . This surface defines the locus of the multiple residue integral defining the twistorial amplitude.
The particle interpretation brings in mind the extension of single particle configuration space E3 to its Cartesian power E3n/Sn for n-particle system in wave mechanics. This description could make sense when point-like particle is replaced with 3-surface or partonic 2-surface: one would have Cartesian product of WCWs divided my Sn. The generalization might be an excellent idea as far calculations are considered but is not in spirit with the very idea of string models and TGD that many-particle states correspond to unions of 3-surfaces in H (or light-like boundaries of causal diamond (CD) in Zero Energy Ontology (ZEO).
Witten's twistor string theory
Witten's twistor string theory is more in spirit with TGD at fundamental level since it is based on the identification of generalization of vertices as 2-surfaces in twistor space.
Generalization of the notion of twistor to 8-D context
- There are several kinds of twistors involved. For massless external particles in eigenstates of momentum and helicity null twistors code the momentum and helicity and are pairs of 2-spinor and its conjugate. More general momenta correspond to two independent 2-spinors.
One can perform twistor Fourier transform for the conjugate 2-spinor to obtain twistors coding for the points of compactified Minkowski space. Wave functions in this twistor space characterized by massless momentum and helicity appear in the construction of twistor amplitudes. BCFW recursion relation allows to construct more complex amplitudes assuming that intermediate states are on mass shells massless states with complex momenta.
One can perform twistor Fourier transformation (there are some technical problems in Minkowski signature) also for the second 2-spinor to get what are called momentum twistors providing in some aspects simpler description of twistor amplitudes. These code for the four-momenta propagating between vertices at which the incoming particles arrive and the differences if two subsequent momenta are equal to massless external momenta.
- In Witten's theory the interactions of incoming particles correspond to amplitudes in which the twistors appearing as arguments of the twistor space wave functions characterized by momentum and helicity are localized to complex curves X2 of twistor space CP3 or its Minkowskian counterpart. This can be seen as a non-local twistor space variant of local interactions in Minkowski space.
The surfaces X2 are characterized by their degree d (of the polynomial of complex coordinates defining the algebraic 2-surface) the genus g of the algebraic surface, by the number k of "wrong" (helicity violating) helicities, and by the number of loops of corresponding diagram of SUSY amplitude: one has d= k-1+ l, g≤ l. The interaction vertex in twistor space is not anymore completely local but the n particles are at points of the twistorial surface X2.
In the addition to the article Classical part of the twistor story a proposal generalizing Witten's approach to TGD is discussed. The following gives a very concise summary of the article.
Consider first various aspects of twistorialization.
- The fundamental challenge is the generalization of the notion of twistor associated with massless particle to 8-D context, first for M4=M4× E4 and then for H= M4 × CP2. The notion of twistor space solves this question at geometric level. As far as construction of the TGD variant of Witten's twistor string is considered, this might be quite enough.
What is especially nice, that twistorialization extends to from spin to include also electroweak spin. These two spins correspond correspond to M4 and CP2helicities for the twistor space amplitude, and are non-local properties of these amplitudes. In TGD framework only twistor amplitudes for which helicities correspond to that for massless fermion and antifermion are possible and by fermion number conservation the numbers of positive and negative helicities are identical and equal to the fermion number (or antifermion number). Separate lepton and baryon number conservation realizing 8-D chiral symmetry implies that M4 and CP2 helicities are completely correlated. At twistorial level this means restriction to amplitudes with k=2(n(F)-n(Fbar)) if
no mixing of M4and CP2 occurs (massivation in M4 sense) . This in quark and lepton sector separately.
- M8-H duality and quantum-classical correspondence however suggest that M8 twistors might allow tangent space description of four-momentum, spin, color quantum numbers and electroweak numbers and that this is needed. What comes in mind is the identification of fermion lines as light-like geodesics possessing M8 valued 8-momentum, which would define the long sought gravitational counterparts of four-momentum and color quantum numbers at classical point-particle level. The M8 part of this four-momentum would be equal to that associated with imbedding space spinor mode characterizing the ground state of super-conformal representation for fundamental fermion.
Hence one might also think of starting from the 4-D condition relating Minkowski coordinates to twistors and looking what it could mean in the case of M8. The generalization is indeed possible in M8 =M4× E4 by its flatness if one replaces gamma matrices with octonionic gamma matrices.
In the case of M4× CP2 situation is different since for octonionic gamma matrices SO(1,7) is replaced with G2, and the induced gauge fields have different holonomy structure than for ordinary gamma matrices and octonionic sigma matrides appearing as charge matrices bring in also an additional source of non-associativity. Certainly the notion of the twistor Fourier transform fails since CP2 Dirac operator cannot be algebraized.
Algebraic twistorialization however works for the light-like fermion lines at which the ordinary and octonionic representations for the induced Dirac operator are equivalent. One can indeed assign 8-D counterpart of twistor to the particle classically as a representation of light-like hyper-octonionic four-momentum having massive M4 and CP2 projections and CP2 part perhaps having interpretation in terms of classical tangent space representation for color and electroweak quantum numbers at fermionic lines.
If all induced electroweak gauge fields - rather than only charged ones as assumed hitherto - vanish at string world sheets, the octonionic representation is equivalent with the ordinary one. The CP2 projection of string world sheet should be 1-dimensional: inside CP2 type vacuum extremals this is impossible, and one could even consider the possibility that the projection corresponds to CP2 geodesic circle. This would be enormous technical simplification. What is important that this would not prevent obtaining non-trivial scattering amplitudes at elementary particle level since vertices would correspond to re-arrangement of fermion lines between the generalized lines of Feynman diagram meeting at the vertices (partonic 2-surfaces).
- In the fermionic sector one is forced to reconsider the notion of the induced spinor field. The modes of the imbedding space spinor field should co-incide in some region of the space-time surface with those of the induced spinor fields. The light-like fermionic lines defined by the boundaries of string world sheets or their ends are the obvious candidates in this respect. String world sheets is perhaps too much to require. Number theoretically string world sheets would be commutative surfaces analogous to space-time surfaces as quaternionic surfaces of octonionic imbedding space.
The only reasonable identification of string world sheet gamma matrices is as induced gamma matrices and super-conformal symmetry requires that the action contains string world sheet area as an additional term just as in string models. String tension would correspond to gravitational constant and its value - that is ratio to the CP2 radius squared, would be fixed by quantum criticality.
Generalization of Witten's construction in TGD framework
The generalization of the Witten's geometric construction of scattering amplitudes relying on the induction of the twistor structure of the imbedding space to that associated with space-time surface looks surprisingly straight-forward and would provide more precise formulation of the notion of generalized Feynman diagrams forcing to correct some wrong details.
- All generalized Feynman graphs defined in terms of Euclidian regions of space-time surface are lifted to twistor spaces. Incoming particles correspond quantum mechanically to twistor space amplitudes defined by their momenta and helicities and and classically to the entire twistor space of space-time surface as a surface in the twistor space of H. Of course, also the Minkowskian regions have this lift. The vertices of Feynman diagrams correspond to regions of twistor space in which the incoming twistor spaces meet along their 5-D ends having also S2 bundle structure over space-like 3-surfaces. These space-like 3-surfaces correspond to ends of Euclidian and Minkowskian space-time regions separated from each other by light-like 3-surfaces at which the signature of the metric changes from Minkowskian to Euclidian. These "partonic orbits" have as their ends 2-D partonic surfaces. By strong form of General Coordinate Invariance implying strong of holography, these 2-D partonic surfaces and their 4-D tangent space data should code for quantum physics. Their lifts to twistor space are 4-D S2 bundles having partonic 2-surface X2 as base.
One of the nice outcomes is that the genus appearing in Witten's formulation naturally corresponds to family replication in TGD framework.
- All elementary particles are many particle bound states of massless fundamental fermions: the non-collinearity of massless momenta explains massivation. The fundamental fermions are localized at wormhole throats defining the light-like orbits of partonic 2-surfaces. Throats are associated with wormhole contacts connecting two space-time sheets. Stability of the contact is guaranteed by non-vanishing monopole magnetic flux through it and this requires the presence of second wormhole contact so that a closed magnetic flux tube carrying monopole flux and involving the two space-time sheets is formed. The net fermionic quantum numbers of the second throat correspond to particle's quantum numbers and above weak scale the weak isospins of the throats sum up to zero.
- Fermionic 2-vertex is the only local many-fermion vertex being analogous to a mass insertion. The non-triviality of the theory could be seen to follow from the analog of OZI mechanism: the fermionic lines inside partonic orbits are redistributed in vertices. Lines can also turn around in time direction which corresponds to creation or annihilation of a pair. 3-particle vertices are obtained only in topological sense as 3 space-time surfaces are glued together at their ends. The interaction between fermions at different wormhole throats is described in terms of string world sheets.
- The earlier proposal was that the fermions in the internal fermion lines are massless in M4 sense but have non-physical helicity so that the algebraic M4 Dirac operator emerging from the residue integration over internal four-momentum does not annihilate the state at the end of the propagator line. Now the algebraic induced Dirac operator defines the propagator at fermion lines. Should one assume generalization of non-physical helicity also now?
- All this stuff must be lifted to twistorial level and one expects that the lift to S2 bundle allows an alternative description of fermions and spinor structure so that one can speak of induced twistor structure instead of induced spinor structure. This approach allows also a realization of M4 conformal symmetries in terms of globally well-defined linear transformations so that it might be that twistorialization is not a mere reformulation but provides a profound unification of bosonic and fermionic degrees of freedom.
The emergence of fundamental 4-fermion vertex and of boson exchanges
The emergence of the fundamental 4-fermion vertex and of boson exchanges deserves a more detailed discussion.
- I have proposed that the discontinuity of the Dirac operator at partonic two-surface (corner of fermion line) defines both the fermion boson vertex and TGD analog of mass insertion (not scalar but imbedding space vector) giving rise to mass parameter having interpretation as Higgs vacuum expectation and various fermionic mixing parameters at QFT limit of TGD obtained by approximating many-sheeted space-time of TGD with the single sheeted region of M4 such that gravitational field and gauge potentials are obtained as sums of those associated with the sheets.
- Non-trivial scattering requires also correlations between fermions at different partonic 2-surfaces. Both partonic 2-surfaces and string world sheets are needed to describe these correlations. Therefore the string world sheets and partonic 2-surfaces cannot be dual: both are needed and this means deviation from Witten's theory. Fermion vertex corresponds to a "corner" of a fermion line at partonic 2-surface at which generalized 4-D lines of Feynman diagram meet and light-like fermion line changes to space-like one. String world sheet with its corners at partonic 2-surfaces (wormhole throats) describes the momentum exchange between fermions. The space-like string curve connecting two wormhole throats serves as the analog of the exchanged gauge boson.
- Two kinds of 4-fermion amplitudes can be considered depending on whether the string connects throats of single wormhole contact (CP2 scale) or of two wormhole contacts (p-adic length scale - typically of order elementary particle Compton length). If string worlds sheets have 1-D CP2 projection, only Minkowskian string world sheets are possible. The exchange in Compton scale should be assignable to the TGD counterpart of gauge boson exchange and the fundamental 4-fermion amplitude should correspond to single wormhole contact: string need not to be involved now. Interaction is basically classical interaction assignable to single wormhole contact generalizing the point like vertex.
- The possible TGD counterparts of BCFW recursion relations should use the twistorial representations of fundamental 4-fermion scattering amplitude as seeds. Yangian invariance poses very strong conditions on the form of these amplitudes and the exchange of massless bosons is suggestive for the general form of amplitude.
The 4-fermion amplitude assignable to two wormhole throats defines the analog of gauge boson exchange and is expressible as fusion of two fundamental 4-fermion amplitudes such that the 8-momenta assignable to the fermion and anti-fermion at the opposite throats of exchanged wormhole contact are complex by BCFW shift acting on them to make the exchanged momenta massless but complex. This entity could be called fundamental boson (not elementary particle).
- Can one assume that the fundamental 4-fermion amplitude allows a purely formal composition to a product of BFF amplitudes? Two 8-momenta at both FFB vertices must be complex so that at least two external fermionic momenta must be complex. These external momenta are naturally associated with the throats of the a wormhole contact defining virtual fundamental boson. Rather remarkably, without the assumption about product representation one would have general four-fermion vertex rather than boson exchange. Hence gauge theory structure is not put in by hand but emerges.
See the chapter Classical part of the twistor story or the article Classical part of the twistor story.