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arXiv:1405.1843 · Journal of Mathematical Physics

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

Yair Zarmi

Published Oct 1, 2014• 3 citations
arXivHTMLPDF

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 ≥ 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.

DOI
10.1063/1.4899085

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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.
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.
The problem of interpreting moving soliton solutions of some nonlinear equations as quantum-like particles is critically analyzed. Purely classical equations (sine-Gordon) as well as nonlinear quantum equations (Schrödinger, Dirac, Kelin–Gordon) are investigated. Three different definitions of energy associated with these equations are considered, and the functional relations replacing the de Broglie postulate, [Formula: see text], are explicitly given. Our analysis shows that fundamental problems generated by the nonlinear approach to quantum mechanics are still unsolved, and these theories have to be considered cautiously.
The structure of soliton solutions of classical integrable nonlinear evolution equations, which can be solved through the Hirota transformation, suggests a new way for the construction of nonlinear quantum-dynamical systems that are based on the classical equations. In the new approach, the classical soliton solution is mapped into an operator, U, which is a nonlinear functional of the particle-number operators over a Fock space of quantum particles. U obeys the evolution equation; the classical soliton solutions are the eigenvalues of U in multi-particle states in the Fock space. The construction easily allows for the incorporation of particle interactions, which generate soliton effects that do not have a classical analog. In this paper, this new approach is applied to the case of the Kadomtsev-Petviashvili II equation. The nonlinear quantum-dynamical system describes a set of M = (2S + 1) particles with intrinsic spin S, which interact in clusters of 1 ≤ N ≤ (M − 1) particles.
Making use of thermal-Green's-function technique, we study the quantum-statistical mechanics of a sine-Gordon system in 1 + 1 dimensions. In the weak-coupling limit, the temperature dependences of the soliton energy, Es{E}_{s}Es​, the soliton inertial mass, and the soliton density are determined. At high temperatures (T>mT>mT>m, where mmm is the mass of the fundamental field), Es{E}_{s}Es​ decreases monotonically as the temperature increases, and Es{E}_{s}Es​ jumps to zero around T=TcrT={T}_{\mathrm{cr}}T=Tcr​ ($\ensuremath{\equiv}{e}^{\ensuremath{-}1}{{E}_{s}}^{0}$), where Es0{{E}_{s}}^{0}Es​0 is the soliton energy at T=0T=0T=0 K. The soliton density agrees with the classical statistical-mechanics results for ${T}_{\mathrm{cr}}>T\ensuremath{\gg}m$, if Es{E}_{s}Es​ in the classical theory is replaced by the temperature-dependent one of the present theory.
We have developed a quantum-mechanical approach to study the critical properties of the quantum sine-Gordon equation describing a massless scalar field in 1+1 space-time dimensions with interaction density proportional to cos(${\mathrm{\ensuremath{\beta}}}_{0}$\ensuremath{\varphi}). The main feature of the present theory is to find an appropriate starting point for perturbation expansion, which can directly treat the infrared divergences encountered in the conventional perturbation theory. Exact critical conditions for the phase transition of the model are derived; and the quantum sine-Gordon Hamiltonian is found to correspond exactly to a free-field model. Near the critical regime, a gap will open near zero momentum in the excitation spectrum, and the corresponding ground-state wave function is a pairing quasiparticle state, analogous to the BCS superconducting state. It is proved that this ground state will possess quasi-long-range and quasi-off-diagonal long-range correlations. Comparisons with the Kosterlitz-Thouless classical theory of the two-dimensional Coulomb gas are also made.
Wigner distributions play a significant role in formulating the phase–space analog of quantum mechanics. The Schrödinger wave functional for solitons is needed to derive it for solitons. The Wigner distribution derived can further be used for calculating the charge distributions, current densities and wave function amplitude in position or momentum space. It can be also used to calculate the upper bound of the quantum speed limit time. We derive and analyze the Wigner distributions for Kink and Sine-Gordon solitons by evaluating the Schrödinger wave functional for both solitons. The charge, current density, and quantum speed limit for solitons are also discussed which we obtain from the derived analytical expression of Wigner distributions.