Geometry of fractional spaces

Calcagni, Gianluca

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Geometry of fractional spaces

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Calcagni, Gianluca
Microscopic Quantum Structure & Dynamics of Spacetime, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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1106.5787
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ATMP16_549.pdf
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Citation

Calcagni, G. (2012). Geometry of fractional spaces. Advances in Theoretical and Mathematical Physics, 16, 549-644. Retrieved from http://arxiv.org/abs/1106.5787.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0012-069B-8

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 will be presented in a companion paper.