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Getting Your Quarks in a Row

A tidy lattice is the key to computing with quantum fields

Brian Hayes

Booking a Flight on Quantum Airlines

A theory in physics is supposed to be more than just a qualitative description; you ought to be able to use it to make predictive calculations. For example, Newton's theory of gravitation predicts the positions of planets in the sky. Likewise QED allows for predictive calculations in its realm of electrons and photons.

Suppose you want to know the probability that a photon will travel from one point to another. For calculations of this kind Richard Feynman introduced a scheme known as the sum-over-paths method. The idea is to consider every possible path the photon might take and then add up contributions from each of the alternatives. This is rather like booking an airplane trip from Boston to Seattle. You could take a direct flight, or you might stop over in Chicago or Minneapolis—or maybe even Buenos Aires. In QED, each such path is associated with a number called an amplitude; the overall probability of getting from Boston to Seattle is found by summing all the amplitudes, then squaring the result and taking the absolute value. The trick here is that the amplitudes are complex numbers—with real and imaginary parts—which means that in the summing process some amplitudes cancel others. (Another complication is that a photon has infinitely many paths to choose from, but there are mathematical tools for handling those infinities.)

A more elaborate application of QED is calculating the interaction between two electrons: You need to sum up all the ways that the electrons could emit and absorb photons. The simplest possibility is the exchange of a single photon, but events involving two or more photons can't be ruled out. And a photon might spontaneously produce an e e + pair, which could then recombine to form another photon. Indeed, the variety of interaction mechanisms is limitless. Nevertheless, QED can calculate the interaction probability to very high accuracy. The key reason for this success is the small value of the electromagnetic coupling constant α. For events with two photons, the amplitude is reduced by a factor of α 2 , which is less than 0.0001. For three photons the coefficient is α 4 , and so on. Because these terms are very small, the one-photon exchange dominates the interaction. This style of calculation—summing a series of progressively smaller terms—is known as a perturbative method.

In principle, the same scheme can be applied in QCD to predict the behavior of quarks and gluons; in practice, it doesn't work out quite so smoothly. One problem comes from the color charge of the gluons. Whereas a photon cannot emit or absorb another photon, a gluon, being charged, can emit and absorb gluons. This self-interaction multiplies the number of possible pathways. An even bigger problem is the size of the color-force coupling constant α c . Because this number is close to 1, all possible gluon exchanges make roughly the same contribution to the overall interaction. The single-gluon event can still be taken as the starting point for a calculation, but the subsequent terms are not small corrections; they are just as large as the first term. The series doesn't converge; if you were to try summing the whole thing, the answer would be infinite.

In one respect the situation is not quite as bleak as this analysis suggests. It turns out that the color coupling constant α c isn't really a constant after all. The strength of the coupling varies as a function of distance. The customary unit of distance in this realm is the fermi, equal to 1 femtometer, or 10 –15 meter; a fermi is roughly the diameter of a proton or a neutron. If you measure the color force at distances of less than 0.001 fermi, α c dwindles away to only about 0.1. The "constant" grows rapidly, however, as the distance increases. As a result of this variation in the coupling constant, quarks move around freely when they are close together but begin to exert powerful restraining forces as their separation grows. This is the underlying mechanism of quark confinement.

Because the color coupling gets weaker at short distances, perturbative methods can be made to work at close range. In an experimental setting, probing a particle at close range requires high energy. Thus perturbative QCD can tell us about the behavior of quarks in the most violent environments in the universe—such as the collision zones at the Large Hadron Collider now revving near Geneva. But the perturbative theory fails if we want to know about the quarks in ordinary matter at lower energy.

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