An Alternative to Collective Coordinates

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.

Excited Kinks as Quantum States

“…Subtracting Q 2 , the two-loop correction to the ground state energy found in Ref. [12], one obtains E 2 − Q 2 , the two-loop correction to the energy required to excite the kink normal mode.…”

“…Recently [12] a method was proposed which allows the practical calculation of higher-loop states. This method, to be reviewed in Sec.…”

The Two-Loop $ϕ^4$ Kink Mass

“…was calculated in Ref. [15]. Here :: a is normal ordering of the operators that create and annihilate plane waves.…”