A Poincaré covariant noncommutative spacetime

We interpret, in the realm of relativistic quantum field theory, the tangential operator given by Coleman and Mandula [All possible symmetries of the [Formula: see text] matrix, Phys. Rev. 159 (1967) 1251–1256] (see also [Much, Pottel and Sibold, Preconjugate variables in quantum field theory and their applications, Phys. Rev. D 94(6) (2016) 065007]) as an appropriate coordinate operator. The investigation shows that the operator generates a Snyder-like noncommutative spacetime with a minimal length that is given by the mass. By using this operator to define a noncommutative spacetime, we obtain a Poincaré invariant noncommutative spacetime and in addition solve the soccer-ball problem. Moreover, from recent progress in deformation theory we extract the idea of how to obtain, in a physical and mathematically well-defined manner, an emerging noncommutative spacetime. This is done by a strict deformation quantization known as Rieffel deformation (or warped convolutions). The result is a noncommutative spacetime combining a Snyder and a Moyal-Weyl type of noncommutativity that in addition behaves covariant under transformations of the whole Poincaré group.