The classical-quantum limit

TL;DR

This paper investigates the limitations of the traditional classical limit in quantum mechanics and proposes a double scaling limit involving decoherence time to derive consistent classical-quantum dynamics.

Contribution

It introduces a double scaling limit with 7e0 and decoherence time to resolve classical limit failures for subsystems, unifying classical-quantum generators.

Findings

Double scaling limit yields irreversible open-system evolution.

Unified classical-quantum generators for various quasiprobability distributions.

Provides a framework for studying effective classical-quantum dynamics.

Abstract

The standard notion of a classical limit, represented schematically by $\hbar\rightarrow 0$, 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 modelling the decoherence of a subsystem by its environment. Denoting the decoherence time $\tau$, we demonstrate that a double scaling limit in which $\hbar \rightarrow 0$ and $\tau \rightarrow 0$ such that the ratio $E_f =\hbar /\tau$ 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 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.