Gauge Properties of the Minkowski Space

Abstract

The dilatations and the 4-parameter special conformal group are interpreted as geometrical gauge transformations of the Minkowski space and new arguments are given as to why the conventional interpretation of the special conformal group as a set of transformations connecting systems with constant relative accelerations cannot be true. The dilatations and the special conformal group are considered to be approximate symmetries in particle physics in the sense that they become "good" groups at very high energies, but are broken for low energies when the rest masses are important. These properties are illustrated in the case of classical point particles by the partially conserved quantities associated with the new symmetries. One of the interesting features is that the transformation by reciprocal radii appears as a new discrete approximate space-time symmetry. This is probably of interest for the problem of P and CP invariance. The geometrical interpretation is analyzed in terms of homogeneous coordinates and the physical significance of these coordinates is discussed. This analysis leads in a straightforward way to the group $O(2,4)\mathrm{}$, which is isomorphic to the full 15-parameter conformal group in Minkowski space, including the full Poincaré group. A conformal-invariant generalization of the usual notion of Einstein causality is given and analyzed. The relationship between tensors and spinors of the group $O(2,4)\mathrm{}$ and the usual space-time spinors and tensors is discussed, employing results obtained by Dirac. Finally the field equations for electrodynamics and the pseudoscalar coupling are written down in terms of the new spinors and tensors. These equations do not contain a bare-mass term. The masses are considered to be consequences of the interaction.