Principles and Possibilities for Bound States in Gauge Theory

Abstract

Bound states differ from scattering yet are not covered in textbooks on Quantum Field Theory. I discuss a perturbative method for QED and QCD based on canonical quantization. Fully fixing temporal gauge $A^0(t,\mathbf{x})=0$ imposes Gauss' law on physical states. As pointed out by Dirac, this implies that electron states include a longitudinal gauge field ${\mathbf{A}}_L$, which determines the instantaneous bound state potential. The situation is analogous for quarks and gluons in QCD. An instantaneous confining potential arises for color singlet $q\bar{q}$ states when a non-vanishing boundary condition on ${\mathbf{A}}_L^a(\mathbf{x}\to \infty )$ is specified in Gauss' constraint. As suggested by Gribov, ${\alpha }_s(Q^2)$ may freeze at a perturbative value when the confining potential dominates. Hadrons can then be calculated perturbatively. At vanishing quark mass there is a $j^{PC}=0^{++}$ state with zero energy which can mix with the perturbative vacuum, giving rise to a spontaneous breaking of chiral symmetry.

Comments: 12 pages, 1 figure. Minor corrections from v1. The bound state equation is considered only in the rest frame