That's what I've been wondering for a long time. According to McMahon, in QFT "fields are promoted to operators". I don't know what this means.
I understand the idea of a physical state being represented by a vector and an observable by an operator and how all that works, but if the field is an operator, upon what does it operate? What kind of mathematical object, and what does this object represent? This seems like a completely different formulation to me.