  The ultimate goal of this project is to develop a systematic method for
deriving quantum master equations that satisfy the requirements of a completely
positive and trace-preserving (CPTP) map, further describing thermal relaxation
processes. In this paper, we assume that the quantum master equation is
obtained through the canonical quantization of the generalized Brownian motion
proposed in [Physics Letters A, 494, 129277 (2024)]. At least classically, this
dynamics describes the thermal relaxation process regardless of the choice of
the system Hamiltonian. The remaining task is to identify the parameters
ensuring that the quantum master equation meets complete positivity. We limit
our discussion to many-body quadratic Hamiltonians and establish a CPTP
criterion for our quantum master equation. This criterion is useful for
applying our quantum master equation to models with interaction such as a
network model, which has been used to investigate how quantum effects modify
heat conduction.
