Crossing Symmetric Dispersion Relations in Quantum Field Theories

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