Hitoshi Kitada (hitoshi@kitada.com)
Tue, 6 Apr 1999 03:25:08 +0900
Dear Ben,
I mention the proof of theorem 2 at an intermediate place below.
Hitoshi
----- Original Message -----
From: Ben Goertzel <ben@goertzel.org>
To: <time@kitada.com>
Sent: Tuesday, April 06, 1999 1:35 AM
Subject: [time 178] Questions
>
> Hitoshi,
>
> A couple questions on reading of "Theory of Local Times".
> I'm not trying to be confrontational, just trying to understand...
>
> The proof of Theorem 2 (p. 11) seems very much a "physicist's proof" and
> the mathematician
> in me is a little dubious
>
> The assumption of independence seems to me to be pulled out of a hat.
> Please explain
> where it comes from. I see that it is true that ~generally~, different
> parts of the universe are going
> to be ~mostly~ independent. But this is different than being able to
> rigorously make the assumption,
> as an axiom for the derivation of a theorem, that the observing local
> system and the observed
> local system are DEFINITELY independent.
>
> In fact, quantum nonlocality means that independence is even ~less~ easily
> assumed than
> one would believe in classical physics.
>
> What am I missing?
>
> In the comments following Theorem 2, you refer to the alternate formulation
> of your
> theory as a vector bundle theory, involving X x R^6
>
> But, I don't see how this solves the problem. Each point x in X is a point
> in GR space, a
> macroscopic point, and the copy of R^6 corresponding to x is the local
> space of which
> x is the centre of mass. Still, this doesn't show that in the REAL
> universe, the different
> copies of R^6 are going to be independent of each other, even though their
> centres of
> mass may be interacting.
Theorem 2 states that axioms 1, 2, 3 are cosistent with axioms 4 and 5.
Proof.
Axioms 1, 2, 3 are concerned only with the internal motions of a local system.
I.e., they are concerned only with the _relative_ coordinates inside the local
system. Relative means that the center of mass is _removed_ . E.g., consider
two partcile system of partcile 1 and 2 with the position vectors r1 and r2.
The relative coordinate x is in this case the difference r1 - r2 of the
psoition vectors of the two partciles. Thus the center X of mass is
independent of the relative coordinate x. I.e. we made a transformation of
coordinate from (r1,r2) to (x,X), and x and X constitute a new coordinate
system (that can be given an orthogonality property in some inner product).
This is the usual Jacobi coordinates. (My idea came from this.)
This procedure separates the motion of the center X of mass of the system and
the relative motion between particles 1 and 2.
We consider two _different_ local systems L1 and L2, and make the separation
of the centers of mass in each system. Axioms 1, 2, 3 impose that the motion
inside each local system obeys QM, while axioms 4 and 5 assume that the two
centers X1 and X2 of L1 and L2 obey GR.
Axiom 1 (and the defintion 2 of local systems) asserts that the Hilbert spaces
H_{n \ell} and H_{n' \ell'} associated to L1 and L2 are different. This means
that we assume that the QM inside L1 and L2 are independent (or as you say,
they are linearly independent, since their Hilbert spaces H_{n \ell} and H_{n'
\ell'} are orthogonal by formula (1) in axiom 1). This is the independence of
L1 and L2, stated in the proof of theorem 2 on page 11. (Difference implies
orthogonality by formula (1).)
As we saw, their centers X1 and X2 of mass are independent of the relative
coordinates x1 and x2 (inside L1 and L2). Thus the motion of X1 and X2 can be
assumed to obey GR's axioms 4 and 5. (Note that there is no relation between
the internal motions of L1 and L2 by the above orthogonality of H_{n \ell} and
H_{n' \ell'}. Nor there is no assumed relation between the centers X1 and X2
of mass of L1 and L2 until we impose axioms 4 and 5.)
Local Lorentz invariance of L1 (in the sense of axiom5) follows from the
independence of x1 from X1.
End of proof.
Note that I impose all only by axioms 1, 2, 3 and 4, 5. Thue there is no
physical relations other than this restriction. Namely no physical intuition
or image should be overlapped on this formulation other than these axioms.
>
> You say that the L^2 representations of the local spaces are independent --
> you mean
> by this standard "linear independence", I assume.
>
> But I just don't get how the local spaces can be truly independent while
> the centres of
> mass interact??
>
> ben
>
>
>
>
>
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