Mathematical Considerations and General Postulates

Abstract

The developments mentioned at the end of the last chapter had restored faith in quantum field theory. On the other hand it could not be overlooked that in spite of the great success of QED there remained ample reasons for dissatisfaction. It was not understood how the theory could be formulated without recourse to the perturbation expansion. The detailed renormalization prescriptions, needed to eliminate all infinities, had become quite complicated and not easily communicable to one who had not acquired familiarity with the procedure the hard way, namely by doing the computations and learning to avoid pitfalls. Apart from QED there existed models for a theory of weak interactions which could be compared in lowest order with experiment but which was not renormalizable, and there were the meson theories of strong interaction where perturbation expansions did not prove to be very useful. This mixture of positive and negative aspects of quantum field theory provided the motivation in the fifties to search for a deeper understanding of the underlying principles and for a more concise mathematical formulation. K.O. Friedrichs described his feelings about the literature on quantum field theory as akin to the challenge felt by an archeologist stumbling on records of a high civilization written in strange symbols. Clearly there were intelligent messages, but what did they want to say? (private conversation 1957). His answer to the challenge was his book on the mathematical aspects of quantum field theory [Friedrichs 1953] where, among other things, he pointed out the existence of inequivalent representations of the canonical commutation relations and discussed examples of these under the heading “myriotic fields”.