Dimensional flow in discrete quantum geometries
aa r X i v : . [ h e p - t h ] A p r PHYSICAL REVIEW D , 084047 (2015) arXiv:1412.8390 AEI-2014-028
Dimensional flow in discrete quantum geometries
Gianluca Calcagni, ∗ Daniele Oriti, † and Johannes Th¨urigen ‡ Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid, Spain Max Planck Institute for Gravitational Physics (Albert Einstein Institute)Am M¨uhlenberg 1, D-14476 Potsdam, Germany (Dated: December 29, 2014)In various theories of quantum gravity, one observes a change in the spectral dimension fromthe topological spatial dimension d at large length scales to some smaller value at small, Planckianscales. While the origin of such a flow is well understood in continuum approaches, in theoriesbuilt on discrete structures a firm control of the underlying mechanism is still missing. We shedsome light on the issue by presenting a particular class of quantum geometries with a flow in thespectral dimension, given by superpositions of states defined on regular complexes. For particularsuperposition coefficients parametrized by a real number 0 < α < d , we find that the spatialspectral dimension reduces to d s ≃ α at small scales. The spatial Hausdorff dimension of such classof states varies between 1 and d , while the walk dimension takes the usual value d w = 2. Therefore,these quantum geometries may be considered as fractal only when α = 1, where the “magic number” D s ≃ time , appearing so often in quantum gravity, is reproducedas well. These results apply, in particular, to special superpositions of spin-network states in loopquantum gravity, and they provide more solid indications of dimensional flow in this approach. I. INTRODUCTION
The identification of good geometric observables isa thorny issue in (quantum) gravitational physics,and it is of particular importance in nonperturbative,background-independent approaches to quantum grav-ity, especially where the fundamental degrees of freedomcharacterizing quantum states and histories of the systemare nongeometric in the standard sense and character-ized by intrinsic discreteness. Examples are loop quan-tum gravity (LQG) [1–3], spin-foam models [4, 5] andgroup field theory (GFT) [6, 7], strictly related to LQG[8, 9]. Here, the major challenge is to find a relation tothe continuum spacetime geometries of classical generalrelativity, i.e., to show that the latter emerge from thefundamental discrete quantum structures of the theoryin some approximation. This emergence has to be ex-pressed in terms of suitable geometry observables, bothclassical and quantum, that should indicate that the de-sired features of smooth spacetimes are recovered. Thisis, in fact, a precondition for extracting physics from suchquantum-gravity formalisms.Effective-dimension observables provide important in-formation about the geometric properties of quantumstates of space and spacetime histories in quantum grav-ity. In particular, the spectral dimension d s , which de-pends on the spectral properties of a geometry throughits definition as the scaling of the heat-kernel trace, hasattracted special attention due to the observation of a di-mensional flow (i.e., the change of spacetime dimensional-ity across a range of scales [10–12]) in various approaches, ∗ [email protected] † [email protected] ‡ [email protected] such as causal dynamical triangulations (CDT) [13], thefunctional renormalization-group approach of asymptoticsafety [14, 15] and Hoˇrava–Lifshitz gravity [16] amongothers.In all these approaches, the spectral dimension ofspacetime exhibits a scale dependence itself, flowing fromthe topological dimension D in the infrared (IR) to D s ≃ D s ≃ / SU (2).There is thus an interplay between two types of dataand their corresponding discreteness: a combinatorialdiscreteness due to the graph substratum for the quan-tum states, and an algebraic discreteness due to the factthat the labels are half-integers corresponding to SU (2)irreducible representations. Quantum effects in the eval-uation of observables are thus to be expected, in general,from both these sources and it is an important limitationto focus only on one of them, as in preliminary studies ofdimensional flow in LQG [26].In a previous work [27], we have already tackled the is-sue of computing the spectral dimension on LQG statesbased on a given graph, dealing both with coherent statesand with their superpositions. There, we showed thatthe underlying discrete structure plays a dominant role.Here, we intend to explore the role of combinatorial dis-creteness and of superpositions of combinatorial struc-tures in greater detail.In this paper, we present a special class of superposi-tions of discrete quantum states characterized by a real-valued parameter α . This parameter will control thescale-dependent values taken by the spectral dimension,and therefore the dimensional flow. These superpositionsare over states based on regular complexes correspond-ing to hypercubic lattices to which a single quantum labelis assigned, uniformly to all cells of a certain dimension.Such states occur indeed in the kinematical Hilbert spaceof the quantum gravity formalisms we just mentioned:LQG, spin-foam models and GFT. Because of the uni-form labeling, these superpositions are also similar to thediscrete geometries in CDT, although we understand theformer not as regularization tools for physically smoothgeometries but as fundamentally discrete structures withtheir own physical interpretation. Contrary to the CDTsetting, we interpret the combinatorial structures we su-perpose as defining quantum gravity states, not histories,and the coefficients in the superpositions to have no im-mediate dynamical content. However, we point out thatthis interpretation enters only minimally in the actualcalculations and it could be generalized.Perhaps surprisingly, superpositions of quantum statessupported on different complexes have not been consid-ered much in the LQG literature so far. Instead, mostanalyses have involved only states based on one and thesame complex. A first simple example of states based onsuperpositions of combinatorial structures are the con-densate states with a homogeneous cosmology interpre-tation introduced recently in the GFT context [28–32].Using a known analytic expression for the spectral di-mension of single members in the superposition [27], wecompute numerically superpositions over up to 10 dis-crete geometries. On these grounds, we find strong ev-idence for a dimensional flow, characterized by the pa-rameter α .Similarly, we find analytic solutions for the walk di-mension and Hausdorff dimension of lattice geometriesand perform again numerical calculations of superposi-tions. For these observables, however, while we recoverthe topological dimension at large scales, we do not findany special properties for superpositions as compared tostates defined on fixed complexes. II. A GENERAL CLASS OF SUPERPOSITIONSTATES
Let us now explain in detail the construction of super-position states of interest, and the calculation of theirspectral, walk and Hausdorff dimension.Most generally speaking, a discrete quantum state ofgeometry |{ j c } , Ci is given by an assignment of quantumnumbers j c to a certain subset of cells c ∈ C of a (com-binatorial) complex C , diagonalizing volume operators ofthese cells d V ( p ) c ′ |{ j c } , Ci ∝ l p ( j c ′ ) |{ j c } , Ci , (1)where we have adopted natural units. An example ofsuch states is spin-network states in LQG, based on the1-skeleton of the dual complex C ⋆ , with the j ’s identifyingirreducible representations of SU (2). In three spacetimedimensions, the spatial ( d = 2) states in the spin networkbasis diagonalize the length operators b L e associated withall edges e ∈ C . Thus, they are labeled by spins j e on thecorresponding dual edges e ⋆ ∈ C ⋆ . The form of the b L e spectra is l ( j e ) = p j e ( j e + 1) + C , with a free parameter C ∈ R due to a quantization ambiguity for the Euclideantheory (as well as for timelike edges in the Lorentzian the-ory, a continuous positive variable being instead assignedto spacelike edges) [33, 34].In four spacetime dimensions ( d = 3), spin-networkstates have the same spectrum for area operators b A f onfaces f ∈ C such that [35, 36] l ( j f ) = [ j f ( j f + 1) + C ] / . (2)Generic quantum-geometry states are superpositionsof the discrete quantum geometries |{ j c } , Ci , which in-deed form a complete spin-network basis of states of theHilbert space in LQG. In particular, this Hilbert spacecan be cast in the form of a direct sum of Hilbert spaces H = L C H C .In the following, we will restrict to superpositions withnonzero coefficients only for states | j, Ci labeled by a sin-gle quantum number j c = j for all cells. Thus, one canconsider the individual states | j, Ci as corresponding toequilateral lattices. Given this class of quantum states,we then consider generic superpositions of the form | ψ i = X j, C a j, C | j, Ci . (3)We also impose a constraint on the overall volume V computed from such superposition states: | ψ, V i = X j, C a j, C δ ( h j, C| b V | j, Ci , V ) | j, Ci , (4)where the delta is a Kronecker delta. We will furtherrestrict the sum to certain regular complexes, i.e., hy-percubic lattices C N based on the canonical vertex set C [0] N := ( Z N ) d of d -tuples of integers modulo N . In thiscase, the fixed volume condition is explicitly V = h j, C N | b V | j, C N i ∝ N d l d ( j ) , (5)which fixes the lattice size N = N ( j ) for a given j (atleast approximately). In general, there are three scalesinvolved in the superposition states | V , j min , j max i := j max X j = j min a j | j, C N ( j ) i , (6)when summing over a finite range from j min to j max : a minimal length scale l ( j min ), an intermedi-ate scale l ( j max ) and the overall volume size V /d ∝ N ( j min ) l ( j min ) = N ( j max ) l ( j max ). Note that a finite vol-ume V bounds also possible cutoffs j max (since N is apositive integer).One can also consider the limit of noncompact geome-tries N ( j min ) → ∞ , where all complexes in the superpo-sition of fixed-volume states (6) converge to the infinitelattice C ∞ . Thus, they are technically the same as super-positions on the fixed complex C ∞ , although the physicalinterpretation is different. Due to the combinatorial sim-plicity, results of infinite-size calculations can be directlyapplied to the finite-volume case.Having defined our superposition states, we can moveon to the evaluation of the geometric observables of in-terest, i.e., dimension estimators. III. EVALUATION OF DIMENSIONOBSERVABLES OF SUPERPOSITION STATESA. Spectral dimension
Let the heat kernel K ( x, x ′ ; τ ) be the solution of thediffusion equation ( ∂ τ − ∆ x ) K = 0 on a space X , withinitial condition K ( x, x ′ ; 0) = δ ( x − x ′ ), where ∆ is theLaplace operator on X . It is a function of the geometry of X via its assigned metric. In the resolution interpretationof [37, 38], the parameter √ τ and its inverse represent,respectively, the length scale and the resolution at whicha geometry is inspected by a pointwise probe deployedat a spatial point x ′ . The trace of the heat kernel overall points is denoted as P ( τ ) = Tr X K ( x, x ′ ; τ ) and called“return probability” from the traditional but somewhat Ratios l ( j ) /l ( j ) for pairs of quantum numbers j , j can benonrational, so that one should take the integer value (floor func-tion) of N ( j ). Physically, it is certainly enough to apply Eq. (4)in such an approximative way. Note also that our results areindependent of the spacing of quantum numbers in the superpo-sition. Thus, one could as well define the states as sums overonly those j ’s for which N ( j ) ∈ N strictly. problematic interpretation in terms of a diffusing pro-cess (see [18, 38] for a discussion and resolutions of suchproblems).While ordinary diffusion takes place on continuousmanifolds, the whole setup, and in particular the defi-nition of the Laplace operator, can be generalized to dis-crete spaces, like (combinatorial) complexes. This wasindeed the subject of [39, 40]. The Laplacian on C , asa differential operator acting on a field φ a on the d -cells c a ∈ C (equivalently, on dual vertices), is then [39] − (∆ C φ ) a = X b ∼ a (∆ C ) ab ( φ a − φ b )= 1 V ( d ) a X b ∼ a V ( d − ab l ⋆ab ( φ a − φ b ) , (7)where the sum runs over d -cells c b adjacent to c a , V ( d ) a is the volume of the cell c a , V ( d − ab is the volume of thecommon bounding ( d − l ⋆ab is the length of itsdual edge. Accordingly, the heat trace on C is given by atrace Tr C over maps of that field space.Both the return probability P ( τ ) and the Laplacian∆ can be turned into operators [ P ( τ ) and b ∆ acting onquantum states of geometry. Quantizing the metric-dependent coefficients (∆ C ) ab which enter in the defini-tion of the discrete Laplacian (7) results in an operator b ∆ C acting on the Hilbert space H C of states on a givencomplex C which returns states together with discreteLaplacians. This can be done in different ways, depend-ing on the geometric variables that are most convenientin the specific quantum geometric context that is chosen.It has been discussed in detail in [39]. In general, theresulting expression will be a complicated function of thequantum labels assigned to the complex, which is how-ever both well-defined and explicitly computable [27].The operators [ P ( τ ) and b ∆ on the full Hilbert space H = L C H C are then defined in terms of the family of Note that only the coefficients of b ∆ C are quantum operators inthe usual sense, i.e., maps from the Hilbert space H C to itself. b ∆ C itself is an operator properly defined only on the coupledHilbert space of geometry and test fields, which we do not intro-duce. We do not consider quantum states of test fields, since therelevant object [ P ( τ ) to define the spectral dimension is a func-tional of pure geometry and, as such, it can eventually be definedas a quantum operator in the strict sense. Let us expand thistechnical point for the interested reader. As a vector space overcomplex numbers, any state in a Hilbert space can be expandedin the elements of a complete basis with complex numerical co-efficients. Elements in the image of the quantum Laplacian b ∆ C are sums over such a basis, but with coefficients that are dis-crete Laplacians instead of complex numbers, that is, maps froma functional space on a complex to itself. These elements areobviously not states in H . Still, we can use b ∆ C to define [ P ( τ )which is the quantum operator acting on pure-geometry statesthat we are interested in here. orthogonal projections π C : H → H C . In this way, theLaplacian acting on generic quantum states of geometryis the formal sum b ∆ := X C π C b ∆ C π C . (8)With the appropriate notion of a trace Tr := P C Tr C π C ,based on the trace Tr C over discrete field space on C in-troduced above, the heat trace is then defined as [ P ( τ ) := Tr e τ b ∆ . (9) [ P ( τ ) is a map from H on itself, and thus a quantumoperator in the strict sense. Then, the spectral dimension d ψ s ( τ ) of a quantum state of geometry | ψ i ∈ H is thescaling of the expectation value of [ P ( τ ) [27]: d ψ s ( τ ) := − ∂∂ ln τ ln h [ P ( τ ) i ψ . (10)Note that it depends only on pure geometry, since therelevant operators are acting on pure-geometry states.For the discrete quantum geometries |{ j c } , Ci it is rea-sonable to assume that they are eigenvectors of b ∆ C , basedon the definition of these labels (1) and on our previouswork [27]. On the states (3) that we are interested inhere, the heat-trace expectation value is thus h [ P ( τ ) i ψ = X C X j a ∗ j, C h j, C| X j ′ a j ′ , C Tr C e τ b ∆ C | j ′ , Ci = X j, C | a j, C | Tr C e τ h j, C| b ∆ C | j, Ci . (11)Some simplifying assumptions are however needed inorder to proceed with systematic computations on ex-tended complexes. In the following, we assume that theexpectation value of the Laplacian b ∆ C scales as h j, C| b ∆ C | j, Ci ab ∝ l − ( j ) (∆ C ) ab , (12)where ∆ C is the combinatorial Laplacian (7) on the com-plex C . This assumption is sensible if the Laplacian canbe expressed as a function of the volumes (1). A similarAnsatz is, in fact, made in [26], although in that workthis is not justified on the basis of a detailed analysis ofthe underlying graph and on the complete expression forthe Laplacian, such as the one presented in [39].We now evaluate the spectral dimension on our super-position states. Under the assumption (12), the expres-sion for the expectation value of the return probabilityfurther simplifies to h [ P ( τ ) i ψ ∝ X j, C | a j, C | Tr C e τl − ( j )∆ C . (13) The above expression can be computed most efficientlyconsidering the limit of infinite lattices, for which an an-alytic expression for the heat trace is available. In [27],we showed that the heat trace on C ∞ = Z d is P C ∞ ( τ ) = [ e τ I ( τ )] d , (14)where I is the modified Bessel function of the first kind.In the same limit, one can give precise formulae for thecontribution to the spectral dimension coming from in-dividual lattices, so that we are in the ideal position toinvestigate the effect of superpositions of the same. Thespectral dimension d j, C ∞ s on a single state | j, C ∞ i equals d for τ ≫ l ( j ) and vanishes for τ ≪ l ( j ): d j, C ∞ s ≃ (cid:26) d , τ ≫ l ( j )0 , τ ≪ l ( j ) . (15)Around the scale τ ≈ l ( j ), there is a peak of approx-imate height 1 . d [27]. We consider these features asdiscretization artefacts, and we conclude that no real di-mensional flow is seen for individual states in the super-position [27].Therefore, we are prompted to extend the search forquantum geometry states that would show true signs ofdimensional flow to superposition states, to which we nowmove. Using the above solution, the spectral dimensionof | V , j min , j max i , Eq. (6) in the limit N ( j min ) → ∞ , isgiven by the scaling of h b P ( τ ) i V ,j min ,j max ∝ j max X j = j min | a j | n e l − ( j ) τ I [ l − ( j ) τ ] o d . (16)For asymptotic power-law spectra l ( j ) ≃ j β , (17)where β > a j and various values of spatial dimension d and cutoffs j max . In all the examples presented here,we use j min = 1; calculations with lower cutoffs of thesame order (e.g., j min = 1 /
2) give similar results. Noticealso that the same finite minimal value for the geomet-ric spectra could be obtained in correspondence with aquantum label j = 0, for choices of quantization mapthat give a nonzero value for C in Eq. (2).The first general class of coefficients to be consideredis of power-law functions, a j ∝ j γ . (18)Defining the parameter α := − γ + 1 β , (19)the spectral dimension of the state under considerationhas the following behaviour depending on the range ofvalues of α . Τ d S FIG. 1. Spectral dimension of a superposition with α = 2 in d = 1 , , , j max = 10 d . • For 0 < α < d :(a) In the IR, i.e., for large length scales τ ≫ l ( j max ), d s ( τ ) = d (Fig. 1). This is of coursea consistency check for the validity of the for-malism, since at large scales we recover thetopological dimension of the space the quan-tum states are supposed to represent. It ishowever already a nontrivial test, as identify-ing quantum states with the right semiclassi-cal continuum properties at large scales is nosmall task in background-independent quan-tum gravity.(b) Below the smallest lattice scale, i.e., for τ ≪ l ( j min ), d s ( τ ) = 0. This is the usual dis-creteness effect which we find also for in-dividual lattice-based states (15), which re-mains at the Planck scale for discrete spec-tra induced by holonomies valued in compactgroups [33, 35, 36]. For noncompact groups,spectra are typically continuous and no vol-ume discreteness effect at Planck scale occurs,as j min → d s ( τ ) = α (Fig. 2). This plateau indi-cates a regime in which the effective dimen-sion is physically smaller than the topologicalone, and thus a proper dimensional flow. Inthe light of our previous results [27], which, asdiscussed, were performed on the same typeof quantum states and in the same formalism,but without considering large superpositionsof lattice structures, we regard this as a trulyquantum effect stemming from the superposi-tion of states | j, Ci with geometric spectra ondifferent scales and based on complexes of dif-ferent size. It is interesting that, at such inter-mediate scales, the effective dimension is in-dependent of the topological one (again, pro- Τ d S FIG. 2. Spectral dimension of superposition states with α =1 / , , / , d =3 with cutoff j max = 10 . Τ d S FIG. 3. Spectral dimension of superpositions with α = 2 in d = 3 for cutoffs j max = 1 , , , (dotted, dashed, dot-dashed, solid curve). vided d > α ) and depends instead only on thespecific choice of quantum states.(d) In particular, for infinite superpositions( j max → ∞ ) this plateau takes the value α and extends indefinitely (Fig. 3). Formally,one can express this behaviour by∆ τ (cid:12)(cid:12) d s = α −→ j max →∞ ∞ . (20)Notice that this only means that the topolog-ical dimension d is obtained further away atlarge τ . Physically, one never takes the infi-nite limit in practice: for large spin labels, theplateau is long but has finite extension ∆ τ .(e) Moreover, these results are independent of thespacing of the quantum labels j . Summingover j ∈ q N for some q ∈ Q slightly changesthe results only at the scale l ( j min ) (Fig. 4).Therefore, neither the IR nor the UV regimedepends on the spacing of the state label j . Τ d S FIG. 4. Spectral dimension of a superposition with α = 2 in d = 3 summing over positive j ∈ q N up to j max = 10 for q = 1 / , , ,
10 (dotted, dashed, dot-dashed, solid curve).
The numerical calculations show, in particu-lar, that this should also be true in the limit q → ∞ , i.e., for positive real j . • For α <
0, no superposition effect occurs andthe profile of the spectral dimension equals ap-proximately the one of the single state | j max , C ∞ i ,Eq. (15): d V ,j min ,j max s ( τ ) ≈ d j max , C ∞ s ( τ ) . (21)This is a numerical result, for which we lack, atpresent, a complete analytical or physical under-standing. Nevertheless, we can offer an intuitive ex-planation. We saw that, in the range 0 < α < d , α is the spectral dimension of the state at sufficientlysmall scales. On a continuous medium, the case α < P ( τ ) ∼ ( √ τ ) − d s ∼ ℓ − d s ∼ (res) d s is theprobability to find the probe anywhere when thegeometry is probed at scales ℓ , i.e., with resolution1 /ℓ . For positive d s , this probability decreases withthe resolution: if 1 /ℓ is too small, there is a chancethat we do not see the probe at all. On the otherhand, a negative d s implies that the coarser theprobe, the greater the chance to find it somewhere.In our case, this pathological behaviour is screenedby discreteness effects and d s is saturated by thelattice with labels j max . The resolution interpre-tation coupled with the LQG interpretation of thespin labels helps in explaining Eq. (15): coarserresolutions can effectively probe only large volumesand the largest volume available for the states (16)is at the upper cutoff j max . Under such conditions,it is natural to expect that the lattice structure completely dominates the profile of the spectral di-mension. • For α > d , no superposition effect occurs and theprofile of the spectral dimension equals approxi-mately the one of the single state | j min , C ∞ i , d V ,j min ,j max s ( τ ) ≈ d j min , C ∞ s ( τ ) . (22)In the continuum limit, α > d would imply a spec-tral dimension larger than the ambient space. Simi-larly to the previous case, one has both the diffusionand the resolution interpretation at hand. In theconventional diffusion interpretation of the spec-tral dimension, the case d s > d may be regardedas physical: the probe effectively sees more than d dimensions and tends to superdiffuse. In the reso-lution interpretation, the probability of finding theprobe somewhere grows more steeply than for thenormal case (Brownian motion) and probes withlarge resolution (small scales ℓ ) become even moreeffective. However, in the present quantum frame-work there is a limit to which one can probe themicroscopic structure of geometry: volume spectraare discrete with minimum eigenvalue determinedby j min . Again, the variation of the spectral di-mension is dominated by lattice effects, this timegoverned by the lower cutoff in the spin labels.A partial understanding of the results with 0 < α < d ,in particular concerning the dependence of the UV valueof d s on the powers β and γ in (19), is provided by thefollowing rewriting of the heat trace (16). A redefini-tion of variables k ( j ) := l − α ( j ) demands a change ofsummation-integration measure by dkdj = ddj l − α ( j ) = − α d ln l ( j ) dj l − α − ( j ) . (23)In particular, for the power-law spectra (17) and the def-inition of α (19) dkdj = − αβ j − αβ − = (2 γ + 1) j γ (24)which is proportional to | a j | for the power-law coeffi-cients (18). Thus, the heat trace on these superpositionsis a uniformly weighted sum in the k -variable [over therange corresponding to (16)]: h b P ( τ ) i V ,j min ,j max ∝ X k h e − k /α τ I ( k /α τ ) i d . (25)Therefore, genuine dimensional flow comes from a subtlebalancing of d and α in this expression, while a negative α yields just a dominant k max = k ( j max ) contributionin the sum. Indeed, we have also calculated the spec-tral dimension directly from (25) for various values of d , α and summing ranges of integer k ’s, obtaining qualita-tively similar results as discussed above for (16). As aconsequence, dimensional flow has some dependence onthe form of the spectrum (17) but only in combinationwith appropriate superposition coefficients.Still maintaining the power-law spectrum (17) (whichis the most reasonable assumption, consistent withknown results in LQG and related approaches), we havecalculated the spectral dimension for various other classesof coefficient functions. In most cases, there are no sur-prising results.(a) For example, exponential coefficients a j ∝ e aj leteither the maximal state j max dominate when a >
0, orthe minimal one j min when a <
0. (b) Gaussian coeffi-cients, on the other hand, result in a dominance of the j at which they are peaked. (c) Trigonometric func-tions add some oscillations to d j max , C ∞ s in the intermedi-ate regime, depending on the relation of the periods tothe spacing of j in the sum. In all these cases, there-fore, the overall behaviour of the spectral dimension isthe same as that found for coefficients given by simplepowers.More interesting is the case of coefficients which arelinear combinations of power functions in j . Then onefinds, for their asymptotic behaviour a j ∼ j γ , the sameeffect as for power-law coefficients. In particular, if thereare several regimes with different approximate scaling γ , γ , . . . , one obtains plateaux in the spectral dimen-sion plot of different values α , α , . . . accordingly. Anexample is shown in Fig. 5. This effect coincides, bothin its qualitative shape and origin, to the one obtainedin the multiscale generalization of the diffusion equationwith different powers of the Laplacian [41]. In general,all coefficient functions with an approximate power-lawbehaviour in some regime give rise to dimensional flowat those scales. Details such as the value of j min and thespacing in j are not relevant for the value of the spectraldimension in these intermediate regimes, in agreementwith the discussion in [41] on the role of regularizationparameters in the profile of d s . The details of regular-ization schemes are nonphysical and affect only transientregimes in d s ( τ ), not the value of the plateaux. B. Walk dimension of superpositions
The spectral dimension is only one of the possible di-mensional observables. Our strategy is well suited to an-alyze other observables as well, and it is interesting to doso because there exist several relations among them, inclassical and continuum spaces. Only a detailed analysisof their combined behaviour can give solid indications onthe nature of the quantum geometries corresponding toquantum gravity states.A closely related observable is the walk dimension d w .It is defined via the scaling of the mean square displace- - Τ d S FIG. 5. Spectral dimension of superpositions with coefficients | c j | = j − + 200 j − summing from j min = 1 / j max = 200for d = 3 and β = 3 (to be able to numerically cover enoughscales with a feasible number of states in the sum). Accordingto (19), two different UV regimes with dimension d s ≈ d s ≈ ment h X i y ( τ ) = Z dx | x − y | K ( x, y ; τ ) ∝ τ /d w , (26)that is d w ( τ ) := 2 (cid:18) ∂ ln h X i y ∂ ln τ (cid:19) − . (27)In the case of the d -dimensional hypercubic lattice C ∞ ,we can choose the origin y = 0, so that h X i C ∞ ( τ ) = X ~n ∈ Z d (cid:12)(cid:12) ~n (cid:12)(cid:12) K ( ~n, τ ) (28)= X ~n ∈ Z d d X j =1 n j e − τ d Y k =1 I n k ( τ ) . (29)This can be evaluated using standard relations of theBessel functions I n , h X i C ∞ ( τ ) ∝ e − dτ d X j =1 X ~n ∈ Z d n j d Y k =1 I n k ( τ ) (30)= e − dτ d X j =1 X n j ∈ Z d n j I n j ( τ ) × d Y k = j " X n k ∈ Z I n k ( τ ) = e − dτ d " τ X n ∈ Z I n − ( τ ) + I n +1 ( τ ) ( e τ ) d − = d τ . (31)Thus, the walk dimension on the lattice is d C ∞ w ( τ ) = 2 , (32)as in the continuum.Quantum superpositions | V , j min , j max i are character-ized by the Laplacian (12), so that along the same linesas (16) one has (cid:10) h X i ( τ ) (cid:11) V ,j min ,j max = j max X j = j min | a j | h X i C ∞ [ l − ( j ) τ ]= d j max X j = j min | a j | l − ( j ) τ (33) ∝ τ . (34)Therefore, also for quantum superpositions the scaling ofthe mean square displacement yields the standard result d V ,j min ,j max w = 2 , (35)independent of the form of the coefficients a j . Notice thatthe dependence on the topological dimension in Eq. (33)is only through a proportionality coefficient. Therefore,Eq. (35) is valid both for space and spacetime. C. Hausdorff dimension of superpositions
The Hausdorff dimension of a quantum state is definedin terms of the scaling of the expectation value of thevolume V ( R ) of a ball with radius R : d ψ h ( R ) := ∂ ln h V ( R ) i ψ ∂ ln R , (36)which can be further expanded like the spectral dimen-sion (11). Using the graph distance and measuring R inunits of the lattice spacing, the volume on the lattice C ∞ is V C ∞ ( R ) = 2 d (cid:18) R + d − d (cid:19) = 2 d d ! d − Y n =0 ( R + n ) , (37)yielding the Hausdorff dimension d C ∞ h ( R ) = R d − X n =0 R + n = R [ ψ ( R + d ) − ψ ( R )] , (38)where ψ is the digamma function. At large scales, d h approaches the topological dimension d , while at smallscales it tends to 1: d C ∞ h ≃ (cid:26) d , R ≫ , R ≪ . (39)On discrete quantum geometries |{ j c } , Ci , we definethe quantum analogue of V ( R ) as follows. Let v ∈ C be agiven vertex in the complex and consider the subcomplex C v ⊂ C of all vertices v which have an expectation valueof their distance to v no larger than the radius, h{ j c } , C| b L vv |{ j c } , Ci ≤
R , (40) where the expectation value of b L vv is the minimum oflengths derived from the sum of edge lengths of possible(combinatorial) paths between v and v .The expectation value of the volume of this subcom-plex |{ j c } , Ci is P v ∈C v h V v i { j c } , C . To obtain the desiredobservable, one must average over all possible centers v : h{ j c } , C| V ( R ) |{ j c } , Ci = X v ∈C X v ∈C v h{ j c } , C| V v |{ j c } , Ci . (41)On the uniform hypercubic lattice states | j, Ci , however,the sum over center vertices v is not necessary due totranslation invariance and because of the local volumesbeing all equal, h j, C| V v | j, Ci ∝ l d ( j ) for all v ∈ C ∞ .Similarly, on | j, Ci the condition (40) simplifies to h j, C| b L vv | j, Ci ∝ l ( j ) N vv (42)where now N vv is the minimal number of edges in a pathfrom v to v .Therefore, the evaluation of V ( R ) on | j, Ci can be ex-pressed in terms of V C ∞ ( R ) as h j, C| V ( R ) | j, Ci ∝ l d ( j ) V C ∞ [ R/l ( j )] ∝ l d ( j ) d − Y n =0 [ R/l ( j ) + n ] . (43)As for the spectral dimension (16), this gives a nontrivialexpectation value for generic superposition states: h V ( R ) i V ,j min ,j max ∝ j max X j = j min | a j | l d ( j ) d − Y n =0 [ R/l ( j ) + n ] . (44)Nevertheless, numerical calculations on the same classesof states as investigated above for the spectral dimensionshow qualitatively similar results to the Hausdorff dimen-sion d j, C ∞ h on single states | j, Ci (Fig. 6). That is, in allinstances there are plateaux as in the pure lattice case(37). Only the scale and steepness of the flow betweenthese plateaux is modified. For example, for power-lawcoefficients (18) the fall-off is much steeper and occurs,as α increases, closer to the scale as in the case of thesingle state | j min , C ∞ i . IV. DISCUSSION
Our calculations have shown that a flow in the spec-tral dimension occurs in quantum gravity, at least for aspecific class of superpositions of regular (both from thecombinatorial perspective and for what concerns the as-signment of additional quantum labels) quantum statesof geometry. These quantum states, although restrictedby the regularity assumption, are exactly of the typeappearing in the related quantum-gravity formalisms of d H FIG. 6. Hausdorff dimension d h of a superposition with α =1 , d = 3 summing up to j max = 10 , compared to d h on single states | , C ∞ i (dashedcurve) and | j max , C ∞ i (dotted curve). loop quantum gravity, spin-foam models and group fieldtheory, but can also simply be seen as quantum states oflattice quantum gravity, in the spirit of quantum Reggecalculus.On the other hand, we see no dimensional flow dueto quantum effects in the Hausdorff and walk dimension.This conclusion is based on the interpretation, which wemaintained throughout the paper, that the flow of a ge-ometric indicator is an artefact of discretization effectswhenever it approximately coincides with the flow forlattices. We will come back to this point.Let us comment a bit further about our results fromthe point of view of loop quantum gravity.Under the assumptions made for the action of thequantum Laplacian on the states (a very simple scalingbehaviour), an important example of states of the type wehave studied are kinematical states in LQG where length( d + 1 = 3) or area and volume operators ( d + 1 = 4)are diagonalized by spin-network states. In this sense,we have identified a class of LQG states with a dimen-sional flow. More precisely, for any 0 < α < d there is aclass of states in the kinematical Hilbert space with a di-mensional flow from the spatial topological dimension d in the IR to a smaller value α in the UV. The UV valuedepends on the exact superposition considered but noton the topological dimension.This result is in contrast with earlier arguments inLQG [26]. There, the author argues for evidence of di-mensional flow for individual spin-network states (thus,for a given graph or complex), and the same result isclaimed in [42, 43] for simple spin-network states withadditional weights given by a 1-vertex spin foam (thus,not yet in a truly dynamical context). The starting pointin [26] is an assumption about the scaling of the expec-tation value of the Laplacian, very similar to (12). Theessential part of the argument is then the further assump-tion that the momenta p of the scalar field defining the spectral dimension are directly related to a length scaleset by the quantum numbers as p ∝ / √ j . The scalingof the Laplacian in j is then translated into a modifieddispersion relation in p and the result depends on theprecise form of the spectrum (2) with C = 0.In our case, no further assumption beyond (12) ismade. Calculations are based on the momenta of thescalar field on the lattice-based geometry, that is, thespectrum of the Laplacian, but the spectral dimensionis computed directly as a quantum geometric observableevaluated on quantum states. As recalled already above,in a previous work using this more direct approach [27]we have found no effects on the spectral dimension forindividual quantum-geometry states of LQG based ongiven graphs or complexes. On the other hand, the gen-uine dimensional flow that we have encountered here forthe states | V , j min , j max i is crucially related to the super-position of spin-network states also with respect to theunderlying combinatorial structures, and it is not solelythe result of the discreteness of geometric spectra. In thisdeeper sense, dimensional flow can indeed be seen as aneffect of quantum discreteness of geometry.We are also in a position to characterize the changeof dimensionality more precisely than a generic “flow” ofgeometry. Quite often in the literature of quantum grav-ity, dimensional flow has been advertised as spacetimebeing “fractal.” However, strictly speaking not all setswith varying dimensionality are fractals. Although nounique operational and rigorous definition of fractal ex-ists, one property all fractals generally possess is a specialrelation among the spectral dimension d s , the Hausdorffdimension d h and the walk dimension d w : d h = d w d s . (45)On the hypercubic lattice superpositions that we haveconsidered, d w = 2 and the above relation simplifies to d h = d s . This is trivially obtained in the IR regime,where both dimensions take the value of the topologicaldimension. In the UV regime above the lattice scale (re-call that below such scale any scaling effect is arguablyunphysical), the Hausdorff dimension takes the classicalvalue d C ∞ h ≃
1. Thus, (45) is only obeyed in the caseof a scaling α = 1 such that also the spectral dimensiontakes this value. Only then can one call the quantum su-perposition | V , j min , j max i an effective one-dimensionalfractal. This is indeed a perfectly allowed choice of quan-tum states and we can conclude that we have identified aparticular class of quantum geometries that corresponds,by all appearances, to a fractal quantum space.However, we should mention a caveat here. For thesegeometries to be safely regarded as fractals, the originof the dimensional flow should be the same in the left-and right-hand side of Eq. (45), which may not be thecase for us: the left-hand side flows due to discretenesseffects, while the right-hand side flows due to physical0quantum effects. This situation might suggest either thatwe should not place particular significance in the fulfill-ment of Eq. (45) or that our discrimination between dis-creteness artefacts and physical effects is somewhat toostrong and should be revised. We no dot attempt to solvethis mild conceptual issue here, which is harmless for ourmain results. Still, it will deserve further attention.Interestingly, the geometry with α = 1 is also the onlyone where the spectral dimension of spacetime reachesthe value D s = d s + 1 ≃ D s = 2 in the UV [45], as well as oflocal theories whose renormalization properties are not atall improved by dimensional flow [46]. Here we provideanother instance pointing towards the same conclusion:the value of d s is governed by a choice of states which, byitself, is not (sufficiently) connected with the dynamicalUV properties of the underlying full theory. V. CONCLUSIONS
We have investigated the effective structure of quan-tum superpositions of regular (hypercubic and homoge-neous in label assignment) states of quantum geometry.It is possible to identify states with a flow of the spec-tral dimension to a dimension α in the UV, provided thesuperposition includes fine enough combinatorial struc-tures and a large enough number of (kinematical) degreesof freedom of quantum geometry, and a particular set ofexpansion coefficients (18) related to α (19).For the Hausdorff and walk dimension, no physicalquantum effects are observed, although discreteness ef-fects do alter the value of the Hausdorff dimension acrossscales. A fractal structure in the strict sense, i.e., where(45) relating the three dimensions is fulfilled also in theUV, is realized in the case α = 1.In particular, these results provide evidence for a di-mensional flow in a certain class of kinematical LQGstates, also available in the spin-foam and group fieldtheory context.The results at hand can be further generalized in vari-ous directions as well as refined within individual theoriesof quantum gravity. In parallel, it becomes feasible to ex-plore the phenomenological consequences of the discov-ered dimensional flow and (when applicable) fractal na-ture of quantum space as a direct effect of the full theory.This possibility is especially interesting in a quantum cos-mological context, where a change of dimensionality canbear its imprint in the early stage of cosmic evolution[47–49]. [1] C. Rovelli, Quantum Gravity (Cambridge UniversityPress, Cambridge, 2004).[2] C. Rovelli,
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