[time 311] Re: [time 297] Mapping p-adic spacetime to its real counterpart

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Wed, 12 May 1999 09:39:56 +0300 (EET DST)

Dear Matti,

        Continuing from where we left off...

> b) There is a deep connection with quantum measurement theory. The
> phases, which are mapped as such to p-adic numbers correspond to
> maximal mutually commuting set of observables formed by the isometry
> charges and generating the group SO(1,1)xSO(2) xU(1)xU(1) of
> the isometry group of imbedding space. Canonical identification map
> commutes with the maximal mutually commuting set of observables.

        It is this property of "maximal mutually commuting" that, IMHO,
is a
key feature. It is there that the "construction" type of behavior of
consciousness (or generically, observation) occurs! I am taking my clues
from Vaughan Pratt's thinking, but extending his discussion into the
realm where the involution transformation is not exact: pg. 3-4 of

Basically this all reduces to the theory of Lie-group representations.
The concepts of mutually commuting algebra of observables and quantum
numbers are inherent to them.


        "The following analogy serves to fix ideas. The numbers +/- 1 are
connected in two ways, algebraic and geometric. The algebraic connection
is via the operation of operation of negation, an involution (- -x = x)
that connects them logically by interchanging them. The geometric
connection is via the interval [-1, 1] of reals lying between these two
numbers, a closed convex space connecting them topologically. We refer
to these connections themselves as respectively as the *duality* and
*interaction* of -1 and 1. The connections themselves might respectively
be understood as mental and physical, but this takes us beyond our
present story.
        We regard each point of the interval as a weighted sum of the
endpoints, assuming nonnegative weight p,q normalized via p+q=1, making
each point the quantity p-q. An important property of interaction is
that it includes the endpoint, namely as the special case where one of
the p or q is zero. An important property of the duality is that it
extends to interaction, namely via the calculus q-p = -(p-q).
        We shall arrange for Cartesian dualism to enjoy the same two basic
connections and the two associated properties, with mind and body in
place of -1 and 1 respectively. Ideally the duality would be a
negation-like involution that interchanges their roles; no information
is lost in this transformation, and the original mind or body is
recovered. And ideally the interaction would turn out to be the
long-sought solution to dualism's main conceptual hurdle. Chu spaces
achieve both of these in a very satisfactory way."

        I am claiming that the ideal case is the special case where there
is no
error or uncertainty, and as Hitoshi has shown, (time_II.ps Section 7,
pg. 17 equations (7.1) and (7.2)) this only occurs in the limit of t ->
+/-\infinity. When we consider finite approximations, which will always
contain error or noise terms, we weaken the ideal "perfect" properties
to integer (or more generally p-adic!) approximations. I am speculating
when I say that it is here that the work of Frieden et al and Kosko
comes into utility. The measure of the departure from the ideal case is,
IMHO, a means to model the "distance" of the "quantum jumps" that you
are taking about! It also points to thermodynamics and other entropies.
This is clear from the fact that in the ideal case the uncertainty, and
thus the entropy, is zero!

There might be deep connection between the concept of
mathematical approximation and how conscious
experience gives only a very limited information content
about initial and final states of quantum jump. Perhaps
conscious experience forms 'pinary cutoff' about the situation.

In p-adic context this is an attractive idea since pinary cutoffs
defines equivalence relation (for reals this is not the case since
decimal expansionis not unique) and one has hierarchy of
increasingly better pinary cutoffs: more and more accurate
cognitive representations adding more and more detail.

I believe in dualism only in the sense that manysheeted spacetime surfaces
decomposes to matter like spacetime sheets and cognitive spacetime
sheets: the latter are almost vacua and can have finite time duration
since classical charges can flow from them back to
the material spacetime sheets. Cognitive spacetime sheets would
correspond to mind in the sense of cognitive representations
but no consciousnes would be involved.

I would underline and emphasize the impossibility
to model quantum jump as a dynamical process. I regard is simply something
totally irreducible. One can predict all kinds of things about quantum
jump and correlate conscious experiences with quantum jumps using
mathematical models, even identify quantum counterparts of sensory qualia.
Also quantum statistical determinism makes modelling of
ensembles consisting of conscious individuals possible.
But basically I believe that quantum jump is not modellable.
I believe in miracles(;-).

> c) Without the special features of SU(3) group (existence of
> completely symmetric structure constants) it would not
> be possible to realize GCI. Neither would this be possible
> if imbedding space were dynamical as in string models.
> d) The p-adic image of the spacetime surface is discrete in generic
> case since only rational phases are mapped to their p-adic counterparts.

        Ah, this catches my attention! :) can we think of this in terms of
harmonics and resonances involve a/b= w, a and b are rational?

Rational frequencies correspond to resonances in model of solar system
based on Newtonian physics, right? But I do not actually know what
actually happens. Now I have rational phases however. Are rational
phase differences somehow special?

> One must complete the image to a smooth surface and the phenomenona
> of p-adic pseudo constants (p-adic differential equations
> allow piecewise constant integration constants) and nondeterminism
> of Kahler action give good hopes that p-adic spacetime surface can
> satisfy the p-adic counterparts of the field equations associated with
> Kahler action. Even the formal p-adic counterpats of the absolute
> mininization conditions can be satisfied since they correspond to purely
> algebraic conditions.

        The non-rational remainder under mappings might be
It may connect to irreducible error in observations...

The discreteness of the image is just what gives hopes of defining
p-adic spacetime surface as
smooth surface. The rationality of cosines and boost velocities could
be translation of the belief that
all our measurements are always expressible in terms of rational
numbers. As physicists we are bound inside the confines of rational

The same phase preserving mapping should work also at the level of
configuration space spinor fields but now basic QM, instead of GCI,
implies it. The nonconstant phases of basis of CH spinor fields
(quantum state basis) are mapped to their p-adic counterparts in the
same manner. This is required by the linearity of QM
and by the requirement that the action of eigen observables realized as
phase multiplication commutes with the identification map.
The phase is Pythagorean only in discrete set of CH and image is


        This I understand! :) It speaks to the "type of triangle" involved
the metric! Umm, Kosko's fuzzy subsethood involves a (generalized)
Pythagorean relation... I'll explore this later...

Discreteness of the image should make it possible to complete the image
of CH spinor field to a smooth p-adic CH spinor field by requiring that
the image is eigenstate of the maximal commuting set of p-adic


        This is a "relative" smoothness, isn't it; relative to value of
p-adic base?
Relative to the p-adic topology used defining its
own concept differentiability. At least first derivatives are required
since Kahler action depends on first derivatives of imbedding space

> e) Similar phase preserving mapping must be applied to the basis
> of configuration space spinor fields in order to achieve consistency
> of canonical identification with linearity of QM and it seems that
> phase preserving canonical identification provides universal solution
> to the mapping problem.

        I think that we have to look carefully at the implications of
from an absolute standard to a local system one, this is not the same as
the usual thinking involving global vs. local symmetries since the
thinking there still tacitly assumes one standard of measure for all.
This is fine for a single LS or X, but not for a model like ours, where
we are distinguishing the disjoint subjectiveness of one observer from


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