Lattice p-Form Electromagnetism and Chain Field Theory

Derek K. Wise

In p-form electromagnetism, we generalize ordinary electromagnetism by promoting the gauge potential A from a mere 1-form to a p-form. This gives an analog of the electromagnetic field which interacts naturally not with charged point particles, but with charged "(p-1)-branes". The most famous example other than ordinary electromagnetism is the Kalb-Ramond field (p=2), which influences the motion of a charged "1-brane", i.e. a string.

The following paper considers a discrete version of p-form electromagnetism, based on ideas from lattice gauge theory and topological quantum field theory.

My description of lattice p-form electromagnetism as a chain field theory gives a meaning to time evolution in the theory — it gives a way to evolve quantum states on a slice S of space into states on a "later" slice S' in a discretized spacetime M.

Talks: If you want a quick overview of my work on lattice p-form electromagnetism, you can take a look at some of the following files, which are slides from talks I've given on this subject. Each talk had a different target audience, so each offers different viewpoints of essentially the same material. They are in reverse-chronological order:

If you like these, you should read my paper above! You can also get an earlier, expository version of the same material, here:

You can see what my advisor, John Baez, had to say about this paper in Week 223 of his column, This Week's Finds in Mathematical Physics.

Maxwell's Equations on a Cubical Lattice

On a related subject, I had the opportunity to speak to physics undergraduates at my alma mater, Abilene Christian University. I wanted to teach them some stuff related to my work on lattice p-form electromagnetism, but without needing to assume much background knowledge. Since students don't usually encounter lattice methods until a course on quantum field theory, I decided to tell give them a head start by telling them about lattice classical electromagnetism! The result was a talk which I called Electricity, Magnetism, and Hypercubes. The main point of the talk was an explanation of how two of Maxwell's equations follow from "the boundary of a boundary is zero", or rather dually from "the oriented sum of an oriented sum is zero". We did this by staring at cubes labelled by components of the electric and magnetic field, like this one:

which is just one cubical face of a hypercubical cell in the lattice. I would love to get the chance to do this lecture as the first in a two part series sometime, since I'd love to explain how the other pair of Maxwell equations works on the lattice. I did have a bit of extra time at the end, and wound up talking about Poincaré duality and how it is related to the source dependent pair of equations. But there wasn't time to do this subject justice.

I think this is a great subject to teach undergrad physics majors. It's easy to explain, leads to interesting and useful mathematics, and primes them for what people are really interested in: lattice quantum field theory. I wrote up a bunch of notes on classical electrodynamics on a lattice when I was preparing for this talk. Eventually these will become a paper which could serve as a nice expository intro to my already expository paper on lattice p-form electromagnetism, above.

© 2005-2006 Derek Wise       Last update: 11 November 2006