## About the twistorial description of light-likeness in 8-D sense using octonionic spinors
The twistor approach to TGD require that the expression of light-likeness of M
M 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. - 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 M
^{4}and E^{4}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 (γ
_{0}=σ_{z}× 1, γ_{k}= σ_{y}× I_{k}) but the octonionic sigma matrices represented by octonions span the Lie algebra of G_{2}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 M^{8}saves the situation. - One obtains the square of p
^{2}=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 M^{8}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 E
^{4}momentum correspond to the imaginary part of four-momentum? - The signature causes the first problem: M
^{8}must be replaced with complexified Minkowski space M_{c}^{4}for to make sense but this is not an attractive idea although M_{c}^{4}appears as sub-space of complexified octonions. - For the extremals of Kähler action Euclidian regions (wormhole contacts identifiable as deformations of CP
_{2}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.
What about twistorialization in the case of M - For H=M
^{4}× CP_{2}the spinor connection of CP_{2}is not trivial and the G_{2}sigma matrices are proportional to M^{4}sigma matrices and act in the normal space of CP_{2}and to M^{4}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 CP
_{2}projection rather than only having vanishing W fields if one requires that octonionic representation is equivalent with the ordinary one. For CP_{2}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 CP_{2}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 CP
_{2}corresponds to CP_{2}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 M^{4}× S^{1}. This would mean enormous technical simplification in the structure of the theory. Whether the spinor harmonics of imbedding space with well-defined M^{4}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 CP
_{2}projection so that string world sheets cannot carry vanishing induced gauge fields. Therefore the strings in TGD are open.
Summarizing
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 CP
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 (M
The M See the chapter Classical part of the twistor story or the article Classical part of the twistor story. |