Outline of axiomatic quantum field theory

by R. F. Streater; Reports on Progress in Physics, 38, 771-846, 1975.

ABSTRACT

In order that fields can be treated as quantum systems, quantum mechanics needs to be carefully formulated. After a brief introduction to the problem, Section 2 describes the Hilbert space approach of von Neumann and the C*-algebra approach of Segal. In both cases symmetry groups and superselection rules can be described, and the latter allows a spontaneously broken symmetry.

Section 3 describes the Wightman theory of fields, with the main results and recent generalizations. Section 4 describes the C*-algebra approach of Haag, which follows the same physical ideas but has become an independent theory capable of explaining superselection rules.

Section 5 describes dispersion relations and the analytic properties in momentum space, with brief reference to the Froissart bound and other consequances of positivity. Finally, Section 6 describes the Euclidean approach leading to a history integral formulation.

The review as a whole is a survey of the work over the period 1954-1974. It is confined to relativistic theories, omitting applications to many-body theory and avoiding the actual solution of the field equations. That nonlinear field equations, at least in two- and three-dimensional space-time, have solutions has recently been proved by Glimm and Jaffe. This suggests that field theory will be solved and used to describe elementary particles within the next few years.

Sept. 1974.

REMARKS

This work was completed at the Institut des hautes etudes, Bures-sur-Yvette, Paris, in the Spring of 1974. I shared a room there with a Chinese physicist, who said that I had "good working habits". In fact, I had something definite to do, and a lot of articles to find and refer to. I came in early and left late. The article was submitted in the early Spring, but the editor said that it was too long, and needed to be reduced by 25 percent. I then spent months rewording it, using shorter words, cutting out elementary explanations, omitting some topics and references. I then resubmitted it, and was told that the editors were very surprised that I had done all this work; nobody else has shortened a commissioned article for them, and they would have published it if I had said that every word was essential. It is a fact that the original was a better article, and is easier to read, than this version.

The optimistic last sentence of the abstract has not turned out to be the case. Some people think that no scalar field theory with nontrivial S-matrix exists. The experimental difficulty in finding the Higgs particle might be said to support this. Quantum fields are still thought to describe elementary particles, but they must be gauge fields (which do not allow an easy axiomatic formulation).

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