High-Energy Bounds on Scattering Amplitudes in Qft on Noncommutative Space-Time

In the framework of Quantum Field Theory (QFT) on noncommutative (NC) space-time with SO(1, 1) × SO(2) symmetry and space-space noncommutativity (θ_0i = 0), we prove that, based on the causality condition usually taken in connection with this symmetry, it is merely impossible to draw any conclusion on the analyticity of the 2 → 2-scattering amplitude in cos Θ, Θ being the scattering angle. A physical choice of the causality condition rescues the situation and as a result an analog of Lehmann's ellipse as domain of analyticity in cos Θ is obtained. However, the enlargement of this analyticity domain to Martin's ellipse and the derivation of the Froissart bound for the total cross section in NC QFT is possible only in the special case when the incoming momentum is orthogonal to the NC plane. This is the first example of a nonlocal theory in which the cross sections are subject to a high-energy bound.