Classical Group Field Theory

TL;DR

This paper extends the classical field-theory formalism to dynamical group field theories (GFTs) defined over copies of SU(2), contrasting local (D=1) and nonlocal (D≥3) models and analyzing colored versus noncolored variants. It derives Noether currents for translations and dilatations, computes the energy-momentum tensor, and reveals that nonlocal interactions generally disrupt local current conservation, while colored GFTs admit covariantly conserved currents after integrating over unused group arguments. Dilatation currents, explicitly dependent on the group coordinate, fail to be covariantly conserved in all models, signaling explicit coordinate-symmetry breaking and challenging scale/conformal invariance in GFTs. The results have implications for Ward–Takahashi identities and renormalization in GFTs and point toward a closer link to metric representations and noncommutative dual formulations.

Abstract

The ordinary formalism for classical field theory is applied to dynamical group field theories. Focusing first on a local group field theory over one copy of SU(2) and, then, on more involved nonlocal theories (colored and non colored) defined over a tensor product of the same group, we address the issue of translation and dilatation symmetries and the corresponding Noether theorem. The energy momentum tensor and dilatation current are derived and their properties identified for each case.