Lattice Gauge Theory

Supercomputers have recently become a crucial tool for the quantum field the-orist. Applied to the formalism of lattice gauge theory, numerical simulations are providing fundamental quantitative information about the interactions of quarks, the fundamental constituents of those particles which experience nuclear interactions. Perhaps most strikingly, these simulations have provided convincing evidence that the interquark forces can prevent the isolation of these constituents. Quarks are the primary constituents of particles subject to the strong nuclear force. Their basic interactions are believed to follow from a generalization of the gauge theory of electromagnetism. Instead of a single photon, this theory involves eight spin-1 quanta, referred to as gluons. Furthermore, these eight gluons are themselves charged with respect to one another. This introduces subtle nonlinear effects which appear even in the pure glue theory. One particularly important consequence of these nonlinearities is that the quark interactions weaken at small separations. This phenomenon, known as " asymptotic freedom " , is essential to many of the successes of the simple quark model. As long as the quarks remain near each other, their interactions are small. In contrast, the behavior of the gauge fields changes dramatically as the quarks are pulled apart. The experimental nonobservance of free quarks has led to the conjecture of the phenomenon of " confinement " , wherein interquark forces increase and remain strong as quarks are pulled apart to arbitrary separations. In this picture, it requires an infinite amount of energy to separate a single quark from the other constituents of a physical particle. This explains why free quarks are not produced in nature. Standard field-theoretical tools are severely hampered in the regime of large distances where these effects come into play. Perturbation theory, the historic mainstay of quantum field theory, begins with free particles and then treats their interaction as a small correction. With confinement, however, the fundamental constituents become increasingly strongly interacting as their separation is increased. In this domain the conventional perturbative approach fails totally. Lattice gauge theory, originally formulated by K. Wilson, provides a novel framework for calculations in this regime. This approach replaces the relativistic continuum of space and time with a discrete space-time lattice. The quarks move through this scaffolding by a sequence of discrete hops between nearest-neighbor sites. The gluon fields lie on the bonds connecting these sites. This lattice is a mathematical trick, introduced for calculational purposes only. It should not be …