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

Yair Zarmi

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arXiv · 1210.6641

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

Yair Zarmi

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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, σ (= +-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.

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Yair Zarmi. 2012-10-24. A nonlinear quantum dynamical system of spin 1/2 particles based on the classical Sine-Gordon Equation. https://arxiv.org/abs/1210.6641

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