Quantum Gravity

THE SEVENTH ROUTE TO GEOMETRODYNAMICS 103Using the result (4.21) for Ha g , one can construct from (3.90) the explicitexpression for H g ⊥. A rather lengthy but straightforward calculation leads to(Hojman et al. 1976; Teitelboim 1980)H g ⊥ =16πG G abcdp ab p cd + V [h ab ] , (4.22)withG abcd = 12 √ h (h ach bd + h ad h bc − h ab h cd ) (4.23)as the (inverse) ‘DeWitt metric’, 5 h denoting the determinant of h ab ,andV = σ√ h16πG ( (3) R − 2Λ) , (4.24)where (3) R is the three-dimensional Ricci scalar. 6 The inverse of (4.23) is called‘DeWitt’ metric because it plays the role of a metric in the space of all metrics(DeWitt 1967a); cf. Section 4.2.5. Due to this it is often referred to as ‘supermetric’.The explict expression reads√hG abcd =2 (hac h bd + h ad h bc − 2h ab h cd ) (4.25)(the last term here is the same in all space dimensions), obeyingG abcd (G cdef = 1 2 δae δf b + ) δa f δb e . (4.26)We recall that the Poisson-bracket relation (3.91) states that H ⊥ transforms asa scalar density under coordinate transformations; this is explicitly fulfilled by(4.22) (G abcd has weight −1, p ab and V have weight 1, so H g ⊥has weight 1). Wethus have∫δH g ⊥ (x) = dy {H g ⊥ (x), Hg a (y)}δNa (y) =∂∂x a (Hg ⊥ (x)δNa (x)) . (4.27)It will be shown in Section 4.2 that H g ⊥ and Hg a uniquely characterize GR, thatis, they follow from the Einstein–Hilbert action (1.1). Finally, we want to remarkthat the uniqueness of the construction presented here ceases to hold in spacedimensions greater than three (Teitelboim and Zanelli 1987).4.1.3 Geometrodynamics and gauge theoriesWe have seen that for vector fields H a is of such a form that the conditionof ultralocality for H ⊥ would be violated, see (4.16). Since vector fields are animportant ingredient in the description of nature, the question arises whether a5 In d space dimensions, the last term reads −2/(d − 1)h ab h cd .6 G and Λ are at this stage just free parameters. They will later be identified with thegravitational constant and the cosmological constant, respectively.