The Groenewold-Moyal plane is the algebra A θ (R d+1 ) of functions on R d+1 with the * -product as the multiplication law, and the commutator [x µ ,x ν ] = iθ µν (µ, ν = 0, 1, ..., d) between the coordinate functions. Chaichian et al. [1] and Aschieri et al. [2] have proved that the Poincaré group acts as automorphisms on A θ (R d+1 ) if the coproduct is deformed. (See also the prior work of Majid [3], Oeckl [4] and Grosse et al [5]). In fact, the diffeomorphism group with a deformed coproduct also does so according to the results of [2]. In this paper we show that for this new action, the Bose and Fermi commutation relations are deformed as well. Their potential applications to the quantum Hall effect are pointed out. Very striking consequences of these deformations are the occurrence of Pauli-forbidden energy levels and transitions. Such new effects are discussed in simple cases.PACS numbers: 11.10. Nx, 11.30.Cp Dedicated to Rafael Sorkin, our friend and teacher, and a true and creative seeker of knowledge. *

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.

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We construct a star product associated with an arbitrary two dimensional Poisson structure using generalized coherent states on the complex plane. From our approach one easily recovers the star product for the fuzzy torus, and also one for the fuzzy sphere. For the latter we need to define the 'fuzzy' stereographic projection to the plane and the fuzzy sphere integration measure, which in the commutative limit reduce to the usual formulae for the sphere.

Recent work [1,2] indicates an approach to the formulation of diffeomorphism invariant quantum field theories (qft's) on the Groenewold-Moyal (GM) plane. In this approach to the qft's, statistics gets twisted and the S-matrix in the non-gauge qft's become independent of the noncommutativity parameter θ µν . 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 of [1]. 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 θ µν = 0, but their interactions with matter fields are sensitive to θ µν . 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 non-trivial effects coming from noncommutativity will be elaborated. This paper contains further developments of [3] and a new formulation based on its approach.