Conditions for the Quantum to Classical Transition: Trajectories vs. Phase Space Distributions

TL;DR

This paper compares two different conditions for the quantum-to-classical transition, showing their equivalence in the semiclassical regime and dominance of different conditions outside this regime.

Contribution

It demonstrates the equivalence of trajectory-based and phase-space density-based conditions for the quantum-classical transition in the semiclassical regime.

Findings

Weak conditions dominate in the semiclassical regime.

Strong conditions dominate when action is comparable to Planck's constant.

Both conditions offer an essentially equivalent local picture in the semiclassical regime.

Abstract

We contrast two sets of conditions that govern the transition in which classical dynamics emerges from the evolution of a quantum system. The first was derived by considering the trajectories seen by an observer (dubbed the ``strong'' transition) [Bhattacharya, et al., Phys. Rev. Lett. 85: 4852 (2000)], and the second by considering phase-space densities (the ``weak'' transition) [Greenbaum, et al., Chaos 15, 033302 (2005)]. On the face of it these conditions appear rather different. We show, however, that in the semiclassical regime, in which the action of the system is large compared to $\hbar$, and the measurement noise is small, they both offer an essentially equivalent local picture. Within this regime, the weak conditions dominate while in the opposite regime where the action is not much larger than Planck's constant, the strong conditions dominate.