Without the free field E 0 (t) in this equation the operator x (t) would be exponentially dampened, and commutators like [ z (t), p z (t)] would approach zero for t ≫ ⁠ 1 / τω 0 ⁠. With the vacuum field included, however, the commutator is iħ at all times, as required by unitarity, and as we have just shown. A similar result is easily worked out for the case of a free particle instead of a dipole oscillator.