Area-scaling of quantum fluctuations

We show that fluctuations of bulk operators that are restricted to some region of space scale as the surface area of the region, independently of its geometry, provided that the operators satisfy certain conditions on their long and short range behavior. Specifically, we consider two point functions of operators that are integrals over local operator densities whose two point functions falls off rapidly at large distances, and does not diverge too strongly at short distances. We show that the two point function of such bulk operators is proportional to the area of the common boundary of the two spatial regions. Consequences of this, relevant to the holographic principle and to area-scaling of Unruh radiation are briefly discussed.