Early-time and late-time quantum chaos

Abstract

We show the relation between the Heisenberg averaging of regularized 2-point out-of-time ordered correlation function and the 2-point spectral form factor in bosonic quantum mechanics. The generalization to all even-point is also discussed. We also do the direct extension from the bosonic quantum mechanics to the noninteracting scalar field theory. Finally, we find that the coherent state and large-[Formula: see text] approaches are useful in the late-time study. We find that the computation of the coherent state can be simplified by the Heisenberg averaging. Therefore, this provides a simplified way to probe the late-time quantum chaos through a coherent state. The large-[Formula: see text] result is also comparable to the [Formula: see text] numerical result in the large-[Formula: see text] quantum mechanics. This can justify that large-[Formula: see text] technique in bosonic quantum mechanics can probe the late time, not the early time. Because the quantitative behavior of large-[Formula: see text] can be captured from the [Formula: see text] numerical result, the realization in experiments should be possible.