  Using the Schwinger-Keldysh path integral, we draw a connection between
localized quantum field theories and more commonly used models of local probes
in Relativistic Quantum Information (RQI). By integrating over and then tracing
out the inaccessible modes of the localized field being used as a probe, we
show that, at leading order in perturbation theory, the dynamics of any finite
number of modes of the probe field is exactly that of a finite number of
harmonic-oscillator Unruh-DeWitt (UDW) detectors. The equivalence is valid for
a rather general class of input states of the probe-target field system, as
well as for any arbitrary number of modes included as detectors. The path
integral also provides a closed-form expression which gives us a systematic way
of obtaining the corrections to the UDW model at higher orders in perturbation
theory due to the existence of the additional modes that have been traced out.
This approach vindicates and extends a recently proposed bridge between
detector-based and field-theory-based measurement frameworks for quantum field
theory [T. R. Perche et al., Particle detectors from localized quantum field
theories, Phys. Rev. D 109, 045013 (2024)], and also points to potential
connections between particle detector models in RQI and other areas of physics
where path integral methods are more commonplace -- in particular, the
Wilsonian approach to the renormalization group and effective field theories.
