Q:

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.

A:

In the book

Bryce DeWitt, The Global Approach to Quantum Field Theory, Oxford 2003

the author amplifies two points (right at the beginning, first page of the preface on page "v" in volume 1):

The relevance of the Peierls bracket for the
spacetime-covariant formulation of QFT;
its implication for a good theory of observers and measurement in QFT, which he attributes to Bohr-Rosenfeld 1933.

There is no doubt about the relevance of the Peierls bracket: This is the covariant form of the Poisson bracket (explained in detail in "Mathematical QFT - 8. Phase space"); and the positive frequency part of its integral kernel is nothing but the vacuum 2-point function (explained in "Mathematical QFT - 9. Propagators").
Chapters 7 and 8 of DeWitt's book (volume 1) mean to lay out a theory of measurement and observers in QFT based on this.  I don't feel quite qualified to review this here, but if you are interested, I would suggest to take a look.

