Could one generalize the notion of twistor to 8-D case using the notion of triality?

The basic problem of the twistorial approach is that one cannot represent massive momenta in terms of twistors in elegant manner. I have proposed a possible representation of massive states based on the existence of preferred plane of M² in the basic definition of theory allowing to express four-momentum as some of two light-like momenta allowing twistor description. One could however ask whether some more elegant representation of massive M⁴ momenta might be possible by generalizing the notion of twistor -perhaps by starting from the number theoretic vision.

Octo-twistors

The basic observation is that massless M⁴ momenta representable as matrices p^kσ_k can be interpreted as non-invertible hyper-quaternions since the chirality condition effectively reduces gamma matrices to sigma matrices representing hyper-quaternionic units obeying instead of anticommutation relations the multiplication table of hyper quaternions. In the same manner massless M⁸ momenta can be mapped to non-invertible hyper-octonions and one could argue that gamma matrices are replaced with hyper-octonion units with modified algebra when chirality condition holds true and having interpretation as separate conservation of lepton and quark numbers. One could map any massive M⁴ momentum to a light-like M⁸ momentum - maybe this association could be made in unique manner. The questions are whether one could assign to a massless M⁸ momentum 8-D spinor and its conjugate such that all three 8-D representations of SO(7,1) together define 26-D object that might be called octo-twistor and obtain 8-D massless Dirac equation as a consequence.

1. In M⁴×CP₂ one has massless states in the sense that mass in 8-D sense vanishes for spinor modes. Same applies in the dual description based on hyper-octonionic M⁸ containing space-time surfaces as hyper-quaternionic surfaces. The expression of M⁴ point as a combination of Pauli sigma matrices can be regarded as hyper quaternion and in the massless case one has expression in terms of spinor and its conjugate: spinor would be a kind of square root of hyper-quaternion existing because an associative algebra is in question. Twistors would be in one-one correspondence with hyper-quaternions for which the inverse fails to exist. Light-like 3-surfaces would be surfaces for which tangent space contains massless hyper-quaternion as on tangent vector allowing representation in terms of a twistor.

2. The points of 8-D Minkowski space can be identified as hyper-octonions. Light-like hyper-octonions are well defined and masslessness means the failure of the number field property. One cannot represent (hyper-)octonions as matrices. Neither can one identify (hyper-)octonions as gamma matrix algebra. Imaginary hyper-octonion units however define a representation of 7-D algebra of space-like gamma matrices γ_k with respect to anti-commutator defined in terms of hyper-octonion multiplication. One can however argue that in the presence of chirality constraint gamma matrices are replaced with 8-D variants of sigma matrices obeying the multiplication rules of hyper-octonion units.

3. Although matrix representation does not exist, one can express the action of hyper-octonion h=h^ke_k on hyper-octonion units as a matrix h_mn=h^kf_kmn. This matrix has a vanishing determinant for light-like hyper-octonions. For N=8 both vector representation and spinor representation and its conjugate for SO(N-1,1) are 8-dimensional. The conjugate of this matrix annihilates an 8-component quantity having h^k as components just as p_kσ^k annihilates in right multiplication the corresponding spinor defining twistor and the natural interpretation is as 8-spinor analogous to Majorana spinor. Any spinor obtained by a phase multiplication satisfies the same condition. An analog of Dirac equation would be in question. Massless Dirac equation implies ΨbarΨ = 0 condition meaning definite M⁸-chirality: that is well-defined quark or lepton number.

4. The condition for the octo-twistor makes sense also for ordinary spinors and the explicit representation can be obtained by using triality. The ansatz is p^k= Ψbarγ^kΨ. The condition p^kp_k=0 gives Dirac equation p^kγ_kΨ=0 and its conjugate solved by Ψ=p^kγ_kΨ₀. The expression of p^k in turn gives the normalization condition Ψbar₀γ^kp_kΨ₀=1/2. Without further conditions almost any Ψ₀ not annihilated by γ^kp_k is possible solution. One can map the spinor basis to hyper-octonion basis and assume Ψ₀→ 1=σ₀. This would give Ψ=p^kγ_kΨ₀ → p^kσ_k so that Ψ and p^k would correspond to each other in 1-1 manner apart from the phase factor of Ψ.

5. A highly unique choice for Ψ₀ is the covariantly constant right-handed neutrino spinor of M⁴× CP₂ giving also rise to super-conformal symmetry. The choice is unique apart from SO(3) rotation but the condition that spin eigenstate is in question for the choice of quantization axis fixed by the choice of hyper-octonion units and also by the definition of the hierarchy of Planck constants fixes Ψ₀ apart from the sign of the spin if reality is assumed. When p^kγ_k Ψ₀=0 holds true for fixed Ψ₀, the ansatz fails so that the gauge choice is not global. There are two gauge patches corresponding to the two signs of the spin of Ψ₀. Right handed neutrino spinor reflects directly the homological magnetic monopole character of the Kähler form of CP₂ so that the monopole property is in well defined sense transferred from CP₂ to M⁴. Note that this argument fails for quark spinors which do not allow any covariantly constant spinor.

6. Could one assign to this spinor a twistor like entity? For ordinary twistors the existence of the antisymmetric tensor ε acting as Kähler form in the space of spinors is what allows to define second spinor and these spinors together form twistor. What is essential is that 2-D spinor and its conjugate as a representation of Lorentz group define twistor. In an analogous manner M⁸ vector, M⁸-spinor, and its conjugate define a triplet of perhaps deserving interpretation as octo-twistor. Together they would form an entity with 24 components when the overall complex phase is eliminated and if no gauge choice fixing Ψ₀ is made apart from the assumption Ψ₀ has real components. If the overall phase is allowed, the number of components is 26 (the momentum constraint of course reduces the number of degrees of freedom to 8). It seems that the magic dimensions of string models are unavoidable! Perhaps it might be a possible to reduce 26-D string theory to 8-D theory by posing triality symmetry and additional gauge symmetry.

7. One could speak also about the imbedding of the light-like hyper-quaternion twistor structure at the light-like 3-surface to the level of imbedding space in the sense that hyper-quaternion units are expressible as combinations of hyper-octonion units and the imbedding space spinor restricted to space-time surface behaves like twistor. Massless momenta in 8-D sense would have representation in terms of octo-twistors and M⁴ momentum with any value of mass could be lifted to light-like M⁸ momentum and one could assign with it octo-twistor as M⁸ or H-spinor.

Could right handed neutrino spinor modes define octo-twistors?

There is no absolute need to interpret induced spinor fields as parts of octo-twistors One can however ask whether this might make sense for the solutions of the modified Dirac equation DΨ = 0 representing right-handed neutrino and expressible as Ψ = DΨ₀.

1. In the modified Dirac equation gamma matrices are replaced by the modified gamma matrices defined by the variation of Kähler action and the massless momentum p^kσ_k is replaced with the modified Dirac operator D. In plane wave basis the derivatives in D reduce to an algebraic multiplication operators in the case of right handed neutrino since right-handed neutrino has no gauge couplings.

2. A non-trivial consistency condition comes from the condition D²Ψ₀=0 giving sum of two terms.
   1. The first term is the analog of scalar d'Alembertian and given by

      G^μνD_μD_νΨ₀ ,

      G^μν=h_klT^{μ k}T^{ν l}
      T^μk= ∂ L_K/&part h^k_α ,

      and has quantum numbers of right handed neutrino as it should.

   2. Second term is given by

      T^μkD_μT^{ν l}Σ_klD_νΨ₀ ,

      and in the general case contains charged components. Only electromagnetically neutral CP₂ sigma matrices having right handed neutrino as eigen state are allowed if one wants twistor interpretation. This is not be true in the general case but might be implied by the preferred extremal property.

   3. This property would allow to choose the induced spinor fields to be eigenstates of electromagnetic charge globally and would be therefore physically very attractive. After all, one of the basic interpretational problems has been the fact that classical W fields in the general situation induce mixing of quarks and leptons with different electro-magnetic charges. If this is the case one could assign to each point of the space-time surface octo-twistor like abstract entity as the triplet (Ψbar₀D,D,DΨ₀). This would map space-time sheet to a 4-D surface (in real sense) in the space of 8-D (in complex sense) leptonic spinors.

Hyper-octonionic Pauli matrices and definition of hyper-quaternicity

Hyper-octonionic Pauli matrices suggest an interesting possibility to define precisely what hyper-quaternionicity means at space-time level.

1. The proposal has been that space-time surface X⁴ is hyper-quaternionic if the tangent space at each point of X⁴ in X⁴ subset M⁸ picture is hyper-quaternionic. What raises worries is that this definition involves in no manner the action principle so that it is far from obvious that this identification is consistent with the vacuum degeneracy of Kähler action. It also unclear how one should formulate hyper-quaternionicity condition in X⁴ subset M⁴×CP₂ picture.

2. The idea is to map the modified gamma matrices γ^α=[(∂L_K)/(∂h^k_α)]γ^k, γ_k = e^A_kγ_A, to hyper-octonionic Pauli matrices σ^α by replacing γ_A with hyper-octonion unit. Hyper-quaternionicity would state that the hyper-octonionic Pauli matrices σ^α obtained in this manner span complexified quaternion sub-algebra at each point of space-time. These conditions would provide a number theoretic manner to select preferred extremals of Kähler action. Remarkably, this definition applies both in case of M⁸ and M⁴×CP₂.

3. Modified Pauli matrices span the tangent space of X⁴ if the action is four-volume because one has [(∂L_K)/(∂h^k_α)]=g^1/2 g^αβ∂h^l_βh_kl. Modified gamma matrices reduce to ordinary induced gamma matrices in this case: 4-volume indeed defines a super-conformally symmetric action for ordinary gamma matrices since the mass term of the Dirac action given by the trace of the second fundamental form vanishes for minimal surfaces.

4. For Kähler action the hyper-quaternionic sub-space does not coincide with the tangent space since [(∂L_K)/(∂h^k_α)] contains besides the gravitational contribution coming from the induced metric also the "Maxwell contribution" from the induced Kähler form not parallel to space-time surface. Modified gamma matrices are required by super conformal symmetry for the extremals of Kähler action and they also guarantee that vacuum extremals defined by surfaces in M⁴×Y², Y² a Lagrange sub-manifold of CP₂, are trivially hyper-quaternionic surfaces. The modified definition of hyper-quaternionicity does not affect in any manner M⁸<--> M⁴×CP₂ duality allowing purely number theoretic interpretation of standard model symmetries.

Could Majorana condition allow some analog in TGD framework?

Majorana condition plays key role in super string models and fixes the dimension of the imbedding space to D=10 in the original formulation of super string models. Hence one can ask whether Majorana condition could have some analog in TGD framework.

1. The number theoretic definition of 8-spinors gives automatically the analogs of Majorana spinors but that these appear only at the twistor level and define the analogs of real twistors encountered in twistor diagrammatics and allowing twistor Fourier transform. Unfortunately, p^kγ_kΨ₀ is not a real spinor in the general case. Despite this there are excellent hopes about an elegant generalization of twistor diagrammatics allowing massive particles. This would be decisive for the twistor formulations of both the low energy theory with massive particles and for the formulation of quantum TGD involving an infinite tower of massive excitations of massless states.

2. Majorana condition in 8-D sense as Ψ^*=-iΨ^TC for quantized spinor fields is not consistent with well-defined baryon and lepton numbers. Neither can one pose it on the modes of the modified Dirac operator as Ψ^* = iγ₀CΨ in Dirac representation with γ₀ = σ³⊗σ₀ and C = -iσ¹⊗σ² since the transformation changes the H-chirality of the spinor.

This representation can be found also from the last section of the chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of "Towards M-matrix".