An Alternative to Collective Coordinates

TL;DR

This paper introduces a simplified method for analyzing soliton excitations that avoids complex canonical transformations by using a nonperturbative operator to construct the theory perturbatively, demonstrated on a scalar kink.

Contribution

It proposes an alternative to collective coordinates that simplifies the treatment of elementary excitations around solitons without canonical transformations.

Findings

Constructed the two-loop ground state of a scalar kink.

Demonstrated perturbative imposition of translation invariance.

Provided a new approach for soliton excitation analysis.

Abstract

Collective coordinates provide a powerful tool for separating collective and elementary excitations, allowing both to be treated in the full quantum theory. The price is a canonical transformation which leads to a complicated starting point for subsequent calculations. Sometimes the collective behavior of a soliton is simple but nontrivial, and one is interested in the elementary excitations. We show that in this case an alternative prescription suffices, in which the canonical transformation is not necessary. The use of a nonperturbative operator which creates a soliton state allows the theory to be constructed perturbatively in terms of the soliton normal modes. We show how translation invariance may be perturbatively imposed. We apply this to construct the two-loop ground state of an arbitrary scalar kink.