Approaches to Quantum Gravity

Spacetime symmetries in histories canonical gravity 75to the one in the standard canonical quantisation scheme. However, the ‘external’one is a novel object: it is similar to the group structure that arises in a Lagrangiandescription. In particular, it explicitly performs changes of foliation. It has beenshown that even though the representations of the history algebra are foliationdependent, the physical quantities (probabilities) are not.5.3 General Relativity historiesThe application of the ideas of continuous-time histories led to a ‘covariant’description of General Relativity in terms of a Lorentzian metric g and its ‘conjugatemomentum’ tensor π, on a spacetime manifold M with the topology ×R[23; 24]. We define the covariant history space cov = T ∗ LRiem(M) as the cotangentbundle of the space of all Lorentzian, globally hyperbolic four metrics on M,and where LRiem(M) is the space of all Lorentzian four-metrics. cov is equipped with a symplectic structure, or else with the covariant Poissonbrackets algebra on cov ,{g μν (X), g αβ (X ′ )}=0 ={π μν (X), π αβ (X ′ )}{g μν (X), π αβ (X ′ )}=δ αβ(μν) δ4 (X, X ′ ),where δ αβ(μν) := 1 2 (δ μ α δ ν β + δ μ β δ ν α ). The physical meaning of π can be understoodafter the 3 + 1 decomposition of M in which it will be related to the canonicalconjugate momenta.5.3.1 Relation between spacetime and canonical descriptionThe representation of the group Diff(M)The relation between the spacetime diffeomorphism algebra, and the Dirac constraintalgebra has long been an important matter for discussion in QuantumGravity. It is very important that, in this new construction, the two algebras appeartogether in an explicit way: the classical theory contains realisations of both thespacetime diffeomorphism group and the Dirac algebra.The history space cov carries a symplectic action of the Diff (M) group ofspacetime diffeomorphisms, with the generator defined for any vector field W onM as V W := ∫ d 4 X π μν (X) L W g μν (X), where L W denotes the Lie derivative withrespect to W . The functions V W satisfy the Lie algebra of Diff (M){V W1 , V W2 }=V [W1 ,W 2 ],where [W 1 , W 2 ] is the Lie bracket between vector fields W 1 , W 2 on M.The spacetime description presented is kinematical, in the sense that we do notstart from a Lagrangian formalism and deduce from this the canonical constraints.