Trkalian fields and Radon transformation

TL;DR

This paper explores the Radon transform of Trkalian fields, revealing their eigenvalue properties, reconstruction methods, and applications to gauge theory and Lundquist solutions, connecting integral transforms with topological and physical properties.

Contribution

It introduces a novel Radon transform framework for Trkalian fields, linking it with differential forms, eigenvalue equations, and gauge theory applications, including topologically massive Abelian fields.

Findings

Radon transform of Trkalian fields satisfies eigenvalue equations.

Reconstruction of fields from Radon transform on a hemisphere is possible.

Radon transform relates to topological mass quantization in gauge theory.

Abstract

We write the spherical curl transformation for Trkalian fields using differential forms. Then we consider Radon transform of these fields. The Radon transform of a Trkalian field satisfies a corresponding eigenvalue equation on a sphere in transform space. The field can be reconstructed using knowledge of the Radon transform on a canonical hemisphere. We consider relation of the Radon transformation with Biot-Savart integral operator and discuss its transform introducing Radon-Biot- Savart operator. The Radon transform of a Trkalian field is an eigenvector of this operator. We also present an Ampere law type relation for these fields. We apply these to Lundquist solution. We present a Chandrasekhar-Kendall type solution of the corresponding equation in the transform space. Lastly, we focus on the Euclidean topologically massive Abelian gauge theory. The Radon transform of an anti-self-dual field is related by antipodal map on this sphere to the transform of the self-dual field obtained by inverting space coordinates. The Lundquist solution provides an example of quantization of topological mass in this context.