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Classical Dynamics for Linear Systems: The Case of Quantum Brownian Motion

It has long been recognized that the dynamics of linear quantum systems is classical in the Wigner representation. Yet many conceptually important linear problems are typically analyzed using such generally applicable techniques as influence functionals and Bogoliubov transformations. In this Letter we point out that the classical equations of motion provide a simpler and more intuitive formalism for linear quantum systems. We examine the important problem of Brownian motion in the independent oscillator model, and show that the quantum dynamics is described directly and completely by a c-number Langevin equation. We are also able to apply recent insights into quantum Brownian motion to show that the classical Fokker-Planck equation is always local in time, regardless of the spectral density of the environment.