Holography= holomorphy vision and functional generalization of arithmetics and p-adic number fields
In TGD, geometric and number theoretic visions of physics are complementary. This complementarity is analogous to momentum position duality of quantum theory and implied by the replacement of a point-like particle with 3-surface, whose Bohr orbit defines space-time surface.
At a very abstract level this view is analogous to Langlands correspondence. The recent view of TGD involving an exact algebraic solution of field equations based on holography= holomorphy vision allows to formulate the analog Langlands correspondence in 4-D context rather precisely. This requires a generalization of the notion of Galois group from 2-D situation to 4-D situation: there are 2 generalizations and both are required.
- The first generalization realizes Galois group elements, not as automorphisms of a number field, but as analytic flows in H=M4× CP2 permuting different regions of the space-time surface identified as roots for a pair f=(f1,f2) of pairs f=(f1,f2): H→ C2, i=1,2. The functions fi are analytic functions of one hypercomplex and 3 complex coordinates of H.
If g1 and g2 are polynomials with coefficients in field E identified as an extension of rationals, one can assign to g º f root a set of pairs (r1,r2) as roots f1,f2)= (r1,r2) and ri are algebraic numbers defining disjoint space-time surfaces. One can assign to the set of root pairs the analog of the Galois group as automorphisms of the algebraic extension of the field E appearing as the coefficient field of (f1,f2) and (g1,g2). This hierarchy leads to the idea that physics could be seen as an analog of a formal system appearing in Gödel’s theorems and that the hierarchy of functional composites could correspond to a hierarchy of meta levels in mathematical cognition.
The inverse corresponds to g-1 for º, which in general is a many-valued algebraic function and to 1/g for times. The maps g, which do not allow decomposition g= hº i, can be identified as functional primes and have prime degree. f:H→ C2 is prime if it does not allow composition f= gº h. Functional integers are products of functional primes gp.
The non-commutativity of º could be seen as a problem. The fact that the maps g act like operators suggest that the functional primes gp in the product commute. Functional integers/rationals can be mapped to ordinary by a morphism mapping their degree to integer/rational.
- The functional powers gpº k of primes gp define analogs of powers of p-adic primes and one can define a functional generalization of p-adic numbers as quantum p-adics. The coefficients Xk Xkºgpk are polynomials with degree smaller than p. The sum +e so that the roots are disjoint unions of the roots of Xkºgpºk.
- The question whether the hierarchy of infinite primes has relevance to TGD has remained open. It turns out that the 4 lowest levels of the hierarchy can be assigned to the rational functions fi: H→ C2, i=1,2 and the generalization of the hierarchy can be assigned to the composition hierarchy of prime maps gp.
- >Could the transitions f→ gº f correspond to the classical non-determinism in which one root of g is selected? If so, the p-adic non-determinism would correspond to classical non-determinism. Quantum superposition of the roots would make it possible to realize the quantum notion of concept.
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.
Source: https://matpitka.blogspot.com/2025/04/holography-holomorphy-vision-and.html
