TY - JOUR

T1 - Nonlinear quantum-mechanical system associated with Sine-Gordon equation in (1 + 2) dimensions

AU - Zarmi, Yair

N1 - Publisher Copyright:
© 2014 AIP Publishing LLC.

PY - 2014/10/28

Y1 - 2014/10/28

N2 - 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 ≥ 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 constructed. 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 an analog in perturbed classical nonlinear evolution equations.

AB - 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 ≥ 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 constructed. 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 an analog in perturbed classical nonlinear evolution equations.

UR - http://www.scopus.com/inward/record.url?scp=84910024053&partnerID=8YFLogxK

U2 - 10.1063/1.4899085

DO - 10.1063/1.4899085

M3 - Article

AN - SCOPUS:84910024053

SN - 0022-2488

VL - 55

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 10

M1 - 103510

ER -