About the topic
The description of the non-equilibrium dynamics of a quantum system presents difficulties which do not arise for classical systems. Model descriptions a la Calderia-Leggett explicitly introduce the system, its environment and the interaction between both. Can one formulate more directly a Langevin equation, which tries to take the interactions with the environment into account by modelling the effects of the environment by stochastic noise and which properties should be prescribed for these noise? In other words: how to distinguish authentic quantum dynamics from semi-classical or purely classical dynamics?
We shall explore these questions by using the quantum harmonic oscillator as a reference system. Its dynamics is phenomenologically characterized by the following conditions: (i) canonical commutation relations between the position and its canonically conjugate momentum, (ii) the Kubo formula of linear-response theory, (iii) the virial theorem of equilibrium quantum statistical mechanics, (iv) the quantum fluctuation-dissipation theorem. These criteria permit to distinguish, in particular, semi-classical dynamics proposed by Bedeau and Mazur from quantum dynamics, which is shown to be identical to the one constructed by Ford, Kac and Mazur. It is checked that the Markov property for the noise and the quantum fluctuation-dissipation theorem are mutually incompatibles.
Applications to magnetic systems, here modeled by the quantum spherical model, will be briefly mentioned.