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.
The Continuity Equation Implies Maxwell's Equations (pdf) The antiderivative of a divergence free multivector (e.g. vector, bivector, spinor) field is shown to fail to be curl free by at most a (generalized) 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 and generalizes to conserved currents on manifolds of arbitrary dimensions. This not only indicates that charge conservation is itself sufficient for an axiomatic foundation for Maxwell's equations; by means of Noether's theorem, this result associates to every symmetry in four dimensions a bivector field satisfying Maxwell's equations, up to specific units.
Gauging duality symmetry of the generalized Maxwell equations (pdf) Duality symmetry of the generalized Maxwell's equations (with magentic sources) is gauged. Because the newly introduced gauge field is indistinguishable from an electromagnetic gauge field, it is suggested that the coupling could describe non-linear interactions between electromagnetic fields.
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)