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 S. Doplicher, R. Haag, and J. E. Roberts (DHR); and we give an alternative proof of Doplicher and Roberts' 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 The latter is based on unpublished ideas due to Roberts and the abstract duality theorem for symmetric tensor * -categories, a self-contained proof of which is given in the appendix.

No abstract

The aim of the paper is twofold. First, we show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category G − LocA of twisted representations. This category is a braided crossed G-category in the sense of Turaev [60]. Its degree zero subcategory is braided and equivalent to the usual representation category Rep A. Combining this with [29], where Rep A was proven to be modular for a nice class of rational conformal models, and with the construction of invariants of G-manifolds in [60], we obtain an equivariant version of the following chain of constructions: Rational CFT ❀ modular category ❀ 3-manifold invariant.Secondly, we study the relation between G−LocA and the braided (in the usual sense) representation category Rep A G of the orbifold theory A G . We prove the equivalence Rep A G ≃ (G−LocA) G , which is a rigorous implementation of the insight that one needs to take the twisted representations of A into account in order to determine Rep A G . In the opposite direction we have G− LocA ≃ Rep A G ⋊ S, where S ⊂ Rep A G is the full subcategory of representations of A G contained in the vacuum representation of A, and ⋊ refers to the Galois extensions of braided tensor categories of [44,48].Under the assumptions that A is completely rational and G is finite we prove that A has g-twisted representations for every g ∈ G and that the sum over the squared dimensions of the simple g-twisted representations for fixed g equals dim Rep A. In the holomorphic case (where Rep A ≃ Vect C ) this allows to classify the possible categories G−LocA and to clarify the rôle of the twisted quantum doubles D ω (G) in this context, as will be done in a sequel. We conclude with some remarks on non-holomorphic orbifolds and surprising counterexamples concerning permutation orbifolds. * Supported by NWO.

No abstract