E_i=0$$ Furthermore, we assume its standard deviation in e.g. the vacuum, $\Omega$, to be finite (which is automatically the case for bounded operators, but we want to include also more general statistical variables) $$\sum\,p_i\cdot E_i^2=(\Delta_{\Omega}E)^2<\infty$$

In a first step we make the simplifying assumption (taking e.g. a bounded function of the energy)

$$|E_i|\leq\Lambda\quad\text{for all}\quad E_i$$

We are interested in the amount of negative (e.g. energy) fluctuations we will observe in measurements. A reasonable quantitative measure of it is $$\sum\,p_i^-\cdot |E_i^-|$$ with $E_i^-,p_i^-$ the negative spectral values and their corresponding probabilities. We then have (with $|E_i^-|/\Lambda\leq 1$) $$\sum\,p_i^-\cdot |E_i^-|/\Lambda\geq  \sum\,p_i^-\cdot |E_i^-|^2/\Lambda^2$$ For the lhs we have $$\sum\,p_i^-\cdot |E_i^-|/\Lambda=\sum\,p_i^+\cdot
  |E_i^+|/\Lambda$$ as the expectation of $E$ was assumed to be zero.

This yields $$\begin{gathered}
\sum\,p_i^-\cdot |E_i^-|/\Lambda=1/2\cdot
  \left(\sum\,p_i^-\cdot |E_i^-|/\Lambda+\sum\,p_i^+\cdot
    |E_i^+|/\Lambda\right)\geq\\ 1/2\cdot
  \left(\sum\,p_i^-\cdot |E_i^-|^2/\Lambda^2+\sum\,p_i^+\cdot
    |E_i^+|^2/\Lambda^2\right)      \end{gathered}$$ I.e. $$\sum\,p_i^-\cdot |E_i^-|\geq 1/2\cdot\sum\,p_i\cdot
  E_i^2/\Lambda=1/2\Lambda\cdot (\Delta_{\Omega}E)^2$$ On the other hand (Cauchy-Schwartz) $$\left(\sum\,p_i^-\cdot |E_i^-|\right)^2=1/4\cdot
  \left(\sum\,p_i^-\cdot |E_i^-|+\sum\,p_i^+\cdot
    |E_i^+|\right)^2\leq 1/4\cdot\sum\,p_i\cdot |E_i|^2$$ We hence arrive at the result

If the observable, $E$, is bounded, so that its spectral values fulfill $|E_i|\leq\Lambda$, we have the estimate $$1/2\Lambda^{-1}\,(\Delta_{\Omega}E)^2\leq
  \left(\sum\,p_i^-\cdot |E_i^-|\right)\leq 1/2\,
  (\Delta_{\Omega}E)$$ with $p_i$ the probabilities that the negative spectral values $E_i$ occur in an observation. That is, we manage to bound a quantity, which 