What does this equation mean?

$\mathcal F_x(\Omega)$ , viewed as a function of $\Omega$ for fixed everything else, has a natural reading as a spectrum: it says how much excitation probability accumulates at each possible gap, for a whole notional bank of detectors sharing one trajectory, one switching profile, and one field state, differing only in $\Omega$ . Restricting to the excitation branch $\Omega$>0 — “did this detector behave as though it absorbed a quantum” is the only branch this article treats as bearing on particle content, since the decay branch $\Omega$<0 mixes genuine field structure with detector-specific spontaneous-emission physics that is present even in flat, empty spacetime with no field excitation at all — normalize: $p_x(\Omega) = \frac{\mathcal F_x(\Omega)}{\displaystyle\int_0^\infty \mathcal F_x(\Omega')\, d\Omega'}, \qquad \Omega > 0$. defined whenever the denominator is finite and nonzero. Given two such normalized spectra, for a detector on worldline x and a detector on worldline y , define their disagreement as the total variation distance between the two probability densities: