On the notion of quantum Lyapunov exponent

Classical chaos refers to the property of trajectories to diverge exponentially as time $t\to\infty$. It is characterized by a positive Lyapunov exponent. There are many different descriptions of quantum chaos. One description related to the notion of generalized (quantum) Lyapunov exponent is based either on qualitative physical considerations or on the so-called symplectic tomography map. The purpose of this note is to show how the definition of quantum Lyapunov exponent naturally arises in the framework of the Moyal phase space formulation of quantum mechanics, and is based on the notions of quantum trajectories and the family of quantizers. The role of the Heisenberg uncertainty principle in the statement of the criteria for quantum chaos is made explicit.