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Is M8 -H Duality Consistent with Fourier Analysis at the Level of M4 x CP2?

Matti Pitkänen


M8-H duality predicts that space-time surfaces as algebraic surfaces in complexified M8 (complexified octonions) determined by polynomials can be mapped to H = M4 x CP2. The polynomials do not involve periodic functions typically associated with the minimal space-time surfaces in H. Since M8 is analogous to momentum space, the periodicity is not needed. However, the representation of the space-time surfaces in H obey dynamics and the H-images of X4M8 should involve periodic functions and Fourier analysis for CP2 coordinates as functions of M4 coordinates. Neper number, and therefore trigonometric and exponential functions are p-adically very special. As a consequence, Fourier analysis extended to allow exponential functions in the case of hyperbolic signature is a number theoretically universal concept making sense also for p-adic number fields. The proposal is that by its non-locality, the map of the tangent space of the space-time surface X4M8 to CP2 point brings in dynamics and therefore requires the expansion of CP2 coordinates as exponential and trigonometric functions of M4 coordinates that is Fourier analysis. The connection with hierarchy of effective Planck constants and the implications for the  quantum model of cognitive process are considered.

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