Coincident-point vacuum field variance
= Coincident-point vacuum field variance
{title2=$\langle X(\mathbf x)^2\rangle=\int d\log k\,\Delta_X(k)$}
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>.