Meaning of Noncommutative Geometry and the Planck-Scale Quantum Group

Abstract

This is an introduction for nonspecialists to the noncommutative geometric approach to Planck scale physics coming out of quantum groups. The canonical role of the `Planck scale quantum group' $C[x]\bicross C[p]$ and its observable-state T-duality-like properties are explained. The general meaning of noncommutativity of position space as potentially a new force in Nature is explained as equivalent under quantum group Fourier transform to curvature in momentum space. More general quantum groups $C(G^\star)\bicross U(g)$ and $U_q(g)$ are also discussed. Finally, the generalisation from quantum groups to general quantum Riemannian geometry is outlined. The semiclassical limit of the latter is a theory with generalised non-symmetric metric $g_{μν}$ obeying $\nabla_μg_{νρ}-\nabla_νg_{μρ}=0$