---

Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark

arXiv:hep-th/9303042v1 8 Mar 1993NBI-HE-93-3January 19934d quantum gravity coupled to matterJ. AmbjørnThe Niels Bohr InstituteBlegdamsvej 17, DK-2100 Copenhagen Ø, DenmarkZ.

Burda1 and J. Jurkiewicz1Inst. of Phys., Jagellonian University.,ul.

Reymonta 4, PL-30 059, Krak´ow 16, PolandC.F. KristjansenThe Niels Bohr InstituteBlegdamsvej 17, DK-2100 Copenhagen Ø, DenmarkAbstractWe investigate the phase structure of four-dimensional quantum gravity coupled toIsing spins or Gaussian scalar fields by means of numerical simulations.

The quan-tum gravity part is modelled by the summation over random simplicial manifolds,and the matter fields are located in the center of the 4-simplices, which constitutethe building blocks of the manifolds. We find that the coupling between spin andgeometry is weak away from the critical point of the Ising model.

At the criticalpoint there is clear coupling, which qualitatively agrees with that of gaussian fieldscoupled to gravity. In the case of pure gravity a transition between a phase withhighly connected geometry and a phase with very “dilute” geometry has been ob-served earlier.

The nature of this transition seems unaltered when matter fields areincluded. It was the hope that continuum physics could be extracted at the transi-tion between the two types of geometries.

The coupling to matter fields, at least inthe form discussed in this paper, seems not to improve the scaling of the curvatureat the transition point.1Supported by the KBN grants no. 2 0053 91 01 and 2P302-169041

1IntroductionLast year a new regularized model of quantum gravity in 4D was introduced [1, 2].The path integral is approximated by a summation over randomly triangulatedpiecewise linear manifolds1. This method is a generalization of the one from twodimensions, which was very successful [4, 5, 7, 6].

In 4D simplicial quantum gravitytwo different phases have been observed, one with a highly connected geometryand a large Hausdorffdimension and one with a low Hausdorffdimension. Basedon numerical simulations it was suggested in [2] that the transition between thetwo types of geometries was of second order and that an interesting continuumlimit might be extracted at the transition point.

This observation has been furthercorroborated in a sequence of papers [8, 9, 10, 11, 12].One obstacle to the above mentioned suggestion is that the average curvaturedoes not scale to zero at the transition point. The average curvature does decrease(albeit slowly) with the volume of the simulated universes and it cannot be com-pletely ruled out that it actually scales to zero in the infinite volume limit.

However,at the moment we consider it as unlikely. This prompts at least a reinterpretationof the meaning of the scaling limit since naive scaling like⟨Rlattice⟩= ⟨Rcont⟩a2(1)(where a is the lattice spacing) cannot be maintained.

Maybe the average curvatureshould be absorbed in a redefinition of the cosmological constant, while the relevantphysical curvature arises only through fluctuations around the “fictitious” averagecurvature. While such an unconventional limit might exist, it seems not to be verynatural to us.

An attempt to improve the situation by adding terms like R2 to theaction was not very successful [10]. At this point we should mention a recent sug-gestion [13] of a different identification of the lattice results with continuum theoryin which one considers the limit of the bare gravitational coupling constant goingto infinity.

This limit might in continuum language correspond to an infrared fixedpoint dominated by the quantum fluctuations of the conformal factor. The scalingrelations derived in [13] agree at the qualitative level quite well with the numericalresults, but they move the interesting region of continuum physics away from thetransition in geometry and to a region in coupling constant space where (1) can besatisfied.

We consider this suggestion as most interesting. In this article we explore1An older, related approach makes use of a fixed triangulation, but allows the variation of thelength of the links.

Contrary, in the present approach one keeps the length of the links fixed, butvaries the connectivity. We refer to [3] for a recent lucid review of the first approach, which we herewill call “Regge gravity”, while we will use the term “simplicial gravity” for the present approach.2

another way to cure the problem with the scaling of the average curvature, namelycoupling of matter fields to gravity. It is of course also of interest in itself to studymatter fields coupled to dynamical random geometries.

In the best of all worldsone could even hope that the quest for correct scaling of gravity observables like theaverage curvature would uniquely determine the matter content of the theory2.The coupling of matter to two-dimensional gravity has revealed a rich and beau-tiful structure as long as the central charge of the field theory is less than or equalto one. This is summarized in the KPZ formulas [14], but was first discovered inthe simplicial gravity approach.

As an example, when the Ising model is coupled to2d simplicial gravity its phase transition changes from being second order to thirdorder [15, 16]. In addition the back-reaction of matter changes the critical exponentγ of gravity at the critical point of the Ising model.

Away from the critical pointthis exponent is unchanged.Unfortunately the analytical methods of 2d have not yet been extended to higherdimensions.The coupling of the Ising model to 3d gravity was investigated bynumerical simulations in [17, 18, 19]. The phase diagram was determined in [19] andthe conclusion was that, although there was a clear coupling between gravity and thespins at the critical point of the spin system, this influence was not sufficiently strongto change the first order transition observed in three dimensions [20, 21] betweenthe two phases of the geometrical system into a more interesting (from the point ofview of continuum physics) second order transition.

In this respect the situation isbetter in 4d where the transition between the two phases of the geometrical systemmay already be of second order, as mentioned above.The rest of this paper is organized as follows: In section 2 we define the model.In section 3 we discuss briefly the numerical method, while section 4 contains ournumerical results. Finally in section 5 we discuss the results obtained.2The modelSimplicial quantum gravity in 4d is described by the following partition function(see e.g.

[2, 10]):Z(κ2, κ4) =XT∈Te−κ4N4+κ2N2(2)where the sum is over triangulations T in a suitable class of triangulations T . Thequantity N4 denotes the number of 4-simplexes in the triangulation and N2 the2But we will of course not seriously pretend, that the present stage of numerical simulations ofquantum gravity is such, that one could really determine the matter content.3

number of triangles. The coupling constant κ2 is inversely proportional to the baregravitational coupling constant, while κ4 is related to the bare cosmological constant.The most important restriction to be imposed on T is that of a fixed topology.

Ifwe allow an unrestricted summation over all topologies in (2) the partition functionis divergent [2]. In the following we will always restrict ourself to consider manifoldswith the topology of S4.Z(κ2, κ4) is the grand canonical partition function.It is defined in a regionκ4 ≥κc4(κ2) in the (κ2, κ4) coupling constant plane.

The only way in which wecan hope to obtain a continuum limit is by letting κ4 approach κc4(κ2) from above.This tentative continuum limit depends only on one coupling constant κ2 and thetransition between the two phases of 4d gravity mentioned above takes place at acritical value of κ2, κc2. It is often convenient to think about the canonical partitionfunction where N4 is kept fixed.

Then κ2 is the only coupling constant and theaspects of gravity which do not involve the fluctuation of the total volume of theuniverse can be addressed in the limit of large N4. For the geometrical system anobservable which has our interest is the average curvature per volume, ⟨R⟩.

Theaverage curvature can for a simplicial manifold be defined by Regge calculus and inthe case of equilateral simplexes one simply has⟨R⟩∝(c4N2/N4 −10)(3)where the constant c4 is the number of 4-simplexes to which each triangle shouldbelong if the manifold were flat. Furthermore one can by an appropriate interpre-tation of the Regge approach introduce the average of the squared curvature pervolume byDR2E∝Pn2 o(n2) [(c4 −o(n2))/o(n2)]210N4(4)where the sum is over triangles n2 and o(n2) is the order of a given triangle i. e. thenumber of 4-simplexes to which this triangle belongs.

The correlator ⟨R2⟩−⟨R⟩2will prove useful as an indicator of a change in geometry.One can now couple matter fields to simplicial quantum gravity. In the case ofIsing spins the partition function will look like:Z(β, κ2, κ4) =XN4e−κ4N4XT∈T (N4)X{σ}eκ2N2eβ P⟨i,j⟩(δσiσi−1).

(5)In this formula T (N4) signifies the subclass of T with volume N4, P{σ} the sum-mation over all spin configurations, whileP⟨i,j⟩stands for the summation over allneighbouring pairs of 4-simplexes. As a function of β there might or might not be4

a phase transition for the spin system, depending on the value of κ2 (assuming thatκ4 = κc4(κ2, β), where κc4 now depends on both κ2 and β).The coupling of scalar fields to simplicial quantum gravity is also straightfor-ward. Here we will ignore self-interaction of the scalar fields and direct couplingbetween the scalar fields and the curvature, and simply consider the following par-tition functionZ(κ2, κ4) =XN4XT∈T (N4)eκ2N2−κ4N4Z Yi,αdφαi√2πngYα=1δ(Xiφαi ) e−12P⟨i,j⟩,α(φαi −φαj )2.

(6)Here i labels the 4-simplexes, α different components of the field φ and ng is the totalnumber of independent Gaussian fields. There is no need for a coupling constant infront of the Gaussian action since it can always be absorbed in κ4 by a rescaling ofthe φ’s.

Of course the gaussian action can in principle be integrated out explicitly,leaving us with an additional weight(Det CT)−ng/2(7)for each triangulation T, where CT is just the incidence matrix for the ϕ5-graphwhich is dual to the triangulation T. In the case of gaussian fields coupled to 2dgravity this fact was used to determine qualitatively the phase diagram of non-critical strings as a function of the number of Gaussian systems, ng [22, 23, 24]. Inprinciple one could try to do the same here.

However, the class of allowed ϕ5 graphsis not so easy to determine as in the case of 2d gravity. In the following we will relyon numerical simulations.3Numerical methodsOne annoying aspect of the above formalism is that we are forced to perform agrand canonical simulation where N4 is not fixed.

The reason is that we have noergodic updating algorithm3 which preserves the volume N4. It is however possibleto perform a grand canonical updating without violating ergodicity and still stay inthe neighbourhood of a prescribed value of N4, which we will denote N4(fix).

Theprocedure involves finetuning of κ4 to its critical value, κc4(κ2, β). We refer to [10]for details.In addition to the updating of the geometry, we also have to update the Isingspin system and the Gaussian systems.

Let us first discuss the Ising spin system.3In 2d gravity we know how to perform a canonical updating, but even there the grand canonicalupdating is occasionally convenient to use [25, 22, 23, 26].5

In order to avoid critical slowing down close to the phase transition between themagnetized and the non-magnetized phase the spin updating is performed by thesingle cluster variant of the Swendsen-Wang algorithm developed by Wolff[27].The cluster updating algorithms have been successfully applied to the Ising modelcoupled to 2d gravity [28, 29, 30, 31] and to the Ising model coupled to 3d gravity[19]. We update the spins once for every sweep, i.e.

after N4(fix) accepted updatingsof the geometry.In the simulations we have scanned the (κ2, β) coupling constant plane by firstfixing κ2 and then varying β in the search for a critical value βc(κ2) where the spinsystem undergoes a transition4. For values of κ2 where we are well inside the phasewith a highly connected geometry where and a large Hausdorffdimension, 5000sweeps are sufficient to achieve equilibrium for bulk quantities when the number ofsimplexes does not exceed N4 = 9000.

This is in agreement with the situation inpure gravity [2, 10]. We have occasionally made longer runs in connection with themeasurement of Binders cumulant (50.000 sweeps) and near critical points either inthe spin or gravity coupling constant.

It seems as if the situation is in all respects asin 2d and 3d gravity. In particular the presence of the spins seems not to slow downthe convergence of bulk geometric observables (in 2d it is known that spins speed itup).

In this phase we have neither seen excessive signs of autocorrelations of spins(the longest of the order of 500 sweeps at the spin transition). This is in agreementwith intuition since the connectivity of the system is large and the maximal distancebetween spins correspondingly small.

The situation is somewhat different when weprobe the phase where the geometry is elongated and where internal distances canbe quite large. Without spin the convergence in geometry is slow in this phase andit is true also after coupling to spins.The Gaussian fields are updated by a heatbath algorithm.

There are two aspects ofthis updating. One type of updating is performed with a fixed background geometryand is standard.

The other one is related to the Metropolis updating of the geomet-rical structure. Since there are slightly unconventional aspects connected with thechange of the fields, when the geometry is changed 5, let us make a few comments.We will not go into details (which are trivial, but clumsy to write down explicitly),but rather sketch the main point: Consider a change in geometry where we take a4-simplex, remove the “interior”, insert a vertex in the “empty” interior and connect4In order that the reader could appreciate the amount of work going into this please note thatwe have to fine-tune κ4 for each value of κ2 and β.5The same aspect is already present in the grand canonical algorithms used in 2d gravity, seee.g.

[25].6

this vertex to the five vertices of the former 4-simplex. With a proper identificationof sub-simplexes we have by this procedure removed one 4-simplex and created fivenew ones.

The inverse “move” is one where we remove a vertex of order five andthe associated five 4-simplexes and replace them by a single 4-simplex. We must becareful to treat the Gaussian fields correctly in such moves.

In the case where weinsert a vertex we will have to introduce five new fields ϕi, i = 1, . .

. 5.

They willinteract quadratically with each other, and each of them will interact with one fieldassociated with a neighbouring 4-simplex untouched by the move. Let us denotethese five fields φi, i = 1 .

. .

5 In addition we have removed a field associated withthe original 4-simplex. We denote it by ϕ0.

It interacted with the five φi’s. Thecorrect probability distribution of the new five ϕi’s isdPnew(ϕi) = Cnew(φi)5Yi=1dϕi e−Snew(ϕi,φi)(8)where the additional part of the action Snew coming from added fields ϕi, determinedfrom (6) isSnew(ϕi) = 12Xi

In a similar way the field ϕ0 which was removed had a Gaussianprobability distribution dPold(ϕ0), just with another actionSold(ϕ0) = 12Xi(ϕ0 −φi)2(10)and an appropriate normalization factor Cold(φi), which again contains the expo-nential of a Gaussian form in φi’s. Assuming that the fields ϕ0, .

. .

, ϕ5 are selectedaccording to Pnew and Pold it is easy to enlarge the condition for detailed balancefor the change in geometry to include the additional change in field content.The geometrical moves fall in three classes (see e.g. [2] for details) of which wehave described one above.

A second class is one where two neighbouring 4-simplexesare removed and replaced by three new ones having in common a link (a 1-simplex),or the inverse move, where three 4-simplexes sharing a link are removed and replacedby two 4-simplexes being neighbours (i.e. sharing a 3-simplex).

Finally the thirdclass of moves is “self-dual”: three 4-simplexes sharing a triangle (a 2-simplex) arereplaced by three others, sharing a different triangle. In all cases one can easilywrite down dPnew and dPold as above and incorporate these probabilities in therequirement of detailed balance needed for performing the purely geometrical move.7

The total updating is now organized in the following way: A sweep over thelattice with an updating of geometry and the above described updating of fieldcontent is followed by a number of sweeps with the geometry fixed and ordinaryheatbath updating of the Gaussian fields.The actual number of such heatbathupdatings for each geometrical updating is chosen so that the fastest convergenceto equilibrium is achieved. For one gaussian field two heatbath updatings for eachgeometrical updating is usually sufficient as long as the geometry is highly connected.In the elongated phase up to 15 gaussian updatings were needed.

The number ofnecessary updatings per sweep increases with the number of Gaussian fields. For 4Gaussian fields 3 updatings per sweep were needed in the highly connected phase ofgravity.4Numerical results4.1Ising spins coupled to gravityPure 4d gravity has two phases and this fact is not changed by the coupling to asingle Ising spin.In the phase where the geometry is highly connected the spin system has a phasetransition.

In fig. 1 we show the absolute value of the magnetization|σ| = 1N4N4Xi=1σi(11)as a function of β for a value of κ2 for which the geometrical system is highlyconnected.

In fig.2 we show Binders cumulant defined byB(β) = 1 −13⟨σ4⟩⟨σ2⟩2(12)and it is seen that the data are consistent with a transition which is second orderor higher. We feel there is no reason to believe that the transition should be ofhigher than second order, since in this phase of the geometrical system the effectiveHausdorffdimension is quite high which should favour mean-field results.

In thephase with elongated geometry the situation is quite different. The magnetizationcurve well inside this phase is shown in fig.3.

There is only a gradual cross over to|σ| ≈1 for large β, and the cross over weakens (slightly) with increasing volume.This is in agreement with the measurements of the Hausdorffdimension, dH, in thisphase which seems to indicate that dH < 2.8

The phase diagram in the (κ2, β) plane is as it appears for a system consistingof 9K simplexes is shown in fig.4. It is in qualitative agreement with the phasediagram of 3d simplicial gravity coupled to Ising spins [19].

The shaded area reflectsthe uncertainty in the location of the transition line separating the two phasesof the geometrical system. This uncertainty is due to a discrepancy between theresults for κc2 arising when one uses different indicators for the change in geometry.One possible indicator is the Hausdorffdimension, dH, another one the correlator⟨R2⟩−⟨R⟩2.

The left boundary of the shaded area results from determining κc2 asthe value of κ2 at the peak of ⟨R2⟩−⟨R⟩2. The right boundary appears when κc2 isdefined as the value of κ2 for which there is a sudden change in Hausdorffdimension.While the left hand boundary is relatively easy to determine (Cf.

figure 7) the rightboundary is difficult to locate precisely due to large fluctuations in geometry andshould only be taken as a rough estimate. The fact that the two boundaries do notcoincide for the size of systems used here should be taken as a clear sign of finitesize effects.

A related phenomenon is seen in the numerical studies of 2d gravitycoupled to Ising spins, where the peak in the specific heat does not coincide withthe peak in the susceptibility due to finite size effects which seem to disappear onlyvery slowly when the size of the system is increased. The lines of phase transition(treating the shaded area as a “line”, which we expect it will be in the infinite volumelimit) divide the coupling constant plane into three regions: The one to the right ischaracterized by no magnetization and elongated geometry, the lower left region ischaracterized by no magnetization and highly connected geometry, while the upperleft corresponds to a magnetized phase and highly connected geometry.

It is difficultto determine the exact position of the bifurcation point since we have here both afluctuating geometry and large spin fluctuations. It is easy to understand that thetransition line separating different geometries will approach the value of κc2 for puregravity when β →∞and β →0.

In these limits the spin fluctuations decouplefrom gravity and the locations of the transition must agree with the one of pure 4dsimplicial gravity.In figure 5 we have shown the behaviour of the average curvature of our manifoldswhen we fix κ2 inside the highly connected phase fix and move vertically in thecoupling constant plane varying β. The value of κ2 is the same as in figure 1 andfigure 2.

The position of the peak in the average curvature exactly coincides withthe value of βc determined from the magnetization curve and the plot of Binderscumulant. This observation allows an easy and not so time consuming determinationof βc(κ2).

The the transition line β = βc(κ2) was determined using this idea. Wenote that this line shows little dependence on κ2.

The dependence of κc2(β) is more9

pronounced. The value of κc2 is smaller for the coupled system than for pure gravity.The shift in κc2 is largest when β = βc showing that the coupling between geometryand spins is indeed largest when the spin system is critical.

This is in agreementwith the intuition we have from the exactly solvable 2d Ising-gravity system. Thetransition line κ2 = κc2(β) shows that effectively the spin system pushes geometrytowards larger κ2 values.

The effect is strongest when β is close to βc(κ2). Onthe other hand we know that for large κ2 values the geometry is such that thesystem cannot be critical.This apparent contradiction seems to be generic forthe interaction between gravity and matter of the kind considered here.

This ishighlighted in a recent paper on multiple spin systems coupled to 2d gravity [32]. In2d the back-reaction of the spin system on gravity is also largest close to criticality,but is such that it counteracts its own criticality by trying to deform the geometryinto generic shapes where it cannot be critical (polymer-like geometries).

It seemsthat we are observing a similar phenomenon here in 4d.It is of course an interesting question whether the coupling between the spins andgravity changes the critical exponent of either of the systems as is the case in 2dimensions. However, since the critical exponents of the pure 4d gravity system areyet not known and since it has proven quite difficult to extract by numerical methodsthe critical exponents of the Ising spins coupled to 2d gravity, we have chosen here themore modest approach to look at the influence of the spin system on bulk geometricquantities like the average curvature.As explained in the introduction this hasspecial interest in relation to the scaling of gravity observables at the transitionbetween geometries.

We will return to this aspect after we have discussed briefly 4dgravity coupled to Gaussian fields.4.2Gaussian fields coupled to gravityIn the case of Gaussian fields we have, as explained above, no coupling constantto adjust. The fields will automatically be critical in the infinite volume limit.

Wehave considered up to four Gaussian fields coupled simultaneously to gravity and forthese systems we can make a statement similar to the one made for the Ising model:The two phases of geometry seem to survive the coupling to Gaussian matter. In fig.6 we have shown the expectation value ⟨φ2⟩of a single component of the Gaussianfield as a function of κ2.

We see a change in ⟨φ2⟩linked to the change in geometry.The value of ⟨φ2⟩increases when we enter into the elongated phase. In fact ⟨φ2⟩also has quite large fluctuations in this phase.10

4.3Behaviour of gravity observables coupled to matterIn the computer simulations we can clearly see the back-reaction of matter on thegeometry for a given choice of coupling constants. It is less obvious, however, thatthis back reaction of matter leads to anything but trivial changes.

Both for the cou-pling of Ising spins and Gaussian fields we still have two phases of the geometry: thehighly connected one and the very elongated one. As mentioned in the introductionone could hope that the inclusion of matter would improve the scaling of the curva-ture at the transition.

We have investigated this in the following way: As remarkedabove there are several indicators of the change in geometry. They result in slightlydifferent values of κc2.

We have chosen here to use the peak of ⟨R2⟩−⟨R⟩2 as anindicator of the transition, mainly because it is easier to identify than the changein Hausdorffdimension. The value of κc2 depends on the matter content as can beseen from figure 7.

In fig. 8 we have plotted the average curvature as a function ofthe distance ∆κ2 from κc2.

It is seen that there is no improvement in the scalingbehaviour of ⟨R⟩(κc2) as a function of the matter content, when we compare withthe situation in pure gravity. In fact the curves look remarkably insensitive to theinclusion of matter and one could at this point wonder whether the back-reactionof matter has any effect on the geometry except to introduce an effective κ2 whichdiffers from the bare parameter.

This is of course enough to explain the peak inthe average curvature observed in figure 5 and it also provides us with an explana-tion why the peak is more narrow for a 9K system than for a 4K system. This isdue to the fact that the change in average curvature across the phase transition ismore sudden for the larger system.

In fig. 7 we have shown ⟨R2⟩−⟨R⟩2 for variousmatter fields coupled to gravity.

We see that the peak grows with the number ofGaussian fields, indicating at least somewhat increased back-reaction with the num-ber of fields. Furthermore we note that the larger the number of Gaussian fields is,the more κc2 is shifted towards smaller values.

Hence systems with a large numberof Gaussian fields favour elongated geometries. The same phenomenon is knownfrom two dimensions where analytic considerations show that the path integral isdominated by elongated geometries when ng is large.

However, there is no indicationthat the presence of matter fields changes the nature of the phase transition of thegeometrical system.Let us comment here on a somewhat surprising feature of 4D simplicial gravity.As mentioned earlier the method of grand canonical simulation requires a finetuningof κ4 to its critical value, κc4. It appears that κc4 depends on κ2 in a universal way.In figure 9 we have shown κc4(κ2) for pure gravity, gravity coupled to Ising spins at11

β = βc and gravity coupled to 1 and 4 Gaussian fields respectively. In reference [10]4D simplicial gravity was simulated using the following actionS = κ4N4 −κ2N2 + hc24Xn2o(n2) c4 −o(n2)o(n2)!2(13)This corresponds to adding to the Einstein Hilbert action a typical higher derivativeterm (Cf.

equation (4)). We have shown also κc4(κ2) for this model when h = 10and h = 20.

For all the systems studied κc4(κ2) is a linear function with a slope ofapproximately 2.5.5DiscussionIt is clear that the numerical exploration of simplicial quantum gravity is still inits infancy. Finite size effects are not under control and it would be most desirableto be able to simulate larger systems.In principle it is possible and it will bepossible in the future.

But even on the small lattices used here one might revealinteresting aspects of the interaction between gravity and matter. Until now wehave only considered the simplest matter systems, spins and Gaussian fields, butnothing prevents us from considering the coupling to for instance non-abelian gaugefields.

It is also in principle possible to to define non-local observables like spin-spincorrelation functions as functions of geodesic distance (see i.e. [19] for a discussionin the case of 3d gravity) and explore their quantum averages.

In this paper we havenot tried to extract any critical exponents of such observables since the experiencefrom 3d is that it is not easy, and we decided in this first investigation to concentrateon bulk quantities.The main result of the simulations is that coupling of matter to discretized grav-ity seems not to influence the geometry in a profound way. Of course it is possiblethat critical indices change (as is the case in 2d gravity).

Our measurements arestill too poor to measure such subleading effects. As mentioned above an interestingeffect would be an improved scaling of the average curvature in the region wherethere is a transition in geometry.

We have not seen any such effect. The tenta-tive conclusion from these first numerical experiments is that matter fields (at leastof the kind we have considered here) will not add very much to our attempts tounderstand the basic structure of four-dimensional quantum gravity.12

References[1] M.E. Agishtein and A.A. Migdal, Mod.

Phys. Lett.

A7 (1992) 1039. [2] J. Ambjørn and J. Jurkiewicz, Phys.Lett B278 (1992) 42.

[3] H. Hamber, Nucl.Phys.B (Proc.Suppl.) 25A (1992) 150-175; H. Hamber, Phasesof simplicial quantum theory in four dimensions, UCI-Th-92-29.

[4] F. David, Nucl. Phys.

B 257 (1985) 45. [5] J. Ambjørn, B. Durhuus and J. Fr¨ohlich, Nucl.

Phys. B 257 (1985) 433.

[6] F. David, Nucl. Phys.

B257 (1985) 543. [7] V. A. Kazakov, I. K. Kostov and A.

A. Migdal, Phys. Lett.

157B (1985) 295. [8] S.

Varsted,UCSD-PhTh-92-03. [9] M.E.

Agishtein and A.A. Migdal, Nucl. Phys.

B385 (1992) 395. [10] J. Ambjørn, J. Jurkiewicz and C.F.

Kristjansen, NBI-HE-92-53 (to appear inNucl. Phys.

B). [11] B. Br¨ugmann, Non-uniform Measure in Four-Dimensional simplicial QuamtumGravity, SU-GP-92/9-1.

[12] B. Br¨ugmann and E. Marinari, 4d Simplicial Quantum Gravity with a Non-Trivial Measure, SU-GP-92/9-2. [13] I. Antoniadis, P.O.

Mazur and E. Mottola, Scaling behavior of quantum four-geometries, Preprint CPTH-A214.1292[14] V. Knizhnik, A. Polyakov and A. Zamolodchikov, Mod. Phys.

Lett. A3 (1988)819.

[15] V.A. Kazakov, Phys.

Lett 119A (1986) 140. [16] D. Boulatov and V.A.

Kazakov, Phys. Lett.

184B (1987) 247. [17] C. Baillie, Phys.

Rev. D46, (1992) 2480.

[18] R. Renken, S. Catterall and J. Kogut, ILL-TH-92-7. [19] J. Ambjørn, Z. Burda, J. Jurkiewicz and C.F.

Kristjansen, Phys. Lett 297B(1992) 253.

[20] J. Ambjørn, D. Boulatov, A. Krzywicki and S. Varsted, Phys.Lett B276 (1992)432. [21] J. Ambjørn and S. Varsted, Nucl.Phys.

B373 (1992) 557.13

[22] J. Ambjørn, B. Durhuus, J. Fr¨ohlich and P. Orland, Nucl.Phys. B270 (1986)457.

[23] J. Ambjørn, B. Durhuus, J. Fr¨ohlich, Nucl. Phys.

B275 (1986) 161. [24] D. Boulatov, V. A. Kazakov, I. K. Kostov and A.

A. Migdal, Phys.Lett. 174B(1986) 87.

[25] J. Jurkiewicz, A. Krzywicki and B. Petersson, Phys.Lett. 168B (1986) 273; 177B(1986) 89.

[26] J. Ambjørn, Ph. De Forcrand, F. Koukiou and D. Petritis, Phys.Lett 197B(1987) 548.

[27] U. Wolff, Phys.Rev.Lett. 62 (1989) 361.

[28] Z.Burda, J.Jurkiewicz and L.K¨arkk¨ainen, Comput. Phys.

Commun. 70 (1992)510.

[29] C.F. Baillie and D.A.

Johnston, Phys. Lett.

B286 (1992) 44;[30] C.F. Baillie and D.A.

Johnston, Mod. Phys.

Lett. A7 (1992) 1519.

[31] S. M. Catterall, J.B. Kogut and R.L. Renken, Phys.

Lett B292 (1992) 277;Phys. Rev.

D45 (1992) 2957. [32] J. Ambjørn, B. Durhuus, T. Jonsson and G. Thorleifsson, NBI-HE-92-35 (toappear in Nucl.

Phys. B)14

Figure CaptionsFig. 1 The absolute value of the magnetization, as defined by (11), as a function ofβ for κ2 = 0.9, i.e.

in the phase with a highly connected geometry. The circlescorrespond to a volume N4 = 4000, the triangles to N4 = 9000.Fig.

2 Binder’s cumulant (12) for κ2 = 0.9 and three volumes: N4 = 4000 (▽),N4 = 6000 (✷) and N4 = 9000 (⃝). The shape corresponds to a transition ofsecond or higher order and the point of intersection to βc(N4 = ∞).Fig.

3 The absolute value of the magnetization, as defined by (11), as a function of βfor κ2 = 1.3, i.e. in the phase with elongated geometry.

The circles correspondto a volume N4 = 4000, the triangles to N4 = 9000.Fig. 4 The phase diagram in the (κ2, β) plane as it appears when N4 = 9000.

Asdiscussed in the text there are reasons to believe that part of the diagram isdistorted by finite size effects and that the in the infinite volume the shadedregion will be replaced by the dashed line.Fig. 5 The effect on the curvature ⟨R⟩−⟨R⟩0 (where ⟨R⟩is defined by (3)) whenwe are in the phase with a large Hausdorffdimension and change β. Thevalue of κ2 = 0.9 and the circles correspond to N4 = 4000 while the trianglescorrespond to N4 = 9000.

⟨R⟩0 denotes the average curvature in the case ofpure gravity (it differs slightly for N4 = 4000 and N4 = 9000 due to finite sizeeffects).Fig. 6 The change in ⟨φ2⟩(a single component field) as a function of κ2 for N4 = 4000.Fig.

7 ⟨R2⟩−⟨R⟩2 for a different matter fields as a function of ∆κ2. Pure gravity(▽), gravity + Ising at βc, (+), gravity + 1 Gaussian field (⃝) and gravity+ 4 Gaussian fields (✷).

(The observables ⟨R⟩and ⟨R2⟩are defined in (3)and (4) respectively.)Fig. 8 ⟨R⟩as a function of ∆κ2 for different matter content.

Pure gravity (▽), gravity+ Ising at βc, (+), gravity + 1 Gaussian field (⃝) and gravity + 4 Gaussianfields (✷).Fig. 9 κc4 as a function of κ2 for different systems.

Pure gravity (×), gravity + Isingat β = βc (⃝), gravity + 1 Gaussian field (✷), gravity + 4 Gaussian fields(△), gravity with higher derivative term for h = 10 (•) and gravity with higherderivative term for h = 20 (+).15

---

출처: arXiv:9303.042 • 원문 보기