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

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 $\mathrm{\hslash }$, 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.