Analysis in space-time bundles. III. Higher spin bundles

AbstractThe universal cosmos M̃ is the unique four-dimensional globally causal space-time manifold to which the Dirac and Maxwell equations (among others) maximally and covariantly extend. A systematic treatment is presented of general fields over M̃, of arbitrary spin; considered are fields induced from all irreducible representation of the isotropy group (scale-extended Poincaré group) to G̃, the connected causal group of M̃. Restricted to any species of such fields, the K̃-invariant canonical Dirac operator (K̃ = maximal essentially compact subgroup of G̃) is shown G̃-covariant for a unique conformal weight. A normalized K̃-finite basis for such fields is constructed. The basis actions thereon of the Dirac operator, infinitesimal generators of G̃, discrete symmetries, second-order Casimir, and the essentially unique third-order noncentral quantum number (enveloping algebra element) invariant under K̃ are derived. Composition series under G̃ of a class of these field spaces—namely, the extension to M̃ of the relativistic fields considered by Bargman and Wigner, or arbitrary spin and conformal weight—are determined, distinguishing by invariance and causality features alone the essentially conventional positive-energy mass 0 subspaces and massive invariant sub-quotient spaces, whose unitarity under G̃ is given a new proof. The “completely positive” subclass (cf. below) of representations is determined. A more detailed treatment of spin one bundles (vector and two-form, of arbitrary conformal weight) is included; the exterior derivative transformations are diagonalized, and the conformally invariant massive spin one scalar product is identified with a mathematical version of the conventional electromagnetic field Lagrangian