What does this equation mean?

Two invariances follow immediately, and one does not. $D_{\rm det}$ does not change if a trajectory is described using a different auxiliary curve parameter before being converted to proper time — proper time itself is not a gauge choice, it is the arc length of the worldline, fixed by the metric once the trajectory and the metric are fixed, so there is no freedom here to exploit. $D_{\rm det}$ also does not change under a Poincaré transformation applied identically to both worldlines and to the field vacuum together, since W is Poincaré invariant and $\tau$ , $\Omega$ , and $\chi$ are all defined intrinsically along each worldline. What does change $D_{\rm det}$ , and is meant to, is any actual physical difference between the two setups. The discipline this article insists on is separating that meant-to variation from an unmeant one: before attributing a nonzero $D_{\rm det}(x,y)$ to “the worldlines disagree about particle content,” four other things have to be checked as identical between the two detectors, because each one moves $D_{\rm det}$ on its own, with nothing to do with x or y as trajectories through spacetime.