Quantum inverse scattering method and correlation functions.

*(English)*Zbl 0787.47006
Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press. xiii, 555 p. (1993).

The subject of this book are solutions of \((1+1)\)-dimensional models in quantum field theory and statistical physics. It consists of four parts.

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.

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.

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 SchrĂ¶dinger model.

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.

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.

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.

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.

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.

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 SchrĂ¶dinger model.

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.

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.

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.

Reviewer: Michael Demuth (Potsdam)

##### MSC:

47A40 | Scattering theory of linear operators |

47N50 | Applications of operator theory in the physical sciences |

81U40 | Inverse scattering problems in quantum theory |

81Txx | Quantum field theory; related classical field theories |

81R12 | Groups and algebras in quantum theory and relations with integrable systems |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |