Some Notes on Nonrenormalizable Field Theory

Abstract

Some nonrecoil, derivative coupling theories, which are exactly soluble, are analyzed in order to discover what it is that produces the nonrenormalizable behavior typical of derivative coupling. This behavior is found to arise as the result of an essential singularity which the operators of the derivative theories possess at the origin in coordinate space and the resulting branch point the Fourier transforms of these operators possess at the origin in the complex coupling constant domain. This branch point causes the breakdown of the expansion of the Fourier transform of the operators in powers of the coupling constant and introduces the nonrenormalizable infinities. It is shown that a coordinate space coupling constant expansion is possible, and that a Fourier transform of the operators of the derivative theories may be defined by analytic continuations and made finite by renormalization.