A pilot model in two dimensions for conformally invariant particle theory

Abstract

The two-dimensional analog of the mathematical model for elementary particles treated in this journal vol 75, 1–57, is developed with greater explicitness than in the four-dimensional case, preliminary to quantization. In Part I the section space of the complex spannor bundle is studied. Invariant subspaces, invariant forms, and irreducible positive-energy factors under the connected conformal group are determined; transformation properties under discrete symmetries are also treated. Relations to the spinor bundle are treated; the Dirac operator is a canonical transform of the quadratic Casimir on the spannor bundle, and the section spaces have the same invariant factors; at the same time, the spinors constitute a deformation of the spannors. Two natural parallelizations of the spannor bundle and their intertwining operator are computed. The irreducible factors of the spannor section space under the conformal group are determined by their restrictions to Minkowski space, and this group acts continuously in the L 2-topology, in contrast to its action on the spinor section space.