Advances in Theoretical and Mathematical Physics

Volume 16 (2012)

Number 4

Momentum transforms and Laplacians in fractional spaces

Pages: 1315 – 1348



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)


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.

Published 30 April 2013