Quantum Gravity

58 COVARIANT APPROACHES TO QUANTUM GRAVITYWe have introduced here the abbreviationS (3) ∆∆∆ ≡δ 3 Sδ〈ϕ i 〉δ〈ϕ k 〉δ〈ϕ l 〉 ∆ i∆ k ∆ l(which also includes an integration), and similar abbreviations for the otherterms. We emphasize that the first functional derivative of S with respect tothe mean field has cancelled in (2.112); this derivative is not zero (as sometimesclaimed) because the mean field is at this stage arbitrary and does not have tosatisfy the classical field equations.In the integral over ∆, the odd terms in ∆ vanish. Moreover, after this integrationthe terms of fourth and sixth power of ∆ yield terms proportional to 2and 3 , respectively. In the highest (‘one loop’) order for Γ loop one thus has toevaluate only the integral over the exponential in (2.112). This yieldsΓ loop = i 2 ln det δ2 S[〈ϕ〉]δ〈ϕ〉δ〈ϕ〉 + O(2 ) ≡ Γ (1) + O( 2 ) , (2.113)where Γ (1) is the one-loop effective action. In general, one has the loop expansionΓ loop =∞∑Γ (L) [〈ϕ〉] , (2.114)L=1where Γ (L) is of order L . We now want to investigate equation (2.108) at onelooporder. Introducing the notationone getsδ 2 S[〈ϕ〉]δ〈ϕ i 〉δ〈ϕ k 〉 ≡ S ik ,δΓ (1)δ〈ϕ j 〉 = i 2 (detS mn) −1 δ(detS mn )δ〈ϕ j 〉= i (S−1 ) mn δS nm2 δ〈ϕ j 〉 ≡−i δS 2 Gmn nmδ〈ϕ j 〉 , (2.115)where G mn denotes the propagator occurring in (2.106) evaluated at this orderof approximation. We introduce the notationδS nmδ〈ϕ j 〉 ≡ S jmn , S ik ≡ F (∇)δ(x, y) ,where we have for simplicity suppressed possible discrete indices attached to thedifferential operator F . (For example, for a free massless scalar field we just have