For 2-2 scattering in quantum field theories, the usual fixed $t$ dispersion relation exhibits only two-channel symmetry. This Letter considers a crossing symmetric dispersion relation, reviving certain old ideas from 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 nonperturbative 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.

中文翻译：

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量子场论中的交叉对称色散关系

对于量子场论中的2-2散射，通常是固定的 $Ť$色散关系仅表现出两个通道的对称性。这封信考虑了一个交叉对称的色散关系，使1970年代的某些旧观念复活了。而不是固定的$Ť$ 色散关系，这需要不同变量中的色散关系 $ž$，与Mandelstam不变量有关 $s$， $Ť$， $ü$ 通过参数三次关系使复数中的交叉对称 $ž$进行几何旋转。产生的色散显然是三通道交叉对称的。对于有效的场论，我们给出了某些已知正性条件的简单推导，其中包括零约束，这会导致两侧边界，并得出一组新的非扰动不等式。我们展示了这些不等式如何使我们能够从II型弦理论中四个dilaton振幅的低能量扩展中找到第一个大规模弦状态。我们还展示了如何从这种方法中获得对所有能量都有效的广义（数字）Froissart界。