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Progress of Theoretical Physics

Boost Operator and Its Application to Quantum Gelfand-Levitan Equation for Heisenberg-Ising Chain with Spin One-Half

Kiyoshi Sogo, Miki Wadati

Published Feb 1, 1983• 75 citations
Publisher

Abstract

The quantum inverse scattering problem for the Ising-like XXZ model is studied. It is found that the boost operator plays an essential role in the analysis. In terms of the boost operator, the bound state operators are defined and the quantum Gelfand-Levitan equation is derived in a natural way. The quantum inverse scattering problem for the XY-like XXZ model is also discussed. In addition, the boost operators for Calogero systems and Toda lattice are explicitly given.

DOI
10.1143/ptp.69.431

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The quantum inverse scattering problem for the Ising-like XXZ model is studied. It is found that the boost operator plays an essential role in the analysis. In terms of the boost operator, the bound state operators are defined and the quantum Gelfand-Levitan equation is derived in a natural way. The quantum inverse scattering problem for the XY-like XXZ model is also discussed. In addition, the boost operators for Calogero systems and Toda lattice are explicitly given.
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 quantum generalization of the Gelfand-Levitan method is presented for the nonlinear Schr\"odinger model. The basic dispersion relation for operator Jost functions is derived, and the Heisenberg field operator is expressed in terms of scattering-data operators. Construction of Green's functions in the zero-density vacuum is discussed. The four-point function is explicitly calculated from the expression for the field operator and compared with the result of a direct Feynman graph summation. In addition it is proved for any number of particles that the Hamiltonian eigenstates constructed from the quantized scattering data are identical with those previously obtained by means of Bethe's ansatz.
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.