On the Transition from the Quantum to the Classical Regime for Massive Scalar Particles: A Spatiotemporal Approach

Abstract

(abridged)If the space-time is presupposed, the coordinate representation of the solutions $\psi(\vec x, t)$ of the Schroedinger equation of a quantum system containing one massive scalar particle has a {\it preferred status}. It is then possible to perform a multipolar expansion of the density matrix $\rho(\vec x, t) = |\psi(\vec x, t)|^2$ (and more generally of the Wigner function) around a space-time trajectory ${\vec x}_c(t)$ to be properly selected. A special set of solutions $\psi_{EMWF}(\vec x, t)$, named {\it Ehrenfest monopole wave functions}(EMWF), is characterized by the conditions that: (i) the quantum expectation value of the position operator coincides at any time with the searched classical trajectory, $< \psi_{EMWF} | {\hat {\vec x}} | \psi_{EMWF}> = {\vec x}_c(t)$: this is possible only when the dipole vanishes; (ii) Ehrenfest's theorem holds for the expectation values of the position and momentum operator: its application to EMWF leads then to a {\it closed Newton equation of motion for the classical trajectory, where the effective force is the Newton force plus non-Newtonian terms (of order $\hbar^2$ or higher) depending on the higher multipoles of the probability distribution $\rho$.} These results can be extended to N particle systems and to relativistic quantum mechanics. There is substantial agreement with Bohr's viewpoint: the macroscopic description of the preparation, certain intermediate steps and the detection of the final outcome of experiments involving massive particles are dominated by these 'classical {\it effective} trajectories'. In the framework of decoherence, one gets a transition from an {\it improper quantum} mixture to a {\it classical statistical} one, when both the particle and the pointer wave functions appearing in the reduced density matrix are EMWF.