Nonlinear quantum-mechanical system associated with Sine-Gordon equation in (1+2) dimensions

TL;DR

This paper constructs a quantum-mechanical system based on the 1+2 dimensional Sine-Gordon equation, revealing soliton solutions and mass-density structures, and allowing for particle interactions with soliton creation and annihilation.

Contribution

It introduces a novel quantum system linked to the 1+2 dimensional Sine-Gordon equation, incorporating soliton solutions and interaction effects not present in classical models.

Findings

Eigenvalues correspond to N-soliton solutions with subluminal velocities.

A projection operator acts as a mass-density generator for multi-particle states.

The system allows for soliton creation and annihilation through particle interactions.

Abstract

Despite the fact that it is not integrable, the 1 + 2-dimensional Sine-Gordon equation has N-soliton solutions, whose velocities are lower than the speed of light (c = 1), for all N greater than or equal to 1. Based on these solutions, a quantum-mechanical system is constructed over a Fock space of particles. The coordinate of each particle is an angle around the unit circle. U, a nonlinear functional of the particle number-operators, which obeys the Sine-Gordon equation in 1+2 dimensions, is construct-ed. Its eigenvalues on N-particle states in the Fock space are the slower-than-light, N-soliton solutions of the equation. A projection operator (a nonlinear functional of U), which vanishes on the single-particle subspace, is a mass-density generator. Its eigenvalues on multi-particle states play the role of the mass density of structures that emulate free, spatially extended, relativistic particles. The simplicity of the quantum-mechanical system allows for the incorporation of perturbations with particle interactions, which have the capacity to annihilate and create solitons - an effect that does not have a classical analog.