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Getting Your Quarks in a Row

A tidy lattice is the key to computing with quantum fields

Brian Hayes

Enter the Lattice

Understanding the low-energy or long-range properties of quark matter is the problem that lattice QCD was invented to address, starting in the mid-1970s. A number of physicists had a hand in developing the technique, but the key figure was Kenneth G. Wilson, now of Ohio State University. It's not an accident that Wilson had been working on problems in solid-state physics and statistical mechanics, where many systems come equipped with a natural lattice, namely that of a crystal.

Introducing an artificial lattice of discrete points is a common strategy for simplifying physical problems. For example, models for weather forecasting establish a grid of points in latitude, longitude and altitude where variables such as temperature and wind direction are evaluated. In QCD the lattice is four-dimensional: Each node represents both a point in space and an instant in time. Thus a particle standing still in space hops along the lattice parallel to the time axis.

It needs to be emphasized that the lattice in QCD is an artificial construct, just as it is in a weather model. No one is suggesting that spacetime really has such a rectilinear gridlike structure. To get rigorous results from lattice studies, you have to consider the limiting behavior as the lattice spacing a goes to zero. (But there are many interesting approximate results that do not require taking the limit.)

One obvious advantage of a lattice is that it helps to tame infinities. In continuous spacetime, quarks and gluons can roam anywhere; even with a finite number of particles, the system has infinitely many possible states. If a lattice has a finite number of nodes and links, the number of quark-and-gluon configurations has a definite bound. In principle, you can enumerate all states.

As it turns out, however, the finite number of configurations is not the biggest benefit of introducing a lattice. More important is enforcing a minimum dimension—namely the lattice spacing a . By eliminating all interactions at distances less than a, the lattice tames a different and more pernicious type of infinity, one where the energy of individual interactions grows without bound.

The most celebrated result of lattice QCD came at the very beginning. The mathematical framework of QCD itself (without the lattice) was formulated in about 1973; this work included the idea that quarks become "asymptotically free" at close range and suggested the hypothesis of confinement at longer range. Just a year later Wilson published evidence of confinement based on a lattice model. What he showed was that color fields on the lattice do not spread out in the way that electromagnetic fields do. As quarks are pulled apart, the color field between them is concentrated in a narrow "flux tube" that maintains a constant cross section. The energy of the flux tube is proportional to its length. Long before the tube reaches macroscopic length, there is enough energy to create a new quark-antiquark pair. The result is that isolated quarks are never seen in the wild; only collections of quarks that are color-neutral can be detected.

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