In the context of noncommutative geometries, we develop a group Fourier transform for the Lie group SU(2). Our method is based on the Schwinger representation of the Lie algebra $\mathfrak{s}\mathfrak{u}(2)$ in terms of spinors. It allows us to prove that the noncommutative ${\mathbb{R}}^3$ space dual to the SU(2) group is in fact of the Moyal type and endowed with the Voros star product when expressed in the spinor variables. Finally, from the perspective of quantum gravity, we discuss the application of these new tools to group field theories for spinfoam models and their interpretation as noncommutative field theories with quantum-deformed symmetries.

• Received 18 July 2012

DOI:https://doi.org/10.1103/PhysRevD.86.105034