Causal Dynamical Triangulations

ENRAGE Newsletter Number 2, June 2007

Causal Dynamical Triangulations, for short tational field. This leads to the prediction that space- CDT, is a new approach to the nonperturba- time geometry has a highly non-trivial microstructure tive quantization of gravity. As in the case of at extremely small scales proportional to the Planck p −3 −35 Euclidean dynamical triangulation approaches, length, lp = ¯hGN c ≈1.616×10 m. CDT provides a regularization of the gravita- There are however obvious problems in construct- tional path integral through a sum over piece- ing a quantum theory of general relativity. It has wise linear geometries where the edge length already been shown in the seventies by ’t Hooft and of the individual building blocks serves as an Veltman that perturbative quantum gravity is non- ultraviolet cutoff. However, in the latter the renormalizable in four dimensions 1. This does not principle of microcausality has been imple- mean that it is impossible to find a predictive theory mented in the path integral by only allow- of quantum general relativity. There are good indica- ing for geometries which have a definite causal tions that one can define a theory of quantum general structure even at the smallest scales. This lead relativity nonperturbatively 2,3. to considerable successes over the Euclidean There are several nonperturbative approaches to model. This summary reviews sum of the re- quantum general relativity of which many are stud- cent results obtained in the case where space- ied by researchers of the ENRAGE network. Some time is two-dimensional and analytical results of those attempts suggest that the ultraviolet diver- can be obtained. gences can be resolved by the existence of a mini- mal length scale, commonly expressed in terms of the Stefan Zohren, Imperial College London Planck length lp. A famous example is loop quantum gravity 4,5,6; in this canonical quantization program Why quantum gravity? the discrete spectra of area and volume operators are interpreted as evidence for fundamental discreteness. Quantum field theory has proven to be a marvelously Other approaches, such as four-dimensional spin-foam successful way to describe three of the four funda- models 7 or causal set theory 8,9, postulate fundamen- mental forces of nature. For gravity however we do tal discreteness from the outset. Unfortunately, most not have a well-defined predictive quantum field theo- of these quantization programs still have problems in retic description yet, but we do have a very successful recovering a sensible classical limit. The latest suc- classical field theoretic description in the form of Ein- cesses in the approach of causal set theory will be pre- stein’s general relativity. Since the other forces are sented in one of the forthcoming ENRAGE newslet- well described by quantized field theories, it seems ters. natural that there also exists a quantum theory of There are also nonperturbative approaches which Einstein’s general relativity. do not introduce a fundamental discreteness scale Another reason for believing that such a theory of from the outset. One example is the exact renor- quantum gravity should exist is the fact that grav- malization group flow method for Euclidean quan- ity is universal in the sense that it couples to all tum gravity in the continuum 3. Another attempt is forms of energy. Hence the energy fluctuations at Causal Dynamical Triangulations (CDT), a covariant small distances due to Heisenberg’s uncertainty re- path integral formulation, in which Lorentzian quan- lations induce also quantum fluctuations in the gravi- tum gravity is obtained as a continuum limit of a su- ENRAGE Newsletter Number 2, June 2007 perposition of simplicial space-time geometries 2. individual triangulations correspond to the individual histories of the path integral, which in general do not resemble smooth manifolds. The CDT approach The foliated structure of the triangulations intro- duces a natural notion of discrete global time t given The CDT program is a quantization scheme for gen- by the label of the successive spatial slices. Note that eral relativity where no supersymmetry or ad hoc one has to be careful in attaching a physical mean- fundamental discreteness is assumed from the out- ing to this global time as we discuss later on. The set. The program is meant to give a rigorous nonper- clear distinction between space-like edges and time- turbative definition of a path integral over all causal like edges enables us to define a Wick rotation on geometries (by this we mean the equivalence class each causal triangulation by analytic continuation of of a Lorentzian metric modulo its diffeomorphisms) α 7→ −α. It is important to realize that the set of weighted by the Einstein-Hilbert action Euclidean triangulations one obtains after the Wick Z rotation is strictly smaller than the set of all Euclidean iS [gµν ] Z = D[gµν ]e EH . (1) triangulations.

From lattice QCD we know that discrete methods are a powerful tool to investigate nonperturbative ef- fects in quantum field theory. In CDT one uses a Recent results in 3+1 dimensions specific discretization similar to Regge calculus where the geometry itself is encoded in a simplicial lattice. Before describing the analytic results in 1+1 dimen- The advantage of using this type of discretization is sions let us mention some of the recent exciting successes of CDT in 3+1 dimensions as a motivation for that one is automatically working with gauge invari- 11 ant degrees of freedom. There is no need to introduce the CDT approach to quantum gravity (see also for coordinates in the construction 10. There is however a general overview). a crucial difference between Regge calculus and dy- In absence of an analytic solution for the 3+1 di- namical triangulations. In the first the dynamics is mensional model, one uses Monte Carlo simulations encoded in the variation of the edge lengths whereas to obtain numerical results. A very important non- in dynamical triangulations the edge lengths are fixed trivial test for every nonperturbative formulation of but the dynamics is encoded in the gluing of the sim- quantum gravity is whether it can reproduce a sen- plicial building blocks. sible classical limit at macroscopic scales. The nu- In the explicit construction of CDT the path in- merical results indicate that the scaling behavior of tegral over all causal geometries is written as a sum the spatial volume as a function of space-time vol- over all causal triangulations T weighted by the Regge ume is that of a four-dimensional universe at large action (the simplicial analog of the Einstein-Hilbert scales, a first indication of sensible classical behav- 12 action including a cosmological constant λ), ior . Moreover, after integrating out all dynamical variables apart from the spatial volume as a function X 1 of proper time, one can derive the scale factor whose Z(λ, G ) = eiSRegge , (2) N C dynamics is described by the simplest minisuperspace causal T T model used in quantum cosmology 13. where CT is the discrete symmetry factor of the tri- Having passed the first consistency checks regard- angulation T . In contrast to previous attempts of ing the macroscopical structure of space-time it is dynamical triangulations the triangulations appear- very interesting what predictions one can make for the ing in causal dynamical triangulations have a definite quantum nature of the microstructure of space-time. foliated structure. In this type of triangulations each One important observable which has been measured (d−1)-dimensional spatial slice is realized as an Eu- is the spectral dimension of space-time which is the clidean triangulation whose simplicial building blocks dimension a diffusion process would feel on the space- 2 2 have all squared edge lengths given by ls = a . The time ensemble. Surprisingly, this quantity depends on successive spatial slices are connected by time-like the scale at which it is measured. More precisely, one 2 2 edges of squared edge lengths lt = −αa with α > 0, observes a dimensional reduction from four at large such that all building blocks in T are d-simplices (see scales to two at small scales within measurement ac- Fig. 1 for an illustration in 1+1 dimensions). Here curacy 14. This gives an indication that nonperturba- the parameter a is a cut-off length that one takes to tive quantum gravity defined through CDT provides zero in order to obtain the continuum limit of the reg- an effective ultraviolet cut-off through a dynamical ularized path integral (2). Note that in this limit the dimensional reduction of space-time.

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t + 2

t + 1 Figure 1: A section of a se- αa2 quence [t, t + 2] of two space- − time strips of a triangulated two- t dimensional spacetime contribut- 2 ing to the regularized path inte- +a gral.

Analytic results in 1+1 dimen- where λ is the bare cosmological constant and N(T ) sions the number of triangles in the triangulation. In two dimensions the sum in (4) can be evaluated and one To better understand the methods used in CDT it is obtains the continuum limit after a suitable choice of useful to look in more detail at the 1+1 dimensional renormalization, yielding the continuum propagator model, since it is exactly solvable 15. Although two-dimensional quantum gravity does not have any propagating degrees of freedom it is √ √ e− 2Λ(Lin+Lout) coth( 2ΛT ) a fertile playground to study certain aspects of dif- GΛ(Lin,Lout; T ) = √ × feomorphism invariant theories. Among the issues sinh( 2ΛT ) that have been addressed within the two-dimensional s √ ! 2Λ 2ΛLinLout framework are the inclusion of a sum over topologies 16 × I1 2 √ , (5) LinLout sinh( 2ΛT ) and the emergence of a background geometry purely from quantum ﬂuctuations 17 which we want to discuss in the following.

Recall that the Einstein-Hilbert action in two di- where I1(x) denotes the modified Bessel function of mensions is given by the first kind and Λ, Li and T are the continuum Z counterparts of λ, li and t. Physically this solution 2 p SEH[g] = d x | det g| Λ − 2πKχ(M) (3) can be interpreted as a fluctuating two-dimensional M “universe” (Fig. 2), where the average spatial length and its fluctuations are determined by the cosmolog- where K = G−1 is the inverse Newton’s constant, √ N ical constant, hLi ∼ h∆Li ∼ 1/ Λ. This means that χ(M)=2−2g the Euler characteristic of the manifold the two-dimensional “universe” is purely governed by M and g the genus of M. Therefore, for fixed spatial quantum fluctuations and we have no notion of a semi- topology, the curvature term in the action contributes classical background. This situation changes when we just a constant phase factor to the path integral. If will discuss the transition to non-compact space-times one allows for topology changes this term becomes im- later on in the text. portant for the quantum dynamics as we will see in the next section. In this section we fix the topology 1 Remarkably, the continuum propagator (5) agrees of space-time to be R×S . The most natural thing with the result of the propagator obtained from a con- to calculate is the propagator from an initial geom- tinuum calculation in the proper-time gauge of 1+1 etry of length lin to a final geometry of length lout dimensional pure gravity 18. This indicates that the in time t. Using the discrete analog of the Einstein- above choice of global time is similar to the one used in Hilbert action (3) one can write down the propagator the proper-time gauge. Another interesting question after Wick rotation as the path integral (2) for fixed is whether the result obtained in (5) is independent boundaries lin and lout, of the choice of foliation. There are good indications X − λ a2 N(T ) that due to the broad universality class of this model Gλ(lin, lout; t) = e , (4) one also obtains the same dynamics (5) for different causal T: choices of time slicing 19. lin→lout

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and we introduce a model with inﬁnitesimal wormholes where one can perform the sum over topologies explicitly 20,21,16.

We deﬁne the sum over topologies in (6) by performing surgery moves directly on the triangulations to obtain regularized versions of higher-genus manifolds 20,21. For the construction of these moves let us concentrate on a single space-time strip of topology [0, 1] × S1 and height ∆t = 1 as illustrated in Fig. 3. The inﬁnitesimal wormholes can be con- structed by identifying two of the time-like edges and subsequently cutting open the geometry along this edge. By applying this procedure repeatedly and obeying certain causality constraints 20,21, more and more wormholes can be created.

To obtain the dynamics of the model it is suﬃ- cient to analyze the one-step propagator. Including the topological term in the action and performing the Wick rotation gives

Figure 2: A typical two-dimensional Lorentzian space- time. The compactified direction shows the spatial X 2 G (l , l ; t = 1) = e−λ a N−2κ g(T ), (7) hypersurfaces of length L and the vertical axis labels λ,κ in out causal T: time T . Technically, the picture was generated by lin→lout a Monte Carlo simulation, where a total volume of N = 18816 triangles and a total time of t = 168 steps was used. Further, initial and final boundary has been where the sum is taken over all possible triangula- identified. tions T of height t = 1 with fixed initial boundary lin and final boundary lout, but arbitrary genus Including topology changes 0 ≤ g(T ) ≤ [N/2] (here N = lin +lout is the number of triangles in T , which coincides with the number A recurring question in the history of quantum gravity of time-like edges). Further, λ is the bare cosmologi- approaches is whether one should allow for topology cal constant and κ is the bare inverse Newton’s con- changes of space-time. In terms of path integrals this stant. The sum over all possible triangulations with translates into whether one should include the sum arbitrary genus can be performed unambiguously and over topologies in the path integral one can obtain the continuum limit after a suitable choice of renormalization and double scaling limit for X Z λ and κ, yielding the continuum propagator 16 of the Z = D[g ]eiSEH[gµν ]. (6) µν form of (5), where the renormalized cosmological con- topol. stant Λ is replaced by the effective cosmological constant Λ = Λ(1 − e−4π/GN ) which both depends on There have been several attempts to solve the path eff the renormalized cosmological constant and Newton’s integral (6) in the setting of Euclidean quantum grav- constant. ity. The big problem however that becomes apparent even in the simplest case of two dimensions is that the In addition to the known geometrical observables full sum over topologies cannot be uniquely defined this model possesses a new type of topological observ- nonperturbatively, since the sum over genera is badly able, namely, the density of wormholes, which can be divergent. One of the main differences between this calculated to give a finite expression approach and the one we propose here is that we re- strict the class of topology changes by means of impos- ing an (almost everywhere) causal structure. In the hN i 1 following we show that this restriction on the topol- n = g = Λ. (8) ogy changes leads to a better defined path integral hV i 4π e GN − 1

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The emergence of background geometry

(i) (ii) In the previous sections we described the quantum wormhole geometry of the 1+1 dimensional universe in the case where the geometry was compact. We have seen that under this conditions the geometry was purely gov- p t+1 erned by quantum fluctuations. Therefore, there was no sensible notion of a semi-classical background. In pt the following we want to describe the transition in which the compact geometries become non-compact. (iii) We will see that in this case we find the emergence of a semiclassical background dressed with small quantum fluctuations. 17 Figure 3: Construction of a wormhole: starting from To determine the background geometry of the 1+1 1 a space-time strip of topology [0, 1]×S as in the pure dimensional universe we calculate the average spatial CDT model (i), one identifies two time-like edged (ii) length at proper time t ∈ [0,T ] and then cuts open the geometry perpendicular to this line (iii). The two resulting saddle points at time 1 hL(t)iX,Y,T = × t and t+1 are labeled with pt and pt+1. GΛ(X,Y ; T ) Z ∞ × dLGΛ(X,L; t) LGΛ(L, Y ; T − t). (9) 0 Here X and Y are the boundary cosmological constants which are the conjugate variables of the boundary lengths L and L . The continuum propagator It is useful to reinterpret the physical system in in out G (X,Y ; T ) with respect to the boundary cosmolog- terms of its physical quantities, namely the cosmolog- Λ ical constants can be obtained form G (L ,L ; T ), ical constant Λ and the density of wormholes in units Λ in out i.e. (5), by inverse Laplace transformation. of Λ, i.e. η = n . These two quantities can be seen to Λ Evaluating the average length at the boundary set the physical scales of the system. Whereas in the t = T and taking the limit T → ∞ gives case without topology changes (5) there was only√ one scale, namely, the cosmological scale hLi ∼ 1/ Λ, in 1 lim hL(T )iX,Y,T = √ . (10) the case with topology changes there is in addition the T →∞ Y + Λ relative scale η between cosmological and topological fluctuations. Both together define the√ effective fluc- Interestingly,√ one observes that there is a special value tuations in length through hLi ∼ 1/ Λeff , where we Y =− Λ of the boundary cosmological constant for can now write Λeff = Λ/(1+η). One can observe that which the boundary length diverges and the geome- the presence of wormholes in spacetime leads to a de- try becomes non-compact. Using this critical value for crease in the “effective” cosmological constant Λeff . the boundary cosmological constant Y one can obtain This connects nicely to former attempts to derive a the boundary length for finite T mechanism, the so-called Coleman’s mechanism, to c √ explain the smallness of the cosmological constant in L (T ) = hL(T )iX,Y =− Λ;T a formal Euclidean path integral formulation of four- 1 1 = √ √ . (11) dimensional quantum gravity in the continuum with Λ coth ΛT − 1 the presence of infinitesimal wormholes 22. The wormholes considered in those models resemble those of our Instead of using boundary cosmological constants one toy model in that both are non-local identifications can also fix the spatial length of the boundaries. Us- of the spacetime geometry of infinitesimal size. The ing the continuum propagator GΛ(L1,L2; T ) we can counting of the wormholes considered here is of course evaluate the average spatial length hL(t)iLin,Lout,T for different, since we are working in a genuinely causal fixed lengths Lin and Lout of the boundary loops. and background independent setup which enables us In the following we want to investigate the quan- to actually perform the sum over topologies explicitly, tum geometry in the case where it becomes non- without assuming any information on a background compact. Therefore we set the boundary length at manifold. t = T to the critical value Lc(T ) as defined in (11)

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and for simplicity we shrink the spatial geometry at Conclusion t = 0 to a point. In the limit T → ∞ one obtains the average length of the spatial geometry at proper time We have presented a summary of some of the recent t ∈ [0,T ] results obtained within the framework of CDT in two spacetime dimensions. Among those were the inclusion of a sum over topologies and the emergence of a hL(t)i ≡ lim hL(t)iL =0,L =Lc(T ),T T →∞ in out background geometry purely from quantum fluctua- 1 √ = √ sinh(2 Λt). (12) tions. In particular, the possibility of obtaining an- Λ alytical results in two spacetime dimensions provides us with a better understanding of some of the phe- Due to the fact that L and T are defined from the con- nomena which have been numerically observed in the tinuum limit of a simplicial geometry there is a rela- four-dimensional context. The reader who wants to tive constant of proportionality that can only be fixed learn more about those recent successes in four di- by comparing with continuum calculations 18 yielding mensions is referred to the following nontechnical in- 11 Lcont(t) = πhL(t)i. From this result the metric for troduction . the background geometry is readily obtained, Bibliography L2 ds2 = dt2 + cont dθ2 4π2 [1] G. ’t Hooft and M. J. G. Veltman, “One loop divergencies in √ the theory of gravitation,” Annales Poincare Phys. Theor. sinh2(2 Λt) = dt2 + dθ2. (13) A 20, 69 (1974). 4Λ [2] J. Ambjørn, J. Jurkiewicz and R. Loll, “Reconstruct- ing the universe,” Phys. Rev. D 72, 064014 (2005), This is nothing but the metric of the Poincaré disc [arXiv:hep-th/0505154]. which can be seen as a Wick rotated version of AdS2 with constant negative curvature R = −8Λ. [3] O. Lauscher and M. Reuter, “Asymptotic safety in quantum Einstein gravity: Nonperturbative renormalizability To better understand the quantum nature of the and fractal spacetime structure,” [arXiv:hep-th/0511260]. geometry it is useful to compute the fluctuations of the spatial length. From expressions analogous to (9) [4] T. Thiemann, “Lectures on loop quantum gravity,” in Quantum gravity: From theory to experimental one can determine the relative fluctuations search, D. Giulini, C. Kiefer, and C. Lämmerzahl, eds., vol. 631 of Lect. Notes Phys., pp. 41–135. 2003. ∆L(t) phL2(t)i − hL(t)i2 √ [arXiv:gr-qc/0210094]. ≡ ∼ e− Λt. (14) hL(t)i hL(t)i [5] C. Rovelli, Quantum gravity. Cambridge University Press, Cambridge, UK, 2004.

Surprisingly, the fluctuations of the spatial geometry [6] H. Nicolai, K. Peeters, and M. Zamaklar, “Loop quantum −1/2 become exponentially small for t Λ . Conclud- gravity: An outside view,” [arXiv:hep-th/0501114]. ing from (13) and (14), one can view the quantum geometry as a version of Wick rotated AdS dressed [7] A. Perez, “Spin foam models for quantum gravity,” Class. 2 Quant. Grav. 20, R43 (2003), [arXiv:gr-qc/0301113]. with small quantum fluctuations. We have shown that in 2D quantum gravity de- [8] L. Bombelli, J.-H. Lee, D. Meyer, and R. Sorkin, “Space- time as a causal set,” Phys. Rev. Lett. 59 (1987) 521. fined through CDT there is a transition from compact geometry to non-compact AdS2-like geometry [9] J. Henson in Approaches to Quantum Gravity: Towards a for a special value of the boundary cosmological con- New Understanding of Space and Time, ed. D. Oriti, Cam- stant. This phenomenon is similar to the Euclidean bridge University Press, (2006), [arXiv:gr-qc/0601121]. case where non-compact ZZ-branes appear in a tran- [10] T. Regge, “General Relativity Without Coordinates,” sition from compact 2D geometries in Liouville quan- Nuovo Cim. 19, 558 (1961). 23 tum gravity. A surprising feature of the CDT re- [11] J. Ambjorn, J. Jurkiewicz and R. Loll, “The uni- sult is that the fluctuations become exponentially verse from scratch,” Contemp. Phys. 47 ,10 (2006), small which enables us to interpret the emerging AdS2 [arXiv:hep-th/0509010]. spacetime as a genuine semiclassical background. It [12] J. Ambjørn, J. Jurkiewicz and R. Loll, “Emergence of a is interesting that similar results have been reported 4D world from causal quantum gravity,” Phys. Rev. Lett. in four-dimensional CDT where numerical simulations 93, 131301 (2004), [arXiv:hep-th/0404156]. indicate the emergence of a semi-classical background [13] J. Ambjørn, J. Jurkiewicz and R. Loll, “Semiclassical uni- from a nonperturbative and background-independent verse from first principles,” Phys. Lett. B 607, 205 (2005), path integral. 2 [arXiv:hep-th/0411152].

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[14] J. Ambjørn, J. Jurkiewicz and R. Loll, “Spectral dimen- [20] R. Loll and W. Westra, “Sum over topologies and double- sion of the universe,” Phys. Rev. Lett. 95, 171301 (2005) scaling limit in 2D Lorentzian quantum gravity,” Class. [arXiv:hep-th/0505113]. Quant. Grav. 23, 465 (2006), [arXiv:hep-th/0306183].

[15] J. Ambjørn and R. Loll, “Non-perturbative Lorentzian [21] R. Loll and W. Westra, “Space-time foam in 2D and the quantum gravity, causality and topology change,” Nucl. sum over topologies,” Acta Phys. Polon. B 34, 4997 (2003), Phys. B 536, 407 (1998), [arXiv:hep-th/9805108]. [arXiv:hep-th/0309012].

[16] R. Loll, W. Westra and S. Zohren, “Taming the cos- [22] S. R. Coleman, “Why there is nothing rather than some- mological constant in 2D causal quantum gravity with thing: A theory of the cosmological constant,” Nucl. Phys. topology change,” Nucl. Phys. B 751, 419 (2006), B 310, 643 (1988). [arXiv:hep-th/0507012]. [23] J. Ambjørn, S. Arianos, J. A. Gesser and S. Kawamoto, [17] J. Ambjørn, R. Janik, W. Westra, and S. Zohren, The geometry of ZZ-branes, Phys. Lett. B 599, 306 (2004), “The emergence of background geometry from quan- [arXiv:hep-th/0406108]. tum ﬂuctuations,” Phys. Lett. B641 (2006) 94–98, [arXiv:gr-qc/0607013]. Stefan Zohren [18] R. Nakayama, “2-D quantum gravity in the proper time gauge,” Phys. Lett. B 325, 347 (1994), Institute for Theoretical Physics [arXiv:hep-th/9312158]. Imperial College London Prince Consort Road [19] M. Arnsdorf, “Relating covariant and canonical approaches to triangulated models of quantum gravity,” Class. London, SW7 2AZ, UK Quant. Grav. 19, 1065 (2002), [arXiv:gr-qc/0110026]. [email protected]