Quantum Gravity

124 HAMILTONIAN FORMULATION OF GENERAL RELATIVITYdirections). The kinetic term in the Hamiltonian constraint is then of a truly hyperbolicnature (in contrast to the general hyperbolicity in the pointwise sense).This holds in particular in the vicinity of closed Friedmann universes which aretherefore distinguished in this respect. A perturbation around homogeneity andisotropy has exhibited this property explicitly (Halliwell and Hawking 1985). It isan open question ‘how far’ one has to go from the hyperbolic case near (3) R>0in order to reach the first points where the signature changes.For Ricci-negative metrics 21 (i.e. all eigenvalues of (3) R ab are negative), onefinds that V h ∩ H h = {0}, and that the projected metric on superspace containsinfinitely many plus and minus signs. For flat metrics, one has V h ∩ H h ≠ {0}.Some of these results can explicitly be confirmed in Regge calculus; cf. Section2.2.6.4.3 Canonical gravity with connections and loops4.3.1 The canonical variablesOne of the key ingredients in the canonical formalism is the choice of the symplecticstructure, that is, the choice of the canonical variables. In the previoussections, we have chosen the three-metric h ab and its momentum p cd .Inthissection, we shall introduce different variables introduced by Ashtekar (1986) followingearlier work by Sen (1982). These ‘new variables’ will exhibit their mainpower in the quantum theory; see Chapter 6. Since they are analogous to Yang–Mills variables (using connections), the name ‘connection dynamics’ is also used.A more detailed introduction into these variables can be found in Ashtekar (1988,1991) and—taking into account more recent developments—Thiemann (2001).The first step consists in the introduction of triads (or dreibeine). They willplay the role of the canonical momentum. Similar to the tetrads (vierbeine)used in Section 1.1.4, they are given by the variables e a i (x) which define anorthonormal basis at each space point. Here, a =1, 2, 3 is the usual space index(referring to the tangent space T x (Σ) at x) andi =1, 2, 3 are internal indicesenumbering the vectors. The position of the internal indices is arbitrary. One hasthe orthonormality conditionfrom which one getsh ab e a i eb j = δ ij , (4.101)h ab = δ ij e a i e b j ≡ e a i e b i . (4.102)This introduces an SO(3) (or SU(2)) symmetry into the formalism, since themetric is invariant under local rotations of the triad. Associated with e a i (x) isan orthonormal frame in the cotangent space Tx ∗ (Σ), denoted by e i a(x) (basisofone-forms). It obeyse i a ea j = δi j , ei a eb i = δb a . (4.103)21 Any Σ admits such metrics.