  In this paper, we investigate the connection between Classical and Quantum
Mechanics by dividing Quantum Theory in two parts: - General Quantum Axiomatics
(a system is described by a state in a Hilbert space, observables are
self-adjoint operators and so on) - Quantum Mechanics properly that specifies
the Hilbert space, the Heisenberg rule, the free Hamiltonian... We show that
General Quantum Axiomatics (up to a supplementary "axiom of classicity") can be
used as a non-standard mathematical ground to formulate all the ideas and
equations of ordinary Classical Statistical Mechanics. So the question of a
"true quantization" with "h" must be seen as an independent problem not
directly related with quantum formalism. Moreover, this non-standard
formulation of Classical Mechanics exhibits a new kind of operation with no
classical counterpart: this operation is related to the "quantization process",
and we show why quantization physically depends on group theory (Galileo
group). This analytical procedure of quantization replaces the "correspondence
principle" (or canonical quantization) and allows to map Classical Mechanics
into Quantum Mechanics, giving all operators of Quantum Mechanics and
Schrodinger equation. Moreover spins for particles are naturally generated,
including an approximation of their interaction with magnetic fields. We find
also that this approach gives a natural semi-classical formalism: some exact
quantum results are obtained only using classical-like formula. So this
procedure has the nice property of enlightening in a more comprehensible way
both logical and analytical connection between classical and quantum pictures.
