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Journal Article

#### Geometry of fractional spaces

##### Fulltext (public)

1106.5787

(Preprint), 839KB

ATMP16_549.pdf

(Any fulltext), 3MB

##### Supplementary Material (public)

There is no public supplementary material available

##### 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: http://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.