Approaches to Quantum Gravity

116 R. PercacciThe requirement of renormalizability played an important role in the constructionof the Standard Model (SM) of particle physics. Given that the SM is not acomplete theory, and that some of its couplings are not asymptotically free, nowadaysit is regarded as an effective QFT, whose nonrenormalizable couplings aresuppressed by some power of momentum over cutoff. On the other hand, any theorythat includes both the SM and gravity should better be a fundamental theory. Forsuch a theory, the requirement of asymptotic safety will have the same significancethat renormalizability originally had for the SM.8.3 The case of gravityWe shall use a derivative expansion of Ɣ k :∞∑ ∑Ɣ k (g μν ; g (n)i) =n=0ig (n)i(k)O (n)i(g μν ), (8.10)where O (n)i= ∫ d d x √ gM (n)iand M (n)iare polynomials in the curvature tensorand its derivatives containing 2n derivatives of the metric; i is an index that labelsdifferent operators with the same number of derivatives. The dimension of g (n)iisd n = d − 2n. The first two polynomials are just M (0) = 1, M (1) = R. Thecorresponding couplings are g (1) =−Z g =− 116πG , g(0) = 2Z g , being thecosmological constant. Newton’s constant G appears in Z g , which in linearizedEinstein theory is the wave function renormalization of the graviton. Neglectingtotal derivatives, one can choose as terms with four derivatives of the metricM (2)1= C 2 (the square of the Weyl tensor) and M (2)2= R 2 . We also note thatthe coupling constants of higher derivative gravity are not the coefficients g (2)irather their inverses 2λ = (g (2)1 )−1 and ξ = (g (2)2 )−1 . Thus,Ɣ (n≤2)k=∫d d x √ gbut[2Z g − Z g R + 12λ C 2 + 1 ξ R2 ]. (8.11)As in any other QFT, Z g can be eliminated from the action by a rescaling of thefield. Under constant rescalings of g μν ,ind dimensions,Ɣ k (g μν ; g (n)i) = Ɣ bk (b −2 g μν ; b d−2n g (n)i). (8.12)This relation is the analog of (8.9) for the metric, but also coincides with (8.3), theinvariance at the basis of dimensional analysis; fixing it amounts to a choice of unitof mass. This is where gravity differs from any other field theory [33; 35]. In usualQFTs such as (8.8), one can exploit the two invariances (8.3) and (8.9) to eliminatesimultaneously k and Z from the action. In the case of pure gravity there is onlyone such invariance and one has to make a choice.