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Zbl 0787.47006
Korepin, V.E.; Bogoliubov, N.M.; Izergin, A.G.
(Bogolyubov, N.M.)
Quantum inverse scattering method and correlation functions. (English)
[B] Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press. xiii, 555 p. \$ 100.00; \sterling 60.00 (1993). ISBN 0-521-37320-4/hbk

The subject of this book are solutions of $(1+1)$-dimensional models in quantum field theory and statistical physics. It consists of four parts.\par In the first part the coordinate Bethe-Ansatz is explained. It reduces many-body scattering matrices to two-body matrices of integrable models. Mainly, four models are studied: the one-dimensional Bose gas, the Heisenberg magnet, the massive Thirring model, and the Hubbard model of interacting fermions on a lattice. The energy and momenta of excitations are evaluated, eigenfunctions are constructed, the thermodynamic limit is considered.\par The quantum inverse scattering method is used in part 2 to link the Bethe-Ansatz with the theory of classical differential equations which can be solved completely by the classical inverse scattering method, a nonlinear generalization of the Fourier transform. The Hamilton structure of integrable models is discussed via the classical $r$-matrix determined by the Yang-Baxter equation, and the concept of the quantum determinant is introduced. The quantum inverse scattering method provides an algorithm for transferring continuous models in quantum field theory to a corresponding lattice version preserving the $R$-matrix. Several integrable models in quantum field theory on lattices are explained and a classification of all integrable models with a fixed $R$-matrix is given.\par In the second half, the main part of the book (part 3 and 4), the quantum correlation functions are described. They are represented as determinants of certain matrices. In the thermodynamic limit there are determinants of Fredholm integral operators. In part 3 the determinant representation for scalar products is analyzed and used to study the norms of Bethe wave functions. Moreover, the current and the field correlators are investigated in some detail. The main example in this part is the nonlinear Schrödinger model.\par In part 4, the determinant representation is used for establishing differential equations of the quantum correlation function. This is closely related to classical nonlinear differential equations which can be solved completely, i.e. which have a Lax representation. The integration of these equations provides an explicit asymptotics for the correlation function. The whole approach is explained for the impenetrable Bose gas. The Riemann-Hilbert problem is studied. The asymptotics for temperature-dependent correlation functions is calculated. Finally, the algebraic Bethe-Ansatz and the conformal approach are sketched.\par The book is well written. Every part and every chapter has its own introduction explaining the motivation, the results and objectives. Every chapter has a conclusion with a summary and some comments. That improves the context between the different parts of the book. The present review is mainly taken from these introductions.\par For understanding this book in theoretical physics the reader should be familiar with the fundamentals in field theory and statistical physics. It starts on a graduate level and is useful for the research work in these fields.
[Michael Demuth (Potsdam)]

MSC 2000:: *47A40 Scattering theory of linear operators
47N50 Appl. of operator theory in quantum physics
81U40 Inverse scattering problems (quantum theory)
81Txx Quantum field theory and related classical field theories
81R12 Relations with integrable systems
47-02 Research monographs (operator theory)
81-02 Research monographs (quantum theory)

Keywords: $(1+1)$-dimensional models in quantum field theory and statistical physics; coordinate Bethe-Ansatz; many-body scattering matrices; two-body matrices of integrable models; one-dimensional Bose gas; Heisenberg magnet; massive Thirring model; Hubbard model of interacting fermions on a lattice; energy and momenta of excitations; eigenfunctions; thermodynamic limit; quantum inverse scattering method; Hamilton structure of integrable models; $R$-matrix; determinants of Fredholm integral operators

Cited in: Zbl 1181.15005 Zbl 1225.82018 Zbl 0941.82016 Zbl 0969.81031 Zbl 0836.35113 Zbl 0880.47003

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