Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory: classical, pre-quantum, quantum, perturbative quantum

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



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).


Original articles

The renormalization process was first suggested in

  • Freeman Dyson, The radiation theories of Tomonaga, Schwinger and Feynman, Phys. Rev. 75, 486, 1949 (pdf)

Review and history

category: reference

Revised on January 15, 2018 09:48:36 by Urs Schreiber (