From Classical to Quantum Mechanics:

From Classical to Quantum Mechanics: " How to translate physical ideas into mathematical language " Abstract 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-adjoints operators and so on...)-Quantum Mechanics properly that specifies the Hilbert space as L 2 (R n); the Heisenberg rule [p i , q j ] = −iℏ δ ij with p = −iℏ∇, the free Hamiltonian H = −ℏ 2 ∆/2m and so on. 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 " ℏ " must be seen as an independent problem not directly related with quantum formalism. The idea of using Hilbert space techniques in Classical Mechanics is not new; it was introduced by Koopman 1 and has been extensively employed in Statistical Mechanics 2,3,4. But in this article, we don't look at this technique as being only a mathematical trick: we consider that the Hilbert space formalism is really a mathematical and physical ground to rebuild Classical Mechanics. At this stage, we show that this non-standard formulation of Classical Mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the " quantization process " , and we show why quan-tization 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 Dynamics and Schrödinger equation. 1 The great advantage of this point of view is that now, quantization is based on concrete physical arguments and not derived from some " pure algebraic rule " (we exhibit also some limit of the correspondence principle). 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 dynamical quantum results are obtained only using classical-like formula. So this procedure has the nice property of enlightening in a more comprehen-sible way both logical and analytical connection between classical and quantum pictures. 2