Covariant chronogeometry and extreme distances: Elementary particles

We study a variant of elementary particle theory in which Minkowski space, M(0), is replaced by a natural alternative, the unique four-dimensional manifold M with comparable properties of causality and symmetry. Free particles are considered to be associated (i) with positive-energy representations in bundles of prescribed spin over M of the group of causality-preserving transformations on M (or its mass-conserving subgroup) and (ii) with corresponding wave equations. In this study these bundles, representations, and equations are detailed, and some of their basic features are developed in the cases of spins 0 and (1/2). Preliminaries to a general study are included; issues of covariance, unitarity, and positivity of the energy are treated; appropriate quantum numbers are indicated; and possible physical applications are discussed.