Generalizations of the Veneziano and Virasoro Models

It is suggested that Virasoro's fixed poles in the $j$ plane of the Veneziano amplitude should be regarded as Gribov-Pomeranchuk poles. The third Veneziano term can probably have other effects associated with a third double-spectral function, such as cuts in the angular-momentum plane. The alternative to the Veneziano formula proposed by Virasoro may therefore not conflict with unitarity. A generalized formula is proposed which contains the Veneziano and Virasoro amplitudes as special cases. The new formula has a double-integral representation which is similar to the beta-function representation for the Veneziano formula. We propose another generalization of the Veneziano formula, in which the signature degeneracy can be broken by an arbitrary amount.