p-Adicization, assuming holography = holomorphy principle, produces p-adic fractals and holograms

Yesterday's chat with Tuomas Sorakivi, a member of our Zoom group, was about the concrete graphical representations of the spacetime surfaces as animations. The construction of the representations is shockingly straightforward, because the partial differential equations reduce to algebraic equations that are easy to solve numerically. For the first time, it seems that GPT has created a program without obvious bugs. The challenges relate to how to represent time=constant 2-D sections of the 4-surface most conveniently and how to build animations about the evolution of these sections.

Tuomas asked how to construct p-adic counterparts for space-time surfaces in H=M⁴× CP₂. I have been thinking about the details of this presentation over the years. Here is my current vision of the construction.

1. By holography = holomorphy principle, space-time surfaces in H correspond to roots (f₁,f₂)=(0,0) for two analytic (holomorphic) functions f_i of of 3 complex coordinates and one hypercomplex coordinate of H (see this). The Taylor coefficients of f_i are assumed to be rational or in an algebraic extension of rationals but even more general situations are possible. A very important special case are polynomials f_i=P_i.
2. If we are talking about polynomials or even analytic functions with coefficients that are rational or in algebraic extension to rationals, then a purely formal p-adic equivalent can be associated with every real surface with the same equations.
3. However, there are some delicate points involved.
   1. The imaginary unit (-1)^1/2 is in algebraic expansion if p modulo 4=3. What about p modulo 4=1. In this case, (-1)^1/2 can be multiplied as an ordinary p-adic number by the square root of an integer that is only in algebraic expansion so that the problem is solved.
   2. In p-adic topology, large powers of p correspond to small p-adic numbers, unlike in real topology. This eventually led to the canonical concept of identification. Let's translate the powers of p in the expansion of a real number into powers of p (the equivalent of the decimal expansion).

      ∑ x_npⁿ ↔ ∑ x_n p^-n ×.

      This map of p-adic numbers to real numbers is continuous, but not vice versa. In this way, real points can be mapped to p-adic points or vice versa. In p-adic mass calculations, the map of p-adic points to real points is very natural. One can imagine different variants of the canonical correspondence by introducing, for example, a pinery cutoff analogous to the truncation of decimal numbers. This kind of cutoff is unavoidable.

   3. As such, this correspondence from reals to p-adics is not realistic at the level of H because the symmetries of the real H do not correspond to those of p-adic H. Note that the correspondence at the level of spacetime surfaces is induced from that at the level of the embedding space.
4. This forces number theoretical discretization, i.e. cognitive representations (p-adic and more generally adelic physics is assumed to provide the correlates of cognition). The symmetries of the real world correspond to symmetries restricted to the discretization. The lattice structure for which continuous translational and rotational symmetries are broken to a discrete subgroup is a typical example.

   Let us consider a given algebraic extension of rationals.

   1. Algebraic rationals can be interpreted as both real and p-adic numbers in an extension induced by the extension of rationals. The points of the cognitive representations correspond to the algebraic points allowed by the extension and correspond to the intersection points of reality as a real space-time surface and p-adicity as p-adic space-time surface.
   2. These algebraic points are a series of powers of p, but there are only a finite number of powers so that the interpretation as algebraic integers makes sense. One can also consider rations of algebraic integers if canonical identification is suitably modified. These discrete points are mapped by the canonical identification or its modification to the rational case from the real side to the p-adic side to obtain a cognitive representation. The cognitive representation gives a discrete skeleton that spans the spacetime surface on both the real and p-adic sides.

Let's see what this means for the concrete construction of p-adic spacetime surfaces.

1. Take the same equations on the p-adic side as on the real side, that is (f₁,f₂=(0,0), and solve them around each discrete point of the cognitive representation in some p-adic sphere with radius p^-n.

   The origin of the generalized complex coordinates of H is not taken to be the origin of p-adic H, but this canonical identification gives a discrete algebraic point on the p-adic side. So, around each such point, we get a p-adic scaled version of the surface (f₁,f₂=(0,0) inside the p-adic sphere. This only means moving the surface to another location and symmetries allow it.

2. How to glue the versions associated with different points together? This is not necessary and not even possible!

   The p-adic concept of differentiability and continuity allows fractality and holography. These are closely related to the p-adic non-determinism meaning that any function depending on finite number of pinary digits has a vanishing derivative. In differential and partial differential equations this implies non-determinism, which I have assumed corresponds to the real side of the complete violation of classical determinism for holography.

   The definition of algebraic surfaces does not involve derivatives but also for algebraic surfaces the roots of (f₁,f₂)=(0,0) can develop branching singularities at which several roots as space-time regions meet and one must choose one representative (see this).

   1. Assume that the initial surface is defined inside the p-adic sphere, whose radius as the p-adic norm for the points is p^-n, n integer. One can even assume that a p-adic counterpart has been constructed only for the spherical shell with radius p^-n.

      The essential thing here is that the interior points of a p-adic sphere cannot be distinguished from the points on its surface. The surface of a p-adic sphere is therefore more like a shell. How do you proceed from the shell to the "interiors" of a p-adic sphere?

   2. The basic property of two p-adic spheres is that they are either point strangers or one of the two is inside the other. A p-adic sphere with radius p^-n is divided into point strangers p-adic spheres with radius p^-n-1 and in each such sphere one can construct a p-adic 4-surface corresponding to the equations (f₁,f₂)=(0,0). This can be continued as far as desired, always to some value n=N. It corresponds to the shortest scale on the real side and defines the measurement resolution/cognitive resolution physically.
   3. This gives a fractal for which the same (f₁,f₂)=(0,0) structure repeats at different scales. We can also go the other way, i.e. to longer scales in the real sense.
   4. Also a hologram emerges. All the way down to the smallest scale, the same structure repeats and an arbitrarily small sphere represents the entire structure. This strongly brings to mind biology and genes, which represent the entire organism. Could this correspondence at the p-adic level be similar to the one above or a suitable generalization of it?
3. Many kinds of generalizations can be obtained from this basic fractal. Endless repetition of the same structure is not very interesting. p-Adic surfaces do not have to be represented by the same pair of functions at different p-adic scales.

   Of particular interest are the 4-D counterparts to fractals, to which the names Feigenbaum, Mandelbrot and Julia are attached. They can be constructed by iteration

   (f₁,f₂)→G(f₁,f₂)= (g₁(f₁,f₂),g₂(f₁,f₂)) →G(G(f₁,f₂)) →...

   so that at each step the scale increases by a factor p. At the smallest scale p^-n one has (f₁,f₂)=(0,0). At the next, longer scale p^-N+1 one has G(f₁,f₂)=(0,0), etc.... One can assign to this kind of hierarchy a hierarchy of extensions of rationals and associated Galois groups whose dimension increases exponentially meaning that algebraic complexity, serving as a measure for the level of conscious intelligence and scale of quantum coherence also increases in the same way.

   The iteration proceeds with the increasing scale and the number-theoretic complexity measured the dimension of the algebraic extension increases exponentially. Cognition becomes more and more complex. Could this serve as a possible model for biological and cognitive evolution as the length scale increases?

   The fundamental question is whether many-sheeted spacetime allows for a corresponding hierarchy at the real side? Could the violation of classical determinism interpreted as p-adic non-determinism for holography allow this?

   See the article TGD as it is towards end of 2024: part I 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.