DIMENSION OF SPACE-TIME

Abstract

In order to make it operationally accessible, it is proposed that the notion of the dimension of space-time be based on measure-theoretic concepts, thus admitting the possibility of noninteger dimensions. It is found then, that the Hausdorff covering procedure is operationally unrealizable because of the inherent finite space-time resolution of any real experiment. We therefore propose to define an operational dimension which, due to the quantum nature of the coverings, is smaller than the idealized Hausdorff dimension. As a consequence of the dimension of space-time less than four, relativistic quantum field theory becomes finite. Also, the radiative corrections of perturbation theory are sensitive on the actual value of the dimension 4–ε. Present experimental results and standard theoretical predictions for the electromagnetic moment of the electron seem to suggest a nonvanishing value for ε.