Trkalian fields and Radon transformation


JOURNAL OF MATHEMATICAL PHYSICS, vol.51, no.3, 2010 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 51 Issue: 3
  • Publication Date: 2010
  • Doi Number: 10.1063/1.3293982
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Istanbul University Affiliated: Yes


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.