We elaborate on the role of quantum statistics in twisted Poincaré invariant theories. It is shown that, in order to have twisted Poincaré group as the symmetry of a quantum theory, statistics must be twisted. It is also confirmed that the removal of UV-IR mixing (in the absence of gauge fields) in such theories is a natural consequence.

Poincar\'e-invariant quantum field theories can be formulated on non-commutative planes if the coproduct on the Poincar\'e 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}.

Recent work indicates an approach to the formulation of diffeomorphism invariant quantum field theories (qft’s) on the Groenewold-Moyal plane. In this approach to the qft’s, statistics gets twisted and the $S$ matrix in the nongauge qft’s becomes independent of the noncommutativity parameter ${\theta }^{\mu \nu }$. Here we show that the noncommutative algebra has a commutative spacetime algebra as a substructure: the Poincaré, diffeomorphism and gauge groups are based on this algebra in the twisted approach as is known already from the earlier work. It is natural to base covariant derivatives for gauge and gravity fields as well on this algebra. Such an approach will, in particular, introduce no additional gauge fields as compared to the commutative case and also enable us to treat any gauge group [and not just $U(N)$]. Then classical gravity and gauge sectors are the same as those for ${\theta }^{\mu \nu }=0$, but their interactions with matter fields are sensitive to ${\theta }^{\mu \nu }$. We construct quantum noncommutative gauge theories (for arbitrary gauge groups) by requiring consistency of twisted statistics and gauge invariance. In a subsequent paper (whose results are summarized here), the locality and Lorentz invariance properties of the $S$ matrices of these theories will be analyzed, and new nontrivial effects coming from noncommutativity will be elaborated. This paper contains further developments of an earlier paper of ours and a new formulation based on its approach.

It is by now well known that the Poincaré group acts on the Moyal plane with a twisted coproduct. Poincaré invariant classical field theories can be formulated for this twisted coproduct. In this paper we systematically study such a twisted Poincaré action in quantum theories on the Moyal plane. We develop quantum field theories invariant under the twisted action from the representations of the Poincaré group, ensuring also the invariance of the $S$-matrix under the twisted action of the group. A significant new contribution here is the construction of the Poincaré generators using quantum fields.