**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Wed, 12 May 1999 09:39:56 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 312] Re: [time 307] rethinking relativity part II"**Previous message:**Matti Pitkanen: "[time 310] Re: [time 309] Rethinking relativity part IV"**In reply to:**Stephen P. King: "[time 309] Rethinking relativity part IV"**Next in thread:**Matti Pitkanen: "[time 313] Re: Mapping p-adic spacetime to its real counterpart"

Dear Matti,

Continuing from where we left off...

[MP]

*> 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

http://boole.stanford.edu/pub/ratmech.ps.gz

[Matti]

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.

[Stephen]

"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!

[Matti]

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(;-).

[MP]

*> 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

how

harmonics and resonances involve a/b= w, a and b are rational?

[Matti]

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?

[MP]

*> 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.
*

[SK]

The non-rational remainder under mappings might be

problematic...

It may connect to irreducible error in observations...

[MP]

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

world.

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

discrete.

[SPK]

This I understand! :) It speaks to the "type of triangle" involved

in

the metric! Umm, Kosko's fuzzy subsethood involves a (generalized)

Pythagorean relation... I'll explore this later...

[MP]

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

observables.

[SPK]

This is a "relative" smoothness, isn't it; relative to value of

the

p-adic base?

[Matti]

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

coordinates.

*> 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

going

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

another!

MP

**Next message:**Matti Pitkanen: "[time 312] Re: [time 307] rethinking relativity part II"**Previous message:**Matti Pitkanen: "[time 310] Re: [time 309] Rethinking relativity part IV"**In reply to:**Stephen P. King: "[time 309] Rethinking relativity part IV"**Next in thread:**Matti Pitkanen: "[time 313] Re: Mapping p-adic spacetime to its real counterpart"

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