CPTH-A214.1292 LA-UR-92-4333 December 1992 SCALING BEHAVIOR OF QUANTUM FOUR-GEOMETRIES Ignatios Antoniadis Centre de Physique Th´eorique 3 Ecole Polytechnique 9 91128 Palaiseau, France 9 1 n Pawel O. Mazur a J Dept. of Physics and Astronomy 2 University of South Carolina 1 Columbia, SC 29208 v 2 and 0 0 Emil Mottola 1 0 Theoretical Division, T-8 3 Mail Stop B285 9 h/ Los Alamos National Laboratory t - Los Alamos, NM 87545 p e h Abstract : v i X We propose that large quantum fluctuations of the conformal factor drastically r modifyclassicalgeneralrelativityatcosmologicaldistancescales, resultinginascale a invariant phase of quantum gravity in the far infrared. We derive scaling relations for the partition function and physical observables in this conformal phase, and suggest quantitative tests of these relations in numerical simulations of simplicial four geometries with S4 topology. In particular, we predict the form of the critical curve in the coupling constant plane, and determine the scaling of the Newtonian coupling with volume which permits a sensible continuum limit. The existing nu- merical results already provide some evidence of this new conformal invariant phase of quantum gravity. At accessible distance scales, from centimeters to light years, gravitational phe- nomena are described quite accurately by the classical Einstein theory. It is very well known that this theory is beset with ultraviolet divergences at the quantum level and is virtually certain to require drastic modification at the Planck scale. A corollary of this severe behavior at ultra-short distances would seem to be mild behavior in the infrared. One does not normally think of quantum fluctuations of the metric field as important at large distances, and indeed in perturbation theory around flat space there is no sign of infrared problems. However, gravitation is a long range force characterized by massless excitations that cannot be shielded. If the classical spacetime is curved on some characteristic distance scale ℓ, fluctua- tions with wavelengths longer than ℓ need not remain small, a fact long ago pointed out in Newtonian theory by Jeans. Infrared behavior of fluctuations depends very muchonthebackgroundgeometry, andmayforcelargemodificationsoftheclassical background. The quantum analog of the classical background is the ground state. Prior to and independent of the question of how to tame the ultraviolet divergences bedev- iling the quantum theory, the question of what is its correct ground state presents itself. This issue is one that intrinsically involves infrared properties, where we may hope to say something sensible in the language of low energy effective lagrangians, without knowledge of Planck scale phenomena. In order to construct a low energy effective lagrangian for gravity that incorpo- rates infrared fluctuations correctly, one must decide first of all what is the relevant order parameter field at large distance scales. Here observational cosmology comes to our aid to suggest that the FRW scale factor, or more generally, the conformal factor of the metric tensor should play an important role. In the classical Einstein theory this scalar part of the metric does not propagate. It is determined in terms of the matter sources and has no independent dynamics of its own. At the quantum level there is a trace anomaly in the energy-momentum tensor of conformally cou- pled matter fields. The existence of the conformal anomaly means that the classical constraints which fix the scalar part of the metric fluctuations in terms of matter sources cannot be maintained upon quantization. In other words, the conformal factor becomes unconstrained in the full quantum theory. The low energy effective lagrangian must be modified accordingly to take ac- count of the trace anomaly and fluctuations of the conformal factor, and this mod- ified theory reanalyzed to discover the correct infrared behavior of the quantum theory of gravity. In two dimensional quantum gravity, otherwise known as non- 1 critical string theory, the effective action induced by the trace anomaly modifies the dynamics of the conformal factor at all distance scales in dramatic fashion. Certainly nothing like the fractal behavior and scaling relations of random surfaces in 2D gravity can arise from the classical two dimensional Einstein-Hilbert action which (being a topological invariant) yields no dynamics whatsoever [1-2]. In an earlier paper we obtained the effective Wess-Zumino action induced by the trace anomaly of conformal matter in four dimensions. We analyzed this con- tinuum effective theory showing that it possesses a non-trivial, infrared stable fixed point, characterized by certain anomalous scaling relations [3]. We argued that this scale invariant phase is the ground state of 4D quantum gravity approached at dis- tance scales much larger than the horizon length of any given classical background. Conformal symmetry, apparently broken by the trace anomaly is restored dynami- cally by the large fluctuations in the conformal factor at these scales. An important consequence of restoration of scale invariance at large distances is the screening of the effective cosmological term as measured by the average scalar curvature in the ′ ground state. In particular, the two-point correlator of Ricci scalars R(x)R(x ) h i falls to zero with a certain universal power of the invariant distance between x and ′ x that depends only on the total number of massless fields in the theory through the effective central charge of the theory. In principle, it should be possible to verify the existence of such a scale in- variant phase of 4D quantum gravity by appropriate numerical simulations. In this letter we suggest that a comparison of the continuum predictions can be made with simplicial simulations of four-geometries with the topology of S4 [4-5]. This topology includes in particular the physically interesting case of Euclidean de Sitter space. We first derive the scaling behavior of the partition function for this topol- ogy, or equivalently the behavior of the fixed volume partition function at large volumes V. The continuum theory predicts the existence of a critical curve for the cosmological term λ as a function of the Newtonian coupling κ. This implies that the corresponding parameters of the dynamical triangulation approach must obey a quadratic relation in the scaling limit with the slope of the linear term completely determined in terms of pure numbers. Moreover in order to obtain the continuum limit corresponding to the scale invariant phase of quantum gravity, the Newtonian coupling of the lattice theory should be scaled to infinity like √V for large V. An indication that this is a proper infinite volume limit to take is that apparently only in this limit does the average scalarcurvaturevanishonthelattice[4]. Aclearandnon-trivialtestoftheexistence 2 of the scale invariant phase is that the entropy exponent must be independent of the rescaled Newtonian parameter. If this turns out to be the case in the numerical simulations, we will have at hand a powerful computational tool for 4D quantum gravity in the far infrared, and should be able to measure the graviton contribution to the effective central charge of the conformal theory. Let us first recapitulate our principal results in the continuum [3,6]. We began with the conformal decomposition of the metric g (x) = e2σ(x)g¯ (x), with g¯ (x) ab ab ab a fixed fiducial metric. By consideration of the general form of the trace anomaly for conformal fields in four dimensions and taking into account the Wess-Zumino integrability condition, we determined the general form of the effective action whose σ variation is the trace anomaly. Treating this effective action as the fundamental quantum action for the σ field at large distances and requiring that general covari- ance be exactly preserved in the vacuum state of this σ theory, we found that the total trace anomaly of the full theory must vanish. In other words the absence of diffeomorphismanomaliesinquantumgravityrequiresthat thevacuumisascalein- variant conformal fixed point where the beta functions of all renormalized couplings are zero. At the fixed point the effective Euclidean action for σ reads: 2 S = d4x√g¯ Q σ∆ σ+ 1 G 2 R σ 3e2ασ α2( σ)2 + R +λe4ασ eff (4π)2 4 2 − 3 − κ ∇ 6 Z n (cid:2) (cid:0) (cid:1) (cid:3) (cid:2) (cid:3) (1o) where G is the Gauss-Bonnet integrand whose integral is the Euler number, 1 χ = d4x√g G , (2) E 32π2 Z and ∆ is the Weyl covariant fourth order operator acting upon scalars: 4 ∆ = 2 +2Rab 2R + 1( aR) . (3) 4 ∇a∇b − 3 3 ∇ ∇a The quantity Q2 plays the role of the central charge at the infrared fixed point and is proportional to the coefficient of the Gauss-Bonnet term in the quantum trace anomaly. The physical metric at the conformal fixed point becomes g (x) = e2ασ(x)g¯ (x), (4) ab ab with α determined by the condition that the Einstein-Hilbert action have its canon- ical scale dimension in this metric. This condition gives a quadratic equation which 3 fixes α in terms of the central charge: 1 1 4 − − Q2 α = . (5) q2 Q2 With α determined in this way the condition that the volume (cosmological) term have dimension 4 requires that the cosmological and Newtonian couplings satisfy the relation1, 18π2 α2 4α2 6α4 λ = f(Q2) , f(Q2) = 1+ + . (6) κ2 Q2 Q2 Q4 " # All of these results were obtained in the continuum by treating the metric g¯ ab as fixed. In other words the transverse, traceless sector of the theory containing the physical spin-2 gravitons was neglected completely. Our basic hypothesis is that these relations remain true in the infrared when the graviton modes are included, up to a possible renormalization of the value of Q2. More precisely we assume that integration over the transverse graviton modes generates an effective action for σ which, when expanded in powers of derivatives, has the same form as (1) but with renormalized coefficients. This assumption we call “infrared conformal dominance.” This is not at all unreasonable from a Wilsonian effective action point of view. Consider the functional integration over transverse gravitons (in other words over conformal equivalence classes of metrics), as well as matter fields, with both in- frared and ultraviolet cut-offs, ℓ and a. At short distances, graviton effects may grow uncontrollably due to the presence of the dimensionful Newtonian coupling κ, so that a cannot be taken to zero in the effective action. Conversely, at large distance scales, the transverse, tracefree fluctuations should be expected to become less important, so that the effective action should be regular as the infrared cutoff ℓ is removed. If this is the case, then an infrared stable renormalization group fixed point of the effective low energy theory is approached as ℓ . Scale → ∞ invariance at this fixed point then requires that the low energy effective action must be of the form (1) when expanded up to four derivatives of σ. The uniqueness of the effective action (1) at the infrared fixed point is still not sufficient to guarantee that the relations (5) and (6) hold after the integration over the graviton modes. For these relations to be unmodified in terms of Q2 one needs to assume that the 1 Note a typographical error in eq. (3.17) of ref. [3] and the difference in the definition of λ. 4 infrared fluctuations of the gravitons are subdominant compared to the fluctuations in the conformal factor. In (1), we have dropped a possible C¯2 σ term. This is justified by scale abcd invariance at the fixed point, which requires the vanishing of the beta function of the Weyl-squared coupling, so that the coefficient of the C¯2 term in the trace abcd anomaly must vanish identically at the fixed point. The same reasoning eliminates the coefficient of a local R2 term in (1), which can be checked explicitly at the level of the σ theory alone. Then the only unknown parameter in the continuum theory is the contribution of gravitons to Q2. We have calculated this contribution to Q2 in perturbation theory in both the Einstein and Weyl theories and found values around 8, which lead to an α 1.2 [6]. ∼ In order to derive the scaling behavior of the partition function of the effective σ theory, subject the σ field to the constant shift [2] ω σ σ + (7) → α and use the translational invariance of the integration measure [ σ] to find: D 2 Z(κ,λ) [ σ]e−Seff[σ] = e−Qα χEωZ(κe−2ω,λe4ω) . (8) ≡ D Z In what follows we restrict to the topology S4 for which χ = 2. It is convenient E to define also the partition function at fixed volume Z(κ;V) [ σ]eλV−Seff[σ] δ d4x√g¯e4ασ V . (9) ≡ D − Z (cid:16)Z (cid:17) Then performing the translation (7) above, we obtain2 2 Z(κ;V) = e−2(Qα +2)ωZ(κe−2ω;e−4ωV) (10) 2 = V−Q2α−1Z˜(κV−12) , where in the second line we put e4ω V. ∝ From these continuum results we turn now to the numerical simulations. The numerical method that has proven most fruitful up until now is that of “dynamical 2 Scaling relations of a similar kind were derived in ref. [7] for conformally self- dual metrics. However, the physical meaning of the second operator identified in this work with the “volume” is unclear to us. The introduction of this new operator is also the reason why the critical relation (6) was not obtained. 5 triangulation,” a variation of Regge calculus in which geometries are constructed by gluing together fundamental simplices of fixed volume [8]. The four-simplices share common faces with their neighbors which are D 1 = 3 dimensional simplices, i.e. − regular tetrahedra of edge length a. The angle between two tetrahedra faces sharing a triangle is 1 θ = arccos = 1.3181161 (11) D in D = 4 dimensions. The volume of a(cid:0)fun(cid:1)damental J-simplex is aJ J +1 V = aJΩ = , (12) J J J! 2J r where a is the lattice spacing, so that the total volume of the simplicial manifold is d4x √g N V (13) 4 4 → Z if N is the total number of 4-simplices in the configuration. 4 As in the Regge approach, space is regarded as flat inside the D = 4 simplices with all the curvature residing on the D 2 = 2 dimensional hinges, i.e. equilateral − triangles with the same fixed edge length. If n is the number of 4-simplices sharing i a given equilateral triangle i, then the deficit angle δ is given by i δ = 2π n θ , (14) i i − and the Einstein-Hilbert action takes the value, d4x √g R δ V = (2πN 10θN )V , (15) i 2 2 4 2 → − Z i X where n = 10N has been used (since each 4-simplex has 10 triangles in its i i 4 boundary). P Dynamics is now specified by giving an action for each simplicial triangulation of the form S( ) = k N ( ), with each triangulationotherwise having equal T T J J J T statistical weight. Actually not all of the N are independent. In order for the sim- P J plicialcomplextoapproximateacontinuousmanifoldthenumbersofJ-subsimplices (for J = 0,1,...,D) must satisfy some relations, called Dehn-Sommerville relations [9]. In addition we have the Euler relation, 4 ( )JN = χ . (16) − J E J=0 X 6 The net result is that only two of the N are independent and the action may be J taken to be of the form, S( ) = k N ( )+k N ( ) . (17) 2 2 4 4 T − T T Comparing this with the Einstein-Hilbert action, and using the substitutions (13) and (15) leads to the following identification of the simplicial action parameters with those of the continuum, √3π a2 k = 2 4 κ (18) 5√3θ a2 √5 k = + λa4 4 4 κ 96 where the numerical coefficients come from eq. (12). Since we are interested in the continuum infrared fixed point of the lattice theory, we do not add any higher derivative couplings to the action. These should correspond to irrelevant operators in the infrared, and in any case the coefficients of possible R2 and Weyl-squared terms in the action vanish at the conformal fixed point. The trace anomaly induced action (1) is not to be added to the lattice action (17) either. It is nonlocal in the full metric (4) and should be generated dynamically by the quantum fluctuations of the simplicial geometries, in analogy with the situation in the two dimensional case. Because the number of triangulations with fixed S4 topology which can be made from a given number N of 4-simplices is exponentially bounded with respect 4 to N [4-5], the partition function of the dynamical triangulation approach, 4 Z (k ,k ) e−S(T) = Z(k ;N )e−k4N4 < e−[k4−k4c(k2)]N4 (19) DT 2 4 2 4 ≡ ∼ XT XN4 XN4 must exist in a region of the coupling constant plane k > kc(k ). By approaching 4 4 2 the boundary of this region from above one can hope to arrive at a continuum limit in which physical correlation lengths go to infinity when expressed in lattice units. In other words, one is searching for a critical curve in the (k ,k ) plane 2 4 corresponding to a second order phase transition of the lattice theory defined by (17) and (19). The first implication of the continuum σ theory for the lattice simulations is that at the infrared fixed point one has the relation (6) 7 which determines the critical curve in the (k ,k ) plane, 2 4 5θ kc(k ) = k +√5f(Q2)k2 , (20) 4 2 π 2 2 where use of eqs. (18) has been made. This relation should hold for the lattice parameters in the infinite volume (con- tinuum) limit. In any finite volume simulation there will be an additive renor- malization of the cosmological term which scales to zero with the lattice length a. Hence the intercept of the curve kc(k ) will not vanish in general, for finite volume. 4 2 However, the slope of the linear term of the critical curve (20) is 5θ = 2.0978469, a π pure number independent of Q2. Actual simulations with N 104 seem to indi- 4 ∼ cate a critical curve kc(k ) which is approximately linear with a slope slightly more 4 2 than 2 [10]. The relation (20) has a quadratic term as well which could be used to determine Q2, in principle if the simulations are run with high statistics and large volumes. However since Q2 is unknown, and possibly large, and f(Q2) 0 for → large Q2, the quadratic term could be difficult to measure. A quite different prediction of infrared conformal dominance and better way to measure Q2 is provided by the finite volume scaling relation (10). Translating this continuum relation to the lattice by using (18) we obtain the following scaling relation for the fixed volume partition function at large volumes, Z(k ;N ) Z˜(k˜ )Nγ−3ek4c(k2)N4 , (21) 2 4 ∼ 2 4 where k˜ = k √N , (22) 2 2 4 and Q2 γ(Q2) = 2 . (23) − 2α The content of the scaling relation (21) is that, when k is scaled to zero with the 2 square root of the volume as in (22) keeping k˜ fixed, the prefactor Z˜ becomes 2 volume independent and the entropy exponent γ depends only on the effective central charge Q. Therefore, a clear test of this scaling relation is that γ must be independent of the parameter k˜ , if our hypothesis of infrared conformal dominance 2 is correct. Otherwise, γ would acquire non-trivial k˜ dependence. If γ is indeed 2 independent of k˜ , a measurement of γ would provide a non-perturbative way to 2 compute the graviton contribution to the central charge. Note also that the entropy exponent goes to in the semiclassical limit Q2 , while γ = 1 for Q2 = −∞ → ∞ 8 4. In the latter case one would expect logarithmic behavior in analogy with the c = 1 case in two dimensions. For Q2 < 4 the exponent α in (5) becomes complex and the theory could exhibit a phase transition with qualitatively new phenomena. Perturbative calculations of the graviton contributions [6] lead to the value γ ∼ 1.3 . − The simplest observable with nice scaling behavior is the average curvature R h i defined by [4]: d4x√gR 2πV 1 ∂ 5θ 2 R = lnZ(k ;N ) , (24) 2 4 h i ≡ d4x√g V N ∂k − π (cid:28)R (cid:29) 4 (cid:20) 4 2 (cid:21) R where the last equality holds for fixed volume and we used the relations (13), (15) and (17). Now inserting the scaling behavior (21) and the expression (20) one obtains: 2πV 1 ∂ R = 2 2√5f(Q2)k + lnZ˜(k˜ ) h i V 2 √N ∂k˜ 2 4 (cid:20) 4 2 (cid:21) (25) 1 0 , ∼ √N → 4 wheretheproportionalityfactordependsonlyonQ2 andtherescaledk˜ . Thisshows 2 that the average curvature scales to zero with the square root of the volume. Hence scaling k 0 with k˜ fixed yields a large volume continuum limit consistent with 2 2 → naive dimensional analysis. This removes the main obstruction to interpretation of the numerical simulations for the continuum theory mentioned in ref. [4], since R = a2 R vanishes as k N−1/2 0. This is consistent with the h ilattice h icontinuum 2 ∼ 4 → linear behavior of R with k observed numerically in ref. [4]. 2 h i Another observable that has been used to search for a continuum limit at a critical value of k is the integrated curvature-curvature correlator, 2 1 ∂2 lnZ(k ;N ) = d4x √g R(x)R(0) V R 2 . (26) −V ∂k2 2 4 h i− h i 2 Z If we substitute (21) into this expression and use (20) and (22), thereby scaling −1 k 0 like N 2, this quantity goes to a constant in the large volume limit, in 2 → 4 contrast to the divergent behavior at a finite value of k that has been suggested in 2 the searches for an ultraviolet fixed point. Finally we would like to mention that considering only the first term of the curvature-curvature correlator in the r.h.s. of (26) is also interesting from our 9