Last week was Wade Cherrington‘s Ph.D. defense – he is (or, rather, WAS) a student of Dan Christensen. The title was “Dual Computational Methods for Lattice Gauge Theory”. The point of which is to describe some methods for doing numerical computations of various physical systems governed by gauge theories. This would include electromagnetism, Yang-Mills theory (which covers the Standard Model and other quantum field theories), as well as gravity. In any gauge theory, the fundamental objects being studied are fields described by $G$-connections, for some (Lie) group $G$. To some degree of approximation, a connection gives a group element for any path in space: $\Gamma : Path(M) \rightarrow G$. Then the dynamics of these fields are described by a Lagrangian, where the action for a field is the integral of the curvature over the whole space $M$: $\int tr(F \wedge \star F)$ (plus possibly some other terms to couple the field to sources).

Now, the point here is to get non-perturbative ways to study these theories: rather than, say, getting differential equations for the fields and finding solutions by expanding a power series. The approach in question is to take a discrete version of this continuum theory, which is finite and can be dealt with exactly, and then take a limit.

So in lattice gauge theory, continuous space is replaced by a – well, a lattice $L$, say $L=\mathbb{Z}^3$, for definiteness (then eventually take the continuum limit as the spacing of the lattice goes to zero). The lattice also include edges joining adjacent points – say the set of edges is $E$. Paths in the lattice are built from these edges. (Furthermore, since an infinite lattice can’t be represented in the computer, the actual computations use a quotient of this – a lattice in a 3-torus, or equivalently, one considers only periodic fields.) Then it’s enough to say that a connection assigns a group element to each edge of the lattice, $\Gamma : E \rightarrow G$.

Of course, to back up, describing connections as functions $\Gamma : E \rightarrow G$, or $Path(M) : \rightarrow G$, often provokes various objections from people used to differential geometry. One is that the group elements assigned don’t have any direct physical meaning – since a physical state is only defined up to gauge equivalence. So if an edge $e$ joins lattice points $a$ and $b$, a gauge transformation $g : L \rightarrow G$ acts on $\Gamma$ to give $\Gamma' : E \rightarrow G$ with $\Gamma'(e) = g(a)\Gamma(e)g(b)^{-1}$. Clearly, for any given edge, there are gauge-equivalent connections assigning any group element you want. As Wade pointed out, one benefit of the dual models he was describing is that their states can be given a definite physical meaning – there are no gauge choices. Another, helpfully, is that they’re easier to calculate with (sometimes). A more physical motivation Wade suggested is that these methods can deal with spin-foam models of quantum gravity, and also matter fields: a realistic look at a theory of gravity should have some matter to gravitate, so this gives a way to simulate them together.

So what are these dual methods? This is described in some detail in this paper by Wade, Dan, and Igor Khavkine. The first step is to find a discrete version of curvature: instead of the action $\int tr(F \wedge \star F)$, we want a sum of face amplitudes. Curvature is described by the holonomy around a contractible loop, so the basic element is a face in the lattice (say $F$ is the set of faces). Given a square face $f \in F$ with edges whose holonomies are $g_1$ through $g_4$ (assuming all faces are oriented in a consistent direction), the holonomy around the face is $g_1 g_2 g_3^{-1} g_4^{-1} = g(f)$. From this, one defines an amplitude for the face, for some function $S(g(f))$ (there are various possibilities – Wade’s example used the heat kernel action mentioned in the paper above), and then the total action $S = \sum_{F} S(g(f))$ is the sum over all faces. Then instead of integrating over an infinite-dimensional, and generally intractable, space of smooth connections, one itegrates over $G^E$, the space of discrete connections.

The duality here is the expansion of this in terms of group characters: a function $S(g)$ can be written as a combination of irreducible characters: $S(g) = \sum_i c_i \chi_i(g)$. Then one can pull this sum over characters outside the integral over $G^E$ (so that local quantities are inside).

There are many nice images on Wade’s homepage (above) showing visualizations of the resulting calculations – one finds sums over certain labellings of the lattice, namely those which can be described by having certain surfaces. In particular, closed (boundaryless), branched (possibly self-intersecting) surfaces with face and edge labels given by representations of $G$ and intertwining operators between them… that is, spin foams. These dual spin foam configurations have the advantage of having a physical interpretation (though I confess I don’t have a good intuition about it) which doesn’t depend on gauge choices.

A variant on this comes about when the action is changed to include a term coupling the Yang-Mills field to fermions (one thinks of quarks and gluons, for example). In this case, the fermion part is described by “polymers” (closed, possibly self-intersecting paths, rather than surfaces), and the coupled system allows the surfaces used in the YM calculations to have boundaries – but only on these polymers. (Again, Wade has some nice images of this on his site. Personally, I find a lot of the details here remain obscure, though I’ve seen a few versions of this talk and related ones, but the pictures give a framework to hang the rest of it on.)

Wade identified two “key” ingredients for doing calculations with these dual spin foams:

1. Recoupling moves for the graphs (as described, for instance, by Carter, Flath, and Saito) which simplify the calculation of amplitudes, and
2. A set of local moves (changes of configuration) which are ergodic – that is, between them they can take any configuration to any other. (The point here is to allow a reasonably random sampling – the algorithm is stochastic – of the configuration space, while making only local changes, requiring a minimum of recomputation, at each step.)

Finally, Wade summed up by pointing out that the results obtained so far agree with the usual methods, and in some cases are faster. Then he told us about some future projects. Some involve optimizing code and adapting it to run on clusters. Others were more theoretical matters: doing for $SU(3)$ what has been done for $SU(2)$ (which will involve developing much of the recoupling theory for 3j- and 6j-symbols); finding and computing observables for these configurations (such as Wilson loops); and modelling supersymmetry and other notions about particle physics.