Approaches to Quantum Gravity

Algebraic approach to Quantum Gravity II 473The best approach here is actually to work with Hopf–von Neumann or C ∗ -algebraversions of these Hopf algebras. For example C[G] might become an algebra builtfrom continuous functions on G with rapid decay at infinity in the noncompactcase. The role of U(g) might become the group C ∗ -algebra which is a completionof the functions on G with convolution product. However, we do not need to makethis too precise at least for the bicrossproduct model. Formally we take a basis {δ u }of δ-functions on G (more precisely one should smear or approximate these). Fordual basis we take the group elements u ∈ U(g) formally as exponential elementsin the completed enveloping algebra. Then∫∫F( f ) = duf(u)u ≈ d n kJ(k) f (k)e ıki e i(24.5)GU⊂R nwhere e i are a basis of g so that the k i are a local coordinate system for the groupvalid in some open domain U and J(k) the Jacobian for this change of variables.There are subtleties particularly in the compact case (e.g. the case of G = SU 2studied in detail in [4] as some kind of ‘noncommutative sampling theory’). If G isa curved position space then the natural momenta e i are noncommuting covariantderivates and in the highly symmetric case of a non-Abelian group manifold theygenerate noncommutative momemtum ‘operators’ U(g) instead of usual commutativecoordinates. So actually physicists have been needing NCG – in momentumspace – for about a century now, without knowing its framework. Indeed, Fouriertransform is usually abandoned in any ‘functional’ form on a nonAbelian group(instead one works with the whole category of modules, 3 j and 6 j-symbols, etc.)but quantum group methods allow us for the first time to revert to Fourier transformas a functional transform, just with noncommutative functions U(g). If this seemsstrange consider that the phase space of a particle on G is T ∗ G = g ∗ × G and hasquantum algebra of observables U(g)⊲