Putting a cap on causality violations in CDT
aa r X i v : . [ g r- q c ] S e p Putting a cap on causality violations in CDT
J. Ambjørn a,b
R. Loll b , W. Westra c and S. Zohren d a The Niels Bohr Institute, Copenhagen UniversityBlegdamsvej 17, DK-2100 Copenhagen Ø, Denmark.email: [email protected] b Institute for Theoretical Physics, Utrecht University,Leuvenlaan 4, NL-3584 CE Utrecht, The Netherlands.email: [email protected], [email protected] c Department of Physics, University of Iceland,Dunhaga 3, 107 Reykjavik, Icelandemail: [email protected] d Blackett Laboratory, Imperial College,London SW7 2AZ, United Kingdom.email: [email protected]
Abstract
The formalism of causal dynamical triangulations (CDT) provides us with anon-perturbatively defined model of quantum gravity, where the sum over histo-ries includes only causal space-time histories. Path integrals of CDT and theircontinuum limits have been studied in two, three and four dimensions. Herewe investigate a generalization of the two-dimensional CDT model, where thecausality constraint is partially lifted by introducing weighted branching points,and demonstrate that the system can be solved analytically in the genus-zerosector.PACS: 04.60.Ds, 04.60.Kz, 04.06.Nc, 04.62.+v.Keywords: quantum gravity, lower dimensional models, lattice models.1 ntroduction
The idea of CDT, by which we mean the definition of quantum gravity theory viacausal dynamical triangulations, is two-fold: firstly, inspired by earlier ideas inthe continuum theory [1, 2], we insist, starting from space-times with a Lorentziansignature, that only causal histories contribute to the quantum gravitational pathintegral. Secondly, we assume the presence of a global time-foliation.The formalism of dynamical triangulations (DT) provides a simple regulariza-tion of the sum over geometries by providing a grid of piecewise linear geometriesconstructed from elementary building blocks (these are d -dimensional simplicesof identical size and shape if we want to construct a d -dimensional geometry,see [3, 4] for reviews). The ultraviolet cut-off is given by the edge length of thebuilding blocks. The causal variant CDT also uses DT as the regularization ofthe path integral. A detailed description of which causal geometries are includedin the grid can be found in references [5, 6].We emphasize that the use of triangulations is merely a technical regulariza-tion of the assumed underlying continuum theory, in the same way a lattice canbe used for regularizing a quantum field theory. By no means do we presupposethat space-time is literally made out of little simplices. Some support for theexistence of an underlying (non-perturbative) continuum quantum field theoryin higher dimensions has been provided in [6]-[11] and seems to be in qualitativeagreement with independent analyses carried out using the renormalization group[12, 13, 14, 15, 16].While the CDT model is defined as a sum over causal space-time histories(each with an appropriate weight), one can ask whether this causality constraintcan be lifted. One motivation for introducing it was that unrestricted summationover space-time histories leads to a dominance of highly singular configurationsin dimensions d >
2, which prevents the existence of a physically meaningfulcontinuum limit of the regularized lattice theory [17, 18, 19]. On the other hand, ifone takes the point of view that a maximal number of possible fluctuations shouldbe included in the path integral (while still leading to a meaningful result), onemay wonder whether it is possible to reintroduce (a subclass of) configurationsinto the sum over geometries which correspond to metric structures with causalityviolations. The question we would like to pose is whether this can be done in acontrolled manner – using the additional time-slicing structure present in CDT– which avoids the problems encountered previously by DT, corresponding to anunrestricted inclusion of all such configurations.Because of the ready availability of analytic tools and the existence of analyt-ical solutions, we will in a first step analyze the situation in two dimensions. Inthis context, the issue has been addressed previously in a 2d toy model, focussing even when keeping the space-time topology fixed
2n the effects of including a class of minimal wormholes in CDT, which can besaid to violate causality only mildly and are much less abundant than generalwormholes [20, 21, 22]. In the present work, we will look at the genus-zero sectorof a generalized model of two-dimensional CDT, which in principle allows for theinclusion of arbitrary space-time topologies, as well as “outgrowths”, that is, thesprouting of baby universes, and associates with them a weight depending on thegravitational coupling constant. A key observation is that requiring the propa-gator of the model to reduce to that of standard CDT when the bare coupling istaken to zero uniquely fixes the scaling of this coupling, leading to a continuumlimit where branching processes occur, but are scarce compared to the situationin DT.The following sections deal with explaining this scaling argument, and withanalytically computing the genus-0 propagator (or loop-loop amplitude) and cor-responding disc amplitude. The computation of higher-genus amplitudes in thisframework is currently under way, and may open new perspectives on the issueof the sum over topologies in theories of quantum gravity, which are only appar-ent in a formulation that has at least some memory of the
Lorentzian structureof space-time built in. However, we have as yet no definite statements to makeabout the properties of general higher-genus amplitudes, the summability of thegenus expansion or a generalization of the model to higher dimensions.
Two-dimensional CDT and Euclidean quantum gravity
Two-dimensional quantum gravity is a wonderful playground for “quantum ge-ometry”, understood as the statistical sum over geometries. The reason for thisis that the action is trivial as long as we ignore topology changes (and even thenit is almost trivial). One can therefore use entirely geometric reasoning to de-rive relations between or properties of “Green’s functions” . In this context it isconvenient to study the proper-time “propagator”, namely, the amplitude of ge-ometries with two space-like boundaries separated by a proper time (or geodesicdistance) t . Although the proper-time propagator is a special amplitude, it hasthe virtue that other amplitudes, like the disc or cylinder amplitudes, can becalculated from it [29, 30, 31, 32, 5]. When the path integral representation ofthis propagator is defined in the Lorentzian domain, using CDT, we can asso-ciate with each of the causal, piecewise linear Lorentzian space-time geometriesa unique Euclidean geometry. After this rotation we perform the sum over the The first analysis of a CDT model with local “decorations” was made in [23]. An early example is the proof [24] that the string tension of bosonic string theory (regular-ized using DT [25, 26]) does not scale, thus providing a simple geometric understanding of theimpossibility of defining bosonic string theory in target space dimensions larger than or equalto 2. Other applications in non-critical string theory can be found in [27, 28]. S × [0 , S [ g µν ] = λ Z d ξ q det g µν ( ξ ) + x I d l + y I d l , (1)where λ is the cosmological constant, x and y are two so-called boundary cos-mological constants, g µν is the metric of a geometry of the kind described above,and the line integrals refer to the lengths of the in- and out-boundaries inducedby g µν . The propagator G λ ( x, y ; t ) is defined by G λ ( x, y ; t ) = Z D [ g µν ] e − S [ g µν ] , (2)where the functional integration is over all causal geometries [ g µν ] such that thefinal boundary with boundary cosmological constant y is separated a geodesicdistance t from the initial boundary with boundary cosmological constant x . Calculating the path integral (2) with the help of the CDT regularization andtaking the continuum limit as the side-length a of the simplices goes to zero leadsto the equation [5] ∂∂t G λ ( x, y ; t ) = − ∂∂x h ( x − λ ) G λ ( x, y ; t ) i , (3)which is solved by G λ ( x, y ; t ) = ¯ x ( t, x ) − λx − λ x ( t, x ) + y , (4)where ¯ x ( t, x ) denotes the solution of the characteristic equation for (3), namely,d¯ x d t = − (¯ x − λ ) , ¯ x (0 , x ) = x. (5) This class of geometries is difficult to define directly in a continuum, gauge-fixed formula-tion; what we have in mind here is the continuum limit of the corresponding discrete sum. The asymmetry between x and y is due to the convention that the initial boundary containsa marked point. Symmetric expressions where neither or both boundaries have marked pointscan be found in [33]. x Figure 1:
Graphical representation of relation 8: differentiating the disc amplitude W λ ( x ) (represented by the entire figure) with respect to the cosmological constant λ corresponds to marking a point somewhere inside the disc. This point has a geodesicdistance t from the initial loop. Associated with the point one can identify a connectedcurve of length l , all of whose points also have a geodesic distance t to the initial loop.This loop can now be thought of as the curve along which the lower part of the figure(corresponding to the loop-loop propagator G λ ( x, l ; t )) is glued to the cap, which itselfis the disc amplitude W λ ( l ). Let l denote the length of the initial and l the length of the final boundary.Rather than considering a situation where the boundary cosmological constant x is fixed, we will take l as fixed, and denote the corresponding propagator by G λ ( l , y ; t ), with similar definitions for G λ ( x, l ; t ) and G λ ( l , l ; t ). All of themare related by Laplace transformations, for instance, G λ ( x, y ; t ) = Z ∞ d l Z ∞ d l G λ ( l , l ; t ) e − xl − yl , (6)where the Laplace-transformed propagator obeys the composition rule G λ ( x, y ; t + t ) = Z ∞ d l G λ ( x, l ; t ) G λ ( l, y ; t ) . (7)Eq. (7) is the simplest example of the use of quantum geometry. While theproperty (7) is evident in the context of CDT where no baby universes are allowed,it is also true in Euclidean quantum gravity (where there is no such constraint), ifone defines the distance between the initial and final loop appropriately [29, 34].Another, slightly more complicated example is illustrated graphically by Fig.1, which implies the functional relation − ∂W λ ( x ) ∂λ = Z ∞ d t Z ∞ d l G λ ( x, l ; t ) lW λ ( l ) . (8)5t encodes the following: let W λ ( l ) denote the disc amplitude, i.e. the Hartle-Hawking amplitude with a fixed boundary length l , and W λ ( x ) the correspondingLaplace-transformed amplitude where x is a fixed boundary cosmological con-stant. Differentiation with respect to the cosmological constant λ means markinga point in the bulk, as shown in the figure. Each configuration appearing in thepath integral has a unique decomposition into a cylinder of proper-time extension t , (where the proper time is defined as the geodesic distance of the marked pointto the boundary), and the disc amplitude itself, as summarized in eq. (8).Starting from a regularized theory with a cut-off a , it was shown in [5] thatthere are two natural solutions to eq. (8). In one of them, the regularized discamplitude diverges with the cut-off a and the geodesic distance t scales canonicallywith the lattice spacing a according to W reg −−→ a → a η W λ ( x ) , η < , (9) t reg −−→ a → t/a ε , ε = 1 . (10)In the other, the scaling goes as W reg −−→ a → const . + a η W λ ( x ) , η = 3 / t reg −−→ a → t/a ε , ε = 1 / , (12)where the subscript “ reg ” denotes the regularized quantities in the discrete latticeformulation. The first scaling (9)-(10), with η = −
1, is encountered in CDT, whilethe second scaling (11)-(12) is realized in Euclidean gravity, i.e. Liouville gravityor gravity defined from matrix models.As demonstrated in [5], it is possible to treat both models simultaneously.Allowing for the creation of baby universes during the “evolution” in proper time t (by construction, a process forbidden in CDT) leads to a generalization of (3),namely, a ε ∂∂t G λ,g ( x, y ; t ) = − ∂∂x h(cid:16) a ( x − λ ) + 2 g a η − W λ,g ( x ) (cid:17) G λ,g ( x, y ; t ) i , (13)where we have introduced a new coupling constant g , associated with the cre-ation of baby universes, and also made the additional dependence explicit in theamplitudes. In [5] it was noted that for g = 1, that is, viewing this creation as apurely geometric process , one obtains Euclidean quantum gravity. This happensbecause according to (9) and (11), we have either η = −
1, which is inconsistent By this we mean that each distinct geometry (distinct in the sense of Euclidean geometry)appears with equal weight in the sum over two-dimensional geometries. = + Figure 2:
In all four graphs, the geodesic distance from the final to the initial loop isgiven by t . Differentiating with respect to t leads to eq. (15). Shaded parts of graphsrepresent the full, g s -dependent propagator and disc amplitude, and non-shaded partsthe CDT propagator. with (13), or we have from (11) that η = 3 / ε = 1 /
2, which is consis-tent with (11). On the other hand, setting g = 0, thereby forbidding the creationof baby universes, leads of course back to (3).In a non-trivial extension of previous work, we will now allow for the possibilitythat the coupling g becomes a non-constant function g = g ( a ) of the cut-off a . Ageometric interpretation of this assignment will be given in the discussion sectionbelow. Since we are interested in a theory which smoothly recovers CDT in thelimit as g →
0, it is natural to assume that η = −
1, like in CDT. Consequently,the only way to obtain a non-trivial consistent equation is to assume that g scalesto zero with the cut-off a according to g = g s a , (14)where g s is a coupling constant of mass dimension three, which is kept constantwhen a →
0. With this choice, eq. (13) is turned into ∂∂t G λ,g s ( x, y ; t ) = − ∂∂x h(cid:16) ( x − λ ) + 2 g s W λ,g s ( x ) (cid:17) G λ,g s ( x, y ; t ) i . (15)The graphical representation of eq. (15) (or (13) for g = 0) is shown in Fig. 2.Differentiating the integral equation corresponding to this figure with respect tothe time t one obtains (15). The disc amplitude W λ,g s ( x ) is at this stage unknown.Note that one could in principle have considered an a priori more generalbranching process, where more than one baby universe is allowed to sprout atany given time step t . However, one observes from the scaling relation (14) thatthe corresponding extra terms in relation (13) would be suppressed by higherorders of a and therefore play no role in the continuum limit.7 + Figure 3:
Graphical illustration of eq. (17). Shaded parts represent the full discamplitude, unshaded parts the CDT disc amplitude and the CDT propagator.
In the next section we will show that quantum geometry, in the sense definedabove, together with the requirement of recovering standard CDT in the limit as g s →
0, uniquely determines the disc amplitude and thus G λ,g s ( x, y ; t ). The disc amplitude
The disc amplitude of CDT was calculated in [5, 35]. In [5] it was determined di-rectly by integrating G λ ( l , l = 0; t ) over all times. This decomposition is unique,since by assumption t is a global time and no baby universes can be created. In[35] it was shown that it could also be obtained from Euclidean quantum gravity(matrix model results) by peeling off baby universes in a systematic way. Byeither method one finds W λ ( x ) = 1 x + √ λ (16)for the disc amplitude as function of the boundary cosmological constant x . Inthe present, generalized case we allow for baby universes, leading to a graphicalrepresentation of the decomposition of the disc amplitude as shown in Fig. 3. Ittranslates into the equation W λ,g s ( x ) = W (0) λ,g s ( x ) + g s ∞ Z d t ∞ Z d l d l ( l + l ) G (0) λ,g s ( x, l + l ; t ) W λ,g s ( l ) W λ,g s ( l )(17)for the full propagator W λ,g s ( x ), where we have introduced a superscript (0) toindicate the CDT amplitudes, that is, W (0) λ,g s ( x ) ≡ W λ,g s =0 ( x ) = W λ ( x ) , (18)8nd similarly for G (0) λ,g s , quantities which were defined in eqs. (16) and (4) respec-tively. The integrations in (17) can be performed, yielding W λ,g s ( x ) = 1 x + √ λ + g s x − λ (cid:16) W λ,g s ( √ λ ) − W λ,g s ( x ) (cid:17) . (19)Solving for W λ,g s ( x ) we find W λ,g s ( x ) = − ( x − λ ) + ˆ W λ,g s ( x )2 g s , (20)where we have definedˆ W λ,g s ( x ) = r ( x − λ ) + 4 g s (cid:16) g s W λ,g s ( √ λ ) + x − √ λ (cid:17) . (21)The sign of the square root is fixed by the requirement that W λ,g s ( x ) → W λ ( x ) for g s →
0, and W λ,g s ( x ) is determined up to the value W λ,g s ( √ λ ). We will now showthat this value is also determined by consistency requirements of the quantumgeometry. If we insert the solution (20) into eq. (15) we obtain ∂∂t G λ,g s ( x, y ; t ) = − ∂∂x h ˆ W λ,g s ( x ) G λ,g s ( x, y ; t ) i . (22)In analogy with (4) and (5), this is solved by G λ,g s ( x, y ; t ) = ˆ W λ,g s (¯ x ( t, x ))ˆ W λ,g s ( x ) 1¯ x ( t, x ) + y , (23)where ¯ x ( t, x ) is the solution of the characteristic equation for (22),d¯ x d t = − ˆ W λ,g s (¯ x ) , ¯ x (0 , x ) = x, (24)such that t = Z x ¯ x ( t ) d y ˆ W λ,g s ( y ) . (25)Physically, we require that t can take values from 0 to ∞ , as opposed to just in afinite interval. From expression (25) for t this is only possible if the polynomialunder the square root in the defining equation (20) has a double zero, which fixesthe function ˆ W λ,g s ( x ) toˆ W λ,g s ( x ) = ( x − α ) p ( x + α ) − g s /α, (26)where α = u √ λ, u − u + g s λ / = 0 . (27)9n order to have a physically acceptable W λ,g s ( x ), one has to choose the solutionto the third-order equation which is closest to 1. Quite remarkably, one can alsoderive (26) from (20) by demanding that the inverse Laplace transform W λ,g s ( l )fall off exponentially for large l . In this region W λ,g s ( x ) equals W (0) λ,g s ( x ) plus aconvergent power series in the dimensionless coupling constant g s /λ / .One can check the consistency of the quantum geometry by noting that using(23) in (8) the integration can be performed to yield ∂W λ,g s ( x ) ∂λ = W λ,g s ( x ) − W λ,g s ( α )ˆ W λ,g s ( x ) , (28)which is indeed satisfied by the solution (20). The loop-loop amplitude
We mentioned above that the loop-loop propagator can be regarded as a buildingblock for other, more conventional “observables” in 2d quantum gravity. One ofthe most beautiful illustrations of this and at the same time a non-trivial exampleof what we have called quantum geometry is the calculation in 2d Euclidean quan-tum gravity of the loop-loop amplitude from the loop-loop proper-time propagator[30]. The full loop-loop amplitude is obtained by summing over all Euclidean 2dgeometries with two boundaries, without any particular restriction on the bound-aries’ mutual position. This amplitude was first calculated using matrix modeltechniques (for cylinder topology) [36].To appreciate the underlying construction, consider a given geometry of cylin-drical topology. Its two boundaries will be separated by a geodesic distance t , inthe sense of minimal distance of any point on the final loop to the initial loop. Itfollows that we can consider the geometry as composed of a cylinder where theentire final loop (i.e. each of its points) has a distance t from the initial one anda “cap” related to the disc amplitude, as illustrated in Fig. 4(a). One can nowobtain the loop-loop amplitude by integrating over all t and all gluings of the cap(we refer to [30] for details). An intriguing aspect of the construction is that thedecomposition of a given geometry into cylinders and caps is not unique. Onecan choose another decomposition consisting of two cylinders of length t and t ,with t + t = t , joined by a cap, as illustrated in Fig. 4(b). As shown in [30], theend result is indeed independent of this decomposition.The whole construction can be repeated for our new, generalized CDT model,in this way defining a loop-loop amplitude. More precisely, although an exactequality of amplitudes corresponding to different decompositions like those de-picted in Fig. 4(a) and (b) is not immediately obvious at the level of the triangula-tions of the discretized theory , the continuum ansatz (29) below is self-consistent, because of the different arrangements of the proper-time slicings W λ t t t t t W λ ( a ) ( b ) Figure 4:
Two different ways of decomposing the loop-loop amplitude into proper-timepropagators and a disc amplitude. Two points touch in the disc amplitude W , pinchingthe boundary to a figure-8, which combinatorially implies a substitution W λ,g s ( l ) → lW λ,g s ( l ) in the formulas. The time variables are related by t + t = t . in the sense that it leads to a non-trivial symmetric expression for the amplitudewith a well-defined g s → G λ,g s ( x, y ), and its Laplace trans-form by G λ,g s ( l , l ), related in the same way as was discussed for the loop-looppropagator (c.f. eq. (6) and the discussion leading up to it). The integral equationcorresponding to Fig. 4 is given by G λ,g s ( l , l ) = Z ∞ d t Z ∞ d l G λ,g s ( l , l ; t ) lW λ,g s ( l + l ) . (29)Laplace-transforming eq. (29), the integrals can be performed using eqs. (23)-(26).After some non-trivial algebra one obtains G λ,g s ( x, y ) = 1 f ( x ) f ( y ) 14 g s (cid:18) [( x + α ) + ( y + α )] ( f ( x ) + f ( y )) − (cid:19) , (30)where we are using the notation f ( x ) = p ( x + α ) − g s /α = ˆ W λ,g s ( x ) / ( x − α ) . (31)In the limit g s → G (0) λ,g s ( x, y ) = 12 √ λ ( x + √ λ ) ( y + √ λ ) , (32)11 result which could of course also have been obtained directly from (29) using (4),(5) and (16). We note that the corresponding expression in the case of Euclidean2d quantum gravity is given by G ( e ) λ ( x, y ) = 12 h ( x ) h ( y )( h ( x ) + h ( y )) , h ( x ) = q x + √ λ, (33)which can be obtained from expressions similar to (23)-(26), only with ˆ W λ,g s ( x )replaced by the Euclidean disc amplitude W ( e ) λ ( x ) = ( x − √ λ/ h ( x ) . (34)We observe a structural similarity between (30) and (34), with the function f ( x )having the same relation to ˆ W λ,g s ( x ) as h ( x ) has to W ( e ) λ ( x ). The existence of well-defined, symmetric expressions for the unrestricted loop-loop amplitudes in ourgeneralized CDT model (at genus 0) and thus in standard two-dimensional CDT,formulas (30) and (32), gives strong support to the claims that (i) the proper-time propagator does indeed encode the complete information on the quantum-gravitational system, and (ii) following the arguments given in [30] concerning thedecomposition invariance of the loop-loop amplitude (c.f. Fig. 4), the continuumtheory is diffeomorphism-invariant. Discussion
The generalized CDT model of 2d quantum gravity we have defined in this paperis a perturbative deformation of the original model in the sense that it has aconvergent power expansion of the form W λ,g s ( x ) = ∞ X n =0 c n ( x, λ ) (cid:16) g s λ / (cid:17) n (35)in the dimensionless coupling constant g s /λ / . This implies in particular thatthe average number h n i of “causality violations” in a two-dimensional universedescribed by this model is finite, a property already observed in previous 2dmodels with topology change [20, 21, 22]. The expectation value of the number n of branchings can be computed according to h n i = g s W λ,g s ( x ) d W λ,g s ( x )d g s , (36)which is finite as long as we are in the range of convergence of W λ,g s ( x ). Asalready mentioned, this coincides precisely with the range where the function12 λ,g s ( x ) behaves in a physically acceptable way, namely, W λ,g s ( l ) goes to zerolike exponentially in terms of the length l of the boundary loop. The same is truefor the other functions considered, namely, G λ,g s ( l , l ; t ) and G λ,g s ( l , l ).The behaviour (36) should be contrasted with that in 2d Euclidean quantumgravity, and is reflected in the different scaling behaviours (10) and (12) for thetime t . These scaling relations show that the effective continuum “time unit” inEuclidean quantum gravity is much longer than in CDT, giving rise to infinitelymany causality violations for a typical space-time history which appears in thepath integral when the cut-off a is taken to zero. This phenomenon was discoveredin the seminal paper [29].As we have already mentioned in the introduction, the calculations presentedhere should be seen as pertaining to the genus-0 sector of a generalized CDTmodel, which also includes a sum over space-time topologies. Although we havenot given a precise definition of the higher-genus amplitudes in this paper, onewould expect them to be finite order by order. If the handles are as scarce as arethe baby universes in the genus-0 amplitudes, it might even be that the sum overall genera is uniquely defined. Whether or not this is so will clearly also dependon the combinatorics of allowed handle configurations.In the context of higher-genus amplitudes, it is natural to associate each han-dle with a “string coupling constant”, because one may think of it as a processwhere (one-dimensional) space splits and joins again, albeit as a function of anintrinsic proper time, rather than the time of any embedding space. An explicitcalculation reveals that in the generalized CDT model this process is related witha coupling constant g s [37], which one may think of as two separate factors of g s ,associated with the splitting and joining respectively.How does the disc amplitude fit into this picture? From a purely Euclideanpoint of view all graphs appearing in Fig. 3 have the fixed topology of a disc.However, from a Lorentzian point of view, which comes with a notion of time, itis clear that the branching of a baby universe is associated with a change of the spatial topology, a singular process in a Lorentzian space-time [38]. One way ofkeeping track of this in a Wick-rotated, Euclidean picture is as follows. Since eachtime a baby universe branches off it also has to end somewhere, we may thinkof marking the resulting “tip” with a puncture. From a gravitational viewpoint,each new puncture corresponds to a topology change and receives a weight 1 /G N ,where G N is Newton’s constant, because it will lead to a change by precisely thisamount in the two-dimensional (Euclidean) Einstein-Hilbert action S EH = − πG N Z d ξ √ gR. (37)Identifying the dimensionless coupling constant in eq. (13) with g ( a ) = e − /G N ( a ) ,13ne can introduce a renormalized gravitational coupling constant by1 G renN = 1 G N ( a ) + 32 ln λa . (38)This implies that the bare gravitational coupling constant G N ( a ) goes to zerolike 1 / | ln a | when the cut-off vanishes, a →
0, in such a way that the producte /G renN /λ / is independent of the cut-off a . We can now identifye − /G renN = g s /λ / (39)as the genuine coupling parameter in which we expand.This renormalization of the gravitational (or string) coupling constant is rem-iniscent of the famous double-scaling limit in non-critical string theory . In thatcase one also has g s ∝ e − /G renN , the only difference being that relation (38) ischanged to 1 G renN = 1 G N ( a ) + 54 ln λa , (40)whence the partition function of non-critical string theory appears precisely as afunction of the dimensionless coupling constant g s /λ / . Acknowledgments
All authors acknowledge support by ENRAGE (European Network on RandomGeometry), a Marie Curie Research Training Network in the European Commu-nity’s Sixth Framework Programme, network contract MRTN-CT-2004-005616.R.L. acknowledges support by the Netherlands Organisation for Scientific Re-search (NWO) under their VICI program.
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