A True Equation to Couple Classical and Quantum Dynamics

Abstract

Starting from the Schr\"odinger-equation of a composite system, we derive unified dynamics of a classical harmonic system coupled to an arbitrary quantized system. The classical subsystem is described by random phase-space coordinates entangled with the quantized variables of the complementary subsystem. Our semiclassical equation is {\it true} in a sense that its predictions are identical to those of the fully quantized composite dynamics. This exact method applies to a broad class of theories, including e.g. the relativistic quantum-electrodynamics and the electron-fonon dynamics.