Stephen P. King (stephenk1@home.com)
Thu, 18 Mar 1999 14:24:35 -0500
Dear Matti and Friends,
Continuing...
> *************
>
> 2. Classical nondeterminism of Kaehler action <-->
> p-adic nondeterminism
>
> I realized the connection when pondering following problem:
>
> *What principle determines the value of the p-adic prime associated with
> given spacetime region?*
>
> The answer to this question relies on the following picture [mblocks].
>
> a) p-Adic spacetime surfaces are in rough sense images of real spacetime
> surfaces obtained by canonical identification mapping real spacetime
> points x, whose imbedding space coordinates h^k have finite number of
> pinary digits, to their p-adic counterparts. p-Adic spacetime
> surface satisfies the p-adic counterparts of the field equations
> associated with the Kahler action.
So you are partitioning an arbitrary subset of, say, a Wheeler
Superspace and mapping them one-to-one with p-adic spacetime manifolds
(if you can excuse my abuse of the term)?
> b) The characteristic feature of the p-adic differential equations is
> the existence of pseudoconstants: pseudoconstants are functions of pinary
> cutoffs of p-adic coordinates, which are constant below some arbitrarily
> small but finite length and time scales. [Pinary cutoff is essentially
> equivalent with decimal cutoff mathematically]. This property makes it
> possible to construct p-adic spacetime surfaces having the properties
> listed in a). In particular, the requirement that p-adic surface is almost
> completely determined as the p-adic image of the real spacetime surface is
> not in conflict with p-adic field equations.
Interesting, the "pinary" cutoff or truncation would make the p-adic
surface finite. This would hold for the sets of possible "classical"
observations among LSs that are logically consistent with each other.
Here I am introducing the idea of a "Cartesian connection": A causes B,
..., N iff B,..., N imply A. (Pratt's discussions of "residuations" and
Barwise and Moss's and Wegner's "infomorphisms" are more detailed
examples of this concept.)
There is a large body of work in logic and computer science studying
the ideas of bounded completeness or equivalent. The property of "almost
completely determined" is very important, I believe, since it goes to
explain why we have such strange things as event horizons and finite
signal velocities in a Universe that is infinite!
> This construction as such does not seem to give any hint about how the
> allowed p-adic prime is determined. In order to get to the core of the
> problem one must consider not only single real spacetime surface but *all
> spacetime surfaces which are identical outside a given spacetime region*
> V^4 and give rise to the same absolute minimum value of the Kaehler
> action. There are a lot of them because of the classical nondeterminism of
> the Kahler action. The idea is simple: require that classical and p-adic
> nondeterminisms are equivalent in the following sense:
>
> **For the physical value of p-adic prime p associated with a given
> spacetime region V^4, the p-adic images of various real spacetime
> surfaces differing from each other only in V^4, must correspond to
> various p-adic spacetime surfaces obtained by varying pseudo-constants
> in the representation of the p-adic spacetime surface.**
As I see it so far, this merely shows the existence of an ensemble-like
structure from which choices can be made, but I will read on... :)
> This requirement fixes the value of the p-adic prime
> since p-adic nondeterminism is characterized by a fractal hierarchy of
> p-adic length scales L(p,k)= p^kL_p and this fractal hierarchy must
> characterize also the classical non-determinism of Kaehler action
> in spacetime region characterized by p.
Interesting, but is there not an uncountable infinity of fractal
hierarchies representable in terms of the possible "computers" that can
"output" them? This follow from considering the way Wegner's Interaction
Machines have nonenumerable possible outputs: "Mathematical models of
interactional supplement inductive specifications of behavior by
enumerable sets of finite structures by coinductive specifications that
express nonenumerable sets of infinite structures representing possible
worlds [same as my "cosmoses] that arise in interactive computations of
finite agents. Inductively defined computing agents [such as minkowski
spacetimes!] model enumerable computations of string processing systems
while coinductively defined finite agents [such as LSs!] model
nonenumerable computations of open stream processing systems." (pg. 2
http://www.cs.brown.edu/people/pw/papers/math1.ps)
What we must understand is that the is no a priori initiality (or
finality) definable with in an Infinite Universe, as explained well by
Hitoshi; thus any particular cosmos has a perceived bound on its local
time defined by the finite interactions of its participants. Perhaps the
unobserved yet predicted "decay" of the Proton is an indication of the
duration of this particular cosmos we can communicate about. :)
> This hypothesis has far reaching consequences:
>
> a) It gives precise quantitative grasp on the nature of the classical
> nondeterminism of Kaehler action and hence also to the dynamics of
> cognitive spacetime sheets.
>
> b) Since classical nondeterminism corresponds also to quantum
> nondeterminism, hypothesis implies that it should be possible to determine
> the p-adic prime characterizing given spacetime region (or spacetime
> sheet) by observing a large number of time developments of this system
> (involving quantum jumps). The characteristic p-adic fractality, that is
> the presence of time scales T(p,k)= p^k T_p, should become manifest in
> the statistical properties of the time developments. For instance,
> quantum jumps with especially large amplitude would tend to occur
> at time scales T(p,k)= p^k T_p. T(p,k) could also provide
> series of characteristic correlation times. Needless to say,
> this prediction means definite departure from the
> nondeterminism of ordinary quantum mechanics
> and only at the limit of infinite p the predictions should
> be identical.
You lost me there. :( I can see that classical and quantum
nondeterminism are similar, from the Cauchy-Shwartz inequality and
perhaps from the ultrametric triangle inequality, I believe there is
more to this that either of us understands now. :) I need to understand
this "characteristic p-adic fractality" better. :) I am somewhat
reminded of Penrose's "single graviton criterion"... and the
"decoherence" thing...
>An interesting possibility is that 1/f noise is direct
> manifestation of classical nondeterminism: if this is the case, it should
> be possible to associate a definite value of p to 1/f noise.
Oh, we definitely agree there! :) I have stated before that the
Universe in-itself is Noise, all possible signals existing
simultaneously, without any order or symmetry or meaning... :)
One way of thinking of Local systems is as digital signal processor or
filters that can "recognize" or "model" or "encode/decode" information
from this Noise...
> ***********
>
> Appendix: How to construct the p-adic counterpart of real
> spacetime surface?
>
> The solution of the problem involves following argument providing a rough
> construction recipe for the p-adic counterpart X^4_p of a real spacetime
> surface X^4. Note that X^4_p belongs to p-adic imbedding space H^4_p
> whereas X^4 belongs to real imbedding space H.
>
> a) One can associate to a real spacetime region V^4 p-adic spacetime
> region V^4_p by mapping certain points h of real spacetime region to
> their p-adic counterparts h_p in p-adic H by canonical identification
>
> x=SUM(n) x_np^(-n)--> SUM(n) x(n)p^n
>
> [Note that this expansion is analogous to decimal expansion.]
>
> applied to the various imbedding space coordinates h^k.
What are the properties of h^k and H? How is their metric, topological
invariants, gauges, etc., defined? Are they contractible by a finite
length computation?
> It seems that the use of canonical identification involves selection of
> preferred coordinates in imbedding space. Since canonical identification
> is well defined for non-negative real coordinatse only these coordinates
> must be non-negative. For instance, the exponentials of the geodesic
> coordinates of M^4_+xCP_2 are good candidates for the preferred
> coordinates. Note that this procedure is General Coordinate Invariant
> at the level of spacetime surface.
"Preferred coordinated" are the beginning of error! We can not
postulate some entity who ab initio creates some particular coordinate
system, or even a "physics" for that mater, if we are to be consistent
with the experimental evidence of QM uncertainty. To do so requires the
introduction of an infinite regress of "because I said so's" that,
unless we use hypersets, leads nowhere!
> b) The trick is to perform pinary cutoff in n:th postive pinary digit
> and taking the limit n--> infinity. This corresponds to cutting
> off the pinary expansions of coordinate variables in some sufficiently
> high pinary digit n. This procedure is completely analogous to decimal
> cutoff. This means that one considers only the p-adic images of the
> points of real spacetime surface for which the pinary expansions of the
> imbedding coordinates contain no pinary digits higher than p^n:
>
> h^k(x) = SUM(r<=n) h^k(r) p^(-r)--> SUM(r<=n) h^k(r) p^(r)
Umm, I am very suspicious by nature of limits, and perhaps it is a
prejudice that I should be rid off... :) But, I am not alone in
insisting on constructable computability for entities that are not
"fundamental." (cf. Pratt, Wegner, Finsler, Brouwer, ...) Iff we are
dealing with fundamental properties, such as that of the Universe, we
must take care for any, *any*, possibility is contained therein! We can
easily become like Max in the movie PI, searching for meaning in
noise...
The practice of decimal cutoff makes sense since the computer is
restrained to a finite number of digits. That analogy works iff we are
mindful of the finite nature of LSs, even though they number of possible
interactions that they can have is nonenumerable as explained earlier...
:)
I am seriously hand waving here, but I think that there is a possible
model to be found in the behavior of tournaments of games that "select"
via pair-wise competition a single winner... I have been trying to find
formalisms to render symbolically my thoughts. The only thread I have so
far is the periodic gossiping idea, but I have not found any one
knowledgeable to discuss this with to the point of "getting somewhere."
:(
> The p-adic images of these real points are two-valued
> since there are two equivalent pinary expansions for these points
> (1=.9999..). The numerically favoured option is to choose the p-adic
> image of the finite pinary expansion as the p-adic image.
> These points provide a discretization of the real and p-adic spacetime
> surfaces becoming increasingly denser as n increases.
Wow, there is some serious controversy involved in this "equality!" The
purist vs. the pragmatists are going to wage war forever on this, like
the debate about angels dancing on pinheads... :) I will side with the
pragmatist for now... :) But, the purists do make a good point that a
difference, however small, is still a difference, but for our purposes,
I think the aphorism: "If you can't tell, it don't mater" works. But, I
tell you, I like Bart Kosko's answer to this by using his subsethood
theorem and the concept of mutual fuzzy entropy to constructively
discriminate to arbitrary precision (!) the similarities and differences
involved.
> c) In this approximation the p-adic images of the real points with pinary
> cutoff form a discrete set in p-adic imbedding space. The task is to
> find smooth p-adic spacetime surface going through all these points
> satisfying p-adic version for the field equations deriving from p-adic
> Kaehler action. By taking the limit, when pinary cutoff is taken to
> infinite (desimal expansion becomes infinitely long) the p-adic spacetime
> surface is fixed uniquely. This is the hope at least!
I say to give up on this hope! There is no unique spacetime surface, we
are just as likely to be able to find its description of the n-ary
expansion of PI! :(
> d) There are good reasons to believe that this hope is fullfilled in
> p-adic context: in real context this would certainly not be the case. The
> reason is the p-adic non-determinism of p-adic differential equations,
> which means that the integration constants are not genuine constants but
> functions depending on the pinary cutoff x_c(n) of their argument and
> having vanishing p-adic derivative. The cutoff can be arbitrarily high
> and the only essential thing is that these pseudoconstants become constant
> in some, arbitrary small, but finite scale. This roughly means that one
> must replace the initial value for p-adic field equation with initial
> values given in an entire spacetime lattice which is gradually made
> infinitely dense.
Umm, again I am skeptical, but follow what you are saying. Please
understand that I an with you in wanting uniqueness, but such logically
eliminates possibility of choice or uncertainty. We are left with an ab
initio apartheid that, frankly, sucks!
> Thus the existence of pseudoconstants together with
> classical nondeterminism of Kaehler action gives good hopes of finding
> p-adic spacetime surface solving the Euler Lagrange equations associated
> with p-adic Kahler action such that the real counterpart of this surface
> in canonical identification coincides with the original 4-surface at
> cutoff points. By p-adic fractality one can increase the value of
> n defining p-adic cutoff without any essential change and at the limit
> n-->infinity one obtains the desired p-adic spacetime surface
> as a solution of p-adic Euler Lagrange equations.
Pseudoconstants, we can deal with, because they can be derived from
interactional computations. But without the possibility of computing
absolutely precise Cauchy hypersurfaces in polynomial times, we are back
where we started.
Let's talk some more! :)
> ****************
>
> References:
>
> [mblocks] The chapter 'Mathematical building blocks'
> of the book 'TGD and p-Adic Numbers'
> at http://www.physics. helsinki.fi/~matpitka/padtgd.html
>
> [timesc] The chapter 'The problem of psychological time' of the book
> 'TGD inspired theory of consciousness with applications to biosystems'
> at http://www.physics. helsinki.fi/~matpitka/cbook.html
>
> ******************************************************************
>
> With Best,
>
> Matti Pitkanen
Onward to the Unknown!
Stephen
This archive was generated by hypermail 2.0b3 on Sat Oct 16 1999 - 00:29:45 JST