Hausdorff dimension of a particle path in a quantum manifold
aa r X i v : . [ g r- q c ] J a n Hausdorff dimension of a particle path in a quantum manifold
Piero Nicolini ∗ and Benjamin Niedner †‡ Frankfurt Institute for Advanced Studies (FIAS),Institut f¨ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨at,Ruth-Moufang-Straße 1, 60438 Frankfurt am Main, Germany (Dated: August 24, 2018)After recalling the concept of the Hausdorff dimension, we study the fractal properties of a quantumparticle path. As a novelty we consider the possibility for the space where the particle propagates tobe endowed with a quantum-gravity-induced minimal length. We show that the Hausdorff dimensionaccounts for both the quantum mechanics uncertainty and manifold fluctuations. In addition thepresence of a minimal length breaks the self-similarity property of the erratic path of the quantumparticle. Finally we establish a universal property of the Hausdorff dimension as well as the spectraldimension: They both depend on the amount of resolution loss which affects both the path and themanifold when quantum gravity fluctuations occur.
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Fractals are often employed to describe the nature of aquantum manifold. Indeed one of the widely expectedfeatures in quantum gravity is the appearance of space-time fluctuations as far as distances comparable with thePlanck length are probed. Fluctuations of this kind im-ply a loss of resolution: Distances smaller than the Plancklength cannot be resolved, and one often speaks of mini-mal length effects when the spacetime passes from a lowenergy differential manifold to its Planck energy quan-tum configuration. Fractals nicely encode the idea ofquantum fluctuations and loss of resolution. Another at-tractive property of fractals is self-similarity, namely, theproperty of being exactly similar to a part of itself. Thisfeature is connected to the concept of scale invariance,which seems to be supported by recent nonperturbativestring theory developments like AdS/CFT and M the-ory. To investigate the properties of a quantum space-time one can employ technical tools which belong to thetheory of fractals. As an example an important issue isthe calculation of the spectral dimension, i.e., the mani-fold dimension perceived by a diffusion process. The wayone formulates the presence of quantum fluctuations ofthe manifold is crucial and gives rise to a variety of ex-pressions for the spectral dimension [1–10]. As soon asthe diffusion starts, small length scales of the manifoldare probed and strong fluctuations emerge. The diffusionprocess is therefore subjected to a loss of resolution andthe spectral dimension turns out to be smaller than theactual topological dimension of the manifold. In the caseof a four-dimensional manifold, one of the crucial featuresis that at the Planck length the spectral dimension equalstwo, supporting the idea of a renormalizable character ofthe gravitational interaction as recently shown in [11]. † Current address: Theoretical Physics Group, Imperial CollegeLondon Prince Consort Road, London SW7 2AZ, UK ∗ [email protected] ‡ [email protected] Another measure of the fractal nature of a manifold isprovided by the Hausdorff dimension. One of the essen-tial features of a fractal is that its Hausdorff dimensionstrictly exceeds its topological dimension [12]. As an ex-ample a quantum particle proceeds along an erratic pathwhose Hausdorff dimension is two, i.e., exceeding the di-mension of a classical trajectory [13]. For the stringyanalogue, it has been shown that the world sheet makesa transition to an excited configuration as far as lengthscales of order ( α ′ ) / are concerned. In this case theworld sheet becomes a fractal surface of dimension three,since the energy in the excited state lets the string ex-plore an additional dimension [14–16] (for further appli-cations see [17]). While the spectral dimension accountsfor the fractal character of the space where the diffusiontakes place, the Hausdorff dimension for a quantum par-ticle is just an indicator of the amount of uncertainty of aquantum path. Nothing is said about the intrinsic uncer-tainty, which any model of quantum spacetime should beendowed with. In other words in [13] the backgroundspace where the particle propagates is still a classicalmanifold. In this paper we want to do a step forward,by implementing the presence of a minimal length in thebackground space where a quantum particle propagates.Therefore this paper has three main goals:1. to provide an example where the Hausdorff dimen-sion can be employed as an indicator of the amountof fluctuations of the manifold rather than of theparticle path;2. to disclose further properties of our method of im-plementing an effective minimal length in the man-ifold other than those we found by studying thespectral dimension in [11];3. to understand whether universal properties exist asa result of the study of both indicators of quantumfluctuations, i.e., the spectral dimension and theHausdorff dimension. FIG. 1: Construction of the Koch curve. At each step, themiddle third of each interval is replaced by the other two sidesof an equilateral triangle.
Given this background we briefly recall the definitionof the Hausdorff dimension. We start by considering theKoch curve in Fig. 1. It is an example of an every-where continuous but nowhere differentiable curve. Wecan construct the Koch curve as a final product of aninfinite sequence of steps. At each step, the middle thirdof each interval is replaced by the other two sides of anequilateral triangle. As a result at each step the length ofthe curve increases by a factor 4 /
3, so the final curve isinfinitely long. However if we assume viewing the curvewith a finite resolution ∆ x , many wiggles smaller than∆ x are neglected. Here the resolution ∆ x has only amathematical meaning related to the diameter of spherescovering the curve [18]. As a result the observed lengthof the curve turns out to be finite, becoming infinite onlyin the limit ∆ x →
0. Let l be the length of the curvewhen the resolution is ∆ x = 0. Improving the resolutionso that ∆ x ′ = (1 / x , we proceed along the next stepof the curve and new wiggles become visible. As a con-sequence we will measure a new length l ′ = (4 / l . Thelength of the curve depends on the resolution at whichthe curve is examined. Therefore we cannot uniquely de-fine the length of the curve in this way. To solve thisproblem Hausdorff proposed a new definition of lengthgiven by L H = l (∆ x ) D H − . (1)Here l is the usual length when the resolution is ∆ x ,and D H is a real number chosen so that L H will be in-dependent of ∆ x , at least in the limit ∆ x →
0. Theparameter D H is called the Hausdorff dimension . Whenthe Hausdorff length L H coincides with the usual length l , the Hausdorff dimension equals the topological dimen-sion d top = 1 of the curve. For the Koch curve we cancalculate the Hausdorff dimension by requiring L H = L ′ H namely, l ′ (∆ x ′ ) D H − = l (∆ x ) D H − . (2)This implies that D H = ln 4 / ln 3. The fact that D H = 1identifies the curve as a fractal.We now switch from mathematics to physics. To dothis we need to connect our “mathematical” resolution∆ x with a physically meaningful quantity. Along thelines of [13] we consider the natural case of quantum me-chanics. Typical paths of a quantum-mechanical particleare highly irregular on a fine scale. According to theHeisenberg uncertainty principle the more precisely theparticle is located in space, the more its path will be-come increasingly erratic. If the localization of the par-ticle is within a region of size ∆ x , an uncertainty willaffect the momentum of order ~ / ∆ x . In the languageof fractals, this is equivalent to saying that paths for aquantum-mechanical particle are not those which admit adefinite slope (velocity and therefore momentum) every-where. For this reason in quantum mechanics we cannotproperly speak of a particle path unless in the statisticalsense. Suppose now to measure the position of a quan-tum particle at a sequence of times t , t = t + ∆ t , ..., t N = t + N ∆ t , with T = t N − t = N ∆ t . Then thelength of the path will be h l i = N h ∆ l i (3)where h ∆ l i = h ψ | ˆ U † (∆ t ) | ˆ x | ˆ U (∆ t ) | ψ i (4)is the average distance which the particle travels in atime ∆ t , with ˆ U ( t ) = exp( − i ˆ p t/ m ~ ) the free particletime evolution operator. Here the wave function of theparticle ψ ( x ) = h x | ψ i takes into account the fact that aposition measurement only localizes the particle withina region of size ∆ x . For later convenience we introducethe following dimensionless quantities: y ≡ x / ∆ x and k ≡ p ∆ x/ ~ . If we consider the case where the averagemomentum of the particle is zero, we obtain h ∆ l i = Z d d x | x | | ψ ( x , ∆ t ) | (5)= (∆ x ) Z d d y | y | (cid:12)(cid:12)(cid:12)(cid:12)Z d d k (2 π ) d/ h k | ψ i e (cid:16) i ky − i ~ ∆ t m (∆ x )2 k (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) . The actual profile of ψ ( k ) ≡ h k | ψ i will not be impor-tant for our discussion. We just need a square integrablefunction, which can localize the particle to a region ofsize ∆ x . Thus a choice for ψ ( k ) is h k | ψ i = (cid:18) π (cid:19) d/ e − k . (6)Note that any other square integrable function can beapproximated arbitrarily well by a linear combination ofsuch Gaussian functions. As a result, one finds h ∆ l i ∝ ~ ∆ tm ∆ x s (cid:18) m (∆ x ) ~ ∆ t (cid:19) . (7) a (cid:13) b (cid:13) c (cid:13) FIG. 2: Schematic view of the geometrical structure of the particle path. In (a) we have the classical regime and in (b) thequantum-mechanical regime. Upon further magnification in (c), the path exhibits the same structure.
Recalling that ∆ x is our resolution parameter, we canvary ∆ x by keeping ∆ t fixed. In the case ∆ x →
0, (7)reduces to h ∆ l i ∝ ~ ∆ t/m ∆ x, (8)a result supported by the uncertainty principle. As aconsequence h l i ∝ ~ T /m ∆ x, (9)a length which is ill defined since it depends on the de-tection resolution ∆ x and diverges in the limit ∆ x → h L H i = h l i (∆ x ) D H − (10)which turns out to be independent of ∆ x if D H = 2.The path of a quantum particle is therefore a fractal ofdimension two. We can check the other interesting prop-erty of fractals: self-similarity. In the case of the Kochcurve self-similarity is evident by increasing the resolu-tion ∆ x ′ = (1 / x . In analogy the path of a quantumparticle is self-similar if h ∆ l i ∝ ∆ x . This relation implies∆ t ∝ m (∆ x ) / ~ (11)which naturally arises in the derivation of h ∆ l i as aconsequence of the uncertainty principle in the energy-momentum relation E = p / m . The whole descriptionof the path of a quantum particle can be generalized tothe case when the particle has some nonzero average mo-mentum p av . In this case the Hausdorff dimension is D H = 1 when the distances being resolved are muchlarger than the particle’s wavelength, i.e., ∆ x ≫ ~ / | p av | .Conversely it is D H = 2 when the distances being re-solved are much smaller than the particle’s wavelength∆ x ≪ ~ / | p av | . In the region between these limits theHausdorff dimension D H is not well defined, since thereis a “phase transition” from the classical to the quantum-mechanical path (see Fig. 2).We are now ready to switch from quantum mechanicsto quantum gravity. One of the most important featuresof quantum gravity is the appearance of an additionalkind of uncertainty which prevents one from measuring positions to better accuracies than the Planck length.Indeed the momentum and the energy required to makesuch a measurement will itself modify the spacetime ge-ometry at these scales [19]. This long held idea has beencorroborated by the noncommutative character of openstring end points on D-branes [20]. The specific feature ofthe presence of a minimal length in a spacetime manifoldcan be taken into account by means of effective theo-ries too. Though they are not the full theory of quan-tum gravity, these effective formulations are particularlyuseful for getting reliable phenomenological scenarios inspecific physical contexts [21, 22]. For instance, an ef-fective minimal length has efficiently been included inthe physics of evaporating black holes, by smearing outthe curvature singularity at the origin and regularizingthe terminal phase of the Hawking emission [23] (for re-views see [24, 25], and the references therein). At thebasis of these approaches there is the possibility of pro-viding a delocalization of point like objects by the actionof a nonlocal operator e ℓ ∆ x [26]. Here ℓ is the mini-mal length and ∆ x is the Laplacian operator acting ona d -dimensional Euclidean manifold. As an example weconsider the Dirac delta δ ( x ) as a standard distributionfor a point like object. By applying the nonlocal operatorone finds δ ( x ) → e ℓ ∆ x δ ( x ) = ρ ℓ ( x ) (12)where ρ ℓ is the modified distribution due to the presenceof the minimal length ℓ . It turns out that the modifieddistribution is ρ ℓ ( x ) = 1(4 πℓ ) d/ e − x / ℓ (13)which is nothing but a Gaussian distribution whose widthequals ℓ . Indeed this is the most narrow distributionwhich is admissible on a manifold endowed with a min-imal length. In [11] it has been shown that the primaryeffect of the presence of a minimal length in the dynamicsof a diffusion process lies in a smearing of point like initialconditions as in (12). We recall that a diffusion process isnothing but a Wick rotated quantum-mechanical prob-ability evolution. It is therefore natural to extend themethod we employed in [11] for the spectral dimensionto the case of the Hausdorff dimension too. In the lattercase, the action of the nonlocal operator e ( ℓ / ~ )∆ x de-termines a modification of the integration measure of themomentum space representation of the wave function h x | ψ i = Z dV ℓ ( p ) h p | ψ i e i ~ px (14)where dV ℓ ( p ) = d d p (2 π ~ ) d e − ℓ ~ p (15)turns out to be squeezed for large momenta only, i.e., | p | & ~ /ℓ . We stress that this kind of deformation of theintegration measure is in agreement with what one re-quires in a generic effective approach to quantum gravity.Indeed the ultraviolet convergence is obtained in generalas dV ( p ) = 1(2 π ~ ) d d d p (1 + f ( p )) (16)where f ( p ) depends on positive powers of the argument[22, 27]. The specific choice f ( p ) = β p correspondsto the case of the generalized uncertainty principle. Thedeformation we are following in (15) is simply equivalentto f ( p ) = e ℓ ~ p / −
1. This choice has two importantvirtues: It keeps the general features of ultraviolet con-vergence required by any effective approach to quantumgravity; it turns out to be the strongest possible suppres-sion of higher momenta.We now have all the ingredients to investigate the frac-tal properties of the path of a quantum particle whichpropagates in a d -dimensional manifold endowed with aminimal length ℓ . We start from the case where the av-erage momentum of the particle is zero. The path willbe affected by both the quantum mechanics uncertaintyencoded in ~ and the quantum gravity uncertainty en-coded in ℓ . As a result the mathematical resolution ∆ x will be related to both ~ and ℓ . We want to understandthe response of the manifold when it is probed at variousregimes of energy, irrespective of the nature of the probe,i.e., the quantum particle. For this reason we stick withthe nonrelativistic formulation. Relativistic effects solelyamount to reducing the Hausdorff dimension to D H = 1in the ultrarelativistic limit where the particle is con-fined to the lightcone [28]. As will be clear from the fol-lowing discussion, this behavior due to relativistic effectsdoes not counteract our conclusions. To this purpose,we recall that the nonrelativistic procedure has alreadybeen used in the context of the Hausdorff dimension ofa quantum string, even if string excitations, which arecrucial for calculating the Hausdorff dimension, occur inthe ultrarelativistic regime [14]. This line of reasoning isin analogy to what happens for the spectral dimension,since the diffusion equation is formally equivalent to aWick rotated Schr¨odinger equation for a nonrelativisticparticle. -2-1012 0 1 2 3 4 D H ∆ x/ℓ D = 2 D = 3 D = 4 FIG. 3: The Hausdorff dimension of a particle path dependingon ∆ x/ℓ for different numbers of spacetime dimensions.
The crucial quantity is h ∆ l i , i.e., the expectation valuefor traveled path length in time lapse ∆ t . To determinethe new value for (4) we need to calculate h y | ˆ U (∆ t ) | ψ i = Z d d k (2 π ) d e − ℓ k x )2 h k | ψ i e i ky e i ~ ∆ t k m (∆ x )2 (17)where h k | ψ i is chosen as in (6). From (3) we obtain thelength of the path: h l i ∝ ~ Tm ∆ x (cid:18) ℓ (∆ x ) (cid:19) − d +12 × s (cid:18) ℓ (∆ x ) (cid:19) m (∆ x ) ~ (∆ t ) . (18)As far as we consider length scales ℓ ≪ ∆ x ≪ p ~ ∆ t/m , the actual length of the path is an ill definedquantity which depends on the choice of the resolutionparameter ∆ x , i.e., h l i ∝ ~ T /m ∆ x , which matches theconventional result in (9). However (18) presents an im-portant new feature. The quantity h l i can never be in-finite. Indeed in the limit ∆ x ≪ ℓ , by keeping ∆ t fixedone finds h l i ∝ ~ Tmℓ (cid:18) ∆ xℓ (cid:19) d . (19)This is due to the presence of ℓ , which nicely works as anatural cutoff in agreement with all the existing literaturebased on this formulation [24]. The problem is that ℓ is actually a minimal length, beyond which we lose thedefinition of position. In other words, for ∆ x ≪ ℓ weare probing the microstructure of the manifold, which isaffected by huge quantum geometry fluctuations. As aresult the very concept of length is no longer meaningful,a fact which is confirmed by the vanishing value of h l i .Again we are left with the only possibility of invokingthe Hausdorff length to have some reliable informationabout the length of the path. By using (10), we obtain L H ∝ (∆ x ) D H − (cid:18) ℓ (∆ x ) (cid:19) − d +12 . (20)In the regime ℓ ≪ ∆ x ≪ p ~ ∆ t/m we find the con-ventional result D H = 2. Conversely for ∆ x ≪ ℓ wefind D H = 1 − d . This means that the Hausdorff di-mension is either vanishing or negative. In fractal geom-etry this is the case of an empty set [29], which physi-cally we could interpret as a “dissolution” of the path asfar as trans-Planckian scales are probed. By requiring ∂L H /∂ (∆ x ) = 0 we can calculate the general form of theHausdorff dimension which reads D H = 2 − d + 11 + (∆ x ) /ℓ . (21)Some comments are in order. First, the Hausdorff di-mension is always smaller than the usual value 2. Thisis reminiscent of what we found when studying the spec-tral dimension in [11]. We recall that in a D -dimensionalEuclidean geometry the heat equation reads∆ K ( x, y ; s ) = ∂∂s K ( x, y ; s ) (22)where s is a fictitious diffusion time of dimension of alength squared, ∆ is the Laplace operator, and K ( x, y ; s )is the heat kernel, representing the probability density ofdiffusion from x to y . We showed that the minimal length ℓ , by introducing quantum gravity fuzziness, prevents thediffusion process to access to all the D topological dimen-sions of the spacetime manifold. More specifically from(22) the spectral dimension for the flat space case turnsout to be D = ss + ℓ D. (23)i.e., it is smaller than the topological dimension of space-time D < D = d + 1. In other words, while the uncer-tainty in quantum mechanics provides an erratic charac-ter to the path and a consequent increase of the Hausdorffdimension, the uncertainty in quantum gravity is respon-sible for resolution loss, whose amount is encoded in thedifference D − D . This fact becomes even more clear ifwe Wick rotate back the diffusion equation (22) and weidentify the diffusion time with (∆ x ) by means of therelation (11). As a result one finds D H = 2 − ( D − D ) . (24)In the special case D = 2, there is just one spatial di-mension and the two indicators coincide: D H = D .Second, one might ask whether the Hausdorff dimen-sion assumes the classical value D H = 1. From (21) thiscondition is met when d = (∆ x ) /ℓ . As a result we have h L H i = h l i ∝ d d/ ( d + 1) d +12 ~ Tmℓ (25)which enjoys the desired feature of being independent ofthe resolution parameter ∆ x . However we cannot inter-pret this result as a restoration of the classical characterof the path. We should better say that we have another example in which ℓ provides a finite value for the properlength h l i of a fractal. Indeed the Hausdorff dimensioncan even descend below the value D H = 1 and beyond.For scales ( d − / x ) /ℓ we have D H = 0. TheHausdorff dimension reaches negative values correspond-ing to the case of empty sets. The full behavior of D H can be seen in Fig. 3.Third, there is the issue of self-similarity. The fact thatthe Hausdorff dimension actually can descend below thetopological value 1 is a sort of “red flag.” This happenswhen we are probing the path with resolution comparablewith the size of the minimal length ∆ x ∼ ℓ . In otherwords, quantum gravity must introduce a length scale.At such a scale the manifold fluctuations are so strongthat the path starts dissolving. As a result we have abreaking of the self-similarity or scale invariance propertyof the path. To check this result we just need to studythe self-similarity condition h ∆ l i ∝ ∆ x . For ℓ ≪ ∆ x ≪ p ~ ∆ t/m we just recover the conventional result as in(11). Conversely for ∆ x . ℓ we get h ∆ l i ∝ ℓ (cid:18) ∆ xℓ (cid:19) d +2 (26)which is different from (11) unless one fixes ∆ x = ℓ . Thisimplies a breaking of scale invariance in the transitionfrom one to the other regime.We are now ready to draw conclusions. In reference tothe three main goals we can say that1. the Hausdorff dimension we calculated in (21) ac-counts for both the quantum mechanics uncertaintyand the amount of fluctuations of the manifoldthrough the term ∆ x/ℓ . In addition D H dependsalso on the number of dimensions d of the manifoldwhere the particle propagates;2. the new feature we discovered through the study of D H is the expected scale invariance breaking as faras one introduces a length scale ℓ in the formalism;3. the universal property we discovered in both indi-cators, i.e., D H and D , is the amount of resolutionloss D − D which affects both the path and the man-ifold when quantum gravity fluctuations occur.We could generalize our calculation to the case of nonva-nishing average momentum p av . However one can provethat there is no additional physical information comingfrom it. Essentially the conclusions we have just drawnare confirmed.As a final point we can summarize our results forthe character of a quantum path in the presence of aminimal length with the following scenario: As longas ∆ x ≫ p ~ ∆ t/m there exists a classical regime inwhich the path is represented by a smooth differentialcurve, whose Hausdorff dimension coincides with thetopological dimension of the curve, i.e., D H = 1; for ℓ ≪ ∆ x ≪ p ~ ∆ t/m there is the quantum mechan-ics regime , in which the path of the particle becomes a (cid:13) b (cid:13) c (cid:13) b b b b b b b d (cid:13) e (cid:13) FIG. 4: Schematic view of the geometrical structure of the quantum particle path in (a) the classical regime, (b) the quantum-mechanical regime, (c),(d) the Planckian regime, and (e) the trans-Planckian regime. strongly erratic and self-similar, approaching the config-uration of a D H = 2 fractal; for smaller length scales, i.e.,∆ x ∼ ℓ , there is the Planckian regime , which is charac-terized by a loss of resolution of the path with conse-quent decrease of the Hausdorff dimension and break-ing of the self-similarity property; finally in the trans-Planckian regime , i.e., ∆ x ≪ ℓ , the path is disintegratedby huge fluctuations of the manifold and the Hausdorffdimension can be vanishing or even negative, correspond-ing to the case of an empty set (see Fig.4).For the sake of truth we have to notice that, even iffascinating and mathematically correct, the interpreta-tion of the trans-Planckian regime in terms of empty setsmight be not fully physically consistent. We recall thatour results are based on an effective approach to quan-tum gravity, which captures the feature of the emergenceof a minimal length. We actually ignored the ultimatefate of a particle path as well as of a manifold in thetrans-Planckian regime. Therefore we must keep our in-terpretation in terms of empty sets just as an indicationfor a possible scenario and assume reliability of our treat-ment only for length scales ∆ x & ℓ . We also add thatour results have been derived in terms of a specific ap-proach to model the presence of a minimal length. Yet our model captures some general feature of any effectiveapproach to quantum gravity by squeezing momentumspace integration measure at higher momenta. In othercontexts, like black hole physics, this approach has ledto model-independent descriptions [23]. 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