Meaning of Noncommutative Geometry and the Planck-Scale Quantum Group [chapter]

Shahn Majid
Lecture Notes in Physics  
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] 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 ) U (g)
more » ... groups C(G ) 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 ∇ µ g νρ − ∇ ν g µρ = 0.
doi:10.1007/3-540-46634-7_10 fatcat:rnteqsoipreh5feq5lx4inaf3q