2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/178068Poincaré-invariant quantum field theories can be formulated on non-commutative planes if the coproduct on the Poincaré group is suitably deformed \cite{Dimitrijevic:2004rf, Chaichian:2004za}.(See also especially Oeckl \cite{Oeckl:1999jun},\cite{Oeckl:2000mar} and Grosse et al.\cite{Grosse:2001mar}) As shown in \cite{Balachandran:2005eb}, this important result of these authors implies modification of free field commutation and anti-commutation relations and striking phenomenological consequences such as violations of Pauli principle \cite{Balachandran:2005eb,Bal3}. In this paper we prove that with these modifications, UV-IR mixing disappears to all orders in perturbation theory from the $S$-Matrix. This result is in agreement with the previous results of Oeckl \cite{Oeckl:2000mar}.Minor Changes in text and abstract, important references addedHigh Energy Physics - TheoryGeneral Relativity and Quantum CosmologyHigh Energy Physics - PhenomenologyMathematical PhysicsQuantum Algebratext