Quantum inverse scattering method: Difference between revisions

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In [[quantum physics]], the '''quantum inverse scattering method''' is a method for solving [[integrable system|integrable model]]s in 1+1 dimensions, introduced by [[Ludvig Faddeev|L. D. Faddeev]] in 1979. {{cite arXiv |eprint=hep-th/9211111|last1=Sklyanin|first1=E. K.|title=Quantum Inverse Scattering Method. Selected Topics|year=1992}}

The quantum inverse scattering method relates two different approaches:

#the [[Bethe ansatz]], a method of solving integrable quantum models in one space and one time dimension;.

#the [[Inverseinverse scattering transform]], a method of solving classical integrable differential equations of the evolutionary type.

This method led to the formulation of [[quantum group]]s. Especially interesting is the [[Yangian]], and the center of the Yangian is given by the [[quantum determinant]].

An important concept in the [[Inverseinverse scattering transform]] is the [[Lax pair|Lax representation]]; the quantum inverse scattering method starts by the [[quantization (physics)|quantization]] of the Lax representation and reproduces the results of the Bethe ansatz. In fact, it allows the Bethe ansatz to be written in a new form: the ''algebraic Bethe ansatz''.<ref>cf.See e.g.for theexample lectures by N.A. Slavnov, {{arXiv|1804.07350}}</ref> This led to further progress in the understanding of quantum [[Integrableintegrable system]]s, forsuch example:as the [[quantum Heisenberg model]], the quantum [[Nonlinear Schrödinger equation]] (also known as the [[Lieb–Liniger model]] or the [[Tonks–Girardeau gas]]) and the [[Hubbard model]].

a) the [[Heisenberg model (quantum)]],

b) the quantum [[Nonlinear Schrödinger equation]] (also known as the [[Lieb–Liniger model]] or the [[Tonks–Girardeau gas]]) and

c) the [[Hubbard model]].

The theory of [[correlation function]]s was developed {{when|date=November 2015}}:, relating determinant representations, descriptions by differential equations and the [[Riemann–Hilbert problem]]. Asymptotics of correlation functions (evenwhich forinclude space, time and temperature dependence) were evaluated in 1991.

Explicit expressions for the higher [[conservation law]]s of the integrable models were obtained in 1989.