Representations of groups of causality-preserving transformations on locally Minkowskian space-times, by actions on classes of wave functions of designated transformation properties, are analyzed, in extension of the conventional theoretical treatment of free relativistic particles. In particular, the constraints of positivity of the enerjy and finiteness of propagation velocity are developed, and te concept of mass is explored, within the indicated framework. The problems of the modelling of space-time (or the "cosmos"), and of elementary particles, seem to have reached a stage where certain very general considerations with clear-cut mathematical interpretations can be brought to bear effectively. The present work aims to develop systematically some of the implications of these considerations, and to specify and derive relevant properties of the possible physical systems. The motivation and intuitive basis are thus physical, but the formulation and techniques are perforce mathematical.In the latter terms, we are concerned with harmonic analysis of vector bundles over locally Minkowskian spaces; spaces of cross sections defined by hyperbolic partial differential equations; and holomorphy features dual to the desideratum of "positivity of the energy," which in turn is related to causality-as is also, in a different way, the hyperbolicity of the cited equations.It seems plausible that stable or quasi-stable free elementary particles should be describable in a familiar theoretical-way by "fields" that are cross-section spaces of the type indicated. If one foregoes the sanctity of the Lorentz group and insists only on the more fundamental features of temporal and spatial isotropy and homogeneity, another possibility emerges, which has found observationally cogent application to astronomy (see refs. 1-3). This macroscopic indication supplements a variety of previous microscopic ones in suggesting the potential physical relevance of a study of the transformation properties of all particle models of the type designated, under symmetry groups implementing the requisite isotropy and homogeneity. The basic representations We recall that a causal manifold is a C-manifold that is endowed with a smooth closed convex proper cone field; the causal group of any such manifold m is the group denoted G(m) of all diffeomorphisms that preserve the cone field. Spatial and temporal homogeneity and isotropy are then naturally definable, and it is known (see refs. 1, 4, and 5) that the only causal 4-manifolds with these properties are locally Minkowskian; i.e., locally, have the same cone-field structure as the future cone-field structure in Minkowski space.There are only three such manifolds that are globally causal in the sense that there exist no closed time-like loops in the manifold: Minkowski space M, the universal covering space M
