X-Message-Number: 14207
Date: Sun, 30 Jul 2000 07:08:11 EDT
Subject: Re: CryoNet #14193 Identity Of Indiscernibles 

 From: "John Clark" <>
 >    >A carbon atom in Boston is not the same as a carbon atom in
  >   >New-York.
> What experiment could you possibly perform to come to that conclusion?
> How would things be different if were not true?
Use common sens: React one with some atoms and the other with different atoms.
For example the NY carbon could be included in an apple and the Boston one in
coal :-)
If the reaction on one atom don't translates to the other, they have different
histories and are different objects.

Now if you want a lot of maths, then there are some basic concepts:
Quantum mechanical spaces are defined at one point in euclidean space.
The vector dual of a point is an hyperplane (a p-1 subspace in a p
dimensions space). Such an object is called a 0-form or a function. Depending
how the plane folds, you have different functions. With the same folding
and different orientations you have different complex arguments. The
infinite sum of such functions defines the so called Hilbert's space of
quantum mechanics. If you move from place to place in euclidean space,
you must define for each function a plane at each euclidean point.
This stack of planes is called a differential 1-form, it is another way to 
a wave. Waves are at the root of the uncertainty principle. So that there
is no uncertainty if there is no wave and there is no wave if there is not an
euclidean displacement.

Two objects are the same *only* if they have the same values for all
functions in the same place. If the place is not the same, they differs
at least by some 1-form.

That is the Schrodinger's view of quantum mechanics, based on
functions. If you move to Heisenberg's operators formalism, then
indeed an operator is p dimensional and cover the whole of the
euclidean space. It is a tangent space and from place to place
tangent spaces differe by a differential form: a wave 1-form, an eggs
crate 2-form, and so on up to a p-form.

I suggest you look at an elementary differential geometry book to
get some familliarity with these concepts. Sorry to have been so

Yvan Bozzonetti.

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