Approaches to Quantum Gravity

78 N. SavvidouRelation between the invariance groupsOne of the deepest issues to be addressed in canonical gravity is the relation ofthe algebra of constraints to the spacetime diffeomorphisms group. The canonicalconstraints depend on the 3 + 1 decomposition and hence on the foliation.The equivariance condition manifests a striking result both in its simplicity andits implications: the action of the spacetime Diff(M) group preserves the set of theconstraints, in the sense that it transforms a constraint into another of the same typebut of different argument. Hence, the choice of an equivariance foliation implementsthat histories canonical field variables related by spacetime diffeomorphismsare physically equivalent. Furthermore this result means also that the group Diff(M)is represented in the space of the true degrees of freedom. Conversely, the space oftrue degrees of freedom is invariant under Diff(M).Hence, the requirement of the physical equivalence of different choices of timedirection is satisfied by means of the equivariance condition.5.3.3 Reduced state spaceGeneral Relativity is a parameterised system in the sense that it has vanishingHamiltonian on the reduced phase space due to the presence of first classconstraints.In the histories framework we define the history constraint surface C h ={t ↦→ C, t ∈ R} as the space of maps from the real line to the single-time constraintsurface C of canonical General Relativity. The reduced state space is obtainedas the quotient of the history constraint surface, with respect to the action of theconstraints.The Hamiltonian constraint is defined as H κ = ∫ dt κ(t)h t , where h t :=h(x t , p t ) is first-class constraint. For all values of the smearing function κ(t),the history Hamiltonian constraint H κ generates canonical transformations on thehistory constraint surface.It has been shown [23; 24] that the history reduced state space red is a symplecticmanifold that can be identified with the space of paths on the canonical reducedstate space red ={t ↦→Ɣ red , t ∈ R}. Therefore the histories reduced state space isidentical to the space of paths on the canonical reduced state space. Consequentlythe time parameter t also exists on red , and the notion of time ordering remainson the space of the true degrees of freedom. This last result is in contrast to thestandard canonical theory where there exists ambiguity with respect to the notionof time after reduction.Moreover, the action functional S commutes weakly with the constraints, so itcan be projected on the reduced state space. It then serves its role in determiningthe equations of motion [23; 24].