ABC of multifractal spacetimes and fractional sea turtles
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Abstract
We clarify what it means to have a spacetime fractal geometry in quantum gravity and show that its properties differ from those of usual fractals. A weak and a strong definition of multiscale and multifractal spacetimes are given together with a sketch of the landscape of multiscale theories of gravitation. Then, in the context of the fractional theory with qderivatives, we explore the consequences of living in a multifractal spacetime. To illustrate the behavior of a nonrelativistic body, we take the entertaining example of a sea turtle. We show that, when only the time direction is fractal, sea turtles swim at a faster speed than in an ordinary world, while they swim at a slower speed if only the spatial directions are fractal. The latter type of geometry is the one most commonly found in quantum gravity. For timelike fractals, relativistic objects can exceed the speed of light, but strongly so only if their size is smaller than the range of particlephysics interactions. We also find new results about logoscillating measures, the measure presentation and their role in physical observations and in future extensions to nowheredifferentiable stochastic spacetimes.
Keywords
Quantum Gravity Hausdorff Dimension Loop Quantum Gravity Dimensional Flow Asymptotically Safe1 Introduction and main results
In the multifaceted quest for a theory of quantum gravity, evidence has been gathered that quantum spacetimes acquire anomalous properties which cannot be described by conventional geometry [1, 2, 3]. Volumes and distances can change depending on their size and length, on the size of the observer, on the scale of the experiment, and so on. In particular, the dimension of spacetime changes with the scale, in a way similar to what happens in multifractal sets. Among the many available examples of this dimensional flow, we count with causal dynamical triangulations [4, 5, 6], asymptotically safe quantum gravity [7, 8], loop quantum gravity and spin foams [9, 10, 11], Hořava–Lifshitz gravity [6, 8, 12], noncommutative geometry [13, 14, 15] and \(\kappa \)Minkowski spacetime [16, 17, 18], nonlocal quantum gravity [19], Stelle’s gravity [20], spacetimes with black holes [21, 22, 23], fuzzy spacetimes [24], random combs [25, 26], random multigraphs [27, 28], causal sets [29] and string theory [30].
 1.
A fine structure: the set has details at every scale.
 2.
An irregular structure: ordinary continuous differential calculus cannot be applied on the set.
 *3.
Selfsimilarity.
 *4.
A noninteger dimension (Hausdorff dimension \(d_\textsc {h}\), spectral dimension \(d_\textsc {s}\) or walk dimension \(d_\textsc {w}\)).
 5.
The relation \(d_\textsc {w}=2d_\textsc {h}/d_\textsc {s}\) holds with \(d_\textsc {s}\le d_\textsc {h}\).
 6.
Properties 1, 2 and 5 hold at any given scale in the dimensional flow.
 A.
Dimensional flow occurs with three properties: [A1] At least two of the dimensions \(d_\textsc {h}\), \(d_\textsc {s}\), and \(d_\textsc {w}\) vary. [A2] The flow is continuous from the infrared (IR) down to an ultraviolet (UV) cutoff (possibly trivial, in the absence of any minimal length scale). [A3] The flow occurs locally, i.e., curvature effects are ignored (this is to prevent a false positive).
 B.
As a byproduct of A, a noninteger dimension (\(d_\textsc {h}\), \(d_\textsc {s}\) or \(d_\textsc {w}\), or all of them) is observed during dimensional flow, except at a finite number of points (e.g., the UV and the IR extrema).
 C.
If, in addition, the relations \(d_\textsc {w}=2d_\textsc {h}/d_\textsc {s}\) and \(d_\textsc {s}\le d_\textsc {h}\) hold at all scales in dimensional flow, then we call the ensuing geometry a weakly multifractal spacetime.
 D.
A geometry is a strongly multifractal spacetime if, in addition of satisfying A–C, it is nowhere differentiable in the sense of integerorder derivatives, at all scales except at a finite number of points (e.g., the UV and the IR extrema).
If the real world has an anomalous geometry, then there must exist some critical time and length scales \(t_*\) and \(\ell _*\) below which fractal properties begin to show up. In other words, geometry must be multiscale or multifractal, as opposed to just fractal. The theory with qderivative is of this form and upper bounds on \(t_*\) and \(\ell _*\) have been derived recently [40, 41]. These bounds (\(\ell _* < 10^{19}\,\mathrm{m}\), \(t_* < 10^{27}\,\mathrm{s}\)) are about 20–30 orders of magnitude smaller than the scales involved in the seaturtle thought experiment (\(\ell _\mathrm{turtle}> 1\,\mathrm{m}\), \(t_\mathrm{turtle}> 1\,\mathrm{s}\)) and there would be no way to discriminate a turtle on such a multifractal from one on plain earth. For the same reason, superluminal motion would be possible only for an object of size \(\lesssim \ell _*\) in an experiment with characteristic time \(\lesssim t_*\), i.e., below the scales of StandardModel interactions. Therefore, even if the multifractional theory with qderivatives can avoid the side effects that superluminal travel [43] entails in Einstein gravity (existence of exotic matter [43, 44, 45] and quantum instability due to a thermal flux of Hawking particles [46]) or in generic Lorentzviolating setups [47], to curb the enthusiasm of SciFi aficionados we already anticipate that our model, or possibly any multifractal spacetime in general, cannot be used as a practical base for a hyperdrive.
 (i)
In order to fully define a multifractional theory, one must choose a frame where geometric coordinates q(x) are written down explicitly. This is not a return to a prerelativistic view of space and time because the frame choice affects the integrodifferential structure of the theory, not the metric structure [39].
 (ii)
There is a limited number of presentation choices; we will see four below.
 (iii)
Different presentations of the same measure correspond to different theories in the same geometric class (i.e., they show the same scaling properties). In Sect. 6.2, this feature is restated under a new perspective based on the famous Itô–Stratonovich dilemma in stochastic processes.
 (iv)
Although they describe the same class of geometries (iii) and they are not many (ii), different presentations may have profound consequences for physical properties such as the propagation speed of bodies or elementary particles, but only in extreme situations of high energy, high curvature or small scales.
 (v)
Therefore, even if the qtheory is invariant under Poincaré transformations on the coordinates q meant as noncomposite objects, the physics is not completely independent of the choice of coordinates x used to describe the system in the frame where physical observables are extracted.
2 Spacetime multifractals in quantum gravity
Let us now examine the list of properties 1–6 and how they apply to quantum gravity or, more generally, to anomalous spacetimes.
2.1 Dimensions
To characterize a set or a geometry, we have various operational definitions of dimension.^{1} In this subsection, we first revisit these definitions in the case of a continuum space, then commenting on theories with discrete structures and finally including also time. In the process, we will get a first glimpse of the type of phenomena we would experience if we lived in a multifractal world, and of how to detect them.
2.1.1 Dimensions of continuous spaces
2.1.2 Dimensions of discrete and combinatorial structures
In several quantumgravity approaches, there is no fundamental continuous spacetime. Nevertheless, it is possible to generalize the above operational definitions to a discrete set and to extract, only in certain regimes, sensible multiscale profiles for the Hausdorff, spectral and walk dimension. A proof of concept is given in [4, 10, 11, 25, 26, 27, 28]. For instance, a generalization of discrete exterior calculus [52, 53, 54] allows one to construct Laplacians on combinatorial structures and hence define diffusion processes thereon. The spectral dimension as well as \(d_\textsc {w}\) and \(d_\textsc {h}\) can be computed for a class of quantumgravity states, in particular those appearing in loop quantum gravity, spin foams, and group field theory [10, 11].
The main requirement we ask in order to have dimensional flow in discrete (pre)geometries is that there exist regimes where all three geometric indicators \(d_\textsc {h}\), \(d_\textsc {s}\), and \(d_\textsc {w}\) are realvalued and positive. These regimes should extend from the infrared down to some effective UV scale below which discreteness or quantum effects destroy some or all of the indicators, for instance if the expectation values on the chosen states become complex. In general, the effective UV scale is determined by the choice of states. The existence of a regime where quantum geometry has welldefined dimensions translates into regularity assumptions on the quantum states [11]. Beyond these assumptions, one can plunge into a wild jungle of quantumgeometry configurations with properties completely different from classical or semiclassical spacetimes.
2.1.3 Dimensions of spacetimes
In a continuous spacetime with D topological dimensions, the definition of Hausdorff dimension is unchanged but for the addition of the Euclideanized time direction. Then the Hausdorff dimension of spacetime is the scaling of the volume of the Dball. For the spectral dimension, one includes (imaginary) time in the operator \(\mathcal {K}(\nabla )\), while in the walk dimension time is included in Euclideanized distances \(X^2(\sigma )=T^2(\sigma )+X_1^2(\sigma )+\cdots \). Similar considerations apply to discrete geometric or pregeometric structures.
In the extension of all the above definitions, one takes the time direction in Euclidean signature. This step is fairly standard when one wants to define the dimension of a geometry with Lorentzian signature. If, for any reason, one cannot or does not want to Euclideanize time, then it is necessary to consider the dimensionality of spatial slices and the time line separately, instead of the whole spacetime. The reader may adopt whichever point of view they might prefer; this does not affect the following.
The spectral dimension is sometimes regarded as a theoretical parameter useful to classify spacetimes but that does not correspond to a physical observable. Elsewhere [8, 20], the author and collaborators had already the occasion to advance a different view: the spectral dimension should be a meaningful observable just like the topological and Hausdorff dimension are. In that case, however, its definition must be well posed at all stages to make sense physically: if we want \(d_\textsc {s}\) to be a physical observable (our working hypothesis here), its definition must provide also an operational way to measure it. For instance, how can we interpret the parameter \(\sigma \) if time is in the operator \(\mathcal {K}\)? Also, in certain cases the form of \(\mathcal {K}\) is such that \(P(x,x',\sigma )\) is not a probability and there is no welldefined underlying diffusion process at all (this is a wellknown problem in transport theory with higherorder or nonlocal operators [42] and in quantum gravity [8, 20]). Mathematically there is no issue whatsoever. If \(\sigma \ne t\), one can enact a fictitious diffusion process with some MonteCarlo time on the geometric or pregeometric structure one wants to explore, let it be a continuous manifold or the graph ensembles of discretized gravity. Even when P is not positive semidefinite and the picture of a diffusing probe fails, to determine \(d_\textsc {s}\) one only needs to consider closed paths and integrate over them with a certain measure. However, this is insufficient to characterize an operational way to physically measure \(d_\textsc {s}\). In two different interpretations of the spectral dimension, valid in any regime where an effective field theory can be formulated, the diffusion equation is a renormalizationgroup running equations depending on the IR cutoff scale \(k=f(\sigma )\) [8] or, alternatively, it stems from the Schwinger representation of the particle propagator and \(\sqrt{\kappa \sigma }=\ell \) is a length scale determining the resolution \(1/\ell \) at which the geometry is probed [20]. The interpretation of the parameter \(\sigma \) is unimportant in the following but it is worth to mention these caveats anyway.
2.2 Fine structure and dimensional flow
The first property of the list 1–6 is that a (multi)fractal set \(\mathcal {F}\) should have “a fine structure.” By this, one means that it is possible to find points of \(\mathcal {F}\) at all scales of observation, no matter how deeply one zooms into the set. For continuous spacetimes (among others: asymptotic safety, Hořava–Lifshitz gravity, nonlocal gravity and multifractional spacetimes), this requirement seems trivially satisfied and not very useful. However, a careful inspection shows that it is neither trivial nor satisfied in general.
One of greatest Einstein’s intuitions was that spacetime points do not have a physical meaning per se unless one attaches an event to them. A spacetime devoid of particle interactions, test particles, light rays or whatever event announced by matter is an empty mathematical construct. To make sense of the idea of “finding spacetime points at all scales,” one should be able to concoct an experiment where the physical probe can be utilized at all scales. Of course, sometimes the same device can give us information on the physics at very different scales, as is the case with the Planck satellite or similar observatories of the cosmic microwave background. But, in general, we do not have a universal instrument and we need to resort to different setups (a telescope, a particle accelerator\(,\ldots \)) to probe the physics at different scales.
Once having defined our ideal probe as a patchwork of instruments and experiments covering all scales of interest, the problem remains to test the spacetime structure at arbitrarily small scales. Apart from obvious technical limitations we have now and probably forever after (we cannot probe the Planck scale directly, nor energies near grand unification), it may even be theoretically impossible to reach an infinite resolution, mainly because of quantum uncertainty. This is the case of asymptotic safety, where, despite the absence of any fundamental length in the theory, a minimal length appears below which one cannot separate two points by a dynamical probe [55].^{3} The plethora of theories based on discrete structures is also unaffected by property 1 because there are no details below the discreteness scale. For instance, causal dynamical triangulations are a discretization of a continuum but, for any practical purpose, one cannot trust any probing of the geometry at scales comparable with the size of the triangulation cell. Loop quantum gravity and spin foams are defined on complexes that induce a minimum physical Plancksize length in the spectra of volume operators [56]. Also, both the underlying discreteness and the combinatorial structure impose an effective UV cutoff limiting the range of scales where one can make sense of the concept of spacetime dimension, while at scales larger than the cutoff they render such dimension anomalous [10, 11].
A much more important property than the fine structure is that the effective geometry must have some quantumtoclassical regime where dimensional flow takes place, otherwise one could not reach a semiclassical continuum limit where the dimension of spacetime is 4. For instance, suppose to find a dimensional flow from \(d_\textsc {s}\sim 2\) in the UV to \(d_\textsc {s}\sim 4\) in the IR (examples of this abound in the literature [1, 2, 3, 4, 7, 11, 12, 30, 33]), while below the UV scale one finds a nongeometric phase where one cannot define the spectral dimension, possibly for discreteness or combinatorial effects (as in [10, 11]). Then below the UV scale the geometry certainly does not show a fine structure (zooming in too much, we enter “inside” the building blocks of the theory, let them be lattice cells, tetrahedra or something else). During dimensional flow, the fundamental degrees of freedom (e.g., quanta of geometry, labeled complexes, and so on) group together into collective modes such that the notion of spacetime dimension makes sense and is measurable. When coarse graining the fundamental degrees of freedom, the resulting effective structure is most likely to be “fine,” which can be tested by finding effective dynamical equations on an effective continuum. However, this test is nontrivial and few are the cases where it can be carried on [4]. Usually, the only datum we know, corroborated by a numerical or analytical study of dimensional flow through all scales from the effective UV cutoff to the IR, is that discreteness effects are present but not dominant in that interval.
From this discussion, we see that theories which have dimensional flow may or may not have a fine structure at all scales. Also, theories which do not have a fine structure do not necessarily have dimensional flow (example: a canonical secondorder scalar field theory on a cubic lattice), while theories which have a fine structure can describe most boring geometries (example: any canonical secondorder field theory on Minkowski spacetime). We conclude that property 1 is not adequate in the context of anomalous (quantum or classical) spacetimes, many of which are not fractal in the standard sense because they do not have a fine structure.
2.3 Irregular structure
An ordinary (multi)fractal set \(\mathcal {F}\) has “an irregular structure” in the sense that it cannot be described by Euclidean geometry. A Euclidean ruler would fail to measure the total length of the Western coast of Britain [57]. Clearly, in a physical context the geometry in the infrared must be “regular,” so that we should consider property 2 only at the microscopic scales of a multifractal spacetime. In gravitational theories, geometric probes are local and curvature effects are usually ignored when one determines the dimension of space (which would be modified by curvature even in a purely classical setting,^{4} according to the Seeley–DeWitt formula [58]). However, even locally there are other effects that make the geometry nonEuclidean, for instance if gravity is quantized or in the presence of a nontrivial integrodifferential structure. In the first case (quantum gravity), the collective effect of quanta of geometry is to push around the probe in an anomalous way, not experienced in a classical space. Often this induces effective operators in the dynamics, which leads to the second case (multifractional spacetimes). A third case consists in frameworks with an underlying discrete nonregular structure, such as the complexes found in loop quantum gravity, spin foams, and group field theory.
All three cases can be realized in so many different ways that establishing the “irregularity” of a geometry is a moot point. If a spacetime shows dimensional flow locally (i.e., ignoring curvature corrections), then there must be some mechanism making it irregular. Conversely, an irregular spacetime does not have to be a (multi)fractal unless it also has a fine structure, just like the rugged surface of a rock may not be a fractal (if we zoom in, we may discover that locally it is smooth).
In Sect. 6, we will consider a more precise characterization of irregularity as one of the requirements to reproduce certain microscopic properties of stochastic processes on fractals.
2.4 Selfsimilarity
Selfsimilarity and selfaffinity are what defines all deterministic fractals. A deterministic fractal \(\mathcal {F}=\bigcup _i \mathcal {S}_i(\mathcal {F})\) is the union of the image of some maps \(\mathcal {S}_i\) which take the set \(\mathcal {F}\) and produce smaller copies of it (possibly deformed, if the \(\mathcal {S}_i\) are affinities). Not all fractals are deterministic, yet they are fractals indeed; sets with similarity ratios randomized at each iteration are of this sort and they are called random fractals.
Since selfsimilarity and selfaffinity are shown by a huge but nonexhaustive class of fractals, it is clear that we cannot use them to characterize spacetimes in an efficient way. The standard Poincaré transformations \({x'}^\mu =\Lambda _\nu ^{\ \mu }x^\nu +a^\mu \) are affinity maps and the dynamics of a covariant field theory on Minkowski spacetime is selfaffine. Yet, it is not a fractal because it has no irregular structure. On the other hand, multifractional spacetimes with ordinary derivatives have dimensional flow but they are neither selfsimilar nor selfaffine [3, 34] and for this reason they can be used only as effective models [39].
Any theory of particle physics and quantum gravity worth of this name is both under analytic control and potentially predictive provided symmetries are enforced. There are no known exceptions to this rule. Whatever these symmetries are (diffeomorphism invariance, conformal invariance, supersymmetry, modular invariance, and more), they constitute a guiding principle and the backbone of the theory; they may or may not give rise to dimensional flow, which is an accidental property of geometry. This point of view is not very different from what happens when a mathematician wants to construct a fractal: first some maps are defined and then the geometry of the set is studied. However, the connection between symmetry and fractality is much more tenuous in physics and symmetry takes precedence over virtually anything else.
2.5 Noninteger dimension
One of the most popular features of fractals is that they have noninteger dimensions. For instance, the Hausdorff dimension \(d_\textsc {h}\) of the middlethird Cantor set is equal to the capacity \(d_\textsc {c}:=\ln N/\ln \lambda =\ln 2/\ln 3\approx 0.63\). Each iteration is made of \(N=2\) copies rescaled by \(\lambda =1/3\). However, there are many fractals with integer dimension, e.g., the Mandelbrot set and its boundary (both with \(d_\textsc {h}=2\)). We refer the reader to [31, 34] for definitions and more examples and counterexamples. Conversely, a set with integer dimension is a fractal only if it has an irregular structure. For instance, we can tell apart the string worldsheet from the boundary of the Mandelbrot set because there is no Virasoro algebra of operators acting on the latter [30].
On the other hand, if we have a continuous dimensional flow we expect to sample over all values of the dimension between the UV and IR terminal points, which implies that the dimension is integer only at a finite number of scales, from a minimum of one (in the infrared, where \(d_\textsc {h}^\mathrm{IR}=d_\textsc {s}^\mathrm{IR}=D\) by default) to a maximum of \(D+1\) (if \(d_\textsc {h}^\mathrm{UV}=0\) or \(d_\textsc {s}^\mathrm{UV}=0\)) if dimensional flow is monotonic from the UV to the IR. (In principle, there can exist extended plateaux where the generalized dimensions have approximately constant, integer values for a continuous range of scales. However, technically such plateaux are inflection or saddle points and there is only one point therein where the dimension can be exactly integer.)
2.6 The \(d_\mathrm{W}=2d_\mathrm{H}/d_\mathrm{S}>2\) relation
For multifractals, relation (8) holds (with \(d_\textsc {w}>2\)) at any given scale.
There are various examples of multiscale processes or geometries similar to fractals but which do not obey Eq. (8) or for which \(d_\textsc {w}<2\). Lévy processes are an example well known to mathematicians (see [42] and references therein). In the case of spacetime geometries, certain noncommutative and nonlocal spacetimes have a spectral dimension which grows in the UV and \(d_\textsc {s}>d_\textsc {h}\) [18, 23]. Also the spacetimes of the multifractional theories with weighted and standard derivatives (Hermitian dual to each other) are not multifractals, since \(d_\textsc {w}=2D/d_\textsc {s}\) and \(d_\textsc {h}\ne D\) [37]. On the other hand, the theory with qderivatives (discussed below) respects Eq. (8) [37] and so does asymptotically safe quantum gravity (where \(d_\textsc {h}=D\)) [61] (see also [8]). We do not have data about the walk dimension in the other approaches mentioned in the introduction.
2.7 ABC of spacetime fractals
From what seen in this section, the most general and powerful characterization of anomalous spacetimes is dimensional flow. In theoretical physics, having a fine or irregular structure is not so much important as having a set of fundamental symmetries in action, but the connection between symmetries and dimensional flow is rarely immediate (exceptions are multifractional theories). Also, relation (8) is not obeyed in several cases.
In Fig. 1, we depict a first snapshot of the landscape of multiscale theories with dimensional flow. Apart from the subclass of multifractal spacetimes, some quantumgravity frameworks are also indicated: asymptotically safe quantum gravity (AS) and loop quantum gravity (LQG). While AS realizes multifractal spacetimes in all known cases (see, e.g., [8]; but the situation may change as the framework evolves), the type of geometry produced by LQG depends on the states chosen in the expectation values of the operators defining the dimensions. In [11], one can see several examples of states corresponding to a multifractal geometry (region A), to multiscale but not multifractal quantum geometries, and to highly quantum geometries which cannot be classified by conventional geometric indicators. The latter case is the corner “?” lying outside the multiscale landscape. A third class of scenarios is the one of multifractional spacetimes, which are not necessarily of quantum gravity.^{5} Of the four multifractional models proposed (with ordinary, weighted, q and fractional derivatives), two do not realize multifractal spacetimes (theories with ordinary and weighted derivatives), one has not been analyzed in full detail yet (theory with fractional derivatives) and the fourth lives on multifractal spacetimes. Region B includes the theory with qderivatives and probably also the one with fractional derivatives.
The theory with qderivatives is useful to describe the renormalizationgroup flow of asymptotic safety in an alternative way [36]; this is represented by an overlap between the AS set and the multifractional one. Used as effective descriptions of geometry, multifractional models can reproduce the dimensional flow of other theories. This connection has not been shown for LQG and is indicated here with the intersections “?” inside the landscape. Many other wellstudied multiscale theories are not shown either (including noncommutative spacetimes and Hořava–Lifshitz gravity, both of which do have an overlap with multifractional models [17, 36], and dynamical triangulations), because the walk dimension has not been calculated yet and we are presently unable to verify that \(d_\textsc {w}=2d_\textsc {h}/d_\textsc {s}\). However, the overwhelming majority of quantumgravity cases have \(d_\textsc {s}\le d_\textsc {h}\), which would put them inside the multifractal region if property C were confirmed.
In the following, we will concentrate on the multifractional theory with qderivatives to illustrate what we could expect to see in a multifractal spacetime.
3 Multifractional theory with qderivatives
3.1 Sketch
Multifractional theories are realizations of anomalous geometries rather than frameworks for quantum gravity, but they can be used also as models describing effective regimes of other proposals and, as in this paper (where gravity does not play any role), to clarify what we mean by fractal spacetimes.
To complete the definition of the theory, we still need two data: the choice of profiles q(x) and the choice of frame. If we want our continuous spacetime to change dimension with the scale, we must be able to tell the difference between “large” and “small” distances, or between “early” and “late” times. For this purpose, we can introduce at least one characteristic length \(\ell _*\) and one characteristic time \(t_*\) in the choice of q(x) (a more general hierarchy is discussed in [33, 35]). It turns out that, in \(D=1\) dimension, dimensional flow can be achieved by exactly the same type of measure \(q(x)\sim x^\alpha \) (called the fractional measure, where \(0<\alpha <1\)) one would obtain if one approximated a fractal dust on a continuum line [35]. To get a multifractal, it is sufficient to add several powerlaw contributions \(x^{\alpha _l}\) with different \(\alpha _l\), each multiplied by a characteristic scale \(\ell _*^{(l)}\).
Having chosen the profiles \(q^\mu (x^\mu )\), let us consider the choice of frame. We must select in which picture physical observables are computed. On an ordinary manifold, the properties of clocks, rods and detectors are the same independently of the scale at which measurements are taken. In contrast, multifractional spacetimes are a framework where physical measurements are performed with instruments which do not adapt with the observation scale even if the geometry does [36]. This adaptation is encoded in the structure of the fractional coordinates (i.e., of the integration measure and of differential operators), where characteristic time, length and energy scales appear.
Specifying units for the coordinates clarifies the point. In \(c=1\) units, time and spatial coordinates scale as \([x^\mu ]=1\) and so do the characteristic scales (\([\ell _*]=1=[t_*]\)) and the geometric coordinates (\([q^\mu (x^\mu )]=1\)). However, in the ultraviolet the variable dependence of \(q^\mu \) has an anomalous scaling \(\simeq [x^\mu ^{\alpha _\mu }]= \alpha _\mu \), which implies that qclocks and qrods adapt with the scale of the experiment. Since our actual clocks and rods are nonadaptive rigid apparatus, observables should be compared with experiments in the fractional picture. A more detailed discussion can be found in [41] but in Sect. 4 we will make an important observation so far overlooked in the literature. One may be confused by the above argument relying on the anomalous scaling of the variable part of q. However, even if x and q have the same measurement units \([x]=1=[q]\) exactly, it is possible to recognize a standard spacetime from an anomalous one by measuring dimensionless quantities such as ratios of observables.
3.2 Multifractal properties
In Sect. 2.6, we mentioned that the effective diffusion equation on fractals is modified not only in the spatial part (Laplacian, diffusion coefficient, friction terms) but also in the time part, via a fractional diffusion operator. Although we have not included these modifications as part of the definition of fractals (to the best of our knowledge, this type of diffusion equation is only an empirical tool to describe transport on fractal media), they teach us that subdiffusion on a spacelike fractal may come from a diffusive process parametrized by an anomalous clock \(\sigma \). From what we know about diffusion in multiscale spacetimes with qderivatives [37], we recognize a diffusion operator \(\partial _{q_\beta (\sigma )}\) in Eq. (14) which is not fractional but it is anomalous nevertheless. Moreover, expanding \(\kappa \nabla _{q(x)}^2\) in x coordinates we find both an effective spacedependent diffusion coefficient \(\sim \kappa /(\partial _x q)^2\) and a firstorder friction term. All the ingredients of fractal diffusion are here, albeit modified with respect to the phenomenological models of [60].
3.3 The problem of presentation
In this subsection, we analyze the effect of changes in the presentation of the geometric coordinates \(q^\mu (x^\mu )\).
In the theory with qderivatives, we repeat exactly the same discussion in the fractional picture, which is one of the coordinate frames \(\{x\}\) where the distance \(\Delta x\) is calculated. However, to each of these fractional frames we must associate an integer frame described by geometric coordinates. Thus, the Cartesian fractional frame \(\{x^1,x^2\}\) is mapped into the integer frame \(\{q^1(x^1),q^2(x^2)\}\) and, after inverting to \(x^i=x^i(q^i)\) (assuming it possible, which is not always the case) the Euclidean norm (17) is mapped into some complicated expression \(\Delta x(q_\mathrm{A}^i,q_\mathrm{B}^i)\) which differs from the geometric Euclidean norm \(\Delta q:=\sqrt{\sum _{i=1}^2q_\mathrm{B}^iq_\mathrm{A}^i^2}\). Below we will calculate the difference and encode it in functions \(\mathcal {X}^i\). But now we redo the mapping to geometric coordinates starting from polar fractional coordinates. The new integer frame is \(\{q_r(r),q_\theta (\theta )\}\), where the relations between \(q_r\) and the \(q^i\) are \(q^1=q_r\cos q_\theta \) and \(q^2=q_r\sin q_\theta \).
Having recalled the rather selfevident fact that arbitrary changes of chart \(\{x^\mu \}\rightarrow \{{x'}^\mu \}\) modify q(x), the question is: On which chart are Eqs. (11) and (12) represented? In the example of the paper sheet, is Eq. (11) the form of q in the integer frame \(\{q^1(x^1),q^2(x^2)\}\) based on Cartesian coordinates \(\{x^1,x^2\}\) or the form of q in the integer frame \(\{q^1(r),q^2(\theta )\}\) based on polar coordinates \(\{r,\theta \}\) (so that \(q^1(r)=r+(\ell _*/\alpha )(r/\ell _*)^{\alpha }\)), or something else? Ordinary Poincaré invariance is violated by factorizable multiscale measures. A change of presentation such as a translation, a rotation of the coordinates or an ordinary Lorentz transformation modify the size of the multiscale corrections \(\mathcal {X}\) and \(\mathcal {T}\) defined below and one realizes that different choices of the fractional frame lead to a different theory in the integer frame. Clearly, \(q^1(r)\ne \sqrt{[q^1(x^1)]^2+[q^2(x^2)]^2}\) due to the nonlinear terms in expressions such as Eq. (11).
Starting from [35], the tacit assumption has been that Eqs. (11) and (12) are based upon the Minkowski frame where all coordinate axes are orthogonal. So far this assumption has not been discussed in detail. We fill this gap here.
First and foremost, a change of presentation changes the theory (i.e., the magnitude of the corrections \(\mathcal {X}\) and \(\mathcal {T}\)) but not its qualitative features. It is well known that inequivalent presentations leave the anomalous scaling of the measure and the dimension of spacetime untouched [34, 35]. Therefore, multifractional scenarios are robust across different presentations. Picking a presentation allows us to make predictions which will change in another presentation, but not by much.
Second, the choice of the Cartesian or Minkowski fractional frame is not so restrictive as it might seem. In the physical examples studied in the literature, the observations studied in the theory with qderivatives involved: (a) the decay rate of the muon [40, 41]; (b) the Lamb shift in the spectrum of hydrogenic atoms [40, 41]; (c) the cosmicmicrowavebackground blackbody spectrum [65]; (d) the cosmicmicrowavebackground temperature spectrum [65]. In the theory with weighted derivatives, we studied (b\(^\prime )=\)(b), (c\(^\prime )=\)(c) and (e\(^\prime \)) the finestructure constant determined from the light of quasars [66]. In (a), the multiscale correction is only time dependent and t is the muon lifetime. In (b), (c), and (c\(^\prime \)) the multiscale correction is energy or temperaturedependent and, as we will argue below, this poses no problem of presentation. In (d), the spectrum is written as a function of the absolute value \(\mathbf{k}\) of comoving spatial momentum; since we use Cartesian momenta \(\{p^1(k^1),p^2(k^2),p^3(k^3)\}\), the expression in the fractional frame is in terms of \(\tilde{k}:=\sqrt{[p^1(k^1)]^2+[p^2(k^2)]^2+[p^3(k^3)]^2}=\mathbf{k}+\cdots \) but, to leading order in the multiscale correction, it is not different from what one would have obtained using a profile \(p(\mathbf{k})\). In (b\(^\prime \)), the multiscale correction depends on the characteristic time t of the electromagnetic processes involved in the Lamb shift. In (e\(^\prime \)), the correction depends on the cosmic time t of emission of light of distant objects since the big bang. All these settings are characterized by an effective onedimensional multiscale correction, either of restframe energies or of a welldefined time variable. There is not much arbitrariness here and the Cartesian or Minkowski chart fits the purpose.
4 Fractional sea turtles
To illustrate the effects of both the presentation choice and the picture selection (i.e., the problem of using nonadaptive instruments in a scaledependent environment), let us consider a \((1+1)\)dimensional nonrelativistic experiment where a wildlife ranger wants to check whether they live in a smooth Minkowski spacetime or, more interestingly, in a multiscale spacetime with binomial measure \(q_*(x\bar{x})\). To this purpose, they observe an adult sea turtle of size \(L \sim 1\,\mathrm{m}\) that, after laying her eggs on a beach, enters the waters at point \(x_\mathrm{A}\) at time \(t_\mathrm{A}\) (event A) and reaches a buoy at point \(x_\mathrm{B}\) at a later time \(t_\mathrm{B}\) (event B). The ranger knows that, according to the theory, measurements are performed in the fractional picture, which is the righthand side of Eq. (10). Here, coordinates x are noncomposite (i.e., their scaling is one and the same at all scales) and the action \(S=\int \mathrm{d}^2x\,v(x)\,\mathcal {L}\) of the system has a nontrivial measure weight \(v(x)=\det \mathrm{d}q^\mu /\mathrm{d}x^\mu \) breaking Lorentz invariance and deforming kinetic terms. The observer will use clocks and rods which do not adapt with the scale of the observed object, a standard analog wrist watch measuring time intervals \(\Delta t\) and a rigid rod measuring distances \(\Delta x\) in metric units. Recall that coordinates are “adaptive” or “nonadaptive” depending on whether they are composite objects or not.
We call the ranger consulting the “fractional twatch” a fractional observer \(\mathcal {O}_x\), to distinguish them from an integer observer \(\mathcal {O}_q\) which would use an adaptive “integer qwatch” measuring intervals \(\Delta q_*(t)\). For \(\mathcal {O}_q\), spacetime is ordinary Minkowski when expressed in terms of the composite (adaptive) coordinates q and the action is \(S=\int \mathrm{d}^2 q\,\mathcal {L}\). In other words, an integer observer is an ordinary observer in an ordinary, nonmultifractal world.
4.1 Preliminaries
Clearly, if all experiments took place at the same spacetime and energy scales, the difference between the fractional and the integer picture would only be in the convention of the observer’s measurement units. Living in a fractional world where a sea turtle takes 20 s to reach a buoy 100 m away would not be physically different from an integer world where the same event takes place in, say, 22 or 18 s. Integer turtles would be slower or faster (albeit not tremendously so) than fractional turtles, but that would just be the normality for integer observers.
4.2 Initialpoint presentation and dimensionless observables
Here is the crucial point at last. The ranger is acquainted with the fact that the ratio \(r_x=(\mathrm{speed})_\mathrm{adult}/(\mathrm{speed})_\mathrm{hatchling}\) of the swimming speed in the fractional picture depends on details such as the physiology of these animals, the temperature and time at the moment of hatching, the temperature of water, and so on, but that the average ratio reaches the universal, empirical value \(\langle r_x \rangle \simeq 10\): in general, adults are about ten times faster than hatchlings.^{7} Repeating their measurements every night and morning of the hatching season and finding a distribution of results definitely peaked at values close to \(r_x=10\), the ranger agrees qualitatively with the turtlespeed law both as a fractional and as an integer observer, \(r_x\simeq r_q\). However, imagine now that the relative experimental uncertainty reached by the ranger is better than in previous experiments, to the point where it is smaller than the correction \(\mathcal {T}'\mathcal {T}\) in (28). Then they also find a systematic discrepancy of the data points in the integer picture and a deviation of \(r_q\) to values smaller than 10. Then the observer must conclude that they are living in a fractional world.
Thus, even if geometric coordinates \(q^\mu (x^\mu )\) have the same units as coordinates \(x^\mu \), the existence of measurable dimensionless quantities such as the ratio r in our fictitious example allows an observer to recognize whether the underlying geometry is standard or multiscale. This is an aspect of the relation between measurements and frame choice not covered in previous discussions [36, 39, 41] and constitutes one of the main messages of the paper.
In practice, the outcome of experiments is less exciting, especially in macroscopic physics such as that describing turtle seafaring. The magnitude of the corrections \(\mathcal {X}\) and \(\mathcal {T}\) are unknown since \(\ell _*\) and \(t_*\) are free parameters of the theory. Therefore, the error bar can at most place an upper bound on such correction and the experiment cannot distinguish between a fractional and an integer world. Unhappy with this situation, the wildlife ranger may decide to change clothes and go to some laboratory or particle accelerator to perform altogether different experiments, this time involving atoms and quantum particles. For instance, they might want to check out the spectral lines of light atoms or the relativistic quantum particles generated in scattering events at high energies. This is precisely the type of observations considered in [40, 65], where upper bounds on \(t_*\) and \(\ell _*\) have been derived recently. These bounds are about 20–30 orders of magnitude smaller than the scales involved in the seaturtle observation (\(t_*<10^{27}\,\mathrm{s}\), \(\ell _*<10^{19}\,\mathrm{m}\)) and there would be no way to discriminate a turtle on such a multifractal from one on ordinary earth.
4.3 Finalpoint presentation
4.4 Space and timelike fractals
In reality, the three pairs of cases (30)–(31), (33)–(34), and (36)–(37) are an idealization of more complicated configurations. The main and most obvious reason is that space and time are entangled and, if the time direction is multiscale, then the corrections \(\mathcal {T}\) and \(\mathcal {X}\) in Eq. (27) compete with opposite signs and they can produce velocities \(v_x=v_q1\mathcal {T}_+\mathcal {X}_<v_q\) even in the finalpoint presentation, provided \(\mathcal {T}_>\mathcal {X}_\). The sign of the overall multiscale correction in Eq. (27) can be generic also in the null presentation (21), depending on the details of the problem.
5 Relativistic motion
For a timelike fractal spacetime, \(f=1+\mathcal {T}>1\) in the initialpoint presentation for any \(\mathcal {T}>0\) or in the finalpoint presentation for \(\mathcal {T}<2\). The Galilean speed \(v_x\) in the fractional picture can exceed the geometric speed of light by a factor f. For a macroscopic object this factor is mild and very close to 1, while for an object at scales \(L\sim \ell _*\) and \(t\sim t_*\) one can break the climit in a more spectacular way. Since \(\ell _*\) and \(t_*\) are at least as small as particlephysics scales, these results imply that one cannot use this multifractal theory to construct useful fasterthanlight spacecraft.
6 Future developments
Our main results have already been outlined in Sect. 1 and we will not repeat them here. We rather comment upon two open subjects to be tackled in the future: the effect of logarithmic oscillations and the role of the presentation in the microscopic structure of multiscale spacetimes.
6.1 Log oscillations
In general, due to the modulation of the oscillations we would not be able to connect relations of the sort \(v_x<v_q\) for nonrelativistic velocities with the local velocities in a coarsegrained subdiffusive stochastic process, even in a purely spacelike or timelike fractal. The discussion of Sect. 4.4 would then need a revision. All these features will deserve to be explored in greater detail.
6.2 Nowhere differentiability: toward stochastic spacetimes
Implicitly, in this paper we have begun to collect some evidence that there is a connection between the presentation choice and the stochastic properties of diffusion in these spacetimes. This interesting point went unnoticed in extant studies [37, 42] and it is worth looking into it in detail.
We have seen in Sect. 4.4 that, depending on the presentation of the measure, in certain time or spacelike systems the velocity of a nonrelativistic body is slower than in an ordinary spacetime. Next, we have argued that this property is plausibly compatible with the microscopic subdiffusion on fractals. The relation \(v_x<v_q\) is valid outside the “box” (38) (i.e., the range of values for \(\mathcal {T}\) and \(\mathcal {X}\)), which is not a region in the parameter space of the theory: both \(\mathcal {T}\) and \(\mathcal {X}\) depend on the measurements taken in the given system. If we require property C, then the only systemindependent configurations are (33), (34) and (36), (37). In that case, we set either the time or the spatial directions to be ordinary (this is part of the definition of the model) and the sign of the correction is unique for any system under consideration and for any regime. The box (38) should then be abandoned as a robust criterion for subdiffusion.

Spacetime spacelike fractals have similar properties than ordinary spatial fractals, for instance the microscopic origin of subdiffusion (i.e., the \(v_x<v_q\) relation), but not in all presentations of the measure.

Diffusion and microscopic properties of spacetime timelike fractals or spacetime fractals with a nontrivial timelike component can differ widely from spacelike ones without breaking the ABC rules.

Physical observables can provide elements to prefer one presentation over another but only in the case of a positive detection of multiscale effects.

Although the dimensions of spacetimes are unaffected by the presentation of the measure, there is some yet unknown nontrivial relation between the presentation and microscopic stochastic properties of diffusion (see also Sect. 3.2).
In the Itô interpretation, \(\tilde{t}_j=t_j\) is the initial point in \(\Delta t_j\) and the function f only depends on the behavior of B(t) up to the time \(t_j\). In this case, the stochastic process X(t) is a martingale: the expectation value of an event at some future time \(\tilde{t}_j\in (t_j,t_{j+1}]\) is equal to the value observed at the present time \(t_j\). In other words, the knowledge of all previously observed values does not help to predict future outcomes.^{11} In the Stratonovich interpretation, \(\tilde{t}_j=(t_{j+1}+t_j)/2\) is taken in the middle of the interval and one symmetrizes between past and future (see the end of Sect. 4.3.6 of [73] for caveats). In this case, X(t) is not a martingale: knowledge of prior outcomes may help to determine future events.
Choice of boundaries in Eq. (48) for different presentations, where \(\Delta x=xx_\mathrm{i}\)
Presentation  \(\bar{x}\)  \(x_0\)  \(x_{n1}\) 

Null  0  \(x_\mathrm{i}\)  x 
Initial point  \(x_\mathrm{i}\)  0  \(\Delta x\) 
Final point  x  \(\Delta x\)  0 
Symmetrized  \(\frac{x_\mathrm{i}+x}{2}\)  \(\frac{\Delta x}{2}\)  \(\frac{\Delta x}{2}\) 
We should bear in mind that the theory with qderivatives is a simplified version of a genuine nowhere differentiable fractal. Barring explicit and most challenging constructions of field theories on fractals [76, 77, 78] or field theories constructed with stochastic measures, the closest thing mimicking nowhere differentiability is fractional calculus. The multiscale theory with fractional derivatives [34, 35] incarnates precisely this possibility and it may be the only multifractional theory obeying property D in the introduction. In fact, the differential operators in the theory with ordinary derivatives are the usual partial derivatives \(\partial _x\). In the theory with weighted derivatives, they are weighted version of the same operators, \(\partial _x\rightarrow (\partial _x q)^{1/2}\partial _x[(\partial _x q)^{1/2}\,\cdot \,]\). In the theory with qderivatives, they are again of integer order but with a different weight distribution, \(\partial _x\rightarrow (\partial _x q)^{1}\partial _x\). These modifications of \(\partial _x\) and the nontrivial integration measure give all these models an “irregular” geometry but in a rather simpleminded way. On the other hand, the derivatives of the fourth multifractional theory are nonlocal integrodifferential operators, which are known to capture the properties of sets not differentiable in the ordinary sense (see [34, 75] and references therein).
Therefore, the theory with fractional derivatives may well be the only one to describe a multifractal geometry in the strong sense. Due to its higher technical challenges, this framework has not been explored as extensively as the other multifractional theories but the preliminary analysis in [34, 35] and work in progress show enticing properties that include an exotic particle content and an improved perturbative renormalizability (absent in the other cases [79]). The arguments presented in this section add fuel to our curiosity and strongly suggest that spacetimes with a fractional integrodifferential structure would be intrinsically stochastic. We hope to report on that soon.
Footnotes
 1.
 2.
 3.
 4.
A classic example is the sphere \(S^2\). Its surface is twodimensional (i.e., isomorphic to a plane) only locally, while \(d_\textsc {s}\ne 2\) at scales comparable with the curvature radius.
 5.
Previously in the literature, this class was often dubbed “multiscale” but, after clarifying the nomenclature, it is better to stick with the name multifractional, leaving the term multiscale to a much wider landscape of theories.
 6.
The anisotropic case where some or all \(\alpha _\mu \) are different is tricky because the components of the random walk \(X^2=X_0^2+X_1^2+\cdots \) would have inhomogeneous scaling and each direction should be considered separately. This complication is responsible of the fact that the dimension of the Cartesian product of fractals may not coincide with the sum of their dimensions [31].
 7.
Disclaimer: This is a fictional situation with no connection with real life and there is no such thing as a speed law for sea turtles. Nevertheless, the numbers given in the text are plausible and the ratio \(r_x\sim 10\) is of the correct order of magnitude for various species of sea turtles and according to extant observations. Consult [67, 68] and references therein for some studies on seaturtle speeds when swimming and walking.
 8.
A nowhere differentiable curve does not admit local tangents and the argument in the text does not apply to an ideal random walk.
 9.
If \(A>1\), then there may even occur pathological situations where \(\Delta q<0\). This happens because the measure is not positive definite for \(A>1\) and it does not correspond, at ultramicroscopic scales, to an ordinary geometry. Still, it is a welldefined geometry, even if highly unconventional.
 10.
Almost surely, i.e., with probability 1.
 11.
For this reason, martingales are used as theoretical models of fair games.
Notes
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