  For 2-2 scattering in quantum field theories, the usual fixed $t$ dispersion
relation exhibits only two-channel symmetry. This paper considers a crossing
symmetric dispersion relation, reviving certain old ideas in the 1970s. Rather
than the fixed $t$ dispersion relation, this needs a dispersion relation in a
different variable $z$, which is related to the Mandelstam invariants $s,t,u$
via a parametric cubic relation making the crossing symmetry in the complex $z$
plane a geometric rotation. The resulting dispersion is manifestly
three-channel crossing symmetric. We give simple derivations of certain known
positivity conditions for effective field theories, including the null
constraints, which lead to two sided bounds and derive a general set of new
non-perturbative inequalities. We show how these inequalities enable us to
locate the first massive string state from a low energy expansion of the four
dilaton amplitude in type II string theory. We also show how a generalized
(numerical) Froissart bound, valid for all energies, is obtained from this
approach.
