In the Nonlinear Schrödinger model, the correspondence between classical soliton and its quantum field theory is investigated. It is shown explicitly that moving soliton as well as soliton at rest arise as the matrix elements of a field operator in the limit n →∞, where n is the particle number making the bound state.
Through the quantum inverse scattering method the nonlinear Schrödinger model is studied for the attractive case. Making use of the bound state operators and the quantum Gelfand-Levitan equation, the matrix elements of the field operator φ( x ) and the density operator φ + ( x )φ( x ) are explicitly calculated. Each matrix element is found to be expressed in a product form. Based on the results, the connection between classical soliton and the quantum field theory is investigated. It is shown that the wave form of the classical soliton is related to the matrix element of the field operator in the limit n →∞, where n is the particle number making the bound state. The powerfulness of the quantum Gelfand-Levitan equation is stressed.
The fabric of the nonlinear quantum field behind optical solitons suggests the many-particle structure of the photon wave function in optical solitons. This wave function derives from the bound states of the original quantum field and is found as the ground-state eigenfunction of a many-body Hamiltonian with a delta-function interaction potential. Drawing on the properties of such a many-particle photon wave function, we show how classical solitons with their signature hyperbolic secant field intensity profiles emerge from quantum nonlinear fields in the limit of very large photon numbers.
We show that spacetime evolution of one-dimensional fermionic systems is described by nonlinear equations of soliton theory. We identify a spacetime dependence of a matrix element of fermionic systems related to the orthogonality catastrophe or boundary states with the τ-function of the modified KP-hierarchy. The established relation allows us to apply the apparatus of soliton theory to the study of nonlinear aspects of quantum dynamics. We also describe a bosonization in momentum space —a representation of a fermion operator by a Bose field in the presence of a boundary state.
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.