Quantum Gravity

92 PARAMETRIZED AND RELATIONAL SYSTEMSLorentzian case (σ = −1, the relevant case here) and the Euclidean case (σ =1).This algebra will play a crucial role in canonical gravity, see Chapters 4–6. It isoften convenient to work with a ‘smeared-out’ version of the constraints, that is,∫∫H[N] = d 3 xN(x)H ⊥ (x) , H[N a ]= d 3 xN a (x)H a (x) . (3.93)The constraint algebra then reads{H[N], H[M]} = H[K a ] , K a = −σh ab (NM ,b − N ,b M) , (3.94){H[N a ], H[N]} = H[M] , M = N a N ,a ≡L N N, (3.95){H[N a ], H[M b ]} = H[K] , K =[N, M] ≡L N M , (3.96)where L denotes here the Lie derivative. Some remarks are in order:1. This algebra is not a Lie algebra, since (3.94) contains the (inverse) threemetrich ab (x) of the hypersurfaces x 0 = constant (i.e. one has structurefunctions depending on the canonical variables instead of structure constants).An exception is two-dimensional space–time (Teitelboim 1984).2. The signature σ of the embedding space–time can be read off directly from(3.94).3. The subalgebra of the diffeomorphism constraints is a Lie algebra as can beseen from (3.96). Equation (3.95) means that the flow of the Hamiltonianconstraint does not leave the constraint hypersurface of the diffeomorphismconstraints invariant. Moreover, this equation expresses the fact that H ⊥transforms under diffeomorphisms as a scalar density of weight one; thisfollows from (3.91) after integration with respect to the shift vector,∫δH ⊥ (y) = d 3 xN a (x){H ⊥ (y), H a (x)} = ∂ a (N a H ⊥ )(y) ,cf. (3.20).4. The algebra is the same as for the corresponding constraints in the caseof the bosonic string, that is, it is in two dimensions equivalent to theVirasoro algebra (3.61). The reason is its general geometric interpretationto be discussed in the following.It turns out that the above algebra has a purely kinematical interpretation.It is just the algebra of surface deformations for hypersurfaces which are embeddedin a Riemannian (or pseudo-Riemannian) space. If a hypersurface is againdescribed by X µ (x), the generators of coordinate transformations on the hypersurfaceare given byX ax ≡ X ,a(x)µ δδX µ (x) ,while the generators of the normal deformations are given by