Outline of axiomatic relativistic quantum field theory

In order that fields can be treated as quantum systems, quantum mechanics needs to be carefully formulated. After a brief introduction to the problem, the Hilbert space approach of von Neumann and the C*-algebra approach of Segal are described. In both cases symmetry groups and superselection rules can be described, and the latter allow a spontaneously broken symmetry. The Wightman theory of fields, with the main results and recent generalizations, and the C*-algebra approach of Haag, which follows the same physical ideas but has become an independent theory capable of explaining superselection rules, are considered. Dispersion relations and the analytic properties in momentum space, with brief reference to the Froissart bound and other consequences of positivity, and finally, the Euclidean approach leading to a history integral formulation, are described. The review as a whole is a survey of the work over the period 1954 to 74. It is confined to relativistic theories, omitting applications to many-body theory and avoiding the actual solution of the field equations. (author)