Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2000-06-21
Lect.Notes Phys. 541 (2000) 227-276
Physics
High Energy Physics
High Energy Physics - Theory
51 pages latex, many figures; see also less technical version, my JMP millenium article
Scientific paper
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 $g_{\mu\nu}$ obeying $\nabla_\mu g_{\nu\rho}-\nabla_\nu g_{\mu\rho}=0$
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