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.
Abstract: We investigate p-form electromagnetism---with the Maxwell and Kalb-Ramond fields as lowest-order cases---on discrete spacetimes, including the regular lattices commonly used in lattice gauge theory, but also more general examples. After constructing a maximally general model of discrete spacetime suitable for our purpose---a chain complex equipped with an inner product on (p+1)-cochains---we study both the classical and quantum versions of the theory, with either R or U(1) as gauge group. We find results---such as a `p-form Bohm--Aharonov effect'---that depend in interesting ways on the cohomology of spacetime. We quantize the theory via the Euclidean path integral formalism, where the natural kernels in the U(1) theory are not Gaussians but theta functions. As a special case of the general theory, we show p-form electromagnetism in p+1 dimensions has an exact solution which reduces when p = 1 to the abelian case of 2d Yang-Mills theory as studied by Migdal and Witten. Our main result describes p-form electromagnetism as a `chain field theory'---a theory analogous to a topological quantum field theory, but with chain complexes replacing manifolds. This makes precise a notion of time evolution in the context of discrete spacetimes of arbitrary topology.
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.
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.