Spinors and Voros star-product for Group Field Theory: First Contact

TL;DR

This paper introduces a spinor-based group Fourier transform for SU(2), demonstrating that the associated non-commutative space has a Moyal-type structure with Voros star-product, with applications to quantum gravity models.

Contribution

It develops a novel spinor-based Fourier transform for SU(2) and shows the non-commutative space is of Moyal type with Voros star-product, advancing non-commutative geometry in quantum gravity.

Findings

Non-commutative space dual to SU(2) is Moyal-type.

Spinor variables enable Voros star-product formulation.

Applications to group field theories and quantum gravity.

Abstract

In the context of non-commutative 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 su(2) in terms of spinors. It allows us to prove that the non-commutative 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 non-commutative field theories with quantum-deformed symmetries.