Light–Cone Quantization: Foundations and Applications

These lecture notes review the foundations and some applications of light– cone quantization. First I explain how to choose a time in special relativity. Inclusion of Poincaré invariance naturally leads to Dirac’s forms of relativistic dynamics. Among these, the front form, being the basis for light–cone quantization, is my main focus. I explain a few of its peculiar features such as boost and Galilei invariance or separation of relative and center–of–mass motion. Combining light–cone dynamics and field quantization results in light–cone quantum field theory. As the latter represents a first–order system, quantization is somewhat nonstandard. I address this issue using Schwinger’s quantum action principle, the method of Faddeev and Jackiw, and the functional Schrödinger picture. A finite–volume formulation, discretized light–cone quantization, is analysed in detail. I point out some problems with causality, which are absent in infinite volume. Finally, the triviality of the light–cone vacuum is established. Coming to applications, I introduce the notion of light–cone wave functions as the solutions of the light–cone Schrödinger equation. I discuss some examples, among them nonrelativistic Coulomb systems and model field theories in two dimensions. Vacuum properties (like chiral condensates) are reconstructed from the particle spectrum obtained by solving the light–cone Schrödinger equation. In a last application, I make contact with phenomenology by calculating the pion wave function within the Nambu and Jona-Lasinio model. I am thus able to predict a number of observables like the pion charge and core radius, the r.m.s. transverse momentum, the pion structure function and the pion distribution amplitude. The latter turns out to be the asymptotic one.