First of all, I'm a mathematician, so forgive me for my possible trivial mistakes and poor knowledge of physics.
In a QFT, we just start with a field (scalar, vectorial, spinorial, gauge etc), so I would like to know what are the observables and the states in this context.
In QFT, the general approach would be by using the Fock space (for the free field case, since I don't really know if this would be true for the interacting one) and getting down, by using the particles associated to the operators $a$ and $a^{\dagger}$, to QM particles (I don't really know if this is true, because the number of particles is not constant and depends on the observer) or by using the wave functional interpretation (a functional on the space of field configurations satisfying Schrödinger equation), though I've heard that this functional is not Lorentz covariant (by the way, any proof?). However, according to this article (David John Baker, Against Field Interpretations of Quantum Field Theory, http://core.ac.uk/download/pdf/11921990.pdf) the wave functional interpretation is equivalent to the Fock space, so, in any case, this interpretation is not physically reasonable.
In AQFT, in contrast, the operators are already given (so we already have the observables). Furthermore, if the Lorentzian manifold is globally hyperbolic, a Cauchy hyper surface would be a possible interpretation for a state.
In other aspect, are the quantized fields of a given QFT really observables in the sense that they measure?
Now, adding gauge fields, everything will be groupoid valued and observables would be defined on quotients by the gauge group. In this context, I haven't really seen anything written about states and I have no idea on how the Fock space would be. The naive approach would be to consider the wave functional interpretation with domain in a groupoid.
Furthermore, if we restrict ourselves to TQFT, CFT or other specific class of field theories, would all this problem be solved?
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