About some number theoretical aspects of TGD

Recently a considerable progress has occurred in the understanding of number theoretic aspects of quantum TGD. I have discussed these aspects in earlier posts but it is useful to collect them together.

1. There are reasons to think that TGD could be formulated purely number theoretically without introduction of any action principle. This would conform with the M⁸-H duality and the generalization of the geometric Langlands correspondence to dimension D=4.

   Number theoretic vision however gives extremely powerful constraints on the vacuum functional suggesting even an explicit formula for it. The condition that this expression corresponds to the exponent of Kähler function expressible as Kähler action fixes the coupling constant evolution for the action.

2. Extensions of rationals, the corresponding Galois groups and ramified primes assignable to polynomials and identifiable as p-adic primes assigned to elementary particles are central notions of quantum TGD. In the recent formulation based on holography = holomorphy principle, it is not quite clear how to assign these notions to the space-time surfaces. The notion of Galois group has a 4-D generalization but can one obtain the ordinary Galois groups and ramified primes? Two ways to achieve this are discussed in this article.

One could introduce a hierarchy of 4-polynomials (f₁,f₂,f₃,f₄) instead oly (f₁,f₂) and the common roots all 4 polynomials as a set of discrete points would give the desired basic notions assignable to string world sheets.

One can also consider the maps (f₁,f₂)→ G( f₁,f₂)= (g₁(f₁,f₂), g₂(f₁,f₂)) and assign these notions to the surfaces (g₁(f₁,f₂), g₂(f₁,f₂))=(0,0).

3. Number theoretical universality is possible if the coefficients of the analytic functions (f₁,f₂) of 3 complex coordinates and one hypercomplex coordinate of H=M⁴× CP₂ are in an algebraic extension of rationals. This implies that the solutions of field equations make sense also in p-adic number fields and their extensions induced by extensions of rationals.

In this article the details of the adelicization boiling to p-adicization for various p-adic number fields, in particular those assignable to ramified primes, are discussed. p-Adic fractals and holograms emerge very naturally and the iterations of (f₁,f₂)→ G(f₁,f₂)= (g₁(f₁,f₂), g₂(f₁,f₂) define hierarchical fractal structures analogs to Mandelbrot and Julia fractals and p-adically mean exponential explosion of the complexity and information content of cognition. The possible relationship to biological and cognitive evolution is highly interesting.

See the article About some number theoretical aspects of TGD.

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.