The spectral density of the vacuum field is fixed by the form of the radiation reaction field, or vice versa: because the radiation reaction field varies with the third derivative of x, the spectral energy density of the vacuum field must be proportional to the third power of ω in order for [ z (t), p z (t)] to hold. In the case of a dissipative force proportional to ẋ, by contrast, the fluctuation force must be proportional to ω ω in order to maintain the canonical commutation relation. This relation between the form of the dissipation and the spectral density of the fluctuation is the essence of the fluctuation-dissipation theorem.