Resolution-Scale Relativity: Fractals, Quantum Laws

• Resolution-Scale Relativity is a framework where scale is a fundamental variable, using fractal, non-differentiable geometries to derive physical laws.
• The theory introduces doubled, complex velocity fields and scale-covariant derivatives that yield quantum-like equations such as Schrödinger and Dirac equations.
• Its applications span from microscopic quantum mechanics to turbulence, dark matter phenomena, and the proton-radius puzzle, offering testable predictions.

Searching arXiv for recent and foundational papers on resolution-scale relativity / scale relativity. Resolution-Scale Relativity is a formulation in which scale or resolution is treated as a fundamental variable of physical description, on the same conceptual footing as position and time. In this framework, there is no absolute scale in nature; only scale ratios have physical meaning, and physical laws are required to be covariant under changes of resolution. The theory is built on the hypothesis that space-time, or the relevant dynamical space, is continuous but non-differentiable and therefore fractal, so that trajectories, fields, and even derivatives become explicitly scale-dependent. Within this setting, complex wave functions, spinors, and quantum-like equations are interpreted as emergent consequences of geodesic motion in a fractal geometry rather than as independent postulates (Nottale et al., 2012, Célérier et al., 2013). The same formalism has also been used outside microscopic quantum mechanics, including harmonic-oscillator dynamics (Bonilla et al., 2017), turbulence in velocity space (Montera, 2013, Nottale et al., 2018), thermodynamics (Carroll, 2011), dark-matter halo phenomenology (Chavanis, 2017), virial analyses of non-differentiable dynamics (LeBohec, 2022), gauge coupling (Célérier et al., 2010), and scale-composition problems such as the proton-radius puzzle (Nottale, 2019).

1. Relativity extended to scale and resolution

Scale Relativity is a generalization of the relativity principle of motion to scales. In this view, a reference system is characterized not only by position, orientation, velocity, or acceleration, but also by its state of resolution. The theory states that there is no absolute scale in nature and that only scale ratios have physical meaning (Nottale, 2019). In its simplest implementation, the natural scale variable is logarithmic, for example $\ln(\varepsilon'/\varepsilon)$, so that multiplicative changes of resolution become additive transformations in scale space (0812.3857).

A central mathematical premise is that a continuous and non-differentiable curve is fractal, in the general meaning that its length is explicitly scale dependent and tends to infinity when the resolution interval tends to $0$ (Nottale et al., 2012, 0812.3857). Coordinate functions therefore become fractal functions, such as $X = X(\mathrm{d}X)$ or $X(t,\varepsilon_t)$, and measured quantities depend on the observational scale. This leads to explicit scale laws, scale differential equations, and, in more elaborate versions, a scale dynamics with its own Lagrangian and Euler–Lagrange equations in scale space (0812.3857, Nottale, 2013).

The theory distinguishes the algebraic differential element from the resolution interval. In the non-relativistic setting, $dt$ can be used both as a differential element and, through its absolute value, as a resolution variable; the physically relevant interface is $\varepsilon_t = |dt|$ (Nottale et al., 2012). In the relativistic setting the same role is played by the invariant parameter $s$, with $\varepsilon_s = |ds|$ and $\varepsilon_x^\mu = \sqrt{2D}\,\varepsilon_s$ for the critical fractal dimension $D_f = 2$ (Célérier et al., 2013). This identification is not a minor convention: it is the scale at which the derivative takes its physical value, independent of how fine the probing becomes (Célérier et al., 2013).

The principle has also been formulated in terms of transformation groups on logarithmic scale variables. In one form, scale transformations obey a Galilean composition law in $0$0; in another, called special scale relativity, they obey a Lorentz-like composition with an invariant limiting scale identified with the Planck length (0812.3857, Nottale, 2019). A plausible implication is that scale itself is treated as a structured transformation domain, not merely as an external control parameter.

2. Non-differentiability, fractality, and two-valued derivatives

The release of differentiability is the technical source of most distinctive structures in resolution-scale relativity. For a non-differentiable function, the derivative depends on the sign of the increment, so the forward derivative and backward derivative no longer coincide (Nottale et al., 2012, Teh et al., 2016). This yields two elementary velocities $0$1 and $0$2, or in relativistic notation a doubled velocity field constructed from $0$3 and $0$4 (Nottale et al., 2012, Célérier et al., 2013).

In the non-relativistic case, the infinitesimal displacement is written as

$0$5

with $0$6, $0$7, $0$8, and, for $0$9,

$X = X(\mathrm{d}X)$0

At the interface $X = X(\mathrm{d}X)$1, this gives

$X = X(\mathrm{d}X)$2

so that

$X = X(\mathrm{d}X)$3

(Nottale et al., 2012). The same structure reappears in the relativistic theory, now for each coordinate $X = X(\mathrm{d}X)$4: $X = X(\mathrm{d}X)$5 with

$X = X(\mathrm{d}X)$6

(Célérier et al., 2013).

Because the two-valuedness is fundamental rather than perturbative, the doubled velocity is naturally represented by complex numbers: $X = X(\mathrm{d}X)$7 (Nottale et al., 2012, Célérier et al., 2013). In this representation, the real part is the symmetric mean velocity and the imaginary part encodes the non-differentiable asymmetry between forward and backward processes (Bonilla et al., 2017, Teh et al., 2016).

A stronger version of the argument appears in the relativistic theory, where additional local symmetry breakings at the fractal level—$X = X(\mathrm{d}X)$8, $X = X(\mathrm{d}X)$9, and $X(t,\varepsilon_t)$0—generate successive doublings of the velocity fields, leading from complex numbers to quaternionic and bi-quaternionic structures (Célérier et al., 2013). This suggests that algebraic enlargement is tied to the hierarchy of broken local symmetries induced by non-differentiability.

3. Covariant derivatives, wave functions, and quantum equations

Once the doubled velocity field is represented by a complex quantity, the usual derivative is replaced by a complex, scale-covariant derivative. In the non-relativistic setting this derivative takes the form

$X(t,\varepsilon_t)$1

(Nottale et al., 2012, Teh et al., 2016, Carroll, 2011). In the relativistic setting, after deriving the Lorentz-invariant fluctuation law

$X(t,\varepsilon_t)$2

the total derivative of a fractal function becomes

$X(t,\varepsilon_t)$3

and for $X(t,\varepsilon_t)$4-stationary functions,

$X(t,\varepsilon_t)$5

(Célérier et al., 2013).

The wave function is introduced as an exponential of the complex action,

$X(t,\varepsilon_t)$6

with $X(t,\varepsilon_t)$7 in the relativistic case (Célérier et al., 2013), and $X(t,\varepsilon_t)$8 in the non-relativistic derivations (Nottale, 2013, Teh et al., 2016). The complex momentum and velocity then satisfy

$X(t,\varepsilon_t)$9

(Célérier et al., 2013), or, in the non-relativistic formulation,

$dt$0

(Nottale, 2013).

The geodesic equation in a fractal geometry,

$dt$1

then yields the Klein–Gordon equation,

$dt$2

(Célérier et al., 2013). In the bi-quaternionic extension, the same construction yields a bi-quaternionic Klein–Gordon equation and, by taking the square root through quaternionic/spinorial algebra, the standard Dirac equation

$dt$3

(Célérier et al., 2013). In the electromagnetic case, a scale-covariant derivative

$dt$4

is derived geometrically from scale transformations, and the electromagnetic Klein–Gordon and Dirac equations are recovered in their standard minimal-coupling forms (Célérier et al., 2010).

In the non-relativistic limit, the same formalism yields the Schrödinger equation

$dt$5

(Nottale et al., 2012, Teh et al., 2016). The paper on the harmonic oscillator makes this correspondence explicit in stationary one-dimensional dynamics, where the scale-relativistic Hamiltonian law reduces to a complex Riccati equation and, through a logarithmic derivative substitution, exactly to the standard harmonic-oscillator Schrödinger problem (Bonilla et al., 2017).

4. Hydrodynamic, stochastic, and energetic representations

Resolution-scale relativity admits several equivalent representations: geodesic, wave-mechanical, fluid-mechanical, and stochastic (Nottale, 2013). Writing

$dt$6

the complex velocity becomes

$dt$7

with

$dt$8

(Nottale, 2013). The generalized Schrödinger equation then separates into a continuity equation,

$dt$9

and an Euler-type equation,

$\varepsilon_t = |dt|$0

with the quantum-type potential

$\varepsilon_t = |dt|$1

(Nottale, 2013, Carroll, 2011).

This potential is a recurring object across the literature. In thermodynamic embedding, it enters the effective energy as

$\varepsilon_t = |dt|$2

and its integral against the density yields a Fisher-information contribution,

$\varepsilon_t = |dt|$3

when $\varepsilon_t = |dt|$4 (Carroll, 2011). In the virial formulation for non-differentiable paths, the same structure appears through the kink velocity $\varepsilon_t = |dt|$5, with

$\varepsilon_t = |dt|$6

and the virial theorem becomes

$\varepsilon_t = |dt|$7

(LeBohec, 2022). This suggests that the quantum-like potential is interpreted not as an external addition but as an energetic manifestation of the non-differentiable component of the dynamics.

The stochastic representation uses forward and backward Fokker–Planck-like equations,

$\varepsilon_t = |dt|$8

with

$\varepsilon_t = |dt|$9

(Nottale, 2013). The paper on non-differentiable mechanics shows that, for stationary systems with null average classical velocity, the generalized Newton equation reduces to a Langevin equation whose numerical integration for the harmonic oscillator reproduces the same statistics as the stationary Schrödinger solutions (Teh et al., 2016). The harmonic-oscillator study independently shows that the scale-relativistic Riccati variable $s$0 is the logarithmic derivative of the usual Hermite-function eigenstates and exactly reproduces the energy spectrum $s$1 (Bonilla et al., 2017).

5. Domains of application

The framework has been extended well beyond the foundational derivation of quantum equations.

In gauge theory, scale covariance and quantum covariance are combined to derive the electromagnetic Klein–Gordon equation,

$s$2

and the standard electromagnetic Dirac equation,

$s$3

(Célérier et al., 2010). The geometric interpretation is that gauge fields encode scale-motion coupling, while spin originates from the deeper broken symmetries of the fractal geometry (Célérier et al., 2010, Célérier et al., 2013).

In turbulence, the theory is applied not to space-time itself but to velocity space. The internal interactions of fluids are assumed to occur at all scales, so the force fields and resulting velocity fields need not be differentiable. The scale-space equation for acceleration,

$s$4

has a solution

$s$5

whose fractal part yields, for $s$6,

$s$7

the Lagrangian Kolmogorov law of fully developed turbulence (Montera, 2013). The same framework predicts a transition between laminar and turbulent behavior when the Reynolds number satisfies $s$8 (Montera, 2013). A Schrödinger-like equation in velocity space,

$s$9

is then proposed for the probability of having a given velocity at a given location (Montera, 2013).

A later development of turbulence in scale relativity derives a macroscopic Schrödinger equation in $\varepsilon_s = |ds|$0-space, identifies the local velocity PDF with $\varepsilon_s = |ds|$1, predicts null minima $\varepsilon_s = |ds|$2, and introduces a new acceleration component

$\varepsilon_s = |ds|$3

expected to diverge near those minima (Nottale et al., 2018). The paper reports that empty zones are detected in observed Lagrangian velocity PDFs, that the new acceleration component is directly identified in laboratory data, and that it accounts for bursts and calm periods of intermittent acceleration (Nottale et al., 2018). This suggests that resolution-scale effects can be empirically visible in a macroscopic chaotic system.

In gravitational astrophysics and cosmology, the formalism has been applied to dark-matter halos through a generalized Schrödinger equation with logarithmic nonlinearity and dissipation,

$\varepsilon_s = |ds|$4

with $\varepsilon_s = |ds|$5 for dark-matter halos (Chavanis, 2017). In this model, the quantum-like potential generates a solitonic core that solves the cusp problem, the logarithmic nonlinearity yields an isothermal halo and flat rotation curves, and the damping term ensures relaxation toward equilibrium via an $\varepsilon_s = |ds|$6-theorem (Chavanis, 2017).

The virial theorem for non-differentiable paths likewise suggests that a quantum-like potential term may contribute in macroscopic chaotic astrophysical systems, and the paper states that this contribution could account for some fraction of the dark matter hypothesis (LeBohec, 2022). A plausible implication is that scale-relativistic corrections are being proposed as effective large-scale contributions rather than as replacements for all standard astrophysical mechanisms.

A different application concerns scale composition itself. In the proton-radius analysis, the theory replaces the usual composition of scale ratios by a Lorentz-like law in logarithmic variables,

$\varepsilon_s = |ds|$7

with $\varepsilon_s = |ds|$8 and $\varepsilon_s = |ds|$9 identified with the Planck length (Nottale, 2019). Using this special scale-relativistic law, the discrepancy between proton radii measured relative to electron and muon scales is reduced from a ratio $\varepsilon_x^\mu = \sqrt{2D}\,\varepsilon_s$0 to $\varepsilon_x^\mu = \sqrt{2D}\,\varepsilon_s$1, i.e. from a $\varepsilon_x^\mu = \sqrt{2D}\,\varepsilon_s$2 discrepancy to agreement within about $\varepsilon_x^\mu = \sqrt{2D}\,\varepsilon_s$3 (Nottale, 2019). This paper presents the proton-radius puzzle as a test of the special scale relativity theory (Nottale, 2019).

The broader review of scale relativity and fractal space-time also reports applications to the QCD coupling, the cosmological constant, extrasolar planets, Kuiper belt objects, stellar cycles, biological evolution, earthquake statistics, and Arctic sea-ice extent, often through power laws with log-periodic corrections generated by scale differential equations (0812.3857).

6. Conceptual scope, analogies, and controversies

Resolution-scale relativity should be distinguished from unrelated uses of “scale-relative” language. In philosophical work on general relativity, a “scale-relative perspective” refers to the local validity of special relativity within general relativity: special relativity is a good approximation when the probing scale $\varepsilon_x^\mu = \sqrt{2D}\,\varepsilon_s$4 is small relative to the curvature scale $\varepsilon_x^\mu = \sqrt{2D}\,\varepsilon_s$5, with

$\varepsilon_x^\mu = \sqrt{2D}\,\varepsilon_s$6

as the small parameter (Linnemann et al., 2023). This use of scale language concerns approximation regimes in curved space-time rather than Nottale’s geometric theory of non-differentiability. Still, both frameworks emphasize that physical descriptions are resolution-dependent and that limiting structures emerge at sufficiently small or large scales (Linnemann et al., 2023). This suggests a family resemblance rather than an identity of theories.

A second distinction concerns the role of stochasticity. Resolution-scale relativity often uses Wiener-process-like fluctuations and diffusion constants, but it does not present itself as standard stochastic mechanics. The repeated claim is that stochasticity is a manifestation of non-differentiable geometry and of the infinite family of geodesics available in a fractal space(-time), rather than an externally postulated subquantum noise (Nottale, 2013, Teh et al., 2016). This is especially explicit in the derivations of the Schrödinger equation and in the turbulence papers, where stochastic terms arise from the scale structure of the medium or of the relevant dynamical space (Teh et al., 2016, Nottale et al., 2018).

A third point concerns empirical status. Some applications, such as the harmonic oscillator correspondence (Bonilla et al., 2017), the turbulence acceleration analysis (Nottale et al., 2018), the proton-radius reinterpretation (Nottale, 2019), and the dark-matter halo phenomenology (Chavanis, 2017), present concrete formulas and quantitative comparisons. Others are broader interpretive proposals. The five-dimensional relativity work on the cosmological constant, for example, is not part of Nottale’s formalism but offers a separate scale-dependent picture in which $\varepsilon_x^\mu = \sqrt{2D}\,\varepsilon_s$7 scales as $\varepsilon_x^\mu = \sqrt{2D}\,\varepsilon_s$8 for small systems and as $\varepsilon_x^\mu = \sqrt{2D}\,\varepsilon_s$9 for large systems (Wesson et al., 2014). This suggests a wider landscape of theories in which scale-dependent laws are treated as physically significant, though not all of them share the same assumptions or mathematical machinery.

The recent exoplanet study explicitly framed in resolution-scale relativity is beyond the publication date of the foundational works and reports that, among 115 systems with four or more planets, 38 systems (33%) comply with the predicted integer-cube period relation with null-hypothesis accidental probability less than $D_f = 2$0, while 16 systems (14%) comply with less than $D_f = 2$1 (Foerster et al., 20 Sep 2025). It also reports a super-fundamental speed $D_f = 2$2 around which system-specific fundamental speeds cluster (Foerster et al., 20 Sep 2025). Since these claims are presented by the paper itself as signatures rather than established laws, a cautious reading is appropriate. A plausible implication is that exoplanet systematics are being explored as a test bed for macro-quantization hypotheses.

Resolution-scale relativity therefore occupies a hybrid position. It is simultaneously a foundational proposal for quantum mechanics, a general theory of scale transformations in fractal geometry, and a source of effective macroscopic models for chaotic, turbulent, astrophysical, and biological systems. Its unifying thesis is that once non-differentiability and scale dependence are made fundamental, complex and spinorial wave functions, quantum-like potentials, and Schrödinger-type dynamics arise as geometric consequences of the relativity of resolution (Nottale et al., 2012, Célérier et al., 2013, 0812.3857).