New Aspects of M8-H Duality

M⁸-H duality (H=M⁴× CP₂) has become one of central elements of TGD. M⁸-H duality implies two descriptons for the states.

1. M⁸-H duality assumes that space-time surfaces in M⁸ have associative tangent- or normal space M⁴ and that these spaces share a common sub-space M²⊂ M⁴, which corresponds to complex subspace of octonions (also integrable distribution of M²(x) can be considered). This makes possible the mapping of space-time surfaces X⁴⊂ M⁸ to X⁴⊂ H=M⁴× CP₂) giving rise to M⁸-H duality.

2. M⁸-H duality makes sense also at the level of 8-D momentum space in one-one correspondence with light-like octonions. In M⁸=M⁴× E⁴ picture light-like 8-momenta are projected to a fixed quaternionic M⁴_T⊂ M⁸. The projections to M⁴_T⊃ M² momenta are in general massive. The group of symmetries is for E⁴ parts of momenta is Spin(SO(4))= SU(2)_L× SU(2)_R and identified as the symmetries of low energy hadron physics.

   M⁴⊃ M² can be also chosen so that the light-like 8-momentum is parallel to M⁴_L⊂ M⁸. Now CP₂ codes for the E⁴ parts of 8-momenta and the choice of M⁴_L and color group SU(3) as a subgroup of automorphism group of octonions acts as symmetries. This correspond to the usual description of quarks and other elementary particles. This leads to an improved understanding of SO(4)-SU(3) duality. A weaker form of this duality S³-CP₂ duality: the 3-spheres S³ with various
   radii parameterizing the E⁴ parts of 8-momenta with various lengths correspond to discrete set of 3-spheres S³ of CP₂ having discrete subgroup of U(2) isometries.

3. The key challenge is to understand why the MacKay graphs in McKay correspondence and principal diagrams for the inclusions of HFFs correspond to ADE Lie groups or their affine variants. It turns out that a possible concrete interpretation for the hierarchy of finite subgroups of SU(2) appears as discretizations of 3-sphere S³ appearing naturally at M⁸ side of M⁸-H duality. Second interpretation is as covering of quaternionic Galois group. Also the coordinate patches of CP₂ can be regarded as piles of 3-spheres and finite measurement resolution. The discrete groups of SU(2) define in a natural manner a hierarchy of measurement resolutions realized as the set of light-like M⁸ momenta. Also a concrete interpretation for Jones inclusions as inclusions for these discretizations emerges.
4. A radically new view is that descriptions in terms of massive and massless states are alternative options leads to the interpretation of p-adic thermodynamics as a completely universal massivation mechanism having nothing to do with dynamics. The problem is the paradoxical looking fact that particles are massive in H picture although they should be massless by definition. The massivation is unavoidable if zero energy states are superposition of massive states with varying masses. The M⁴_L in this case most naturally corresponds to that associated with the dominating part of the state so that higher mass contributions can be described by using p-adic thermodynamics and mass squared can be regarded as thermal mass squared calculable by p-adic thermodynamics.
5. As a side product emerges a deeper understanding of ZEO based quantum measurement theory and consciousness theory. 4-D space-time surfaces correspond to roots of octonionic polynomials P(o) with real coefficients corresponding to the vanishing of the real or imaginary part of P(o).

These polynomials however allow universal roots, which are not 4-D but analogs of 6-D branes and having topology of S⁶. Their M⁴ projections are time =constant snapshots t= r_n,r_M≤ r_n 3-balls of M⁴ light-cone (r_n is root of P(x)). At each point the ball there is a sphere S³ shrinking to a point about boundaries of the 3-ball.

What suggests itself is following “braney” picture. 4-D space-time surfaces intersect the 6-spheres at 2-D surfaces identifiable as partonic 2-surfaces serving as generalized vertices at which 4-D space-time surfaces representing particle orbits meet along their ends. Partonic 2-surfacew would define the space-time regions at which one can pose analogs of boundary values fixing the space-time surface by preferred extremal property. This would realize strong form of holography (SH): 3-D holography is implied already by ZEO.

This picture forces to consider a modification of the recent view about ZEO based theory of consciousness. Should one replace causal diamond (CD) with light-cone, which can be however either future or past directed. “Big” state function reductions (BSR) meaning the death and re-incarnation of self with opposite arrow of time could be still present. An attractive interpretation for the moments t=r_n would be as moments assignable to “small” state function reductions (SSR) identifiable as “weak” measurements giving rise to to sensory input of conscious entity in ZEO based theory of consciousness. One might say that conscious entity becomes gradually conscious about its roots in increasing order. The famous question “What it feels to be a bat” would reduce to “What it feels to be a polynomial?”! One must be however very cautious here.

See the article New Aspects of M⁸-H Duality.

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

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