The following comments are inspired by Ben Goertzel's blog post Wild-ass shit: P-adic physics, p-adic complex probabilities and the Eurycosm
. The reading of Ben's posting helps considerably in understanding of my comments.
Before continuing a pedantic remark (as a theory builder I keep watch that the one and only interpretation prevails!;-)). p-Adic physics is important in TGD but it does not replace real number based physics. Real physics would still describe the sensory world. p-Adic physics would take care of cognition. All of this would be integrated to adelic physics of sensory experience and cognition.
p-Adic physics at space-time level
The idea about cognitive representation would be realized at the fundamental level, even for elementary particles. And also at space-time level: real and p-adic space-time surfaces are in some sense completions of algebraic 2-D objects to 4-D ones. p-Adic space-time sheets would serve as correlates for cognitive, real ones for sensory.
p-Adic space-time surfaces would be very closely related to real space-time surfaces. They would have a set of common points with preferred imbedding space coordinates in extension of rationals and by strong form of holography (SH) they would be obtained from string world sheets and partonic 2-surfaces (in the following just "2-surfaces") as preferred extremal and highly unique in real sector: one has effective 2-dimensionality. The 2-surfaces would be number theoretically universal in the sense that the parameters characterizing them - say coefficients appearing in polynomials defining rational functions, would be in algebraic extension of rationals and make sense for reals and for corresponding extension of p-adics for every p.
Entanglement would be number theoretically universal. Even in strong sense so that superpositions coefficients of Hilbert space states would have interpretation in all number fields, reals and p-adics. Entanglement entropy for this entanglement would make sense in both real and various p-adic senses. In real sector it would measure the ignorance of the outsider about the state of system, in p-adic sector the conscious information of system about itself.
The notion of p-adic uncertainty discussed by Ben Goertzel is very interesting. In TGD framework it would relate to cognitive uncertainty. Here the non-well ordered character of p-adic numbers is the key notion: p-adic numbers with same real number valued norm (power of p) cannot be well-ordered using p-adic norm alone: p-adic interval has no boundaries meaning problems with definite integral, which give an extremely powerful guideline in attempts to define p-adic physics and fuse them with real physics.
- One can refine the notion of resolution. Instead of mapping p-adic number to its norm one maps by what I call canonical identification (or some of its variants) to real number by formula ∑ xnpn→∑ xnp-n. The map is continuous and one can also perform pinary cut at right hand side so that this is like taking N first pinary digits as significant digits. Very natural.
- Second aspect of the p-adic uncertainty relates to the fact that the notion of angle as a purely p-adic notion fails. Trigonometric and exponential functions can be defined by same formulas as in real case but they are not periodic and exponential function does not converge. The only cure that I know is to consider algebraic extensions induced by those of rationals. In particular, root of unity values of imaginary exponents for allowed angles as m×2π/n. Angles - or rather corresponding trigonometric functions, are discretized. Same happens for hyperbolic angles due to the completely unique feature of e: ep is ordinary p-adic number.
This number theoretic analog of Uncertainty Principle (only discrete set of phases exists p-adically but not the corresponding angles) seems to have nothing to do with Uncertainty Principle but better to be cautious. One ends up with the notion of p-adic geometry as a kind of collection of Leibniz monads labelled by points, which are number theoretically universal and correspond to algebraic extension of rationals. Each monad is in p-adic sectors p-adic continuum and differential calculus and field equations make sense. In the real sector the notion of manifold is replaced with its number theoretic version involving also the discretization as labels for manifold charts. Riemann geometry would have finite measurement resolution.
The discrete spine of the space-time surface consisting of algebraic points would help to algebraically continue
the 2-surfaces to space-time surface as preferred extremal and give additional conditions. In p-adic sectors p-adic pseudo constants (non-vanishing functions with vanishing derivative) would make the continuation easy: it is easy to imagine. In real sector the continuation need not be possible always: all that is imaginable is not realizable.
- Inclusions of hyperfinite factors provide a third view about measurement uncertainty. This would also be a number theoretically universal notion. Hyperfinite factors define infinite-D spaces for which finite-D approximation can be made arbitrarily precise and this strongly suggests at connection with p-adicity with pinary digits can be classified by their significance and ultrametricity holds.
p-Adic variants of lattices (Knut and Skilling)
Any lattice with basis vectors whose components belong to algebraic extension of rationals makes sense also p-adically in corresponding extension of p-adic numbers. One would have p-adic integer multiples of the basic vectors and in p-adic sense it would be a continuum and one could do p-adic differential calculus. Definite integral is the problem and has led me to the notion of p-adic/adelic monadology allowing both smooth physics (field equations) and discretization in terms of finite measurement resolution. Concerning discrete data processing - about which I know very little - the possibility to bring in p-adic differential calculus as a tool might be interesting.
Here the problem is that Hilbert space with p-adic coefficients allows zero norm states since SUM x_n^2 can vanish. The manner to get rid of this problem is number theoretic universality. The coefficients x_n belong to algebraic extension of rationals and make sense in the induced extensions of p-adics for all primes. This might look like a technical detail but is of fundamental importance. Also from the point of view of cognitive consciousness. One also ends up with the notion of negentropic entanglement in p-adic sectors of the adelic Universe.
Ultrametricity (UM) emerged originally from modelling of spin glass energy landscape having fractal structure: valleys inside valleys. In TGD the huge vacuum degeneracy of K&aml;hler action inspired the view that small deformations of the vacuum extremals can be regarded as 4-D version of spin glass energy landscape. Twistor lift of TGD strongly suggests that this degeneracy is lifted by cosmological constant (extremely small in recent cosmology) and is is plausible that the landscape remains and p-adic physics for various values of p would be natural manner to characterize the landscape at the level of "World of Classical Worlds" - the TGD analog for the world of euryphysics of Goertzel.
The observation about distribution of distances among sparse vectors ultrametricity (see this) is very interesting. I confess that I do not understand exactly how it comes out since I have no intuition about data representations.
What is however interesting is following.
- In very high-dimensional situations the notion of nearest neighbor must be defined modulo resolution. This brings in mind p-adicity and discretization. There is no point to keep well-ordering when measurement resolution is poor.
- According to the reference, the notions of metric distance and Riemannian geometry might not be useful in very high dimensions. This in turn suggest that the notion of the monadic/adelic geometry and discretized distances associated with it might be more useful approach. There is also a connection with hierarchies mentioned to make possible the unexpected effectiveness of deep learning. Associated with a given p-adic number field there is hierarchy of measurement resolutions (powers of p), p-adic number fields form a hierarchy, and there are sub-hierarchies corresponding to primes near powers of some prime, in particular 2. This fits nicely to the idea about p-adic physics as physics of cognition.
- There might be a deep connection with inclusions of hyperfinite factors and p-adic description of these sparse data sets. Finite resolution makes the notion of nearest neighbor obsolete for distances smaller than the resolution and therefore p-adicization, which does not allow well-ordering of points below the resolution would be natural.
- One could also start directly from the ultrametric distance function defined in the space of data points. Define some height function (energy is good metaphor for it) in the data space and define the distance between valleys A and B (minima of height function) along given path from A to B as maximum of height function at that path: that is as the height of the highest mountain which you must climb over. The distance from A to B is the distance along the path for which this distance is minimal. The challenge would be to assign some kind of energy or action to a data point. Can one imagine some universal height function? Could it correspond to some kind of physical energy or its exponent maybe characterizing the probability for the occurrence of the data point as analog of Boltzmann weight. p-Adic thermodynamics for data points?
Example: in chemical reaction kinetics this distance would correspond to the minimum of energy for intermediate states leading from configuration A to B.
Algebraic extensions of rationals and data processing
Algebraic extensions have some algebraic dimension. Could one perform data processing by mapping discretized real spaces of arbitrary dimension to these extensions, which are also complex. Arbitrarily high-D structures would be mapped to a subset of complex numbers and endowed with natural p-adic topology and metric. These structures interpreted in terms of 2-D string world sheets and partonic 2-surfaces could be lifted also to space-time level by SH : one would have representation of data at space-time level!
For a summary of earlier postings see Latest progress in TGD.
Articles and other material related to TGD.