  For the purpose of understanding the quantum behavior such as quantum
decoherence, fluctuations, dissipation, entanglement and teleportation of a
mesoscopic or macroscopic object interacting with a general environment, we
derive here a set of exact master equations for the reduced density matrix of
$N$ interacting harmonic oscillators in a heat bath with arbitrary spectral
density and temperature. Two classes of problems of interest to us which these
equations can be usefully applied to are that of the quantum dynamics of
nanoelectromechanical oscillators and the entanglement evolution of
multi-partite macroscopic states such as quantum superposition of mirrors in a
high Q cavity. To address a key conceptual issue for macroscopic quantum
phenomena we examine the conditions for an assumption often implicitly made in
these studies to be valid, namely, that the quantum behavior of a macroscopic
object in an environment can be accurately represented by only treating the
dynamics of its center-of-mass variable. We also mention how these results can
be used to calculate the uncertainty principle governing a macroscopic object
at finite temperature.
