  In a previous article [H. Bergeron, J. Math. Phys. 42, 3983 (2001)], we
presented a method to obtain a continuous transition from classical to quantum
mechanics starting from the usual phase space formulation of classical
mechanics. This procedure was based on a Koopman-von Neumann approach where
classical equations are reformulated into a quantumlike form. In this article,
we develop a different derivation of quantum equations, based on purely
classical stochastic arguments, taking some elements from the
Bohm-Fenyes-Nelson approach. This study starts from a remark noticed by
different authors [M. Born, Physics in My Generation (Pergamon Press, London,
1956); E. Prugovecki, Stochastic Quantum Mechanics and Quantum Spacetime
(Reidel, Dordrecht, 1986)], suggesting that physical continuous observables are
stochastic by nature. Following this idea, we study how intrinsic stochastic
properties can be introduced into the framework of classical mechanics. Then we
analyze how the quantum theory can emerge from this modified classical
framework. This approach allows us to show that the transition from classical
to quantum formalism (for a spinless particle) does not require real
postulates, but rather soft generalizations.
