Approaches to Quantum Gravity

76 N. SavvidouWe start instead from the histories canonical General Relativity and we show thatthis formalism is augmented by a spacetime description that carries a representationof the Diff(M) group.In the standard canonical formalism we introduce a spacelike foliation E : R × → M on M, with respect to a fixed Lorentzian four-metric g. Then the spacelikecharacter of the foliation function implies that the pull-back of the four metric ona surface is a Riemannian metric with signature +++. In the histories theorywe obtain a path of such Riemannian metrics t ↦→ h ij (t, x), each one defined onacopyof t with the same t label. However, a foliation cannot be spacelike withrespect to all metrics g and in general, for an arbitrary metric g the pullback of ametric E ∗ g is not a Riemannian metric on .This point reflects a major conceptual problem of Quantum Gravity: the notionof ‘spacelike’ has no apriorimeaning in a theory in which the metric is a nondeterministicdynamical variable; in the absence of deterministic dynamics, therelation between canonical and covariant variables appears rather puzzling. InQuantum Gravity, especially, where one expects metric fluctuations the notion ofspacelikeness is problematic.In histories theory this problem is addressed by introducing the notion of a metricdependent foliation E[·], defined as a map E[g] :LRiem(M) ↦→ FolM, thatassigns to each Lorentzian metric a foliation that is always spacelike with respectto that metric. Then we use the metric dependent foliation E[g] to define the canonicaldecomposition of the metric g with respect to the canonical three-metric h ij ,the lapse function N and the shift vector N i .Definedinthiswayh ij is always aRiemannian metric, with the correct signature. In the histories theory therefore, the3 + 1 decomposition preserves the spacetime character of the canonical variables,a feature that we expect to hold in a theory of Quantum Gravity.The introduction of the metric-dependent foliation allows the expression of thesymplectic form in an equivalent canonical form, on the space of canonical GeneralRelativity histories description can , by introducing conjugate momenta forthe three-metric π ij , for the lapse function p and for the shift vector p i . Thus weprove that there exists an equivalence between the covariant history space covand the space of paths on the canonical phase space can =× t (T ∗ Riem( t ) ×T ∗ Vec( t ) × T ∗ C ∞ ( t )), where Riem( t ) is the space of all Riemannian threemetricson the surface t , Vec( t ) is the space of all vector fields on t ,andC ∞ ( t ) is the space of all smooth scalar functions on t .Canonical descriptionThe canonical history space of General Relativity can is a suitable subset of theCartesian product of copies of the phase space Ɣ of standard canonical General