We study the Gaussian approximation to the quantum fluctuations of the metric
of the four dimensional anti-De Sitter spacetime. The associated massless
scalar field has a quartic self interaction, for which we construct the
generating functional of the $n$-point correlation functions. The concomitant
infrared divergence is cured by a mass renormalization provided by the
cosmological constant, which is also responsible for the renormalization of the
coupling constant of the field theory.Comment: 9 pages, 1 figur

Camargo, H. A.

de Urreta, E. J. Gonzalez

Socolovsky, M.

The European Physical Journal Plus

English

arXiv

We study the Gaussian approximation to the quantum fluctuations of the metric of the fourdimensional anti-de Sitter spacetime. The associated massless scalar field has a quartic self-interaction, for which we construct the generating functional of the n-point correlation functions. The concomitant infrared divergence is cured by a mass renormalization provided by the cosmological constant, which is also responsible for the renormalization of the coupling constant of the field theory.Fil: Camargo, H. A.. Universidad Nacional Autonoma de Mexico. Facultad de Ciencias; MéxicoFil: González de Urreta, Emiliano José. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Socolovsky, Miguel. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Universidad Nacional Autónoma de México. Instituto de Ciencias Nucleares; Méxic

González de Urreta, Emiliano José

Socolovsky, Miguel

CONICET Digital

DOI 10.1140/epjp/i2016-16172-9Regular ArticleEur. Phys. J. Plus (2016) 131: 172 THE EUROPEANPHYSICAL JOURNAL PLUSWeyl geometry, anti-de Sitter space, and Φ4-theoryH.A. Camargo1,a, E.J. Gonzalez de Urreta2,b,c, and M. Socolovsky2,3,d1 Facultad de Ciencias, Universidad Nacional Autónoma de México Circuito Exterior, Ciudad Universitaria, 04510,México D. F., Mexico2 Instituto de Ciencias, Universidad Nacional de General Sarmiento Juan Maŕı a Gutiérrez 1150 (B1613GSX), Los Polvorines,Pcia. de Buenos Aires, Argentina3 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México Circuito Exterior, Ciudad Universitaria, 04510,México D. F., MexicoReceived: 21 April 2016Published online: 26 May 2016 – c© Società Italiana di Fisica / Springer-Verlag 2016Abstract. We study the Gaussian approximation to the quantum fluctuations of the metric of the four-dimensional anti-de Sitter spacetime. The associated massless scalar field has a quartic self-interaction,for which we construct the generating functional of the n-point correlation functions. The concomitantinfrared divergence is cured by a mass renormalization provided by the cosmological constant, which isalso responsible for the renormalization of the coupling constant of the field theory.1 IntroductionThe equivalence of a Weylian geometry with vanishing 1-form or, equivalently, with zero gauge curvature, with aRiemannian geometry, allows us to interpret a conformally flat metric as a massless scalar field in Minkowski space.In particular, we consider the four-dimensional anti-de Sitter spacetime, and study the Gaussian approximation to thequantum fluctuations of the metric. This is done through the computation of the propagator between two conformalmetrics along a finite interval in one of the two patches of the AdS space. A change of the field variable, however, showsthat the system is equivalent to a massless Φ4-theory in one spacetime dimension. We then construct the generatingfunctional of the n-point correlation functions, which give the full quantum fluctuations of the AdS metric around itsclassical value. The infrared divergence of the scalar theory, due to its massless character, is cured by a finite massrenormalization provided by the cosmological constant, which plays the role of an IR cutoff; this is already exhibitedby the two-point function calculated to first order in the coupling constant, which is also finitely renormalized by thefour-point function with the same cutoff, at zero momenta, and to second order in the perturbation expansion.2 Weyl and Riemann geometriesAs is well known [1] any Riemannian or pseudo-Riemannian (in particular Lorentz) manifold (MnR, gμν) can can beconsidered as a Weylian manifold with zero Weyl field φ locally given by the 1-form φ = φμdxμ, (MnW , gμν , 0), withMnR = MnW ≡ Mn as differentiable manifolds. Of particular interest is the case when Mn is Lorentzian and conformallyflat, i.e.gμν = eϕημν , (1)where ϕ is a real valued scalar function on Mn and ημν is the Lorentz metric; so(MnW , gμν , 0) = (Mn, eϕημν , 0). (2)a e-mail: hugo.camargo@correo.nucleares.unam.mxb Post doctoral felloship, CONICET, Argentinac e-mail: egurreta@ungs.edu.ard e-mail: msocolov@ungs.edu.arPage 2 of 8 Eur. Phys. J. Plus (2016) 131: 172However, any Weyl manifold belongs to the equivalence class of manifolds [(Mn, g′μν , φ′)] defined by(Mn, g′′μν , φ′′) ∼ (Mn, g′μν , φ′), (3)if and only ifg′′μν = λ−1g′μν (4a)andφ′′ = φ′ − λ−1dλ, (4b)where λ is a smooth positive real valued function defined on Mn. (4a) is a conformal transformation of the metric,while (4b) is a gauge transformation of the Weyl field. Both (4a) and (4b) constitute what is called a Weyl transfor-mation, denoted by w. A choice of a representative of the class is called a choice of gauge or gauge fixing. Locally, theWeyl connection Γανμ is given by the sum of the Levi-Civita connection,(ανμ)=12gαβ(∂νgμβ + ∂μgνβ − ∂βgνμ), (5)and the Weyl tensor,Wανμ =12(δαν φμ + δαμφν − gνμgαρφρ). (6)It is easily verified thatΓανμ =(ανμ)+ Wανμ (7)is invariant under w, and so are the curvature tensorRρσμν = ∂μΓρνσ − ∂νΓ ρμσ + Γ ρμβΓ βνσ − ΓρνβΓβμσ (8)and the Ricci tensorRνσ = Rμνμσ. (9)However, sincegμνw−→ g′μν = λgμν , (10)the curvature scalarR = gμνRμν (11)is not an invariant but transforms to λR, i.e.Rw−→ R′ = λR, (12)leaving the Einstein tensorGμν = Rμν +12gμνR (13)invariant.Applying (3), (4a) and (4b) to (2) with φ′ = 0 and λ = eϕ one obtains(Mn, eϕημν , 0) ∼ (Mn, ημν ,−dϕ). (14)That is, a conformally flat Lorentzian manifold is equivalent to a flat Weylian manifold (Mn, ημν , φ′′) with Weyl fieldφ′′μ = −∂μϕ ≡ −ϕ,μ. (15)Since φ′′ = −dϕ, its curvature vanishes:F = dφ′′ = 0. (16)So, classically, the whole gravitational information in (Mn, eϕημν) is encoded in the scalar ϕ living in Minkowskispace Minkn.For any constant scalar curvature Riemannian or pseudo-Riemannian manifold, with curvature tensorRμνρσ =Rn(n − 1)(gμρgνσ − gμσgνρ), (17)it is easy to verify that the Weyl curvature tensor [2] Cμνρσ, defined byRβσμν = Cβσμν −R(n − 1)(n − 2)(gμβgνσ − gνβgμσ) +1(n − 2)(gμβRνσ − gνβRμσ − gμσRνβ + gνσRμβ), (18)vanishes everywhere, and, since a necessary and sufficient condition for a metric to be conformally flat is that its Weylcurvature tensor vanishes everywhere [3], then all maximal symmetric spaces are conformally flat.Eur. Phys. J. Plus (2016) 131: 172 Page 3 of 8Fig. 1. Classical massless conformal scalar field.3 Anti-de Sitter spaceAs a concrete example, we consider the anti-de Sitter space in four space-time dimensions (AdS4), whose metric isgiven byds2 =(az)2(dt2 − dy21 − dy22 − dz2) =(az)2ημνdxμdxν , (19)with x0 = t, xi = yi ∈ (−∞,+∞), i = 1, 2, x3 = z ∈ (−∞, 0) ∪ (0,+∞) ((−∞, 0) and (0,+∞) defining the leftand right patches of AdS4, respectively), [t] = [yi] = [z] = [a] = [length], and a related to the attractive cosmologicalconstant Λ throughΛ = − 3a2. (20)It is clear that from (1) and (19), eϕ = (az )2 = a2|z|2 and thereforeϕ ≡ ϕ0(z) = 2 ln(a|z|), (21)which is plotted in fig. 1. z = 0 corresponds to the boundary with radial coordinate r = +∞ and z = ±∞ are thecoordinate values of the “right” and “left” horizons, ϕ0(z) obeys the non-linear partial differential equationημν∂μ∂νϕ +12ημν(∂μϕ)(∂νϕ) +4a2eϕ = 0, (22)with Lagrangian densityL = 1a2(eϕημν(∂μϕ)(∂νϕ) −4a2e2ϕ)(23)and actionS =∫d4xL(z). (24)Equation (22) is obtained from the vacuum Einstein equation for a conformally flat metric eϕημν in the presence of acosmological constant Λ:∂μ∂νϕ − ημνηαβ∂α∂βϕ −12(∂μϕ)(∂νϕ) −14ημν(∂αϕ)(∂αϕ) + Λημνeϕ = 0, (25)through its contraction with the inverse Minkowski metric ημν .From the statement immediately after eq. (16), the quantum fluctuations of the metric (19) are given by thequantum theory of the field ϕ with action (24) in Mink4. An important element which must then be studied in thiscontext, is the propagator from the conformal metric g(1)μν (z) = eϕ1(z)ημν to the conformal metric g(2)μν (z) = eϕ2(z)ημν ,Page 4 of 8 Eur. Phys. J. Plus (2016) 131: 172with z1 ≤ z ≤ z2, 0 /∈ [z1, z2], ϕi(z) = ϕ0(z) + ψi(z), ψi(zk) = 0, i = 1, 2, k = 1, 2, which is represented by theFeynman path integral over the field ϕ, i.e.K(g(2)μν , g(1)μν ) ≡ K(2, 1) =∫ g(2)μνg(1)μνDϕeiS[ϕ]. (26)From the decompositionϕ(z) = ϕ0(z) + ψ(z), (27)(26) becomesK(2, 1) =∫ ψ2ψ1DψeiR z2z1d4xL(ϕ0+ψ) =∫ ψ2ψ1Dψe−4iR z2z1d4x eψz4(1−zψ′+ z24 ψ′2+eψ)=∫ ψ2ψ1Dψe−4iV∞R z2z1dz eψz4(1−zψ′+ z24 ψ′2+eψ), (28)where we assumed that the quantum fluctuation ψ only depends on z, with ψ′ = dψdz , and V∞ =∫ +∞−∞ dtdx1dx2 is aninfinite volume transverse to z which will be considered below.4 Semiclassical quantizationThe path integral (28) can not be done exactly. So, we consider here its Gaussian or semiclassical (quadratic) approx-imation, which we call K2(2, 1). Fromeψ(1 − zψ′ + z24ψ′2)+ e2ψ = 2 + 3ψ − zψ′ + 52ψ2 − zψψ′ + z24ψ′2 + cubic terms, (29)we haveK2(2, 1) =∫ ψ2ψ1Dψe−4iV∞R z2z1dzz4(2+3ψ−zψ′+ 52 ψ2−zψψ′+ z24 ψ′2) =∫ ψ2ψ1DψeiS , (30)whereS = −4(V∞a3)∫ z2z1dzL2(z), (31)withL2(z) =a3z4(2 + 3ψ − zψ′ + 52ψ2 − zψψ′ + z24ψ′2). (32)The Lagrange equation for L2,ddz(∂L2∂ψ′)− ∂L2∂ψ= 0, (33)isψ′′ − 2zψ′ − 4z2ψ = 0 (34)(ψ′′ = d2ψdz2 ) which, in terms of the dimensionless coordinateẑ =za, (35)becomesψ̈ − 2ẑψ̇ − 4ẑ2ψ = 0, (36)with ψ̇ = dψdẑ and ψ̈ =d2ψdẑ2 . Its general solution isψ̄(ẑ) = C1ẑ4 +C2ẑ, C1, C2 ∈ R. (37)Eur. Phys. J. Plus (2016) 131: 172 Page 5 of 8In terms of z,ψ̄(z) =C1a4z4 +C2az, (38)leading to∫ z2z1dzz4(2 + 3ψ̄ − zψ̄′ + 52ψ̄2 − zψ̄ψ̄′ + z24(ψ̄′)2)=[− 23z3− C1a4z − C2az4+C212a8z5 − 34C22a2z5]z2z1. (39)Since ψ̄ has to obey the same boundary conditions as ψ, i.e. ψ̄(z1) = ψ̄(z2) = 0, the unique solution (38) satisfyingthese conditions with z1 	= z2 is ψ̄(z) = 0, i.e. C1 = C2 = 0. So∫ z2z1dzz4(2 + 3ψ̄ − zψ̄′ + 52ψ̄2 − zψ̄ψ̄′ + z24(ψ̄′)2)=23(1z31− 1z32), (40)which gives the actionS̄(ψ̄ = 0) = −83(V∞a3)a3(1z31− 1z32). (41)V∞/a3 is a dimensionless infinite transverse volume. Since the problem is essentially 1-dimensional —only the zcoordinate is relevant— we define the action per unit dimensionless transverse volumeσ =S̄V∞/a3= −83a3(1z31− 1z32). (42)The van-Vleck determinant [4,5] associated with σ isΔ = det(∂2σ∂ẑ12∂2σ∂ẑ1∂ẑ2∂2σ∂ẑ2∂ẑ1∂2σ∂ẑ22)=(32)2a10(z1z2)5, (43)which, together with the exponentiation of σ, gives the desired propagator (k2) for the 1-dimensional problem, i.e.that ignoring the irrelevant transverse coordinates t, x1 and x2:k2(2, 1) =√iΔ2πeiσ =√i2π32a5(z1z2)5/2e− 83 ia3„1z31− 1z32«. (44)5 Φ4-theoryA simple change of field variable, ϕ → Ω, shows that the Lagrangian (23) corresponds to a Φ4-theory for the realmassless scalar field Ω in one spacetime dimension, that corresponding to the z-coordinate. (Quantum field theory in(0+1) dimensions is discussed in ref. [6], where the space coordinate z is replaced by a time coordinate t.) In fact, witheϕ = Ω2, (45)i.e. the field Ω gives the standard form for the conformal factor in (1), the Lagrangian density (23) becomesL = 4a2(ημν(∂μΩ)(∂νΩ) −1a2Ω4). (46)As for ϕ, [Ω] = [L]0 and [L] = [L]−4. Assuming again that Ω = Ω(z), L becomesL = − 4a2((dΩdz)2+1a2Ω4), (47)with actionS =∫d4xL = −4a(V∞a3)∫dz((dΩdz)2+1a2Ω4). (48)In terms of the dimensionless coordinateζ =za, (49)Page 6 of 8 Eur. Phys. J. Plus (2016) 131: 172Ω = Ω(ζ) and the classical action becomesS = −4(V∞a3) ∫dζ((dΩdζ)2+ Ω4). (50)Again, defining an action per unit dimensionless transverse volume, one obtainsσ =∫dζ l = −4∫dζ((dΩdζ)2+ Ω4), (51)with Lagrangian densityl = −4((dΩdζ)2+ Ω4)(52)and classical equation of motionΔζΩ + 2Ω3 = 0, Δζ = −d2dζ2, (53)with solutionΩ0(ζ) = ζ−1 → 0 as ζ → ±∞ (54)(cf. (21)).The Feynman “propagator” is the Green’s function of the free one-dimensional Klein-Gordon equation with a deltafunction source:(Δζ − iε)ΔF (ζ) = −δ(ζ), (55)namely [7]ΔF (ζ) =∫ +∞−∞dξ2πe−iξζξ2 + iε. (56)As it stands, ΔF (ζ) is infrared (IR) divergent (as ξ → 0); this is so because the bare mass of the Ω field is zero. Anatural IR cutoff isμ̃2 = 3a−2, (57)i.e. minus the cosmological constant Λ. In dimensionless units,μ2 = a2μ̃2 = 3 (58)and so the IR finite propagator isΔF (ζ)′ =∫ +∞−∞dξ2πe−iξζξ2 − μ2 + iε =∫ +∞−∞dξ2πe−iξζ(ξ − ξ+)(ξ − ξ−), (59)with ξ± = ±(μ− iε2μ ) = ±(√3− iε2√3). According to ζ < 0 or ζ > 0, we integrate in the upper or lower complex ξ-plane,respectively, obtainingΔF (ζ)′ = −i2√3e±i√3ζ . (60)Continuing to ζ = 0,ΔF (0)′ = −i2μ= − i2√3, (61)which, as will be shortly seen, provides a mass renormalization to the bare theory.To construct the generating functional Z of the n-point correlation functions, one must to introduce a source factorJ(ζ) to the Lagrangian l, defininglJ = −4((dΩdζ)2+ Ω4 − JΩ). (62)In order to take the theory to the standard λ4!Φ4 form, where λ is the bare coupling constant, we redefine again thefield function asω(ζ) = 2√2Ω(ζ), (63)Eur. Phys. J. Plus (2016) 131: 172 Page 7 of 8and the source function,j(ζ) =√2J(ζ), (64)obtaining the actionσ[ω, j] =∫dζ(l0 + lint + jω), (65)withl0 = −12(∂ζω)2, lint = −3/24!ω4, (66)i.e. λ = 3/2.Then Z[j] is given by (see [7] p. 200)Z[j] =exp(i∫dζ lint(−i δδj(ζ)))Z0[j]exp(i∫dζ ′ lint(−i δδj(ζ′)))Z0[j]|j=0, (67)whereZ0[j] = exp(− i2∫dζ∫dζ ′j(ζ)ΔF (ζ − ζ ′)′j(ζ ′))(68)is the generating functional of the free theory, i.e. that with λ = 0.The n-point correlation functions are given byπ(ζ1, . . . , ζn) = 〈vac|Π(ω̂(ζ1) . . . ω̂(ζn))|vac〉 = (−i)nδnZ[j]δj(ζ1) . . . δj(ζn)|j=0, (69)where Π denotes the path ordered product of the quantum field operators ω̂(ζk) associated with the classical fieldsω(ζk), for k = 1, . . . , n. The normalizationZ[0] = 1, (70)i.e. the presence of the denominator in (67), guarantees the cancellation of vacuum diagrams in π(ζ1, . . . , ζn) in theformal perturbation expansion of the exponential function in (67).6 Mass and coupling constant renormalizationWe shall not proceed to the derivation of the diagrams involved in the perturbation series, since the result is completelyanalogous to the usual spacetime λ4!Φ4-theory, where in (69) π is replaced by τ and Π by the time ordered productoperator T . Instead, we want to emphasize three points, which suggest a possible further generalization to conformallyflat metrics other than the four dimensional AdS case:i) The quantum fluctuation of the classical metric hides a Φ4-theory in a one-dimensional spacetime.ii) The Φ4-theory is massless at the classical limit but, at the quantum level, it suffers a finite mass renormalizationgenerated by the cosmological constant in the AdS spacetime, which in the Φ4-theory plays the role of a natural IRcutoff. In particular, to first order in λ, from (66), (67), (68) and (69), for the 2-point function one obtainsπ(ζ1, ζ2) = i∫ +∞−∞dξ2πe−iξ(ζ1−ζ2)ξ2 − m2r + iε, (71)wherem2r = δm2 =iλ2ΔF (0)′ =λ4μ=λ4a√−Λ=√38. (72)iii) The coupling constant of the scalar field is also finitely renormalized by the IR cutoff μ2. In fact, to secondorder of the perturbation expansion, the connected 4-point function in momentum space is given byπ4(p) = −iλ + 3(−iλ)2I(p), (73)whereI(p) =12π∫ +∞−∞dξ(ξ2 − μ2 + iε)((p − ξ)2 − μ2 + iε) =∫ 10dt2π∫ +∞−∞dξ(ξ2 − μ2 + p2t(1 − t) + iε)2 , (74)Page 8 of 8 Eur. Phys. J. Plus (2016) 131: 172with p = p1 + p2 (pi, i = 1, 2: ingoing momenta) (see [7] p. 315). ThenI(0) =12π∫ +∞−∞dξ(ξ2 − μ2 + iε)2 =12π∫ +∞−∞dξ(ξ − ξ+)2(ξ − ξ−)2:= Limζ → 0I(0, ζ), (75)withI(0, ζ) =12π∫ +∞−∞dξe−iξζ(ξ − ξ+)2(ξ − ξ−)2, (76)and ξ± = ±(μ − iε2μ ) = ±(√3 − iε2√3). Making the complex integrations as for the case of ΔF (ζ)′, one obtainsI(0) =i4μ3. (77)So, the renormalized coupling constant resultsiπ4(0) = λr = λ(1 +3λ4μ3)= λ(1 +3λ4a3(−Λ)3/2)=32(1 +√38). (78)7 Final commentsIt is interesting to see the natural appearance of a massless scalar field derived from a conformally flat metric infour dimensions via a Weyl transformation. This unexpected result suggests several questions which could serve asmotivation for further research. Firstly, since this result is valid for all conformally flat metrics in four spacetimedimensions, one could wonder what the consequences of the appearance of a massless scalar field are for all Robertson-Walker geometries, which have all been shown to be conformally flat [8] and which describe the large-scale structure ofa homogeneous and isotropic Universe. Could this massless scalar field be interpreted as a natural-appearing inflatonfield? If so, what are the implications of this fact for the field of Quantum Cosmology? Secondly, is it possible toobtain a similar result in higher dimensional conformally flat spacetimes within the framework of General Relativity?In particular, can one construct a quantum theory of such a field in five-dimensional Anti-de Sitter spacetime? Thiswould be an interesting question related to the well known AdS/CFT correspondence, and which could help physiciststo better understand this duality. And finally, can one obtain a similar higher-dimensional result considering alternativetheories of gravity such as Einstein-Gauss-Bonnet gravity or Lovelock theories? We leave all these questions open tofurther research.References1. C. Romero, J.B. Fonseca-Neto, M.L. Pucheu, Found. Phys. 42, 224 (2012) arXiv:1101.5333v1 [gr-qc].2. S. Carroll, Spacetime and Geometry. An Introduction to General Relativity (Addison Wesley, San Francisco, 2004) p. 130.3. R. d’Inverno, Introducing Einstein’s Relativity (Oxford University Press, Oxford, 1992) p. 89.4. J.V. Narlikar, T. Padmanabhan, Phys. Rep. 100, 151 (1983).5. L.S. Schulman, Techniques and Applications of Path Integration (J. Wiley, New York, 1981) pp. 88, 95, 96.6. A.D. Boozer, Eur. J. Phys. 28, 729 (2007).7. L.H. Ryder, Quantum Field Theory (Cambridge University Press, Cambridge, 1996) p. 184.8. M. Ibison, J. Math. Phys. 48, 122501 (2007) arXiv:0704.2788 [gr-qc].

Weyl geometry, anti-de Sitter space, and Φ4-theory

H. A. Camargo

E. J. Gonzalez de Urreta

M. Socolovsky

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