From: israel@unixg.ubc.ca (Robert B. Israel)
Newsgroups: sci.math,sci.physics
Subject: Re: Bayes' theorem and QM
Date: 30 Dec 92 22:32:04 GMT
Organization: The University of British Columbia
NNTP-Posting-Host: unixg.ubc.ca

In <1ht1arINNf8a@chnews.intel.com> bhoughto@sedona.intel.com (Blair P. Houghton) writes:

>In article <1992Dec24.101452.16194@oracorp.com> daryl@oracorp.com (Daryl McCullough) writes:
>>It would be nice if the volume of a region of space were always
>>well-defined, but it is not. A demonstration due to Banach and Tarski
>>showed that it is possible (mathematically, rather than physically
>>possible) to decompose a sphere into a finite number of pieces and
>>then recombine them by rotations and translations to get two complete
>>spheres. It is easy to see that these pieces cannot possibly have
>>volumes, since that would lead to the conclusion that the volume of
>>two spheres equals the volume of one sphere.
>>
>>So one cannot always assume that *every* set has a measure (or volume,
>>or probability).

>I'll buy the explanation (conditioned on the conjecture
>that such a situation might exist, which as yet isn't
>anything more than conjecture), but not the example.

I'm jumping in here without seeing McCullough's article, so I don't know
what you're calling "explanation", "conjecture", "situation" and "example".
But what Banach and Tarski have is a theorem.  If you don't buy it, you
don't buy the Axiom of Choice.

>The two spheres thus formed do indeed have the same volume
>sum as the original sphere, but each has less volume than
>the original, or else they weren't composed of a finite
>number of pieces[*], or else when constructed they contained
>gaps, internally.

The pieces are _very_ strange sets.  But they fit together perfectly,
without gaps.  The spheres are solid.

>That, or their "volume" is fractal, being the effective
>volume of a convoluted surface, and therefore isn't
>actually of order 3 but of some order less than 3 and
>greater than 2. 

The pieces are much stranger than fractals.  They (or at least
some of them) are non-measurable sets.  Impossible to really
visualize, but think of dense "clouds" of points.

>                 This way you get a finite number of pieces
>but some pieces are composed of an infinite number of
>infinitesimal objects connected by infinitesimal objects,
>e.g. concentric spheres of rational diameters connected
>by a single diameter, leaving concentric spheres of
>irrational diameters connected by a single chord displaced
>from the diameter by an irrational, infinitesimal amount (i.e.,
>tangent to the smallest spherical shell of irrational diameter).

"Irrational, infinitesimal"? No such animal.  We're dealing with
real numbers here.

>And it might even be "possible" physically; imagine a large
>crystal of NaCl carved to be spherical.  Now separate
>disintegrate it so that you have two spheres, one only of
>Na+ ions and one only of Cl- ions, with each ion in its
>original position wrt the center of mass of its own kind.
>Their masses are each now less than the original, but their
>"volumes" are each at least as large (likely larger and
>expanding at a fantastic rate, unless you do something
>interesting like containing them inside inert, solid shells
>for the purpose of argument).

>But that's beside the point.

The construction is extremely non-physical.  It is talking about
mathematical spheres, composed of mathematical points.  Collections
of atoms are something else entirely.

>Are there any constraints on the mathematics of "quantum
>probabilities" that are similar to Sigma algebras for
>"classical probabilities?"

>				--Blair
>				  "Beside the point is of
>				   course another point."
-- 
Robert Israel                            israel@math.ubc.ca
Department of Mathematics             or israel@unixg.ubc.ca
University of British Columbia
Vancouver, BC, Canada V6T 1Y4
Newsgroups: sci.math.num-analysis