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In this paper we derive an exact master equation for two coupled quantumharmonic oscillators interacting via bilinear coupling with a commonenvironment at arbitrary temperature made up of many harmonic oscillators witha general spectral density function. We first show a simple derivation based onthe observation that the two-harmonic oscillator model can be effectivelymapped into that of a single harmonic oscillator in a general environment plusa free harmonic oscillator. Since the exact one harmonic oscillator masterequation is available [Hu, Paz and Zhang, Phys. Rev. D \textbf{45}, 2843(1992)], the exact master equation with all its coefficients for this twoharmonic oscillator model can be easily deduced from the known results of thesingle harmonic oscillator case. In the second part we give an influencefunctional treatment of this model and provide explicit expressions for theevolutionary operator of the reduced density matrix which are useful for thestudy of decoherence and disentanglement issues. We show three applications ofthis master equation: on the decoherence and disentanglement of two harmonicoscillators due to their interaction with a common environment under Markovianapproximation, and a derivation of the uncertainty principle at finitetemperature for a composite object, modeled by two interacting harmonicoscillators. The exact master equation for two, and its generalization to $N$,harmonic oscillators interacting with a general environment are expected to beuseful for the analysis of quantum coherence, entanglement, fluctuations anddissipation of mesoscopic objects towards the construction of a theoreticalframework for macroscopic quantum phenomena.Comment: 35 pages, revtex, no figures, 2nd version, references added, to  appear in PR

Exact master equation and quantum decoherence of two coupled harmonic oscillators in a general environment