What is an observer in QFT?

Question

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.

This post imported from StackExchange Physics at 2017-10-11 16:32 (UTC), posted by SE-user Slereah

Comments

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.

This post imported from StackExchange Physics at 2017-10-11 16:32 (UTC), posted by SE-user Solenodon Paradoxus

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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

This post imported from StackExchange Physics at 2017-10-11 16:32 (UTC), posted by SE-user Bruce Greetham

Answer 1

As long as only scattering experiments are involved, the observer prepares the in-state at time $-\infty$ and takes a measurement on the out-state at time $+\infty$. In this setting, the observer is completely outside the QFT formalism. 

A correct account of an observer in relativistic QFT would have to model it as a very massive (many-particle) part of the quantum field localized in some region, in the spirit of the nonrelativistic treatment by in the work by Allahverdyan et al. reviewed by me here.  I haven't seen anything like this for the relativistic case. 

On the other hand, papers by Peres and Terno (e.g., https://arxiv.org/abs/quant-ph/0212023) discuss relativistic quantum mechanics (not QFT) for multiple observers in different Lorentz frames.

Comments

I generally agree with Arnold's opinion and comments, and I would like just underline that "the observer" makes sure that the incident particles have the necessary energy/momentum and other properties, that the target is also has the necessary properties, that the collisions happen in a deep vacuum (no other obstacles), that the measuring devices detect the scattering products correctly, etc., etc. The observer makes a huge preparatory work, accompanying work, and result processing work. In other words, he makes a permanent work keeping the necessary and sufficient conditions for a given experiment. Only with all that we may be sure of the initial and final states, withing the experimental uncertainties established with the observer. As you can see, "observer" includes experimentalists, theorists, staff and stuff, all working to make sure that a (thus simplified) QFT is applicable to the studied processes.

Answer 2

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):

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.

Comments

Does the Peierls bracket exist for interacting systems too? In particular, does the statement about 2-point functions still apply? It sounds very interesting.

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The positive frequency part of the Peierls bracket is the vacuum 2-point function of a free (linear) quantum field theory corresponding to the associated linear classical theory. Did you mean this? I don't know of any nonlinear version of this statement, and your PhysicsForums article is too big to see quickly the precise statement you intend here.

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The Peierls bracket exist generally, also for interacting field theories. I recommend Khavkine 14.

Answering whether it's relation to the vacuum 2-point function goes beyond perturbation theory would seem to require non-perturbative theory which does not exist at this point.

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Unfortunately, the book by deWitt has no significant contribution to the original question; no definition of a model observer is given. Chapters 7 and 8 of volume 1 are about classical measurement and heuristic quantum measurement of a primitive form. All problems are swept under the many-worlds carpet where no measurement is ever done - since there is no mechanism for splitting the worlds, and without splitting there are no definite measurement results. (See also my critique of Everett's many-worlds = relative state interpretation.)