The DHR Analysis from Algebraic Quantum Field Theory

Algebraic Quantum Field theory, or AQFT for short, is an attempt to give a mathematically rigorous formulation of quantum field theories. In AQFT the focus is on observables which are described by abstract algebras. At first glance the formalism seems empty of a lot of the rich structure common to quantum field theories. Where are the gauge symmetries, and where are the (unobservable) fields? For a very restricted class of quantum field theories, the Doplicher Haag Roberts (DHR) analysis sheds light on these and more questions. Given an AQFT from this restricted class, the analysis constructs a Hilbert space where the observables are presented by self-adjoint operators on this space. There is also a symmetry group having a unitary action on this Hilbert space such that the operators that are invariant under this action are exactly the observables. Finally there is an algebra of operators on this space deserving of the name field operators. The whole construction is essentially unique. Although tremendous obstacles have to be overcome before any such reconstruction program can be applied to physically interesting theories like the Standard Model, this analysis provides a crucial first step. The analysis may also help shed light on the role of inequivalent representations in AQFT.