The Doplicher-Roberts reconstruction theorem

Sander Wolters (RU)

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 presented as self-adjoint elements of abstract C*-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? The DHR analysis demonstrates that at least for a very restricted class of quantum field theories we can construct this structure in an essentially unique way from the algebra of observables. The DHR analysis relies to a large extent on the Doplicher Roberts reconstruction theorem. This theorem tells us when an abstract linear tensor category can be identified as the category of finite dimensional complex representations of a compact group. This group, which is determined up to isomorphism, acts as a (global) gauge group in the DHR analysis. The DR Reconstruction Theorem is a generalization of the classical Tannaka-Krein theorem which demonstrates how a compact group can be reconstructed from its finite dimensional representations.

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