Research work · 1963
Quasiparticles and the Born Series
Steven Weinberg
Physical Review
Abstract
Perturbation theory always works in nonrelativistic scattering theory, unless composite particles are present. By "composite particle" is meant a bound state or resonance, or one that would exist for an interaction of opposite sign; in fact, this provides a precise definition of resonances. It follows that if fictitious elementary particles (quasiparticles) are first introduced to take the place of all composite particles, then perturbation theory can always be used. There are several ways of accomplishing this, one of which corresponds to the $\frac{N}{D}$ method. In order to prove these results it is necessary to make a detailed study of the eigenvalues of the scattering kernel, and as a by-product we obtain new proofs of the applicability of the Fredholm theorems to scattering theory, of the convergence of the Born series at high energy, of the Bargmann-Schwinger theorem on the number of bound states, of the Pais-Jost theorem on the identity of the Jost function with the Fredholm determinant, and of Levinson's theorem. We also give explicit formulas for binding energies and phase shifts in potential theory, using first-order perturbation theory after insertion of a single quasiparticle; these formulas work well for the lowest bound state and the $S$-wave scattering length of the Yukawa potential, and give precisely 13.6 eV for the hydrogen atom binding energy.