How to interpret the field function Φ in QFT?

[A. Neumaier]

OK, I see. You are right, interpreting this expression in QM it corresponds to a density (like a charge density) in space.

I would never call this density a "density operator" b/c a density operator is already defined in non-rel. QM and its meaning is something totally different

The correct name is (operator-valued) density field.

 

[A. Neumaier]

When reading about the basics of QFT I found there is a so called "density operator" which gives the particle density at a given location. It is the combination of annihilation and creation operator in the position space.

Can the density operator be understood as another possible answer? It basically tells where the particles are in the space. When I use the scattering theory it seems I get similar answer: where the particles go (spatially) after they interact (if I understand that correctly). Is the density operator used in the scattering theory?

Yes, the expectation of the density field multiplied by the mass is what becomes in the case of macroscopically many particles the thermodynamical mass density, and tells about where the mass is concentrated.