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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Solvable quantum field theories and polynomial conserved quantities for the quantum nonlinear Schrödinger equation

M. Omote, M. Sakagami, Ryu Sasaki B, Itaru Yamanaka

Published Apr 15, 1987• 14 citations
Publisher

Abstract

The quantum version of the infinite set of polynomial conserved quantities for the nonlinear Schr\"odinger equation in 1+1 dimensions is discussed. In the conventional formulation of the theory, which describes a quantum many-particle system with a \ensuremath{\delta}-function potential, it is shown that an infinite set of quantum polynomial commuting operators does not exist. However, by adopting a rather unconventional quantization method, we construct an infinite set of quantum polynomial commuting operators explicitly in a parallel way with the Korteweg--de Vries and the modified Korteweg--de Vries equation cases discussed in a previous paper. Based on these results a new scheme of solvable quantum field theories is proposed.

DOI
10.1103/physrevd.35.2423

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The quantum version of the infinite set of polynomial conserved quantities for the nonlinear Schr"odinger equation in 1+1 dimensions is discussed. In the conventional formulation of the theory, which describes a quantum many-particle system with a \ensuremath{\delta}-function potential, it is shown that an infinite set of quantum polynomial commuting operators does not exist. However, by adopting a rather unconventional quantization method, we construct an infinite set of quantum polynomial commuting operators explicitly in a parallel way with the Korteweg--de Vries and the modified Korteweg--de Vries equation cases discussed in a previous paper. Based on these results a new scheme of solvable quantum field theories is proposed.
The quantum-mechanical implications of the inverse-scattering-transform method and its relationship to the structure of Bethe's ansatz are discussed in the context of the nonlinear Schr\"odinger equation (many-body problem) associated with the classical (quantum) field theory $\mathcal{L}=(\frac{i}{2}){\ensuremath{\varphi}}^{*}{\stackrel{}{\ensuremath{\partial}}}_{0}\ensuremath{\varphi}\ensuremath{-}{|{\ensuremath{\partial}}_{1}\ensuremath{\varphi}|}^{2}\ensuremath{-}c{|\ensuremath{\varphi}|}^{4}$. We review the transformation of the classical problem to action and angle variables and the derivation of an infinite number of polynomial conservation laws. The values of the conserved constants are given by the moments of the classical action variable. It is suggested that there exists a corresponding set of conserved polynomial operators in the quantum field theory and that they reflect the conservation of velocity content which characterizes the solution of the many-body scattering problem (Bethe's ansatz). This implies that the quantized action variable is just the occupation-number density operator in the asymptotic momentum-(velocity-) parameter space of Bethe's ansatz, and that Bethe's wave functions are eigenstates of all conserved operators with eigenvalues given by the moments of the NNN-particle distribution in asymptotic momentum space. These statements are verified for the first four operators, including one which has not previously been studied.
We study differential and integral relations for the quantum Jost solutions associated with an integrable derivative nonlinear Schrodinger (DNLS) model. By using commutation relations between such Jost solutions and the basic field operators of DNLS model, we explicitly construct first few quantum conserved quantities of this system including its Hamiltonian. It turns out that this quantum Hamiltonian has a new kind of coupling constant which is quite different from the classical one. This modified coupling constant plays a crucial role in our comparison between the results of algebraic and coordinate Bethe ansatz for the case of DNLS model. We also find out the range of modified coupling constant for which the quantum NNN-soliton state of DNLS model has a positive binding energy.
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A lattice version of quantum nonlinear Schrodinger (NLS) equation is considered, which has significantly simple form and fullfils most of the criteria desirable for such lattice variants of field models. Unlike most of the known lattice NLS, the present model belongs to a class which does not exhibit the usual symmetry properties. However this lack of symmetry itself seems to be responsible for the remarkable simplification of the relevant objects in the theory, such as the Lax operator, the Hamiltonian and other commuting conserved quantities as well as their spectrum. The model allows exact quantum solution through algebraic Bethe ansatz and also a straightforward and natural generalisation to the vector case, giving thus a new exact lattice version of the vector NLS model. A deformation representing a new quantum integrable system involving Tamm-Dancoff like qqq-boson operators is constructed.
The quantum inverse scattering method is applied to the nonlinear Schrodinger model of fermions with attractive interactions. Both scattering and bound-state operators are constructed. Among state operators, there are generators of infinitely many conserved quantities and the creators of the eigenstates of these quantities. The commutation relations and the eigenvalues of physical interest are calculated. The quantum Gelfand-Levitan equations are established. The Fourier transform of the connected part of the 4-point Green function and the two body S matrix are calculated explicitly.
Canonical coordinates for the Schrödinger equation are introduced, making more transparent its Hamiltonian structure. It is shown that the Schrödinger equation, considered as a classical field theory, shares with Liouville completely integrable field theories the existence of a recursion operator which allows for the infinitely many conserved functionals pairwise commuting with respect to the corresponding Poisson bracket. The approach may provide a good starting point to get a clear interpretation of Quantum Mechanics in the general setting, provided by Stone–von Neumann theorem, of Symplectic Mechanics. It may give new tools to solve in the general case the inverse problem of quantum mechanics whose solution is given up to now only for one-dimensional systems by the Gel’fand-Levitan-Marchenko formula.