Detector-based measurements for QFT: Two issues and an algebraic QFT proposal

We present and investigate two issues within the measurement scheme for QFT presented by Polo-G\'omez et al. [Phys. Rev. D 105, 065003 (2022)]. We point out some discrepancies that arise when the measurement scheme is applied to contextual field states and show that $n$-point function assignments based on local processing regions sometimes lead to inconsistencies. To solve these issues, we modify the measurement scheme to use nonrelativistic detectors to induce an update rule for algebraic states in the Haag-Kastler formulation of quantum field theory. In this way, $n$-point functions can be consistently evaluated across any region having a definite causal relation with measurements.