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.

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    $\begingroup$ I like to think of "observer/system" separation in the context of boundary formalism, where quantum fields live on the compact bulk region of spacetime bounded by a 3-surface where boundary states live. These states describe the interaction with the outside "observer", though in this picture the term "observer" completely loses its original meaning. $\endgroup$ – Solenodon Paradoxus May 3 '17 at 22:53
  • $\begingroup$ Nima Arkani-Hamed speaks very eloquently on the general question of observers in quantum field theory and quantum gravity. See for example pirsa.org/displayFlash.php?id=10080010 $\endgroup$ – Bruce Greetham May 9 '17 at 17:07
  • $\begingroup$ I answered this at physicsoverflow.org/40030 $\endgroup$ – Arnold Neumaier Nov 28 '17 at 16:55

In the book

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

  1. The relevance of the Peierls bracket for the spacetime-covariant formulation of QFT;

  2. 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.


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