Matti Pitkanen (matpitka@pcu.helsinki.fi)
Fri, 19 Mar 1999 08:49:01 +0200 (EET)
On Thu, 18 Mar 1999, Stephen P. King wrote:
> 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)?
I am partitioning the 8-dimensional imbedding space M^4xCP_2
containing X^4 as a surface to small cubes of side p^(-n).
X^4 is replaced with the discrete point set by replacing points
of X^4 belonging to a given cube with the point
whose coordinates are with pinary cutoff.
This is like replacing imbedding space with lattice. On p-adic side
this process has very nice interpretation since these small cubes
correspond to equivalence classes for equivalence relation
|x-y|_p<p^(-n)<---> x and y equivalent.
>
> > 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!
>
I see that the key point is following: the points of spacetime with finite
number of pinary digits have the cardinality of integers whereas the
points of entire surface have cardinality of reals. p-Adic nondeterminism
says that solution of field equation can be fixed arbitrarily in a
set having cardinality of integers. This allows to map real
differentible surface to its p-adic differentiable countepart. For all
numerical purposes the p-adic and real surfaces are 'identical'.
> > 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... :)
In standard physics there is no classical nondeterminism. It is the
properties of Kahler action which make this possible.
What the *hypothesis* means that classical nondetermism and p-adic
nondetermism are closely related things: this gives direct
quantitative/topological grasp to classical nondeterminism of Kahler
action. Also it in principle allows to decide what is the p-adic prime
associated with given (sufficiently small) spacetime region.
The lack of precise criterion of this kind has been the basic problem of
p-adic TGD for a long time.
>
> > 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?
I do not quite understand your question because I do not have
thought TGD from computational perspective. In any
case, fractal hierarchies labelled by preferred
p:s are predicted and each p represents particular kind of fractality
which its characteristic lengt and time scales.
Even infinite p:s are predicted and ordinary real topology corresponds
in excellent approximation to infinite p p-adic topology.
>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)
Huh! This requires careful reading! I realize that I know nothing
about computer thinking.
> 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. :)
BTW, TGD predicts that proton is stable(:-).
> > 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. :)
The connection is much more deeper in TGD. The entanglement of
particle states with the branches of multifurcations of spacetime surface
makes possible quantum jump in which this entanglement is reduced and
one branch is selected. My original naive expectation that quantum
nondetermihism would directly correspond to classical nondeterminism
turned out to be wrong. In fact, I gave up the idea for years and only for
months ago realized that quantum entanglement is the crucial piece
of the puzzle.
I need to understand
> this "characteristic p-adic fractality" better. :) I am somewhat
> reminded of Penrose's "single graviton criterion"... and the
> "decoherence" thing...
One manner to understand this is to plot graphs of p-adic fractals.
There are some of them on my homepage. What is done is to construct
real function f_R= Iof_poI^(-1). I is canonical identification mapping
p-adics to reals and f_p is p-adically analytic function, such as x^2
and plot the graphs of this function (I is the map SUM x_np^n -->
SUMz_n p^(-n)).
p-Adic might indeed provide clues to the understanding of decoherence:
the decomposition of
spacetime surface to regions obeying effectively physics in different
number fields suggests strongly decoherence in the sense
that QM applies in finite region characterized by some p. But I believe
that this is only effective although probably very practical description.
>
> >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...
I would see 1/f noise as signature of the presence of cognitive spacetime
sheets which control the physics of material spacetime sheets. Extremely
small energy fluxes between cognitive and material spacetime sheets would
control the initial value sensitive dynamics of material spacetime sheets.
Perhaps our computers and all electric circuits are conscious in some
primitive sense: 1/f is typically present. I do not believe that
this cognition has anything to do with ordinary AI based philosophy:
the presence of Kahler electric fields is a necessary prequiste for
cognition in TGD based picture and biosystems and electronic systems
contain a lot of electric fields. [I think I have sent posting
about electric fields and life?]
>
> > ***********
> >
> > 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?
h^k denote just the *coordinates* of the 8-dimensional imbedding space
H=M^4_+xCP_2 where spacetime surfaces reside. M^4_+ is the future
lightcone of Minkowski space. CP_2 is compact coset space SU(3)/U(2).
Both are constant curvature spaces: all their points are metrically
equivalent. The simplest Riemann manifolds one can imagine are in
question: 'the laws of physics in each point are the same'.
> > 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!
You are right here. This troubled me. I analyzed this
during last two days and found that one General Coordinate invariance
at the level of H is achieved in the following sense:
coordinate transformations must preserve the finite number of pinary
digits property. New coordinates are polynomials of old coordinates
with coefficients possessing finite number of pinary digits.
Computationally this is not a restriction: this is just what we
always do in practice. It however means a delicate number theoretical
spontaneous breaking of GCI: there is infinite number of nonequivalent
equivalence classes of coordinatizations of H. A possible physical
interpretation is that the interaction of spacetime region with
surrounding spacetime breaks the unrestricted general coordiante
invariance through boundary conditions.
In fact, the consideration of isometries leads to preferred coordinate
system concept naturally. In coordinates in which maximal
subgroup H_max of H-isometries is realized linearly the
isometries respecting finite pinary digits property are represented
as affine transformations x--> Ax+B, where A is matrix with entries
having finite number of pinary digits and B is vector with the same
property. For M^4 the geodesic coordinates are the preferred coordinates.
For CP2 only U(2) subgroup of SU3 acts linearly so that number
theoretical symmetry breaking occurs. This has however physical
interpreation. Allowed SU3 transformations must leave the boundary of
spacetime region V^4 invariant since transformation are not allowed
to affect external world: in the ideal case this region correspond to a
surface at which CP2 coordinates are constant so that
allowed isometries must leave point of CP2 invariant and thus
form the group U2. Everything is OK!
> > 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...
I would be happy if I could make the argument without the limit
considerations.
In practice limit is not needed: the incredibly rapid convergence
of perturbation series in powers of p means that a couple of lowest
powers are needed for physical values of p.
The limit construction might provide a technical manner to prove the
hypothesis. The beauty of limit is that at each step it is 'obvious' that
p-adic nondeterminism makes it possible to find the requireed p-adic
spacetime surface and by p-adic fractality each n looks the same when
zoom is made. Without p-adic nondeterminism the argument would
pathetically fail at each step.
> 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 think that p-adics might provide kind of a mathematization for the
concept of cutoff. Recall only the fact that |x-y|<p^(-n) defines
equivalence relation, call it E_n, in p-adic context but not for reals.
In the approximation sequence one replaces p-adic numbers R_p
by the set R_p/E_n of equivalence classes and goes to the limit.
> 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! :(
What would be needed is proof that the p-adic surface *exists*. In
practical calculations only few first steps in the construction are
quite enough. For instance, in case of electron the first step leads
directly form electron Compton length to the length scale 10^4 Planck
lengths: no need to continue to make the next step until we have
accelerators of size of galaxy!
>
> > 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.
As I said, the argument is only a plausibility argument showing that
p-adic surface exists. It might be refined to a proof by a competent
mathematician. At practical side: the argument would provide construction
recipe.
The big question is:
Can one really construct solution of p-adic field equations of p-adic
Kahler action going through the p-adic images of the points
of real surface having finite number of pinnary digits. Does p-adic
nondetermism really make this miracle possible.
This would be almost like specifying the time development of dynamical
completely before using ideal computer and requiring that this time
development satisfies the dynamics dictate by given dynamical model!
By the way, one of the original big ideas related to p-adics was
that in p-adic world engineering is possible whereas in real world
everything is dictated at the moment of big bang.
MP
> 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
>
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