On the localized quantum oscillators in a common heat bath

To remedy the failure of minimal coupling method in describing the quantum dynamics of two localized Brownian oscillators interacting with a common medium, a scheme is introduced to modeling the medium by a continuum of complex scalar fields or equivalently two independent real scalar fields. The starting point is a Lagrangian of the total system and quantization is achieved in the framework of canonical quantization. The equations of motion, memory or response functions and fluctuation-dissipation relations are obtained. An induced force between oscillators is obtained originating from the fluctuations of the medium. Ohmic regime and Drude regularization is discussed and the positions of oscillators are obtained approximately in large time limit and weak coupling regime.