Solitons are solutions of nonlinear wave equations. A single soliton is just like a normal dispersionless wave in that it does not change its shape in the course of time. The soliton is known for its remarkable stability. When a soliton encounters another soliton of arbitrary size or velocity it can change beyond recognition for a short or long period, but ultimately it will revert to its original shape. This type of wave is observed in many fields of physics. In this paper I shall concentrate on giving a short explanation of the various ways in which mathematical physicists have tried to understand the amazing soliton. The reader is not expected to have studied advanced mathematics but he or she must be prepared to work through some of the algebra, which is sometimes rather tedious.
After a brief review of experimental data on lepton pair production, I describe the perturbative calculation of this process in QCD and the exponentiation of soft gluon effects. Results of numerical calculations for the “\(K\)-factor” are compared with experiment.
I present recent results for lattice gauge theories obtained by mean field theory with radiative corrections (MFT). After a quick survey of the strong and weak sides of other methods on the lattice, an introduction to MFT is given. The utility of MFT is discussed. I report on some work in progress, and motivate some future projects.
The formulation of fermion path integrals as transition amplitudes is discussed for gauge theories. Fermion states are specified by products of Grassmann variables, which form an orthogonal and complete basis. The expression for the Hamiltonian operating on the products is derived. In an application to Green functions, the fermion propagator is derived for a variety of boundary conditions at \(t=\pm \infty \).
