Temporal crossover from classical to quantal behavior near dynamical critical points

The behavior of a class of dynamical systems is analyzed as a function of time and of Planck’s constant ħ when the latter is small compared with the (classical) action of the system. The case considered is that the classical system (ħ=0) is near a dynamical critical point, and there is a definite scaling of the variables of the classical motion with time. It is shown that the parameter ħ is a relevant variable in the renormalization-group sense, which means that as one scales to longer times, ħ scales to larger values. This is just a way of saying that quantum effects become progressively more important with time, and even if they can initially be ignored, there comes a time $t^{\mathrm{*}}$∝${ħ}^{\mathrm{-}1/\gamma }$ after which the system can no longer be treated classically, i.e., $t^{\mathrm{*}}$ characterizes the crossover away from classical to quantal behavior. This is similar to the effect of noise, which also smears out the deterministic classical phase-space path and destroys the sharp stochastic phase transition; however, unlike noise, the quantum exponent γ is simply related to the classical ones. We present arguments that this is the consequence of a property of the system’s operators in the Heisenberg picture. The cases of a period-doubling cascade to chaos and the disappearance of the last Kol’mogorov-Arnol’d-Moser trajectory in the standard map are specifically discussed. The results are shown to be consistent with numerical calculations.