# Algebraic Quantum Field Theory

February 14, 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 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 The latter is based on unpublished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
*-categorie s, a self-contained proof of which is given in the appendix.

Keywords:

Superselection Theory