Multiscale spacetimes from first principles
aa r X i v : . [ g r- q c ] M a r PHYSICAL REVIEW D , 064057 (2017) arXiv:1609.02776 Multiscale spacetimes from first principles
Gianluca Calcagni Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid, Spain (Dated: September 7, 2016)Assuming only a smooth and slow change of spacetime dimensionality at large scales, we find,in a background- and model-independent way, the general profile of the Hausdorff and the spectraldimension of multiscale geometries such as those found in all known quantum gravities. Examples ofvarious scenarios are given. In particular, we derive uniquely the multiscale measure with log oscil-lations of theories of multifractional geometry. Predictivity of this class of models and falsifiabilityof their abundant phenomenology are thus established.
I. INTRODUCTION AND MAIN RESULT
The quest for a consistent theory of quantum gravita-tion unraveled the unexpected existence of generic prop-erties of geometry found across all known frameworks[1]. First, that in a quantum setting geometry is anoma-lous and areas and volumes can behave very differentlyfrom their classical counterparts. Second, that the di-mension of space or spacetime changes with the scale ofobservation. A relation between this dimensional flow or multiscaling and the ultraviolet (UV) properties of thetheories was long since suspected but nowadays it hasbecome clear that there is no direct universal link be-tween the change of dimensionality and the microscopicfiniteness of the quantum forces. However, it remainsunclear why dimensional flow can often be described bysimilar asymptotic expressions in different theories. Inthis paper, we give an answer purely based on the in-frared (IR) properties of the flow and independent of thedynamics, both for the Hausdorff dimension d H and forthe spectral dimension d S of spacetime. First flow-equation theorem.
Assume that(I) dimensional flow of spacetime in dimension d = d H or d = d S is described by a continuous scaleparameter ℓ ;(II) this flow is slow at scales larger than a referencescale ℓ ∗ separating the IR from the UV;(III) effective spacetime is noncompact.Then, at mesoscopic scales the most general real-valuedspacetime dimension is d ( ℓ ) ≃ D + b (cid:18) ℓ ∗ ℓ (cid:19) c + (log oscillations) , (1) where, to leading order, the log-oscillatory part is of theform ( ℓ /ℓ ) c ˜ F ω ( ℓ ) , c may differ from c , and the modu-lation factor ˜ F ω is given by ˜ F ω ∝ cos (cid:20) ω ln (cid:18) ℓℓ ∞ (cid:19)(cid:21) or ˜ F ω ∝ sin (cid:20) ω ln (cid:18) ℓℓ ∞ (cid:19)(cid:21) . (2) As a consequence, and up to an overall normalization,for d = d H V ( ℓ ) ≃ ℓ D (cid:20) − b H c H (cid:18) ℓ ∗ ℓ (cid:19) c H (cid:21) + (log oscillations) (3) represents a generic Euclidean(ized) D -volume of linearsize ℓ . For d = d S , P ( ℓ ) ≃ ℓ D (cid:20) b S c S (cid:18) ℓ ∗ ℓ (cid:19) c S (cid:21) + (log oscillations) (4) is the return probability of spacetime. Exact expressionswill be given in the text. Terminology will be explained shortly. We will com-pare the flow in d H and d S of a number of popular theoriesof quantum gravity and find that they all realize the uni-versal behavior predicted here. The sole and surprisinglysimple reason is that dimensional flow is always slow inthe IR and always reaches the topological dimension asan asymptote. This universal multiparametric form ofthe change in spacetime dimensionality must be comple-mented by information from the dynamics (which fixesthe numerical values of the parameters) but it can havea great impact on model-independent phenomenology.In parallel, we will also settle a long-standing issueconcerning a class of theories of anomalous geometryknown as multifractional spacetimes (see [2] and refer-ences therein). These theories, three in total, had a dou-ble purpose originally. On one hand, to quantize gravityperturbatively in such a way that dimensional flow beunder analytic control at all scales; this objective is stillunder pursue [2]. On the other hand, by virtue of thesupposed universality of certain properties of the flow,multifractional theories can be interpreted not only asstand-alone proposals, but also as effective descriptions ofother independent multiscale theories of quantum grav-ity, most of which take considerable effort to produceusable phenomenology. The disarmingly easier way ofmultifractional theories to make contact with observa-tions makes them an ideal testing ground of the typeof phenomena we would expect if geometry was, as inall quantum gravities, multiscale. The measure beingthe same in all theories and their dynamics being rela-tively simple (except in the case with so-called fractionalderivatives), it was possible to obtain a number of con-straints from observations ranging from particle-physicsand atomic scales to astrophysics and cosmology [2].All theoretical and phenomenological results of multi-fractional theories rely on the assumption that the barespacetime geometry (i.e., before considering the dynam-ics) is multifractal. In early papers, it was argued thatdimensional flow can be well captured by a multifrac-tal measure [2] which, when coordinates factorize andthe continuum approximation is taken, is of the form Q µ dq µ := dq ( x ) dq ( x ) · · · dq D − ( x D − ) in D integertopological dimensions, where each q µ ( x µ ) is a real poly-nomial with noninteger exponents and logarithmic oscil-lations, decorated with a number of length scales ℓ , , , ··· .Omitting the label µ in the quantities x µ , α µ , and ℓ µn be-low, the measure in each direction is q ( x ) = x + + ∞ X n =1 ℓ n α n (cid:12)(cid:12)(cid:12)(cid:12) xℓ n (cid:12)(cid:12)(cid:12)(cid:12) α n F n ( x ) , (5) F n ( x ) = 1 + A n cos (cid:18) ω n ln (cid:12)(cid:12)(cid:12)(cid:12) xℓ ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + B n sin (cid:18) ω n ln (cid:12)(cid:12)(cid:12)(cid:12) xℓ ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , where α n > α n +1 , A n , B n are constants comprised be-tween 0 and 1, ω n are frequencies, and ℓ ∞ · · · ℓ n ℓ n − · · · ℓ ℓ ≡ ℓ ∗ (6)are the fundamental scales of the geometry. ℓ ∞ can beidentified with the Planck length [2] and, for this reason,it is placed at the bottom of the hierarchy. Papers onthe phenomenology of these models invoke the simplestmultifractional example, the binomial measure q α ( x ) = x + ℓ ∗ α (cid:12)(cid:12)(cid:12)(cid:12) xℓ ∗ (cid:12)(cid:12)(cid:12)(cid:12) α F ω ( x ) (for each direction) , (7a)where F ω ( x ) = 1 + A cos (cid:18) ω ln (cid:12)(cid:12)(cid:12)(cid:12) xℓ ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + B sin (cid:18) ω ln (cid:12)(cid:12)(cid:12)(cid:12) xℓ ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . (7b)At scales ℓ ∼ ℓ ∞ , the measure approximately enjoysthe discrete scale invariance q α ( λ ω x ) ≃ λ αω q α ( x ) underthe dilation x → λ ω x , where λ ω := exp( − π/ω ). Atscales ℓ ∞ ≪ ℓ ∼ ℓ ∗ = ℓ , the log oscillations can becoarse grained and spacetime becomes effectively contin-uous, with Hausdorff dimension d UVH ≃ P µ α µ . At scales ℓ ≫ ℓ ∗ , one recovers standard Minkowski spacetime with d H ≃ D . The binomial measure (7) encodes the geom-etry at scales larger than the largest characteristic scale ℓ = ℓ ∗ , thus removing temporarily the need to considerthe polynomial multifrequency measure (5).Despite its practical advantages, the whole multifrac-tal spacetime paradigm has been criticized for relying ontoo strong a statement. Why using measures such as (5)or (7) and not others? If other measures realizing di-mensional flow are possible, then are not multifractionaltheories ad hoc and nonpredictive? We now state theanswer, which is the second result of this paper: Second flow-equation theorem.
Assume (I)–(III) asbefore and that(IV) the measure is factorizable.Then, d = X µ d ( µ ) , (8) where for each direction d ( µ ) is given by Eq. (1) with D = 1 , b → b µ , and c → c µ . For d = d H , the mostgeneral real measure is given by Eq. (5) at all scales andby Eq. (7) at mesoscopic scales. For a trivial flow, ℓ n = 0 for all n . If c >
0, Eq. (1) recovers the correct dimension at largescales, either from above ( b >
0) or from below ( b < d S found in quantum gravities. On the other hand, (5) isan infinite perturbative expansion around the IR pointand it may be inconvenient for writing analytic expres-sions of d ( ℓ ) ≃ P µ α ,µ + O ( ℓ ) in the deep UV. Inciden-tally, Eq. (1) shows the existence of log oscillations in themathematical definition of d S , well known in examples offractal geometry [3] but conjectured to be more general.In quantum gravity, these oscillations are either absentor often coarse grained with an averaging procedure, justlike for the Hausdorff dimension [2]. In general, they al-ways appear when heat kernels and correlation functionsare calculated on fractals. In multifractional theories,they are a long-range modulation of the geometry comingfrom the UV discrete scale invariance and they can leavea large-scale imprint on cosmological spectra as well asplay a role in renormalization [2]. Still, it is interesting tonote that highly quantum states in loop quantum grav-ity (LQG), spin foams, and group field theory (GFT) canshow a “pathological” (in the sense of very nonclassical)behavior such as a complex-valued d S , where complexphases do not combine into real-valued oscillations [2, 4]. II. ASSUMPTIONS
Before proving the claims in italics, let us discuss thehypotheses.(I) A continuous parameter ℓ exists in all quantumgravities with a notion of distance, even when thereis no fundamental notion of continuous spacetime.In continuous spacetimes, ℓ is an arbitrary lengthidentified with the length scale at which we areprobing the geometry, while in discrete and com-binatorial settings (for instance, LQG and GFT) ℓ is measured in units of a lattice spacing or of thelabels of combinatorial structures (e.g., complexes)[4]. For the spectral dimension, the probing is of-ten imagined to take place via a diffusion processwhere ℓ plays the role of evolution parameter, in-volving a quantity P ( ℓ ) (possibly coming from theexpectation value of operators on quantum states[4]) called return probability. The interpretation of ℓ as a distance or as the inverse of resolution is pos-sible even when geometry is fundamentally discreteand requires nonsmooth forms of calculi [4]. There-fore, (I) is not restrictive and is fulfilled in all casesof interest to our knowledge. The Hausdorff dimen-sion is defined as the scaling of the Euclideanizedvolume V ( ℓ ) of a D -ball of radius ℓ (or of a hyper-cube of edge ℓ ), while d S is the scaling of the returnprobability: d H ( ℓ ) := d ln V ( ℓ ) d ln ℓ , (9a) d S ( ℓ ) := − d ln P ( ℓ ) d ln ℓ . (9b)(II) Assumption (II) states that the profile d ( ℓ ) is flatnear the IR endpoint d IR = const. This means that d IR is reached as an asymptote at ℓ → + ∞ , which isalways the case if ℓ can be arbitrarily large. Then, d IR = d ( ∞ ). We have no counterexample entailinga “maximum attainable length” and (II) is generalenough.(III) Hypothesis (III) is that spatial sections are non-compact, so that in physical situations the IR di-mension of spacetime coincides with the topologicaldimension. Then, d IR = d ( ∞ ) = D . In compacttopologies, the ℓ → + ∞ limit corresponds to vol-umes wrapping around (or diffusion paths windingonto) space and the relation d ( ∞ ) = D is altered.A typical example is the 2-sphere, which is isomor-phic to R only locally. In all calculations of d H and d S , curvature effects must be ignored to pre-vent false positives [5].(IV) Concerning the factorizability assumption in thesecond theorem, there have been attempts in thepast to describe field theories on irregular ge-ometries with nonfactorizable measures, but theirrange of applicability to physical situations wasseverely limited or nonextant [6]. The purely tech-nical choice of defining multifractional theories withfactorizable measures Q µ dq µ has been successfulin extracting observational constraints. Here, itwill allow us to consider each spacetime directionseparately, the total dimension of (Euclideanized)spacetime being Eq. (8). Of course, it may be thatNature, if multiscale, is not represented by factoriz-able geometries, in which case we have to look intoother proposals. The first theorem will cover mostof them.If spacetime is fundamentally continuous or embeddedin a continuum (as in multifractional theories), then thefirst theorem applies exactly. If spacetime is continuous only after some coarse-graining, averaging, condensation,or semiclassical procedure (as in LQG/GFT), then Eq.(5) is a good description of geometry only if the scale ℓ ∞ is larger than the UV cutoff ℓ uv of the effective theory. III. PROOF
The proof of the flow-equation theorems uses only theproperties of dimensional flow listed above; it does notdepend on the dynamics of the theory. According to Eqs.(9), the most natural parametrization of dimensional flowis via logarithmic scales. Therefore, it is convenient toemploy the variable y := ln ℓ (10)and the constants y n := ln ℓ n , y ∗ := y , and y ∞ := ln ℓ ∞ ,corresponding to the characteristic scales of the geome-try. Integrating Eqs. (9), we have V ( y ) ∝ exp (cid:20)Z dy d H ( y ) (cid:21) , (11a) P ( y ) ∝ exp (cid:20) − Z dy d S ( y ) (cid:21) . (11b)Given that we will find the same formal expression for d H and d S , dependent on a set of parameters λ j = c j , b j , ω j , . . . , at a formal level we can relate the D -volume V ( y, λ H j ) and the return probability P ( y, λ S j ) by P ( y, λ S j ) = 1 V ( y, λ S j ) . (12)Since λ H j = λ S j in general, in any concrete theory one can-not say that the return probability is simply “the inverseof the volume” V ( y, λ H j ). However, in this section we areinterested in the functional form of these expressions, in-dependently of the dynamically determined values of theparameters λ H j and λ S j . Therefore, for the purpose of theproof Eq. (12) is a valid tool.We also introduce two useful quantities: the difference δ n ( y ) := d ( n ) ( y ) − d ( n − ( y ) (13)(with d ( − := d IR ) of the dimension d = d H , d S calcu-lated at adjacent orders in an expansion we will introduceshortly, and the difference δ ( n ) ( y ) := d ( n ) ( y ) − d IR (14)between d at order n and the IR value.At zero order, δ = δ (0) . The case of trivial flow cor-responds to the simple equation δ = 0 , (15)i.e., d (0) = d IR . This is the first example of relationsbetween the dimension d ( ℓ ), its variation with respectto the scale ℓ , and its IR value. As we will see, theserelations are organized as an order expansion of the linear flow equation with derivative order n n X j =0 c j ∂ jy δ n = 0 , (16)where the c j are constants. The solution δ n is labelledby n . We do not see any immediate justification for gen-eralizing this equation to nonlinear terms or nonconstantcoefficients.The rest of the section consists in solving the flow equa-tion (16) order by order. A. First theorem In D dimensions for the general case, d IR = D is thetopological dimension, which coincides with the dimen-sion at large scales. At n = 0 order, integrating Eq. (15)one gets d = d (0) = D . For d = d H , this correspondsto V ∝ exp( Dy ) = ℓ D , ordinary Euclidean space withLebesgue measure d̺ ( x ) = d D x . For d = d S , one getsthe return probability P ∝ ℓ − D : V ∝ ℓ D , d = d H = D , n = 0 , (17) P ∝ ℓ D , d = d S = D , n = 0 . (18)The next order brings information about the derivativeof d with respect to y . By virtue of assumption (II), d ′ = ∂ y d is approximately zero at sufficiently large scales,which means that the profile d ( y ) is almost flat. To geta nontrivial description of geometry, we must combinethe information coming from d (0) ≃ d IR with that from d ′ ≃
0, in such a way that the dimension and its firstderivative are nonzero separately with increasingly goodapproximation. This is achieved by the first-order flowequation [ n = 1 and c = 1 in Eq. (16)] δ ′ + c δ = 0 . (19)Integrating and noting that δ (1) = δ , we find δ (1) ( y ) = δ ( y ) = b exp [ − c ( y − y ∗ )] , (20)where b and y ∗ are arbitrary constants. This expressionreproduces the second term in Eqs. (1), (3), and (4),where c = c and ℓ ∗ = e y ∗ is the first fundamental scaleof the geometry encountered when running from the IR.In fact, d ( ℓ ) = d IR + δ (1) , so that d H(1) = D + b H (cid:18) ℓ ∗ ℓ (cid:19) c H , (21a) d S(1) = D + b S (cid:18) ℓ ∗ ℓ (cid:19) c S , (21b) while from Eqs. (11) and (20) one has V ∝ exp Z dy [ D + δ ( y )]= exp (cid:26) Dy − b H c H exp [ − c H ( y − y ∗ )] (cid:27) = ℓ D exp (cid:20) − b H c H (cid:18) ℓ ∗ ℓ (cid:19) c H (cid:21) (22a) ≃ ℓ D (cid:20) − b H c H (cid:18) ℓ ∗ ℓ (cid:19) c H (cid:21) , (22b) P ∝ ℓ D exp (cid:20) b S c S (cid:18) ℓ ∗ ℓ (cid:19) c S (cid:21) (22c) ≃ ℓ D (cid:20) b S c S (cid:18) ℓ ∗ ℓ (cid:19) c S (cid:21) . (22d)In general, b H = b S and c H = c S . These coefficients aredetermined by the dynamics of the model.At n = 2, the general solution of the flow equation(here the factor of 2 is for convenience) δ ′′ + 2 c δ ′ + c δ = 0 (23)is δ ( y ) = e − c ( y − y ) [ b + exp( − p c − c y ) + b − exp( p c − c y )]. The dimension is d (2) = d (1) + δ = d IR + δ + δ . (24)The case c − c > c = c = √ c ) or (exactlyor in the mesoscopic/IR limit y ≫
1) a trinomial profile d (2) ∝ D + b ( ℓ/ℓ ) − Dc + ˜ b ( ℓ/ℓ ) − Dc ]. Setting thus ω := c − c > c for the first-ordercoefficient, we get the expression of δ (2) = d (2) − d IR = δ + δ : δ (2) ( y ) = b e − c ( y − y ∗ ) + e − c ( y − y ) (cid:0) b + e − iωy + b − e iωy (cid:1) . (25)This is the most general second-order solution, with c > c (for consistency, the n = 2 correction cannot dominateover the n = 1 one) and complex roots. The dimensionwhere running occurs (Hausdorff and/or spectral) is d (2) = D + b (cid:18) ℓ ∗ ℓ (cid:19) c + (cid:18) ℓ ℓ (cid:19) c (cid:0) b + ℓ − iω + b − ℓ iω (cid:1) , (26)where the set b = b H , c = c H , c = c , H , β ± = β ± , H , and ω = ω H for d = d H may be different from the parameters b = b S , c = c S , c = c , S , β ± = β ± , S , and ω = ω S for d = d S . The volume and return probability read V ∝ ℓ D exp (cid:20) − b H c H (cid:18) ℓ ∗ ℓ (cid:19) c H − (cid:18) ℓ ℓ (cid:19) c , H ˜ F H ( ℓ ) (cid:21) (27a) ≃ ℓ D (cid:20) − b H c H (cid:18) ℓ ∗ ℓ (cid:19) c H − (cid:18) ℓ ℓ (cid:19) c , H ˜ F H ( ℓ ) (cid:21) , (27b) P ∝ ℓ D exp (cid:20) b S c S (cid:18) ℓ ∗ ℓ (cid:19) c S + (cid:18) ℓ ℓ (cid:19) c , S ˜ F S ( ℓ ) (cid:21) (27c) ≃ ℓ D (cid:20) b S c S (cid:18) ℓ ∗ ℓ (cid:19) c S + (cid:18) ℓ ℓ (cid:19) c , S ˜ F S ( ℓ ) (cid:21) , (27d)where ˜ F H ( ℓ ) = ˜ b + , H ℓ − iω H + ˜ b − , H ℓ iω H , (28a)˜ F S ( ℓ ) = ˜ b + , S ℓ − iω S + ˜ b − , S ℓ iω S , (28b)and ˜ b ± = b ± / ( c ± iω ). If ˜ b + = ˜ b ∗− ∝ e iωy ∞ , thenone obtains the real-valued logarithmic oscillations (2)in Eqs. (1), (3), and (4). B. Second theorem
For multifractional theories, the length V ( µ ) along the µ th direction in a continuous space is just the integral V ( µ ) ( ℓ ) = R ℓℓ uv d̺ ( x µ ) = ̺ ( ℓ ) − ̺ ( ℓ uv ) of the spacetimemeasure ̺ ( x µ ) from a UV cutoff ℓ uv ; the constant ̺ ( ℓ uv )does not affect Eq. (9a) and is zero in these theories. Thevolume of a hypercube is simply V ∝ Q µ V ( µ ) (the D -volumes of other objects differ in their normalization butnot in the general scaling), while the return probabilityis the product of D profiles: P ∝ Q µ P ( µ ) . Therefore,the dimension of spacetime is the sum of the dimensionsalong each direction, Eq. (8) with d IR ,µ = 1, and Eqs.(11) remain valid.An expression we will need is the approximate profile d H ( y ) from Eq. (7) at mesoscopic scales. Working in thepositive half-line x > µ everywhere, for ̺ = q α we have δ H ( y ) := d H ( y ) − d IRH ≃ − ( α − − e ( α − y − y ∗ ) F ω ( y ) ,F ω ( y ) = 1 + A − e iω ( y − y ∞ ) + A + e − iω ( y − y ∞ ) , (29)where A ± = ( A ± iB )[1 ± iω/ (1 − α )] / δ H is real-valued).At n = 0 order, the results (17) and (18) are recovered.For n = 1, to reproduce the dimensional flow of d H inmultifractional theories without log oscillations, we mustcompare Eq. (20) with Eq. (29) when F ω = 1: then, b H ,µ = − ( α − µ − , c H ,µ = 1 − α µ . (30)This is just a redefinition of labels, since the constants y ∗ and c H are mutually independent just like ℓ ∗ and α .Signs are also unconstrained: we can set α < b H < c H >
0) only in the case we want a dimensional flowwhere d UVH < d
IRH . This requirement may be desirable forphenomenology but it plays no role here. The coefficients in the spectral dimension are more model-dependent. Forinstance, in the theory with weighted derivatives d S = D is constant ( b S ,µ = 0), while in the theory with q -derivatives d S ≃ d H at all the plateaux of dimensionalflow [2].Integrating the solution (20) and expanding in ℓ ≫ ℓ ∗ (consistently with the mesoscopic-scale approximationentailed in the n = 1 truncation), we can get V and P from Eq. (11). Taking into account factorizability, onehas d H(1) = D + D − X µ =0 b H ,µ (cid:18) ℓ ∗ ℓ (cid:19) c H ,µ , (31a) d S(1) = D + D − X µ =0 b S ,µ (cid:18) ℓ ∗ ℓ (cid:19) c S ,µ , (31b)and V ∝ ℓ D exp " − D − X µ =0 b H ,µ c H ,µ (cid:18) ℓ µ ∗ ℓ (cid:19) c H ,µ (32a) ≃ ℓ D " − D − X µ =0 b H ,µ c H ,µ (cid:18) ℓ µ ∗ ℓ (cid:19) c H ,µ , (32b) P ∝ ℓ D exp " D − X µ =0 b S ,µ c S ,µ (cid:18) ℓ µ ∗ ℓ (cid:19) c S ,µ (32c) ≃ ℓ D " D − X µ =0 b S ,µ c S ,µ (cid:18) ℓ µ ∗ ℓ (cid:19) c S ,µ . (32d)Notice that the scale ℓ is always taken to be the samealong all directions. When geometry is time-spaceisotropic, these expressions simplify to Eqs. (22a)–(22d)with b H → Db H , b S → Db S . (33)This shows that the second flow-equation theorem is littlemore than a corollary of the first.We have just obtained, for d = d H , the measure ̺ ( x ) ≃ x + ( ℓ ∗ /α )( x/ℓ ∗ ) α in the absence of log oscil-lations. The binomial measure (7) with F ω = 1 is theapproximation of the full log-oscillating measure at scalesabove ℓ ∞ . Therefore, if the description in terms of flowequations is correct and self-consistent, there is a verynatural way in which we can obtain log oscillations: toconsider higher-order versions of (16). In fact, it is nec-essary to go to second order, n = 2, as we saw in theprevious subsection. We do not repeat the calculationhere: at the end of the day, Eqs. (31) and (32) are aug-mented by a factor ( ℓ /ℓ ) c ˜ F ω µ ( ℓ ) in the sums over µ ,where ˜ F ω µ ( ℓ ) = ˜ b + ,µ ℓ − iω µ + ˜ b − ,µ ℓ iω µ . (34)Again, in the fully isotropic configuration ω µ = ω , ˜ b ± ,µ =˜ b ± , and when ˜ b + ∝ ˜ b ∗− ∝ e iωy ∞ , one has the modulationfactors (2). Going to third and fourth order, matchingthe parameters order to order (exponents, frequencies,and so on) and taking the fully isotropic configuration,one can easily obtain the modulation factor (7b). Com-paring with Eq. (29), the binomial measure (7) corre-sponds to b and c reparametrized as in Eq. (30) and with b ± = − ( α − − A ± e (1 − α ) y ∗ ± iωy ∞ , c = ω + (1 − α ) ,and c = 1 − α . There is one independent constant less,since c = c , but this choice maximizes the chance toget nontrivial effects from log oscillations at scales ∼ ℓ ∗ .Therefore, Eq. (7) is the most general n = 4 real-valuedfactorizable solution of the flow equations for d H suchthat large-scale effects of log oscillations are maximizedat scales ∼ ℓ ∗ .Note, however, that the effect of log oscillations can bemaximized even in the IR by setting c = 0. Moreover,the most general fully isotropic case where different-orderparameters are not matched is Eq. (34), which has log-average h ˜ F ω i = 0, contrary to the modulation factor (7b)typically used in multifractional theories and such that h F ω i = 1. This fact could play a very important role inreinterpreting the microscopic structure of these space-times [2]. C. All orders
The method of the flow equation starts from the IR andhits first the largest scale ℓ = ℓ ∗ , and then the lowestscale ℓ ∞ in the hierarchy (6) of the multiscale paradigm.There is no contradiction in having found the largest andthe smallest scale of the hierarchy at n = 2. Log oscil-lations are an independent structure with respect to thepolynomial behavior of the measure [2] and they mod-ulate the geometry even at scales much larger than ℓ ∞ .Scales below ℓ ∗ are too small to be constrained by ex-periments, which can only say something about ℓ ∗ . Thelatter “screens” the microscopic structure of the measurebut it does not prevent the logarithmic modulation tomanifest itself in subtle ways [2]. Then, at n = 2 thederivative expansion has both the expected range in thedimensional flow ( ∼ ℓ ∗ ) and enough sensitivity to catchthe modulation structure.The second-order solution is an approximation of morecomplicated measures. The general solution of the n th-order flow equation (16) is a complex superposition δ n = n − X i =0 b i,n exp( k i,n y ) , n X j =0 c j k ji,n = 0 (35)of exponentials with n complex wavenumbers k i,n satisfy-ing a characteristic equation for all i . Physical solutionsdo not have exactly n distinct roots because they shouldbe (and can always be, by choosing the c j and k i ) real-valued (a condition lifted in Ref. [2]), positive semidefi-nite, and with the correct IR asymptote. Getting thusnew scales, frequencies, and amplitudes in the dimen-sion d = P + ∞ n =0 δ n , we enter the realm of the polynomial multimodal measure (5) for d = d H , or of a polynomialmultimodal return probability for d = d S . IV. COMPARISON WITH QUANTUMGRAVITIES
We conclude by comparing Eq. (1) with several theo-ries of quantum gravity. Asymptotic safety [7], causal dy-namical triangulations (CDT) [8], spacetimes near blackholes [9, 10], nonlocal gravity and string field theory[11, 12] all have trivial dimensional flow in the Hausdorffdimension ( d H = D ). Noncommutative spacetimes usu-ally have d H = D [13], but in the case of κ -Minkowskiwith cyclic-invariant action b < c = 1 [2]. Fi-nally, states of LQG and GFT describing general dis-crete quantum geometries display the kink profile of thebinomial measure (7) without log oscillations [4]. In theanalytic example of the lattice C ∞ = Z D − , the Haus-dorff dimension reads d H = 2 + O ( ℓ ) in the UV ( ℓ ismeasured in units of the lattice spacing), while in the IR d H = D − ( D − D − / (2 ℓ ) + O ( ℓ − ), giving b < c = 1.Concerning the spectral dimension near the IR, theprofile (1) reproduces the one in the multifractional the-ories with weighted and q -derivatives [2]. In particular,in the theory with q -derivatives b = α − c = 1 − α for each direction. The log-oscillatory modulation is alsorecovered. In asymptotic safety, ℓ is the IR cutoff gov-erning the renormalization-group equation of the metric[7]. The multiscale profile of the spectral dimension iscalculated analytically at each plateau and numericallyin transition regions. The author is unaware of any semi-analytic approximation giving b and c in (1). The sameholds for Hoˇrava–Lifshitz gravity. The rest of the modelslisted from now on have c = 2, without exception. InCDT, b < b = ( D + 1) / b = − D [14]. In nonlocal gravity with e (cid:3) operators as in string field theory, b < b = −
36 in D = 4) [12]. The noncommutativeexamples of [13] are the following: in D = 3 Einsteingravity with quantized relativistic particles, b = − / κ -Minkowski space with bicovariant Lapla-cian and AN(3) momentum group manifold, D = 4 and b = −
2; with AN(2) momentum group manifold, D = 3and b = − /
2; with bicrossproduct Laplacian, D = 4and b = 1. In LQG and GFT, one can check numericallythat b > c = 2.More examples can be found in Ref. [2]. In this com-panion paper, we will also discuss in greater detail someof the physical implications of the flow-equation theo-rems, including the resolution of the so-called presen-tation problem, a major contribution towards a defini-tion of the theory with fractional derivatives, and conse-quences for the priors on the length hierarchy, complexdimensions, the big-bang singularity, and renormaliza-tion. ACKNOWLEDGMENTS
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