Analysis in space-time bundles IV. Natural bundles deforming into and composed of the same invariant factors as the spin and form bundles

Abstract

We study two interesting new bundles over the universal cosmos M̃ (or maximal isotropic space-time), which may be physically applicable. The treatment is from a homogeneous vector bundle point of view and uses the notation and some of the results of the treatment in Papers I–III ( S. M. Paneitz and I. E. Segal, J. Funct. Anal. 47 (1982), 78–142; 49 (1982), 335–414; 54 (1983) , 18–22)) of conventional bundles over M̃. The “spannor” bundle deforms into essentially the usual spinor bundle as a conformally invariant parameter that may be interpreted as the space curvature becomes arbitrarily small. From a Minkowski space standpoint, however, the spannors involve a nontrivial action of space-time translations that deforms into a trivial action in the spinor limit and also have more complex transformation properties under discrete symmetries. Also studied are the “plyors,” consisting of the dual to the bundle product of the spannors with themselves. Composition series for the spannor and plyor section spaces are treated, relative to the conformal group, and irreducible subquotients are identified with certain that occur in conventional bundles. In particular, factors corresponding to the Maxwell and massless Dirac equations, and which may represent certain of the observed elementary particles, are determined. A gauge and conformally invariant nonlinear coupling between spannors and plyors, constituting essentially a generalization of that used in quantum electrodynamics, is developed, and an associated invariant nonlinear partial differential equation is derived. Covariant and causal quantization for spannors (as fermions) and plyors (as bosons) is formulated algebraically. The present treatment is basically mathematical, but physical motivations and possible interpretations are briefly noted.