zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Quantum inverse scattering method and correlation functions. (English) Zbl 0787.47006
Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press. xiii, 555 p. $ 100.00; £60.00 (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 Schrödinger 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.

47A40Scattering theory of linear operators
47N50Applications of operator theory in quantum physics
81U40Inverse scattering problems (quantum theory)
81TxxQuantum field theory and related classical field theories
81R12Relations of groups and algebras in quantum theory with integrable systems
47-02Research monographs (operator theory)
81-02Research monographs (quantum theory)