Quantum inverse scattering method: Difference between revisions

Revision as of 19:30, 20 October 2023

Revision as of 19:32, 20 October 2023

Line 15:

This method led to the formulation of [[quantum group]]s, in particular the [[Yangian]]. The center of the Yangian, given by the [[quantum determinant]] plays a prominent role in the method.

An important concept in the [[inverse 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>See for example lectures by N.A. Slavnov {{arXiv|1804.07350}}</ref> This led to further progress in the understanding of quantum [[integrable system]]s, such as the [[quantum Heisenberg model]], the quantum [[Nonlinearnonlinear Schrödinger equation]] (also known as the [[Lieb–Liniger model]] or the [[Tonks–Girardeau gas]]) and 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 which include 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.

Line 26:

==Procedure==

The steps can be summarized as follows {{harvs|last=Sklyanin|first=Evgeny|year=1992}}:

# Take an [[R-matrix|''R''-matrix]] which solves the [[Yang–Baxter equation]].

# Take a [[representation (group theory)|representation]] of an algebra <math>\mathcal{T}_R</math> satisfying the RTT{{What|date=October 2023}} relations.

# Find the spectrum of the [[generating function]] <math>t(u)</math> of the [[center (group theory)|centre]] of <math>\mathcal{T}_R</math>.

# Find correlators.