Kenneth G. Wilson

The renormalization group and the ε expansion

The modern formulation of the renormalization group is explained for both critical phenomena in classical statistical mechanics and quantum field theory. The expansion in ϵ = 4− d is explained [ d is the dimension of space (statistical mechanics) or space-time (quantum field theory)]. The emphasis is on principles, not particular applications. Sections 1–8 provide a self-contained introduction at a fairly elementary level to the statistical mechanical theory. No background is required except for some prior experience with diagrams. In particular, a diagrammatic approximation to an exact renormalization group equation is presented in sections 4 and 5; sections 6–8 include the approximate renormalization group recursion formula and the Feyman graph method for calculating exponents. Sections 10–13 go deeper into renormalization group theory (section 9 presents a calculation of anomalous dimensions). The equivalence of quantum field theory and classical statistical mechanics near the critical point is established in section 10; sections 11–13 concern problems common to both subjects. Specific field theoretic references assume some background in quantum field theory. An exact renormalization group equation is presented in section 11; sections 12 and 13 concern fundamental topological questions.

The case for a single-chip multiprocessor

Advances in IC processing allow for more microprocessor design options. The increasing gate density and cost of wires in advanced integrated circuit technologies require that we look for new ways to use their capabilities effectively. This paper shows that in advanced technologies it is possible to implement a single-chip multiprocessor in the same area as a wide issue superscalar processor. We find that for applications with little parallelism the performance of the two microarchitectures is comparable. For applications with large amounts of parallelism at both the fine and coarse grained levels, the multiprocessor microarchitecture outperforms the superscalar architecture by a significant margin. Single-chip multiprocessor architectures have the advantage in that they offer localized implementation of a high-clock rate processor for inherently sequential applications and low latency interprocessor communication for parallel applications.

The Renormalization group and the epsilon expansion

The modern formulation of the renormalization group is explained for both critical phenomena in classical statistical mechanics and quantum field theory. The expansion in ϵ = 4− d is explained [ d is the dimension of space (statistical mechanics) or space-time (quantum field theory)]. The emphasis is on principles, not particular applications. Sections 1–8 provide a self-contained introduction at a fairly elementary level to the statistical mechanical theory. No background is required except for some prior experience with diagrams. In particular, a diagrammatic approximation to an exact renormalization group equation is presented in sections 4 and 5; sections 6–8 include the approximate renormalization group recursion formula and the Feyman graph method for calculating exponents. Sections 10–13 go deeper into renormalization group theory (section 9 presents a calculation of anomalous dimensions). The equivalence of quantum field theory and classical statistical mechanics near the critical point is established in section 10; sections 11–13 concern problems common to both subjects. Specific field theoretic references assume some background in quantum field theory. An exact renormalization group equation is presented in section 11; sections 12 and 13 concern fundamental topological questions.

Quarks and Strings on a Lattice

Three lectures describe the lattice version of the color gauge theory of quarks. The string interpretation of the theory is emphasized. The strong coupling expansion is defined by a set of Feynman rules. The dominant diagrams are identified. The result is that for strong quark-gluon coupling, the lattice spacing is about 1/5 x 10−13cm, the nucleon has a mass of 1720 MeV/c2 while the N* mass is 1750 MeV/c. The π and ρ masses are fitted to experiment. The relativistic limit is explained for free field theories on a lattice. For the colored quark theory only a few aspects of the relativistic continuum limit are discussed. It is shown how short wavelength string fluctuations are suppressed. It is shown that the classical limit of the lattice theory is the relativistic continuum color gauge theory.

The renormalization group and the ϵ expansion

The modern formulation of the renormalization group is explained for both critical phenomena in classical statistical mechanics and quantum field theory. The expansion in ϵ = 4− d is explained [ d is the dimension of space (statistical mechanics) or space-time (quantum field theory)]. The emphasis is on principles, not particular applications. Sections 1–8 provide a self-contained introduction at a fairly elementary level to the statistical mechanical theory. No background is required except for some prior experience with diagrams. In particular, a diagrammatic approximation to an exact renormalization group equation is presented in sections 4 and 5; sections 6–8 include the approximate renormalization group recursion formula and the Feyman graph method for calculating exponents. Sections 10–13 go deeper into renormalization group theory (section 9 presents a calculation of anomalous dimensions). The equivalence of quantum field theory and classical statistical mechanics near the critical point is established in section 10; sections 11–13 concern problems common to both subjects. Specific field theoretic references assume some background in quantum field theory. An exact renormalization group equation is presented in section 11; sections 12 and 13 concern fundamental topological questions.

Nonperturbative QCD: A weak-coupling treatment on the light front.

In this work the determination of low-energy bound states in Quantum Chromodynamics is recast so that it is linked to a weak-coupling problem. This allows one to approach the solution with the same techniques which solve Quantum Electrodynamics: namely, a combination of weak-coupling diagrams and many-body quantum mechanics. The key to eliminating necessarily nonperturbative effects is the use of a bare Hamiltonian in which quarks and gluons have nonzero constituent masses rather than the zero masses of the current picture. The use of constituent masses cuts off the growth of the running coupling constant and makes it possible that the running coupling never leaves the perturbative domain. For stabilization purposes an artificial potential is added to the Hamiltonian, but with a coefficient that vanishes at the physical value of the coupling constant. The weak-coupling approach potentially reconciles the simplicity of the Constituent Quark Model with the complexities of Quantum Chromodynamics. The penalty for achieving this perturbative picture is the necessity of formulating the dynamics of QCD in light-front coordinates and of dealing with the complexities of renormalization which such a formulation entails. We describe the renormalization process first using a qualitative phase space cell analysis, and we then set up a precise similarity renormalization scheme with cutoffs on constituent momenta and exhibit calculations to second order. We outline further computations that remain to be carried out. There is an initial nonperturbative but nonrelativistic calculation of the hadronic masses that determines the artificial potential, with binding energies required to be fourth order in the coupling as in QED. Next there is a calculation of the leading radiative corrections to these masses, which requires our renormalization program. Then the real struggle of finding the right extensions to perturbation theory to study the strong-coupling behavior of bound states can begin.

Operator product expansions and composite field operators in the general framework of quantum field theory

The short distance behavior of field operator products is analyzed. It is shown that under certain conditions operator product expansions can be derived which give complete information on the short distance behavior and lead to the construction of composite field operators.

Proof of a Conjecture by Dyson

We prove a mathematical conjecture by Dyson which he used in a study of the statistical distribution of energy levels in complex nuclei.

Limit cycles in quantum theories.

Renormalization group limit cycles and more chaotic behavior may be commonplace for quantum Hamiltonians requiring renormalization, in contrast to experience based on classical models with critical behavior, where fixed points are far more common. We discuss the simplest quantum model Hamiltonian identified so far that exhibits a renormalization group with both limit cycle and chaotic behavior. The model is a discrete Hermitian matrix with two coupling constants, both governed by a nonperturbative renormalization group equation that involves changes in only one of these couplings and is soluble analytically.

Grand challenges to computational science

Abstract Computational Science is at the very beginning of centuries of growth, comparable to the four centuries of experimental advances since Galileo. The Grand Challenges to Computational Science are unsolved scientific problems of extraordinary breadth and importance which will demand continuing computational advances throughout the forthcoming computational era. Supercomputers can be used to see phenomena not directly accessible to experiment in key scientific and engineering areas such as atmospheric science, astronomy, materials science, molecular biology, aerodynamics, and elementary particle physics. However, the benefits of supercomputers will be greatly increased if some major difficulties are overcome. In this paper, I address some of the tougher requirements on current grand challenge research to ensure that it has enduring value. The problems of algorithm development, error control, software productivity, and the fostering of technological advances are especially important.