Thursday, November 18, 2004

Fermions on the lattice, part one of four

This is a "fessing up" post, before I talk about all the great things that lattice QCD can do. I'm going to talk in general about fermions on the lattice, and discuss the ways in which you can simulate them. This will take us down three roads, and at the end we'll find pros and cons to each approach. I'll focus on an outstanding issue with staggered fermions at the end.

The formulation of fermion fields on the lattice has been a huge problem. Compared to Wilson's elegant formulation of gauge fields on the lattice fermions look downright ugly. To top it off, they're slow, so it's hard to deal with them. We'll look briefly at the latter problem, the speed first, and then talk about why lattice fermions are ugly.

Fermion fields in quantum field theory are represented by anti-commuting numbers, known as Grassman numbers. This is fine when you're working with a pencil and paper, but it's impossible to put a Grassman number into a computer. Computers are good for reals, integers and complex numbers (really two reals), but they can't do Grassman numbers.

This is not quite as big a problem as it sounds. The reason is every fermion action that people use can be written as
S_{F} = \bar{\psi} M[U] \psi
where $M[U]$ is a complex matrix functional of the gauge fields. This is happy, since we can do path integrals of this form,
\int D\bar{\psi} D\psi exp(-S_{F}) = \det(M[U])
and we're done! We now have fermion fields on the lattice. The trouble is that we need to evaluate the determinant. And that is mad slow. So slow, in fact, that until the mid-nineties it was essentially impossible. This led people to just drop it, using something called the quenched approximation. This causes uncontrollable ten or twenty percent errors, but it makes things faster. Physically, it amounts to neglecting dynamical fermions in your simulation.

As people started to remove the quenched approximation another problem came up. Evaluating the determinant gets slower as you go to smaller and smaller quark masses. This forces you to work at unphysical values for the up and down quark masses. That's the situation today. Simulations are forced to work at masses that are larger than the physical up and down quark masses. Then you have to extrapolate to the physical limit (which is near m=0).

That's the present situation. It is possible to simulate with dynamical fermions, but, for the most part, you have to simulate at unphysical masses.

Now why are lattice fermions ugly? Well, it turns out that if you take the Dirac action
S = \bar{\psi} ( \gamma \cdot D + m) \psi
and naively discretize it (by replacing D with a covariant difference operator) you get an action which describes 16 degenerate types of fermions. These degenerate types are called "tastes". So the naive discritization of lattice fermions induces 16 tastes. Obviously we don't want to simulate 16 degenerate quarks, we want to simulate one quark, so we need to figure out some way of getting rid of the other 15 tastes. In the next installment we'll see how the extra tastes show up, and how to remove them by adding a Wilson term to the action.