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?