Cosmology with General Curvature-Dependent Lagrangian

Do not worry about your diﬃculties in Mathematics. I can assure you mine are still greater. Albert Einstein (1879-1955)

29 Cosmology with General Curvature-Dependent Lagrangian

According to a sixty-year-old idea of Andrei Sakharov [1], the gravitational properties of spacetime should not be considered a consequence of a fundamental action associated with the metric of spacetime. Instead they may be viewed as an emerging property caused by the bending stiffness of all quantum fields. This idea of induced gravity has inspired many subsequent generalization attempts of Einstein’s theory. More than four decades ago, Buchdahl set up a nonlinear Lagrangian for the metric tensor which was supposed to govern the evolution of the cosmos [2]. This led Starobinsky to work with a simple specific version of it that was restricted to linear and quadratic terms in curvature and Ricci tensors [3]. Later Adler [4] proposed to consider Einstein gravity as a scale-breaking effect in a quantum field theory. Other recent versions of induced gravity were inspired by string theories [5]. Another early generalization [6] of Einstein gravity was inspired by theories of the elasticity and plasticity [7] of solid materials. When a steel tape is bent, it initially resists linearly to the deformations, following Hooke’s law of elasticity. If the deformation becomes strong, however, defects are created which lead to tiny cracks in the material and make a further bending increasingly easier. If gravity arises in a similar way from the bending stiffness of all matter fields, it may be assumed to lose stiffness in a similar way. For small curvature R, the Lagrange density starts out following a Hooke-like law which is linear in R, so that gravity is described by the Einstein-Hilbert action. When the curvature becomes larger, however, the Lagrangian density (R) grows at a smaller pace and may even saturate at a finite value that is observableL as a cosmological constant Λ. Many present-day discussions generalize our proposal for (R) and discuss it un- L der the name of f(R)-gravity (see the review article [8]). Whatever specific function is used, all such Lagrangians start out linearly for small R, and continue in various ways for larger R.

1511 1512 29 Cosmology with General Curvature-Dependent Lagrangian

29.1 Simple Curvature-Saturated Model

As a warm-up, let us investigate the physical consequences of the simple Lagrangian density proposed in [6]. It interpolates smoothly between the Einstein-Hilbert La- grangian in (4.352) for small R and a pure cosmological constant for large R. It will be referred to as curvature-saturated model [6] and its Lagrangian density has the form 1 R cs(R)= . (29.1) L −2κ √1+ l4R2

Here κ is related to Newton’s gravitational constant by Eq. (4.354). It contains a length parameter l that may be much larger than the Planck length lP in Eq. (4.355). 2 2 According to present cosmological data, l is larger than lP by roughly a factor 10122/2. This is seen, for example, in Eq. (19.157) of the textbook [9], or in the data of Ref. [11], according to which cs(R) seems to saturate at the cosmological constant Λ: L

−122 1 cs(R) = Λ 10 . (29.2) R→∞ 4 L − ≈ − lP

Let us derive the cosmological consequences of such an cs(R), and compare the ansatz (29.1) with other possible model functions of R. TheL mathematical techniques for this were developed in Refs. [12, 13, 14, 15, 16]. One of the motivations for a renewed interest in a more detailed consideration of cosmology with non-linear curvature terms comes from the so-called M-theory (see Ref. [17] “Brane New World”). In that theory, a conformal anomaly becomes relevant, and this turns out to have similar consequences as the R2-term in Starobin- sky’s model. The latest results concerning the effective Λ-term in such models are contained in Ref. [18]. A general argument why we should expect the full gravitational action to in- terpolate between the linear behavior in R and a saturation at large curvature can be seen in the treatment of the vacuum energy in Ref. [19]. A study of finite-size effects on the quantum field states in a closed Friedmann universe [10] shows that instead of a continuous distribution of the energy levels one has a discrete spectrum. If the radius a of the spatial part of spacetime shrinks to small values, the scalar curvature R becomes large. The spacings between the energy levels increase, and after a certain threshold, all fields wind up in the ground state. It is this general behavior that is summarized in an effective Lagrangian density (R). The con- Lcs crete form of (R) cannot yet be fully determined, but the discussion suggests that the R-dependenceL at large curvature should approach a cosmological constant Λ. The model (29.1) represents the simplest analytic function connecting this large-− R behavior with the correct weak-field shape àla Einstein-Hilbert. This chapter is organized as follows. First we calculate the consequences of the Lagrangian density (29.1). In particular we want to see how the cosmological 29.1 Simple Curvature-Saturated Model 1513 evolution is influenced by the R-dependence of the effective gravitational constant κeff (R). This is defined by the derivative

1 d (R) L . (29.3) 2κeﬀ (R) ≡ − dR

For the Lagangian density (29.1), it reads explicitly:

3 4 2 κeﬀ (R)= κ√1+ l R . (29.4)

It corresponds to an eﬀective Newton constant

3 4 2 Geﬀ (R)= GN √1+ l R , (29.5) that tends to inﬁnity for large R.

Then we apply two diﬀerent conformal transformations to cs(R). One of them will make (R) asymptotically equivalent to the Gurovich-ansatzL [20, 21]: Lcs R (R)= c R 4/3. (29.6) LG −2κ − 1| |

The other makes use of Bicknell’s theorem that will be discussed in Section 29.4. That theorem establishes a conformal relation of the cs(R)-theory to Einstein’s theory by coupling it minimally to a scalar field φ with aL suitably chosen potential V˜ (φ). Up to now, only the second of these conformal transformations has been employed in the literature.1 The physical consequences of different conformal transformations are quite different since the resulting metrics are not related to each other by any coordinate transformation. Our approach differs fundamentally from that derived from the limiting curvature hypothesis (LCH) in Refs. [23], where the gravitational Lagrangian reads

Λ (R)= R 1 R2/Λ2 1 . (29.7) LLHC − − 2 − − q Its derivative with respect to R diverges for R Λ. This divergence was supposed to prevent a curvature singularity to appear in→ cosmological evolution. This goal not completely reached by the model (29.7), since other curvature invariants may still diverge. Some of them are discussed in Ref. [24]. All these considerations have remained restricted to isotropic models. In more general spacetimes one faces the problem that sometimes a curvature singularity exists, even though all polynomial curvature invariants stay ﬁnite. In contrast to Eq. (29.7), which limits large curva- tures, our model (29.1) favors increasing curvature values.

1See, for instance, Ref. [22] or references cited therein. 1514 29 Cosmology with General Curvature-Dependent Lagrangian

(R) −Lcs

1/2κl2

1/2κl2 −

Figure 29.1 Curvature-saturated Lagrangian as a function of the curvature scalar R.

29.2 Field Equations of Curvature-Saturated Gravity

The curvature-saturated Lagrangian (29.1) interpolates between the Einstein- Hilbert Lagrangian R = , (29.8) LEH −2κ which is experimentally well confirmed at weak fields, and a pure cosmological constant Λ at strong fields: 1 cs(R) = Λ= . (29.9) L R→∞ − −2κl2 The R-dependence is plotted in Fig. 29.1. From our curvature-saturated Lagrangian (29.1) we obtain, via the derivative with respect to R, an effective gravitational constant (29.5). If one considers the Newtonian limit for a general Lagrangian (R) which may tend asymptotically to a nonvanishing cosmological constant, the potentialL between two point masses contains a Newtonian 1/r-part plus a Yukawa-like part exp( r/rY ) stemming from the nonlinearities of the Lagrangian. Details will be given in the− Appendix. At distances much larger than r , but much smaller than 1/ R , only the 1/r-term survives, Y | | and the coupling strength of the 1/r-term is givenq by the effective gravitational constant Geff (R). For a general Lagrangian (R) such as ours in (29.1), the calculation of the field equation is somewhat tedious,L since the Palatini formalism, which simplifies calculations in Einstein’s theory, is no longer applicable. Recall that in this formalism, metric and affine connections are varied independently, and the latter are identified with the Christoffel symbols only at the end. Fortunately, the following indirect procedure leads rather efficiently to the correct field equations. We define

d d2 ′ L, ′′ L, (29.10) L ≡ dR L ≡ dR2 29.2 Field Equations of Curvature-Saturated Gravity 1515 and calculate the covariant energy-momentum tensor of the gravitational ﬁeld. It is given by the variational derivative of (R) with respect to the metric g (x): L µν f 2 δ (R)√ g T µν L − , (29.11) ≡ −√ g δgµν − f where g denotes the determinant of gµν (x). For dimensional reasons, T µν has the following structure:

f ′ ′ ′ ′ T µν = α Rµν + β Rgµν + γ gµν + δ gµν + ǫ ;µν, (29.12) L L L L L with the 5 real constants α,β,...,ǫ. The index separated by a semicolon abbreviates the covariant derivative, i.e., vν;µ(x) Dµvν (x), and the symbol denotes the Laplace-Beltrami operator (1.390) in≡ four spacetime dimensions. The 5 constants can be uniquely determined, up to a common overall constant, by the covariant conservation law f µν T ;ν =0. (29.13) Recalling the coupling of matter to gravity in Eq. (5.66), we obtain the generalization of the Einstein-Hilbert equation (5.71):

m 1 ′ ′ ′ T µν = (2 Rµν gµν +2 gµν 2 ;µν ) . (29.14) −2κ L −L L − L ′ The calculation is straightforward if one carefully distinguishes between ( );µ ′ L and ( ;µ), which diﬀer from one another by a multiple of the scalar curvature R. InsertingL the curvature-saturated Lagrangian (29.1) into (29.10), we have

d 1 −3/2 ′ = L = 1+ l4R2 , (29.15) L dR −2 and we ﬁnd from (29.14):

m 1 Rµν Rgµν 1 1 T µν = +gµν . κ 4 2 3/2 − 4 2 1/2 " 4 2 3/2#−" 4 2 3/2#  (1 + l R ) 2(1+ l R ) (1 + l R ) (1 + l R ) ;µν (29.16)   Setting l = 0, the right-hand side reduces to 1/κ times the Einstein tensor. For the trace of the energy-momentum tensor, the equation (29.16) yields

m 4 3 µ 1 R +2l R 1 T µ = 3 . (29.17) κ ((1 + l4R2)3/2 − "(1 + l4R2)3/2 #)

This implies that in vacuum, the only constant curvature scalar is R = 0. Therefore, this model does not possess a de Sitter solution. Furthermore, we can see from Eq. (29.16) that a curvature singularity does not necessarily imply a divergence in the energy-momentum tensor. 1516 29 Cosmology with General Curvature-Dependent Lagrangian

29.3 Eﬀective Gravitational Constant and Weak-Field Behavior

Let us compare the effective gravitational constant Geff of our curvature-saturated model with those of other models discussed in the literature. From (29.5), we see that Geff (R) has the weak-field expansion

3 4 2 Geﬀ (R)= GN 1+ l R + ... , (29.18) 2 and the strong-ﬁeld expansion

6 3 3 Geﬀ = GNl R 1+ 4 2 + ... . (29.19) | | 2l R The full R-behavior is plotted in Figs. 31.1.

(R)

Figure 29.2 Eﬀective gravitational constant as a function of the scalar curvature R.

The weak-ﬁeld expansion of (R) is given by Lcs ∞ R R 2k+1 cs(R)= = bk R (29.20) L −2κ√1+ l4R2 −2κ − kX=1 with real coeﬃcients b , where b = l4/32πG. k 1 − 29.4 Bicknell’s Theorem

This theorem was published four decades ago [25]. It relates general R-dependent Lagrangians of the type (29.15) to ordinary Einstein’s theory coupled minimally to a scalar ﬁeld φ with a suitably chosen interaction potential V˜ (φ). More details are given in Ref. [12]. The full Lagrangian density is given by 1 + φ φ,µ V˜ (φ) . (29.21) LEH 2 ,µ − 29.4 Bicknell’s Theorem 1517

The relation of V˜ (φ) with (R) is expressed most simply by introducing a ﬁeld Lcs with a diﬀerent normalization ψ 2/3 φ, in terms of which the potential reads ≡ q R V˜ (φ) V (ψ)= (R)e−2ψ e−ψ, (29.22) ≡ Lcs − 2 with R being the inverse function of

ψ(R) = ln[2 ′ (R)]. (29.23) Lcs The metric in the transformed Lagrangian (29.21) is a conformally transformed version of gµν: ψ g˜µν = e gµν . (29.24)

For our particular Lagrangian (29.15) we have from (29.23):

3 ψ(R)= ln(1 + l4R2). (29.25) −2

Now we restrict our attention to the range R> 0 and ψ < 0; the other sign can be treated analogously. Then (29.25) is inverted to

1 −2ψ/3 R = 2 e 1, (29.26) l q − and (29.22) becomes

1 −5ψ/3 −ψ −2ψ/3 V (ψ)= 2 (e e ) e 1. (29.27) 2l − q − In the range under consideration, this is a positive and monotonously increasing function of ψ shown in Fig. 29.3, with the large-ψ behavior − 1 V (ψ)= e−2ψ. (29.28) 2l2

This is the typical exponential potential used in cosmological theories with power- law inflation. As mentioned before, there exists no exact de Sitter inflation. For ψ 0, also V (ψ) tends to zero like a power 4 2/3ψ3/2. → q If V (ψ) has a quadratic minimum at some ψ0 with a positive value V0 = V (ψ0), then there exists a stable de Sitter inflationary phase. As a pleasant feature, the potential V (ψ) has no maximum which would have given rise to undesired tachyons. From Eq. (29.25) one can see that for weak fields, the function ψ starts out like R2. A theory whose Lagrangian density behaves like R + R2 would have ψ R. ∼ 1518 29 Cosmology with General Curvature-Dependent Lagrangian

V (ψ) 140 120 100 80 60 40 20 ψ 0.2 0.4 0.6 0.8 1 −

Figure 29.3 Potential V (ψ) associated with the curvature-saturated action (29.1) via Bicknell’s theorem.

Appendix 29A Newtonian Limit in a Nonﬂat Background

The Newtonian limit of a theory of gravity in a nonflat background is defined as being the weak-field m slow-motion limit of fields whose energy-momentum tensor T µν (x) is dominated by the component m T 00(x) in comoving time. In textbooks, the limit is formed in a flat background, and sometimes it is believed that the flatness is a necessary assumption. This is, however, not true, and we show here briefly how the limit can be calculated in a nonflat background. Note that the result is different from what is usually called a Newtonian cosmology [26]. To have a concrete example, we take as a background the de Sitter spacetime. The slow-motion assumption allows us to work with static spacetime and matter, assuming the energy-momentum tensor to be m 0 0 T ij = ρ δ iδ j , (29A.1) where ρ is the energy density, and time is assumed to be synchronized. The length element in de Sitter spacetime in its static form can be written as

dr2 ds2 = (1 kr2)dt2 + + r2dΩ2, (29A.2) − − 1 kr2 − where x0 = t, x1 = r, x2 = χ, x3 = θ and dΩ2 = dχ2 + sin2 χdθ2 is the metric of the 2–sphere. In this Appendix, we have changed the signature of the metric from (+ ), which is usual in cosmology, to ( + ++), which leads to the standard deﬁnition of the Laplacian.− −− The parameter k characterizes− the following physical situations: For k = 0, we have the usual ﬂat background. By setting k = 0 we can therefore compare the equations with well-known results. The case k> 0 corresponds to a positive cosmological constant Λ. In the calculations, we must observe that the time coordinate t fails to be synchronized for k = 0, but it is obvious from the context how to obtain the synchronized time from it. In the coordinates6 (29A.2), there is a horizon at r = r 1/√k. So, 0 ≡ our approach makes sense in the interval 0

dr2 ds2 = (1 kr2)(1 2ϕ)dt2 + + r2dΩ2 (1+2ψ), (29A.3) − − − 1 kr2 − where ϕ and ψ depend on the spatial coordinates only. The weak-field assumption allows us to linearize the metric with respect to ϕ and ψ. An extended matter configuration can be obtained by superposition of point particles, so we only need to solve the problem for a δ-source at r = 0. Appendix 29A Newtonian Limit in a Nonflat Background 1519

Assuming this to be spherically symmetric, we may take ϕ = ϕ(r) and ψ = ψ(r) in Eq. (29A.3). For the metric components we get: 1+2ψ g = (1 kr2)(1 2ϕ), g = , g = r2(1+2ψ), g = g sin2 χ. (29A.4) 00 − − − 11 1 kr2 22 33 22 · − The inverted components are, up to linear order in ϕ and ψ, 1+2ϕ 1 2ψ g00 = , g11 = (1 kr2)(1 2ψ), g22 = − , g33 = g22 sin−2 χ, (29A.5) −1 kr2 − − r2 − which gives the Christoﬀel symbols kr Γ 0 = ϕ′ , (29A.6) 01 − − 1 kr2 − Γ 1 = (1 kr2) kr +2kr(ϕ + ψ) ϕ′(1 kr2) , (29A.7) 00 − − − − kr Γ 1 = ψ′ + , (29A.8) 11 1 kr2 − 1 Γ 2 = Γ3 = ψ′ + , (29A.9) 12 13 r Γ 1 = r(1 kr2) ψ′r2(1 kr2), (29A.10) 22 − − − − 1 2 1 Γ33 = sin χ Γ22, (29A.11) 3 Γ32 = cot χ, (29A.12) Γ 2 = sin χ cos χ, (29A.13) 33 − and the Ricci tensor reads 2ϕ′ R = 3k(1 kr2) ϕ′′(1 kr2)2 (1 kr2)+6k(ϕ + ψ)(1 kr2) 00 − − − − − r − − + kr(1 kr2)(5ϕ′ ψ′), (29A.14) − − 2 3k kr R = 2ψ′′ + ϕ′′ ψ′ + + (ψ′ 3ϕ′), (29A.15) 11 − − r 1 kr2 1 kr2 − − − R = 3kr2 ψ′′r2(1 kr2) ψ′(2r 4kr3) + (ϕ′ ψ′)(r kr3), (29A.16) 22 − − − − − − 2 R33 = R22 sin χ. (29A.17)

Before we discuss these equations, we consider two obvious limits: For k = 0, we see that

R = ϕ′′ 2ϕ′/r = ∆ϕ, (29A.18) 00 − − − which leads to the usual Newtonian ﬁeld equation ∆ϕ = 4πGρ. For ϕ = ψ = 0, we get for the Ricci tensor: −

0 1 2 3 R0 = R1 = R2 = R3 =3k, (29A.19) and thus the de Sitter spacetime with R = 12k for k> 0. Returning to the general case we have R 2 4 =6k 12kψ + (ϕ′′ 2ψ′′)(1 kr2)+ ϕ′ 5krϕ′ ψ′ +7krψ′, (29A.20) 2 − − − r − − r and thus R 4 R 0 = 3k +6kψ +2ψ′′(1 kr2) 6krψ′ + ψ′. (29A.21) 0 − 2 − − − r 1520 29 Cosmology with General Curvature-Dependent Lagrangian

The other components have a similar structure and can be calculated easily from the above equations. The first term 3k on the right-hand side will be compensated by the Λ-term. The usual gauging to ψ 0 and−ϕ 0 for r is impossible since the above approximation is no more → → → ∞ applicable for r > r0. As an alternative gauge we add to ψ and ϕ suitable constant values so that they are approximately zero in the region under consideration. So we may disregard the term 6kψ. All remaining terms with k can be obtained from those without k by multiplying with factors of 2 2 the type 1 + ǫ where ǫ kr , k = 1/r0, with r0 being of the order of magnitude of the world radius. In a first approximation,≈ this gives only a small correction to the gravitational constant. In a second approximation, there are deviations from the 1/r-behavior. An analogous discussion 2 2 for the Lagrangian R + l R tells us that, in a range where l r r0, the potential behaves like (1 c e−r/l)/r, as in flat space. ≪ ≪ − 1 Notes and References

The individual citations refer to: [1] A. Sakharov, Dokl. Akad. Nauk SSSR 177, 70 (1967); Vacuum quantum fluctuations in curved space and the theory of gravitation, reprinted in Gen. Relat. Grav. 32, 365 (2000). See also the editor’s note in H.J. Schmidt, Gen. Relat. Grav. 32, 361 (2000). [2] H.A. Buchdahl, Monthly Notices Roy. Astron. Soc. 150, 1 (1970). [3] A.A. Starobinsky, Physics Letters B 91, 99 (1980). [4] S.L. Adler, Rev. Mod. Phys. 54, 729 (1982). [5] See, for example, J. Hwang, H. Noh, Conserved cosmological structures in the one-loop superstring effective action (astro-ph/9909480); G. Ellis, D. Roberts, D. Solomons, P. Dunsby, (gr-qc/9912005). [6] H. Kleinert and H.J. Schmidt, Gen. Rel. Grav. 34, 1296 (2002) (http://klnrt.de/311). [7] H. Kleinert, Gauge Fields in Condensed Matter, World Scientific, 1989, Vol. I (http://klnrt.de/b1), and Vol. II (http://klnrt.de/b2). [8] S. Capozziello and V. Faraoni, Beyond Einstein Gravity, Springer, Berlin, 2010. [9] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Finan- cial Markets, 5th ed., World Scientific, 2009, p. 1–1579 (http://klnrt.de/b5). [10] A discussion of the Friedman universe is contained in Subsection 19.2.3 of the textbook [9].

[11] J.D. Barrow and D.J. Shaw, Phys. Rev. Lett. 106, 101302∼ (2011). [12] H.J. Schmidt, Comparing selfinteracting scalar fields and R + R3 cosmological models, As- tron. Nachr. 308, 183 (1987); On the critical value of the curvature scalar, Proc. Conf. General Relativity GR11, Stockholm 1986, p. 117. [13] H.J. Schmidt, New exact solutions for power-law inflation Friedmann models, Astron. Nachr. 311, 165 (1990); Gen. Relat. Grav. 25, 87, 863 (1993). [14] V. Müller and H.J. Schmidt, Gen. Relat. Grav. 17, 769 and 971 (1985). [15] H.J. Schmidt, Stability and Hamiltonian formulation of higher derivative theories, Phys. Rev. D 49, 6354 (1994); Phys. Rev. D 54, 7906 (1996) (gr-qc/9404038). See also H.J. Schmidt, Phys. Rev. D 50, 5452 (1994); Phys. Rev. D 52, 6198 (1995). [16] H.J. Schmidt, A new duality transformation for fourth-order gravity, Gen. Relat. Grav. 29, 859 (1997). Notes and References 1521

[17] S.W. Hawking, T. Hertog, and H. Reall, Phys. Rev. D 62, 043501 (2000). [18] V. Sahni and A.A. Starobinsky, Int. J. Mod. Phys. D 9, 373-443 (2000). [19] H. Kleinert and A. Zhuk, Theor. Math. Phys. 108, 1236 (1996) (http://klnrt.de/218); Theor. Math. Phys. 109, 307 (1996) (http:// klnrt.de/de/252). See also Zentralblatt MATH 933, 674 (2000). [20] V. Gurovich, Sov. Phys. Dokl. 15, 1105 (1971). [21] V. Gurovich, I. Tokareva, Gen. Relat. Grav. 31, 21 (1999). [22] J. Fabris, S. Reuter, Quantum cosmology in higher derivative and scalar-tensor gravity, Gen. Relat. Grav. 32, 1345 (2000). [23] Y. Anini, in Current topics in mathematical cosmology, Eds.: M. Rainer and H.J. Schmidt (WSPC Singapore 1998), p. 183. [24] R. Brandenberger, V. Mukhanov, A. Sornborger, Phys. Rev. D 48, 1629 (1993). [25] G. Bicknell, J. Phys. A 7, 341, 1061 (1974). This original paper did not attract much attention when it was published. A decade later, the results were rediscovered simultaneously by several researchers and have since enjoyed many applications. See, for instance, Ref. [22]. Also Chapter 6 of the 1996 Thesis of Henk van Elst at the Queen Mary and Westfield College London. Further see K.I. Maeda, Phys. Rev. D 39, 3159 (1989); L. Amendola, Phys. Rev. D 60, 043501 (1999); D. Barraco, E. Dominguez, R. Guibert, Phys. Rev. D 60, 044012 (1999); S. Capozziello, G. Lambiase, Gen. Relat. Grav. 32, 295 (2000). Different authors attribute the theorem to different persons.

[26] See the internet page (http://abyss.uoregon.edu/∼js/ast123/lectures/lec03.html). 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, although 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 background 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

The individual citations refer to:

[1] H. Weyl, Raum, Zeit, Materie, Springer, Berlin, 1918. [2] K. Lanczos, Ann. Phys. 74, 518 (1924). [3] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966). [4] P.C. Hohenberg, Phys. Rev. 158, 383 (1967). [5] H. Kleinert, Phys. Lett. A 83, 294 (1981) (http://klnrt.de/85). [6] S. Coleman, Commun. Math. Phys. 31, 259 (1973). [7] H. Kleinert, Phys. Lett. A 114, 263 (1986) (http://klnrt.de/128); Mod. Phys. Lett. A 3, 531 (1987) (klnrt.de/170); Phys. Lett. B 189, 187 (1987) (http://klnrt.de/162). [8] H. Kleinert, Phys. Lett. A 83, 1 (1981) (http://klnrt.de/73). [9] H. Kleinert, Phys. Lett. B 102, 1 (1981) (http://klnrt.de/85). [10] H. Kleinert, Phys. Lett. B 174, 335 (1986) (http://klnrt.de/149); Phys. Rev. Lett. 58, 1915 (1987) (http://klnrt.de/164). [11] A. Polyakov, Nucl. Phys. B 286, 406 (1986). [12] H. Kleinert, Phys. Lett. B 196, 355 (1987) (http://klnrt.de/165). [13] A broad survey on the status of quantum gravity is given in the textbook: C. Kiefer, Quantum Gravity, 3rd ed., Internat. Ser. Monogr. Phys. 155, Oxford U. Press, Oxford, 2012. [14] S. Carlip, Class. Quantum Grav. 15, 2629 (1998). [15] R. Geroch and J.B. Hartle, Found. Phys. 16, 533 (1986). [16] I.L. Buchbinder, S.D. Odintsov, and I.L. Shapiro, Eﬀective Action in Quantum Theory, (IOP Publishing Ltd, London, 1992). Notes and References 1543

[17] A. Pais and G.E. Uhlenbeck, Phys. Rev. 79, 145 (1950). Differential equations with higher derivatives were first discussed in detail by M. Ostrogradski, Mémoires sur les équations differentielles relatives au problème des isopérimètres, Mem. Ac. St. Petersbourg VI 4, 385 (1850). [18] G.K. Savvidy, Phys. Lett. B 71, 133 (1977); S.G. Matinyan and G.K. Savvidy, Nucl. Phys. B 134, 539 (1978). [19] H.B. Nielsen and M. Ninomiya, Nucl. Phys. B 163, 57 (1980); B 169, 309 (1980); H. Flyvbjerg, Nucl. Phys. B 176, 379 (1980); S.L. Adler, Phys. Rev. D 23, 2905 (1981); P. Cea and L. Cosmai, Phys. Rev. D 48. 3364 (1993) (arXiv:hep-lat/9303003). [20] M. Kaku, Phys. Rev. D 27, 2819 (1983); E.T. Tomboulis, Phys. Re. Lett. 52, 1173 (1983). [21] K.S. Stelle, Phys. Rev. D 16, 953 (1977). [22] S. Weinberg, in Understanding the Fundamental Constituents of Matter, ed. by A. Zichichi, Plenum Press, New York, 1978; S. Weinberg, in General Relativity, ed. S.W. Hawking and W. Israel, Cambridge University Press, 1979, p. 790. [23] M. Niedermaier, Nucl. Phys. B 833, 226 (2010); Class. Quant. Grav. 24, R171-230 (2007) (arXiv:gr-qc/0610018). [24] J. Maldacena, Einstein Gravity from Conformal Gravity (arXiv:1105.5632). [25] P.D. Mannheim, Found. Phys. 42, 388 (2012) (arXiv:1101.2186). [26] C.M. Bender and P.D. Mannheim, J. Phys. A 41, 304018 (2008) (arXiv:0807.2607); Phys. Lett. A 374, 1616 (2010) (arXiv:0902.1365); Phys. Rev. D 78, 025022 (2008) (arXiv:0804.4190); Phys. Rev. Lett. 100, 110402 (2007) (arXiv:0706.0207); See also the preprint (arXiv:0707.2283). [27] A.A. Starobinsky, Phys. Lett. B 91, 99 (1980). [28] For constraints on inflationary models deduced from the Planck space observatory see the the group’s publication: P.A.R. Ade et al., Astron. Astrophys. 571, A22 (2014) (arXiv:1303.5082). For a general survey of Planck results see P.A.R. Ade et al., Planck Collaboration, (arXiv:1303.5062). [29] G. ’t Hooft, (arXiv:1104.4543) [in particular see Eq. (4.4) therein]. See also the papers (arXiv:0909.3426); (arXiv:1011.0061); (arXiv:1009.0669). [30] A. Sakharov, Dokl. Akad. Nauk SSSR 177, 70; reprinted in Gen. Rel. Grav. 32, 365 (2000). See also the editors note on that paper in H.J. Schmidt, Gen. Rel. Grav. 32, 361 (2000). [31] C.G. Callan, S. Coleman, and R. Jackiw, Ann. Phys. 59, 42 (1970); S. Coleman and R. Jackiw, Ann. Phys. 67, 552 (1970); J. Polchinski, Nucl. Physics B 303, 226 (1988); D.Z. Freedman, I.J. Muzinich, and E.J. Weinberg, Ann, Phys. 87, 95 (1974). [32] A. Landete, J. Navarro-Salas, and F. Torrenti, (arxiv:1311.4958v2); G. Tsoupros, Gen. Rel. Grav. 37, 399 (2005). [33] I. Antoniadis and N.C. Tsamis, SLAC Publication, SLAC-PUB-3297 (1984); Phys. Lett. B 144, 55 (1984). [34] H. Kleinert, EJTP 11, 1 (2014) (www.ejtp.com/articles/ejtpv11i30p1.pdf). [35] P. Jizba, H. Kleinert, F. Scardigli, Eur. Phys. J. C 75:245, (2015) (arXiv:1410.8062). [36] H. Ohanian, preprint (arXiv:1502.00020). 1544 30 Einstein Gravity from Fluctuating Conformal Gravity

[37] H. Ohanian, Weyl Gauge-Vector and Complex Dilaton Scalar for Conformal Symmetry and its Breaking, Univ. of Vermont preprint, Burlington, 2015. [38] C. Brans and R.H. Dicke, Phys. Rev. 124, 925 (1961). [39] Z. Haba, Renormalization in Quantum Brans-Dicke Gravity, preprint (arXiv:hep-th/020 5130). [40] D.M. Weld, J. Xia, B. Cabrera, and A. Kapitulnik, Phys. Rev. D 77, 062006 (2008). [41] G. Amelino-Camelia, J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, and S. Sarkar, Nature 393, 763 (1998). [42] See the parallel session DE1 in the Proceedings of the Fourteenth Marcel-Grossmann Meet- ing in Rome, July 12–18, 2015, World Scientiﬁc. Singapore, ed. by Remo Ruﬃni. [43] A. Edery, Y. Nakayama, (arXiv:1502.05932v1). [44] H. Kleinert, J. Math. Phys. 27, 3003 (1986) (arXiv:0807.2607). [45] Y.K. Ha, Int. J. Mod. Phys. (Conference Series) 7, 219 (2012) (arXiv:1106.6053). [46] See the discussion triggered by Dyson on the internet (http://backreaction. blogspot.de/2013/06/quantum-gravity-phenomenology-neq.html). In order for the light to shine so brightly, the darkness must be present. Francis Bacon (1561-1626)

31 Purely Geometric Part of Dark Matter

Plots of orbital velocities of stars of galaxies as a function of their distance from the center show a great surprise: They do not decrease with distance in a way that would be expected from an ordinary gravitational ﬁeld created by the visible masses. This led F. Zwicky [1] in 1933 to postulate the existence of dark matter. In fact, the observed velocity curves ask for large amounts of invisible matter in each galaxy. The presently best theoretical ﬁt to the data is shown in Figs. 31.1 and 31.2, and the reader is referred to the original publication dealing with this issue [2]. If a Friedmann model [3] is used to explain the evolution of the universe, one needs a large percentage of dark matter, roughly 23% of the mass energy of the

Figure 31.1 (a) Details of the fits to the velocity data. Filled triangles refer to the northern half, open squares to the southern half of the galaxy. Straight lines are the best linear fits for R 25 arcmin to all data shown, once to the northern and once to the ≥ southern data set. (b) The straight line is the best linear fit to all unbinned data for R > 25 arcmin for the M33 rotation curve: Filled circles are from data displayed in (a), open circles follow Newton’s theory, for comparison.

1545 1546 31 Purely Geometric Part of Dark Matter

Figure 31.2 Velocity curve (points) of the galaxy M33 and comparison with a best ﬁt model calculation (continuous line). Also shown is the halo contribution (dashed-dotted line), the stellar disk (short dashed line), and the gas contribution (long dashed line). universe. If dark matter is added to the so-called dark energy, which accounts for roughly 70% of the energy, one ﬁnds that the visible matter is practically negligible. See the contribution of the various components of matter in Fig. 31.3. This is the reason for ignoring visible atoms completely in the most extensive computer simula- tions of the evolution of cosmic structures [4], the so-called Millennium Simulation.

Figure 31.3 Various types of matter in the universe

In Fig. 31.4 we show the decomposition of the black-body radiation when the universe became first transparent. There are many speculations as to its composition. In this chapter we want to give the simplest possible explanation of at least part of it [5]. We argue that this part is not made of elementary particles, but of the singular worldlines and worldsurfaces in the solutions of Einstein’s vacuum field equation Gµν = 0. In this way, the Einstein-Hilbert action governing spacetime also governs the fluctuations of these singular configurations of worldlines and worldsurfaces. Thus the spacetime action contains on the one hand the well-known 1547

Figure 31.4 Various contributions to Dark Matter when the Universe became transparent (Credit: NASA/WMAP Science Team). action controlling the mechanics of relativistic point particles Mc ds, on the other − hand it contains the action used to describe elementary particlesR as closed bosonic “strings”. Note that in the present context the “string” action appears in the physical number four of dimensions, such that it appears here in a novel physical context that is different from that in presently fashionable string theory where the unphysical number of 26 dimensions plays a special role. Let us remember that all static electric fields in nature may be considered as originating from nontrivial solutions of the Poisson equation for the electric potential φ(x) as a function of x =(t, x): ∆φ(x) ∇ ∇ φ(x)=0. (31.1) ≡ · The simplest of them has the form e/r, where r = x . It is attributed to pointlike electric charges, whose size e can be extracted from| | the pole strength of the singularity of the electric field E which points radially outward and has a strength e/r2. This becomes explicitly visible by performing an area integral over the E field around the singularity. Applying the famous Gauss integral theorem, d3x ∇ E = d2a E, (31.2) ZV · ZA · the area integral is equal to the volume integral over ∇ E = ∆φ(x). Thus a field · − that solves the homogeneous Poisson equation can have a nonzero volume integral 3 V d x ∆φ(x)= 4πe. This fact can be expressed in a local way with the help of a − (3) x Dirac-deltaR function δ ( ) as ∆φ(x)= 4πe δ(3)(x). (31.3) − In the sequel, it will be useful to re-express the Gauss theorem (31.2) in a one- dimensional form as R 2 2 dr∂rr Er(r)= R Er(R). (31.4) Z 1548 31 Purely Geometric Part of Dark Matter

2 This is valid for all R, in particular for small R, where Er(R)=4πe/R . Hence we can express the combination of Eqs. (31.2) and (31.2) in the radial form

R R 3 2 d x ∇ E = 4π drr ∇ ∇e/r =4πe dr ∂rδ(r). (31.5) Z · − Z · Z Thus we ﬁnd the electric charge e from the one-dimensional equation

R e dr ∂rδ(r)= e. (31.6) Z This shows once more the radial part of the homogeneous Maxwell equation in the presence of a pointlike singularity: ∇ E =4πe δ(3)(x). · For gravitational objects, the situation is quite similar. The Einstein equation in the vacuum, Gµν = 0, possesses simple nontrivial solutions in the form of the Schwarzschild metric deﬁned by

ds2 = B(r)c2dt2 A(r)dr2 r2(dθ2 + sin2 θdϕ2), (31.7) − − with B(r)=1 r /r, A(r)=1/B(r), where r 2G M/c2 is the Schwarzschild − S S ≡ N radius and GN Newton’s gravitational constant. Its Einstein tensor has the component

G t = A′/A2r (1 A)/Ar2, (31.8) t − − which vanishes in the vacuum. Let us now allow for singularities in spacetime and calculate the volume integral d3x √ gG t. Inserting (31.8) we find d3x B/A[A′/Ar (1 A)/r2]. If this V − t V − − isR evaluated with the gravitational singularitiesR q in the same way as in the electro- R magnetic case in Eqs. (31.2)–(31.5), we find that it is equal to dr∂r(r r/A)= R − (2GN/c)M dr∂rδ(r)=(2GN/c)M. Thus we obtain the nonzeroR integral R 3 t d x√ gGt = κcM, (31.9) ZV − where κ is defined in terms of the Planck length lP, as

κ 8πl2 /h¯ =8πG /c3. (31.10) ≡ P N From (31.9) we identify the mass of the object as being M. If the mass point moves through spacetime along a trajectory parametrized by xµ(τ), it has an energy-momentum tensor

m ∞ T µν (x)= Mc dτ x˙ µ(τ)x ˙ ν (τ)δ(4)(x x(τ)), (31.11) −∞ Z − where a dot denotes the τ-derivative. 1549

We may integrate the associated solution of the homogeneous Einstein equation Gµν = 0 over spacetime, and ﬁnd that its Einstein-Hilbert action

1 4 EH = d x√ gR (31.12) A −2κ Z − vanishes. The situation is quite diﬀerent, however, if we allow spacetime to be perforated by singularities. For line-like singularities, the Einstein-Hilbert action m µν µ (31.12) will look as if it contains a δ-function-like source obeying G = κ T µ (x). If we insert here the equation of motion of a point particlex ˙ 2(τ) = 1, we arrive m µ 2 (4) from the ﬁeld equation R = G = κ T µ (x) = Mc dτ x˙(τ) δ (x x(τ)) at − − − − an action which is proportional to the classical action ofR a point-like particle:

worldline EH Mc ds. (31.13) A ∝ − Z A slight modification of (31.13), that is the same at the classical level, but different for fluctuating orbits, describes also the quantum physics of a spin-0 particle via a path integral over all orbits (see the discussion in Section 19.1 of the textbook [6], in particular Eq. (19.10)). Thus Einstein’s action for a singular world line in spacetime can be used to define also the quantum physics of a spin-0 point particle. In addition to pointlike singularities, the homogeneous Einstein equation will possess also surface-like singularities in spacetime. These may be parametrized by xµ(σ, τ), and their energy-momentum tensor has the form

∞ T µν(x) dσdτ(x ˙ µx˙ ν x′µx′ν)δ(4)(x x(σ, τ)), (31.14) −∞ ∝ Z − − where a prime denotes a σ-derivative. In the associated vanishing Einstein tensor, the δ-function on the surface manifests itself in the nonzero spacetime integral [7]

4 µ 2 ′ 2 ′2 ′2 d x√ gGµ d a dσdτ (xx ˙ ) x˙ x . (31.15) − ∝ ≡ A − Z Z Z q By analogy with the line-like case we obtain, for such a singular ﬁeld conﬁguration, an Einstein-Hilbert action (31.12)

worldsurface 1 2 h¯ 2 EH d a = 2 d a. (31.16) A ∝ −2κ ZA −16πlP ZA The prefactor on the right-hand side has been expressed in terms of the Planck length lP. The important observation is now that, apart from a numerical proportionality factor of order unity, the right-hand side of Eq. (31.16) is precisely the Nambu-Goto action [8, 9] of a bosonic closed string in the true physical spacetime dimension four

h¯ 2 NG = 2 d a. (31.17) A −2πls ZA 1550 31 Purely Geometric Part of Dark Matter

In this formula, ls is the so-called string length parameter ls. It measures the string tension, and corresponds in spacetime to a certain surface tension of the worldsurface. This length scale can be related to the rather universal slope parameter α′ = dl/dm2 of the Regge-trajectories [10]. These are found in plots of the angular momenta against the squares of the meson masses m2. The relation between the ′ string length parameter ls and slope parameter of the Regge theory is ls =hc ¯ √α . Note that now there is no extra mass parameter M, this being in contrast to the situation in world lines. The masses of the elementary particle come from the eigen- modes of the string vibrations. The original string model was proposed to describe color-electric flux tubes and their Regge trajectories whose slopes α′ lie around 1 GeV−2. However, since the tubes are really fat objects, as fat as pions, only very long flux tubes are approximately line-like. Short tubes degenerate into spherical “MIT-bags” [11]. The flux-tube role of strings was therefore abandoned, and the action (31.17) was re- interpreted in a completely different fashion, as describing the fundamental particles of nature, assuming lS to be of the order of lP. Then the spin-2 particles of (31.17) would interact like gravitons and define Quantum Gravity. However, the ensuing “new string theory” [12] has been criticized by many authors [13]. One of its most embarrassing failures is that it has not produced any experimentally observable results. The particle spectra of its solutions have not matched the existing particle spectra. The arisal of the string action proposed here has a chance of curing this problem. If “strings” describe “dark matter”, there would be no need to reproduce other observed particle spectra. Instead, their celebrated virtue of extracting the interaction between gravitons from the properties of their spin-2 quanta can be used to fix the proportionality factor between the Einstein action (31.16) and the string action (31.17). It must be kept in mind that just as Mc ds had to be modified for fluctuating − paths [6] (recall Eq. (19.10) in that textbook),R also the Nambu-Goto action (31.17) needs a modification for fluctuating surfaces. That was found by Polyakov [14] when studying the consequences of the conformal symmetry. He replaced the action (31.17) by a new action that is equal to (31.17) at the classical level, but contains in D = 26 dimensions another spin-0 field with a Liouville action. 6 Since the singularities of Einstein’s fields possess only gravitational interactions, their identification with “dark matter” seems very natural. All visible matter con- sists of singular solutions of the Maxwell equations as well as the field equations of the standard model. A grand-canonical ensemble of these and the singular solutions without matter sources explain the most important part of all matter in the Friedmann model of cosmological evolution. But the main contribution to the energy comes from the above singularities of Einstein’s equation. Soon after the universe was created, the temperature was so high that the configurational entropy of the surfaces overwhelmed completely the impeding Boltzmann factors. Spacetime was filled with these surfaces in the same way as superfluid helium is filled with world-surfaces of vortex lines. 1551

Vortex lines in superfluid helium are known to attract material particles such as frozen helium. This phenomenon provides us with an important tool to visual- ize vortex lines and tangles thereof [15]. In spacetime, we expect that any stable neutral particle will be attracted by its singularities. These would be the elusive objects which people have been looking for in elaborate searches of WIMPs (weakly interacting massive particles) in particle physics [16]. Thus they should rather be called GIMPs (gravitationally interacting massive particles). In helium above the temperature of the superfluid phase transition, these lie so densely packed that the superfluid behaves like a normal fluid [17, 18]. The Einstein- Hilbert action of such a singularity-filled turbulent geometry behaves like the action of a grand-canonical ensemble of world surfaces of a bosonic closed-string model. Note once more that here these are two-dimensional objects living in four spacetime dimensions. There is definitely a need to understand their spectrum by studying the associated Polyakov action. To be applicable in the physical dimension four one should not circumvent the accompanying Liouville field. Or one must find a way to take into account the fluctuations of the gravitational field around the field near the singular surface. It should be realized that in the immediate neighborhood of line- and surface- like singularities, the curvature will be so high that Einstein’s linear approximation (1/2κ)R to the Lagrangian must break down. It will have to be corrected by some − nonlinear function of R. This starts out like Einstein’s, but continues differently, similar to the action discussed in Chapter 29, that was suggested a long time ago [19]. Many modifications of this idea have meanwhile been investigated further [20]. After the big bang, the universe expanded and cooled down, so that large singular surfaces shrunk by emitting gravitational radiation. Their density decreased, and some phase transition made the cosmos homogeneous and isotropic on the large scale [21]. But it remained filled with gravitational radiation and small singular surfaces that had shrunk until their sizes reached the levels stabilized by quantum physics. The statistical mechanics of this cosmos can be described by analogy with a spacetime filled with superfluid helium. The specific heat of that is governed by the zero-mass phonons and by rotons. Recall that in this way Landau discovered the fundamental excitations called rotons [23], whose existence was deduced by him from the temperature behavior of the specific heat. In the universe, the role of rotons is played by the smallest surface-like singularities of the homogeneous Einstein equation. They must be there to satisfy the cosmological requirement of dark matter. The situation can also be illustrated by a further analogy with many-body sys- tems. The defects in a crystal, whose “atoms” have a lattice spacing lP, simulate precisely the mathematics of a Riemann-Cartan spacetime, in which disclinations and dislocations define curvature and torsion [17, 18, 24]. Thus we may imagine a model of the universe as a “floppy world crystal” [25], a liquid-crystal-like phase [26] in which a first melting transition has led to correct gravitational 1/r-interactions between disclinations. The initial hot universe was filled with defects and thus it was in the “world-liquid”-phase of the “world crystal”. After cooling down to the 1552 31 Purely Geometric Part of Dark Matter present liquid-crystal state, there remained plenty of residual defects around, which form our dark matter. In the process of cooling down over a long time, the dark matter fraction can have decreased so that the expansion of the universe could have become faster and faster over the millenia. Thus it is well possible that the baby universe had so much dark-matter content that it practically did not expand at all for a long time. It would have been closer to the steady-state universe advocated in 1931 by Einstein and in the 1940’s by Hoyle, Bondi, and Gold [27]. This would relief us from the absurd- sounding assumption that the entire universe came once out of a tiny beginning in which all matter of the world was compressed into a sphere of Planck radius. We know that the cosmos is now filled with a cosmic microwave background (CMB) of photons of roughly 2.725 Kelvin, the remnants of the big bang. They 2 −5 contribute a constant Ωradh = (2.47 0.01) 10 to the Friedmann equation of motion, where h = 0.72 0.03 is the± Hubble× parameter, defined in terms of the Hubble constant H by h± H/(100 km/Mpc sec). The symbol Ω denotes the ≡ 2 energy density divided by the so-called critical density ρc 3H /8πGN = 1.88 −26 2 3 2≡ × 10 h kg/m [28]. The baryon density contributes Ωradh = 0.0227 0.0006, or 2 ± 720 times as much, whereas the dark matter contributes Ωdarkh =0.104 0.006, or 4210 as much. Let us assume, for a moment, that all massive strings are± frozen out and that only the subsequently emitted gravitons form a thermal background [29]. Since the energy of massless states is proportional to T 4, the temperature of this 1/4 background would be TDMB 4210 8TCMB 22K. We expect the presence of other singular solutions of Einstein’s≈ equation≈ to≈ change this result. There is an alternative way of deriving the above-described properties of the fluctuating singular surfaces of Einstein’s theory. One may rewrite Einstein’s theory as a gauge theory [17, 18] and formulate it on a spacetime lattice [30]. Then the singular surfaces are built explicitly from plaquettes, as in lattice gauge theories of asymptotically-free nonabelian gauge theories [31]. In the abelian case, the surfaces are composed as shown in Ref. [32], for the nonabelian case, see [33]. An equivalent derivation could also be given in the framework of loop gravity [34]. But that would require a separate study beyond this chapter. Summarizing we have seen that the Einstein-Hilbert action governs not only the classical physics of gravitational fields but also, via the fluctuations of its line- and surface-like singularites, the quantum physics of dark matter. A string-like action, derived from it for the fluctuating surface-like singularities, contains interacting spin-2 quanta that define a finite Quantum Gravity. It is curious to note that almost hundred years ago, Einstein published in 1919 a remarkable paper entitled “Do Gravitational Fields Play a Significant Role in the Composition of Material Elementary Particles?” [35]. He thus asked this question long before any of the numerous elementary particles treated in this book was discovered. After their discovery, physicists thought for a long time that the answer is negative. However, as we understand increasingly well the structure of the universe, the answer is rather an “almost yes”. Moreover, if we restrict his question in the Notes and References 1553 title from ‘universe’ to the ‘dark-matter part of the universe’, which makes up about one third of the universe, the answer seems to be “yes”. For other models of dark matter see [36].

Notes and References

[1] F. Zwicky, ”Die Rotverschiebung von extragalaktischen Nebeln”. Helv. Phys. Acta. 6, 110 (1933); See also his textbook Morphological Astronomy, Springer, Berlin, 1957. Note that Zwicky triggered the launching of the satellite ’Artificial Planet No. Zero’. It was 12 days after the Sputnik went in orbit in October 1957. In contrast to Sputnik, his satellite remained indefinitely in orbit, whereas Sputnik eventually fall back to earth. [2] E. Corbelli and P. Salucci, R. Astro. Soc. 311, 441 (2000) (arxiv:9909252v1). See also the improved data in E. Corbelli, D. Thilker, S. Zibetti, C. Giovanardi, and P. Salucci, A&A (Astronomy and Astrophysics) 572, 23 (2014) (arxiv.org:1409.2665v2). [3] A discussion of the Friedmann universe is contained in Subsection 19.2.3 of the textbook [6]. [4] V. Springel, S.D.M. White, A. Jenkins, C.S. Frenk, N. Yoshida, Liang Gao, J. Navarro, R. Thacker, D. Croton, J. Helly, J.A. Peacock, S. Cole, P. Thomas, H. Couchman, A. Evrard, J. Colberg, F. Pearce, Nature 435, 629 (2005). [5] H. Kleinert, EJTP 8, 27 (2011)(http://www.ejtp.com/articles/ejtpv9i26 p27.pdf). [6] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Finan- cial Markets, 5th ed., World Scientific, 2009, p. 1–1579 (http://klnrt.de/b5). [7] For this take the delta function on the surface as defined by P.A.M. Dirac, Phys. Rev. 74, 817(1948) [see his Eq. (15)], and make use of the distributional form of Gauss’s integral theorem as formulated in H. Kleinert, Int. J. Mod. Phys. A 7, 4693 (1992) (http://klnrt.de/203.pdf), or in the textbook [18] on p. 253. [8] Y. Nambu, in Proc. Int. Conf. on Symmetries and Quark Models, Wayne State University 1969 (Gordon and Breach 1970) p. 269. [9] T. Goto, Progr. Theor. Phys. 46, 1560 (1971). [10] P.D.B. Collins, An Introduction to Regge Theory and High-Energy Physics, Cambridge Uni- versity Press, Cambridge, U.K., 1977. [11] L. Vepstas and A.D. Jackson, Physics Reports 187, 109 (1990). [12] M. Green, J. Schwarz, and E. Witten, Superstring Theory, Cambridge University Press, 1987. [13] See the list of critics in the Wikipedia article on “string theory”’. In particular B. Schroer, String theory deconstructed, (arXiv:hep-th/0611132). [14] A.M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers, New York, 1987. [15] D.R. Poole, C.F.Barenghi, and Y.A.Sergeev, and W.F. Vinen, Phys. Rev. B 71, 064514 (2005); R.E. Packard, Physica 109&110B, 1474 (1982). 1554 31 Purely Geometric Part of Dark Matter

[16] T.J. Sumner, Experimental Searches for Dark Matter, (http://relativity.living reviews.org/Articles/lrr-2002-4); LUX collaboration, (arXiv:1605.03844, arXiv:1602.03489,arXiv:1512.03506). [17] H. Kleinert, Gauge Fields in Condensed Matter, Vol. I, Superflow and Vortex Lines, World Scientific, Singapore, 1989, pp. 1–744 (http://klnrt.de/b1). [18] H. Kleinert, Multivalued Fields, World Scientific, Singapore, 2008, pp. 1–497 (http:// klnrt.de/b11). [19] H. Kleinert and H.-J. Schmidt, Gen. Rel. Grav. 34, 1295 (2002) (klnrt.de/311). [20] See the review by T.P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82, 451 (2010) (arxiv:0805.1726). [21] There are parallels with the work of Kerson Huang and collaborators in K. Huang, H.-B. Low, R.-S. Tung, (arXiv:1106.5282v2), (aXiv:1106.5283v2). However, their turbulent “baby universe” is filled with tangles of vortex lines of some scalar field theory, whereas our spacetime contains only singularities of Einstein’s homogeneous field equation. A bridge may be found by recalling that the textbook [17] explains how tangles of line-like defects can be described by a complex disorder field theory, whose Feynman diagrams are direct pictures of the worldlines. Thus, if Huang et al. would interpret their scalar field as a disorder field of the purely geometric objects of my theory, the parallels would be closer. Note that in two papers written with K. Halperin [22], Huang manages to make his scalar field theory asymptotically free in the ultraviolet (though at the unpleasant cost of a sharp cutoff introducing forces of infinitely short range). This property allows him to deduce an effective dark energy in the baby universe. With our purely geometric tangles, such an effect may be reached using a lattice gauge formulation of Einstein’s theory sketched at the end of the textbooks [17] and [18]. [22] K. Halpern and K. Huang, Phys. Rev. Lett., 74, 3526 (1995); Phys. Rev. 53, 3252 (1996). [23] S. Balibar, Rotons, Superfluidity, .and Helium Crystals, AIP lecture 2016 (http://www.lps.ens.fr/balibar/LT24.pdf). [24] B.A. Bilby, R. Bullough, and E. Smith, Proc. Roy. Soc. London, A 231, 263 (1955); K. Kondo, in Proceedings of the II Japan National Congress on Applied Mechanics, Tokyo, 1952, publ. in RAAG Memoirs of the Unified Study of Basic Problems in Engineering and Science by Means of Geometry , Vol. 3, 148, ed. K. Kondo, Gakujutsu Bunken Fukyu-Kai, 1962. [25] H. Kleinert, Ann. d. Physik, 44, 117 (1987) (http://klnrt.de/172). [26] H. Kleinert and J. Zaanen, Phys. Lett. A 324, 361 (2004) (http://klnrt.de/346). [27] See the Wikipdia article in https://de.wikipedia.org/wiki/Steady-State-Theorie and the papers C. O’Raifeartaigh, C. ORaifeartaighB. McCann, W. Nahm, S. Mitton, (arXiv:1402.0132); C. O’Raifeartaigh, S. Mitton, (arXiv:1506.01651). [28] K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075021 (2010). [29] We do this although the weakness of gravitational interactions may be an obstacle to thermal equilibration. See S. Khlebnikov and I. Tkachev, Phys. Rev. D 56, 653 (1997); J. Garcia-Bellido, D.G. Figueroa, A. Sastre, Phys. Rev. D 77, 043517 (2008) (arXiv:0707.0839). [30] She-Sheng Xue, Phys. Lett. B 682, 300 (2009); Phys. Rev. D 82, 064039 (2010). [31] K. Wilson, Physical Review D 10, 2445 (1974). Notes and References 1555

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