Title: Out-of-time-order Operators and the Butterfly Effect,
Abstract: Out-of-time-order (OTO) operators have recently become popular diagnostics of
quantum chaos in many-body systems. The usual way they are introduced is via a
quantization of classical Lyapunov growth, which measures the divergence of
classical trajectories in phase space due to the butterfly effect. However, it
is not obvious how exactly they capture the sensitivity of a quantum system to
its initial conditions beyond the classical limit. In this paper, we analyze
sensitivity to initial conditions in the quantum regime by recasting OTO
operators for many-body systems using various formulations of quantum
mechanics. Notably, we utilize the Wigner phase space formulation to derive an
$\hbar$-expansion of the OTO operator for spatial degrees of freedom, and a
large spin $1/s$-expansion for spin degrees of freedom. We find in each case
that the leading term is the Lyapunov growth for the classical limit of the
system and argue that quantum corrections become dominant at around the
scrambling time, which is also when we expect the OTO operator to saturate. We
also express the OTO operator in terms of propagators and see from a different
point of view how it is a quantum generalization of the divergence of classical
trajectories.