Quantum Gravity

112 HAMILTONIAN FORMULATION OF GENERAL RELATIVITYp N ≡ ∂Lg∂Ṅ =0,pg a ≡ ∂Lg =0. (4.62)∂Ṅ aBecause lapse function and shift vector are only Lagrange multipliers (similarto A 0 in electrodynamics), these are constraints (called ‘primary constraints’according to Dirac (1964), since they do not involve the dynamical equations).Second,p ab ≡ ∂Lg∂ḣab= 116πG Gabcd K cd =√h (K ab − Kh ab) . (4.63)16πGNote that the gravitational constant G appears here explicitly, although no couplingto matter is involved. This is the reason why it will appear in vacuumquantum gravity; see Section 5.2. One therefore has the Poisson-bracket relation13 {h ab (x),p cd (y)} = δ(a c δd b) δ(x, y) . (4.64)Recalling (4.48) and taking the trace of (4.63), one can express the velocities interms of the momenta,ḣ ab = 32πGN √(p ab − 1 )h 2 ph ab + D a N b + D b N a , (4.65)where p ≡ p ab h ab . One can now calculate the canonical Hamiltonian densityH g = p ab ḣ ab −L g ,for which one gets the expression 14√h(H g =16πGNG abcd p ab p cd (3) R − 2Λ)− N− 2N b (D a p ab ) . (4.66)16πGThe full Hamiltonian is found by integration,∫∫H g ≡ d 3 x H g ≡ d 3 x (NH g ⊥ + N a Ha) g . (4.67)The action (4.61) can be written in the form∫ ()16πG S EH = dtd 3 x p ab ḣ ab − NH g ⊥ − N a Hag . (4.68)Variation with respect to the Lagrange multipliers N and N a yields the constraints1513 This is formal at this stage since it does not take into account that √ h>0.14 This holds modulo a total divergence which does not contribute in the integral because Σis compact.15 These also follow from the preservation of the primary constraints, {p N ,H g } = 0 ={p g a,H g }.