Relativistic Bound States

In this contribution, I will give a brief survey of present techniques to treat the bound state problem in relativistic quantum field theories. In particular, I will discuss the Bethe–Salpeter equation, various quasi–potential equations, the Feynman–Schwinger representation, and similarity transformation methods for Hamiltonian approaches in light–front quantization. Finally, I will comment on a related similarity transformation in the usual equal–time quantized theory. One of the most striking and ubiquitous phenomena in nature is the binding of matter. It is found on essentially all length scales, from galaxies to quarks, and is even speculated to be crucial for the description of physics much below the length scale of hadronic physics. We currently have a good understanding of the binding mechanism in crystals, molecules, and atoms. The non–relativistic Schrödinger equation provides a satisfactory description at these scales, and small corrections due to relativistic and virtual particle creation effects can accurately be treated in perturbation theory. However, at smaller length scales relativistic effects become increasingly important, and the description through the non–relativistic Schrödinger equation is not accurate enough. It is hence inevitable to employ quantum field theory (QFT), the best currently known description of microscopic physics. In the following, I will give a brief survey of the most popular equations which have been employed for bound state calculations in relativistic QFT. Attention will be restricted to equations which are rooted in field theory, in particular, I will leave out the generalizations of relativistic one–particle quantum mechanics to two or more particles, as well as equations with phenomenological input. Equations that are derived from field theory can naturally be divided into two classes, those based on the manifestly covariant Lagrangian, path–intregal formulation of QFT, and those built on a Hamiltonian formulation in a perturbatively defined Fock space. In the Lagrangian approach to QFT, which is particularly convenient for the calculation of transition amplitudes, a bound state of two constituents appears as a pole in the 2–particle scattering amplitude, or of the field–theoretic 4–point correlation function. The pole is interpreted as the pole in the propagator of the bound state as illustrated in Fig. 1, and its position determines the bound state mass (via the usual pole mass definition). From the figure it is clear that near the pole the 4–point function has the form 1 e–mail: [email protected]