Jhep03(2015)151

Home , Loop quantum gravity

JHEP03(2015)151 2 on / 3 ∼ Springer March 3, 2015 March 26, 2015 : February 2, 2015 : December 2, 2014 : : Accepted Revised Published 10.1007/JHEP03(2015)151 Received doi: 4 on large distance scales to Published for SISSA by ∼ [email protected] , . 3 1411.7712 The Authors. c Models of Quantum Gravity, Lattice Models of Gravity

We calculate the spectral dimension for a nonperturbative lattice approach , [email protected] Faculty of Physics, Astronomy andul. Applied prof. Computer Stanislawa Science, Lojasiewicza Jagiellonian 11, Krakow, University, PLE-mail: 30-348, Poland Open Access Article funded by SCOAP ArXiv ePrint: lattice spacing within thewhich might physical prove phase useful ofquantum when gravity. the studying CDT renormalization parameter group flow space in is models alsoKeywords: of outlined, lattice Abstract: to quantum gravity, knowndimension as of causal spacetime dynamical smoothly triangulationssmall decreases (CDT), distance from scales. showing This that novelargument result the may against provide the a possible asymptotic resolution safety to a scenario. long-standing A method for determining the relative D.N. Coumbe and J. Jurkiewicz reduction in causal dynamical triangulations Evidence for asymptotic safety from dimensional JHEP03(2015)151 ) is (1.1) 1.1 ) is divergence , meaning that d 1.1 − -dimensional space- ]. Equation ( d 1 [ ] = 2 L N . In G N -dimensional space, showing G d 5 13 ]. Interestingly, eq. ( 1 dp, 8 L ] N G [ as − – 1 – A L p Z 14 0, because the integral will grow without bound as the loop- < 9 13 ] 4 N G scales with loop order 1 p 2, meaning that gravity as a perturbative quantum field theory can be renor- ≤ is a process dependent quantity that is independent of 3.2.1 Investigating systematic errors in phase A increases in the perturbative expansion [ d A L Remarkably, a number of seemingly independent approaches to quantum gravity have 3.1 Searching for3.2 a continuum limit Systematic in errors CDT 3.3 Statistical errors two. This raisesregulator the via exciting the mechanism possibility ofand that dynamical predictive dimensional spacetime theory reduction, could of possibly quantum yielding act gravity. a as finite its ownreported ultraviolet that the dimension of spacetime exhibits a scale dependence. Causal dynamical where clearly divergent for [ order free for malizable by power counting if the dimension of spacetime is equal to, or smaller than, number of counterterms ofperturbative ever quantum increasing field theoretic dimension. treatmentthat of One momentum gravity can in clearly see this from the not give consistent resultsincongruity when stems applied from to the quantuminteractions general fact of relativity. that nature gravity The by is origintime its distinguished of Newton’s dimensionful from this gravitational coupling the coupling constant other hasin a fundamental the mass case dimension of of [ 4-dimensional spacetime higher-order loop corrections generate a divergent 1 Introduction Three of the four fundamentalnotable interactions exception of being nature gravity. have The beengravity central is successfully difficulty that quantised, in the the formulating computational a techniques theory applied of so quantum successfully to the other forces do 4 Discussion and conclusions 3 Measurements of the spectral dimension in CDT Contents 1 Introduction 2 Asymptotic safety JHEP03(2015)151 - ]. n 2 ]. Another ]. -dimensional 14 12 n ], grav- Hoˇrava-Lifshitz 3 -simplices, where the geometry ] all provide evidence that the n 7 , 6 -simplices of fixed edge length, where n -dimensional simplicial manifold. A key n ]. These local update moves can result in the -dimensional simplices that are glued together 15 n – 2 – -dimensional building blocks will always result in a ], and string theory [ n 5 ]. In response to these problems a causality condition was 11 , 10 ], exact renormalization group methods [ 2 ]. Since a scale dependent dimension may have important implications -dimensional analogue of a triangle. However, the original EDT model 2 n 2)-dimensional faces, forming a − -dimensional building blocks is a non-trivial test of the theory; a test that CDT n ], which defines a spacetime of locally flat n 9 , ], and the likely identification of a second-order phase transition line suggests the 8 13 ], loop quantum gravity [ 4 At first glance one might think that performing a weighted sum over geometries The introduction of the causality condition in the CDT approach to quantum gravity In close analogy to the sum over all possible paths in Feynman’s path integral ap- One of the original formulations of lattice gravity is Euclidean dynamical triangulations -simplex is the n deletion or insertion of verticesstructure within that simplices, has and so self-similar ita properties is fractal. at possible different to A scales; obtain fractalspace meaning a geometry from geometric the admits geometry non-integer can dimensions,has be so passed by recovering demonstrating that a four-dimensional geometry emerges on large scales [ for the renormalizability of quantum gravity it forms theconstructed central by focus of gluing this together dimensional work. geometry, however, this istions not the necessarily dynamics is the contained case.is in updated For the by dynamical connectivity a of triangula- set the of local update moves [ CDT [ exciting possibility that the theorykey may result have is a that well within defineddependent, the dynamically continuum de reducing limit Sitter-like from [ phase approximately of fourtwo on CDT on small large the scales scales dimension [ to appears approximately to be scale geometries that can bethat foliated define in the this path way integral are measure. included in thehas ensemble produced of triangulations a numberfour-dimensional of de promising Sitter results, like in phase contrast was to shown to the emerge original within EDT the version. parameter A space of time geometries are defined byalong locally their flat ( ingredient of CDT isbetween the introduction space-like of and a time-liketion causality links condition, of on in the the which lattice lattice. one into distinguishes space-like In hypersurfaces, this each way with one the can same define fixed a topology. folia- Only take a continuum limitadded, [ giving rise to the method of causal dynamicalproach triangulations to (CDT) quantum [ mechanics, CDTof is quantum an gravity attempt via to a construct sum a over nonperturbative different theory spacetime geometries. In CDT, such space- (EDT) [ a quickly ran into significant problems.phases, The neither parameter of space of which couplings resembledtwo contained phases 4-dimensional just were semi-classical two separated general by relativity, a and first the order critical point, making it unlikely that one could ity [ dimension of spacetime appearsto to approximately reduce two from on approximatelysubstantial four microscopic evidence on scales. in macroscopic support scales Individuallya of these compelling dimensional results argument reduction; do that collectively, demands however, not they further constitute form attention. triangulations (CDT) [ JHEP03(2015)151 . ) σ µν ( g r (1.4) (1.5) (1.6) (1.7) (1.2) (1.3) P with a M -dimensional d diffusion steps σ diffusion steps. One σ , . ) , ) = 0 σ 4 ζ, ζ, σ / ( . , ζ, σ ) has the simple solution, 0 ) 0 g ]) starting from the . ζ 2 0 ζ . i ( 1.3 K d/ ) g 17 ) )) ζ, ζ , σ , ( ( ζ K and 2 σ 2 r ν πσ 2 g by taking the logarithmic derivative with ζ 1 d d/ P )), 5 (4 log h ln (r) S σ − ζ d µ 0, ln (V (r)) D log is the covariant derivative of the metric det (g ( 5 0 d → ) = – 3 – → 2 p exp σ µν r 5 r ζ ( g − det (g ( d r . = lim d -dimensional closed Riemannian manifold − P = σ ) = p d H ) ζ Z S d D d D 1 V ] generalises the concept of dimension to non-integer val- , ζ, σ R 0 , ζ, σ ζ 0 16 ( ζ = ) = in the limit ( g σ g V r ( K K r P ∂ ∂σ ). ζ ( ) is strictly only valid for an infinitely flat Euclidean space. However, one µν g 1.7 ) is the geodesic distance between 0 is known as the heat kernel describing the probability density of diffusion from ζ, ζ ( scales with radius g in a fictitious diffusion time 2 g K d T ζ ) for a random walk over the ensemble of triangulations after D Equation ( The quantity that is measured in the numerical simulations is the probability In the case of infinitely flat Euclidean space, eq. ( The spectral dimension, on the other hand, is related to the probability of return The Hausdorff dimension [ σ to ( r . Specifically, the probability that the random walk will return to the origin approaches 0 can still use this definitioncurved, of or the finite spectral volume, by dimension factoringσ to in compute the appropriate the corrections fractal for dimension large of diffusion a times and so we can extractrespect the to spectral the dimension diffusion time, giving The probability of return to the origin in asymptotically flat space is given by, where that the diffusion processover will the spacetime return volume to a randomly chosen origin after ζ The diffusion process is takensmooth over metric a diffusion equation, where P can derive the spectral dimension (following refs. [ angulations to be computed numerically, typicallydimension this and is the done spectral by dimension. computing the Hausdorff ues, and can be definedsion by considering how the volume of a sphere with topological dimen- The CDT approach to quantum gravity allows the fractal dimension of the ensemble of tri- JHEP03(2015)151 for 4 (2.1) (2.2) (2.3) ) via σ ( /N r 1 P ]. Curvature 2 → ]) against the ) σ 19 ( r is the mass of the P is the temperature. , so that M T 2 2 − d ], among others. r 21 ∼ , where S M d N T ] (see also Shomer [ 1 G − 18 d The spectral dimension allows one to ∼ R 3 1 . . . − 1 2 3 ∼ S d d − − − r d d d /D ,E 2 E E 4 1 , which determines the behaviour of – 4 – N − ∼ ∼ g d ) S S −4 σ RT ( ], the concept of the renormalizability of gravity might ∼ 1 S ]. 22 ) for which σ ( S -dimensional Schwarzschild solution in asymptotically flat spacetime d D , will dominate the diffusion in this region, causing n λ ]. One can therefore factor in the appropriate finite volume corrections by The 2 ] for a critique of the reasoning that leads to this scaling. The Bekenstein-Hawking area law tells us that [ 3 20 4 S is the radius of the spatial volume under consideration and /D 2 4 R N For gravity, however, one expects that the high energy spectrum will be dominated by However, there exists an argument due to Banks [ See ref. [ Although this assumption hasAsymptotically been safe questioned black by holes Percacci are and actually Vacca Schwarzschild-de [ Sitter black holes whose entropy is given The curvature of the space on which the diffusion process occurs should also be corrected for due to 2 3 4 1 the fact that it willcorrections change are the not probability that estimated the in diffusion this process work. will return to the origin [ by the Cardy-Verlinde formula,entropy and which asymptotic may safety itself [ resolve the apparent contradiction between black hole has a black holeblack with hole. event horizon of radius where It follows that the entropy of a renormalizable theory must scale as black holes. as a conformal field theory. Itthe follows quantities from dimensional considered, analysis, and and the theno fact extensive dimensionful scaling that scales of a other finite than the temperature temperature, conformal that field the theory entropy has S and energy E scale as which would define a continuum limit. possibility of asymptotic safety. Thegies argument expected compares for the a density theory ofable of states quantum gravity at field to high theory that is ener- of a perturbationa a of conformal renormalizable a field field conformal theory. field theory theory Since must by a relevant have operators, renormaliz- the same high energy asymptotic density of states As first suggested bybe Weinberg generalised [ to includeIn the this nonperturbative scenario regime gravityrelevant via would couplings the end be asymptotic on nonperturbatively an safetysuch renormalizable ultraviolet scenario. as fixed if point CDT, a (UVFP). an finite In UVFP a number lattice would of theory appear of as gravity, a second order critical point, the approach to probe the geometry of spacetimedimensions coincide over with varying the distance standard scales. measurewhen of The the the dimension, Hausdorff manifold the and topological is spectral dimension, non-fractal. 2 Asymptotic safety diffusion time is much greaterthat than the the zero volume. modeits The of mathematical eigenvalues the explanation Laplacian forσ this is omitting values of unity as the ratio of the volume and the diffusion time approaches zero, i.e. when the JHEP03(2015)151 ) 6), . 2.3 (3.1) ∆ = 0 , 2 . = 2 (phase B) and the 0 κ ] correctly claim the short 2 crumpled phase 25. With a fit to the functional . 0 , ± σ b + 80 . c (phase A), a − a ). Assuming the argument leading to eq. ( 0) = 1 – 5 – ) = 2.2 → σ ( ]. As the authors of ref. [ σ ( S 2 S (phase C). The thicker transition lines represent previously D D = 54 [ c 10, and . 0 ± de Sitter phase branched polymer-type phase = 119 and 02 . b 02, . ) = 4 A schematic representation of the phase diagram of 4-dimensional CDT. We observe = 4 a → ∞ σ ( S form giving distance spectral dimension is thus consistentmeasurement with the is integer for 2. just However, the a fact single that this point in the parameter space, coupled with the relatively 3 Measurements of theThe spectral canonical dimension point in in CDT has the physical previously de been SitterD phase shown of to CDT, exhibit namely ( a scale dependent spectral dimension, yielding This scaling disagrees withis that valid of then eq. one ( isquantum field led theory. to This conclude is thatresolution a gravity potentially of serious cannot which obstacle be is for formulated provided asymptotic as safety, in a a the possible renormalizable following section. measured phase transition points and thediagram thinner are lines an the interpolation. 4work, Superimposed locations as on within the indicated phase phase by Crelative the at lattice black spacing. which squares. the The spectral arrows dimension indicate is the determined apparent in direction this of decreasing Figure 1. three main phases: a physically interesting JHEP03(2015)151 2 , as 3 1 sim- N , (3.2) 4 1 , N 4 0, within N . 2 on short / at each point i ∆ = 2 2 , , 3 4 . We find that the . . For three of these N 2 , and is typically of 1 1 + , = 4 4 1 , 0 4 N 92. In the absence of any κ . N h 0), we have also calculated . i = 1 2 , ). and table 3 2 N 3.2 , ). ∆ = 2 | + , /d.o.f 1 ]. We implement an effective linear 1 4 2 , . 2 384,000 207,000 267,000 367,000 4 χ 1 , target N 4 simplices with the average total number of h 02 afterwards. We choose to fix = 4 . N . However, using the full covariance matrix 0 1 , for technical convenience. We have checked κ 4 − 2 = 0 2 1 N 1 , , , 3 4 4 – 6 – N N N | 300,000 160,000 160,000 160,000 + 1 = 6) and ( , . 4 0) 6) 6) 6) N . . . . δS ∆) 2 0 0 0 data in figure simplices we also obtain a sharply peaked number of , , , , , 0 ]. In this way we can be sure that we are investigating the 4 4 6 2 1 ∆ = 0 , . . . . κ K 2 , 4 ( 4 (4 (4 (3 (2 . 6 line, in addition to a fourth point N . 4 , 2 6) 160 . we obtain a relatively large . 0 2 for each point studied in the parameter space. , χ i = 2 2 2 . , 0 3 sweeps for the largest ensembles [ κ N 8 , with each unit defining a sweep. The number of sweeps required to reach + 6 10 1 , 05 during thermalization and 4 . A table comparing the number of ∼ N h = 0 ) for multiple lattice volumes. This multi-volume study and related discussion can σ = 80. The attempted Monte Carlo moves that update the geometry were performed Both correlated and uncorrelated fits to the data give similar results, as demonstrated The main results of this work are presented in figure In the calculation of the spectral dimension presented in this work we take the starting We calculate the spectral dimension as a function of diffusion time for three different ( values along the ∆ = 0 t S 0 and that the spectral dimensiondistance smoothly scales decreases to and a for value sufficiently consistent fine with lattice 3 spacings. by the fits to thein (2 the estimation of that for a given numbersimplices, of and hence asampled well-defined in average phase total C four-volume of the parameter space (seelong table distance spectral dimension is consistent with the semiclassical dimensionality of 4, the order four-volume fixing constraint with opposed to the total four-volume bulk properties of theto geometry a with maximum each diffusion. ofof 500 The diffusion diffusion steps. process isin Simulations followed units were out of performed 10 witha a thermalized time configuration extension grows approximately linearly with D be found in the subsection on systematic errors (section point of our diffusionplices, to as be is in done the in time e.g. slice ref. containing [ the maximal number of attempting a more comprehensive study of the spectral dimensionκ in phase C ofthe CDT. physical phase of CDT,points, as indicated namely by the ( black squares in figure Table 1. simplices large statistical error makes definitive conclusionsimportant difficult. consequences Since this for result the has potentially renormalizability of gravity, we revisit this calculation, JHEP03(2015)151 6 is . rel 6 and 0) for a . → rel σ a 0.10 0.11 0.57 1.00 ( ∆ = 0 S , 2 500 D . for four different = 2 σ 0 0 is calculated using using the fit function . κ (0) from 2 S 7.7 8.0 4.5 0.1 → ∞ ∆ = 2 400 , D =0.6 =0.6 =0.6 =2.0 σ 4 . ∆ ∆ ∆ ∆ = 4 0 s.d. of =2.2, =3.6, =4.4, =4.4, κ = 0 and 0) are determined from a fit-function 0 0 0 0 Correlated fit κ κ κ κ 494] for the point σ , 300 → Uncorrelated fits ) for 060 058 093 266 σ [50 . . . . ( σ 0 0 0 0 ( σ S ∈ (0) S D ± ± ± ± σ S D D ) curves corresponding to points along the ∆ = 0 ]. The fifth column gives the number of standard σ 2 540 534 576 970 – 7 – . . . . 0) from the integer 2. The rescaling factor ( 200 ) and 1 1 1 1 S ) and short distance spectral dimension → D as a function of the diffusion time σ S 12 16 32 17 → ∞ ) ( simplices. . . . . D S 0 0 0 0 → ∞ σ 1 include the total statistical and systematic error estimate. ∞ , ( D 4 ( σ ± ± ± ± S 2 ( S N D S D 14 12 31 05 100 . . . . D 4 4 4 4 ) and using the fit range 1 , 3.1 4 as first proposed in ref. [ ∆) values. N 0 , σ 300,000 160,000 160,000 160,000 2 b 2 3 4 κ + c 2.5 1.5 3.5 − 0) 6) 6) 6)

. . . . simplices. The light blue error bands come from uncorrelated ﬁts to the data using the The spectral dimension

S a ∆) 2 0 0 0 D 1 , , , , , , A table of the long 0 4 4 4 6 2 ). The uncorrelated fit shows only the central value for comparison. Errors presented here . . . . κ 492] for the other three points. We extrapolate to N ( , (4 (4 (3 (2 3.1 [60 ∈ deviations (s.d.)determined of by the the method values of of best overlap of the rescaled spectral dimension curves. several different ( of the form Table 2. 300,000 functional form of eq. ( σ of eq. ( are statistical only. Errors in table Figure 2. points in the de Sitterline phase are of CDT. calculated The using 160,000 JHEP03(2015)151 , ) ν ]. ]), 2.3 1 27 28 − E d R ) agree if, ∼ should be 1 ]. However, − 2.3 rel S the spectral by using the d 24 a R 2 1 ) and ( 2.2 0. Here we outline one 490] in steps of 4 for the → , ) of the form ]. a 23 2.3 [50 ∈ ) as our fit ansatz, and attempt ) is incorrect for the class of black hole σ for each curve until they overlap, (0) quoted in table which is precisely the value we find 2.3 3.1 rel S 5 a ]. A central motivation of the present D . The implication being that as one of the black hole, whereas to obtain eq. ( for a semiclassical black hole. The authors of The idea that the value of the short dis- 2 26 E 1 2 6 , ]. This result appears to have some tension with , which should be taken into account when a = 20 25 ) and ∞ BH ( ν – 8 – = 2 [ by a factor S d σ D decreases (similar results were reported in ref. [ is equal to 3/2; a d ) seems to suggest the rescaling factor when ) over the data range 1.3 3.1 CFT ν depends on the energy = 490] in steps of 4 for the other three points. The errors quoted R , BH ν [60 (0) in this work, at least for some of the points we sampled in phase C of CDT. S ∈ D . Equation ( σ 3 for a conformal field theory, and 1 6) and and ∆ the lattice spacing . d ] the authors argue that the scaling relation of eq. ( − d 0 0 , 20 are determined by varying the fit function and the fit range as discussed above κ 2 = . 2 ] then point out that cft ν Moving along the black line in the direction of the arrows in figure We now return to the holographic argument against the asymptotic safety scenario pre- 20 This counter-argument relies on the plausible assumption that the relevant dimensionIn in the ref. holographic [ 5 6 must be treated as a constant. This leads to a modified version of eq. ( the values we obtain for scaling argument is also the spectral dimension asconsidered, suggested due by to e.g. the ref.R fact [ that with ref. [ increases since it takes aOne greater can number then of rescale the diffusionas diffusion steps shown time before in the figure proportional same to dimension the square is of obtained. the lattice spacing CDT, and in particular insuch the method search that for could a be fixedof point used the at to running which determine spectral the dimension relative at lattice different spacing values via of adimension the comparison curves bare parameters. flatten out, as shown in figure as a second-order criticalThe point, divergent the correlation approach lengthallow to characteristic one which of to would a takephysical define the second-order units. a lattice phase continuum spacing transition Hence, limit. touseful would developing when zero a investigating while renormalization method group keeping to flow observable within determine quantities the the fixed physical de lattice in Sitter spacing phase of may prove work was then to measure thedynamical small triangulations distance (CDT); spectral a dimension formulation using known the to causal version have3.1 of a semiclassical phase Searching [ forIn a a continuum lattice limit formulation in of an CDT asymptotically safe field theory, the fixed point would appear tance spectral dimension might resolvephy the was tension first between proposed asymptotic in safetya and the hologra- context detailed of study Euclidean of dynamical thefor triangulations particular a [ semiclassical region phase of revealed parametersemiclassical an space effective spacetime considered dimension on the inconsistent macroscopic with best scales four candidate [ dimensional in table and adding the statistical error in quadrature. sented in the introduction. Weand wish only to if, highlight the the spacetimefor dimension fact the that small eqs. distance ( spectral dimension of CDT. use an uncorrelated versionto of the more fit accurately functionrange. estimate of We systematic eq. obtain ( the errorsuncorrelated central by fit values function of varying of the eq.point fit (2 ( functions and the fit better theoretical guidance as to the correct functional form of the spectral dimension we JHEP03(2015)151 , 0 ), . rel rel for a 2.0 0.6 0.6 0.6 1.3 b σ/a rel + D= D= D= D= a c ∆ = 2 , − 4 . 4.4, 2.2, 3.6, 4.4, a = = = = 0 0 0 0 = = 4 Κ Κ Κ Κ S 0 κ D ], we are led to the ∆) points the canonical 27 , Σ 0 κ 500 6 and . that each curve must be , as suggested by eq. ( rel a a ∆ = 0 , 4 . 400 = 4 0 κ ∆) value. Interestingly, going from the The factor , 0 7 κ 6) we find qualitatively similar behaviour to . , as well as the long and short distance spectral – 9 – between different points in the parameter space is 300 rel 6 curve. a ∆ = 0 . rel , a 6 . for each ( 2 = 3 ∆ = 0 , 0 2 . κ 200 = 2 ] between the same two points in parameter space, although the 0 6) to ( κ . 27 ∆ = 0 , 100 2 . Rescaled spectral dimension fits according to the functional form = 2 chosen such that the curves give the best overlap. 0 κ rel a Due to finite computational power it is only ever possible to simulate with finite lattice This is obviously a matter of preference and one is free to make any of the ( 7 S 0 D systematic errors, the main sources being finite-size effects andvolumes, however, discretization one can errors. quantify finite-size effects by calculating an observable for several value against which the others are compared. conclusion that simulations for the bare parameters have a lattice spacing already in the sub-Planckian3.2 regime. Systematic errors Approximating continuous spacetime with a discrete and finite lattice inevitably introduces point ( that observed in ref.exact [ quantitative agreement strongly dependsthe change on in the the arguments rescaling parameter used.proportional to If the we change assume in that and the by square using the of values the of lattice the spacing absolute lattice spacing reported in ref. [ which the best overlapis occurs. set to The unity curvesrescaled for are by the normalized to such obtain thatin agreement the lattice with spacing. scale the factor The otherdimension, rescaling curves factor are will displayed then in be table related to the change Figure 3. with determining the cut-off scalerather in than physical units. the actual The data fit because curves are it used is in easier the to comparison determine the rescaling factor 1.5 3.0 2.5 2.0 4.5 4.0 3.5 JHEP03(2015)151 . is 4 . 0. 6, . 5 ). S 2 → 3.1 /D 2 a 4 6 there . N 6, and for . . σ and table ∆ = 0 4 , 2 50 for the coarse ∆ = 0 2 , . 2 ≤ . and σ ) can be significantly 0 as demonstrated in = 2 . 2 with fixed ∆ = 0 3 . σ = 2 ( 2 0 0 κ S κ > , where a, b, c and d are D is not met for these larger d 0 ∆ = 2 , κ S )) , i.e. taking the limit 4 σ for which a . /D 2 S 4 + values over which the fit function D c = 4 N ( 0 σ finite-size effects begin to dominate, κ b/ S ( σ ) curves appear to stop growing, becom- /D − 2 σ 4 ( a 6 and N . S value, and thus the condition greater than approximately 350, as evidenced D σ suggest that finite-size effects are mostly under ) = σ σ 5 – 10 – ) increases quite rapidly when comparing the rela- ( ∆ = 0 ) in the fit to our data we also estimate a contri- , S ∞ 4 ( . and D 3.1 6 for 4 . S 4 , D 6 . 0 the much finner lattice spacing results in a much smaller . ) and 50 for the coarsest lattice, namely ∆ = 0 = 3 range presented. Furthermore, as we move to points in the , cσ ∼ 0 2 values. . σ κ − ∆ = 2 σ , σ 4 . = 2 to zero. Finite-size effects can be seen to play a significant role for 0 exp ( κ b S = 4 0 D becomes much greater than − κ a σ 60 for the finner lattices, from the fit to the functional form of eq. ( is smaller for an equivalent S ≤ D ) = σ σ ( ) is applied. Furthermore, due to the absence of any solid theoretical motivation ) reaching a maximum and then beginning to decrease. However, this is not true S indicates that the value of σ . To reduce discretization errors we omit values of 3.1 ( 5 D 6 show the spectral dimension as a function of diffusion time for different lattice volumes S 60 for the finer lattices We obtain a more complete estimation of the systematic error associated with our spec- Errors associated with using a discrete lattice to approximate continuum physics, dis- For the point 5 D ∼ for using the functionalbution form to of the eq. systematic ( errorforms associated with using theunconstrained alternative fit asymptotic parameters, functional the values of which are given in tables σ figure lattice, and tral dimension measurements by varyingof the eq. range ( of Hence, discretization errors becomespacing. increasingly Large insignificant discretization errors as are onemension. typically associated decreases For with a the the small small lattice number scaledifferent of spectral when diffusion considering di- steps an the even or behaviour oddtions of become number of negligible diffusion for steps. These odd-even oscilla- control for the largest lattice volumes at each point,cretization as errors, presented in can figure bedown to estimated the by continuum. using Oneulations an estimates at discretization effective successively errors smaller field by values theory performing of numerical and the sim- extrapolating lattice cut-off so as to not underestimateFigure the large distance spectraltively smaller dimension, lattice as volumes of suggested 160K, ining 240K the figure and larger 270K at 270K this and point,ing 300K but statistically ensembles comparable. that the when Figures compar- parameter space corresponding tothe finner value lattice of spacings, i.e. only met for much larger absolute lattice volume, and so one should be careful to simulate with a large enough volume is because when eventually driving the 80K ensemble at by of the 120K andlattice 160K volumes ensembles within as the the condition estimate the lattice volume required such thatand finite-size effects become negligible. Figures at three different pointsexists in a the statistically parameter significant80K space. difference and 160K between For ensembles the the for point spectral large dimension diffusion curves times. for As the mentioned in the introduction this different lattice volumes and extrapolating to the infinite volume limit. Thus, one can JHEP03(2015)151 240k 160k 160k 120k 80k 300k 270k 160k 120k 80k ======4,1 4,1 4,1 4,1 4,1 4,1 4,1 4,1 4,1 4,1 simplices. 0. Since this . 1 , 4 2.0, N 2.0, N 0.6, N 0.6, N 0.6, N 2.0, N 2.0, N 0.6, N 0.6, N 0.6, N N D= D= D= D= D= D= D= D= D= D= ∆ = 2 , 4 4.4, 4.4, 4.4, 4.4, 4.4, 4.4, 4.4, 2.2, 2.2, 2.2, . ======0 0 0 0 0 0 0 0 0 0 Κ Κ Κ Κ Κ Κ Κ Κ Κ Κ = 4 6 ensembles appear to be . 0 κ Σ Σ 500 500 4 at ∆ = 0 . = 4 0 κ 400 400 2 and . – 11 – = 2 0 κ 300 300 200 200 100 100 A multi-volume study of the spectral dimension at two different points in the parameter A multi-volume study of the spectral dimension at the point S S 0 0 D D Figure 5. point in the parameter space correspondsa to much a larger very lattice small volume latticedimension spacing of in 270K due Planck or to units 300K one the so must much use as smaller to absolute not underestimate lattice the volume large for distance a spectral given number of 1.5 1.0 2.5 2.0 3.0 space of CDT. Finite-sizeunder effects control for for the the larger 160K lattices. 2.0 1.5 Figure 4. 3.0 2.5 4.0 3.5 JHEP03(2015)151 2.0 0.6 0.6 0.6 D= D= D= D= ) we omit σ 4.4, 2.2, 3.6, 4.4, ( 000. = = = = , S 0 0 0 0 Κ Κ Κ Κ D 60 values for the 0) extension for values . = 160 6) . 2 1 σ < σ , , 0 4 Σ , 4 . 6 N 90 . 000 simplices, and for the (4 , (3 6) with = 160 . 6, and omit 0 1 . , 80 , a b c d 4 a b c d 6 . 4.00 1337.74 648.10 1.25 4.12 2.46 0.0013 - N 3.74 2.14 0.0078 - 4.01 479.57 339.14 2.43 ∆ = 0 0. . , 2 . 70 6) with 6) and (3 . . 0 = 2 ∆ = 2 0 , , , 0 4 4 2 . . . κ – 12 – 6) 6) . = 4 . 000. 0 60 0 , 0 , , κ 2 4 . . (2 (4 = 300 1 6 and , . 4 50 N a b c d 3.74 1.73 0.013 - 4.20 108.17 21.69 0.62 ∆ = 0 a b c d , 3.99 1213.56 586.11 1.24 3.83 2.18 0.0016 - 4 . d 4 0) with ) . , 40 d )) 2 6 ) . cσ , σ )) 4 . cσ − and ∆. Note the oscillations have a larger amplitude and σ + = 3 0 − c + κ 0 ( κ c exp ( ( b/ 30 b Odd-even oscillations in the small scale spectral dimension for four different values of exp ( ( b/ The fit parameters a, b, c and d for the two alternative fit functions used in estimating The fit parameters a, b, c and d for the two alternative fit functions used in estimating b ( − Fit-function − − a Fit-function − a a a 50 values for the coarsest lattice, namely S 20 D Figure 6. the parameters of the bare parametersσ < that correspond tofinner lattices finner lattices. In the calculation of 1.5 1.0 2.5 2.0 3.5 3.0 Table 3. the systematic error for the bare parameters (2 Table 4. the systematic error forbare the parameters bare (4 parameters (4 JHEP03(2015)151 , 6 . 29 0.6 D= ∆ = 0 , 0 . 8.0, = . This result = 8 0 Κ 7 0 κ by comparing our σ Σ 500 60 for this lattice volume. σ > 400 492], as can be seen in figure , [60 – 13 – 300 ∈ σ ] is also valid in the full four-dimensional theory then it 30 , 200 29 range studied σ simplices. 1 , 4 100 N The spectral dimension in phase A of CDT, calculated at the point For each point in the parameter space we check that the ensemble is thermalized using ] to be 4/3. Although such a result is yet to be established in the full four-dimensional S 0 D one begins taking measurements.number of Once configurations thermalization used hasmean in been approaching the achieved, the increasing correct calculation value the of with the an observable increasingly will smalltwo just statistical methods. error. result Firstly, in we the begin with a thermalized smaller volume and allow it to evolve numerical results for the spectral dimensioncomparison in also phase suggests A discretization with the effects constant are value small 4/3. for 3.3 Such a Statistical errors If one calculates anerroneous observable result. using a lattice It that is is therefore not important thermalized one to will check obtain all an lattices are thermalized before constant 4/3 over the suggests that the geometrywith in branched phase polymer A systems. oftwo-dimensional CDT models If at [ we least assumewould shares that be some the possible universal analytical properties to value get of a 4/3 sense found of in how small we can reliably take of Euclidean quantum gravity has been30 determined from purely analytic considerationstheory [ the geometric propertiesto in phase be A analogous ofdetermine four-dimensional to the CDT the spectral are dimension largely branched in expected polymer phase phase A of of CDT EDT. and find In a this value consistent work with we the numerically Figure 7. using 160,000 3.2.1 Investigating systematicUsing errors two-dimensional in toy phase models A the spectral dimension in the branched polymer phase 1.1 1.0 1.4 1.3 1.2 1.7 1.6 1.5 JHEP03(2015)151 1 2 , / 4 N 6 as a ]. The . 2 (0) and S D ∆ = 0 , 4 . = 4 0 κ decreases, because for larger a 0), plotting them as a function of → σ (0) for the point ( S S . and ∆ within the physical phase of CDT ]. D ]. This is the principal result of this work. D 000 simplices. For a configuration number 2 0 2 , 24 κ ]. Secondly, after the ensemble has reached a 25 on small distance scales is reported [ . ) and 0 ) and 15 – 14 – = 160 ± ∞ . From these results we conclude that the small ( 1 , 80 4 S . 2 → ∞ N D σ ( S D 000 there is no statistical difference in the mean values of ) agree, and thus it may resolve the tension between asymptotic safety , show the values of range if there exists no statistically significant difference between the first 20 2.3 σ 8b ∼ and ) and ( 1 on large distance scales, to 1 . ) and extract values for ) when comparing the first and second half of the data set, and we thus conclude this 8a 0 2.2 ∞ 3.1 Our studies indicate that as one increases Here we apply a best fit to the spectral dimension data using the functional form of ± ( S 02 . such trajectories in the parametervalues space of the the lattice bare spacing dimension couplings is it obtained. takes One can athe then relative greater rescale lattice spacing number the for diffusion of each time curvegive diffusion until by “the the a steps best variance factor is overlap”. before that minimised, This is i.e. the method until related same the for to curves determining the relative lattice spacing may prove than with the integer 2,We as wish previously thought to [ pointeqs. ( out that thisand holography, value as of originally the proposed in dimension ref. is [ preciselythe the spectral dimension value curves for flatten which out. The implication being that as one moves along of this result could havework important implications we for give the a renormalizabilityvalue of more gravity. at detailed In several this study different ofOur values the of results running the are spectral bare summariseddistance dimension parameters in spectral by and table dimension calculating for in its multiple lattice the volumes. de Sitter phase of CDT is more consistent with 3 The aim of this workfound is in to make the a CDT more4 detailed approach study to of quantum dimensionalsmall gravity, reduction in distance previously spectral which dimension a is dimensional of reduction particular interest, from as a more precise determination using a single-elimination jackknifedimension procedure. measurements are The determinederrors total in by error quadrature, adding and estimate are the of presented total our in systematic table spectral and4 statistical Discussion and conclusions function of Monte Carlo timegreater using than D ensemble is thermalized forare such calculated a using configuration thermalized range. lattices as All detailed results above. presented Statistical in errors this are work estimated eq. ( Monte Carlo time. We concludeover that a a specific particular ensemble ofand triangulations second is half thermalized of the datafigures range, after passing the first thermalization test. As an example, simplices increases very slowly, eventuallydefining reaching thermalization a as plateau. defined Thisconfiguration in is that ref. the satisfies [ same the above methodas condition of we a then function plot the ofthe observable Monte to first be Carlo and measured time second and half of check the that data there set is over statistical which we agreement perform between the measurement. towards a larger target volume. During thermalization, the width of the distribution of JHEP03(2015)151 ) 4 . 3.1 = 4 to its 0 0 κ κ (0) = 2. In S D MCtime MCtime , namely eq. ( ]), and aid in the 2 28 80 000 80 000 8 standard deviations with ), and thus tuning appearing to decrease slightly ∼ 1 rel 70 000 70 000 a appears to result from maximising rel a 2 inferred by previous measurements of 60 000 60 000 ∼ on ∆, with (0) rel S – 15 – a 0) = 2. D → 50 000 50 000 σ ( ]. We find a tension of S 2 D 40 000 40 000 to the C-A transition and then studying the effect of varying ∆ (0) as a function of Monte Carlo time for the bare parameters 0 S κ (0) shows no sign of changing even for points in the parameter space D S D 30 000 30 000 ) and ∞ ( L L S 2, differs from the value of 0 492], respectively. H / • 6. The data range we believe to be thermalized is divided into two data sets that are , D S . 3 H S D 4. 20 000 [60 1.5 ∼ would be a natural next step. 20 000 4.2 4.1 D 1.48 1.52 ∈ ) presented in this work exhibit a monotonic decrease to a value that is consistent σ rel The novel value of the short distance spectral dimension of CDT obtained in this work, The most rapid decrease in the rescaling factor σ (0) ( within phase C of the CDT phase diagram (see figure a S S 0 that appear to bepresent, probing the have sub-Planckian some regime, tensionlight suggests with of that such renormalization our suggestive results, group comparisonsarguments at predictions it leading least may that to at be the worth result revisiting the renormalization group D the spectral dimension ofthe integer CDT value [ 2 forD our finest lattices.with Furthermore, the 3/2, and fact that that the measurements of κ critical value at the first-orderto transition be a dividing significantly phase weaker dependence Cas of and ∆ phase increases. A. Tuning Thereon also seems useful when studying the renormalizationsearch group for flow a in putative CDT second (e.g. order ref. critical [ point at which one may take a continuum limit. Figure 8. and ∆ = 0 compared with each other for statisticaland agreement fit to within range 2 used standardand to deviations. obtain The fit these function results are the same as those used in figure JHEP03(2015)151 B JHEP , Phys. , (2009) 242002 Phys. Lett. 26 , Phys. Rev. Lett. , (2014) 355402 A 47 ] for points corresponding 27 , Cambridge Univ. Press, ]. J. Phys. , Class. Quant. Grav. (0) = 2 as expected from renormal- ]. , S SPIRE D IN ][ SPIRE ]. IN [ ]. – 16 – Spectral dimension of the universe SPIRE IN SPIRE ][ IN (1988) 291 ][ Four-dimensional simplicial quantum gravity arXiv:0902.3657 ), which permits any use, distribution and reproduction in Nonlocality in string theory ]. Due to the current absence of such further investigations [ Fractal spacetime structure in asymptotically safe gravity ]. ]. ], for all values of the bare couplings investigated in this 32 B 310 The Hagedorn transition and the number of degrees of freedom of 27 , ]. 31 SPIRE SPIRE IN IN hep-th/0505113 ][ ][ CC-BY 4.0 [ SPIRE IN hep-th/0508202 (2009) 161301 This article is distributed under the terms of the Creative Commons [ Nucl. Phys. [ General relativity, an Einstein centenary survey Fractal structure of loop quantum gravity , (0) remains consistent with 3/2 as one probes the manifold on yet smaller Spectral dimension of the universe in quantum gravity at a Lifshitz point 102 S D (1992) 42 (2005) 171301 (2005) 050 arXiv:1310.4957 arXiv:0812.2214 string theory [ 278 Rev. Lett. [ Cambridge U.K. (1997). 95 10 Determining the absolute lattice spacing by measuring fluctuations about de Sitter G. Calcagni and L. Modesto, J. Ambjørn and J. Jurkiewicz, P. Hoˇrava, L. Modesto, J.J. Atick and E. Witten, J. Ambjørn, J. Jurkiewicz and R. Loll, O. Lauscher and M. Reuter, S. Weinberg, [7] [8] [4] [5] [6] [2] [3] [1] any medium, provided the original author(s) and source are credited. References We also thankMielczarek A. for Gorlich, useful J.2012/06/A/ST2/00389 discussions. from Gizbert-Studnicki, the D.N.C F. National Science acknowledges Saueressig, Centre the Poland. J.Open support Cooperman of Access. and theAttribution grant J. License DEC- ( Acknowledgments D.N.C is especiallyWe grateful thank to R. J. Loll Laiho for carefully for reading suggestive a early draft discussions version on and this providing work. useful comments. plings really are probing theto sub-Planckian regime, whether and possibly givedistance a scales, definitive answer or as whether itization group begins predictions to [ increase to a definitive comparison betweendistance renormalization spectral group dimension and is CDT currently predictions incomplete. of However, the this short work is in progress. space, as presented inwork would ref. allow one [ tospectral more dimension thoroughly in assess determining the the reliabilitythe change of absolute in using lattice lattice the spacing. spacing rescaling of Furthermore, viato determining the our the finest method lattices presented would in indicate ref. whether [ measurements at these values of the bare cou- JHEP03(2015)151 , Phys. D 89 Phys. (2005) , , D . 4 ]. Phys. Rev. Class. , D 72 Class. Quant. , ]. , , ]. ]. Phys. Rev. SPIRE ]. , IN (1919) 157 SPIRE ][ ]. SPIRE SPIRE IN 79 IN IN Phys. Rev. SPIRE ][ Renormalization group quantum gravity with a , ]. IN ][ ][ ]. ]. D SPIRE ][ 4 IN ]. ][ SPIRE dynamical triangulation is first SPIRE ]. SPIRE IN IN Math. Ann. D IN ][ 4 ][ ]. ][ SPIRE Euclidian arXiv:1405.4585 ]. Focusing on the fixed point of IN SPIRE [ The nonperturbative quantum de Sitter Planckian birth of the quantum de Sitter ][ IN A second-order phase transition in CDT hep-lat/9601024 ]. [ SPIRE ][ arXiv:1307.2270 hep-lat/9401026 [ [ SPIRE IN arXiv:0712.2485 [ IN ][ (in German), ]. SPIRE ][ arXiv:0807.4481 ]. IN [ Phase structure of four-dimensional simplicial – 17 – hep-th/9805108 ][ Reconstructing the universe [ (2014) 165003 (1996) 293 hep-lat/9603024 SPIRE arXiv:1108.3932 (2013) 100 (1994) 277 [ SPIRE [ IN ]. ]. 31 IN [ ][ 10 arXiv:1306.1546 [ (2008) 091304 B 472 arXiv:1104.5505 Spectral geometry as a probe of quantum spacetime [ B 328 SPIRE Asymptotic safety, emergence and minimal length SPIRE Lower bound on the spectral dimension near a black hole (1998) 407 Structural aspects of asymptotically safe black holes ]. (2008) 063544 IN IN arXiv:1108.4686 100 [ JHEP Exploring Euclidean dynamical triangulations with a non-trivial Evidence for asymptotic safety from lattice quantum gravity (1996) 238 [ Black hole thermodynamics under the microscope Nonperturbative Lorentzian quantum gravity, causality and topology ][ arXiv:1008.3621 , [ arXiv:0911.0401 (2011) 211303 SPIRE D 78 B 536 [ IN [ Nucl. Phys. B 389 Phys. Lett. (2014) 015006 , Further evidence that the transition of 107 , arXiv:1212.1821 (2011) 161301 arXiv:1401.3299 Class. Quant. Grav. 31 [ Dimension und Maß äußeres , , A pedagogical explanation for the non-renormalizability of gravity (2011) 084029 TASI lectures on holographic space-time, SUSY and gravitational effective field 107 (2010) 245026 Phys. Rev. Phys. Rev. Lett. , , hep-th/0505154 Nucl. Phys. arXiv:1007.4001 [ , Phys. Lett. 27 (2009) 124036 , , D 84 measure term universe flow in CDT Rev. Lett. non-trivial measure term Quant. Grav. Rev. arXiv:0709.3555 (2014) 084002 Grav. D 80 theory Phys. Rev. Lett. 064014 change universe quantum gravity simplicial gravity order S. Catterall, J.B. Kogut and R. Renken, J. Ambjørn, A. J. Görlich, Jurkiewicz and R. Loll, J. Ambjørn, A. J. Görlich, Jurkiewicz, A. Kreienbuehl and R. Loll, J. Laiho and D. Coumbe, J. Ambjørn, L. Glaser, A. and Görlich J. Jurkiewicz, D. Coumbe and J. Laiho, B. Koch and F. Saueressig, S. Carlip and D. Grumiller, K. Falls and D.F. Litim, R. Percacci and G.P. Vacca, D. Benedetti and J. Henson, T. Banks, A. Shomer, J. Ambjørn, J. Jurkiewicz and R. Loll, F. Hausdorff, J. Ambjørn and R. Loll, J. Ambjørn, A. J. Görlich, Jurkiewicz and R. Loll, J. Ambjørn, S. Jordan, J. Jurkiewicz and R. Loll, P. Bialas, Z. Burda, A. Krzywicki and B. Petersson, B.V. de Bakker, [9] [27] [28] [24] [25] [26] [22] [23] [20] [21] [17] [18] [19] [15] [16] [12] [13] [14] [10] [11] JHEP03(2015)151 ] (2012) 055022 ]. 14 ]. SPIRE hep-lat/9710024 The spectral dimension of [ IN SPIRE ][ IN ][ New J. Phys. , (1998) 549 B 515 arXiv:1206.0657 phase-diagram of QEG and its spectral hep-th/9801099 [ [ 2 – 18 – R The Nucl. Phys. , (1998) 010 The spectral dimension of the branched polymer phase of Quantum Einstein gravity (2012) 024018 ]. 02 D 86 SPIRE JHEP IN , ][ Phys. Rev. , ]. quantum gravity SPIRE D IN arXiv:1202.2274 dimension two-dimensional quantum gravity [ 2 [ S. Rechenberger and F. Saueressig, J. Ambjørn, D. Boulatov, J.L. Nielsen, J. Rolf and Y.M. Watabiki, Reuter and F. Saueressig, T. Jonsson and J.F. Wheater, [32] [30] [31] [29]