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Wikipedia describes many variants of quantum field theory:

  • conformal quantum field theory
  • topological quantum field theory
  • axiomatic/constructive quantum field theory
  • algebraic quantum field theory

Are these approaches to the same thing or actually different?

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up vote 29 down vote accepted








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Even though the answer says QFTs should be studied by doing, I really like this very clear explanation of the "taxonomy". It is a pleasure to read a short definition and the purpose or field of application of each QFT version stated like this :-). – Dilaton Jan 21 '12 at 11:00

Irving Segal, the obnoxious and disliked genius mathematician of MIT, once went around asking physicists « ¿what is a quantum field? » As he tells the story, only Enrico Fermi gave him an answer (after pausing for a little thought). « The occupation number formalism.»

What does this mean. What Fermi meant is that for a given particle, say an electron, there is a quantum field (in this case, the electron field. For a photon, it would be the electromagnetic field of Maxwell.) This particle could have many different states. Well, make a list of each possible state and put in the appropriate place in the list the number of particles in the Universe which are in that state, i.e., as we say, « occupy » that state. That is a quantum field. All the rest is sound and fury, ahem, all the rest is the attempt to study this concept mathematically, as prof. Motl explained very well. I wish this helped....

If I remember correctly, the beginning of Feynmans's notes on Quantum Electrodynamics, the first successful quantum field theory, give a short exposition of the occupation number formalism.

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As Joseph says, Luboš touches everything in his usual way.

Less usefully than Luboš' Answer, but perhaps more specific for a mathematician, for something hypertopical you might try a Stonybrook workshop that ended yesterday on "Mathematical Foundations of Quantum Field Theory". has videos of talks by some of the players. The Buchholz PDF (no video in this case) looks a nice summary of AQFT at first glance, and Jaffe talks about a number of the variants you mention. You'll have to choose amongst the other talks according to your mathematical tastes, but the stated aim of the conference is

to review the current state of the field, agree on what has been accomplished and what could be accomplished by a systematic application of the known ideas and techniques, try to identify where new ideas and techniques could have the most impact, and agree on a list of important problems and questions whose resolution would at the least serve as benchmarks to measure our progress, and at best significantly advance the field.

Math papers usually leap into their own particular approach, citing a few papers to place themselves. I can't think of a paper that really taxonomizes and places all the approaches you mention relative to each other.

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protected by Qmechanic Sep 25 '15 at 6:43


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