Coincident-point vacuum field variance ($⟨X(\mathbf{x}{)}^2⟩=\int d\log k{\Delta }_X(k)$)

For a free field in the annihilation-operator vacuum, the coincident two-point function is its mode-power integral. Unsmeared vacuum products generally have an ultraviolet divergence. A finite band or spatial smearing defines a regulated variance without altering the meaning of a mode-by-mode dimensionless power spectrum.