Alternative to collective coordinates

Jarah Evslin,Hengyuan Guo

doi:10.1103/physrevd.103.l041701

Jarah Evslin, Hengyuan Guo

Open Access

https://doi.org/10.1103/physrevd.103.l041701

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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.

Highlights

When a theory is reformulated in terms of collective coordinates, some phenomena involving large numbers of elementary quanta, such as plasma waves, can be treated in perturbation theory [1]
In [2], following the spirit of [1], the collective coordinates are related to the elementary fields by a canonical transformation
Decomposing fields in terms of the plane wave operators, Bogoliubov transforming to the normal mode operators and normal mode normal ordering one finds that the one-loop Hamiltonian is a sum of quantum harmonic oscillators plus a free quantum mechanical particle for the center of mass

Summary

INTRODUCTION

When a theory is reformulated in terms of collective coordinates, some phenomena involving large numbers of elementary quanta, such as plasma waves, can be treated in perturbation theory [1]. In [2], following the spirit of [1], the collective coordinates are related to the elementary fields by a canonical transformation This transformation allows a straightforward quantization of the system. Intuition from large N [6] suggests that hadrons are quantum solitons and so their masses, form factors, and general matrix elements may be calculated by solving for the corresponding quantum state. In this case, we will propose a much simpler alternative to collective coordinates which allows one to pass to higher numbers of loops using reasonably elementary computations

REVIEW

IMPOSING TRANSLATION INVARIANCE

OLD-FASHIONED PERTURBATION THEORY

CONCLUSIONS

Z d2k0 jðωk01 16Q0 ð2πÞ2 1 Z dk0 jf00ðxÞj2