Unitarity and the spin-family solution

Abstract

In continuation of the research on a second, strong-coupling type of solution of a suitable field equation of motion, namely the spin-family solution, the demands of unitarity are investigated. One begins with a proposal for an asymptotic condition for general spin, and derives the relation expressing unitarity for the two-point function in terms of a complete set of asymptotic states with a general value of spin (integral or half-integral). This asymptotic unitarity condition is then applied to the proposed spin-family solution up to the second-order term. One obtains a number of relations involving the residues at the poles of the propagator. To satisfy these, one must have a positive-definite metric on the mass shells. Off the mass shells the indefinite metric associated with the unstable members of the spin family, appears to require a background term in the propagator.The notion of local unitarity is then introduced in which the unitarity for the two-point function assumes an especially simple form, which is capable of explicitly showing the contributions with indefinite metric and due to the unstable poles in the local interaction domain. The form generalizes that for a free field and suggests that the spin-family solution should be approached as a generalized free field. This will generalize with respect to spin and momentum squared, and will be an infinite-component field off the mass shells.