Hitoshi Kitada (hitoshi@kitada.com)
Thu, 9 Sep 1999 18:46:43 +0900
Dear Stephen,
Stephen P. King <stephenk1@home.com> wrote:
Subject: [time 717] Re: [time 716] Time operator => Ensembles of clocks?
> Dear Hitoshi,
>
> Hitoshi Kitada wrote:
> >
> > Dear Stephen,
> >
> > stephen p. king <stephenk1@home.com> wrote:
> >
> > Subject: [time 711] Re: [time 708] Time operator => Ensembles of clocks?
> snip
>
> > > [SPK]
> > > > > Matti, are you saying that the dynamical law is a priori to time?
How?
> > > > > I see the "dynamical law" as defining a pattern of behavior of a
system
> > > > > as it evolves in its time. When we say that we localize it in time,
we
> > > > > are refering, to be consistent, to the time of the localizing agent,
not
> > > > > the system in question's time. There is no "time" for all unless we
are
> > > > > merely considering the trivial case when all systems are
synchronized...
> > >
> > > Did this make sense?
> >
> > Yes, but sounds obvious.
>
> Can we use the concepts of entrainment and phase locking to think of
> how LS's interact?
I do not understand what your question is.
>
> > I see LS's as fundamental clocks, and thus it
> > > should be possible to consider an "ensemble of clocks" as given by a
> > > ensemble of LSs. But, I am very sketchy in my thinking of this. :-(
> > >
> > > > > Hitoshi, are the QM orbits constructed in a Hilbert space such that
> > > > > they are strictly orthogonal to each other? This, to me, says that
the
> > > > > LS are independent and thus have independent space-time framings of
> > > > > their observations. Does this affect your argument?
> > > [HK]
> > > > No. E.g., consider two orbits Psi(x,t) = exp(-itH/h)Psi(x,0) and
Phi(x,t) =
> > > > exp(-itH/h)Phi(x,0) in the same LS. The inner product of these wrt the
usual 3
> > > > dimensional Hilbert space is
> > > >
> > > > (Psi(t), Phi(t)) = (Psi(0), Phi(0)).
> > > >
> > > > This is not equalt to zero unless the initial states are orthogonal.
> > > >
> > > > But two orbits in different LS's are of course orthogonal by
definition.
> [SPK]
> > > Ok, this is that I suspected. I am trying to work backwards from the
> > > notion that the ordits of LS's are orthogonal to each other. I am
> > > wondering if it is possiple to think of LS's as subjective observers and
> > > the orthogonality condition as making them independent of each other.
> [HK]
> > Yes, this is the assertion of my axiom 1.
>
> Yes, I am trying to see how we can recover a local approximation of
> Riemannian geometry as the way that the individual posets of
> observations of each of the LS's are "ordered". We need to show that it
> is necessary and sufficient that the observations of any LS, e.g. a set
> of classical center of mass particles, are arranged in a way that we can
> think of relations among them in terms of a Riemannian metric. It could
> be that we only need a Minkowski metric, because, I think that gravity
> is defined in terms of the differences between the local space-times of
> different LS's.
> Remember that I am thinking of space-time in a way that is very similar
> to how Schommers thinks of it: "...we have argued that physically real
> processes do not take place $in$, but are $projected on$ space-time. The
> coordinated [metrics, connections, etc.] and time are not accessible to
> empirical tests and we can only observe distances between bodies and
> time intervals in connection to processes: There is no exception to this
> law. Thus, we can conclude that the phenomenon space-time comes into
> being through bodies and processes. ... Objects and processes of the
> real world [the poset of observations that a finite number of LS's can
> agree upon] are perceived by an interaction process with our sense
> organs; this reality is pictured by our perceiving apparatus, and the
> phenomenon space-time belongs to the perceiving apparatus."
> pg. 263, Quantum Theory and Pictures of Reality...
>
> I am going further than Schommers in that I am saying that the
> so-called "real world" is not an absolute, it is a relative concept,
> e.g. a given set of LS's that share a common metric such that a
> synchronized frame of motions can be defined among them constitutes a
> "world". In information theoretical terms we can think of a world as
> that a given set of LS's can agree upon within the information encoded
> in the configurations of their QM orbits. (This is tentative and maybe
> very wrong!)
In a rough sense, I agree. You need to formulate your thought if you wanted to
be understood by others, i.e. by the so-called scholars or physicists, but
this might be unnnecessary/useless, seeing them being involved only in their
own special thoughts. (I do not mean speciality is good or should be
evaluated, but the reverse ;-) )
>
> [SPK]
> > > This implies to me that a space of n-dimensions can be defined by the
> > > set of LS's, where each LS defines a dimension.
> [HK]
> > Each LS defines some finite dimensions according to the number of
particles
> > it contains.
>
> Ok, but can be define a space were each orthogonal "dimension" or
> independent basis vector is an LS?
>
> [SPK]
> > > Question: Would this
> > > space have "continuous" dimensions like a Von Neumann space?
> [HK]
> > I do not think so.
>
> So the dimensionality of a LS space would have integer valued
> dimensions, not Real valued dimensions, each orthogonal?
Yes.
>
> > > Now, as to your question about how we have a subject/object dichotomy,
> > > re: "So I am interested in how/why the two different views could be
> > > possible." I believe that each LS defines an observer, specifically a
> > > "subject". This "subject" has something that it is not as an "object".
> > > This is very important. So the "object" of the "subject" is the
> > > "~subject". Does this make sense?
> >
> > Again sounds obvious/trivial.
>
> OK...
>
> [SPK]
> > It follows that ~(~subject) = subject.
> > > I am seeing the scattering propagator (orbit?)of the LS as defining the
> > > subjective actions of an LS and that the mapping of such to that of the
> > > ~(orbit) as defining the objective actions, e.g. the LS observes
> > > situations that are "not" the behaviour of the scattering propagator or
> > > orbit.
> [HK]
> > Neglecting some of your abuse of words, I agree.
>
> How should I say this as to not abuse words?
The word "scattering propagator" does not seem to exist: it should be replaced
by the word "unitary propagator" like exp(-itH/h). There is a word "scattering
operator" S that is defined by
S = W_{+}^* W_{-},
where
W_{+ or -} = lim_{t -> + infinity or - infinity} exp(itH/h) exp(-itH_0/h).
Here H is the Hamiltonian of the system
H = H_0 + V,
where H_0 is the unperturbed Hamiltonian and V is the perturbation.
Then the first part of your statement may be
> I am seeing the unitary propagator exp(-itH/h) of the LS as defining the
> subjective actions of an LS.
On the second part:
> and that the mapping of such to that of the
> ~(orbit) as defining the objective actions,
you may need to articulate what the mapping is. But seeing the following part:
e.g. the LS observes
> situations that are "not" the behaviour of the scattering propagator or
> orbit.
I think that you mean by "objective actions" the phenomena outside an
observer.
I thus think that you just rephrased what we have been discussing about the
inside of an LS and the observation of the outside of an LS. I.e., this
statement of yours does not seem to add any new to the previous arguments.
>
> [SPK]
> > The trick is to see how it is that the class or set of {~(orbit)}
> > > is finite.
> [HK]
> > If {~(orbits)} means the complement of the set of "orbits," it would be
> > infinite.
>
> Would it be finite if we only consider orbits that we can think as
> synchronized with each other?
How are these systems synchronized? The number of the LS's is countable if the
totality of the particles inside the universe is countable, but there is no
defnite finite limit to it as you say:
Umm, this does not seem to limit the set
> to a finite number! Perhaps it is the information theoretic aspect that
> does this...
>
> [SPK]
> > > All I have right now is a metaphor, the metaphor of a dictionary. I see
> > > the "meaning" as given by the n-ary relations that exist between
> > > objects. In a dictionary, the meaning of each word is given by the
> > > relationship it has with a finite number of other words. Particularly,
> > > the relational structure that the words in the set "{definition}" have
> > > with each other. I think that Pratt's CABAs are a formal way of thinking
> > > of this notion.
> > > Now, how do we think of the communication or interaction or, qua Bill's
> > > thinking, interference, between LSs? In order for two systems to
> > > communicate there must be at least one aspect of the systems that they
> > > share or have in common.
> [HK]
> > The common part can be played by an observer of the two systems.
>
> Umm, this looks like Peter's "secondary observer". It should be
> possible to generate a mathematical model of the observations of three
> LS's of each other such that it follows some approximation of the
> Lorentz or Poincare transformation group. I think that the mutual
> observations of two LSs should give us the Lorentz transformations when
> we look at the respective behavior of the centers of mass of the LS's!
>
> [SPK]
> > We can think of this as a common element in
> > > their poset of their observations.
> > > (BTW, Hitoshi, does the notion of a poset of observations make sense to
> > > you?
> [HK]
> > What order do you define among the observations? The class of LS's is a
> > partially ordered set (poset) with respect to the set inclusion relation.
In
> > this sense, i.e. in the sense that the order is a partial order, the
totality
> > of LS's in my sense does not have a clear hierarchy as Matti's world seems
to
> > have.
>
> Yes, the poset is with respect to set inclusion relation, but I was
> thinking more in terms of Kosko's set subsethood relation. I see Matti's
> p-adic ordering as "vertical" and this one as "horizontal". The former
> is the inclusion relations within the particular set of synchronized
> LS's and the latter is phylogenetic. The key distinction is that between
> the subsethood relation and the ultrametric. Umm, I do not know how to
> explain this further right now...
>
> > > Also, do you see any big problems with Schommers work?)
> >
> > No, as far as I saw it. It is an interesting book containing speculative
> > thinkings.
>
> Do you think that his redefinition of SR and QM to be useful?
I need to remember what his redefinition was. Would you let me remember? It is
a cumbersome task to resee the book :-) My impression was what he wrote is not
new. Just a kind of review of QM and relativity written as a good reading like
a good novel.
>
> Later,
>
> Stephen
>
Best wishes,
Hitoshi
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