Five papers on the mathematical aspects of the quantum theory of fields previously printed in Communications on Pure and Applied Mathematics are republished in the present volume.
As stated in the introduction to these papers, they represent an attempt towards a mathematically consistent treatment of basic notions and simple problems in the quantum theory of fields. Specifically, we shall deal with problems in which the forces acting on the field depend linearly on the field quantity and in which, therefore, this quantity satisfies a linear equation; these linear problems are treated rather exhaustively. Many more or less straightforward conclusions, which in the physical literature are implied but not always explicitly stated, are here developed in great detail.
It is true that nonlinear interactions occupy at present the center of interest in the quantum theory of fields. A mathematically satisfactory' treatment of such nonlinear problems could not. however, be given. It may be doubted whether nonlinear interaction in the form so far assumed admits such a treatment; in fact it has frequently been suggested that fundamentally different laws of nonlinear interaction should be adopted.
The Introduction contains a description of the content of the five parts. Since these five parts were written successively, the actual content of the later parts does not always exactly coincide with what has been announced in the introduction. The point of view has gradually changed, and some inconsistency has crept into the notations. The most important deviations are listed in “Corrections and Comments.” In particular, Part VI has been omitted.
The appendix contains a Lorentz invariant treatment of the problem dealt with in Part V, on the basis of work by B. Zumino which will appear in forthcoming papers.