# Hi, I'm Luke Burns

I study math and physics, and code some. All of my work is on Github.

Below is a list of math and physics projects that are bopping around in my brain. Each project has a Github repository with a full version history, TeX source, open issues, and a most recent pdf build. You're welcome to open an issue, or fork and submit a pull request with improvements.

When does the continuity equation imply Maxwell's equations? (pdf) A fundamental result of electrodynamics is that Maxwell's equations imply that charge is locally conserved. Here we show that local charge conservation implies the existence of fields satisfying Maxwell's equations.

Can duality symmetry of Maxwell's equations be gauged? (pdf) 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.

An extension of the Dirac equation (pdf) 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.