Bethe-Salpeter Equation

Abstract

We treat the Bethe-Salpeter equation as a problem in singular integral equations. As such, it has three outstanding features: its algebraic structure, the fixed propagator singularities in the direct channel, and the possible singularities in the potential, which are usually moving singularities. We exploit the algebraic structure in order to give insight into the possible correctness classes for the equation. We give explicit prescriptions for the removal of fixed singularities in a wide class of equations. We show under what circumstances these prescriptions can be adapted to maintain such desirable features as symmetry of the kernel. Moving singularities arise in physically realistic kernels; they are the crossed-channel singularities. The basic mathematics of such singularities is well known and is related to the Riemann-Hilbert problem, but this is useless in off-shell methods because it cannot cope with the integration over the space parts of 4-momenta. Instead, we adopt a method (proposed by one of us elsewhere) based on analyticity in energy variables. The resulting formalism is too complicated to be applied in full generality. We therefore consider the example of the single-particle exchange potential in detail, and show how the moving singularities can be eliminated, exhibiting the resulting equations explicitly in a form to which our theory of fixed singularities can immediately be applied. All our arguments are exact.