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Ilya Shapiro, Quantum Gravity from the QFT Perspective April - 2019 Lecture 4 Quantum Gravity from the QFT perspective Ilya L. Shapiro Universidade Federal de Juiz de Fora, MG, Brazil Partial support: CNPq, FAPEMIG ICTP-SAIFR/IFT-UNESP – 1-5 April, 2019 Ilya Shapiro, Quantum Gravity from the QFT perspective April - 2019 Lecture 4. Renormalization in QG Gauge symmetry and renormalization. Power counting. • Quantum GR vs higher derivative theory (HDQG). • Superrenormalizable QG. • Ghosts in general HDQG. • Ambiguities and 1-loop results. • Bibliography K. Stelle, Phys. Rev. D (1977). I.L. Buchbinder, S.D. Odintsov, I.Sh., Effective Action in Quantum Gravity. (1992 - IOPP). M. Asorey, J.L. Lopez, I.Sh., hep-th/9610006; In.J.M.Ph. A12(1997). Ilya Shapiro, Quantum Gravity from the QFT perspective April - 2019 We start from some covariant action of gravity, S = d 4x√ g (g ) . − L µν Z (g ) can be of GR, (g )= κ−2(R + 2Λ) or some other. L µν L µν − Gauge transformation x ′µ = x µ + ξµ. The metric transforms as ′ δg = g (x) g (x)= ξ ξ . µν µν − µν −∇µ ν − ∇ν µ It is customary to parametrize metric as gµν (x)= ηµν + κhµν (x) , then 1 δh = (∂ ξ ∂ ξ ) h ∂ ξα h ∂ ξα ξα∂ h = R ξα . µν −κ µ ν − ν µ − µα ν − να µ − α µν µν , α The gauge invariance of the action means δS α Rµν , α ξ = 0 . δhµν · Ilya Shapiro, Quantum Gravity from the QFT perspective April - 2019 It proves useful to divide hµν into irreducible components k k ω = µ ν , θ = η ω . µν k 2 µν µν − µν The (Rivers) projectors are 1 1 P(2) = (θ θ θ θ ) θ θ ; µν , αβ 2 µα νβ − µβ να − 3 µν αβ 1 P(1) = (θ ω + θ ω + θ ω + θ ω ); µν , αβ 2 µα νβ να µβ µβ να νβ µα − 1 − P(0 s) = θ θ , P(0 w) = ω ω ; µν , αβ 3 µν αβ µν , αβ µν αβ and the transfer operators (ws) 1 (sw) 1 P = θµν ωαβ , P = ωµν θαβ . µν , αβ √3 µν , αβ √3 Ilya Shapiro, Quantum Gravity from the QFT perspective April - 2019 Then for the propagator of the gravitational field we get P(2) 0 0 0 0 P(1) 0 0 hµν hαβ = − h i 0 0 P(0 s) P(sw) 0 0 P(sw) P(0−s) Also, one can present the quantum metric as ⊥ 1 h = h + η ϕ + 2 ε , µν µν 4 µν ∇(µ ν) where ⊥ h = P(2) αβ h , 2 ε = P(1) αβ h , µν µν αβ ∇(µ ν) µν αβ and ⊥ ε = ε + ∂ σ , ϕ = h σ , h = hµ . µ µ µ − µ Performing gauge transformation, we get ⊥ ⊥ ⊥ ⊥ δh = 0 , δϕ = 0 , δσ = 2ξ, δε = ξ , where ξ = ξ +∂ξ . µν − µ − µ µ µ Ilya Shapiro, Quantum Gravity from the QFT perspective April - 2019 An important observation is that one can use another parametrization for quantum metric, e.g., κΦµν = √ ggµν ηµν , − − or even quantize in a very different variables. Here we shall use the parametrization with hµν . An important requirement is that relevant observables should not depend on the choice of parametrization and, in particular, on the gauge fixing condition (gauge dependence). In general, this dependence is not easy to deal with, but in some cases (one-loop divergences) we can do it easily. Ilya Shapiro, Quantum Gravity from the QFT perspective April - 2019 Gauge invariant renormalizability Gauge invariant renormalizability of the theory means that the divergences (and therefore counterterms), in all loop orders, have the same symmetry as the classical action. It does not guarantee the multiplicative renormalizability, that requires also a “correct” power counting. The papers on renormalizability in quantum Gravity (QG) include K. Stelle, Phys. Rev. D (1977). I.L. Buchbinder, S.D. Odintsov, I.Sh., Effective Action in Quantum Gravity (IOPP, 1992). B.L. Voronov and I.V. Tyutin, Sov. Nucl. Phys. (1981,1984). P.M. Lavrov, I.Sh., Gauge invariant renormalizability of quantum gravity, arXiv:1902.04687 The last work deals with the QG theory of a general form and use Batalin-Vilkovisky formalism. In what follows we use notations of Stelle. Ilya Shapiro, Quantum Gravity from the QFT perspective April - 2019 As far as we intend to arrive at the Feynman rules, we can start from the Faddeev-Popov approach, ¯ Z (J) = DhDCDC Det (Yαβ ) Z i i exp iS(h)+ χα Y χβ + C¯ α Mβ Cβ + iJµν h . × 2 αβ 2 α µν n α o β δχ where the ghost part is Mα = Rµν , α . δhµν The useful choice of the gauge fixing condition and the weight function depends on the theory. The most popular gauges are the Fock-deDonder one χµ = ∂ Φµν , κΦµν = √ ggµν ηµν , ν − − 1 harmonic gauge χ = ∂ν h β∂ h , β = , µ µν − µ 2 and background gauges ′ χ = ν h β h , where g g = g + h . µ ∇ µν − ∇µ µν → µν µν µν Ilya Shapiro, Quantum Gravity from the QFT perspective April - 2019 Consider the total action with gµν (x)= ηµν + κhµν (x) , 1 1 S = S(h)+ χα Y χβ + C¯ α Mβ Cβ . t 2 αβ 2 α It is not gauge invariant, but possesses BRST invariance, δRµν , α δRµν ,β γ Rρσ,β Rρσ , α = Rµν ,γ f αβ , δhρσ − δhρσ where for a gravity theory f γ (x, y, z) = δγ δ(x z)∂ δ(x y)+ δγδ(x y)∂ δ(x z) . αβ α − β − β − α − Then, α δBRST hµν = κRµν , α C δµ , 1 δ Cα = f α Cβ Cγ δµ = Cβ ∂ Cα δµ . BRST 2 βγ α · ¯ β δBRST Cα = Yαβ χ (h) δµ , Ilya Shapiro, Quantum Gravity from the QFT perspective April - 2019 It proves useful to introduce two extra sources. Consider the total action ¯ µν ¯ α σ β S = St + K Rµν , αC + Lσ∂β C C . One can easily prove that it satisfies the Noether identity ¯ ¯ ¯ ¯ ¯ δS δS δS δS β δS + + Yαβ χ = 0 . µν ¯ σ ¯ δK δhµν δLσ δC δCα One can perform the BRST transformation in Z and since it does not change, we arrive at the Slavnov-Taylor identities for Z , ¯ Z (J, β,¯ β, K , L) = DhDCDC Det (Yαβ ) Z exp iS¯ + iJµν h + iC¯ βµ + iβ¯ Cµ . × µν µ µ n o Ilya Shapiro, Quantum Gravity from the QFT perspective April - 2019 As a next step we define W = i ln Z and the mean fields − δW δW δW h Cσ C¯ µν = µν , = , ρ = ρ . h i δJ h i δβ¯σ h i δβ The corresponding effective action σ ¯ µν Γ(hµν , C , Cρ, K , Lσ) = W (Jµν , β¯ , βρ, K µν , L ) h Jµν C¯ βρ Cσβ¯σ . σ σ − µν − ρ − One can prove that it satisfies the identity ˜ δΓ δΓ δΓ δΓ β δΓ + + Yαβ χ = 0 . µν ¯ σ ¯ δK δhµν δLσ δC δCα Ilya Shapiro, Quantum Gravity from the QFT perspective April - 2019 Finally, we introduce modified quantities 1 S˜ = S¯ χα Y χβ − 2 αβ and 1 Γ=Γ˜ χα Y χβ , − 2 αβ which satisfy the identities δS˜ δS˜ δS˜ δS˜ 0 µν + = . δK δhµν δLσ δC¯ σ and δΓ˜ δΓ˜ δΓ˜ δΓ˜ 0 µν + = , δK δhµν δLσ δC¯ σ along with the so-called ghost equation δχα δΓ˜ δΓ˜ = 0 . ρσ ¯ δhρσ δK − δCα Ilya Shapiro, Quantum Gravity from the QFT perspective April - 2019 Consider the loop expansion ∞ ˜ ˜ ~k ˜(k) ˜(k) ˜(k) ˜(k) Γ = S + Γ , where Γ = Γdiv + Γfin . Xk=1 Assuming that the orders 1, 2, ..., k 1 are already renormalized, one can show that k - order terms do− satisfy the identities k ˜(k−i) ˜(i) ˜(k−i) ˜(i) ˜(k) δΓ δΓ δΓ δΓ Γdiv = µν + , ( ) E δK δhµν δLσ δC¯ σ ∗ Xi=0 n o where the nilpotent 2 = 0 operator is defined as E δS˜ δ δS˜ δ δS˜ δ δS˜ δ = µν + µν + + . E δhµν δK δK δhµν δLσ δC¯ σ δC¯ σ δLσ Obviously, (*) is equivalent to Γ˜(k) = 0 . E div Ilya Shapiro, Quantum Gravity from the QFT perspective April - 2019 Due to the nilpotence of and E Γ˜(k) = 0 E div the most general local solution is Γ˜(k) = A(h ) + X(h , Cσ, C¯ , K µν , L ) , div µν E µν ρ σ where A is some covariant functional and X is an arbitrary local functional of its variables. One can prove that the X -terms can be removed by E Renormalization of the field hµν together with some gauge transformation.• ¯ ν Renormalization of the FP ghosts Cµ and C together with some• BRST transformation. Finally, the problem is reduced to the possible form of the covariant local functional A(hµν ). Ilya Shapiro, Quantum Gravity from the QFT perspective April - 2019 Let us use the notion of power counting to explore A(hµν ). The universal formula for the superficial degree of divergence is D + d = (4 rl ) 4n + 4 + K . − − ν ν Xlint X Here D is the superficial degree of divergence for a given diagram, d is the number of derivatives on external lines of the diagram, rl is the power of momenta in the propagator of internal line, n is the number of vertices and Kν is the power of momenta in a given vertex. On the top of that one can use topological relation between number of loops p, vertices n, and internal lines lint = p + n 1 . − Ilya Shapiro, Quantum Gravity from the QFT perspective April - 2019 As the first example consider quantum GR. 1 S = d 4x√ g (R + 2Λ) . − 16πG − Z For the sake of simplicity we consider only vertices with maximal Kν .
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