Analysis in space-time bundles. I. General considerations and the scalar bundle

Abstract

Mathematical features involved in the systematic development of elementary particle theory on an alternative cosmos (space-time) are presented. Bundles representative of physical fields are studied in the unique variant of Minkowski space M 0 enjoying similar properties of causality and symmetry. The “universal cosmos” M̃ is the universal cover of the causal compactification of M 0 . The bundles studied are induced from representations of the scale-extended Poincaré group, which forms the isotropy group in M̃ of the universal cover G ∽ = S ∽ U(2,2) of the connected component of the group of all causal transformations on M̃. Discrete symmetries and higher-dimensional cases are also discussed. The primary focus is on the temporal evolution, especially stability (involving positivity of the energy), wave equations (implicative of finite propagation velocity), and the unitarity and/or composition series of associated actions of G̃. General spin bundles on M̃ are treated, parallelized, and correlated with bundles on M 0 . Associated covariant wave equations and the spectral resolution of fundamental quantum numbers are studied in detail in the scalar case.