# Hi, I'm Luke Burns

I program a bunch and study physics a bunch. I recently attended the Recurse Center, where I built a p2p event emitter, toyed with a model for open peer review, studied some category theory, and worked on an extension of the Dirac equation. All of my work is on Github.

Below is a list of projects I am currently actively working on. 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.

Does the continuity equation imply Maxwell's equations? (pdf) The antiderivative of a divergence free multivector field is shown to be curl free up to a harmonic function. This result implies that *any* vector valued current density $J$ that is divergence free possesses a bivector valued antiderivative $F$ that satisfies $\partial F = J$ under suitable boundary conditions. In four dimensions, this is Maxwell's equation. This reinforces an existing result indicating that charge conservation can serve as a foundation for an axiomatic formulation of electrodynamics.

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.

An introduction to geometric algebra (pdf)

Is it possible to construct spinor valued gauge fields? (pdf)