Approaches to Quantum Gravity

24Algebraic approach to Quantum Gravity II:noncommutative spacetimeS. MAJID24.1 IntroductionIn this chapter we present noncommutative geometry (NCG ) not as a ‘theory ofeverything’ but as a bridge between any future, perhaps combinatorial, theory ofQuantum Gravity and the classical continuum geometry that has to be obtained insome limit. We consider for the present that NCG is simply a more general notionof geometry that by its noncommutative nature should be the correct setting for thephenomenology and testing of first next-to-classical Quantum Gravity corrections.Beyond that, the mathematical constraints of NCG may give us constraints on thestructure of Quantum Gravity itself in so far as this has to emerge in a natural wayfrom the true theory.Also in this chapter we focus on the role of quantum groups or Hopf algebras[10] as the most accessible tool of NCG, along the lines first introduced for Planckscale physics by the author in the 1980s [13; 14; 15; 16]. We provide a full introductionto our theory of ‘bicrossproduct quantum groups’, which is one of thetwo main classes of quantum group to come out of physics (the other class, theq-deformation quantum groups, came out of integrable systems rather than QuantumGravity). The full machinery of noncommutative differential geometry suchas gauge theory, bundles, quantum Riemannian manifolds, and spinors (at leastin principle) has also been developed over the past two decades; these topics aredeferred to a third article [11]. This should allow the present article to be readwithout prior knowledge of either NCG or quantum groups. The first article in theseries is about the philosophical basis [17].As is well-known, quantum groups are a generalised notion of symmetry. Thereis a theorem that all bicrossproduct quantum groups indeed have associated to themnoncommutative spaces on which they canonically act. Thus the bicrossproductPoincaré quantum group denoted U(so 3,1 )⊲◭C[R 3 >⊳R] has associated to it theproposal [23]Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, ed. Daniele Oriti.Published by Cambridge University Press. c○ Cambridge University Press 2009.