4d Deformed Special Relativity from Group Field Theories

TL;DR

This work derives a scalar field theory of Deformed Special Relativity on κ-Minkowski spacetime directly from a four-dimensional SO(4,1) group field theory for BF theory, using a non-perturbative spin-foam framework. By analyzing classical flat solutions and 2D perturbations, the authors show how matter emerges as an effective non-commutative field theory governed by a κ-Poincaré symmetry, first on the full SO(4,1) momentum space and then localized to the AN_3 sector to realize κ-Minkowski dynamics. The construction provides three complementary routes to a DSR matter sector: (i) SO(4,1) GFT with dynamical decoupling of Lorentz modes leading to AN^c_3 DSR dynamics, (ii) a direct AN_3^c GFT BF theory yielding a DSR action on AN^c_3, and (iii) a partially projected/modified GFT approach to emphasize AN_3 localization. This establishes a concrete bridge between fundamental quantum gravity models and effective non-commutative field theories, with potential implications for quantum gravity phenomenology and emergent spacetime symmetries.

Abstract

We derive a scalar field theory of the deformed special relativity type, living on non-commutative kappa-Minkowski spacetime and with a kappa-deformed Poincare symmetry, from the SO(4,1) group field theory defining the transition amplitudes for topological BF-theory in 4 space-time dimensions. This is done at a non-perturbative level of the spin foam formalism working directly with the group field theory (GFT). We show that matter fields emerge from the fundamental model as perturbations around a specific phase of the GFT, corresponding to a solution of the fundamental equations of motion, and that the non-commutative field theory governs their effective dynamics.