The Index of discontinuous Vector Fields: Topological Particles and Vector Fields

TL;DR

This paper introduces topological particles and radiation as connected components of vector field defects, defining an index that remains invariant under transformations and changes only with radiation, with implications for physical pseudo-vector fields.

Contribution

It defines a new topological index for vector field defects, demonstrating its invariance and conservation properties, and predicts zero index for physical pseudo-vector fields.

Findings

The index is conserved during particle interactions.

The index remains invariant under coordinate transformations.

Physical pseudo-vector fields have an index of zero.

Abstract

We define the concepts of topological particles and topological radiation. These are nothing more than connected components of defects of a vector field. To each topological particle we assign an index which is an integer which is conserved under interactions with other particles much as electric charge is conserved. For space-like vector fields of space-times this index is invariant under all coordinate transformations. We propose the following physical principal: For physical vector fields the index changes only when there is radiation. As an implication of this principal we predict that any physical psuedo-vector field has index zero. The definition of the index is quite elementary. It only depends upon the concepts of continuity, compactness, the Euler-Poincare number, and the idea of inward pointing. The proof that this definition is well defined takes up most of the paper. The paper concludes with a list of properties of the index.