From the Theory of Observables to the Theory of Quantum Fields

Abstract

The results of Chapter 1 provide a consistent and developed picture of local quantum theory conceived as the ‘axiomatic algebraic approach’ in quantum field theory. The picture, however, does not look finished since the information available on many properties of local nets (such as types of the algebras R (O), their modular structure, duality, etc.) is only fragmentary. Another unsatisfactory point concerns applications of the formalism. The whole complex of the results of Chapter 1 is still too poor for the needs of describing concrete models and processes of interaction of elementary particles. It lacks a wide range of notions and properties inherent in quantum field systems but not reflected in Axioms I-VI (cf. Introduction). As a result, the Haag-Araki or Haag-Kastler theory, like other axiomatic approaches, provides an essentially incomplete formulation of quantum field theory and should not be considered as a self-contained ‘axiomatic theory’ in the strict sense of axiomatic theories in mathematics. It is rather a starting ground, a base set of firmly established facts, which still needs to be expanded and complemented (obviously, on some other principles, not purely axiomatic any more).