Radon transform on the cylinder and tomography of a particle on the circle

TL;DR

This paper introduces a tomographic probability distribution on the phase space cylinder for particles on a circle, relating it to the Radon transform, and extends the approach to multidimensional tori with geometric insights.

Contribution

It presents a new tomographic framework for particles on a circle and torus, connecting it explicitly to the Radon transform and exploring its geometric interpretation.

Findings

Explicit relations between tomographic and phase space densities

Connection of the map to Radon transform on the plane

Generalization to multidimensional tori and geometric interpretation

Abstract

The tomographic probability distribution on the phase space (cylinder) related to a circle or an interval is introduced. The explicit relations of the tomographic probability densities and the probability densities on the phase space for the particle motion on a torus are obtained and the relation of the suggested map to the Radon transform on the plane is elucidated. The generalization to the case of a multidimensional torus is elaborated and the geometrical meaning of the tomographic probability densities as marginal distributions on the helix discussed.