Can TGD Predict the Value of Newton's Constant?

Abstract

Newton's constant G cannot be a fundamental constant in TGD framework, where CP₂ radius R and Kahler coupling strength as the analog of fine structure constant are the fundamental constants. Dimensionally G corresponds to R₂/ħ. This gives guidelines for predicting G. TGD predicts a hierarchy of effective Planck constants h_eff/h₀=n, where n is the order of Galois group of Galois extension defining extension of rationals. Dimension n factorizes to a product n = n₁ n₂... for extension E₁ of extension E₂ of .... rationals. M⁸-H correspondence allows to associate the Galois group with an irreducible polynomial characterizing space-time surface as an algebraic surface in M⁸. The gradual increase of extension by forming a functional composite of a new polynomial with the already existing one(P - > P_new o P) would be analogous to the evolution of genome: earlier extensions would be analogous to conserved genes. The proposal modifying the earlier proposal is G = R²/ n_gr ħ₀, where n_gr is the order of Galois group G_gr at the bottom" of the hierarchy of extensions, and one has ħ= 6h₀. One would have n = n₁ n₂......n_gr. G_gr "at the bottom" is proposed to represented number theoretically geometric information about the imbedding space by providing a discretization for the product of maximal finite discrete sub-group of isometries and tangent space rotations of imbedding space. By M⁸-H duality these sub-groups should be identical for H and M⁸. The prediction is that maximal G_gr is product of icosahedral group I with 3 copies of coverings I^-. Rather remarkably, the prediction for G is correct if one assumes that the value of R is what p-adic mass calculation for electron mass gives. Since the hierarchy of Planck constants relates to number theoretical physics proposed to describe the correlates of cognition, the connection with cognition strongly suggests itself. Icosahedral and tetrahedral geometries occur also in the TGD based model of genetic code in terms of bio-harmony, which suggests that genetic code represents geometric information about imbedding space symmetries. These connections are discussed in detail.