Recovering Classical Langevin Dynamics by Coupling the System to Quantum Noise

We consider a general Hamiltonian system and couple it to a large heat bath consisting of an infinite number of harmonic oscillators at fixed temperature. By linearly coupling the canonical momentum of the system to the heat bath and taking the classical limit as well as a properly defined Markov limit, we recover the purely relaxational Langevin dynamics of a non-conserved order parameter. Using the same formalism and additionally fixing the symmetry of the Hamiltonian system, one may also derive the ensuingLangevin dynamics of conserved modes, as originally constructed from phenomenological hydrodynamics arguments. The formalism can be extendedto systems with additional conserved modes to obtain reversible mode coupling terms in the classical limit. We exemplify this procedure forthe critical dynamics of isotropic ferro- and antiferromagnets.