Einstein Gravity from Fluctuating Conformal Gravity

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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 invariant 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 àla 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 transformations, 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). This can be achieved by inserting into the functional integral (30.14) a gauge-fixing functional analogous to one of the various possible choices (14.353). For example we may use the functional

i 4 µ 1 2 F4[h] = exp d x[∂ hµν (x) 2 ∂ν h(x)] , (30.16) (−2ζ Z − ) where ζ is an arbitrary parameter. To find the effect of this choice upon the measure of functional integration, we proceed as in Section 14.16 and insert into F4[h] the gauge-transformed hµν -field àla (4.380),

Λ hµν = hµν + ∂µΛν + ∂ν Λµ. (30.17) 1526 30 Einstein Gravity from Fluctuating Conformal Gravity

This yields

Λ i 4 µ 1 2 F4[h ] = exp d x[∂ hµν(x) 2 ∂ν h(x)+ ∂ Λν] . (30.18) (−2ζ Z − )

Integrating this over all Λµ(x) determines the normalization functional for the measure [compare (14.365)]:

4 Λ 4 −2 Φ4[h]= ΛµF4[h ] Det(∂ ) . (30.19) Z D ∝

With the gauge fixing functional F4[h]/Φ4[h], the measure (30.15) is free of the l components gµν. Finally, we account for the conformal invariance of the theory. For this we insert s a δ-functional ensuring the vanishing of the scalar part gµν of the metric. In the linear approximation, this is done by inserting a gauge-fixing functional for the scalar part of the field hs h µ ∂ ∂−2∂ hµν [recall (4.367)]. As an example we insert ≡ µ − µ ν

i 4 2 s 2 i 4 2 µ µ ν 2 F4′ [h] = exp d x(∂ h ) = exp d x[∂ hµ ∂ ∂ hµν ] . (30.20) (−2ξ Z ) (−2ξ Z − )

Here ξ is another arbitrary parameter. Subjecting F4′ [h] to an inﬁnitesimal Weyl transformation (30.7), which reads

h h +2α(x)η , (30.21) µν → µν µν we obtain the transformed functional

α i 4 2 µ µ ν 2 2 F4′ [h ] = exp d x[∂ hµ (x) ∂ ∂ hµν (x) 6∂ α] . (30.22) (−2ξ Z − − ) This can be integrated functionally over all α(x) to yield the normalization functional

−1/2 4 Φ ′ [h] Det (∂ ). (30.23) 4 ∝

Thus, by analogy with the quantum electrodynamic gauge-ﬁxing factor F4[A]/Φ4 in (14.366), we obtain in the weak-ﬁeld approximation the proper measure of quantum gravity in the functional integral (30.14) as

F4[h] F4′ [h] (2) l s 5/2 4 ′ hµν hµν hµν hµν = Det (∂ ) hµνF4[h]F4 [h]. (30.24) D D D ≡D Φ4 Φ4′ D A great advantage of the conformal action (30.2) is that the coupling constant αc is dimensionless so that, by pure power counting, the Feynman diagrams of its perturbation expansions can be renormalized order by order. In addition, the β- function governing the flow of the coupling constant is asymptotically free [16]. It has been explained in the beginning of this chapter that the fluctuations of the field around the Minkowski metric are controlled by a term with four derivatives. As a negative consequence, the propagator in four dimensions d4k eikx/k4, when R 30.2 Quantization 1527 treated àla Feynman, contains modes that move faster than light (tachyons) [17]. They are a signal of the fact that the weak-field expansion of the metric around the Minkowski metric ηµν is not an expansion around a state of minimal energy. There exist field configurations of lower energy, and a consistent perturbative expansions must be based on oscillations around that. These are necessarily free of tachyons, as discussed in Section 18.7. Similar stability arguments have been used in QCD to find an optimal field configuration around which one can perform a perturbation expansion of the theory. In QCD, a first search for a lower minimum was successfully performed by Savvidy [18]. Improvements of his result were found by various authors [19]. If QCD is formulated on a lattice and studied numerically, the formation of a nontrivial vacuum can be followed most explicitly [20]. The instability of Minkowski spacetime against small fluctuations and the associated tachyon problem in quantized gravity have prevented many authors from accepting the conformal action as a viable extension of Einstein’s theory. The problem has been discussed in detail by Stelle [21], who showed that tachyonic states are a serious obstacle for making a unitary quantum field theory. One possible way out of the dilemma is by reducing the requirement of renormalizability of the quantum field theory to the more modest requirement of asymptotic safety [22, 23]. Another proposal is based on the choice of suitable boundary conditions at large infinity [24]. A third proposal has been made by Bender and Mannheim to use a different quantization procedure [25] of the gravitational fields. They point out that, by studying fluctuation in complex phase space rather than in the real phase space we live in, the tachyonic states disappear [26]. This is a simple generalization of the obvious fact that the unstable potential V (x)= x2/2 is stable if x is permitted to fluctuate only along the imaginary axis ix. Of course,− this consideration does not produce a truly stable minimum of the energy, over which the metric fluctuations would necessarily represent only physical particles which move slower than light. However, if we focus attention upon some important effects of quantum fluctuations, a true stability is not really needed. The reason is that quantum mechanical integrals are of the Fresnel type and can be calculated not only for a stable minimum but also for the inverted potential. In any quantum theory whose action involves four derivatives, an analogous situation exists in the extended phase space consisting of coordinates and their derivatives. In the complex version of it, a quantum theory can be constructed after slight modification of Feynman’s rule for calculating propagators of the theory. These will be discussed in detail in Appendix 30B. In this chapter we want to emphasize the positive consequences of the four derivatives in (30.2). At small k, they make the quantum fluctuations of the gravitational field so violent, that they spontaneously generate Newton forces which will govern the long-distance attraction between celestial bodies. In particular, we shall demon- strate that this type of mechanism spontaneously creates an effective gravitational action that has been proposed long time ago by Starobinsky [27]:

1 4 1/2 2 2 St = d x ( g) (R ξ R ) . (30.25) A −2κ Z − − 1528 30 Einstein Gravity from Fluctuating Conformal Gravity

Here ξ is Starobinsky’s parameter that is determined by the inflational scale of the Planck satellite data to be ξ/√κ 105 (see Ref. [28]). ∼ To derive this we begin by expanding the conformal action around a flat back- ground up to quadratic order in hµν (x)= gµν(x) ηµν , using the weak-field expansions (4.362), (4.363), (4.364), (4.366), and (4.367):−

1 1 4 µν 4 4 λ µν ρ 1 4 s 4 s conf = 2 d x h ∂ hµν d x ∂ hλµH ∂ hρν d x h ∂ h .(30.26) A −8ec 2 Z − Z − 6 Z 1 2 s Here we have introduced the tensors Hµν = 2 ∂µ∂ν ∂ ηµν and the scalar ﬁeld h = µ −2 µν − hµ ∂µ∂ ∂ν h , as before in (4.367). We have also omitted constant contributions from− total derivatives. After some algebra, relegated to Appendix 30A on page 1535, the action (30.26) may be rewritten as

1 4 2 λµ (2) 2 λκ conf = 2 d x ∂ h Pµν,λκ(i∂)∂ h , (30.27) A −16ec Z (2) where Pµν,λκ(i∂) is the projection operator into the spacetime-dependent symmetric tensor ﬁelds of spin 2. It is composed of products of transverse projection operators P t (i∂)= η ∂ ∂ /∂2 as µν µν − µ ν

(2) 1 t t t t 1 t t P (i∂) 2 [P (i∂)P (i∂)+ P (i∂)P (i∂)] 3 P (i∂)P (i∂) (30.28) µν,λκ ≡ µλ νκ µκ νλ − µν λκ (2) [recall Eq. (4G.4)]. Inserting Pµν,λκ(i∂) into (30.27), performing some integrations by part and discarding pure volume terms, we find indeed agreement with (30.26). It has been shown in Ref. [29] that a conformally invariant action can be derived, in the spirit of Sakharov [30], from the fluctuations of the conformal factor in the partition function of all matter fields of spin s. Ina D-dimensional Riemann spacetime with metric gµν, the conformal action receives contributions from the fields of various spins s. Integrating out the conformal transformations of the various spin fields, one obtains the conformal action (30.2) with the inverse coupling constant

1 1 1+ N N N N N = 0 + 1/2 + 1 233 3/2 + 53 2 , (30.29) 4e2 8π2(4 D) 120 40 10 − 720 45 ! c − where Ns is the number of fields of spin s. In four dimensions, this is divergent and requires a conformal action (30.2) to supply a counter term to end up with a finite coupling constant that can eventually be chosen to fit experiments. Apart from these divergent terms, there are also finite contributions to the conformal action which come from loop diagrams. If all fields of the total action are massless, the classical energy momentum tensor is traceless, as long as we work with the so-called improved energy-momentum tensor of Callan, Coleman, and Jackiw [31]. Apart from that classical part, there are contributions caused by loop diagrams. These are the famous conformal anomalies of gravitational theories. For example, a large part of the matter of the universe comes from spin-1/2 Dirac particles. These are believed to be originally created without a mass, so that their 30.2 Quantization 1529 classical energy-momentum tensor is traceless. But if we calculate loop diagrams, each massless Dirac field contributes an anomaly of the gravitational action of the form [32]:

anom 1 11 µνλκ = 2 CµνλκC 6 R , (30.30) A 2880π 2 − where the symbol denotes the Laplace-Beltrami operator (1.390) in four spacetime dimensions. A similar effective action comes from the loops of other massless particles. The sum of the anomalies of all fields has to be zero, if we want to make sure that quantum gravity can truly be renormalized. It will be important to know at the end which are precisely the fields in the ultimate theory of field and matter. As usual, we assume that all particle masses arise from a spontaneous symmetry breakdown in scalar field theories of the Higgs type. The precise set of scalar fields in the theory is not yet fully known and will certainly be subject to change in the future. To simulate their typical effect upon gravity we shall couple, pars pro toto, a charged scalar fields ϕ to the gravitational field in an almost minimal way. For this we add, to the conformal action (30.2), the generic Higgs-type action (27.83) that contains the simplest conformally invariant term proportional to R, namely R ϕ 2. Formulated in curved spacetime with a metric g (x), this has the form | | µν [33, 34, 35, 36]:

m 2 4 1/2 1 µν ∗ ∗ R 2 m 2 g 4 = d x( g) g Dµϕ Dνϕ + ϕ ϕ ϕ . (30.31) A Z − 2 12| | − 2 | | − 4| | Here D ∂ + ieA denotes the covariant derivatives defined as in (27.84), with e µ ≡ µ µ denoting the respective coupling constant, for example the electromagnetic one. We have omitted the action 1 F F µν of the gauge fields themselves that was written − 4 µν down in Eq. (27.83), where the field tensor Fµν collects the covariant curls of all relevant gauge fields, as explained in (27.88). The masses m2 break the conformal invariance of the theory. Since the Higgs actions are supposed to generate the correct mass terms of all matter fields, we have constructed them to be invariant under the Weyl transformations

g (x) Ω2(x)g (x), ϕ(x) Ω−1(x)ϕ(x). (30.32) µν → µν → Under these the gradient term of the scalar ﬁeld transforms as follows [43]:

gµνD∗ ϕ∗D ϕ Ω−2gµνD∗ (Ω−1ϕ∗)D (Ω−1ϕ) µ ν → µ ν Ω−4[D∗ ϕ∗Dµϕ ϕ∗(D ϕ)Ω−1∂µΩ (D∗ ϕ∗)ϕΩ−1∂µΩ+ ϕ∗ϕΩ−2∂ Ω∂µΩ] → µ − µ − µ µ Ω−4[D∗ ϕ∗Dνϕ+ϕ∗ϕ Ω−1 Ω ∂ (ϕ∗φ ∂µ ln Ω)]. (30.33) → µ − µ In the action, the last term can be neglected since it is a total derivative. The scalar curvature term in (30.31) changes under Weyl transformations like

R Ω−2(R 6Ω−1 Ω). (30.34) → − 1530 30 Einstein Gravity from Fluctuating Conformal Gravity

Therefore a scalar action is properly Weyl invariant if it contains the invariant combination

4 1/2 µν ∗ ∗ 1 2 d x( g) g Dµϕ Dνϕ + R ϕ . (30.35) Z − 6 | | After the electroweak phase transition, where some of the Higgs ﬁelds acquire a nonzero vacuum expectation value, each of them contributes a term of the type

4 1/2 1 2 d x( g) R ϕi . (30.36) Z − 6 | | For a smooth expectation value of ϕ 2 in spacetime, this is a term that produces stiffness in spacetime proportionally| to| the scalar curvature R. When comparing this with the Einstein-Hilbert action (30.1), we see that it has, unfortunately, the wrong overall sign in front of R. Hence the associated gravitational field is unstable. The weak-field gravitational fluctuations of the metric would contain tachyons, and this cannot be, as discussed in Section 18.7. The universe would collapse. Fortunately, there is a possibility of rescuing stability. This can be done with the help of a modified Higgs-like action which is not made Weyl-invariant by an R/12-term in (30.31), but with the help of a gauge field wµ(x) introduced by Weyl in his original work. This can be used to make any derivative of scalar or fermion fields covariant under Weyl transformation. A minimal way of achieving this is by considering a single scalar field φ and forming the action

w w 4 1/2 1 µν w w g 4 1 µν WT = d x( g) g Dµ φDν φ φ F µν F . (30.37) A Z − 2 − 2 − 4gw

Here wµ is the Weyl gauge ﬁeld. It was introduced by him to convert a globally Weyl-invariant ﬁeld into a locally Weyl-invariant form. The symbol

Dw D Ω−1(x)∂ Ω(x)= D w (30.38) µ ≡ µ − µ µ − µ denotes the associated covariant derivative, and F w ∂ w ∂ w the four-curl. µν ≡ µ ν − ν µ If the Weyl transformation is written in an exponential form as Ω(x) = eα(x), the Weyl ﬁeld changes by

w w + ∂ α(x). (30.39) µ → µ µ The gauge field w(x) has the virtue that it makes the action (30.37) Weyl-invariant without an extra R-term. In the action (30.37), the φ-fluctuations are stable. The important observation is now that by a combination of the actions (30.37) and the Higgs-type actions (30.35) it is possible to generate spontaneously an Einstein-Hilbert action whose spacetime fluctuations around Minkowski spacetime are stable. To achieve this we simply add to (30.37) a small admixture of a Higgs- type action with the opposite sign to that of (30.35):

1 4 1/2 µν R 2 WT′ = d x( g) g DµφDνφ φ . (30.40) A 2 Z − − − 6 30.2 Quantization 1531

While doing this we must make sure that the accompanying gradient term of the scalar field remains smaller than that in (30.37) to maintain the stability of the φ- fluctuations. The admixture of (30.40) will bring in an R-term with the correct sign to ensure stable metric fluctuation. The admixture must be small enough to ensure that the total action still has φ-stability. But it must be large enough to ensure that the metric fluctuations follow Einstein’s theory. In order to achieve both goals we choose the mixing to be hyperbolic. We multi- ply action (30.37) by a factor cosh2 θ that is larger than unity, and the admixture of (30.40) by a factor sinh2 θ that is smaller than unity. Thus we form the combination

2 2 = cosh θ sinh θ ′ . (30.41) Amixture AWT − AWT To this we add the curvature terms (30.36) coming from the Higgs ﬁelds. After this, the ﬁeld φ is governed by an action

1 R g cosh2 θ = d4x( g)1/2 gµνD φD φ (φ2 sinh2 θ 2 ϕ 2) φ4 , (30.42) Amix 2 − µ ν − 6 − | i| − 2 Z Xi where the sum over i runs over all Higgs fields. In going from (30.41) to (30.42), we w w µν have omitted the action (1/4gw) F µν F of the Weyl field in (30.37), as well as the accompanying mixing− factor cosh2 θ. We can now choose a small enough coupling constant g so that φ2 sinh2 θ be- 2 h i comes much larger than the sum of the ϕ i ’s of all Higgs-fields. Then the effective gravitational action spontaneously generatedh| | i by the scalar field expectations reads

2 2 2 ( φ sinh θ 2 i ϕi ) 4 1/2 1 4 1/2 ind = h i − | | d x( g) R = d x( g) R. (30.43) A − 12 P Z − −2¯κ Z − This corresponds to an induced gravitational constant

1 ( φ2 sinh2 θ 2 ϕ 2) h i − i | i| , (30.44) κ¯ ≡ 6 P which can be adjusted to be equal to the experimental value 1/κ. Hyperbolic mixing angles of this type have been introduced before in Refs. [34]– [37]. If the Higgs fields are neglected and only a single scalar field φ is assumed to be present, the theory is similar to the modified gravity in the Brans-Dicke formulation [38], for which a quantization has been proposed in Ref. [39]. Let us study the consequences of the generated R-term (30.43) in the conformal 2 2 gravity. The new terms coming from the expectation values φ and ϕi carry mass dimensions and therefore break the conformal symmetryh ofi the action.h| |i The action governing the fluctuations of the metric comes from the sum of (30.27) and (30.43), in which we have madeκ ¯ equal to the experimental gravitational coupling κ:

1 (2) 1 ′ 4 2 λµ 2 λκ 4 µν conf = 2 d x ∂ h Pµν,λκ∂ h + d x hµν G . (30.45) A −16ec Z 4κZ 1532 30 Einstein Gravity from Fluctuating Conformal Gravity

The last term in this action can also be composed in terms of the projection operators discussed in Appendix 4G. From these we derive Eq. (5.77), thus obtaining the action

1 (2) 1 (2) ′ 4 2 λµ 2 λκ 4 µν 2 λκ conf = 2 d x ∂ h Pµν,λκ∂ h d x h Pµν,λκ∂ h A −16ec Z − 8κZ 1 4 µν s 2 + d x h Pµν,λκ∂ hλµ. (30.46) 4κ Z This governs the fluctuations of h (x) for smooth average values of φ2 and ϕ 2 . µν h i h| i| i We can now integrate out the gravitational field fluctuations hµν(x). For this purpose, we rewrite the action (30.46) as

2 1 (2) 2e 1 ′ 4 λµ 4 c 2 λκ 4 µν s λκ conf = 2 d x h Pµν;λκ ∂ + ∂ h + d x h Pµν;λκh . A −16ec Z κ ! 4κZ (30.47)

Integrating out the spin-2 part yields the eﬀective action

D 2 (2) d k 4 2ec 2 Γ = D log k k . (30.48) Z (2π) − κ ! The divergent momentum integral can be made ﬁnite by standard counterterms. The logarithm contains the denominator of the propagator i G(2)(k)= (30.49) k2(k2 2e2/κ) − c of the graviton in this theory. It shows that the nonzero average values of φ 2 and 2 | | ϕi in (30.43) and (30.46) change the initial free-ﬁeld propagator associated with |the| conformal action (30.27), (2) ∗ P (k) h(2)(k)h(2) (k) =8e2 µν,λκ (30.50) h µν λκ i c k4 into the propagator (30.49) associated with the action (30.47). In spacetime, the propagator requires calculating a Fourier transform

4 d k i ′ G(2)(x x′)= e−ik(x−x ). (30.51) − (2π)4 k2(k2 2e2/κ) Z − c For this we have to specify the boundary conditions. This is commonly done by prescribing the path in the complex plane along which the integral over k0 should encircle the zeros in the denominator of the integral. If this is done in the same way as in Feynman’s QED, one would ﬁnd [compare (30B.15)]

4 d k i ′ G(2)(x x′)= e−ik(x−x ). (30.52) − (2π)4 (k2 + iǫ)(k2 2e2/κ + iǫ) Z − c 30.2 Quantization 1533

2 2 This would imply that the nonzero average values of φ and ϕi in (30.43) and (30.46) would change the free-ﬁeld propagator (30.50)| into| | |

(2) (2)∗ (2) 1 1 hµν (k)hλκ (k) =4κPµν,λκ(k) . (30.53) h i k2 + iǫ − k2 2e2/κ + iǫ! − c The negative sign in front of the second term is a signal of the “wrong quanti- 2 zation” in which the states with the nonzero square mass 2ec/κ have a negative norm. As announced before, this can be corrected by exchanging the Feynman rules for calculating the propagators with new iǫ-prescriptions (discussed in detail in Appendix 30B; see in particular the poles in Fig. 30.2). Thus, after a proper quantization the massive contributions to the propagator will appear with an opposite sign with respect to (30.53), and with an opposite iǫ-term, so that (30.53) turns into [compare with (30B.38) and (30B.41)]:

(2) (2)∗ (2) 1 1 hµν (k)hλκ (k) =4κPµν,λκ(k) + . (30.54) h i k2 + iǫ k2 2e2/κ iǫ ! − c − The spin-0 part adds to this the propagator 1 hs (k)hs∗ (k) =2κP s (k) . (30.55) h µν λκ i µν,λκ k2 + iǫ As a cross check we verify that the sum of the first term in (30.54) plus the spin-0 term (30.55) is equal to the fluctuations (5.98) in Einstein gravity. At long distances, this guarantees the Newton potential G M M /r between celestial bodies − N 1 2 of masses M1 and M2 [recall the derivation in (5.100)]. 2 At very short distances of the order of rY = κ/2ec, the gravitational potential 1/r is modified by the addition of a repulsive Yukawaq potential. It becomes 1/r + e−r/rY /r. Experimentally, such quantum corrections to the gravitational forces are ex- tremely hard to measure. If two bodies are brought together to atomic distances, there are immediately other forces which are much stronger than gravity and will dominate any measurement. Even if the bodies are carefully kept neutral, to avoid Coulomb interactions, there are molecular forces of the van-der-Waals type, which will win at the nanometer scale. It is therefore not astonishing that present experimental limits for the Newton forces are quite rough. They do not reach below 5 micrometers [40]. So far, one must be quite inventive to find observations which allow better tests [41]. The data produced by the Planck satellite [28] seem to be attractive candidates for this purpose, albeit only after a generous use of a theory that is still being discussed in the literature [42]. Let us see what phenomenological R2-term emerges from the action developed so far as a correction in the Starobinsky action (30.25). It can be found from the weak-field approximation to the scalar curvature in the actions (30.31) and (30.43). In the former, we simply take the effective potentials

1 2 R 2 gi 4 V (ϕi)= mi + ϕi + ϕi . (30.56) 2 6 | | 4 | | 1534 30 Einstein Gravity from Fluctuating Conformal Gravity

2 For negative mi , these give an unstable contribution to the Einstein action (30.43) for each Higgs ﬁeld ϕi. From the minima, we obtain the condensation energies

1 2 2 Vmin = mi + R/6 . (30.57) − 4gi Xi The sum of these yields the eﬀective action of the R2-type

2 eﬀ 1 4 1/2 R R2 ϕ = d x( g) . (30.58) A 4gi − 36 Xi Z A further term proportional to R2 is found from the action (30.42):

2 4 eff 1 4 1/2 R sinh θ R2 φ = d x( g) 2 . (30.59) A 4g Z − 36 cosh θ Adding this to (30.58), we find a total effective action of the R2-type:

2 4 eff 1 4 1/2 R 1 sinh θ 1 R2 = d x( g) 2 + . (30.60) A 2 − 36 2g cosh θ 2gi ! Z Xi This can be brought into agreement with the R2-correction term in the Starobinsky action (30.25), if we choose the dimensionless parameters g and gi to satisfy the condition ξ2 1 1 sinh4 θ 1 = 2 + . (30.61) κ 36 2g cosh θ 2gi ! Xi Then we obtain the correct prefactor of the R2-term in Starobinsky’s action (30.25), thus reproducing optimally the data of the Planck satellite. Let us also calculate loop corrections to these results. For this we assume all coupling constants gi in the scalar Higgs actions (30.31) to be small, at most of the 4 2 order of ec . Then we can ignore higher order perturbations of the order of gi , and find that the gravitational fluctuations produce an effective potential as a function of ϕi:

4 2 gi 4 3ec 4 ϕi 25 V (ϕi)= ϕi + ϕi log | | . (30.62) 4 | | 64π2 | | µ2 − 6 ! Here we may introduce a dimensionally transmuted coupling constant M deﬁned by the equation 4 2 gi 3ec µ 11 = 2 log 2 + . (30.63) 4 64π Mi 3 ! Then Eq. (30.62) becomes

4 2 3ec 4 ϕi 1 V (ϕi)= 2 ϕi log | |2 . (30.64) 64π | | Mi − 2! 30.3 Outlook 1535

This potential has a minimum at ϕi = ϕi,f which satisfies ϕ 2 = M 2. (30.65) h| i,f | i i To see what this minimum implies for the effective potential (30.64), we expand around ϕ 2 = M 2 up to the second order in ∆ ϕ 2 M 2 and find | i| i i ≡| i| − i 3e4 V (ϕ )= V (0) RM 2 R∆ + c ∆2 + (∆3), (30.66) i − i − i 64π2 i O i 4 4 2 where V (0) 3M ec /128π . The minimum with respect to ∆i lies at ∆i = 2 4 ≡ − 32π /3ec. It has the value 2 2 32π 2 Vmin = V (0) RMi 4 R . (30.67) − − 6ec The sum of second terms over i changes the spontaneously generated Einstein action.

30.3 Outlook

We have discussed a quantum field theory based on a conformally invariant action for the gravitational field. The standard problem has appeared that small fluctuations around Minkowski spacetime possess tachyonic excitation. It implies that spacetime is unstable with respect to these fluctuations. To solve this problem, one must find a new nontrivial spacetime configuration that is analogous to a confining field configuration of QCD. That started from an approximate configuration found by Savvidy and others. In gravity, a solution of this problem must be left to the future. In this chapter we have dealt with a more modest problem. We have used the fact that by extending the phase space of the spacetime of our universe into certain complex directions and by allowing fluctuations to take place only along the new axes, we have succeeded in constructing a conformal quantum field theory. Its restricted phase space fluctuations have only physical properties. As shown at the end of Appendix 30B, the perturbative expansions contain no tachyons and are capable of generating spontaneously an Einstein-Hilbert action. The resulting theory reproduces the presently known forces between celestial bodies, and specifies their fluctuation corrections. We did not enter into a discussion of the very fundamental problem whether it makes sense to consider quantum fluctuations of a field system that is intrinsically classical [13, 45]. Indeed, famous quantum field theorists like Freeman Dyson have expressed severe doubts about this [46].

Appendix 30A Some Algebra

Inserting the projection operator P (2) (i∂) of Eq. (30.28) into (30.27), we ﬁnd with ∂ˆ ∂/√∂2: µν,λκ ≡ d4x ∂2hµν P (2) (ˆq)∂2hλκ = d4x ∂2hµν ∂2[h 2∂ˆ (∂ˆλh )+ ∂ˆ ∂ˆ (∂ˆλ∂ˆκh )] µν,λκ µν − µ λν µ ν λκ Z Z 1536 30 Einstein Gravity from Fluctuating Conformal Gravity

1 d4x ∂2hµν (g ∂ˆ ∂ˆ )(g ∂ˆ ∂ˆ )∂2hλκ − 3 µν − µ ν λκ − λ κ Z = d4x ∂2hµν ∂2[h 2∂ˆ (∂ˆλh )+ ∂ˆ ∂ˆ (∂ˆλ∂ˆκh )] 1 d4x ∂2h ∂2h µν − µ λν µ ν λκ − 3 s s Z Z = d4x ∂2hµν ∂2h + d4x (∂ hµν )∂2[2(∂λh ) ∂ˆ ∂ˆκ(∂λh )] 1 d4x ∂2h ∂2h µν µ λν − ν λκ − 3 s s Z Z Z = d4x ∂2hµν ∂2h 2 d4x (∂ hµν )∂2H κ(∂λh ) 1 d4x ∂2h ∂2h . (30A.1) µν − µ ν λκ − 3 s s Z Z Z Using (4.366), we see that

d4x( g)1/2R Rµν − µν Z = 1 d4x(∂2h ∂ ∂λh ∂ ∂λh +∂ ∂ h)(∂2h ∂ ∂λh ∂ ∂λh +∂ ∂ h) 4 µν − µ λν − ν λµ µ ν µν − µ λν − ν λµ µ ν Z 1 4 2 2 λ 2 λ ν µ κ 2 = 4 d x[(∂ hµν ) + 2(∂µ∂ hλν ) + 2(∂ ∂ hλν )(∂ ∂ hµκ) + (∂µ∂ν h) Z +4∂2(∂µh )(∂λh )+2∂2(∂µ∂ν h )h 4∂2(∂λ∂ν h )h] µν λν µν − λν = 1 d4x[(∂2h )2 ∂ν∂κ(∂λh )(∂µh )+∂2(∂µh )(∂λh )+[(∂ ∂ η ∂2)hµν ]2] 4 µν − λν µκ µν λν µ ν − µν Z = 1 d4x[(∂2h )2 2(∂λh )Hνκ(∂µh ) + (∂2h )2]. (30A.2) 4 µν − λν µκ s Z Hence

2 d4x( g)1/2(R Rµν 1 R2) = 1 d4x[(∂2h )2 2(∂λh )Hνκ(∂µh ) 1 (∂2h )2] − µν − 3 2 µν − λν µκ − 3 s Z Z 1 4 2 µν (2) 2 λκ = 2 d x ∂ h Pµν,λκ∂ h . (30A.3) Z Note that the linearized Einstein tensor (4.376) can be expressed in terms of the projection operator (2) s Pµν,λκ(i∂) of Eq. (30.28) and the spin-zero projection operator Pµν,λκ(i∂) of Eq. (5.76). The result was stated in Eq. (5.77). We may use that formula to find a more convenient expression for the weak-field Einstein-Hilbert action (4.378). Inserting formula (5.77) for the Einstein tensor Gµν into the weak-field action (4.378), we obtain:

f 1 4 1/2 µν 1 4 1/2 µν 1 (2) 2 λκ 1 2 = d x( g) hµν G = d x( g) h 2 P ∂ h + 3 hs∂ hs . (30A.4) A 4κ − 4κ − − µν,λκ Z Z h i µν s 2 λκ The last term in the brackets may also be written as h Pµν,λκ∂ h .

Appendix 30B Quantization without Tachyons

The simplest prescription for quantizing a scalar ﬁeld with an action

1 = dt x( ∂2 ω2)x (30B.1) A 2 − t − 0 Z was found by Feynman. He inverted the differential operator between the fluctuating field variable in Fourier space and defined

i 1 i i G(ω)= = . (30B.2) ω2 ω2 2ω ω ω − ω + ω − 0 0 − 0 0 Appendix 30B Quantization without Tachyons 1537

ω0 ω ω − 0

Figure 30.1 Calculation of Feynman propagator in Green function (30B.3). Compare with Fig. (7.1) on p. 495.

From this he calculated the Fourier transformation and settled the boundary conditions by cir- cumventing the poles in the complex ω-plane at ω0 and ω0 as shown in Fig. 30.1. The result of the ω-integration is [recall (1.319)] − 1 GF(t) = dω e−iωtG(ω) 2π Z e−iω0t eiω0t = θ(t) + θ( t) . (30B.3) 2ω0 − 2ω0

Equivalently, we can place the poles at the slightly shifted positions ω0 iǫ and ω0 + iǫ with an inﬁnitesimal positive ǫ. Then the Feynman propagator is given by the integral− − 1 i GF(t) = dω e−iωt (30B.4) 2π ω2 ω2 + iǫ Z − 0 1 1 i i = dω e−iωt e−iωt. (30B.5) 2π 2ω ω ω + iǫ − ω + ω iǫ Z 0 − 0 0 − Let us now perform the quantization in the canonical formulation of the model. Then we replace the action (30B.1) by = dt (ip x˙ H) , (30B.6) Acan x − Z where H is the Hamiltonian: p2 ω2 H = + 0 x2. (30B.7) 2 2 The quantization proceeds via the canonical path integral x (x′ t x 0) = pD eiAcan . (30B.8) | D 2π Z The variables x(t) and p(t) can be expressed in terms of creation and annihilation operators a† and a as [compare (7.12) and (7.10)]: 1 x = ae−iω0t + a†eiω0t , √2ω0 1538 30 Einstein Gravity from Fluctuating Conformal Gravity

iω p = − 0 ae−iω0t a†eiω0t =x. ˙ (30B.9) √2ω0 − In terms of these, the Hamiltonian reads [compare 7.30) and 7.31)] 1 H = x˙ 2 + ω2x2 = (a†a + 1 )ω . (30B.10) 2 0 2 0 Let us now quantize a theory whose ﬂuctuating variable has four time derivatives in its action 1 = dt x( ∂2 ω2)( ∂2 ω2)x. (30B.11) A 2 − t − 1 − t − 2 Z For this we have to calculate the Fourier transform of i G(ω)= . (30B.12) (ω2 ω2)(ω2 ω2) − 1 − 2 The integrand can be decomposed as 1 i i G(ω)= (30B.13) ω2 ω2 ω2 ω2 − ω2 ω2 1 − 2 − 1 − 2 and further as 1 i i G(ω) = 2ω (ω2 ω2) ω ω − ω + ω 1 1 − 2 − 1 1 1 i i . (30B.14) − 2ω (ω2 ω2) ω ω − ω + ω 2 1 − 2 − 2 2 If one had to calculate a propagator according to the Feynman prescription, one would ﬁnd 1 GF(t) = dω e−iωtG(ω) 2π Z θ(t) e−iω1t e−iω2t θ( t) eiω1t eiω2t = + − . (30B.15) ω2 ω2 2ω − 2ω ω2 ω2 2ω − 2ω 1 − 2 1 2 1 − 2 1 2 This propagator arises from the following quantization procedure. We rewrite the action (30B.11) as 1 = dt x¨2 ω2 + ω2 x˙ 2 + ω2ω2x2 , (30B.16) A 2 − 1 2 1 2 Z and identify the velocityx ˙ as a new variable v. Then we replace the action (30B.16) by

= dt (ip x˙ + ip v˙ H) , (30B.17) Acan x v − Z with the Hamiltonian p2 1 1 H = v + p v + ω2 + ω2 v2 ω2ω2x2. (30B.18) 2 x 2 1 2 − 2 1 2 The quantization of this canonical system was performed a long time ago in Ref. [44]. The amplitude is the result of a path integral v x (x‘ v′; t x v;0) = p D v p D eiAcan . (30B.19) | D v 2π D x 2π Z The operator form of this quantization was obtained by the substitution (assuming ω2 >ω1)

x = x1 + x2, 1 p = (ω2x˙ ω2x˙ ), x −ω2 ω2 2 1 − 1 2 2 − 1 v =x ˙ 1 +x ˙ 2, 1 p = (ω2x + ω2x ). (30B.20) v −ω2 ω2 1 1 2 2 2 − 1 Appendix 30B Quantization without Tachyons 1539

ω ω ω − 2 1 ω ω − 1 2

Figure 30.2 Calculation of Feynman propagator without tachyons in Green function (30B.12).

† If we express the real variables x1 and x2 in terms of creation and annihilation operators a and a as in (30B.9), but with a slightly more convenient normalization of these, we obtain [compare Section 2.22.2)]

−iω1t † iω1t −iω2t † iω2t x = a1e + a1e + a2e + a2e , 1 p = [iω ω2(a e−iω1t a†eiω1t)+ iω2ω (a e−iω2t a†eiω2t)], x 2(ω2 ω2) 1 2 1 − 1 1 2 2 − 2 2 − 1 v = iω (a e−iω1t a†eiω1t) iω (a e−iω2t a†eiω2t), − 1 1 − 1 − 2 2 − 2 1 p = [ω2(a e−iω1t + a†eiω1t)+ ω2(a e−iω2t + a†eiω2t)], (30B.21) v −2(ω2 ω2) 1 1 1 2 2 2 2 − 1 and the Hamiltonian operator takes the form

1 H = ω a†a ω a†a + (ω + ω ). (30B.22) 1 1 1 − 2 2 2 2 1 2

† The states created by powers of a2 have an energy that can be lowered arbitrarily which makes the ground state unstable. Let us now quantize the same theory without such unwanted states [26]. Once this is done for each momentum eigenmode and integrated over all momenta, one obtains a quantum ﬁeld theory without tachyons. We simply use an ω-integration with a contour as shown in Fig. (30.2). The pole at ω = ω2 is circumnavigated for negative t in the anticlockwise sense, and this cancels the negative sign in the last term of (30B.15). A similar sign change happens with the second term in (30B.15) for positive t from the pole at ω = ω . Thus the integral yields the positive-deﬁnite − 2 expression (assuming that ω2 >ω1):

1 Gpos(t) = dω e−iωtG(ω) 2π Z θ(t) e−iω1t eiω2t θ( t) eiω1t e−iω2t = + + − + . (30B.23) ω2 ω2 2ω 2ω ω2 ω2 2ω 2ω 2 − 1 1 2 2 − 1 1 2 1540 30 Einstein Gravity from Fluctuating Conformal Gravity

The corresponding quantization comes about after a unitary transformation of the canonical pairs of variables z,pz and v,pv by an operator

= iαp p iβvz. (30B.24) Q v z − This yields α e−QveQ = v cosh( αβ) p sinh( αβ), − β z p r p −Q Q β e pve = pv cosh( αβ)+ z sinh( αβ), rα p α p e−QzeQ = z cosh( αβ) p sinh( αβ), − β v p r p −Q Q β e pze = pz cosh( αβ)+ v sinh( αβ). (30B.25) rα p p We shall choose α and β to have the ratio β = ω2ω2, (30B.26) α 1 2 and satisfy

2 2 2ω1ω2 ω1 + ω2 sinh( αβ) = 2 2 , cosh( αβ)= 2 2 , (30B.27) ω2 ω1 ω2 ω1 p − p − so that α 2 sinh( αβ)= 2 2 . (30B.28) β ω2 ω1 r p − Then the transformations (30B.25) can also be written as

ω2 + ω2 2 e−QveQ = v 1 2 p , ω2 ω2 − z ω2 ω2 2 − 1 2 − 1 ω2 + ω2 2ω2ω2 e−Qp eQ = p 1 2 + z 1 2 , v v ω2 ω2 ω2 ω2 2 − 1 2 − 1 ω2 + ω2 2 e−QzeQ = z 1 2 p , ω2 ω2 − v ω2 ω2 2 − 1 2 − 1 ω2 + ω2 2ω2ω2 e−Qp eQ = p 1 2 + v 1 2 , (30B.29) z z ω2 ω2 ω2 ω2 2 − 1 2 − 1 (30B.30) and the canonical form of the action (30B.18) becomes

= dt (ip z˙ + ip v˙ H) , (30B.31) Acan z v − Z with the Hamiltonian 2 2 pv pz 1 2 2 2 1 2 2 2 H = + 2 + ω1 + ω2 v + ω1ω2z . (30B.32) 2 2ω2 2 2 This is a positive-deﬁnite operator. The canonical variables in H can be expressed in terms of two variables with frequencies ω1 and ω2 >ω1 deﬁned by

z = x1 + ix2, Appendix 30B Quantization without Tachyons 1541

1 p = ( ω2x iω2x ), v 2(ω2 ω2) − 1 1 − 2 2 2 − 1 v =x ˙ ix˙ , 1 − 2 1 p = (ω2x˙ + iω2x˙ ). (30B.33) z 2(ω2 ω2) 2 1 1 2 2 − 1 The corresponding creation-annihilation operator forms are

−iω1t † iω1t −iω2t † iω2t z = a1e + a1e + i(a2e + a2e ), 1 p = [ ω2(a e−iω1t + a†eiω2t) iω2(a e−iω2t + a†eiω2t)], v 2(ω2 ω2) − 1 1 1 − 2 2 2 2 − 1 v = iω (a e−iω1t a†eiω1t)+ ω (a e−iω2t a†eiω2t), − 1 1 − 1 2 2 − 2 1 p = [iω2ω (a e−iω1t a†eiω1t)+ ω2ω (a e−iω2t a†eiω2t)]. (30B.34) z 2(ω2 ω2) 2 1 1 − 1 1 2 2 − 2 2 − 1 This brings the Hamiltonian (30B.32) to the form

1 H = ω a†a + ω a†a + (ω + ω ), (30B.35) 1 1 1 2 2 2 2 1 2 showing once more that the eigenstates all have a positive energy. All results in this Appendix can immediately be taken over from quantum mechanics to quantum ﬁeld theory. We simply replace the frequencies ω0, ω1 and ω2 by the momentum-dependent 2 2 2 2 2 2 frequencies ω0(p)= p + m , ω1(p)= p + m1 and ω2(p)= p + m1, and sum the expres- sions over all momenta, using the phase space integral formula = V dp3/(2π)3. p p pp Then the quantum mechanical Green function (30B.3) turns into the quantum ﬁeld-theoretic P R one:

d3p e−iω0(p)x0+ipx d3p eiω0(p)x0+ipx G(x)= θ(x ) + θ( x ) . (30B.36) 0 (2π)3 2ω (p) − 0 (2π)3 2ω (p) Z 0 Z 0 Remember now that there is another, covariant way of expressing this Green function, namely in the Feynman way (7.146):

4 d p i ′ GF (x x′)= e−ip(x−x ), (30B.37) m − (2π)4 p2 m2 + iǫ Z − where we have used a superscript to indicate the Feynman iǫ boundary condition and a subscript for the mass. Indeed, the p0-integral reproduces the Heaviside functions and forces the energy to be equal to the p-dependent values ω0(p). By the appropriate treatments of the poles in the energy integral, the positive Green function (30B.23) turns into the tachyon-free expression

θ(x ) d3p e−iω1(p)x0+ipx eiω2(p)x0+ipx GNT(x) = 0 + m2 m2 (2π)3 2ω (p) 2ω (p) 2 − 1 Z 1 2 θ( x ) d3p eiω1(p)x0+ipx e−iω2(p)x0+ipx + − 0 + . (30B.38) m2 m2 (2π)3 2ω (p) 2ω (p) 2 − 1 Z 1 2 Also here exists another, covariant way of expressing the Green function, namely in the form:

d4p i GNT(x) = e−ipx (2π)4 (p2 m2 + iη)(p2 m2 iη) Z − 1 − 2 − 1 d4p i i = e−ipx + eipx . (30B.39) m2 m2 (2π)4 p2 m2 + iǫ p2 m2 iǫ 2 − 1 Z − 1 − 2 − 1542 30 Einstein Gravity from Fluctuating Conformal Gravity

Again, the p0-integral reproduces the Heaviside functions and forces the energy to be equal to the p-dependent values ω1(p) and ω2(p). After reversing the direction of the energy integral in the second expression, this can be rewritten as 1 d4p i i GNT(x) = e−ipx + eipx , (30B.40) m2 m2 (2π)4 p2 m2 + iǫ p2 m2 + iǫ 2 − 1 Z − 1 − 2 or as 1 GNT(x) = GF (x)+ GF ( x) . (30B.41) m2 m2 m1 m2 − 2 − 1 Now the spectral decomposition has explicitly a sum over propagators which carry only physical, non-tachyonic states with positive norm. Since a perturbatively deﬁned quantum gravity can be expanded via Wick’s theorem into a sum of free-particle diagrams in GNT(x), all diagrams involve only physical propagators.

Notes and References

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