Quantum Gravity

100 HAMILTONIAN FORMULATION OF GENERAL RELATIVITY∫δF = −dx ′ {F, H a (x ′ )}δN a (x ′ ) , (4.3)whereδN a = −σh ab (δN I δN II,b − δN II δN I,b ) . (4.4)Inserting for calculational purposes a delta function into (4.3) and performing apartial integration, one has∫− dx ′ {F, H a (x ′ )}δN a (x ′ )=∫−σ dx ′ dx ′′ ∂∂x ′b δ(x′ − x ′′ ){F, H a (x ′′ )}h ab (x ′′ )δN I (x ′′ )δN II (x ′ )∫−σ dx ′ dx ′′ ∂∂x ′b δ(x′ − x ′′ ){F, H a (x ′ )}h ab (x ′ )δN I (x ′′ )δN II (x ′ ) . (4.5)Setting this equal to (4.2) and using the arbitrariness of δN I (x ′′ )δN II (x ′ ), onefinds{F, {H ⊥ (x ′ ), H ⊥ (x ′′ )}} = −σδ ,b (x ′ − x ′′ )h ab (x ′′ ){F, H a (x ′′ )}−σδ ,b (x ′ − x ′′ )h ab (x ′ ){F, H a (x ′ )} . (4.6)Inserting the Poisson bracket (3.90) on the left-hand side, one finds the condition∂∂x ′a δ(x′ − x ′′ ) ( {F, h ab (x ′ )}H b (x ′ )+{F, h ab (x ′′ )}H b (x ′′ ) ) =0. (4.7)As this should hold for all F , the generators H a themselves must vanish asconstraints, H a ≈ 0. This result from the principle of path independence onlyfollows because the Poisson bracket (3.90) depends on the metric h ab .SinceH a ≈ 0 must hold on every hypersurface, it must be conserved under a normaldeformation. From (3.91) one then finds that H ⊥ must also vanish, H ⊥ ≈ 0.We have thus shown that the algebra of surface deformations, together with theprinciple of path independence (equivalent to the principle of embeddability),enforces the constraintsH ⊥ ≈ 0 , H a ≈ 0 . (4.8)This result follows for all number of space–time dimensions different from two.For two dimensions (after a suitable definition of the H µ ), the metric does notappear on the right-hand side of the algebra (Teitelboim 1984). This leads tothe possibility of having path independence without constraints, resulting inpotential Schwinger terms in the quantum theory (cf. Sections 3.2 and 5.3.5).4.1.2 Explicit form of generatorsHow does one find the explicit form of H ⊥ and H a ? The constraints H a generatecoordinate transformations on a hypersurface. If the transformation law of certainfields is given, H a follows (or, conversely, the transformation properties are