A critical re-examination of M

^{8}-H duality hypothesis led to what might be called a brealthrough in the understanding of M

^{8}-H duality. Several intuitive beliefs turned out to be wrong at the level of details as precise calculation became possible. The improved understanding led to a construction of scattering amplitudes at M

^{8} level as counterpart for momentum space representations of scattering amplitudes wheteras the earlier breakthrough was related to their construction at the level of H=M

^{4}× CP

_{2} (see

this).

Since the article grew rather long, I decided to divide it into two parts. Below are the absracts of these articles.

**A critical re-examination of M**^{8}-H duality hypothesis: part I

This article is the first part of an article representing a critical re-examination of M^{8}-H duality, which is one of the cornerstones of Topological Geometrodynamics (TGD). The original version of M^{8}-H duality assumed that space-time surfaces in M^{8} can be identified as associative or co-associative surfaces. If the surface has associative tangent or normal space and contains a complex or co-complex surface, it can be mapped to a 4-surface in H=M^{4}× CP_{2}.

Later emerged the idea that octonionic analyticity realized in terms of real polynomials P algebraically continued to polynomials of complexified octonion could fulfill the dream. The vanishing of the real part Re_{Q}(P) (imaginary part Im_{Q}(P)) in the quaternionic sense would give rise to an associative (co-associative) space-time surface.

The realization of the general coordinate invariance motivated the notion of strong form of holography (SH) in H allowing realization of a weaker form of M^{8}-H duality by assuming that associativity/co-associativity conditions are needed only at 2-D string world sheet and partonic 2-surfaces and possibly also at their light-like 3-orbits.

The outcome of the re-examination was a positive surprise. Although no interesting associative space-time surfaces are possible, every distribution of normal associative planes (co-associativity) is integrable. Another positive surprise was that Minkowski signature is the only possible option. Equivalently, the image of M^{4} as real co-associative sub-space of O_{c} (complex valued octonion norm squared is real valued for them) by an element of local G_{2,c} or its subgroup SU(3,c) gives a real co-associative space-time surface. The conjecture is that the polynomials P determine these surfaces as roots of Re_{Q}(P). These surfaces also possess co-complex 2-D sub-manifolds allowing the mapping to H to H by M^{8}-H duality as a whole. SH would not be needed and would be replaced with number theoretic holography determining space-time surface from its roots and selection of real subspace of O_{c} characterizing the state of motion of a particle. The equations for Re_{Q}(P)=0 reduce to simultaneous roots of ordinary real polynomials defined by the odd and even parts of P having interpretation as complex values of mass squared mapped to light-cone proper time constant surfaces in H.

Octonionic Dirac equation as analog of momentum space variant of ordinary Dirac equation forces the interpretation of M^{8} as an analog of momentum space and Uncertainty Principle forces to modify the map M^{4}⊂ M^{8}→ M^{4}⊂ H from identification to inversion. Contrary to the earlier expectations the space-time surface in M^{8} would be analogous to Fermi ball and mass squared sections would correspond to Fermi spheres. This leads to the idea that the formulation of scattering amplitudes at M^{8} levels provides the counterpart of momentum space description of scattering whereas the formulation at the level of H provides the counterpart of space-time description.

See the article A critical re-examination of M^{8}-H duality hypothesis: part I.

**A critical re-examination of M**^{8}-H duality hypothesis: part II

This article is the second part of an article representing a critical re-examination of M^{8}-H duality. This re-examination has yielded several surprises. The first surprise was that space-time surfaces in M^{8} must and can be co-associative so that they can be constructed also as as images of a map defined by local G_{2,c} (octonionic automorphisms) transformation applied to co-associative sub-space M^{4} of complexified octonions O_{c} in which the complexified octonion norm squared reduces to the real M^{4} norm squared. An alternative manner to construct them would be as roots for the real part Re_{Q}(P) of an octonionic algebraic continuation of a real polynomial P.

The outcome was an explicit solution expressing space-time surfaces in terms of ordinary roots of the real polynomial defining the octonionic polynomials. The equations for Re_{Q}(P)=0 reduce to simultaneous roots of the real polynomials defined by the odd and even parts of P having interpretation as complex values of mass squared mapped to light-cone proper time constant surfaces in H.

The second surprise was that space-time surface in M^{8} can be mapped to H as a whole so that the strong form of holography (SH) is not needed at the level of H being replaced with much stronger number theoretic holography at the level of M^{8}.

The third surprise was that octonionic Dirac equation as an analog of momentum space variant of ordinary Dirac equation forces the interpretation of M^{8} as an analog of momentum space and Uncertainty Principle forces to modify the map M^{4}⊂ M^{8}→ M^{4}⊂ H from identification to inversion. One obtains both massless quarks and massive quarks corresponding to two different number-theoretically characterized phases.

This picture combined with zero energy ontology (ZEO) leads also to a view about the construction of the scattering amplitudes at the level of M^{8} as analog of momentum space description of scattering amplitudes in quantum field theories. Local G_{2,c} element has properties suggesting a Yangian symmetry assignable to string world sheets and possibly also partonic 2-surfaces. The representation of Yangian algebra using quark oscillator operators would allow to construct zero energy states at representing the scattering amplitudes. The physically allowed momenta would naturally correspond to algebraic integers in the extension of rationals defined by P. The co-associative space-time surfaces (unlike generic ones) allow infinite-cognitive representations making possible the realization of momentum conservation and on-mass-shell conditions.

The new view about M^{8}-H duality differs from the earlier one rather dramatically so that a summary of the differences is added to the end of paper.

See the article A critical re-examination of M^{8}-H duality hypothesis: part II.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.