Recently I've started to study Quantum Field Theory and I think that since then I've been missing the whole point with the theory, and as a result I've been getting really confused.
Let's start with Classical Field Theory and its most well known example: Electrodynamics.
What is the basic problem in Electrodynamics? Well it is to understand how charges affect each other. This problem has two steps: the first is to understand how charges are affected by others by mean of fields and the second is to understand how these fields are generated and how they evolve in time.
Historically we can say that one started discovering the influences on charges as forces and then introduced the field viewpoint. The whole historical process on how one got to the Maxwell equations for the fields is covered in EM textbooks like Griffiths.
That is all fine, but now let's get to QFT. At first sight my thought was:
In analogy to Classical Field Theory, the basic problem in QFT is probably obviously be to find the fields $\phi(x)$, which are now operator-valued, how they evolve, and how they interact with other systems.
This would be the natural generalization of what we see in Classical Electrodynamics textbooks.
This lead me to a bunch of confusion when I started reading books and finding out that authors never, ever, end up defining or finding the fields $\phi(x)$. In truth, based on this discussion actually they suppose the field exists and obeys some commutation relations and start working with this.
This is in sharp contrast with CFT, where the whole point is to find out what is the functional dependency $\phi(x)$ of the field on events $x\in M$ on spacetime.
This led me to question the following: what is really the main objective of Quantum Field Theory, if it is not finding the field and its time evolution $\phi(x)$ as one does, e.g., in Electrodynamics? What is the whole point with the theory by the way? What is really the problem being tackled and what kind of thing is being sought?
If possible, I think some historical explanation could clarify this matter, but every help is appreciated.