Covariant field theories on alternative spacetimes

A complete classification of the unitary irreducible representations of the 2 + 1 dimensional Poincare groups is given. It is shown, in particular, that only two types of ''spin'' are available for massless field theories. Several examples of covariant field theories in three dimensions and the generalized ''Foldy-Wouthuysen transformations'' which allow one to connect these theories with unitary irreducible representations of the spacetime symmetry groups are given. Elementary particles and quantum field theories in 3 + 2 de Sitter space and in conformal space are discussed. The importance of working in a realistic spacetime rather than using Euclidean or spherical models is demonstrated. The ''parton-like'' representations Di and Rac give rise to gauge theories of scalar and spinor fields, and a theory of interacting massless particles with all spins. This theory (in four dimensional de Sitter space) is constructed on the basis of a conformally invariant field theory in three-dimensional spacetime. A manifestly covariant quantum electrodynamics is constructed. The setting is again a realistic spacetime and a complete Gupta-Bleuler quantization scheme is carried out. Conformal invariance of the quantum field theory (as opposed to either the classical field theory or to a theory defined by its Feynman rules) requires a richer Gupta-Bleuler structure than has been considered previously. Yet the essential features of this structure are preserved