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High temperature and white noise approximations are frequently invoked whenderiving the quantum Brownian equation for an oscillator. Even if this whitenoise approximation is avoided, it is shown that if the zero point energies ofthe environment are neglected, as they often are, the resultant equation willviolate not only the basic tenet of quantum mechanics that requires the densityoperator to be positive, but also the uncertainty principle. When thezero-point energies are included, asymptotic results describing the evolutionof the oscillator are obtained that preserve positivity and, therefore, theuncertainty principle.

Zero-point energies, the uncertainty principle and positivity of the quantum Brownian density operator