  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.
