The ultrametric topology of discretized "world of classical worlds"

For more than a month, I have been preparing an article related to the role of finite fields in TGD. I however feel that "Finite Fields and TGD" is now in rather "final" shape and I expect that no new ideas will emerge anymore.

One of the unexpected outcomes is that I now understand the discretization of the "world of classical worlds" (WCW) using polynomials P(x) with integer coefficients as representations of 4-surfaces.

1. The polynomials P(x) satisfy strong additional conditions implying that finite fields can be regarded as basic building bricks in the mathematical structure of TGD besides all other basic number fields.
2. Polynomial P(x) defines a set of mass shells via its roots containing a set of 3-surfaces and defining holographic data fixing space-time surface almost uniquely by M⁸-H correspondence. The holography is forced by general coordinate invariance and there is no need to postulate it as a separate principle.
3. The discretization of WCW by polynomials, assumed to correspond to extrema or even maxima for the exponent of K\"ahler function of WCW, replaces WCW with a discrete set. Polynomials P(x) can have fixed degree k, or degree smaller than some maximum degree k_max, or satisfy some more general condition.
4. These restrictions reflect the basic feature of the spin glass energy landscape, namely that annealing as repeated heating and cooling allows to build localized thermodynamic equilibria localized inside some valley since thermal excitations are not able to kick the system out of the valley (failure of ergodicity).

   Elementary particles with D=P would result during cosmic evolution as repeated annealing when the degeneracy d(D) of polynomials with fixed value of D=P is fixed.

WCW has the fractal structure of a spin glass energy landscape containing valleys inside valleys inside... valleys. This discretized WCW is expected to have ultrametric topology and to decompose to sectors with p-adic topologies. The challenge is to understand what this means.

1. Spin glass energy landscape is realized number-theoretically in terms of the polynomials P(x) with integer coefficients and contains always a finite number of space-time surfaces, a hierarchy of analogues of Riemann zeta emerge defined as

   ζ= ∑ d(D) D^-k .

   These zeta functions have interpretation as partition functions and provide probabilistic description of the number-theoretically discretized WCW. The analogy with Riemann zeta suggests that k=1 corresponds to point at which convergence fails.

   1. Discriminant D provides a concrete realization for how the ultrametric distance function emerges.
   2. d(D) is the number of space-time surfaces with the same D, degeneracy.
   3. k corresponds naturally to the degree or more generally, maximal degree, of polynomials contributing to sub-WCW. k can be also interpreted as an analogue of inverse temperature. k=1 would correspond to linear polynomials defining trivial algebraic extensions.
2. p-Adic length scale hypothesis P ≈ p^k, p=2 or small prime, for preferred ramified primes D=P turns out to be equivalent with the proposal for logarithmic coupling constant evolution for Kähler coupling strength fixed to high degree by number theoretical constraints. Therefore two separate hypotheses fuse to a single one.

Consider now the structure of WCW as an analogue of the spin glass energy landscape.

1. Number theoretically, WCW decomposes to subsets for which a given ramified prime P appears as a prime factor of discriminant D characterizing the polynomial and coding information of ordinary primes that split or are ramified in the extension defined by P(x).
2. D=P space-time regions correspond to particles and those with several ramified primes to interaction regions with external particles corresponding to various primes P_i dividing D: these interaction regions are shared by several regions characterized by P as a factor of D.
3. Elementary particles correspond to D=P regions for which one has an especially large number of 4-surfaces with D=P: that is the degeneracy factor d(D) appearing in the analog of Riemann zeta is large so that annealing leads with a high probability to this state. One can say that space-time surfaces define number theoretical analogs of Feynman graphs consisting of particle lines and vertices.

In the standard ontology, one can predict scattering rates but particle densities cannot be predicted without further assumptions. In ZEO both can be predicted since there is a complete democracy between particles and particle reactions. Physical event as a superposition of deterministic time evolutions becomes the basic notion and both particles and particle reactions correspond to physical events.

The statistical model represents the probabilities of physical events within the quantization volume defined by CD. Particle characterized by D=P and corresponds to a scattering event with a single incoming and outgoing particle, and the statistical model predicts the densities of various particles as probabilities of D=P events. Genuine particle reaction corresponds to D= ∏ P_i and the model gives the probabilities of observing these events within CD.

See the article Finite Fields and TGD or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles related to TGD.