algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
field theory: classical, pre-quantum, quantum, perturbative quantum
quantum mechanical system, quantum probability
interacting field quantization
The origins of perturbative quantum field theory go back to informal ideas on quantum electrodynamics due to Julian Schwinger, Shin'ichirō Tomonaga, Richard Feynman and Freeman Dyson, culminating in the informal idea of “renormalization” due to (Dyson 49).
While highly succesful, the conceptual nature of this original formulation, in particular of the process of “removal of UV-divergences”, had remained mysterious (see Scharf 95, section 0.0 for survey):
$[$ the theory is $]$ an ugly and incomplete one (Dirac 51)
I think that the renormalization theory is simply a way to sweep the difficulties of the divergences of electrodynamics under the rug. I am, of course, not sure of that. (Feynman 66, Nobel lecture)
…is technically called ‘renormalization.’ But no matter how clever the word, it is what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory $[...]$ is self-consistent. It’s surprising that the theory still hasn’t been proved self-consistent one way or the other by now; $[...]$ What is certain is that we do not have a good mathematical way to describe the theory of quantum electrodynamics: such a bunch of words… (Feynman 85, Chap. 4. “Loose Ends”)
These conceptual mysteries were resolved in a mathematically rigorous formulation of ("re"-)normalization in perturbative QFT by (Epstein-Glaser 73), based on (Bogoliubov-Shirkov 59 and Stückelberg 51), now known as causal perturbation theory; laid out, together with other rigorous approaches, in the seminal Erice summer school proceedings (Velo-Wightman 76) and later developed by (Scharf 95, Scharf 01) and eventually grown into perturbative AQFT (see there for more).
The renormalization process was first suggested in
Paul Dirac, Proc. Roy. Soc. A 209, 291 (1951)
Richard Feynman, Nobel lecture, reproduced in Science 153, 699 (1966)
Richard Feynman, QED: The Strange Theory of Light and Matter, 1985
Silvan Schweber, QED and the men who made it: Dyson, Feynman Schwinger and Tomonaga, Princeton Series in Physics, 1994
Günter Scharf, section 0.0 of Finite Quantum Electrodynamics -- The Causal Approach, Berlin: Springer-Verlag, 1995, 2nd edition