# Hi, I'm Luke Burns

I study math and physics, and code some. My work is on Github . Below is a list of projects of mine. You're welcome to open an issue, or fork and submit a pull request with improvements.

**Maxwell's equations are universal for locally conserved quantities**

A fundamental result of classical electromagnetism is that Maxwell's equations imply that electric charge is locally conserved. Here we show the converse: Local charge conservation implies the local existence of fields satisfying Maxwell's equations. This holds true for any conserved quantity satisfying a continuity equation. It is obtained by means of a strong form of the PoincarĂ© lemma presented here that states: Divergence-free multivector fields locally possess curl-free antiderivatives on flat manifolds. The above converse is an application of this lemma in the case of divergence-free vector fields in spacetime. We also provide conditions under which the result generalizes to curved manifolds.

**An extension of the Dirac equation**

A minimal extension of the Dirac equation is shown to describe a pair of massless, electrically charged fermions. Solutions exhibit rotation in the particle's spin plane, analogous to the zitterbewegung of massive Dirac theory and identical to electric and magnetic fields of circularly polarized electromagnetic waves.

**Can duality symmetry of Maxwell's equations be gauged?**

It is argued that duality symmetry of the generalized Maxwell's equations (with magentic sources) *can* be gauged in the usual way, and that arguments against incorrectly assume a particular form of potential or Lagrangian.