Fermions and noncommutative theories

TL;DR

This paper explores the symmetry properties of an extended space-time incorporating noncommutative geometry, proposing a generalized Dirac equation with fermions dependent on both coordinates and noncommutativity parameters.

Contribution

It introduces a framework where noncommutativity parameters are independent degrees of freedom and proposes a generalized Dirac equation in this extended space.

Findings

The extended $x+ heta$ space-time has a symmetry group $P'$ containing the Poincaré group.

The generalized Dirac equation maintains invariance under the extended symmetry group $P'$.

The framework uses the minimal canonical extension of the DFR algebra.

Abstract

By using a framework where the object of noncommutativity $\theta^{\mu\nu}$ represents independent degrees of freedom, we study the symmetry properties of an extended $x+\theta$ space-time, given by the group $P$', which has the Poincar\'{e} group $P$ as a subgroup. In this process we use the minimal canonical extension of the Doplicher, Fredenhagen and Roberts algebra. It is also proposed a generalized Dirac equation, where the fermionic field depends not only on the ordinary coordinates but on $\theta^{\mu\nu}$ as well. The dynamical symmetry content of such fermionic theory is discussed, and we show that its action is invariant under $\cal P$'.