X-Message-Number: 14106
Date: Sun, 16 Jul 2000 07:29:55 EDT
Subject: beyond nanotech, a taste of quantum world

I have said before why the first quantification is the logical
next step beyond nanotech, the technological limit of
classical mechanics. Today, QM (quantum mechanics)
remains a mysterious subject outside specialized circles.
Here I'll give a taste of its many forms, the objective be to
bring it into the domain we think we can grasp in a practical
way (going from science to technology, with a look at cryonics)

Everybody know (or must!) about newtonian dynamics based on
the equation : Force = Mass x Acceleration. 
Unfortunately, that formalism becomes very cumbersome and
intractable when there are more than 2 or 3 moving parts in a

Lagrange, in its sejourns on the border of the Bourget lake in
South-East France was the first to give an alternative description
based on the difference between kinetic and potential energy. In his
honnor, that difference is called the Lagrangian. Richard Feynmann
(the inventor of nanotechnology) has pushed that formalism into
the so called path integral representation.

After Lagrange and its sentimental problems, came Hamilton, its
physics representation was baser on the sum of the kinetic and
potential energy, a quantity now called the hamiltonian H.

An even more general and complex formalism was produced
by Jacobi. Poisson has produced a full set of formalisms based
on its brackets (the Poisson's brackets define the amount of torsion
or Lie's derivative in a system).

At the end of 19th century, it was realized that Hamilton's formalism
was in fact four infinite classes of formalisms. Today physics users
when looking at a problem, choose their own hamilton derived
formalism best suited to the question at hand. Other formalisms,
don't have that adaptability or are too complicated (Jacobi).
Whatever the choice we may make, the result is the same in a
linear space.

To go from these expressions of classical physics to the quantum
domain, we need only to add the constrain of an elementary unit
of action (the product of energy by time).

Schrodinger, who was both a greath mathematician and a nazi,
has selected the hardest way: the Jacobi formalism, so he could
put the then new domain of QM on the most general setting and
demonstrate its kindship to the superior race. Today users throw
in the h constant of action in the H hamiltonian and don't bother.

This is the thing to do when working at the function *or* operator
level with a single time parameter.

The limit is the "or". When it turns to *and*, with a time intervention
at both, function and operator level, this last one is no more in the
linear domain and different representations give different answers.
They no more describe the same reality.

That must comes as no surprise: The tensor forms of QM are
known too (and mostly) as squeezed states and there are many
of them for a given tensor rank.

Well, if you get until here, you have now a fairly good look at the
frame of basic theoretical physics. Now you know that, outside
relativity, the 20th century revolution in physics summarized
simply into adding h to H :-)

Yvan Bozzonetti.

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