Support for Edward Witten’s Claim that Non-Commutative Geometry is Essential for Planck-Scale Quantum-Gravity

Support for Edward Witten’s Claim that Non-Commutative Geometry is Essential for Planck-Scale Quantum-Gravity Two Arguments: Meaning of Noncommutative Geometry and the Planck-Scale Quantum Group 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]I/C[p] and its observable-state T-dualitylike 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?)I/U(g) and Uq(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 gµν obeying ∇µgνρ − ∇νgµρ = 0. Intersecting Quantum Gravity with Noncommutative Geometry We review applications of noncommutative geometry in canonical quantum gravity. First, we show that the framework of loop quantum gravity includes natural noncommutative structures which have, hitherto, not been explored. Next, we present the construction of a spectral triple over an algebra of holonomy loops. The spectral triple, which encodes the kinematics of quantum gravity, gives rise to a natural class of semiclassical states which entail emerging fermionic degrees of freedom. In the particular semiclassical approximation where all gravitational degrees of freedom are turned of, a free fermionic quantum field theory emerges. We end the paper with an extended outlook section.