Holography= holomorphy hypothesis, small primes viz. large p-adic primes and p-adic length scale hypothesis

Prime polynomials and complexity hierarchy

We have had highly interesting Facebook discussions with Robert Paster relating to the p-adic mathematics (see this). Robert Paster finds the so called Universal Witt Vectors (UWVs) as a more promising approach than p-adic numbers whereas I am a proponent of adelic physics in which all p-adic number fields and also a hierarchy of their extensions induces by the extensions of rationals are in a central role.

UWVs generalize the Witt Vectors (WVs) (see this), which provide a representation of p-adic number field in terms of Witt polynomials with order, which is power of prime p and pinary expansion can be represented in terms of the hierarchy if Witt vectors W_k for which sum and product are element-wise operations. The UWVs are defined for any positive integer n and Wn codes for all factors of integer n and reduces to a collection of W_p^k when n is restricted to n=_p^k: it is denoted by W_k in this case.

An interesting question is whether UWVs as a number theoretic notion might have some role in TGD.

These discussions led to some progress in the longstanding attempts to understand how small primes and p-adic primes emerging from p-adic mass calculations and very large compared to them could relate and what the interpretation of the small primes could be. In the sequel I will discuss this aspect rather than UWVs of WVs.

The polynomials (P₁,P₂) and also the rational functions (g₁=P₁/Q₁,g₂=P₂/Q₂) appearing the holography= holomorphy vision (see this) form a well-defined complexity hierarchy.

1. In the general case, the space-time surfaces (P₁,P₂)=(0,0) have several disjoint components. This is the case if (f₁,f₂) is a composite function of form f=g(h): in other words one has (f₁,f₂)= (g₁(h₁,h₂),g₂(h₁,h₂)). The space-time surfaces correspond to roots h_i=r_i, which are disjoint.

To avoid disjoint union of space-time surfaces f_i must be a prime polynomial with respect to functional composition. For the polynomials of a single variable, this is the case if the degree of the polynomial is prime but this is not a necessary condition for primeness. As already found, this condition generalizes to the polynomials of 3 complex variables considered in the recent case.

Space-time surfaces of these kinds are excellent candidates for fundamental objects and the polynomial in question would have prime degree with respect to each of the 3 complex coordinates of H: this would make 3, presumably small primes. The composites formed of maps g and of these fundamental function pairs f would define cognitive representations of the surface defined by f as kind of statements about statements. An interesting question is whether these surfaces could correspond to elementary particles.

If this is the case, p∼ 2^k and k would tell about cognitive abilities of an electron and not so much about the system characterized by the function pair (f₁,f₂) at the bottom. Could the 2^k disjoint space-time surfaces correspond to a representation of p∼ 2^k binary numbers represented as disjoint space-time surfaces realizing binary mathematics at the level of space-time surfaces? This representation brings in mind the totally discontinuous compact-open p-adic topology. Cognition indeed decomposes the perceptive field into objects.

See the article A more detailed view about the TGD counterpart of Langlands correspondence or the chapter About Langlands correspondence in the TGD framework.

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.