Advances in Theoretical and Mathematical Physics

Geometry of fractional spaces

Gianluca Calcagni

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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.

Article information

Source
Adv. Theor. Math. Phys., Volume 16, Number 2 (2012), 549-644.

Dates
First available in Project Euclid: 23 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.atmp/1358950889

Mathematical Reviews number (MathSciNet)
MR3019412

Zentralblatt MATH identifier
1268.28009

Citation

Calcagni, Gianluca. Geometry of fractional spaces. Adv. Theor. Math. Phys. 16 (2012), no. 2, 549--644. https://projecteuclid.org/euclid.atmp/1358950889


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