[time 741] Schommers' Ideas I: Mach's Principle

Stephen P. King (stephenk1@home.com)
Fri, 10 Sep 1999 15:00:45 -0400

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

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

        The connection between Schommers' 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:


        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:

                              "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,
                               means that the rankings constitute a
                               so-called "fixed point" of the matrix.
                               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... :-)



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