By three methods we may learn wisdom: First, by reflection, which is noblest; Second, by imitation, which is easiest; And third by experience, which is the bitterest. Confucius (551 BC- 471 BC) 30 Einstein Gravity from Fluctuating Conformal Gravity Among the many possible gravitational theories, set up in the previous section to generalize the Einstein-Hilbert action 1 4 1/2 EH = d x ( g) R, (30.1) A −2κ Z − there exists a special one that has a chance of leading to a renormalizable quantum field theory. Its action is based on the conformally invariant expression1 1 4 1/2 λµνκ conf = 2 d x ( g) CλµνκC , (30.2) A −8ec Z − where Cλµνκ is the conformal Weyl tensor composed of the Riemann curvature tensor R and the Ricci tensor R gνλR as [1] λµνκ µκ ≡ µνλκ 1 1 C = R + (g R g R g R +g R ) R (g g g g ).(30.3) λµνκ λµνκ 2 λν µκ − λκ µν − µν λκ µκ λν − 6 λν µκ − λκ µν Inserting this decomposition into (30.2), we obtain 1 4 1/2 λµνκ µν 1 2 conf = 2 d x ( g) (RλµνκR 2Rµν R + 3 R ). (30.4) A −8ec Z − − Using the fact that the Lanczos integral [2] 4 1/2 λµνκ µν 2 d x ( g) (RλµνκR 4Rµν R + R ) (30.5) Z − − is a topological invariant that does not contribute to the equations of motion, the conformal action (30.4) can be rewritten as 1 4 1/2 µν 1 2 conf = 2 d x ( g) (Rµν R 3 R ). (30.6) A −4ec Z − − 1 2 Note that 1/ec has the dimension ¯h. 1522 1523 We have neglected surface terms which do not contribute to the equation of motion. The action (30.2), and thus also (30.6), does not contain any mass scale and is in- variant under local conformal transformations of the metric introduced by Hermann Weyl [1] as: g (x) Ω2(x)g (x) e2α(x)g (x). (30.7) µν → µν ≡ µν Based on this action it will eventually be possible to construct a theory of all matter with its weak, electromagnetic, and strong interactions. For this we shall have to extend it with the actions of various extra fields. Before we come to this, we shall study the behavior of spacetime itself as it would follow from the action (30.6) alone. Up to this point, it is not yet clear that we should choose a negative overall sign of the action (30.2). Since the fluctuations are quantum-mechanical and the path integral in the linear approximation is of the Fresnel type, there is no convergence criterion to settle the sign. Recall that in the Einstein-Hilbert action (4.352), the sign was fixed by the fact that the linearized action (4.372) corresponded in the Euclidean formulation to a strictly positive energy. In the quantum mechanical formulation, this led to the attractive Newton law (5.99) between masses. For the conformally-invariant action, the determination of the sign is nontrivial. Only later, after the calculation of the gravitational potential in Eqs. (30.53) and (30.54), it will turn out that no matter whether we quantize the gravitational field without tachyons (i.e., correctly) or with them (i.e., wrongly), we must choose the negative overall sign. In either quantization, the long-range forces are Newton-like. For simplicity, we shall first focus attention upon the vicinity of the Minkowski metric ηµν . For this we set gµν(x) = ηµν + hµν(x), as in (4.350), and expand the action (30.6) in powers of the field hµν (x), assuming all ten components of hµν (x) to be much smaller than 1. A functional integral over eiAconf takes into account all quantum fluctuations of hµν(x). The important observation made in this chapter is that these fluctuations are so violent that they are capable of spontaneously generating an Einstein-Hilbert term in the effective action. This will become clear after Eq. (30.44). The Einstein-Hilbert term introduces a nonzero mass scale into the conformal action (30.2). This mass plays the role of a dimensionally transmuted coupling constant. It appears in the same way as in scalar QED in Section 17.12. The formal reason for this is that the conformal action has, in the weak-field limit, a term that is quadratic in the fields hµν(x) and contains four field derivatives. 4 ikx 4 These lead to propagators of hµν (x) which behave like d k e /k . This makes them logarithmically divergent at small k. Such divergenciesR are quite familiar from two-dimensional many-body field theories. There they prevent the existence of long-range ordered states [3, 4, 5]. In quantum field theories of elementary particles, they lead to a no-go theorem for fields of spinless particles of zero mass [6]. A similar phenomenon has been observed long ago in the context of bio-membranes [7], pion condensation [8], ferromagnetism [9], later in string theories with extrinsic curvature [10, 11], and after that in a gravity-like theory [12]. The fourth power of derivatives in the free graviton action makes the long-wavelength fluctuations so violent that the theory spontaneously creates a new mass term. In the case of 1524 30 Einstein Gravity from Fluctuating Conformal Gravity biomembranes and stiff strings, this is responsible for the appearance of a surface tension. In conformal gravity, the violent long-wavelength fluctuations produce an effective action `ala Einstein-Hilbert that is responsible for the correct long-range behavior of gravitational forces. 30.1 Classical Conformal Gravity Conformal Gravity has become a fashionable object of study. The initial motivation was to investigate it as an interesting extension of Einstein gravity [25]. It is a purely metric theory that exhibits general coordinate invariance and satisfies the equivalence principle of standard gravity. In addition, it is invariant under the local κ Weyl transformation (30.7) of the metric. Let us take Rµνλ from Eq.(4.360), and write it as a covariant curl R κ = ∂ Γ ∂ Γ [Γ , Γ ] κ (30.8) µνλ { µ ν − ν µ − µ ν }λ κ κ of four 4 4 -matrices built from the Christoffel symbols Γν λ Γνλ . Then the × { } ≡κ curvature tensor may be viewed as a set of 4 4 -matrices (Rµν )λ . These have a form completely analogous to the field strength× tensor (28.11) of the SU(3)-invariant nonabelian gauge theory of strong interactions (QCD). This similarity is part of the esthetical appeal of the conformal action (30.2). 2 As in the QCD action (28.10), the coupling constant ec is dimensionless, and this ensures that all Feynman diagrams in a perturbation expansion of the theory can be made finite by counter terms that have the same form as those in the initial action. This is what makes the conformal action an attractive candidate for a quantum theory of gravitation. Adding to (30.2) a source term for the coupling of the gravitational field with matter, which reads [compare with (5.1) and (5.66)] m m 1 4 1/2 µν = 2 d x ( g) gµν T , (30.9) A − Z − m we obtain, from a variation of the total action conf + with respect to the metric A A gµν, the field equation [25]: m 1 µν µν 2 B = T , (30.10) 2ec where Bµν on the left-hand side is the symmetric and conserved Bach tensor defined by 1 µν −1/2 δ conf 2 B 2( g) A . (30.11) 2ec ≡ − δgµν Functional differentiation of (30.2) yields the expression Bµν 2Cµλνκ + CµλνκR , (30.12) ≡ ;λ;κ λκ 30.2 Quantization 1525 whose explicit form is 1 1 Bµν= gµνR;κ +Rµν;κ Rµκ;ν Rνκ;µ 2RµκRν + gµνR Rλκ 2 ;κ ;κ − ;κ − ;κ − κ 2 λκ 2 2 2 1 + gµν R;κ R;µ;ν + RRµν gµνR2. (30.13) 3 ;κ − 3 3 − 6 Note that the Bach tensor is traceless so that Eq. (30.10) can be satisfied only if the energy-momentum tensor is also traceless. 30.2 Quantization Let us now study the quantum field theory associated with the action (30.2). It is formally defined by the functional integral (¯h = c = 1) [13]: iAconf Z = gµν e . (30.14) Σi D Xi Z Here gµν denotes the measure of the functional integral. Its proper treatment re- quiresD fixing of both gauge symmetries, the first with respect to coordinate transfor- mations, the second with respect to conformal transformations (30.7). Alternatively, we can use the decomposition (4.435) of the field tensor hµν into its different irre- ducible parts under Lorentz tranformations, and extend them to the metric tensor to factorize g g(2) gl gs . (30.15) D µν ≡D µν D µνD µν The sum in (30.14) comprises a sum over topologically distinct manifolds Σi, al- though it is not clear how to do that, even in principle, since so far four-dimensional manifolds cannot even be classified [13, 14, 15]. Due to the invariance of the theory under general coordinate transformations, only the spin-2 content of the field is physical. This fact may be taken care of by l inserting into this measure a δ-functional for the unphysical field components gµν . In order to do this in a covariant way we adapt the Faddeev-Popov method of quantum electrodynamics to the present field theory. We go to the weak-field limit and fix the gauge of hµν (x) by requiring the divergence to satisfy the Hilbert condition (4.399).