A nonlinear quantum dynamical system of spin 1/2 particles based on the classical Sine-Gordon Equation

TL;DR

This paper introduces a nonlinear quantum dynamical system for spin 1/2 particles based on the classical Sine-Gordon equation, using the Hirota transformation to connect classical soliton solutions with quantum operators.

Contribution

It presents a novel construction of a quantum system that maps classical soliton solutions onto quantum operators, enabling simple incorporation of particle interactions.

Findings

Soliton solutions are mapped onto eigenvalues of a quantum operator U.

Multi-particle states are eigenstates of U with eigenvalues as classical solitons.

The framework allows easy inclusion of particle interactions with soliton effects.

Abstract

The Hirota transformation for the soliton solutions of the classical Sine-Gordon equation is suggestive of an extremely simple way for the construction of a nonlinear quantum-dynamical system of spin 1/2 particles that is equivalent to the classical system over the soliton sector. The soliton solution of the classical equation is mapped onto an operator, U, a nonlinear functional of the particle-number operators, that solves the classical equation. Multi-particle states in the Fock space are the eigenstates of U; the eigenvalues are the soliton solutions of the Sine-Gordon equation. The fact that solitons can have positive as well negative velocities is reflected by the characterization of particles in the Fock space by two quantum numbers: a wave number k, and a spin projection, {\sigma} (= +-1). Thanks to the simplicity of the construction, incorporation of particle interactions, which induce soliton effects that do not have a classical analog, is simple.