Friday, July 28, 2006

The most recent view about p-adicization program

The generalization of the number concept obtained roughly by glueing reals and various p-adic numbers and their algebraic extensions together along common rationals and possibly also common algebraics is the starting point of TGD vision about fusion of real physics and various p-adic physics. I call the process of assigning to real physics p-adic physics p-adicization and the heuristic idea is that it corresponds to an algebraic continuation. I realized that the recent progress in the understanding of the formulation of quantum TGD at parton level leads also to a considerable progress in p-adicization.The following text gives a brief summary about the most recent view about what p-adicization might be. This view might be characterized as minimalism and would involve geometrization of only the reduced configuration space consisting of the maxima of Kähler function.

1. p-Adicization at the level of space-time

The minimum amount of p-adicization correspond to the p-adicization for the maxima of the Kähler function. The basic question is whether the equations characterizing real space-time sheet make sense also p-adically. Suppose that TGD indeed reduces to almost topological theory defined by Chern-Simons action for the light-like 3-surfaces interpreted as orbits of partonic 2-surfaces (see this, this, and this). If this is the case, then the starting point here would be the algebraic equations defining light-like partonic 3-surfaces via the condition that the determinant of the induced metric vanishes. If the coordinate functions appearing in the determinant are algebraic functions with algebraic coefficients, p-adicization should make sense. This of course, means the assumption of some preferred coordinates and the construction of solutions of equations leads naturally to such coordinates (see this).

If the corresponding 4-dimensional real space-time sheet is expressible as a hyper-quaternionic surface of hyper-octonionic variant of the imbedding space as number-theoretic vision suggests, it might be possible to construct also the p-adic variant of the space-time sheet by algebraic continuation in the case that the functions appearing in the definition of the space-time sheet are algebraic.

2. p-Adicization of second quantized induced spinor fields

Induction procedure makes it possible to geometrize the concept of a classical gauge fields and also of the spinor fields with internal quantum numbers. In the case of the electro-weak gauge fields induction means the projection of the H-spinor connection to a spinor connection on the space-time surface.

In the most recent formulation induced spinor fields appear only at the 3-dimensional light-like partonic 3-surfaces and the solutions of the modified Dirac equation can be written explicitly (see this, this, and this) as simple algebraic functions involving powers of the preferred coordinate variables very much like various operators in conformal theory can be expressed as Laurent series in powers of a complex variable z with operator valued coefficients. This means that the continuation of the second quantized induced spinor fields to various p-adic number fields is a straightforward procedure. The second quantization of these induced spinor fields as free fields is needed to construct configuration space geometry and anti-commutation relation between spinor fields are fixed from the requirement that configuration space gamma matrices correspond to super-canonical generators.

3. Should one p-adicize at the level of configuration space?

If Duistermaat-Heckman theorem holds true in TGD context, one could express configuration space functional integral in terms of exactly calculable Gaussian integrals around the maxima of the Kähler function defining what might be called reduced configuration space CHred. The huge super-conformal symmetries raise the hope that the rest of S-matrix elements could be deduced using group theoretical considerations so that everything would become algebraic. If this optimistic scenario is realized, the p-adicization of CHred might be enough to p-adicize all operations needed to construct the p-adic variant of S-matrix.

The optimal situation would be that S-matrix elements reduce to algebraic numbers for rational valued incoming momenta and that p-adicization trivializes in the sense that it corresponds only to different interpretations for the imbedding space coordinates (interpretation as real or p-adic numbers) appearing in the equations defining the 4-surfaces. For instance, space-time coordinates would correspond to preferred imbedding space coordinates and the remaining imbedding space coordinates could be rational functions of the latter with algebraic coefficients. Algebraic points in a given extension of rationals would thus be common to real and p-adic surfaces. It could also happen that there are no or very few common algebraic points. For instance, Fermat's theorem says that the surface xn+yn=zn has no rational points for n>2..

This picture is probably too simple. The intuitive expectation is that ordinary S-matrix elements are proportional to a factor which in the real case involves an integration over the arguments of an n-point function of a conformal theory defined at a partonic 2-surface. For p-adic-real transitions the integration should reduce to a sum over the common rational or algebraic points of the p-adic and real surface. Same applies to p1→ p2 type transitions.

If this picture is correct, the p-adicization of the configuration space would mean p-adicization of CHred consisting of the maxima of the Kähler function with respect to both fiber degrees of freedom and zero modes acting effectively as control parameters of the quantum dynamics. If CHred is a discrete subset of CH ultrametric topology induced from finite-p p-adic norm is indeed natural for it. 'Discrete set in CH' need not mean a discrete set in the usual sense and the reduced configuration space could be even finite-dimensional continuum. Finite-p p-adicization as a cognitive model would suggest that p-adicization in given point of CHred is possible for all p-adic primes associated with the corresponding space-time surface (maximum of Kähler function) and represents a particular cognitive representation about CHred.

A basic technical problem is, whether the integral defining the Kähler action appearing in the exponent of Kähler function exists p-adically. Here the hypothesis that the exponent of the Kähler function is identifiable as a Dirac determinant of the modified Dirac operator defined at the light-like partonic 3-surfaces suggests a solution to the problem. By restricting the generalized eigen values of the modified Dirac operator to an appropriate algebraic extension of rationals one could obtain an algebraic number existing both in the real and p-adic sense if the number of the contributing eigenvalues is finite. The resulting hierarchy of algebraic extensions of Rp would have interpretation as a cognitive hierarchy. If the maxima of Kähler function assignable to the functional integral are such that the number of eigenvalues in a given algebraic extension is finite this hypothesis works.

If Duistermaat-Heckman theorem generalizes, the p-adicization of the entire configuration space would be un-necessary and it certainly does not look a good idea in the light of preceding considerations.

  1. For a generic 3-surface the number of the eigenvalues in a given algebraic extension of rationals need not be finite so that their product can fail to be an algebraic number.

  2. The algebraic continuation of the exponent of the Kähler function from CHred to the entire CH would be analogous to a continuation of a rational valued function from a discrete set to a real or p-adic valued function in a continuous set. It is difficult to see how the continuation could be unique in the p-adic case.

The chapter TGD as a Generalized Number Theory I: p-Adicization Program of the book "TGD as a Generalized Number Theory" contains the most recent view about p-adicization.

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