Conformal Group in Space-Time

Representations of the conformal group and their physical interpretations are discussed in order to provide a basis for interesting applications of this group in particle physics. The conformal group seems to be of particular interest in connection with the ultraviolet singularities in field theory, for it is associated with test functions which vanish on the light cone and may therefore provide an appropriate regularization of Green's functions in coordinate space. In momentum space this property corresponds to an indefinite metric in Hilbert space with the transformation by reciprocal radii as the metric operator. Two examples of representations are investigated in detail; the first one belongs to the case of spin zero and mass zero, and the second one describes a system with spin zero but nonvanishing mass. Possible physical applications of these representations are illustrated by a simple measuring process which determines the value of the electron mass.