The simplest models, which become solvable in the limit of a large number of field components, deal with a field which has N components forming an O(N) vector in an internal symmetry space. A model of this kind was first considered by Stanley [Sta68] in statistical mechanics and is known as the spherical model. The extension to quantum field theory was made by Wilson [Wil73] both for the four-Fermi and ϕ4 theories.

Within the framework of perturbation theory, the four-Fermi interaction is renormalizable only in d = 2 dimensions and is nonrenormalizable for d > 2. The 1/N-expansion resums perturbation-theory diagrams after which the four-Fermi interaction becomes renormalizable to each order in 1/N for 2 ≤ d < 4. An analogous expansion exists for the nonlinear O(N) sigma model. The ϕ4 theory remains “trivial” in d = 4 to each order of the 1/N-expansion and has a nontrivial infrared-stable fixed point for 2 < d < 4.

The 1/N-expansion of the vector models is associated with a resummation of Feynman diagrams. A very simple class of diagrams – the bubble graphs – survives to the leading order in 1/N. This is why the large-N limit of the vector models is solvable. Alternatively, the large-N solution is nothing but a saddle-point solution in the path-integral approach. The existence of the saddle point is a result of the fact that N is large.

The large-N reduction was first discovered in 1982 by Eguchi and Kawai [EK82], who showed that the SU(N) Yang–Mills theory on a d-dimensional space-time is equivalent at N = ∞ to the one at a point. This construction is based on an extra symmetry of the reduced model which should not be broken spontaneously.

Soon after that it was recognized that this symmetry is, in fact, broken for d > 2. Two ways were proposed to cure the construction: the quenching prescription [BHN82] and the twisting prescription [GO83a]. Each of these two prescriptions results in a reduced model which recovers multicolor QCD both on the lattice and in the continuum.

While the reduced models look like a great simplification, since the space-time is reduced to a point, they still involve an integration over d infinite matrices which is, in fact, a continual path integral. For some years it was not clear whether or not this is a real simplification of the original theory which can make it solvable, so the point of view on the reduced models was that they are just an elegant representation at large N.

The recent interest in reduced models has arisen from the matrix-model formulation [BFS97, IKK97] of M-theory combining all types of superstring theories. The novel point of view on the reduced models is that they are equivalent [CDS98] to gauge theories on noncommutative space.

In the summer of 1986 I left Stanford University, after 26 years on the faculty, to assume the job of Scientific Director at the Continuous Electron Beam Accelerator Facility (CEBAF) now known as the Thomas Jefferson National Accelerator Facility (TJNAF). This facility, funded by the Department of Energy and located in Newport News, Virginia provides a high-energy, high-intensity, high-duty-factor electron accelerator for studying the internal structure of nuclei and nucleons. It has long been a top priority for the field of nuclear physics in the United States. Each year I gave a physics lecture series at the site. The initial series on electron scattering was based on a set of lectures I had given at Argonne National Laboratory in the winter of 1982–1983. As Scientific Director, I was continually called upon to make presentations on this topic. This book is based both on the lecture series on electron scattering, and on the many presentations I have given on this subject over the years.

The scattering of high-energy electrons from nuclear and nucleon targets essentially provides a microscope for examining the structure of these tiny objects. The best evidence we have on what nuclei and nucleons actually look like comes from electron scattering. An intense continuous electron beam with well-defined energy provides a powerful tool for structure investigations. Inclusive experiments, where only the final electron is detected, examine static and transition charge and current densities in the target. Coincidence experiments, where other particles are detected together with the scattered electron, provide valuable additional information.

In electron scattering experiments where the momentum of the initial and final electron are well-defined, a virtual quantum of electromagnetic radiation is produced which interacts with the target.