Infinite Multiplets and Crossing Symmetry. I. Three-Point Amplitudes

This paper investigates what remains of crossing symmetry in theories that are conventional local field theories in all but one respect: that infinite irreducible representations of the homogeneous Lorentz group are used. Only vertex functions are studied here; results for scattering amplitudes will be reported in a sequel. It is found that: (i) Form factors for scattering ($t<\mathrm{}0\mathrm{}$) and form factors for annihilation ($t>\mathrm{}4\mathrm{}{m}^2$) are strongly related to each other by the requirement that the interaction Lagrangian density be local, but they are not connected by analytic continuation. (ii) In the case of half-integral-spin fields, the empirical fact that the parities of particles and antiparticles are opposite makes it necessary to use a pair of conjugate irreducible representations, rather than a single unitary irreducible representation. An analog of the Dirac equation allows one to avoid parity doubling and to ensure a proper physical interpretation, provided that quantization is carried out with anticommutators.