Lectures on Bound states

TL;DR

This paper discusses the concept of lowest order bound states in quantum field theories, exploring classical solutions, eigenstates, and confinement mechanisms in QED and QCD, with implications for hadron spectra and chiral symmetry.

Contribution

It introduces a framework for defining and analyzing lowest order bound states using classical field solutions and eigenstate methods in QED and QCD, including confinement and spectrum properties.

Findings

Positronium states are eigenstates of the QED Hamiltonian at Born level.

Bound states in 1+1 dimensions exhibit a continuous mass spectrum with mixed features.

Classical solutions of Gauss' law suggest a string-breaking mechanism in QCD.

Abstract

Even a first approximation of bound states requires contributions of all powers in the coupling. This means that the concept of "lowest order bound state" needs to be defined. In these lectures I discuss the "Born" (no loop, lowest order in $\hbar$) approximation. Born level states are bound by gauge fields which satisfy the classical field equations. As a check of the method, Positronium states of any momentum are determined as eigenstates of the QED Hamiltonian, quantized at equal time. Analogously, states bound by a strong external field $A^\mu(\boldsymbol{x})$ are found as eigenstates of the Dirac Hamiltonian. Their Fock states have dynamically created $e^+e^-$ pairs, whose distribution is determined by the Dirac wave function. The linear potential of $D=1+1$ dimensions confines electrons but repels positrons. As a result, the mass spectrum is continuous and the wave functions have features of both bound states and plane waves. The classical solutions of Gauss' law are explored for hadrons in QCD. A non-vanishing boundary condition at spatial infinity generates a constant \order{\alpha_s^0} color electric field between quarks of specific colors. Poincar\'e invariance limits the spectrum to color singlet $q\bar q$ and $qqq$ states, which do not generate an external color field. This restricts the \order{\alpha_s^0} interactions between hadrons to string breaking dynamics as in dual diagrams. Light mesons lie on linear Regge and parallel daughter trajectories. There are massless states which may be significant for chiral symmetry breaking. Since the bound states are defined at equal time in all frames they have a non-trivial Lorentz covariance.