The Effect of low Momentum Quantum Fluctuations on a Coherent Field Structure

TL;DR

This paper investigates how low momentum quantum fluctuations influence the stability and interactions of solitons in the Sine-Gordon model, combining analytical and numerical methods to explore quantum effects on classical field structures.

Contribution

It introduces a non-perturbative approach to study quantum fluctuations in the Sine-Gordon equation, highlighting the stability and scattering of solitons under quantum effects.

Findings

Quantum fluctuations stabilize solitons.

No fusion or splitting due to integrability.

Good agreement between approximate and numerical solutions.

Abstract

In the present work the evolution of a coherent field structure of the Sine-Gordon equation under quantum fluctuations is studied. The basic equations are derived from the coherent state approximation to the functional Schr\"odinger equation for the field. These equations are solved asymptotically and numerically for three physical situations. The first is the study of the nonlinear mechanism responsible for the quantum stability of the soliton in the presence of low momentum fluctuations. The second considers the scattering of a wave by the Soliton. Finally the third problem considered is the collision of Solitons and the stability of a breather. It is shown that the complete integrability of the Sine-Gordon equation precludes fusion and splitting processes in this simplified model. The approximate results obtained are non-perturbative in nature, and are valid for the full nonlinear interaction in the limit of low momentum fluctuations. It is also found that these approximate results are in good agreement with full numerical solutions of the governing equations. This suggests that a similar approach could be used for the baby Skyrme model, which is not completely integrable. In this case the higher space dimensionality and the internal degrees of freedom which prevent the integrability will be responsable for fusion and splitting processes. This work provides a starting point in the numerical solution of the full quantum problem of the interaction of the field with a fluctuation.