Criticizing the notion of twistor space of M4
Twistor lift of TGD involves representation of space-time surfaces as 6-surfaces in twistor space of H having structure of S2 bundle over space-time surface resulting in dimensional reduction. These 6-surfaces would be holomorphic and thus minimal surfaces represented in terms of polynomials having same degree as the corresponding M8 octonionic polynomial by number theoretic universality.
- I have assumed that what I call geometric twistor space of M4 is simply M4× S2. It however turned out that one can consider standard twistor space CP3 with metric signature (3,-3) as an alternative. This option reproduces the nice results of the earlier approach but the philosophy is different: there is no fundamental length scale but the hierarchy of causal diamonds (CDs) predicted by zero energy ontology (ZEO) gives rise to the breaking of the exact scaling invariance of M8 picture. This forces to modify M8-H correspondence so that it involves map from M4 to CP3 followed by a projection to hyperbolic variant of CP2.
M4 in H would not be replaced with conformally compactified M4 (M4conf) but conformally compactified causal diamond cd (cdconf) of M4 for which a natural identification is as CP2 with second complex coordinate replaced with hypercomplex coordinate. The sizes of twistor spaces of cdconf using CP2 size as unit would reflect the hierarchy of size scales for CDs. The consideration on the twistor space of M8 in similar picture leads to the identification of corresponding twistor space as HP3 - quaternionic variant of CP3: the counterpart of CD8 would be HP2.
- Octotwistors can be expressed as pairs of quaternionic twistors. Octotwistor approach suggests a generalization of twistor Grassmannian approach obtained by replacing the bi-spinors with complexified quaternions and complex Grassmannians with their quaternionic counterparts. Although TGD is not a quantum field theory, this proposal makes sense for cognitive representations identified as discrete sets of spacetime points with coordinates in the extension of rationals defining the adele implying effective reduction of particles to point-like particles.
- The outcome of octo-twistor approach together with M8-H duality leads to a nice picture view about twistorial description of massive states based on quaternionic generalization of twistor Grassmannian approach. A radically new view is that descriptions in terms of massive and massless states are alternative options, and correspond to two different alternative twistorial descriptions and leads to the interpretation of p-adic thermodynamics as completely universal massivation mechanism having nothing to do with dynamics. As a side product emerges a deeper understanding of ZEO based quantum measurement theory and consciousness theory relying on the universal roots of octonionic polynomials of M8, which are not 4-D but analogs of 6-D branes. By M8-H duality the finite sub-groups of SU(2) of McKay correspondence appear quite concretely in the description of the measurement resolution of 8-momentum.
What about super-twistors in TGD framework?
- The parallel progress in the understanding SUSY in TGD framework in turn led to the identification of the super-counterparts of M8, H and of twistor spaces modifying dramatically the physical interpretation of SUSY. Super-spinors in twistor space would provide the description of quantum states. Super-Grassmannians would be involved with the construction of scattering amplitudes. Quaternionic super Grassmannians would be involved with M8 description.
- The great surprise from physics point of view is that in fermionic sector only quarks are allowed by SO(1,7) triality and that anti-leptons are local 3-quark composites identifiable as spartners of quarks. Gauge bosons, Higgs and graviton would be also spartners and assignable to super-coordinates of imbedding space expressible as super-polynomials of quark oscillator operators. Super-symmetrization means also quantization of fermions allowing local many-quark states.
- SUSY breaking would be caused by the same universal mechanism as ordinary massivation of massless states. The mass formulas would be supersymmetric but the choice of p-adic prime identifiable as ramified prime of extension of rationals would depend on the state of super-multiplet. ZEO would make possible symmetry breaking without symmetry breaking as Wheeler might put it.
For a summary of earlier postings see Latest progress in TGD.