is difficult to measure directly, by quantities, which are usually more easily accessible.

We can generalize this result to situations where the $E_i$ are not exactly bounded by some $\Lambda$ but are bounded in at least an essential way. We assume that there exists some $\Lambda$ so that $$\sum_{|E_i|>\Lambda}\,p_i\cdot |E_i|^2<\varepsilon_{\Lambda}$$ We then have $$\begin{gathered}
\sum\,p_i^-\cdot |E_i^-|/\Lambda=1/2\,\left(\sum\,p_i^-\cdot |E_i^-|/\Lambda+\sum\,p_j^+\cdot |E_j^+|/\Lambda\right)\geq\\
1/2\,\left(\sum_{|E_i^-|\leq\Lambda}\,p_i^-\cdot |E_i^-|/\Lambda+\sum_{|E_j^+|\leq\Lambda}\,p_j^+\cdot |E_j^+|/\Lambda\right)\geq\\
1/2\,\left(\sum_{|E_i^-|\leq\Lambda}\,p_i^-\cdot |E_i^-|^2/\Lambda^2+\sum_{|E_j^+|\leq\Lambda}\,p_j^+\cdot |E_j^+|^2/\Lambda^2\right)\geq\\
1/2\,\left(\sum\,p_i\cdot
  |E_i|^2/\Lambda^2-\varepsilon_{\Lambda}/\Lambda^2\right)\end{gathered}$$

Under the above assumption of an essentially bounded $E$ we have $$\sum\,p_i^-\cdot |E_i^-|\geq 1/2\Lambda\,\left((\Delta_{\Omega}E)^2-\varepsilon_{\Lambda}\right)$$

Another, rigorous, but not quantitative, argument can be derived from axiomatic quantum field theory (see e.g. [@Wigh1]). It follows from the so-called [Reeh-Schlieder theorem]{} that there are no [local observables]{} or [fields]{} which can annihilate the vacuum (where by local we mean that the objects commute for space-like separation). I.e., we have for any local $A$ (with $A=A^*$) $$A\,\Omega\neq 0\quad\Rightarrow\quad
  (A\Omega|A\Omega)=(\Omega|A^2\Omega)\neq 0$$ We take now as local observable the energy density integrated over a certain spatial region, $V$, $$H_V:=\int_V\,h_{00}({\mathbf}{x},0)\,d^3\!x$$ One usually normalizes $h_{00}(x)$ to $$(\Omega|h_{00}(x)\,\Omega)=0\quad\Rightarrow\quad (\Omega|\int_V\,h_{00}({\mathbf}{x},0)d^3\!x\,\Omega)=0$$

The classical expression of the energy density, being derived in Lagrangian field theory, is positive. The corresponding quantized expression, after a necessary [Wick-ordering]{} (see e.g. [@Bjo] or [@Itz]) is however no longer positive definite 