that this may be problematical in a theory containing gravity). Eigenstates, however, have the peculiar property that the standard deviation is necessarily zero, $$\Delta_{\Omega}\,H=\langle (H-\langle
  H\rangle_{\Omega})^2\rangle_{\Omega}^{1/2}=\langle H^2\rangle_{\Omega}^{1/2}=0$$ According to the standard interpretation of quantum theory combined with spectral theory, $H^2\geq 0$, this implies that in each individual observation process the total energy of the vacuum which is, according to conventional wisdom, the (hypothetical) sum or superposition of local (small scale) fluctuations, happens to be exactly zero. In other words, the elementary fluctuations have to exactly compensate each other in an apparently fine-tuned way. Put differently

If there are positive local energy fluctuations, there have to be at the same time by necessity negative energy fluctuations of exactly the same order. That is, at each moment, the global pattern of energy fluctuations in the quantum vacuum is an array of rigidly correlated positive and negative local excitations.

Remark: Note the similarity of this independent observation to what we have said above in connection with the holographic hypothesis.\
It should be mentioned that Hawking in [@Haw1] invoked exactly this picture of a particle pair excitation near the event horizon with the virtual particle, having negative energy, falling into the BH while the 