Transition From Quantum To Classical Mechanics As Information Localization

Abstract

Quantum parallelism implies a spread of information over the space in contradistinction to the classical mechanical situation where the information is "centered" on a fixed trajectory of a classical particle. This means that a quantum state becomes specified by more indefinite data. The above spread resembles, without being an exact analogy, a transfer of energy to smaller and smaller scales observed in the hydrodynamical turbulence. There, in spite of the presence of dissipation (in a form of kinematic viscosity), energy is still conserved. The analogy with the information spread in classical to quantum transition means that in this process the information is also conserved. To illustrate that, we show (using as an example a specific case of a coherent quantum oscillator) how the Shannon information density continuously changes in the above transition . In a more general scheme of things, such an analogy allows us to introduce a "dissipative" term (connected with the information spread) in the Hamilton-Jacobi equation and arrive in an elementary fashion at the equations of classical quantum mechanics (ranging from the Schr\"{o}dinger to Klein-Gordon equations). We also show that the principle of least action in quantum mechanics is actually the requirement for the energy to be bounded from below.