Properties of dynamical fractal geometries in the model of Causal Dynamical Triangulations
PProperties of dynamical fractal geometries in the modelof Causal Dynamical Triangulations
J. Ambjorn a,b , Z. Drogosz c , A. Görlich c , J. Jurkiewicz c a The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100Copenhagen Ø, Denmark b IMAPP, Radboud University, Nijmegen, PO Box 9010, the Netherlands c Institute of Theoretical Physics, Jagiellonian University, Lojasiewicza 11, 30-348 Krakow,Poland
Abstract
We investigate the geometry of a quantum universe with the topology of thefour-torus. The study of non-contractible geodesic loops reveals that a typicalquantum geometry consists of a small semi-classical toroidal bulk part, dressedwith many outgrowths, which contain most of the four-volume and which havealmost spherical topologies, but nevertheless are quite fractal.
Keywords: quantum gravity, Causal Dynamical Triangulations, Monte Carlosimulations, topological observables
1. Introduction
Causal Dynamical Triangulations (CDT) is a model that attempts to applymethods of Quantum Field Theory in the context of a quantum model of geo-metric degrees of freedom . As will be described below, the model comes with a(proper) time, whereas the description of the geometries in the spatial directionsis genuinely coordinate independent. The existence of the time coordinate hasbeen instrumental for the construction of an effective mini-superspace action ofthe quantum theory, where we have integrated over the spatial geometries. Inparticular, it allowed us to talk about the emergence of a semi-classical mini-superspace geometry, as well as quantum fluctuations thereof [15, 16, 17, 18, 19].There is no reason not to expect a similar emergence of geometry in the spatialdirections. However, bearing in mind the importance of the proper time coor-dinate in our analysis of the mini-superspace geometry, it might be preferableto reintroduce some aspects of coordinates in the spatial directions, too. In Reviews of the model can be found in [1, 2]. The main idea is to have a lattice modelof quantum gravity where one in a non-perturbative way can test the idea of asymptoticsafety [3, 4, 5, 6, 7, 8]. Models of Dynamical triangulations (DT) were earlier attempts in thisdirection [9, 10, 11], which however did not work, but see [12, 13, 14] for recent attempts torevive that class of lattice models.
Preprint submitted to Elsevier July 28, 2020 a r X i v : . [ h e p - t h ] J u l igure 1: Left: illustration of a torus with outgrowths. The blue and red lines represent twonon-equivalent and non-contractible loops. The green loop is the shortest loop passing throughthe green point in the same direction as the blue line. Right: embedding of a triangulationof the two-torus consisting of 150000 triangles into the Euclidean plane (picture from [21]).Shown in red is the shortest non-contractible loop. some sense this is against the spirit of General Relativity which is coordinateindependent, but coordinates can be very useful.In our recent paper [20] we discussed one possibility of reintroducing co-ordinates in CDT in the spatial directions. In many of the former studies offour-dimensional CDT, the spatial topology was chosen to be that of a three-sphere S , and, given a spatial geometry as it appears in the path integral,we know of no simple way of reintroducing useful spatial coordinates in thatcase. However, in [20] the spatial topology of the Universe was chosen to bethat of a three-torus T . A d -dimensional manifold with a toroidal topologycan be viewed as consisting of an elementary cell, which is periodically repeatedinfinitely many times in all d directions. Although the choice of the elementarycell is not unique, the possibility of introducing such an object enables the useof its boundaries as a reference frame, with respect to which a Cartesian-likesystem of coordinates determined by the geodesic distance to the boundariesmay be constructed. One conclusion drawn from the analysis in [20] was thatthe geometry of a typical triangulation which appears in the CDT path integralis surprisingly fractal. Before trying to extract any emergent spatial geometryfrom such triangulations it is thus important to understand the spatial geome-try of a typical quantum configuration better. For that purpose we have foundit advantageous to use topological observables: closed non-contractible geodesicloops, connecting the same geometric object in different copies of the elementarycell. The distribution of the length of shortest loops with a given set of wind-ing numbers passing through particular elements of geometry yields informationabout geometric structures, and this kind of analysis has been used successfullyin the study of two-dimensional Euclidean quantum gravity [21].The left part of Fig. 1 provides a two-dimensional illustration of what we2re looking for in the case of the higher dimensional tori. We imagine that wehave an underlying toroidal structure, but there can be many outgrowths, whichcan be viewed as quantum fluctuations of an underlying “semi-classical” toroidalstructure. A point in an outgrowth will have a long non-contractible geodesicloop passing through it, while a point on the “semi-classical” toroidal part willhave a short non-contractible geodesic loop. In this way one can map out thegeometry of the toroidal universe in considerable detail, as will be describedbelow. An example point in an outgrowth is marked with a green dot in theleft part of Fig. 1. and the green line is a non-contractible geodesic loop for thispoint.In the right part of Fig. 1 we have shown how a two-dimensional toroidalquantum configuration looks. The configuration is a two-dimensional triangula-tion made of 150000 equilateral triangles, generated by Monte Carlo simulationsof two-dimensional Euclidean quantum gravity. By a conformal mapping thetriangulation can be mapped to an elementary cell in the plane. What is shownis a piecewise-linear approximation to this mapping (plus an affine mapping tomake it a square). The figure illustrates how such a quantum configuration con-sists of mountains (outgrowths) and valleys . By far the most two-volume (thegreatest number of triangles) is contained in the outgrowths, as can readily beseen from the picture. In four-dimensional CDT, we consider paths that connectcenters of simplices, i.e., which consist of edges of the dual triangulation (seeSec. 2). In the two-dimensional case, the red line shown in the right side ofFig. 1 consists of links of the direct lattice. The picture shows quite preciselythe fractal structure of two-dimensional quantum gravity. It is known that theHausdorff dimension of spacetime in two-dimensional quantum gravity is four and not two, as one might perhaps naively expect [22, 23, 24]. On a regular torusconsisting of N triangles one would expect a shortest loop of length approxi-mately N / links. However, here we see that the length is much closer to N / .In particular, this implies that the number of triangles in the valleys scale as N / , and not proportionally to N . The area of the valleys will thus disappearin an N → ∞ limit where the continuum area V ∝ N a is kept fixed, a beingthe length of a link in the triangulation before it was projected onto the plane.Therefore, in the two-dimensional case the valleys are not semi-classical, but aquantum phenomenon. We expect the situation to be different in the case of afour-dimensional CDT torus, the reason being that the Hausdorff dimension ofa typical CDT configuration is four, i.e. the same as the canonical dimension ofthe spacetime. We might then have a picture where the valleys of T constitutea semi-classical configuration which can act as a starting point for a descriptionof a semi-classical spatial geometry. This is one of the points we will investigatein this article.The rest of the article is organized as follows: in Sec. 2 we shortly definethe CDT model of quantum gravity, in order to fix the notation (we refer to[1, 2] for more detailed definitions). In Sec. 3 we define certain characteris-tics which are special for spacetimes with toroidal topologies. Sec. 4 describeshow the Monte Carlo simulations are performed, whereas Sec. 5 reports on themeasurements of the shortest loops of winding number one. In Sec. 6 these3easurements are generalized to loops with higher winding numbers. In Sec. 7we generalize even the possibilities of higher winding numbers, acknowledgingthe fact that our winding numbers are depending of our chosen reference frameand that a true geometric winding can be any linear combination of our label-ing of windings. Sec. 8 discusses if simplices in the outgrowths and simplices inthe valleys have different geometric neighborhood, possibly signifying that thevalleys can be viewed as semi-classical, while the outgrowths might be viewedas quantum fluctuations. Finally, Sec. 9 contains a discussion of the results andour conclusions.
2. The model
The basic idea in CDT is to calculate the quantum amplitude of the tran-sition between two physical states. The amplitude is defined as a path integralover field configurations, which in this case are spacetime geometries, Z = (cid:90) D [ g µν ] e iS EH [ g µν ] . (1) S EH is the Einstein-Hilbert action S EH [ g µν ] = 116 πG (cid:90) M . x (cid:112) − det g ( R − , (2)where R is the scalar curvature and Λ is the cosmological constant. This ex-pression is formal and requires regularization and a precise definition of boththe integration measure over g µν and the domain of integration over spacetimes.In CDT it is assumed that we will take into account only spacetimes that ad-mit a global time foliation: M = Σ × I . The term causality in the context ofthe model means that the topology of space Σ is preserved in time evolution.An additional assumption is that the spatial topology of the Universe is closed.Corresponding to I there is an initial and a final global time for the geome-tries considered, and the amplitude (1) is the transition amplitude between thespatial geometries at the initial and final global times. This amplitude can becalculated analytically if spacetime is two-dimensional [25], but in the case ofthree- or four-dimensional spacetime we have to rely on numerical simulations,and thus a discretization of spacetime geometries.The spacetime geometries are discretized using a method based on an ideaof Regge [26], and the diffeomorphism invariant integral over metrics (1) isregularized by a sum over a set of simplicial manifolds with a correct topologicalstructure. For each spacetime of this kind it is possible to perform Wick rotationto Euclidean signature, after which the exponent in the sum becomes real andthe complex amplitudes become real probabilities (see [27] for details): P ( T ) ∝ e − S ( T ) . (3)This formulation is well-suited to numerical simulations, which, as mentioned,are the main tool used in the analysis. The foliation of spacetime defines an4rdering on the slices (leaves) Σ , each of which can in a natural way be assignedan integer time parameter t .In the 3+1-dimensional case, the spacetime is built out of four-dimensionalsimplices. Each of them is the convex hull of five vertices that lie on two neigh-boring slices Σ . There are thus two types of four-simplices: { , } -simplices withfour vertices on a slice t and one vertex on a slice t ± , and { , } -simplices withthree vertices on a slice t and two vertices on a slice t ± . Each simplex abutsalong its three-dimensional faces on five other simplices, called its neighbors.All space-like links, i.e., line segments which connect two vertices on the sametime slice, are of length a s , and all time-like links, i.e., line segments whichconnect two vertices on neighboring time slices, are of length a t . Those lengthsare unchanging, and their ratio squared is the asymmetry factor: α = a t /a s .The Regge action (the Hilbert-Einstein action on a piecewise linear manifold)for a causal triangulation depends only on global quantities: S EH ( T ) = − ( K + 6∆) N + K (cid:16) N { , } + N { , } (cid:17) + ∆ · N { , } , (4)where N , N { , } and N { , } denote the total number of vertices and of { , } -and { , } -simplices in the configuration. The three dimensionless coupling con-stants, K , K and ∆ , are related respectively to the inverse of the gravitationalconstant G − , the cosmological constant Λ , and the asymmetry factor α .To describe a configuration fully, one has to do the following: • choose the initial and final states. To avoid the problem of making such achoice, we customarily adopt the periodic boundary conditions with somenumber of time slices T ; • label all the vertices and all the four-simplices; • list all the vertex labels together with corresponding time parameters; • list all the four-simplex labels together with the quintuples of their verticesand their neighbors placed opposite to the vertices.The same data are contained in the dual description, which is a graph (called thedual lattice) whose vertices correspond to the four-simplices of the configuration,and whose links correspond to interfaces between the four-simplices.As mentioned, no analytic solution for the model exists in 3+1 dimensions.Therefore, we probe the trajectory space by random generation of configura-tions with desired topology and scrutinize the results. The configurations arenot created one-by-one from scratch, but instead they are generated in largenumber by performing a Monte Carlo simulation, which starts from a very sim-ple triangulation and lets it gradually evolve by means of 7 types of geometricmoves. The moves modify the configuration locally in a topology-preservingway and are ergodic, which means that by performing them it is possible toobtain any triangulation with the same topology. In every simulation many bil-lions moves are performed, which allows us to overcome auto-correlation and to5enerate independent configurations. Moves are performed at random in a waysatisfying the detailed balance condition and with correct probabilities derivedfrom the action. We set the values of the couplings K and ∆ before startingthe simulation in order to study the model at a chosen point in the couplingconstant space (cf. [28]). The number of triangulations grows exponentiallywith N = N { , } + N { , } for a fixed topology . Summing over all triangula-tions with a fixed N , using as a weight e − S EH ( T ) for each triangulation T , willresult (to leading order in N ) in an expression Z N ( K , ∆ , K ) ∝ e ( K crit ( K , ∆) − K ) N , (5)and the full discretized version of (1) is then Z ( K , ∆ , K ) = (cid:88) N Z N ( K , ∆ , K ) . (6)In general we are interested in the limit where the average value of N → ∞ ,which corresponds to the limit K → K crit + . In simulations, taking this limitis replaced by studying the properties of a sequence of spacetimes, each with afixed N and N → ∞ .
3. Topology, boundaries and coordinates
In the original formulation of CDT in four dimensions it was assumed thatthe spatial topology of time slices was spherical ( S ). For technical reasons,related to the computer simulations, it was assumed that time was periodic.However, this periodicity played no role in the initial study of universes with S topology as along as the time period was sufficiently large. The existenceof the time foliation sufficed to analyze the phase structure of the model. Thephase diagram is surprisingly complex when one considers the extreme simplic-ity of the action (4), which only depends on the global quantities N , N { , } and N { , } . Of the four different phases, only one, the so-called C phase (alsocalled the de Sitter phase) seems to be related in a simple way to semi-classicalspacetimes, and we will here only discuss results obtained when the couplingconstants are chosen such that the system is in this phase. The simplest ob-servable measured was the spatial volume profile N ( t ) , defined as the numberof spatial tetrahedra on a time slice t . A typical system with a sufficientlylarge number of slices T consisted of a blob and a cut-off size stalk (necessaryto satisfy the periodic time boundary conditions mentioned above). Owing tothe invariance with respect to (discrete) translations in time, the position of theblob could be arbitrarily shifted in time. We used this possibility to center itaround a fixed time position. It was shown that both the average volume (cid:104) N ( t ) (cid:105) and its fluctuations can be derived from the discretized version of the effectivemini-superspace action [30] for the isotropically homogeneous 4D universe. The There is no analytical proof of this, only numerical evidence [29]. N ( t ) corresponds to a collective state, where all degrees offreedom are integrated out. The second difference is the sign of the effectiveaction, opposite to the one found in General Relativity. In CDT the solution ofclassical equations of motion gives a stable classical vacuum state, where at each t we have all possible geometric realizations with a particular value of N ( t ) .The existence of this highly nontrivial classical General Relativity limit of themodel was one of the most important results in the early studies of CDT.It is an interesting question whether the semi-classical limit can be extendedto include degrees of freedom in spatial directions. The simplicity of the S topology, however, makes such analysis very difficult or even impossible to per-form because of the background independence. We do not have any referencesystem with respect to which observables could be measured. This may be dif-ferent if we decide to formulate the model with a richer spatial topology. Inthe analysis presented in this article we chose Σ = T , and for technical rea-sons (ease of computer implementation and eschewal of the need for selectingthe initial and the final states) we imposed periodic boundary conditions in thetime direction: M = T × T .Thus, each configuration is topologically a Cartesian product of four circles.Each closed curve within the configuration is homotopically equivalent to acombination of those circles, and the coefficients of that linear combination arethe four winding numbers of the loop. Let us call them the winding numbers inthe x , y , z and t directions.One can equivalently consider the four-torus as an infinite periodic system.All the N simplices of the torus are contained in an elementary cell, whichrepeats itself infinitely many times in four directions. The elementary cell canbe defined in various ways, each of which is equivalent to a choice of a set offaces between neighboring four-simplices to form the cell’s four boundaries. Aloop within the torus corresponds in this picture to a path joining the samesimplices in two different copies of the elementary cell. We can assign a setof four numbers to each copy of the elementary cell in such a way that thedifferences between them for any two copies are identical to the four windingnumbers of the corresponding loop.Arguably it is the most convenient to look at loops in the dual picture, andso henceforth we will most often use the word “loop” to mean not a spacetimecurve but an ordered set of connected simplices whose image in the dual latticeis a non-contractible directed cycle. The length of a loop is the number of linksin the cycle. (For simplicity we assume that all links have the same length.)Similarly, a geodesic between two simplices will mean a line connecting themwhose image in the dual lattice has minimal length.The distance (the minimal number of links in the dual lattice) from a simplexto each of the boundaries of the elementary cell serves as pseudo-Cartesiancoordinates of the simplex. This definition was studied in a previous paper7
10 20 30 400.000.020.040.060.080.10 sum of distances p p Figure 2: Distributions of a sum of distances from simplices to the two opposite boundaries inthe x direction (red), the y direction (green) and z direction (blue) for systems with N { , } =80k (left) and N { , } = 160k (right). The distributions scale consistently with the Hausdorffdimension d H = 4 . The dotted lines refer to simplices adjacent to the boundary, x = 1 or x (cid:48) = 1 respectively for the two sides of the boundary. [20]. In a regular hypercubic lattice a sum of distances from any simplex to thetwo opposite boundaries is a constant, equal to the geodesic distance betweenthe boundaries. On a random lattice generated by CDT this is, however, notthe case. We observe a nontrivial distribution of these values (see Fig. 2),which may indicate either that the shape of the elementary cell is far frombeing rectangular, or that quantum fluctuations of the geometry can be viewedas ”mountains“ and in effect simplices close to the top of the mountains havea larger geodesic distance to the boundaries than those lying in the valleysbetween the mountains. Results indicate that both effects may be important.The latter effect is supported by the difference visible in Fig. 2 between thedistributions P ( x + x (cid:48) ) for all simplices (solid lines) and simplices adjacent toa boundary (dotted line, x = 1 or x (cid:48) = 1 ). Boundaries are chosen to locallyminimize their area, thus they prefer the central region of a torus ( valleys ) andomit outgrowths ( mountains ). Therefore, simplices adjacent to one boundaryare closer to the second boundary than an average simplex.In this paper we will try to perform a closer analysis of relevant structuresproduced in simulations to understand the properties of the quantum landscape .Of a primary interest will be correlations between valleys, which in this picturecan be interpreted as a semi-classical background geometry.
4. Description of simulations
The starting point of all the Monte-Carlo simulations was a single sim-ple configuration, which contained 4096 simplices in regularly placed four-dimensional hypercubes. The considered configuration consisted of T = 4 timeslices. Interfaces between some neighboring simplices were chosen as the bound-aries of the elementary cell (cf. Fig. 3). The precise shape of the initial config-uration and the initial position of the boundaries can be chosen freely as longas they have the correct topology. 8 igure 3: A schematic view of a single time slice in the initial configuration. Visible are the hypercubes (each of which is divided into 16 simplices) and the starting position of theboundaries. The boundaries are encoded as an additional set of numbers assigned to everydual link in a triangulation. Each link { ij } in a dual lattice is characterized bya set of four numbers n µij = ± , , where nonzero values mean that the linkcrosses the corresponding boundary in a positive or negative direction. Here, µ = 1 , . . . , enumerates directions. These numbers have an obvious property n µij = − n µji , and their sums along a closed loop reproduce the winding numbersof the loop.In order to keep the size of the boundary small, after each performed movea procedure that changes the position of boundary if more than two faces of asingle simplex belong to it was invoked in the region affected by the move.The simulations were performed at the canonical point in the phase spaceof toroidal CDT, i.e., in the C phase, for the parameters K = 2 . and ∆ = 0 . and for N { , } = 160000 . For the analysis we chose a typical, well-thermalizedconfiguration. The total number of simplices of the configuration we analyzedwas equal to N = N { , } + N { , } = 370724 .
5. Shortest loops
In a previous article [20] we introduced the idea of analyzing the shortestloops of non-zero winding numbers passing through a given simplex to gainunderstanding of the shape of the system. We described the distribution oflengths of loops with low winding numbers and noted the universality of itsshape. We also noted the strong correlations of distribution of loop lengths indifferent directions. 9
10 20 30 40 500.000.020.040.060.080.10 height p p Figure 4: Distributions of distances from simplices to their copies in neighboring elementarycells (heights) for systems with N { , } = 80k (left) and N { , } = 160k (right). These arelengths of minimal loops with winding numbers { , , , } (red), { , , , } (green) and { , , , } (blue) shifted in r by a shift of order one. The dotted lines refer to simplicesadjacent to the boundary. To recapitulate, in order to find the shortest loop of a given winding num-ber passing through a simplex, we treat the four-torus as an infinite periodicsystem and follow step by step the front of a diffusion wave beginning at thechosen simplex (using a diffusion wave in a system infinite in four directions isapplicable to the case of low winding numbers; otherwise this method becomescomputationally inefficient and should be modified, cf. Sec. 6). The number ofloops with a given winding number that pass through a simplex grows (eventu-ally) exponentially fast with the loop length. Thus, while it is feasible to list allthe shortest loops of a given type, in the case of longer loops we usually haveto pick one sample loop, representing their universal properties.Fig. 5 presents the connections between simplices forming the shortest loopsof winding numbers {1,0,0,0} in the configuration. It is evident that such veryshort loops are rare: in a configuration containing N = 370724 simplices thereare only 20 simplices belonging to loops of length 18. Moreover, loops of lengthfrom 19 to 21 often differ from each other only by a few simplices; the number ofseparate short loops of the same length – the number of distinct deepest valleys – is very small. The results in the other spatial directions are similar.Geometry of the random manifold generated by computer simulation ishighly fractal. It is tempting to interpret the distribution of the lengths ofloops with a unit winding number in spatial or time directions, having its max-imum at a length above 30, as a signal that in most cases the starting simplexis located inside one of the fractal structures (mountains, outgrowths), whereasthe (rarer) simplices belonging to the loops with a length that is minimal orclose to minimal correspond to the (relatively simple) basic structure of valleysin the configuration. With this interpretation, the length of a loop starting froma particular simplex reflects the position of the simplex relative to the valleys.And so, for a particular configuration we assign to each simplex a unique setof four numbers: lengths of loops with a unit winding number in a particulardirection and zero in the other three directions. Following these numbers along10 igure 5: The dual-lattice graphs of all the connections between simplices of lowest x-heights:18 (purple), 19 (red), 20 (yellow) and 21 (blue). In general, loops longer than the minimallength almost always contain fragments belonging to shorter loops, but for small heights thereexist also a small number of loops built only of simplices with equal x-height. simplices belonging to any particular loop, we may see how far the simplices be-longing to the loop are from the base, and how this distance changes along theloop. The four numbers assigned to a simplex can be called its heights , as theyreflect its position above the basic structure. For the sake of brevity, we can usethe names x-height, y-height, z-height, t-height for the length of loops in theunit directions. It was checked that, as expected, the height values of the fiveneighbors of any simplex differ from its own height by ± , ± or 0. In generalany loop starting deep inside a fractal structure is expected to move closer tothe base and then eventually climb back to simplices in the same fractal. Thereare only few loops whose simplices are all of equal height. This property ispossessed by the shortest loop in the configuration and a few dozen other shortloops, which are, so to speak, the “locally deepest valleys”. To summarize, the height of a simplex is defined as the length of the shortest loop passing throughit. It is determined separately for each topological dimension of the torus. Fora given simplex and direction, there might exist, and often do, several shortest11oops, all with the length equal to the height of the simplex. In further analy-sis we pick only one of them for each simplex. Usually, through each simplexpass also many loops (of the same length or longer) that are minimal for othersimplices (see the discussion near the end of Sec. 7).
6. Loops with higher winding numbers
The short loops contain important information about the underlying struc-ture of the manifold and about the distribution of valleys. The choice of fourdirections ( x, y, z and t ) is nevertheless to a large extent arbitrary. It reflectsa particular structure of the initial configuration and the memory about theinitially chosen elementary cell. It may give a false impression that the elemen-tary cell remains geometrically hypercubic during the thermalization. We canextend the analysis of minimal loop length distributions to include simplices incells with an arbitrary set of winding numbers { n µ } . The analysis shows thatthe network of minimal loops contains not only loops in the four basic direc-tions, as discussed above, but also loops with nontrivial winding numbers. It isimportant to note, that the winding numbers of a loop does not depend on aparticular choice of boundaries or, equivalently, the elementary cell. a v e r aged i s t an c e Figure 6: Distance from a starting simplex to its copy in cell { n, , , } minus distance fromthe same simplex to its copy in cell { n − , , , } , averaged over all the simplices of theconfiguration. Already for n = 8 the minimal value of 18 is reached. Using a four-dimensional diffusion wave in a system treated as infinite in fourdirections is a simple method to ensure that we find the shortest loop of a givenwinding number, regardless of the shape of the dual lattice, as the diffusion wavecannot “miss” any short path. One could continue the diffusion to find loopswith any higher winding numbers. However, the number of visited simplices ata distance R from the initial simplex grows as R , which means that eventually,12or large R , the procedure would become too time- and memory-consuming,and computationally inefficient. We should, therefore, modify the boundaryconditions in such a way that the number of simplices visited by the diffusionwave grows more slowly. One example is to consider the four-torus as a systeminfinite in only one direction – for example the x-direction – and strictly periodicin all other directions. This way we can measure loops with winding numbers { n, , , } , n = 1 , , . . . . A small modification of this idea is to assume that cellboundaries in all directions except x are impenetrable for the diffusion wave.In both of the methods the number of visited simplices in the R th shell, for R large enough, stabilizes and becomes independent of the distance R . Wewill observe a faster growth only up to the range where the diffusion wavereaches the boundaries of the system in the finite directions. A similar resultcan be obtained if we put the impenetrable walls in all directions except x atboundaries between cells number ± and ± . This choice improves the behaviorof the diffusion process in case the initial simplex is near the boundary of theelementary cell. h x h y h z h z Figure 7: Heights in the 4 basic directions of consecutive simplices along loops of windingnumbers { , , , } (blue), { , , , } (orange), { , , , } (green), { , , , } (red) startingfrom a simplex in an outgrowth. The dashed lines indicate the minimal and maximal heightsin the configuration, and the dotted line indicates the height of the initial simplex. In Fig. 6 we present results of using this method (with impenetrable wallsin all directions except x at boundaries between cells number ± and ± )to measure the average lengths of loops with winding numbers { n, , , } , n = 1 , , . . . , . Similar measurements were performed in y- and z-directions. Itturns out that the difference between the distance to the copy number n andthe distance to the copy number n − of a given simplex decreases rapidly13ith increasing n , down to the length of the minimal shortest loop in the givendirection (lowest height of a simplex), and then it remains constant. The expla-nation is simple: in order to minimize the length of a loop with a high windingnumber n in a given direction, it becomes advantageous for the loop to connectthe initial simplex to a simplex of the lowest possible height, then trace theshortest loop of unit winding number n times, and finally return to the initialsimplex. As can be seen in Fig. 6, already for n = 8 the shortest loop of theconfiguration is a part of all the loops of winding number n . The figure showsthe distances averaged over all simplices in the configuration. We note thatthe average for loops with a unit winding number in the x direction for thisparticular configuration is 34.64, which is well above the minimal length of 18,showing that most simplices belong to fractal structures.Fig. 7 shows the heights of consecutive simplices along loops with a growingsequence of winding numbers in the x-direction, starting from a simplex lyingfar within such a fractal structure, called also an outgrowth . We can see that asthe winding number of the loop increases, usually the minimal x-height of thesimplices belonging to the loop decreases, ultimately down to 18, which is thelength of the shortest { , , , } loops in the configuration. As there are no twocompletely separate x-loops of length 18, this means that all the loops of a highwinding number pass many times mostly through the same set of simplices.The graphs showing the heights in the y, z and t directions for the same setof loops demonstrate that although there is a correlation between height in allthe directions, the correlation is not perfect, especially after the loop leaves theoutgrowth. The repeating saw-like pattern is a loop of a high winding numbertracing one of the shortest loops of unit winding number several times. Theheights of simplices belonging to a loop that is shortest in the x-direction arenot minimal in the other three directions. However, even though in the otherdirections the height fluctuates, it still remains close to the minimal height, asthe simplices in the semi-classical region have low heights in all directions.
7. Alternative boundary conditions
The other method, mentioned in Sec. 6, is searching for a shortest pathconnecting a simplex to its copy in another elementary cell in a system that isinfinite in one direction and has periodic boundary directions imposed in theother three directions. The values of the winding numbers in the other directionsare irrelevant for this method, i.e., we find loops with winding numbers of theform { n, a, b, c } , a, b, c being any integers, instead of only { n, , , } .14inding numbers Lengthx y z t1 0 0 0 181 -1 1 0 162 -1 1 0 270 1 0 0 190 1 0 1 16-1 1 -1 0 16-2 3 -1 1 431 0 0 0 181 -1 1 0 162 -1 3 0 47 Table 1: Lengths of the shortest loops in the three basic spatial directions.
It turns out that also for these boundary conditions paths starting at varioussimplices tend to converge and follow a handful of very short loops. However,those loops are almost never the shortest loops with unit winding numbers, e.g. { , , , } . Rather than that, those loops have winding numbers of the form { n, a, b, c } yet are shorter than n times the length of the shortest { , , , } loop in the configuration (see Table 1; we used data from a diffusion in a four-dimensional infinite system to find the precise winding numbers). One couldconjecture that as we probe loops of higher winding numbers we should even-tually find loops with even smaller ratio of total length to the winding number,but in fact it turns out that even loops of winding numbers of order 50 utilizethe loops described in Table 1, so it appears that those loops are minimal. Itseems likely that the torus is twisted in such a way that it is impossible to re-define the elementary cell so as to make loops in the basic directions always anoptimal choice as components for loops of higher winding numbers.Figs. 8-10 show the heights in the basic directions of the consecutive simplicesalong the shortest loops described above. We can see that the heights, while notminimal, are quite low compared to the average (which is, as mentioned before,above 30). This signifies that these paths too belong to the base (“bulk”) regionof the torus and are not composed of generic simplices, which mostly belong tothe fractal structures. 15 h x h y h z h t Figure 8: The four basic heights of simplices belonging to one of the loops from Table 1: aloop of length 27 with winding numbers { , − , , } . h x h y h z h t Figure 9: The four basic heights of simplices belonging to one of the loops from Table 1: aloop of length 43 with winding numbers {− , , − , } .
10 20 30 40202530 step number along the path h x h y h z h t Figure 10: The four basic heights of simplices belonging to one of the loops from Table 1: aloop of length 47 with winding numbers { , − , , } . The algorithm we used creates a diffusion wave starting from a chosen sim-plex. For each simplex reached in consecutive steps it stores one of the sim-plices from which it came. In this way, we obtain at the same time not onlythe lengths of loops of winding number n in the chosen direction but also thelists of simplices along those loops. For each simplex in the configuration, wefound and wrote down one shortest path connecting it with its nearest copyin the x-direction. In this way we obtained the lists of simplices belonging to370724 shortest loops. We repeated the same process in the y- and z-directions.Next, we removed from each list the initial and final simplex of each loop, andwe counted the number of appearances of each simplex in the lists. A log-logscale histogram is shown in Fig. 11. The maximum value was more than 40000,which corresponded to one of the bulk simplices, and the minimal value waszero, which occurred numerous times and corresponded to simplices at the farends of the outgrowths. The latter simplices are not a part of any geodesicsapart from those that start within them. We were able to fit to the histograma power law curve with the exponent very close to − , which seems to bear acertain significance. This functional relationship is different than in the casesof, e.g., a branched polymer or a regular lattice. We have not yet found analgorithmic method of constructing a toroidal graph with behavior described bythe same exponent. We plan to investigate this point in a future article.We also noted that the heights of the simplices and the number of shortestloops passing through them are strongly correlated. We sorted the simplices inthe order of descending number of loops passing through them, divided the listinto blocks containing 1000 simplices each, and within each block averaged the17eights in each of the three spatial directions. With this ordering of simplices,the heights turned out to be increasing functions of the ordinal number of sim-plices in the list, modulo statistical fluctuations (see Fig. 12). The fluctuationsin all three directions were strongly correlated. We fitted a power law to thecurves. The exponent is probably the same for all the directions, and the con-stant factor depends on the shape of the torus – it is higher for the directionsin which the torus is more elongated (and so the average height of the simplicesis greater).This shows that the number of loops passing through a given simplex canserve as another indicator of its position within the torus. Most of the geodesicsbetween distant points pass through the bulk simplices in the semi-classicalregion and do not enter the outgrowths, which are the regions of quantumfluctuations. If a geodesic passes through a simplex in the outgrowth, it usuallymeans that it had its beginning even deeper in the same outgrowth. const ⨯ N paths -
500 1000 5000 1 × × N paths N s i m Figure 11: A log-log scale histogram of the number of simplices crossed by a given number ofshortest loops. x h y h z he i gh t Figure 12: The x-, y-, and z-heights of simplices. The simplices were sorted in the orderof descending number of loops passing through them, and then the heights were averagedover blocks containing 1000 simplices each. The fitted functions are h x = 21 . k . , h y =21 . k . and h z = 22 . k . .
8. Neighborhoods of bulk- and outgrowth-simplices
As observed, geodesics between distant simplices tend to pass through sim-plices of low heights and avoid simplices of middle and greater heights. It istempting to interpret the former as belonging to a semi-classical “bulk”, and thelatter to “outgrowths”, which are a result of quantum fluctuation. If so, thenthose regions should differ also in other properties apart from such non-localones as height and the number of loops passing through the simplices. This isin fact the case. The neighborhoods of bulk simplices and outgrowth simpliceslook considerably different, which allows for a construction of local quantitiesdistinguishing between them.Figs. 13 - 15 show subgraphs of the dual lattice, depicting all the simplices ofdistance up to 6 from a starting simplex, together with the connections betweenthem. The simplices – vertices of the dual lattice – are represented by circleswhose size is a decreasing function of distance from the starting simplex. Colorsof the circles indicate heights, with red corresponding to the shortest and violetto the longest loops in a given figure. The heights are also noted as numbers inthe circles. Figures in this section are based on data obtained using the version of algorithm withperiodic boundary conditions in directions y, z and t. Heights defined in this alternative way Figure 13: Six concentric shells around a simplex of height equal to 16. The colors and thenumbers indicate the height of a simplex, and the size of a vertex its distance from the startingsimplex. The central simplex lies in the bulk region.
In Fig. 13 the starting simplex had height equal to 16. As that is the minimalvalue, we can see that in the neighborhood of the simplex height tends graduallyto increase together with distance from it. By the same token, loop lengthdecreases with increasing distance from a simplex with height equal to 40, whichis among largest in the configuration – Fig. 15. Even a cursory glance at thegraphs, moreover, suffices to note a strong dependence of the total number ofsimplices of distance up to 6 from the starting simplex on its loop-length. As have similar values and interpretation to x-heights defined previously, though they are notdirectly equivalent, e.g., here the minimal value is 16, which occurs in simplices that form theshortest loop with winding numbers {1,-1,1,0} (see Table 1).
030 3029 32 3130 2929 30 3230 312732 31 3231 3332293029 3029 3233313030 313126 28 2827 30 3132 32 30 3330 31 34 34 3531 2929 302831 28 2829 3129 333334 333128 31 323030 33292726 26 27 28 30 282728 2928 30 293130 3331 2932 333233 32 3433 36 373430 30292928 292730313233 29 2831 2929 28 29 34323535 35 3333 3129 3030 31 3332 33 3429 31 3534 32312827 2626252626 2627 28 27 2829 323132 32 302828262928 29 28 3130 29 29323130 31 3229 3129 333134 33 3432 3636 36
Figure 14: Six concentric shells around a simplex of height equal to 30. The colors and thenumbers indicate the height of a simplex, and the size of a vertex its distance from the startingsimplex. The central simplex lies in the middle of an outgrowth. expected, “outgrowth” regions have an elongated shape and a lower Hausdorffdimension than the “bulk” region, which is a sign of their fractality.Fig. 16 shows the shortest {1,-1,1,0} loop and its neighborhood. Comparingit also with Table 1, we note that it is among the very shortest non-trivial loopsin the configuration. It is the only loop with that set of winding numbers andlength 16. It is readily seen that as we count loops with the same windingnumbers and length 17, 18, . . . in the vicinity of the marked loop, the numbergrows approximately exponentially. 21
041 3939 39 4039 40 383738 38393739 39 4038 403739 3939 3639 3837 3738 383938 38 39 3737 3839 4038 3940 3840 3638 383539 38 3940 36 3737 36 3836 37 35 37 38 393938 38 39 38 3637 3840 39 39383939 393840 37 383636 35 33383838 39383938 3436 38 3738 393636 36 34 3936363636 37 36 37 3733 38 39 38 38 3940 403838 3637 3739 39 40 39 38 39393836 3938 37 37 39 37
Figure 15: Six concentric shells around a simplex of height equal to 40. The colors and thenumbers indicate the height of a simplex, and the size of a vertex its distance from the startingsimplex. The central simplex lies near the deep end of an outgrowth.
9. Conclusion
The detailed measurements performed on a typical toroidal configurationwhich appears in the CDT path integral in the C phase shows the following:the spatial T part consists of a relatively small bulk region, the toroidal center ,which we have denoted semi-classical, and numerous fractal outgrowths of al-most spherical topology (with a single small boundary) which contains most ofthe simplices. A lower-dimensional illustration of this is shown in the left partof Fig. 1. Introducing the lengths of the shortest non-contractible loops in thecoordinate directions as the heights associate with a given simplex allowed usto classify the simplex as belonging to the toroidal center or to an outgrowth.Further, the number of simplices in the outgrowths where the height is a lo-cal maximum is not small. The interpretation of this is that the outgrowths22 Figure 16: The shortest {1,-1,1,0} loop together with its neighborhood. The colors and thenumbers indicate the height of a simplex, and the size of a vertex its distance from the loop. are quite fractal, again in the way illustrated in the left part of Fig. 1. Animportant feature of the length distributions of non-contractible loops associ-ated to the simplices is that they scale as N / , where N denotes the size ofthe triangulation, i.e., the number of four-simplices, as shown in Fig. 2. Themost likely consequence of such a “canonical” scaling is that the volume of thetoroidal center, although small compared to the volume of the outgrowths, willalso scale with N . It might thus be justified to think of it as semi-classical, incontrast to the toroidal center-part of the two-dimensional configuration shownin the right part of Fig. 1, which vanishes in the large N limit, as discussed inthe Introduction.We conclude that there is a well defined geometric structure underlying thetypical path integral configuration of T in CDT. It is somewhat more fractalthan we had hoped for in the sense that the outgrowths contain most of the sim-plices, but it invites to use a classical scalar field to define a coordinate system, aprocedure common in classical General Relativity. By imposing suitable bound-23ry conditions on the scalar field one can make its values record the structureof the toroidal center well, whereas the field is almost constant in an outgrowth.It thus emphasizes what we have denoted the semi-classical part of the config-uration and might be a good choice of coordinates if one wants to construct asemi-classical action. Work in this direction will be reported elsewhere. Acknowledgement
The authors thank Jakub Gizbert-Studnicki and Daniel Németh for manyfruitful discussions. JA acknowledges support from the Danish Research Coun-cil grant
Quantum Geometry , grant 7014-00066B. ZD acknowledges supportfrom the National Science Centre, Poland, grant 2019/32/T/ST2/00390. AGacknowledges support by the National Science Centre, Poland, under grant no.2015/17/D/ST2/03479. JJ acknowledges support from the National ScienceCentre, Poland, grant 2019/33/B/ST2/00589.
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