I'm just tasting a bit of QFT and want to get started. I got stuck right at the start: what is a quantum field, and how should I look at it? This question can be a follow up of the What is a field, really? question.
For me a field can be seen as a function. So saying "vector field" is the shorthand of saying a "function that maps from whatever domain to vectors of some dimensions".
What's the type of the value this function maps to when speaking of quantum fields?
In this another question. They says it's operator valued. As far as I can remember operators can be seen as "ket-bras" so the matrix product of an infinitely long column and row vector. So an infinitely large matrix. Or in other words a function that takes 2 parameters and gives back a scalar. Is that right?
So if I'm right a quantum field is a "scalar field field". So a function that maps space-time coordinates to functions that maps 2 scalars to a scalar.
In this yet another question. It's said it's a scalar valued in path integral formalism. I'm yet to understand that formalism so far. But for now I don't see how a scalar and a function valued field be equivalent...
On the other hand just out of curiosity, how many fields do we need to deal with in QFT? In classical electromagnetics we had two vector fields the electric and magnetic and the Maxwell equations that describes the time evolution. After some googling I can see there are boson fields, fermion fields, higgs field, whatever field... Are all of these quantum fields (so operator valued)? Do we have the field equations that describe all the relationship between these fields (just like the Maxwell-equations do in EM)?