**Stephen P. King** (*stephenk1@home.com*)

*Fri, 10 Sep 1999 16:58:57 -0400*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 743] Holy trinity summarizes the mathematical structure of quantum TGD"**Previous message:**Stephen P. King: "[time 741] Schommers' Ideas I: Mach's Principle"

Hi All, please discart my last post, it had some editing errors. My

apologies. ;-)

**

Dear Hitoshi et al,

I am writing up this series of extended quotes to try to see if

we can

use the ideas of Wolfram Schommers to stimulate further thinking,

particularly in light of some of the previous posts. :-)

All quotes are from "Quantum Theory and Pictures of Reality, W.

Schommers (ed.) Springer-Verlag (April 1989), except were noted.

First to set the stage:

"Space-Time and Quantum Theory: A formulation in Accordance with Mach's

Principle

Space and time are $absolute$ quantities in Newton's mechanics.

COncerning the term "absolute" note the following:

1) Absolute space was invented by Newton for the explanation of

inertia. However, we do not know of any other phenomenon for which the

absolute space would be responsible. So, the hypothesis of absolute

space can only be proven by the phenomenon (inertia) for which it has

been introduced. This is unsatisfactory and artificial.

2) The term "absolute" not only means that space is physically

real,

but also "independent in its physical properties, having a physical

effect, but not itself influenced by physical conditions". This must

also be considered as unsatisfactory.

[SPK note: we can relate this consideration to Hitoshi and Lance's

discussion in http://www.kitada.com/time_III.html]

This is why Mach eliminated space as an active cause in the

system of

mechanics (Mach's Principle). According to him, a particle does not move

in unaccelerated motion relative to space, but relative to the center of

all the other masses in the Universe; in this way, the series of causes

of mechanical phenomena was closed, in contrast to Newton's mechanics.

[SPK note: Mach seemed to assume a finite Universe.]

In fact, absolute space and, of course, absolute time, must be

considered as metaphysical elements because they are, in principle, not

accessible to empirical tests: there is no possibility of determining

the space coordinates x_1, x_2, x_3 and the time t. We can only say

something about the distances in connection with masses, and time

intervals in connection with physical processes.

Is Mach's principle fulfilled within the special theory of

relativity

(STR)? Definitely not; Newton's three-dimensional space is merely

extended within STR to a four-dimensional space-time, without overcoming

the absoluteness. In other words, instead of Newton's absolute space,

within STR we have an absolute space-time. This is why Einstein was led

to formulate the general theory of relativity GTR). However, within GTR

the absolute character of space-time is not completely eliminated. De

Sitter's and Goedel's solutions of Einstein's field equations lead to

the following results:

1) Space-time can exist without any mass (de Sitter).

2) Within GTR absolute rotations are possible: the whole

Universe (all

the masses) can rotate within an absolute space-time (Goedel).

...

Quantum theory (QT) and the theory of relativity have been

developed

independently of each other. Whereas the basics of classical mechanics

(STR, GTR) are given by space-time properties, the basics of QT have no

connection with certain space-time features. For example, the "principle

of superposition" belongs to the first-principles of QT and does not

result from certain space-time properties.

Usual QT is formulated within Newton's space and time

(Schroedinger's

theory) and also within the framework of the absolute space-time of STR

(e.g., Dirac's theory of the electron). Thus, in any case, QT is based

on an absolute space-time picture and, therefore, in these formulations

Mach's principle is not fulfilled. The following important question

arises: Can quantum phenomena be treated in accordance with Mach's

principle?"

pg. 232.

As an aside, it is important to understand the necessity of

Mach's

principle! If we do not have any hope of measuring a quantity in our

model of our physical reality, should we really consider it as

defensible? Lee Smolin and others have considered similar ideas as those

expressed by Schommers above and arrived at the same conclusion.

Something has to give!

We find in Mach's Principle a way out of the Newtonian dilemma,

but I

should say that we must carefully qualify certain terms and notions. We

must specify that the center of mass of a set of particles against which

"a particle moves in unaccelerated motion" is not the "totality of

Existence" Universe, which is all inclusive and thus neutral with

respect to qualities and quantities, but is a finite set of subsets of

the Universe. These can be considered as the set of all "classical

particles" that any given LS can observe within any particular moment of

its time, e.g., within the interval {t, \delta + t} with the definition

of the variable t given by Hitoshi's LS theory.

Now, when we consider that each particle's unaccelerated motion

as

being defined relative to a finite set of subsets of the Universe, we

are relativizing or "contextualizing" the idea of the state of motion,

thus

it is necessary to think of the observations of motions of particles in

terms of equivalence classes of observations of motions instead of

particular motions "themselves". This follows from both from Schommers

thinking and from Peter Wegner's discussion of how computational objects

observe each other in terms of $classifications$, to vit:

"A Classification *A* = (A, \SUM_A, |=_A) consists of a set A of

objects or tokens, a set \SUM_A of classes of tokens, and a binary

relation |=_A that relates tokens to classes. ...

Classifications model the observation of system behavior and

reflect

the fact that observers perceive only the observational equivalence

classes to which objects belong and not the objects themselves.

Classifications are a form of Chu spaces, whose properties have been

extensively studied by Pratt and others. Classifications are a

specialization of Chu spaces to a particular class of interpretations"

[sic]

http://www.cs.brown.edu/~pw/papers/math1.ps, pg. 20.

The connection between Schommers', Wegner's and Pratt's notions

will become

evident as we continue this discussion. :-)

The idea of a "center of mass" of an equivalence class, seems to

me to

relate to the idea of a "fixed point" when we consider that the "point"

within the equivalence class that does not change in the context of a

specific unitary transformation, like exp(-itH/h), of the equivalence

class has the same key properties as a fixed point in the traditional

sense, with the obvious exception that an equivalence class is not an a

priori given geometric space, or is it?! There is a connection here to a

way of defining "trajectories" but the math is over my head.

I found these references on the web:

http://www.cudenver.edu/~hgreenbe/glossary/fixedpts.html

http://markun.cs.shinshu-u.ac.jp/Mirror/JFM/Vol4/treal_1.html

I need to get the mathematicians of the Time List to let me know

if I

am making any sense here... :-)

A clue may be this passage from:

http://www.nsplus.com/sciencebooks/reviews/fivegoldenrules.html

"The going gets decidedly tougher when

Casti moves on to Brouwer's fixed point

theorem. He begins with a simple

question: what is the best way of

ranking football teams? He shows that

this question boils down to finding the

solution to a matrix equation: what

vector representing the rankings will,

when multiplied by a certain matrix,

give the vector again? In the jargon,

this means that the rankings constitute a

so-called "fixed point" of the matrix.

But does it exist ?

This is the question Brouwer's theorem

answers: it shows almost at a glance

whether a unique solution exists, and

thus whether the search for it is worth

the candle or not. The Dutchman L. E. J.

Brouwer was the topologist who

developed the arguments that led to this

theorem which shows, as Casti says,

"We'll know that the needle really is in

the haystack before we invest time,

energy, and money in trying to find it.""

It is my thinking that the particular ordering of posets of

observations over an equivalence class of can be considered as closely

analogous to Casti's "football team rankings"! But, I may be seeing

patterns in the wood grain... :-)

Later,

Stephen

**Next message:**Matti Pitkanen: "[time 743] Holy trinity summarizes the mathematical structure of quantum TGD"**Previous message:**Stephen P. King: "[time 741] Schommers' Ideas I: Mach's Principle"

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