Statistical Geometry and Fundamental Particles

To take account of the fact that the existence of a fundamental length $a$, the classical electron radius, sets a limit to the accuracy with which the position of a point can be measured, it is proposed to introduce two spaces, an "abstract" space consisting of points, and an "observable" space in which one deals with elementary volumes correlated to the points of the former by means of a statistical distribution function in the form of a three-dimensional Gaussian error function. Such a function is not Lorentz invariant, but one can obtain Lorentz covariance in the observable space by carrying out the usual Lorentz transformation in the abstract space. If one assumes that the usual equations for wave fields, in which the fundamental particles are regarded as points, are valid in the abstract space, then one can obtain corresponding equations in the observable space, with the particles behaving as if they had finite volumes. The difficulties associated with infinite self-energies and singularities in the interactions between particles, as calculated by the usual perturbation method, disappear, but the difficulty associated with the divergence of the series of successive orders of perturbations remains.