Emergence of Non-commutative Space-time in Pre-geometric Gravity

Abstract

We develop a non-commutative extension of pre-geometric gravity based on a BF theory with gauge group SO(1, 4) or SO(2, 3). The theory is formulated on a topological manifold without presupposing any spacetime metric, with coordinates promoted to non-commutative operators via the Moyal star product. Four constrained BF structures reconstruct the complete gravitational sector, including the Einstein-Hilbert action, cosmological constant, and topological terms, after spontaneous symmetry breaking. We construct a non-commutative dictionary relating pre-geometric fields to emergent geometric quantities, and show how the Seiberg-Witten map organizes θ-deformed corrections while preserving internal gauge symmetries. Matter couplings are consistently implemented, providing a framework for embedding the Standard Model. We demonstrate that the noncommutativity parameter can emerge dynamically from the B-field through an adjoint Higgs field, tying the noncommutative scale to the symmetry breaking scale. This mechanism naturally regulates UV/IR mixings through the topological UV completion, potentially offering a UV finite completion of non-commutative quantum gravity. Importantly, the breaking of spacetime Lorentz invariance in favor of an exact internal gauge Lorentz symmetry leads to emergent violations of the spin-statistics theorem, with strong phenomenological implications including energy-dependent Pauli Exclusion Principle Violations that could be tested in precision experiments.