Jost–Lehmann–Dyson representation, analyticity in the angular variable, and upper bounds in noncommutative quantum field theory

We prove the existence of an analogue of the Jost–Lehmann–Dyson representation in noncommutative quantum field theory for the case where the noncommutativity affects only the spatial variables. Using this representation, we show that there is a certain class of elastic scattering amplitudes that have an analytic continuation to the complex $\cos\vartheta$ plane with the Martin ellipse as the related analyticity domain. Using the analyticity in the angular variable and the unitarity as a basis, we establish an analogue of the Froissart–Martin bound for the total cross section in the noncommutative case.