Quantum inverse scattering method: Difference between revisions

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{{Short description|Method used to solve integrable many-body quantum systems}}

In [[quantum physics]], the '''quantum inverse scattering method''' (QISM) or the '''algebraic Bethe ansatz''' is a method for solving [[integrable system|integrable model]]s in 1+1 dimensions, introduced by [[Leon Takhtajan]] and [[Ludvig Faddeev|L. D. Faddeev]] in 1979.<ref>{{cite journal |last1=Takhtadzhan |first1=L A |last2=Faddeev |first2=Lyudvig D |title=THEThe QUANTUMQuantum METHODMethod OFof THEthe INVERSEInverse PROBLEMProblem ANDand THEthe HEISENBERGHeisenberg XYZXyz MODELModel |journal=Russian Mathematical Surveys |date=31 October 1979 |volume=34 |issue=5 |pages=11–68 |doi=10.1070/RM1979v034n05ABEH003909|bibcode=1979RuMaS..34...11T }}</ref>

It can be viewed as a quantized version of the classical [[inverse scattering method]] pioneered by [[Norman Zabusky]] and [[Martin Kruskal]]<ref>{{cite journal |last1=Zabusky |first1=N. J. |last2=Kruskal |first2=M. D. |title=Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States |journal=Physical Review Letters |date=9 August 1965 |volume=15 |issue=6 |pages=240–243 |doi=10.1103/PhysRevLett.15.240|doi-access=free |bibcode=1965PhRvL..15..240Z }}</ref> used to investigate the [[Korteweg–de Vries equation]] and later other [[integrable system|integrable]] [[partial differential equations]]. In both, a [[Lax matrix]] features heavily and [[scattering|scattering data]] is used to construct solutions to the original system.

While the classical inverse scattering method is used to solve integrable partial differential equations which model [[continuous media]] (for example, the KdV equation models shallow water waves), the QISM is used to solve [[many-body problem|many-body]] quantum systems, sometimes known as [[spin chain]]s, of which the [[Heisenberg spin chain]] is the best-studied and most famous example. These are typically discrete systems, with particles fixed at different points of a lattice, but limits of results obtained by the QISM can give predictions even for [[quantum field theory|field theories]] defined on a continuum, such as the quantum [[sine-Gordon model]].

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==References==

{{Reflist}}

*{{cite journal |last1=Chakrabarti |first1=A. |title=RTT relations, a modified braid equation and noncommutative planes |journal=Journal of Mathematical Physics |year=2001 |volume=42 |issue=6 |pages=2653–2666 |doi=10.1063/1.1365952 |url=https://pubs.aip.org/aip/jmp/article-abstract/42/6/2653/803265/RTT-relations-a-modified-braid-equation-and|arxiv=math/0009178 |bibcode=2001JMP....42.2653C }}

*{{cite arXiv |eprint=hep-th/9211111|last1=Sklyanin|first1=E. K.|title=Quantum Inverse Scattering Method. Selected Topics|year=1992}}

*{{Citation | last1=Faddeev | first1=L. | title=Instructive history of the quantum inverse scattering method |doi=10.1007/BF00994626 | mr = 1329554 | year=1995 | journal=Acta Applicandae Mathematicae | volume=39 | issue=1 | pages=69–84| s2cid=120648929 }}