- #36

LittleSchwinger

Gold Member

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I think here you are discussing the lack of a rigorous existence proof, correct me if I am wrong. If so I think one should distinguish between work in rigorous QFT and algebraic field theory, as I don't think what you mention is really a problem for C*-algebra methods.The problem with AQFT, however is that for decades nobody was able to describe interacting particles in (1+3) dimensions!

Algebraic Field Theory involves using C*-algebras to extract physical predictions. This can be done at a non-rigorous level, just as in normal particle physics applications of QFT one uses operator-valued distributions without concerning oneself with their operator domain or smearing class of functions.

Recent papers by Witten on the arxiv about how deSitter space uses a Type ##II_{1}## algebra or many papers on QFT in curved spacetime are examples were C*-algebras are used in a non-rigorous or semi-rigorous way. In these "physicist" approaches to C*-algebras it has been possible to handle particles in 3+1 dimensions for around forty years and in the last ten years extract the same physical information one would from normal Feynman diagram techniques. It's become increasingly necessary to use C*-algebraic methods for discussing issues of quantum information in curved spacetime once we lack the symmetries giving a preferred Hilbert space structure or anything like particle states or global states.

Existence proofs in 3+1D are a separate issue and are equally an issue/non-issue for regular operator or path integral approaches to field theory as they are for C*-algebra methods.