We discuss some conventional aspects of certain model theories based both on traditional field equations and on previous solutions obtained by unconventional means. We show that the conventional approach, as the cutoff necessary to define it is removed, cannot reproduce the no-cutoff results of the unconventional approach, nor can any meaningful limit emerge unless the interaction vanishes. The no-cutoff results permit explicit solutions in the lowest "sectors" of Hilbert space and allow for a discussion of renormalization constants, two-point functions, and analyticity of the eigenvalues in the coupling constants: that is, the validity of a certain perturbation expansion. The presence of two asymptotic "one-particle" states is noted and discussed. Finally, we comment briefly on the failure of the conventional field equations, and the relevance of reducible representations of the canonical commutation relations for problems in quantum field theory.

• Received 22 June 1966

DOI:https://doi.org/10.1103/PhysRev.152.1198