 One can raise serious questions about these quantum-to-classical studies, however. The semi-classical systems are derived using various asymptotic procedures (Berry 1987 and 2001), but these procedures do not yield the actual classical systems that are supposed to be the limiting cases of the quantum systems. More importantly, the actual kinds of limiting relations between the quantum and classical domains are different than are typically considered in semi-classical approaches (§6.3 below). The actual relationship between the mathematical results and actual quantum and classical physical systems is tenuous at best leaving us, again, with the worry that chaos is an artifact of the mathematics (§5). One of the reasons the quantum chaos conjecture remains unproven likely is that inappropriate notions of “the classical limit” are being used. Even though the energy level statistics for quantum billiards in the semi-classical counterparts to classical billiard systems share universal properties, the actual behavior of the trajectories in the two systems is substantially different (under Schrödinger evolution Hilbert  space vectors never diverge from one another). 