A Continuous Transition Between Quantum and Classical Mechanics. I

Partha Ghose, Manoj K. Samal

T. Yoneda, Aya Nagasato, Tatsuo Akamine +1 more

Anthony Duncan A, Michel Janssen B

Abstract This chapter elaborates and reflects on the authors’ use of the “arch-and-scaffold” metaphor to characterize the transition from classical mechanics, via the old quantum theory, to modern quantum mechanics. It discusses in detail why this metaphor does justice to both continuous and discontinuous aspects of this transition, as well the various ways in which the metaphor applies to the contents of both volumes of the work. It contrasts the account of this transition based on the “arch-and-scaffold” metaphor with an account based on Kuhn’s notion of a paradigm shift. This shift occurs, resulting in the understanding in which the continuous aspects of the transition from classical to quantum mechanics are lost.

Walter Greiner

Hervé Bergeron

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.

Walter Greiner