Vector Currents and Current Algebra. I. Duality and Zero-Width Models

In this paper we begin an investigation of the properties of vector currents consistent with the hadron bootstrap, assuming duality and zero resonance widths. First, on general grounds and independently of current algebra, we show that two-current amplitudes must have fixed singularities in the angular momentum plane. Then we discuss some general consequences for current amplitudes of duality and the zero-width approximation. Throughout we treat amplitudes for one or two vector currents and an arbitrary number $N$ of spinless hadrons.