In a quantum gravity theory, it is expected that the classical notion of spacetime disappears, leading to a quantum structure with new properties. A possible way to take into account these quantum effects is through a noncommutativity of spacetime coordinates. In the literature, there is not a clear way to describe at the same time a noncommutativity of spacetime and the phase-space noncommutativity of quantum mechanics. In this paper we address this issue by constructing a Drinfel'd twist in phase space which deals with both quantizations. This method can be applied to a noncommutativity which involves only space, leaving time aside. We apply our construction to the so-called $λ$-Minkwoski and $\mathbb{R}^3_λ$ noncommutative spaces.

v2 matches the version accepted for publication on Phys. Rev. D. It includes additional clarifications and references. v3 includes some missing terms in a couple of equations. The conclusions are essentially unaffected