algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
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 1949.
While highly successful, the conceptual nature of this original formulation, in particular of the process of “removal of UV-divergences”, had remained mysterious (see Scharf 1995, section 0.0 for survey):
Dirac 1951: “[the theory is] an ugly and incomplete one”
Feynman 1966, Nobel lecture: “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 1985, Chap. 4. “Loose Ends”: “…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…”
These conceptual mysteries were resolved in a mathematically rigorous formulation of ("re"-)normalization in perturbative QFT by Epstein & Glaser 1973, based on Bogoliubov & Shirkov 1959 and Stückelberg 1951, now known as causal perturbation theory; laid out, together with other rigorous approaches, in the seminal Erice summer school proceedings (Velo & Wightman 1976) and later developed by Scharf 1995 2001 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 S. Schweber: QED and the men who made it: Dyson, Feynman Schwinger and Tomonaga, Princeton University Press (1994) [ISBN:9780691033273, jstor:j.ctv10crg18, pdf]
Günter Scharf, section 0.0 of Finite Quantum Electrodynamics – The Causal Approach, Springer (1995)
Last revised on September 25, 2025 at 13:33:58. See the history of this page for a list of all contributions to it.