From classical to quantum mechanics: “How to translate physical ideas into mathematical language”

In this paper, we investigate the connection between Classical and QuantumMechanics by dividing Quantum Theory in two parts: - General Quantum Axiomatics(a system is described by a state in a Hilbert space, observables areself-adjoint operators and so on) - Quantum Mechanics properly that specifiesthe Hilbert space, the Heisenberg rule, the free Hamiltonian... We show thatGeneral Quantum Axiomatics (up to a supplementary "axiom of classicity") can beused as a non-standard mathematical ground to formulate all the ideas andequations of ordinary Classical Statistical Mechanics. So the question of a"true quantization" with "h" must be seen as an independent problem notdirectly related with quantum formalism. Moreover, this non-standardformulation of Classical Mechanics exhibits a new kind of operation with noclassical counterpart: this operation is related to the "quantization process",and we show why quantization physically depends on group theory (Galileogroup). This analytical procedure of quantization replaces the "correspondenceprinciple" (or canonical quantization) and allows to map Classical Mechanicsinto Quantum Mechanics, giving all operators of Quantum Mechanics andSchrodinger equation. Moreover spins for particles are naturally generated,including an approximation of their interaction with magnetic fields. We findalso that this approach gives a natural semi-classical formalism: some exactquantum results are obtained only using classical-like formula. So thisprocedure has the nice property of enlightening in a more comprehensible wayboth logical and analytical connection between classical and quantum pictures.Comment: 47 page