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 ^–15meter; 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.

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.

Date: 2015-12-24; view: 1049