Approaches to Quantum Gravity

13Loop quantum gravityT. THIEMANN13.1 IntroductionThe modern version of canonical Quantum Gravity is called loop quantum gravity(LQG), see [1; 2] for textbooks and [3; 4; 5; 6] for recent reviews. At present,there is no other canonical approach to Quantum Gravity which is equally welldeveloped. LQG is a Quantum Field Theory of geometry and matter which isbackground independent and takes fully into account the backreaction of (quantum)matter on (quantum) geometry. Background independence means that thereis no preferred spacetime metric available, rather the metric is a dynamical entity 1which evolves in tandem with matter, classically according to the Einstein equations.These precisely encode the backreaction. This is therefore an entirely newtype of QFT which is radically different from ordinary QFT. One could even saythat the reason for the fact that today there is not yet an established theory of QuantumGravity is rooted in the background dependence of ordinary QFT. Thereforeordinary QFT (quantum mechanics) violates the background independence of classicalGR while classical GR violates the quantum principle of QFT. This is thepoint where the two fundamental principles of modern physics collide. LQG triesto overcome this obstacle by constructing a background independent QFT.In order to see in more detail where the background metric finds its way into thevery definition of an ordinary QFT, recall the fundamental locality axiom of thealgebraic approach [7]. There one deals with nets of local algebras A(O) definedover regions O of a spacetime (M, g 0 ) where M is a differential manifold andg 0 a Lorentzian metric on M. The locality axiom now demands that if O, O ′are spacelike separated with respect to g 0 (that is, no causal geodesics of (M, g 0 )can connect points of O, O ′ ) then the elements of the two algebras A(O), A(O ′ )1 Describing an infinite number of physical degrees of freedom.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.