[Paper Review] Geometry of fractional spaces

Gianluca Calcagni|arXiv (Cornell University)|Jun 28, 2011

Advanced Mathematical Theories and Applications153 references6 citations

TL;DR

This paper introduces fractional flat space—a continuous geometry with constant non-integer Hausdorff and spectral dimensions—using fractional calculus to define its differential structure, distances, volumes, and symmetries. The key contribution is establishing that fractional spaces are fractals when the ratio of spectral to Hausdorff dimension exceeds one, providing a rigorous geometric foundation for field theories on multi-fractal spacetimes.

ABSTRACT

We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool is fractional calculus, which is cast in a way convenient for the definition of the differential structure, distances, volumes, and symmetries. By an extensive use of concepts and techniques of fractal geometry, we clarify the relation between fractional calculus and fractals, showing that fractional spaces can be regarded as fractals when the ratio of their Hausdorff and spectral dimension is greater than one. All the results are analytic and constitute the foundation for field theories living on multi-fractal spacetimes, which are presented in a companion paper.

Motivation & Objective

• To develop a geometric framework for fractional flat space with constant non-integer Hausdorff and spectral dimensions.
• To clarify the relationship between fractional calculus and fractal geometry, showing when fractional spaces can be interpreted as fractals.
• To provide analytic tools—differential structure, metric, volume, and symmetries—for field theories on multi-fractal spacetimes.
• To establish the spectral dimension of fractional space through diffusion processes and Laplacian operators.
• To lay the groundwork for quantum field theories on spacetimes with scale-dependent geometry, particularly in the context of quantum gravity.

Proposed method

• Uses fractional calculus, reformulated for geometric applications, to define derivatives, integrals, and differential operators on fractional spaces.
• Applies Lebesgue–Stieltjes measures and fractional integrals to model anomalous scaling and diffusion properties.
• Defines the metric and distance in fractional space via the fractional gradient and norm, generalizing Euclidean geometry.
• Introduces a geometric notation and volume form based on fractional calculus, preserving symmetry and structure.
• Computes the spectral dimension using diffusion equations with fractional Laplacians, linking it to the walk dimension.
• Analyzes the conditions under which fractional spaces are fractals, based on the ratio of spectral to Hausdorff dimension.

Experimental results

Research questions

• RQ1 How can fractional calculus be systematically used to define a geometric structure for a continuous space with non-integer dimensions?
• RQ2 Under what conditions does a fractional space exhibit fractal properties, particularly in terms of scaling and diffusion?
• RQ3 What is the relationship between the Hausdorff dimension, spectral dimension, and walk dimension in fractional spaces?
• RQ4 How does the spectral dimension of fractional space behave under different diffusion processes, and what does this imply for field theory?
• RQ5 In what way do fractional spaces serve as a continuum approximation to fractals, and when is this approximation valid?

Key findings

• Fractional flat space is defined with constant non-integer Hausdorff and spectral dimensions, generalizing Euclidean space via fractional calculus.
• The spectral dimension of fractional space is given by $ d_{ m S} = \beta d_{ m H} $, where $ \beta $ is the order of the diffusion operator and $ d_{ m H} $ is the Hausdorff dimension.
• Fractional spaces are fractals if and only if the walk dimension $ d_{ m W} \geq 2 $, which occurs when $ \beta \leq 1 $, corresponding to anomalous diffusion.
• For $ \beta = 1 $, the spectral dimension equals the Hausdorff dimension, recovering normal diffusion and integer-dimensional behavior.
• When $ \beta = 1/\alpha $, the spectral dimension matches the topological dimension $ D $, indicating a transition to standard geometry at large scales.
• The spectral dimension remains constant and non-vanishing across all $ \beta $, with no convergence to embedding dimension in the limit of low lacunarity, distinguishing fractional spaces from fractal approximations.