The Classical-Quantum Limit

The standard notion of a classical limit, represented schematically by h 0, provides a method for approximating a quantum system by a classical one. In this work, we explain why the standard classical limit fails when applied to subsystems, and show how one may resolve this by explicitly modeling the decoherence of a subsystem by its environment. Denoting the decoherence time by t, we demonstrate that a double scaling limit in which h 0 and t 0 such that the ratio E f = h /t remains fixed leads to an irreversible open -system evolution with well-defined classical and quantum subsystems. The main technical result is showing that, for arbitrary Hamiltonians, the generators of partial versions of the Wigner, Husimi, and Glauber-Sudarshan quasiprobability distributions may all be mapped in the above -mentioned double scaling limit to the same completely positive classical -quantum generator. This provides a regime in which one can study effective and consistent classical -quantum dynamics.