Stochastic Lorentz observers and the divergences in quantum field theory

Abstract

Sharp space-time coordinatesx are replaced by stochastic space-time coordinatesX with meansx and dispersion measured by a universal lengthλ as the kinematical basis of Lorentz covariant field theories, classical or quantum. It is found that 4-dimensional distributions are not possible, but 3-dimensional (unsharp rods, sharp clocks) and 1-dimensional (unsharp clocks, sharp rods) distributions are allowed. We find that Lorentz invariance and the new feature of nonzero dispersion together imply that a given physical field be represented by an infinity of (in general, slightly different) tensor fields, one for each equivalent observer. Some consequences of this idea in quantum field theory are investigated. It results, generally speaking, in high momentum cut offs in some expressions. For free fields we remark that the invariant functions turn out continuous and bounded; a « skin effect» violation of microcausality is present in the exterior of the light cone. For interacting fields we mention only that the iteration solutionS-matrix is unambiguous (no subtraction formalism) and has only finite matrix elements. The renormalization idea that self energy parts, etc., only renormalize bare particle parameters must, strictly speaking, be given up. Correction to « point particle» scattering arise in the form of form factor functions of momentum transfersq. These arenot functions ofq2 only, as in currentad hoc theories. The « standard deviation» λ is estimated as nucleon Compton wave length or smaller.