Tuesday, February 20, 2024

A fresh look at M8-H duality and Poincare invariance

M8-H duality is a proposal to integrate geometric and number theoretic visions of TGD. M8-H duality has several questionable features. For various reasons it seems that M8 must be replaced with its complexification M8c interpreted as complexified octonions Oc. This however leads to several problems. The modified variant of M8-H duality identifying M8 as a quaternionic sub-space of octonions O with a number theoretic norm defined by Re(o2), rather than oo*, solves these problems.

The proposal has been that octonionic polynomials P(o) define the number theoretic holography. Their roots would define 3-D mass shells for which mass squared values are in general complex and the initial data for the holography would correspond to 3-surfaces at these mass shells. Also this assumption has problems. There is however no need for this assumption: the holography on the H side is induced by the M8-H duality!

The hierarchy of polynomials defines a hierarchy of algebraic extensions defining an evolutionary hierarchy central for all applications of TGD and one must have it. Luckily, the recent realization that a generalized holomorphy realizes the holography at the H side as roots for pairs of holomorphic functions of complex (in generalized sense) coordinates of H comes to rescue. It can be strengthened by assuming that the functions form a hierarchy of pairs of polynomials.

Twistor lift strongly suggests that M4 and space-time surfaces allow a Kähler structure and what I call Hamilton-Jacobi structure. These structures force a breaking of Poincare and even Lorentz invariance unless they are dynamically generated. It indeed turns out that M8-H duality generates them dynamically.

See the article A fresh look at M8-H duality and Poincare invariance or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

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