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Getting Your Quarks in a Row

A tidy lattice is the key to computing with quantum fields

Brian Hayes

Lattice QCD for Novices

When I first heard about lattice QCD, I found the idea instantly appealing. Other approaches to particle physics require mastery of some very challenging mathematics, but the lattice methods looked like something I could get a grip on—something discrete and finite, where computing the state of a quantum system would be a matter of filling in columns and rows of numbers.

Those early hopes ended in disappointment. I soon learned that lattice QCD does not bring all of quantum field theory down to the level of spreadsheet arithmetic. There is still heavy-duty mathematics to be done, along with a great deal of heavy-duty computing. Nevertheless, I continue to believe that the lattice version of the weird quantum world is easier to grasp than any other. My conviction has been reinforced by the discovery of an article, "Lattice QCD for Novices," published 10 years ago by G. Peter Lepage of Cornell University. Lepage doesn't offer lattice QCD in an Excel spreadsheet, but he does present an implementation written in the Python programming language. The entire program fits in a page or two.

Lepage's lattice model for novices has just one space dimension as well as a time dimension; in other words, it describes particles moving back and forth along a line segment. And what the program simulates isn't really a quantum field theory; there are no operators for the creation and annihilation of particles. All the same, reading the source code for the program gives an inside view of how a lattice model works, even if the model is only a toy.

At the lowest level is a routine to generate thousands of random paths, or configurations, in the lattice, weighted according to their likelihood under the particular rule that governs the physical evolution of the system. Then the program computes averages for a subset of the configurations, as well as quantities that correspond to experimentally observable properties, such as energy levels. Finally, more than half the program is given over to evaluating the statistical reliability of the results.

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