Geometry of fractional spaces (1106.5787v3)

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.

• The paper establishes fractional calculus as a tool to characterize geometric properties in spaces with non-integer dimensions.
• The paper reveals that dimensional flow yields fractal-like behavior, with Hausdorff and spectral dimensions transitioning continuously.
• The paper outlines implications for quantum gravity and field theory, suggesting new avenues for constructing renormalizable models.

Overview of "Geometry of Fractional Spaces"

The paper by Gianluca Calcagni presents a comprehensive investigation into the novel concept of fractional spaces, a class of spaces characterized by continuous geometries with non-integer Hausdorff and spectral dimensions. Using fractional calculus as a foundational tool, the paper delineates the differential structure, distances, volumes, and symmetries of these fractional spaces, effectively positioning them as fractal-like when specific dimensional conditions are met.

Key Points and Numerical Results

1. Fractional Calculus and Geometry: The paper establishes fractional calculus as a means to explore geometric properties traditionally associated with fractals. This includes expressing distances and forms in these fractional spaces through modified metric definitions and integration apparatuses.
2. Dimensional Analysis: The exploration of dimensional flow in fractional spaces reveals how dimensions can smoothly transition to non-integer values, a common characteristic in fractal geometries. Specifically, conditions are set such that the spaces exhibit fractal attributes when the ratio of Hausdorff to spectral dimensions exceeds one.
3. Spectral Properties: Calcagni introduces a method to calculate the spectral dimension through a diffusion process, revealing a constant nature for these dimensions across scales due to the inherent continuous nature of fractional spaces. Even in these generalized spaces, numerical robustness is maintained as the house-dimensionality consistently correlates with how volume scales.
4. Comparison with Traditional Spaces: Fractional spaces are juxtaposed with conventional Euclidean spaces by highlighting differences in scaling laws and measurement weights, which allow fractional systems to exhibit properties akin to those found in deterministic and random fractals.

Implications and Speculative Outlook

The paper lays foundational work for prospective field theories operating in spaces with fractional characteristics, proposing potential avenues for understanding phenomena in quantum gravity and cosmology. Several implications are drawn:

• Theories of Quantum Gravity: The formulation of effective quantum gravity models could benefit from incorporating fractional space concepts, especially in models where conventional approaches struggle to maintain renormalizability.
• Fractal Cosmology: Fractional spaces could introduce new perspectives for tackling issues like inflation and the cosmological constant problem due to their unique scaling properties and dimensional flow.
• Field Theory Development: The paper indirectly suggests future development of perturbative quantum field theories that could support non-trivial dimensional flows and potentially lead to renormalizable gravitational theories within fractional spaces.

Calcagni’s exploration ensures that fractional space constructs remain mathematically rigorous while implying their applicability in understanding high-energy physical theories. It opens pathways for theorists to tackle problems involving spacetime, offering a complementary perspective potentially harmonizing with existing quantum gravity formulations. Future research will likely explore the applications of fractional spaces, extending this foundational framework to more complex and dynamic models.