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Characterization of wedge-local observables in integrable models with bound states

Karim Shedid Attifa

Published Dec 18, 2023• 0 citations
OpenAlex

Abstract

Integrable quantum models (IQMs) are a class of interacting quantum field theories in 1+1 dimensions in which the Hilbert space can be described as a Fock space with generalized statistics. Because of this Fock space description, any observables can be expanded in the creation- and annihilation operators with the coefficients of the expansions being the connected matrix elements. One example of such an observable in IQMs without bound states is the so-called wedge-local field (a free field analog with the interacting creation- and annihilation operators) which is wedge-local in the sense that it strongly commutes with its reflection when smeared with a function localized in the left wedge. In this thesis, we study wedge-local observables in integrable quantum models and the effect of bound states on the properties of the connected matrix elements of these observables. This allows us to shine light on the connection between the localization of an observable and the functional analytic properties of its connected matrix elements in models with critical phenomena, in this case models which exhibit bound states. In scalar integrable quantum models with bound states (e.g. the Bullough-Dodd model), we proof that the connected matrix elements of quadratic forms which satisfy a wedge-locality condition are boundary values of meromorphic functions, so-called form factors, which satisfy a number of properties. Conversely, we show that any expansion in such form factors defines a quadratic form which satisfies the wedge-locality condition. In contrast to models without bound states, such an expansion does not terminate due to a recursion relation between the form factors and the wedge-local field, a wedge-local operator which creates one-particle states from the vacuum, does not fall into the class of the characterization theorem. Further, we conjecture a candidate for the wedge-local field in the Z(4)-Ising model. This model is interesting because it has a ”higher-order” bound state structure and no established wedge-local field is known in such models. However, the conjectured candidate comes with some difficulties. Most notably, it is ill-defined as an unbounded operator.

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Integrable quantum models (IQMs) are a class of interacting quantum field theories in 1+1 dimensions in which the Hilbert space can be described as a Fock space with generalized statistics. Because of this Fock space description, any observables can be expanded in the creation- and annihilation operators with the coefficients of the expansions being the connected matrix elements. One example of such an observable in IQMs without bound states is the so-called wedge-local field (a free field analog with the interacting creation- and annihilation operators) which is wedge-local in the sense that it strongly commutes with its reflection when smeared with a function localized in the left wedge. In this thesis, we study wedge-local observables in integrable quantum models and the effect of bound states on the properties of the connected matrix elements of these observables. This allows us to shine light on the connection between the localization of an observable and the functional analytic properties of its connected matrix elements in models with critical phenomena, in this case models which exhibit bound states. In scalar integrable quantum models with bound states (e.g. the Bullough-Dodd model), we proof that the connected matrix elements of quadratic forms which satisfy a wedge-locality condition are boundary values of meromorphic functions, so-called form factors, which satisfy a number of properties. Conversely, we show that any expansion in such form factors defines a quadratic form which satisfies the wedge-locality condition. In contrast to models without bound states, such an expansion does not terminate due to a recursion relation between the form factors and the wedge-local field, a wedge-local operator which creates one-particle states from the vacuum, does not fall into the class of the characterization theorem. Further, we conjecture a candidate for the wedge-local field in the Z(4)-Ising model. This model is interesting because it has a ”higher-order” bound state structure and no established wedge-local field is known in such models. However, the conjectured candidate comes with some difficulties. Most notably, it is ill-defined as an unbounded operator.
Recently, large families of two-dimensional quantum field theories with factorizing S-matrices have been constructed by the operator-algebraic methods, by first showing the existence of observables localized in wedge-shaped regions. However, these constructions have been limited to the class of S-matrices whose components are analytic in rapidity in the physical strip. In this work, we construct candidates for observables in wedges for scalar factorizing S-matrices with poles in the physical strip and show that they weakly commute on a certain domain. We discuss some technical issues concerning further developments, especially the self-adjointness of the candidate operators here and strong commutativity between them.
We present a new viewpoint on the construction of pointlike local fields in integrable models of quantum field theory. As usual, we define these local observables by their form factors; but rather than exhibiting their n-point functions and verifying the Wightman axioms, we aim to establish them as closed operators affiliated with a net of local von Neumann algebras, which is defined indirectly via wedge-local quantities. We also investigate whether these fields have the Reeh-Schlieder property, and in which sense they generate the net of algebras. Our investigation focuses on scalar models without bound states. We establish sufficient criteria for the existence of averaged fields as closable operators, and complete the construction in the specific case of the massive Ising model.
In the context of constructing interacting quantum field theories in the operator-algebraic approach, wedge-local fields play an important role. After the work of Lechner to construct factorizing scattering matrix models with scalar S-matrices without bound states, we recently extended this construction to a class of models with a richer particle spectrum and which are believed to have bound states. These include the Z(N)-Ising model and the affine-Toda field theories. This construction is done by exhibiting wedge-local fields which arise as a deformation of Lechner's fields with the so called "bound state operator". I will review the main passages of this construction and explain the open problems.
In quantum field theory, the rigorous construction of local observables in the presence of nontrivial interaction is a crucial problem. In a class of integrable quantum field theories, a very abstract existence proof has recently been given by Lechner. We give an explicit characterization of these local observables in terms of the properties of the coefficient functions in an expansion by interacting creators and annihilators. Some results on the operator domains of these local observables are given. Using these, we constructed explicit examples of local observables in the quantum Ising model.
Abstract We present a new viewpoint on the construction of pointlike local fields in integrable models of quantum field theory. As usual, we define these local observables by their form factors; but rather than exhibiting their n-point functions and verifying the Wightman axioms, we aim to establish them as closed operators affiliated with a net of local von Neumann algebras, which is defined indirectly via wedge-local quantities. We also investigate whether these fields have the Reeh–Schlieder property, and in which sense they generate the net of algebras. Our investigation focuses on scalar models without bound states. We establish sufficient criteria for the existence of averaged fields as closable operators, and complete the construction in the specific case of the massive Ising model.
In the bootstrap approach to integrable quantum field theories in the (1+1)-dimensional Minkowski space, one conjectures the two-particle S-matrix and tries to study local observables. The massless sine-Gordon model is conjectured to be equivalent to the Thirring model, and its breather-breather S-matrix components (where the first breather corresponds to the scalar field of the sine-Gordon model) are closed under fusion. Yet, the residues of the poles in this breather-breather S-matrix have wrong signs and cannot be considered as a separate model. We find CDD factors which adjust the signs, so that the breather-breather S-matrix alone satisfies reasonable assumptions. Then we propose candidates for observables in wedge-shaped regions and prove their commutativity in the weak sense.