More abstract than Wightman, more concrete than Haag-Kastler

Peter Morgan; Yale University

The Wightman axioms introduce a concrete Hilbert space as a representation space of the Poincaré group and Wightman fields as unbounded operator-valued distributions, whereas the Haag-Kastler axioms introduce abstract C*-algebras of bounded operators associated with regions of space-time that also act on a representation space of the Poincaré group. The construction here adopts a common practice in physics of using a manifestly Poincaré invariant formalism instead of using a representation space of the Poincaré group. This allows us to work in a relatively elementary mathematics of unbounded abstract measurement operators that is similar to quantum optics and ordinary quantum mechanics when that is sufficient to describe experimental apparatuses, but we can use a direct limit to construct a representation of the Poincaré group when that is mathematically necessary.  A construction in terms of manifest covariance in part seems worthwhile because it lets us make more contact with elementary approaches to signal analysis, which hopefully will give new, more accessible ways to teach and to explain quantum field theory.