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Physica Scripta

On the Poisson-Bracket algebra for the scattering data of a non-ultralocal field theory with soliton solutions

Asesh Roy Chowdhury, Shibani Sen

Published Jul 1, 1987• 2 citations
Publisher

Abstract

A preliminary investigation is done about the Poisson-Bracket algebra obeyed by the scattering data associated with a non-ultralocal field theory sustaining soliton solutions. The result is quite distinct from the usual Poisson-bracket relations usually observed in ultralocal field theory. This important difference is reflected in the changed Zamoldachikov algebra of the R ( K )-reflection operator, crucial in the quantization progress of the nonlinear system in two dimensions. Our results though not conclusive, yet is useful in the future progress of quantization of such field theories.

DOI
10.1088/0031-8949/36/1/001

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A preliminary investigation is done about the Poisson-Bracket algebra obeyed by the scattering data associated with a non-ultralocal field theory sustaining soliton solutions. The result is quite distinct from the usual Poisson-bracket relations usually observed in ultralocal field theory. This important difference is reflected in the changed Zamoldachikov algebra of the R ( K )-reflection operator, crucial in the quantization progress of the nonlinear system in two dimensions. Our results though not conclusive, yet is useful in the future progress of quantization of such field theories.
We classify nonultralocal Poisson brackets for 1-dimensional lattice systems and describe the corresponding regularizations of the Poisson bracket relations for the monodromy matrix. A nonultralocal quantum algebras on the lattices for these systems are constructed. For some class of such algebras an ultralocalization procedure is proposed. The technique of the modified Bethe-Anzatz for these algebras is developed and is applied to the nonlinear sigma model problem. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The form of the complex Hamiltonian evolution equations (linear and nonlinear) is deduced here from that of the classical Hamiltonian systems. Their properties and those of the associated continuous Poisson bracket are studied. It is shown that if any space-localized solution of the above type equations is expanded in any Hilbert space basis the expansion coefficients are canonically conjugate variables and obey Hamiltonian systems of complex equations. The last is proved also for sets of “canonically conjugate” functionals of general type. As an application, commutation relations which are equivalent in form to Dirac’s quantization rules (the constant in place of ℏ is undetermined) are obtained. The question of how a nonlinear field theory may contain quantum mechanics as a special case is discussed in view of this paper’s results.
We show that any rule for quantization of functions on the phase space of a dynamical system is equivalent to a deformation of the Poisson bracket, in the sense of Lie algebras. All Schrödinger-type quantizations are equivalent mathematically but not physically. We examine whether a quantization rule exists in which the Poisson bracket of the free Hamiltonian and any function not more than quadratic in momentum gets quantized into the commutator of the quantized versions of these two observables. We show that if the (Riemannian) configuration manifold is irreducible, this condition uniquely specifies the quantization of these functions, in the cases when it can be satisfied. However, it can only be satisfied for systems whose configuration manifolds are one-dimensional, or have vanishing Ricci tensor, or are spaces of constant curvature. Thus the condition will not serve as a general physical principle which could be invoked to fix a quantization rule.