Lehmann-Symanzik-Zimmermann S-Matrix elements on the Moyal Plane

Field theories on the Groenewold-Moyal(GM) plane are studied using the Lehmann-Symanzik-Zimmermann(LSZ) formalism. The example of real scalar fields is treated in detail. The S-matrix elements in this non-perturbative approach are shown to be equal to the interaction representation S-matrix elements. This is a new non-trivial result: in both cases, the S-operator is independent of the noncommutative deformation parameter $\theta_{\mu\nu}$ and the change in scattering amplitudes due to noncommutativity is just a time delay. This result is verified in two different ways. But the off-shell Green's functions do depend on $\theta_{\mu\nu}$. In the course of this analysis, unitarity of the non-perturbative S-matrix is proved as well.