Vector Fields and the Unity of Mathematics and Physics

TL;DR

This paper argues that magnetic monopoles should not exist based on the invariance of vector field indices under space-time transformations, and explores the implications of this concept in both mathematics and physics.

Contribution

It introduces the Law of Vector Fields and the concept of index invariance, providing new proofs of classical theorems and suggesting applications in physics.

Findings

Magnetic monopoles are argued to be non-existent due to index invariance.

The Law of Vector Fields generalizes classical theorems like Gauss-Bonnet.

New proofs of mathematical theorems are derived using the index concept.

Abstract

We give an argument that magnetic monopoles should not exist. It is based on the concept of the index of a vector field. The thrust of the argument is that indices of vector fields are invariants of space-time orientation and of coordinate changes, and thus physical vector fields should preserve indices. The index is defined inductively by means of an equation called the Law of Vector Fields. We give extended philosophical arguments that this Law of Vector Fields should play an important role in mathematics, and we back up this contention by using it in a mechanical way to greatly generalize the Gauss--Bonnet theorem and the Brouwer fixed point theorem and get new proofs of many other theorems. We also give some other suggestions for using the Law and index in physics.