The twistor approach to TGD require that the expression of light-likeness of M4 momenta in terms of twistors generalizes to 8-D case. The light-likeness condition for twistors states that the 2× 2 matrix representing M4 momentum annihilates a 2-spinor defining the second half of the twistor. The determinant of the matrix reduces to momentum squared and its vanishing implies the light-likeness. This should be generalized to a situation in one has M4 and CP2 twistor, which are not light-like separately but light-likeness in 8-D sense holds true (allowing massive particles in M4 sense and thus
generalization of twistor approach for massive particles).
The case of M8=M4× E4
M8-H duality suggests that it might be useful to consider first the twistorialiation of 8-D light-likeness first the simpler case of M8 for which CP2 corresponds to E4. It turns out that octonionic representation of gamma matrices provide the most promising formulation.
In order to obtain quadratic dispersion relation, one must have 2× 2 matrix unless the determinant for the 4× 4 matrix reduces to the square of the generalized light-likeness condition.
This is not the only possibility that I have considered.
- The first approach relies on the observation that the 2× 2 matrices characterizing four-momenta can be regarded as hyper-quaternions with imaginary units multiplied by a commuting imaginary unit. Why not identify space-like sigma matrices with hyper-octonion units?
- The square of hyper-octonionic norm would be defined as the determinant of 4× 4 matrix and reduce to the square of hyper-octonionic momentum. The light-likeness for pairs formed by M4 and E4 momenta would make sense.
- One can generalize the sigma matrices representing hyper-quaternion units so that they become the 8 hyper-octonion units. Hyper-octonionic representation of gamma matrices exists (γ0=σz× 1, γk= σy× Ik) but the octonionic sigma matrices represented by octonions span the Lie algebra of G2 rather than that of SO(1, 7). This dramatically modifies the physical picture and brings in also an additional source of non-associativity. Fortunately, the flatness of M8 saves the situation.
- One obtains the square of p2=0 condition from the massless octonionic Dirac equation as vanishing of the determinant much like in the 4-D case. Since the spinor connection is flat for M8 the hyper-octonionic generalization indeed works.
- Is it enough to allow the four-momentum to be complex? One would still have 2× 2 matrix and vanishing of complex momentum squared meaning that the squares of real and imaginary parts are same (light-likeness in 8-D sense) and that real and imaginary parts are orthogonal to each other. Could E4 momentum correspond to the imaginary part of four-momentum?
- The signature causes the first problem: M8 must be replaced with complexified Minkowski space Mc4 for to make sense but this is not an attractive idea although Mc4 appears as sub-space of complexified octonions.
- For the extremals of Kähler action Euclidian regions (wormhole contacts identifiable as deformations of CP2 type vacuum extremals) give imaginary contribution to the four-momentum. Massless complex momenta and also color quantum numbers appear also in the standard twistor approach. Also this suggest that complexification occurs also in 8-D situation and is not the solution of the problem.
The case of M8=M4× CP2
What about twistorialization in the case of M4× CP2? The introduction of wave functions in the twistor space of CP2 seems to be enough to generalize Witten's construction to TGD framework and that algebraic variant of twistors might be needed only to realize quantum classical correspondence. It should correspond to tangent space counterpart of the induced twistor structure of space-time surface, which should reduce effectively to 4-D one by quaternionicity of the space-time surface.
- For H=M4× CP2 the spinor connection of CP2 is not trivial and the G2 sigma matrices are proportional to M4 sigma matrices and act in the normal space of CP2 and to M4 parts of octonionic imbedding space spinors, which brings in mind co-associativity. The octonionic charge matrices are also an additional potential source of non-associativity even when one has associativity for gamma matrices.
Therefore the octonionic representation of gamma matrices in entire H cannot be physical. It is however equivalent with ordinary one at the boundaries of string world sheets, where induced gauge fields vanish. Gauge potentials are in general non-vanishing but can be gauge transformed away. Here one must be of course cautious since it can happen that gauge fields vanish but gauge potentials cannot be gauge transformed to zero globally: topological quantum field theories represent basic example of this.
- Clearly, the vanishing of the induced gauge fields is needed to obtain equivalence with ordinary induced Dirac equation. Therefore also string world sheets in Minkowskian regions should have 1-D CP2 projection rather than only having vanishing W fields if one requires that octonionic representation is equivalent with the ordinary one. For CP2 type vacuum extremals electroweak charge matrices correspond to quaternions, and one might hope that one can avoid problems due to non-associativity in the octonionic Dirac equation. Unless this is the case, one must assume that string world sheets are restricted to Minkowskian regions. Also imbedding space spinors can be regarded as octonionic (possibly quaternionic or co-quaternionic at space-time surfaces): this might force vanishing 1-D CP2 projection.
- Induced spinor fields would be localized at 2-surfaces at which they have no interaction with weak gauge fields: of course, also this is an interaction albeit very implicit one! This would not prevent the construction of non-trivial electroweak scattering amplitudes since boson emission vertices are essentially due to re-groupings of fermions and based on topology change.
- One could even consider the possibility that the projection of string world sheet to CP2 corresponds to CP2 geodesic circle at which also the induced gauge potentials vanish so that one could assign light-like 8-momentum to entire string world sheet, which would be minimal surface in M4× S1. This would mean enormous technical simplification in the structure of the theory. Whether the spinor harmonics of imbedding space with well-defined M4 and color quantum numbers can co-incide with the solutions of the induced Dirac operator at string world sheets defined by minimal surfaces remains an open problem.
- String world sheets cannot be present inside wormhole contacts, which have 4-D CP2 projection so that string world sheets cannot carry vanishing induced gauge fields. Therefore the strings in TGD are open.
To sum up, the generalization of the notion of twistor to 8-D context allows description of massive particles using twistors but requires that octonionic Dirac equation is introduced. If one requires that octonionic and ordinary description of Dirac equation are equivalent, the description is possible only at surfaces having at most 1-D CP2 projection - geodesic circle for the most stringent option. The boundaries of string world sheets are such surfaces and also string world sheets themselves if they have 1-D CP2 projection, which must be geodesic circle if also induce gauge potentials are required to vanish. In spirit with M8-H duality, string boundaries give rise to classical M8 twistorizalization analogous to the standard M4 twistorialization and generalize 4-momentum to massless 8-momentum whereas imbedding space spinor harmonics give description in terms of four-momentum and color quantum numbers. One has SO(4)-SU(3) duality: a wave function in the space of 8-momenta corresponds to SO(4) description of hadrons at low energies as opposed to that for quarks at high energies in terms of color. The M4 projection of the 8-D M8 momentum must by quantum classical correspondence be equal to the four-momentum assignable to imbedding space-spinor harmonics serving as building bricks for various super-conformal representations. This is nothing but Equivalence Principle (EP) in the most concrete form: gravitational four-momentum equals to inertial four-momentum.
EP for internal quantum numbers is clearly more delicate. In twistorialization also helicity is brought and for CP2 degrees of freedom M8 helicity means that electroweak spin is described in terms of helicity.
Biologists have a principle known as "ontogeny recapitulates phylogeny" (ORP) stating that the morphogenesis of the individual reflects evolution of the species. The principle seems to be realized also in theoretical physics - at least in TGD Universe. ORP would now say that the evolution of theoretical physics via the emergence of increasingly complex notion of particle reflects the structure physics itself. Point like particles are really there as points at partonic 2-surfaces carrying fermion number: their 1-D orbits correspond to the boundaries of string world sheets; 2-D hyper-complex string world sheets in flat space (M4× S1) are there and carry induced spinors; also complex (or co-complex) partonic 2-surfaces (Euclidian string world sheets) and carry particle numbers; 3-D space-like surfaces at the ends of causal diamonds (CDs) and the 3-D light-like orbits of partonic 2-surfaces are there; 4-D space-time surfaces are there as quaternionic or co-quaternionic sub-manifolds of 8-D octonionic imbedding space: there the hierarchy ends since there are no higher-dimensional classical number fields. ORP would thus also realize evolution of mathematics at the level of physics.
The M4 projection of the 8-D M8 momentum must by quantum classical correspondence
be equal to the four-momentum assignable to imbedding space-spinor harmonics serving as building bricks
for various super-conformal representations. This is nothing but Equivalence Principle in the most concrete form: gravitational four-momentum equals to inertial four-momentum.
See the chapter Classical part of the twistor story or the article Classical part of the twistor story.