## You are here

# Title: Bound States in Perturbation Theory

Even a first approximation of bound states requires contributions of all powers in the coupling. This means that the approximation of “lowest order” needs to be defined. 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 arbitrary momenta are determined as eigenstates of the QED Hamiltonian. The Dirac bound states of a relativistic electron in a classical external field A^\mu(\xv) are found in a similar way. Their virtual $e^+e^-$ pair contributions are expressed in terms of the Dirac wave function. The linear potential of D=1+1 dimensions confines electrons but repels positrons. This leads to a continuous mass spectrum and wave functions that 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. These singlet states generate no long range color fields, which restricts the \order{\alpha_s^0} interactions between hadrons to string breaking dynamics as in dual diagrams. Light mesons lie on linear Regge and daughter trajectories. There are massless states which may be significant for chiral symmetry breaking. Being defined at equal time in all frames the bound states have a non-trivial Lorentz covariance.