Tuesday, August 30, 2011

p-Adic fractality, canonical identification, and symmetries

The original motivation for the canonical identification

I: ∑ xnpn →∑ xnp-n,

and its variants - in particular the variant mapping real rationals with the defining integers below a pinary cutoff to p-adic rationals - was that it defines a continuous map from p-adics to reals and produces beautiful p-adic fractals as a map from reals to p-adics by canonical identification followed by a p-adically smooth map in turn followed by the inverse of the canonical identification.

The first drawback was that the map does not commute with symmetries. Second drawback was that the standard canonical identification from reals to p-adics with finite pinary cutoff is two-valued for finite integers. The canonical real images of these transcendentals are also transcendentals. These are however countable whereas p-adic algebraics and transcendentals having by definition a non-periodic pinary expansion are uncountable. Therefore the map from reals to p-adics is single valued for almost all p-adic numbers.

On the other hand, p-adic rationals form a dense set of p-adic numbers and define "almost all" for the purposes of numerics! Which argument is heavier? The direct identification of reals and p-adics via common rationals commutes with symmetries in an approximation defined by the pinary cutoff an is used in the canonical identification with pinary cutoff mapping rationals to rationals.

Symmetries are of extreme importance in physics. Is it possible to imagine the action of say Poincare transformations commuting with the canonical identification in the sets of p-adic and real transcendentals? This might be the case.

  1. Wick rotation is routinely used in quantum field theory to define Minkowskian momentum integrals. One Wick rotates Minkowski space to Euclidian space, performs the integrals, and returns to Minkowskian regime by using the inverse of Wick rotation. The generalization to the p-adic context is highly suggestive. One could map the real Minkowski space to its p-adic counterpart, perform Poincare transformation there, and return back to the real Minkowski space using the inverse of the rational canonical identification.

  2. For p-adic transdendentals one would a formal automorph of Poincare group as IPI-1 and this Poincare group could seen as a fractal counterpart of the ordinary Poincare group. Mathematician would regard I as the analog of intertwining operator, which is linear map between Hilbert spaces. This variant of Poincare symmetry would be exact in the transcendental realm since canonical identification is continuous. For rationals this symmetry would fail.

  3. For rationals which are constructed as ratios of small enough integers, the rational Poincare symmetry with group elements involving rationals constructed from small enough integers would be an exact symmetry. For both options the use of preferred coordinates, most naturally linear Minkowski coordinates would be essential since canonical identification does not commute with general coordinate transformations.

  4. Which of these Poincare symmetries corresponds to the physical Poincare symmetry? The above argument does not make it easy to answer the question. One can however circumvent it. Maybe one could distinguish between rational and transcendental regime in the sense that Poincare group and other symmetries would be realized in different manner in these regimes?

Note that the analog of Wick rotation could be used also to define p-adic integrals by mapping the p-adic integration region to real one by some variant of canonical identification continuously, performing the integral in the real context, and mapping the outcome of the integral to p-adic number by canonical identification. Again preferred coordinates are essential and in TGD framework such coordinates are provided by symmetries. This would allow a numerical treatment of the p-adic integral but the map of the resulting rational to p-adic number would be two valued. The difference between the images would be determined by the numerical accuracy when p-adic expansions are used. This method would be a numerical analog of the analytic definition of p-adic integrals by analytic continuation from the intersection of real and p-adic worlds defined by rational values of parameters appearing in the expressions of integrals.

For background see the chapter p-Adic numbers and generalization of number concept of "Physics as Generalized Number Theory".


L. Edgar Otto said...


Poincare ideas were on my mind last night and it sounds like you are aware of the same questions I ask myself. I wish I had access to the ideas of differential geometry or something like it- in the meantime I have to keep developing or rediscovering such wheels.

I may not get around to this post- or I should at least post the notes in a picture.

Now, in this sort of inverse of some value, even if this justifies and non-Poincarean concept of scale for say hierarchies of Planck's, at that infinitesimal place might we find the whole structure again, a sort of echoes of numbers? I never saw much explanation of that idea, a sort of closed bootstrapping of elements.

The post will be called "Minimum Structural Quantization"

The PeSla

matpitka@luukku.com said...

Wick rotation is a notion that is usually taken as a trick. I have however begun to feel that it involves a deep mathematical principle which practical physicists have lost. Even a blind hen can find the grain!

The idea is that one performs the analog of Wick rotation defined as a map to another space, performs symmetry operation or calculates integral or octonion real-analytic map or something else, and returns back to the original space using the analog for the inverse of Wick rotation. Wick rotation might be called intertwining operator by a mathematician working with Hilbert spaces and group representations.

Examples from TGD.

*"Wick rotation" could map the real 8-D imbedding space to a p-adic one in the case of p-adic variants of symmetries. Here the symmetry would be realized. Also the other way around is possible.

*In the case of p-adic integral "Wick rotation" could map p-adic space to real one where integral could be calculated using the standard numerical methods of ordinary integration.

*In the case of conformal field theory Wick rotation makes strings in Minkowskian space to strings in Euclidian space.

I had already given up hopes that the old ultra-romantic idea that conformal invariance could generalize to octonionic one. Wick rotation however revived this idea and showed that it could be equivalent with the recent view about what quaternionicity for space-time surfaces means.

Wick rotation would map 8-D imbedding space to its octonionic counterpart in which space-time surfaces would correspond to quaternionic surfaces obtained by putting the quaternionic "imaginary part" of octonion analytic function to zero.

The complex counterpart would be a curve of complex plane at which imaginary part of analytic function vanishes. In 2-D electrostatic problem the curve would typically defined the boundary of conductor.

TGD would be exactly solvable and the solution would be obtained using same trick as used in simple 2-D electrostatic problems but by replacing complex numbers with octonions!!

Maybe Wick rotation is ugly duckling which might transform to a beautiful swan with some help from mathematicians.

L. Edgar Otto said...


interesting and educational posts to which I strive to address such issues.

Is the swan logically black or white?

While this more relativistic idea of rotation is useful I think it needs more of a generalization - I think you might find my last post of interest where I post Plyctals or the folding and cutting of cubes.

The PeSla

matpitka@luukku.com said...

Your question probably contains some deep idea which I did not catch;-).

L. Edgar Otto said...


The logic question about swans was that used as and example it turned out there were black swans after all in Western Australia.

Or the math question is that are there hierarchies of the repeating patterns of numbers beneath a certain threshold?

I tried to explain this best I could on recent posts- including the last one of the four dimensional case of multidimensional unfolding of structures. (I did not post my brief but even more enlightening excursion into the 5D case.)

But I need a break awhile I think. I look forward again to any breakthroughs you may have- we can only set down the seeds of wisdom but who knows if they can grow or if it matters?

The PeSla (wonder if such complex topology is know then why does not the body of mathematicians make it clear to us say in the yang 6d case- they do not know do they?)