Dow could p-adicization and hyper-finite factors could relate?

Factors of type I are von Neumann algebras acting in the ordinary Hilbert space allowing a discrete enumerable basis. Also hyperfinite factors of type II₁ (HFFs in the sequel) play a central role in quantum TGD (see this and this). HFFs replace the problematic factors of type III encountered in algebraic quantum field theories. Note that von Neumann himself regarded factors of type III pathological.

HFFs have rather unintuitive properties, which I have summarized in (see this and this).

1. The Hilbert spaces associated with HFFs do not have a discrete basis and one could say that the dimension of Hilbert spaces associated with HFFs corresponds to the cardinality of reals. However, the dimension of the Hilbert space identified as a trace Tr(Id) of the unit operator is finite and can be taken equal to 1.
2. HFFs have subfactors and the inclusion of sub-HFFs as analogs of tensor factors give rise to subfactors with dimension smaller than 1 defining a fractal dimension. For Jones inclusions these dimensions are known and form a discrete set algebraic numbers. In the TGD framework, the included tensor factor allows an interpretation in terms of a finite measurement resolution. The inclusions give rise to quantum groups and their representations as analogs of coset spaces.

p-Adic numbers represent a second key notion of TGD.

1. p-Adic number fields emerged in p-adic mass calculations (see this, this and this and this). Their properties led to a proposal that they serve as correlates of cognition. All p-adic number fields are possible and can be combined to form adele and the outcome is what could be called adelic physics (see this and this).
2. Also the extensions of p-adic number fields induced by the extensions of rationals are involved and define a hierarchy of extensions of adeles. The ramified primes for a given polynomial define preferred p-adic primes. For a given space-time region the extension is assignable to the coefficients for a pair of polynomials or even Taylor coefficients for two analytic functions defining the space-time surface as their common root.
3. The inclusion hierarchies for the extensions of rationals accompanied by inclusion hierarchies of Galois groups for extensions of extensions of .... are analogous to the inclusion hierarchies of HFFs.

Before discussing how p-adic and real physics relate, one must summarize the recent formulation of TGD based on holography = holography correspondence.

1. The recent formulation of TGD allows to identify space-time surfaces in the imbedding space H=M⁴× CP₂ as common roots for the pair (f₁,f₂) of generalized holomorphic functions defined in H. If the Taylor coefficients of f_i are in an extension of rationals, the conditions defining the space-time surfaces make sense also in an extension of p-adic number fields induced by this extension. As a special case this applies to the case when the functions f_i are polynomials. For the completely Taylor coefficients of generalized holomorphic functions f_i, the p-adicization is not possible. The Taylor series for f_i must also converge in the p-adic sense. For instance, this is the case for exp(x) only if the p-adic norm of x is not smaller than 1.
2. The notion of Galois group can be generalized when the roots are not anymore points but 4-D surfaces (see this). However, the notion of ramified prime becomes problematic. The notion of ramified prime makes sense if one allows 4 polynomials (P₁,P₂,P₃,P₄) instead of two. The roots of 3 polynomials (P₁,P₂,P₃) give rise to 2-surfaces as string world sheets and the simultaneous roots of (P₁,P₂,P₃,P₄) can be regarded as roots of the fourth polynomial and are identified as physical singularities identifiable as vertices (see this).

   Also the maps defined by analytic functions g in the space of function pairs (f₁,f₂) generate new space-time surfaces. One can assign Galois group and ramified primes to h if it is a polynomial P in an extension of rationals. The composition of polynomials P_i defines inclusion hierarchies with increasing algebraic complexity and as a special case one obtains iterations, an approach to chaos, and 4-D analogs of Mandelbrot fractals.

Consider now the relationship between real and p-adic physics.

1. The connection between real and p-adic physics is defined by common points of reals and p-adic numbers defining a discretization at the space-time level and therefore a finite measurement resolution. This correspondence generalizes to the level of the space-time surfaces and defines a highly unique discretization depending only on the pinary cutoff for the algebraic integers involved. The discretization is not completely unique since the choice of the generalized complex coordinates for H is not completely unique although the symmetries of H make it highly unique.
2. This picture leads to a vision in which reals and various p-adic number fields and their extensions induced by rationals form a gigantic book in which pages meet at the back of the book at the common points belonging to rationals and their extensions.

What it means to be a point "common" for reals and p-adics, is not quite clear. These common numbers belong to an algebraic extension of rationals inducing that of p-adic numbers. Since a discretization is in question, one can require that these common numbers have a finite pinary expansion in powers of p. For points with coordinates in an algebraic extension of rationals and having p-adic norm equal to 1, a direct identification is possible. In the general case, one can consider two options for the correspondence between p-adic discretization and its real counterpart.

1. The real number and the number in the extension have the same finite pinary expansions. This correspondence is however highly irregular and not continuous at the limit when an infinite number of powers of p are allowed.
2. The real number and its p-adic counterpart are related by canonical identification I. The coefficients of the units of the algebraic extension are finite real integers and mapped to p-adic numbers by x_R=I(x_p)= ∑ x_np^-n → x_p= ∑ x_npⁿ. The inverse of I has the same form. This option is favored by the continuity of I as a map from p-adics to reals at the limit of an infinite number of pinary digits.

   Canonical identification has several variants. In particular, rationals m/n such that m and n have no common divisors and have finite pinary expansions can be mapped their p-adic counterparts and vice versa by using the map m/n→ I(m)/I(n). This map generalizes to algebraic extensions of rationals.

The detailed properties of the canonical identification deserve a summary.

1. For finite integers I is a bijection. At the limit when an infinite number of pinary digits is allowed, I is a surjection from p-adics to reals but not a bijection. The reason is that the pinary expansion of a real number is not unique. In analogy with 1=.999...for decimal numbers, the pinary expansion [(p-1)/p]∑_{k≥ 0}p^-k is equal to the real unit 1. The inverse images of these numbers under canonical identification correspond to x_p=1 and y_p= (p-1)p∑_{k≥ 0} p^k. y_p has p-adic norm 1/p and an infinite pinary expansion. More generally, I maps real numbers x= ∑_n<Nx_np^-n +x_Np^-N and y=∑_n<Nx_np^-n +(x_N-1)p^-N +p^-N-1(p-1)∑_{k≥ 0}p^-k to the same real number so that at the limit of infinite number of pinary digits, the inverse of I is two value for finite real integers if one allows the two representations. For rationals formed from finite integers there are 4 inverse images for I(m/n)= I(m)/I(n).
2. One can consider 3 kinds of p-adic numbers. p-Adic integers correspond to finite ordinary integers with a finite pinary expansion. p-Adic rationals are ratios of finite integers and have a periodic pinary expansion. p-Adic transcendentals correspond to reals with non-periodic pinary expansion. For real transcendentals with infinite non-periodic pinary expansion the p-adic valued inverse image is unique since x_R does not have a largest pinary digit.
3. Negative reals are problematic from the point of view of canonical identification. The reason is that p-adic numbers are not well-ordered so that the notion of negative p-adic number is not well-defined unless one restricts the consideration to finite p-adic integers and the their negatives as -n=(p-1)(1-p)n=(p-1)(1+p+p²+...)n. As far as discretizations are considered this restriction is very natural. The images of n and -n under I would correspond to the same real integer but being represented differently. This does not make sense.

   Should one modify I so that the p-adic -n is mapped to real -n? This would work also for the rationals. The p-adic counterpart of a real with infinite and non-periodic pinary expansion and its negative would correspond to the same p-adic number. An analog of compactification of the real number to a p-adic circle would take place.

Both hyperfinite factors and p-adicization allow a description of a finite measurement resolution. Therefore a natural question is whether the strange properties of hyperfinite factors, in particular the fact that the dimension D of Hilbert space equals to the cardinality of reals on one hand and to a finite number (D=1 in the convention used) on the other hand, could have a counterpart in the p-adic sector. What is the cardinality of p-adic numbers defined in terms of canonical identification? Could it be finite?

1. Consider finite real integers x=∑_n=0^N-1x_npⁿ but with x=0 excluded. Each pinary digit has p values and the total cardinality of these numbers of this kind is p^N-1. These real integers correspond to two kinds of p-adic integers in canonical identification so that the total number is 2p^N-2. One must also include zero so that the total cardinality is M=2p^N-1. Identify M as a p-adic integer. Its padic norm equals 1.
2. As a p-adic number, M corresponds to M_p=2p^N+(p-1)(1+p+p²+...)= p^N+p^N+1 +(p-1)(1+p+...-p^N)). One can write M_p=p^N+p^N+2+ (p-1)(1+p+...-p^N-p^N+1). One can continue in this way and obtains at the limit N→ ∞ p^{N→ ∞}(1 + p+ ...)+ (p-1)(1+ p+ ... + p^N-1). The first term has a vanishing p-adic norm. The canonical image of this number equals p at the limit N→ ∞. The cardinality of p-adic numbers in this sense would be that of the corresponding finite field! Does this have some deep meaning or is it only number theoretic mysticism?

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.