Advances in Theoretical and Mathematical Physics

Volume 16 (2012)

Number 4

Momentum transforms and Laplacians in fractional spaces

Pages: 1315 – 1348

DOI: http://dx.doi.org/10.4310/ATMP.2012.v16.n4.a5

Authors

Gianluca Calcagni (Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Golm, Germany)

Giuseppe Nardelli (Dipartimento di Matematica e Fisica, Università Cattolica, Brescia, Italy; INFN, Gruppo Collegato di Trento, Università di Trento, Povo (Trento), Italy)

Abstract

We define an infinite class of unitary transformations between position and momentum fractional spaces, thus generalizing the Fourier transform to a special class of fractal geometries. Each transform diagonalizes a unique Laplacian operator. We also introduce a new version of fractional spaces, where coordinates and momenta span the whole real line. In one topological dimension, these results are extended to more general measures.

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