2000/06/21 by S. Majid
Physics and Astronomy · #hep-th
published as Lect.Notes Phys. 541 (2000) 227-276 · 51 pages latex, many figures; see also less technical version, my JMP millenium article
arxiv created 2000/06/21 · arxiv updated 2009/11/30
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^⋆)\bicross 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