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.

1. 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 f_i are analytic functions of one hypercomplex and 3 complex coordinates of H.

2. Second realization is for the spectrum generating algebra defined by the functional compositions g_º f, where g: C²→ C² is analytic function of 2 complex variables. The interpretation is as a cognitive hierarchy of function of functions of .... and the pairs (f₁,f₂) which do not allow a composition of form f=g_º h correspond to elementary function and to the lowest levels of this hierarchy, kind of elementary particles of cognition. Also the pairs g can be expressed as composites of elementary functions.

   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 r_i 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.

3. The quantum generalization of integers, rationals and algebraic numbers to their functional counterparts is possible for maps g: C²→ C². The counterpart of the ordinary product is functional composition _º for maps g. Degree is multiplicative in _º. In sum, call it +_e, the degree should be additive, which leads to the identification of the sum +_e as an element-wise product. The neutral element 1_º of _º is 1_º=Id and the neutral element 0_e of +_e is the ordinary unit 0_e=1.

   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= h_º i, can be identified as functional primes and have prime degree. f:H→ C² is prime if it does not allow composition f= g_º h. Functional integers are products of functional primes g_p.

   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 g_p in the product commute. Functional integers/rationals can be mapped to ordinary by a morphism mapping their degree to integer/rational.

4. One can define functional polynomials P(X), quantum polynomials, using these operations. In P(X), the terms p_nº X^_ºn, p_n and X should commute. The sum ∑_e p_nXⁿ corresponds to +_e. The zeros of functional polynomials satisfy the condition P(X)=0_e=1 and give as solutions roots X_k as functional algebraic numbers. The fundamental theorem of algebra generalizes at least formally if X_k and X commute. The roots have representation as a space-time surface. One can also define functional discriminant D as the _º product of root differences X_k-_e X_l, with -_e identified as element-wise division and the functional primes dividing it have space-time surface as a representation.

What about functional p-adics?

1. 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.

2. Large powers of prime appearing in p-adic numbers must approach 0e with respect to the p-adic norm so that gPºn must effectively approach Id with respect toº. Intuitively, a large n in gPºn corresponds to a long p-adic length scale. For large n, gPºn cannot be realized as a space-time surface in a fixed CD. This would prevent their representation and they would correspond to 0e and Id. During the sequence of SSFRs the size of CD increases and for some critical SSFRs a new power can emerge to the quantum p-adic.
3. Universal Witt polynomials W_n define an alternative representation of p-adic numbers reducing the multiplication of p-adic numbers to elementwise product for the coefficients of the Witt polynomial. The roots for the coefficients of W_n define space-time surfaces: they should be the same as those defined by the coefficients of functional p-adics.

There are many open questions.

1. 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 f_i: H→ C², i=1,2 and the generalization of the hierarchy can be assigned to the composition hierarchy of prime maps g_p.
2. >Could the transitions f→ g_º 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.

3. What is the interpretation of the maps g^-1 which in general are many-valued algebraic functions if g is rational function? g increases the complexity but g^-1 preserves or even reduces it so that its action is entropic. Could selection between g and g^-1 relate to a conscious choice between good and evil?
4. Could one understand the p-adic length scale hypothesis in terms of functional primes. The counter for functional Mersenne prime would be g2ºn/g1, where division is with respect to elementwise product defining +e? For g2 and g3 and also their iterates the roots allow analytic expression. Could primes near powers of g2 and g3 be cognitively very special?

See the article A more detailed view about the TGD counterpart of Langlands correspondence 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.