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Axiomatic quantum field theory (e.g. the wightman formalism and constructive quantum field theory) is an important subject. When I look into textbooks and papers, I mostly find that the basic constructions involve functional analysis and operator algebra. Now, modern developments in the field of quantum field theory involve subjects such as algebraic geometry, topology, and knots. Mostly modern developments deal with supersymmetric quantum field theory.

My question is: What is the status of current research on the mathematical foundations of the dynamics of "non topological" quantum field theory? Are the older approaches to the subjects, such as the ones in Glimm and Jaffe's and Wightman's books, obsolete?

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    $\begingroup$ Paging @Urs Schreiber... $\endgroup$
    – David Roberts ♦
    May 20, 2020 at 5:26
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    $\begingroup$ The answer to the second question "Are the older approaches to the subjects, such as the ones in Glimm and Jaffe's and Wightman's books, obsolete?" is: No. $\endgroup$
    – Abdelmalek Abdesselam
    May 20, 2020 at 13:58

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The Axiomatic Quantum Field Theory from the 1950's has been rebranded as Algebraic QFT (keeping the same abbreviation), and the emphasis has shifted somewhat from quantum fields to local observables. The Wightman axioms for fields are then replaced by the Haag–Kastler axioms for the algebra of observables. For two relatively recent overviews see Current trends in axiomatic quantum field theory (1998) and Algebraic quantum field theory (2006). A concise summary is at nLab. An alternative approach to AQFT is FQFT (Functorial QFT), which focuses on states rather than observables.

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  • $\begingroup$ Thanks. But algebraic QFT fails to capture topological and geometric phenomena in QFT, doesn't it? For example, it does not address the mathematical nature of quantum dualities, such as S-duality. $\endgroup$
    – Physicsstudent000
    May 20, 2020 at 11:34
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    $\begingroup$ @Physicsstudent000 : AQFT -> blob homology -> causally local net of observables -> S-matrix. $\endgroup$
    – Eric Towers
    May 20, 2020 at 16:49
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    $\begingroup$ @Physicsstudent000 You might possibly find this article interesting: arxiv.org/pdf/1511.00316.pdf Maybe it addresses your objections? $\endgroup$
    – Severin Bunk
    May 22, 2020 at 7:01
  • $\begingroup$ Sorry if I ask a stupid question. I understand that Euclidean field theory usually uses path integral as its way of quantizing fields. So why the finding of Osterwalder (focusing on the Euclidean formalism of QFT) would be classified into AQFT? From nLab, I understand that it is FQFT which is the formalization of path integral quantization. $\endgroup$
    – Qi Tianluo
    Jul 15, 2021 at 15:34

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