Collective Coordinate Methods and Their Applicability to $\varphi^4$ Models

Abstract

Collective coordinate methods are frequently applied to study dynamical properties of solitons. These methods simplify the field equations - typically partial differential equations - to ordinary differential equations for selected excitations. More importantly though, collective coordinates provide a practical means to focus on particular modes of otherwise complicated dynamical processes. We review the application of collective coordinate methods in the analysis of the kink-antikink interaction within the $\varphi^4$ soliton model and illuminate discrepancies between these methods and the exact results from the field equations.