Removal of singularities from collective-variable theory by incorporating relativistic invariance

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Abstract
The equations of motion which govern the evolution of collective variables introduced into a nonlinear Klein-Gordon system are obtained by projecting the nonlinear field equation onto vectors in Hilbert space. For systems in the zero-kink sector, which admit breather and kink-antikink dynamics, the magnitude of some vectors onto which the field equation is projected become zero at particular instants in time. At these instants in time, the projections are not defined, and consequently the collective-variable equations of motion themselves are not defined: the transformation from the original field variables to the collective variables becomes singular. Although the singularity problem has already appeared in various contexts in the literature, it has not been resolved in the context of a collective-variable theory which contains collective variables in addition to that of the center of mass. For such cases, we show the singularities are removed by correctly taking into account the relativistic invariance of the underlying theory and give an example Ansatz which admits a well-defined collective-variable theory which would otherwise have been singular. We comment on the increased complexity of the present Ansatz relative to the utility of collective-variable theory.