and with $$\langle P\rangle_0=\langle Q\rangle_0 =0$$ in the groundstate, $\psi_0$, we have $$\hbar\cdot\omega/2=\langle H\rangle_0=1/2m\cdot
  \langle (P-\langle P\rangle_0)^2\rangle_0+ m\omega/2\cdot\langle (Q-\langle Q\rangle_0)^2\rangle_0$$ with $$\langle (P-\langle P\rangle_0)^2\rangle_0\cdot \langle
  (Q-\langle Q\rangle_0)^2\rangle_0\geq \hbar^2/4$$ which follows from $[P,Q]=-i\hbar$. In the same way we have in (matter-free) QED: $$H=const\cdot ({\mathbf}{E}^2+{\mathbf}{B}^2)$$ with $$\langle {\mathbf}{E}\rangle_0=\langle {\mathbf}{B}\rangle_0
  =0$$ so that again $\langle H\rangle_0$ is a sum over pure vacuum fluctuations of the non-commuting quantities ${\mathbf}{E}$ and ${\mathbf}{B}$. One should however note that in the quantum field context products of fields at the same space-time point have to be Wick-ordered (in order to be well-defined). It is, on the other hand, frequently argued that with gravity entering the stage, these eliminated zero-point energy fluctuations have to be taken into account again. In our view, this problem is not really settled.

We now come to an important point. It is our impression that in some heuristic discussions (vacuum fluctuations as virtual particle-antiparticle pairs) the consequences of the fact that the vacuum state is an exact eigenstate of the Hamiltonian in a Hilbert space representation of some quantum field theory are not fully taken into account. I.e., we have $$H\,\Omega=0$$ (provided the ground state energy is for convenience normalized to zero; note however 