Momentum constraints in collective-variable theory

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Abstract
We present an analysis of the constraints used in collective-variable treatments of kink-bearing nonlinear Klein-Gordon equations, which appear in field theory and in continuum and discrete condensed-matter systems. In particular, we introduce into the collective-variable theory a family of momentum constraints that includes the momentum constraints that have been used in the literature so far. We derive the collective-variable Hamiltonian and show that there is a single member of the family of constraints for which the kinetic energy of the collective mode separates from the other variables in the theory so that a truly particle-like description of kink dynamics results. We discuss the general structure of the Hamiltonian collective-variable equations of motion and also present a simple derivation of the collective-variable theory beginning from a Lagrangian. We obtain, therefore, the correct choice of momentum constraint within the family for both Hamiltonian and Lagrangian approaches to multiple collective-variable theories.