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In nonrelativistic quantum mechanics the state of a system is characterized by a vector of a Hilbert space.

To characterize a state we need a complete set of commuting observables, and once we have that we label the state using some set of quantum numbers.

Now, in field theory I have been told that the Hilbert space of possible states is the space of all field configurations. I see that, but nothing has yet been mentioned to me of something analogous to quantum numbers.

So, my question is, how is a possible field configuration labeled in QFT? Do we have something analogous to a complete set of commuting observables? do we have quantum numbers?

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The problem is that entities like the Dirac field are not observable. We cannot measure the value of the electron field at a particular point in space; it is impossible to experimentally find a specific configuration of an electron field in space. What we can measure are results of scattering experiments and cross sections. For this reason the QFT is optimized to allow the calculation of these measurable quantities. So a typical calculation in QFT is: what is the probability that when electron collides with a positron, the output will be a Z boson?

You can somehow label a possible field configuration in QFT. It is done by a wave functional. It gives a "probability amplitude" for every field configuration. But this approach is rarely used and it is not addressed in most books.

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