An Algebraic Approach to Quantum Field Theory

It is shown that two quantum theories dealing, respectively, in the Hilbert spaces of state vectors H 1 and H 2 are physically equivalent whenever we have a faithful representation of the same abstract algebra of observables in both spaces, no matter whether the representations are unitarily equival...

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Bibliographic Details
Published in:Journal of mathematical physics 1964-07, Vol.5 (7), p.848-861
Main Authors: Haag, Rudolf, Kastler, Daniel
Format: Article
Language:English
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Summary:It is shown that two quantum theories dealing, respectively, in the Hilbert spaces of state vectors H 1 and H 2 are physically equivalent whenever we have a faithful representation of the same abstract algebra of observables in both spaces, no matter whether the representations are unitarily equivalent or not. This allows a purely algebraic formulation of the theory. The framework of an algebraic version of quantum field theory is discussed and compared to the customary operator approach. It is pointed out that one reason (and possibly the only one) for the existence of unitarily inequivalent faithful, irreducible representations in quantum field theory is the (physically irrelevant) behavior of the states with respect to observations made infinitely far away. The separation between such ``global'' features and the local ones is studied. An application of this point of view to superselection rules shows that, for example, in electrodynamics the Hilbert space of states with charge zero carries already all the relevant physical information.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.1704187