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

TL;DR

This paper demonstrates that in scalar field theories on the Moyal plane, the non-perturbative S-matrix remains unaffected by noncommutative deformation, equating to the commutative case, with unitarity preserved.

Contribution

It proves that the non-perturbative S-matrix on the Moyal plane is independent of the noncommutative parameter, a novel result verified through multiple methods.

Findings

S-matrix equals the interaction representation S-matrix.

S-operator is independent of the noncommutative parameter.

Unitarity of the non-perturbative S-matrix is established.

Abstract

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.