Why we use fields instead of wave functions? I'm starting studying QFT and I have a problem about a concept that maybe it's stupid but I don't understand exactly.
Why we use fields instead of wave functions? I tried to answer myself, and I found on internet that a reason is because it's so difficult to describe a wave functions of many particles, and since particles behaves both like particle and waves we can use a field description. But this answer doesn't make me satisfied. Can someone explain me precisely why we use fields, abandoning the classical QM? Or also some text to understand?
 A: The short answer is: we don't. Everything, be it a particle with a finite dimensional configuration space or a field with an infinite dimensional configuration space, has a wavefunction (see a previous answer). The fact that one can begin QFT without explicit reference to the wavefunction (or wavefunctional as it's more properly called) is very much analogous to how the ladder operator formalism that allows you to solve the harmonic oscillator without knowing about Hermite polynomials.
So fields arise for a simpler reason: QM shouldn't be something we only apply to point particle dynamics. Classical field theories have been written down since the 1800s so it is important to be able to quantize those too. A more subtle point is that QFT is also needed to model experiments that scatter a fixed number of particles. But this is because pair production can happen in a relativistic theory so the number of particles in the final state can become arbitrarily large as the center of mass energy goes up.
