Discrete p-form Electromagnetism

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.

p-form electromagnetism on discrete spacetimes

Derek K Wise 2006 Class. Quantum Grav. 23 5129-5176

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:

My talk at Loops '05, October 11, 2005 — a 20 minute talk given to an audience of experts in quantum gravity.
A joint Math/Physics colloquium at UCCS, September 1, 2005.
My oral exam at UCR, which occurred on June 15, 2005. The talk itself was 40 minutes, to allow ample time for my committee to grill me afterwards.

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

Lattice p-form Electromagnetism and Chain Field Theory

gr-qc/0510033

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.