Algebraic quantum field theory

In J. Butterfield & J. Earman (eds.), Handbook of the philosophy of physics. Kluwer (2006)
Algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools -- the theory of operator algebras, category theory, etc.. Given the rigor and generality of AQFT, it is a particularly apt tool for studying the foundations of QFT. This paper is a survey of AQFT, with an orientation towards foundational topics. In addition to covering the basics of the theory, we discuss issues related to nonlocality, the particle concept, the field concept, and inequivalent representations. We also provide a detailed account of the analysis of superselection rules by Doplicher, Haag, and Roberts (DHR); and we give an alternative proof of Doplicher and Robert's reconstruction of fields and gauge group from the category of physical representations of the observable algebra. The latter is based on unpublished ideas due to J. E. Roberts and the abstract duality theorem for symmetric tensor *-categories, a self-contained proof of which is given in the appendix.
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Gábor Hofer-Szabó & Péter Vecsernyés (2013). Bell Inequality and Common Causal Explanation in Algebraic Quantum Field Theory. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 44 (4):404-416.
Emanuele Rossanese (2013). Trope Ontology and Algebraic Quantum Field Theory: An Evaluation of Kuhlmann's Proposal. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 44 (4):417-423.

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