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An algebraic approach to form factors

Max R. Niedermaier

Published Dec 19, 1994• 11 citations
OpenAlex

Abstract

An associative *-algebra is introduced (containing a TTR-algebra as a subalgebra) that implements the form factor axioms, and hence indirectly the Wightman axioms, in the following sense: Each T-invariant linear functional over the algebra automatically satisfies all the form factor axioms. It is argued that this answers the question (posed in the functional Bethe ansatz) how to select the dynamically correct representations of the TTR-algebra. Applied to the case of integrable QFTs with diagonal factorized scattering theory a universal formula for the eigenvalues of the conserved charges emerges.

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Introduction to quantum integrability, factorized S-matrix theory and TBA

Francesco Ravanini

In this series of lectures given at KIAS in Nov 2018, I briefly introduce the concepts of integrability, at classical and quantum level. Then I elaborate on the factorised scattering theory emerging in 1+1 integrable QFT. The Thermdynamic Bethe Ansatz (TBA) is a well known method to get exact information from the scattering theory, in order to identify the underlying QFT. Here I illustrate how to get TBA in the case of the prototypical Sine-Gordon model, particularly interesting to show the techniques when the S-matrix is not diagonal.

•Jan 1, 2018•0 citations
An associative *-algebra is introduced (containing a TTR-algebra as a subalgebra) that implements the form factor axioms, and hence indirectly the Wightman axioms, in the following sense: Each T-invariant linear functional over the algebra automatically satisfies all the form factor axioms. It is argued that this answers the question (posed in the functional Bethe ansatz) how to select the dynamically correct representations of the TTR-algebra. Applied to the case of integrable QFTs with diagonal factorized scattering theory a universal formula for the eigenvalues of the conserved charges emerges.
Form factor sequences of an integrable QFT can be defined axiomatically as solutions of a system of recursive functional equations, known as "form factor equations". We show that their solution can be replaced with the study of the representation theory of a novel algebra F(S). It is associated with a given two-particle S-matrix and has the following features: (i) It contains a double TTS algebra as a subalgebra. (ii) Form factors arise as thermal vector states over F(S) of temperature 1=2ß. The thermal ground states are in correspondence to the local operators of the QFT. (iii) The underlying `finite temperature structure' is indirectly related to the "Unruh effect" in Rindler spacetime. In F(S) it is manifest through modular structures (j; ffi) in the sense of algebraic QFT, which can be implemented explicitly in terms of the TTS generators. 1. Introduction and survey The form factor approach to integrable quantum field theories (QFTs) has several remarkable features. First it yiel...
Abstract We apply the nested algebraic Bethe ansatz to the models with g l ( 2 | 1 ) and g l ( 1 | 2 ) supersymmetry. We show that form factors of local operators in these models can be expressed in terms of the universal form factors. Our derivation is based on the use of the RTT -algebra only. It does not refer to any specific representation of this algebra. We obtain thus determinant representations for form factors of local operators in the cases where an explicit solution of the quantum inverse scattering problem is not known.
In this series of lectures given at KIAS in Nov 2018, I briefly introduce the concepts of integrability, at classical and quantum level. Then I elaborate on the factorised scattering theory emerging in 1+1 integrable QFT. The Thermdynamic Bethe Ansatz (TBA) is a well known method to get exact information from the scattering theory, in order to identify the underlying QFT. Here I illustrate how to get TBA in the case of the prototypical Sine-Gordon model, particularly interesting to show the techniques when the S-matrix is not diagonal.