Quantum Gravity

PATH-INTEGRAL QUANTIZATION 59F = −✷.) This may look confusing, but in DeWitt’s condensed notation discreteand continuous indices appear on the same footing. Then,G mn ≡G(x, y) , F(∇)G(x, y) =−δ(x, y) ,and (2.115) readsδΓ (1)δ〈ϕ(x)〉 = − i ∫δ 3 Sdydz G(y, z)2δ〈ϕ(z)〉δ〈ϕ(y)〉δ〈ϕ(x)〉 ≡−i 2 Gmn S nmk .(2.116)From (2.108) we then get the effective field equation up to one-loop order,δSδ〈ϕ(x)〉 − i 2 Gmn S nmk = −J(x) . (2.117)The second term on the left-hand side thus yields the first quantum correctionto the classical field equations. According to the Feynman rules, it correspondsto a one-loop diagram.So far, the formalism applies to any quantum field. Let us now switch toquantum gravity where ϕ(x) corresponds to (g µν (x), φ(x)), where φ(x)representsa non-gravitational field. For the classical part in (2.117) we get, writing S =S EH + S m ,δS EHδ〈g µν (x)〉 + δS √m −gδ〈g µν (x)〉 = 16πG (R µν − 1 √ −g2 g µνR) −2 T µν , (2.118)where the right-hand side is evaluated at the mean metric 〈g µν 〉. For the quantumcorrection in (2.117) we get the sum of a ‘matter loop’ and a ‘graviton loop’,− i 2 Gmn S nmk ≡− i ∫dzdy2− i ∫dzdy2δ 3 Sδ〈g µν (x)〉δ〈φ(z)〉δ〈φ(y)〉 G m(z,y)δ 3 Sδ〈g µν (x)〉δ〈g αβ (z)〉δ〈g γδ (y)〉 Gαβ,γδ (z,y) . (2.119)Using again the expression for the energy–momentum tensor as a variationalderivative of the matter action, one recognizes that the matter loop is given by√ ∫i −g2 2dzdyδ 2 T µνδ〈φ(z)〉δ〈φ(y)〉 G m(z,y) .A similar expression is obtained for the graviton loop if one replaces T µν bythe energy–momentum tensor t µν for weak gravitational waves. Expanding theexpression for the matter energy–momentum tensor in powers of φ −〈φ〉 andperforming its expectation value with respect to the matter state, one recognizesthat the matter terms in (2.117) are equivalent to this expectation value. The