Light-Cone Quantization: Foundations and Applications

Abstract

Abstract. These lecture notes review the foundations and some applications of lightcone 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.

Baryons as relativistic three-quark bound states

“…[161][162][163][164][165][166] for reviews) take a different route in this respect. Motivated by the triviality of the vacuum in light-cone quantization, the goal is to calculate the eigenstates of the gauge-fixed QCD Hamiltonian directly by solving large-scale eigenvalue problems.…”

Pion light-cone wave function and pion distribution amplitude in the Nambu–Jona-Lasinio model

“…The pion distribution amplitude has been evaluated with QCD sum rules [9,10,11,12,13,14], in standard [15] (only the second ξ-moment) and transverse lattice approaches [16,17,18], and in chiral quark models [19,20,21,22,23,24,25,26,27,28,29]. In chiral quark models the results are not always compatible to each other, and even their interpretation has not always been the same.…”

“…In chiral quark models the results are not always compatible to each other, and even their interpretation has not always been the same. While in same cases there are problems with chiral symmetry and proper normalization [20,21,25], in other cases [22,23,24,25,26,28,29] it is not clear how to associate the scale at which the model is defined, necessary to define the starting point for the QCD evolution. Nevertheless, there is a precise way to identify the low energy scale, Q 0 , at which the model is defined, namely the one at which the quarks carry 100% of the total momentum [30,31].…”

“…Nevertheless, there is a precise way to identify the low energy scale, Q 0 , at which the model is defined, namely the one at which the quarks carry 100% of the total momentum [30,31]. The fact that several calculations [20,21,23,24,25,26,28] produce a PDA strongly resembling the asymptotic form suggests that their working scale is already large, and the subsequent QCD evolution becomes unnecessary, or numerically insignificant. This also tacitly assumes that these models already incorporate the QCD radiative corrections.…”

“…This is the main objective of this work. In the NJL model PDA has already been estimated by several authors [23,24,28]. The work of Refs.…”