Particle spectrum in model field theories from semiclassical solutions of the field equations

Abstract

Bound-state spectra (of mesons or of solitons-antisolitons) are studied for the sine-Gordon and quartic-coupling nonlinear boson models in one-plus-one dimensions using methods developed previously by the authors for the example of the nonlinear Schr\"odinger equation. In addition to reproducing the spectra first derived by Dashen, Hasslacher, and Neveu (DHN) up to the order of the first quantum correction, we have also calculated (in the semiclassical approximation) all bound-state form factors as well as the matrix elements of the field between any bound state and any continuum state for the scattering of a meson on a bound state. For the sine-Gordon theory the results have been obtained in two ways: first, by an algorithm, derived in due course, for transcribing an exact classical solution into a quantum operator; second, by a systematic expansion about the weak-coupling limit which requires the techniques used previously for the nonlinear Schr\"odinger equation. Only this latter technique is available for the quartic model, but its (more complicated) application here leads to an explanation of why in leading order the same form of bound-state spectra are obtained for the two models. Compared to the work of DHN, aside from methodology, the main new results are the matrix elements of the field operators, but we also present a complete quantum interpretation of all their classical calculations as well as an explanation of why our methods are equivalent.