quantum mechanics - What is an observer in QFT?

In non-relativistic quantum mechanics, an observer can be roughly describe as a system with wavefunction $\vert \psi^O \rangle$ which, upon interaction with another system $\vert \psi^S\rangle$ (in some way that measures the observable $\hat A$) evolves into the following system

$$\vert \psi^O \rangle \otimes\vert \psi^S \rangle \to \sum_\alpha a_\alpha \vert \psi^O_\alpha \rangle \otimes \vert \phi_\alpha \rangle $$

with $\hat A \vert \phi_\alpha \rangle = A_\alpha \vert \phi_\alpha \rangle$ and $a_\alpha = \langle \phi_\alpha\vert \psi^S \rangle$ the probability of measuring the system in the state $\alpha$. $\vert \psi^O_\alpha \rangle$ is the way the observer will be when it has interacted with the system in the state. From the "point of view" of the observing system, the state will be

$$\vert \psi^O_\alpha \rangle \otimes \vert \phi_\alpha \rangle$$

for some $\alpha$.

The basic example works fairly well because the two systems can be decomposed in two fairly distinct rays of the Hilbert space. But in the case of a quantum field theory, how does one define an observer? Any "realistic" object (especially for interactive QFTs) will likely be a sum of every state of the Fock space of the theory, hence I do not think it is trivial to separate the system and the observer into a product of two wavefunctionals.

Is there a simple way of defining observers in QFT? Perhaps by only considering wavefunctionals on compact regions of space? I can't really think of anything that really delves into the matter so I don't have a clue.