A new approach to the construction of interacting quantum field theories on two-dimensional Minkowski space is discussed. In this program, models are obtained from a prescribed factorizing S-matrix in two steps. At first, quantum fields which are localized in infinitely extended, wedge-shaped regions of Minkowski space are constructed explicitly. In the second step, local observables are analyzed with operator-algebraic techniques, in particular by using the modular nuclearity condition of Buchholz, d'Antoni and Longo.
Besides a model-independent result regarding the Reeh-Schlieder property of the vacuum in this framework, an infinite class of quantum field theoretic models with non-trivial interaction is constructed. This construction completes a program initiated by Schroer in a large family of theories, a particular example being the Sinh-Gordon model. The crucial problem of establishing the existence of local observables in these models is solved by verifying the modular nuclearity condition, which here amounts to a condition on analytic properties of form factors of observables localized in wedge regions.
It is shown that the constructed models solve the inverse scattering problem for the considered class of S-matrices. Moreover, a proof of asymptotic completeness is obtained by explicitly computing total sets of scattering states. The structure of these collision states is found to be in agreement with the heuristic formulae underlying the Zamolodchikov-Faddeev algebra.
We calculate factorizing twists in evaluation representations of the quantum affine algebra U_q(\hat sl_2). From the factorizing twists we derive a representation independent expression of the R-matrices of U_q(\hat sl_2). Comparing with the corresponding quantities for the Yangian Y(sl_2), it is shown that the U_q(\hat sl_2) results can be obtained by `replacing numbers by q-numbers'. Conversely, the limit q -> 1 exists in representations of U_q(\hat sl_2) and both the factorizing twists and the R-matrices of the Yangian Y(sl_2) are recovered in this limit.
Using notation inherited from the six-vertex model, we construct diagrams that represent the action of the factorizing $F$-matrices associated to the finite length XXZ spin-1/2 chain. We prove that these $F$-matrices factorize the tensor $R^σ_{1... n}$ corresponding with elements of the permutation group. We consider in particular the diagram for the tensor $R^{σ_c}_{1... n}$, which cyclically permutes the spin chain. This leads us to a diagrammatic construction of the local spin operators $S_i^{\pm}$ and $S_i^{z}$ in terms of the monodromy matrix operators.
A recently proposed criterion for the existence of local quantum fields with a prescribed factorizing scattering matrix is verified in a non-trivial model, thereby establishing a new constructive approach to quantum field theory in a particular example. The existence proof is accomplished by analyzing nuclearity properties of certain specific subsets of Fermionic Fock spaces.
We consider quantum spin chains arising from $N$-fold tensor products of the fundamental evaluation representations of $Y(sl_n)$ and $U_q(\hat{sl_n})$. Using the partial $F$-matrix formalism from the seminal work of Maillet and Sanchez de Santos, we derive a completely factorized expression for the $F$-matrix of such models and prove its equivalence to the expression obtained by Albert, Boos, Flume and Ruhlig. A new relation between the $F$-matrices and the Bethe eigenvectors of these spin chains is given.
Starting from a given factorizing S-matrix $S$ in two space-time dimensions, we review a novel strategy to rigorously construct quantum field theories describing particles whose interaction is governed by $S$. The construction procedure is divided into two main steps: Firstly certain semi-local Wightman fields are introduced by means of Zamolodchikov's algebra. The second step consists in proving the existence of local observables in these models. As a new result, an intermediate step in the existence problem is taken by proving the modular compactness condition for wedge algebras.
The subject of this thesis is a novel construction method for interacting relativistic quantum field theories on two-dimensional Minkowski space. The input in this construction is not a classical Lagrangian, but rather a prescribed factorizing S-matrix, i.e. the inverse scattering problem for such quantum field theories is studied. For a large class of factorizing S-matrices, certain associated quantum fields, which are localized in wedge-shaped regions of Minkowski space, are constructed explicitely. With the help of these fields, the local observable content of the corresponding model is def…
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.
The subject of this thesis is the rigorous construction of QFT models with nontrivial interaction. Two different approaches in the framework of AQFT are discussed. On the one hand, an inverse scattering problem is considered. A given factorizing S-matrix is thereby taken as the starting point of the construction. The particle spectrum taken into account involves an arbitrary number of massive particle species, transforming under a global gauge group. Starting from known wedge-local auxiliary fields, the transition to local theories is shown. To his end, we make use of the modular nuclearity co…
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 concernin…
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…
Considering the model of a scalar massive Fermion, it is shown that by means of deformation techniques it is possible to obtain all integrable quantum field theoretic models on two-dimensional Minkowski space which have factorizing S-matrices corresponding to two-particle scattering functions S_2 satisfying S_2(0) = -1. Among these models there is for example the Sinh-Gordon model. Our analysis provides a complement to recent developments regarding deformations of quantum field theories. The deformed model is investigated also in higher dimensions. In particular, locality and covariance proper…
In a large class of factorizing scattering models, we construct candidates for the local energy density on the one-particle level starting from first principles, namely from the abstract properties of the energy density. We find that the form of the energy density at one-particle level can be fixed up to a polynomial function of energy. On the level of one-particle states, we also prove the existence of lower bounds for local averages of the energy density, and show that such inequalities can fix the form of the energy density uniquely in certain models.
We consider scalar two-dimensional quantum field theories with the factorizing S-matrix which has poles in the physical strip. In our previous work, we introduced the bound state operators and constructed candidate operators for observables in wedges. Under some additional assumptions on the S-matrix, we show that, in order to obtain their strong commutativity, it is enough to prove the essential self-adjointness of the sum of the left and right bound state operators. This essential self-adjointness is shown up to the two-particle component.
We present a solution method for the inverse scattering problem for integrable two-dimensional relativistic quantum field theories, specified in terms of a given massive single particle spectrum and a factorizing S-matrix. An arbitrary number of massive particles transforming under an arbitrary compact global gauge group is allowed, thereby generalizing previous constructions of scalar theories. The two-particle S-matrix $S$ is assumed to be an analytic solution of the Yang-Baxter equation with standard properties, including unitarity, TCP invariance, and crossing symmetry. Using methods from …
We study representation theory of Drinfel'd twists, in terms of what we call F matrices, associated to finite dimensional irreducible modules of quantum affine algebras, and which factorize the corresponding (unitary) R matrices. We construct explicitly such factorizing F matrices for irreducible tensor products of the fundamental representations of the quantum affine algebra sl2 and its associated Yangian. We then apply these constructions to the XXX and XXZ quantum spins chains of finite length in the framework of the Algebraic Bethe Ansatz.
The Drinfeld twists or factorizing F-matrices for the open XXZ spin chain with non-diagonal boundary terms are constructed. It is shown that in the F-basis the two sets of pseudo-particle creation operators simultaneously take completely symmetric and polarization free form. The explicit and completely symmetric expressions of the two sets of Bethe states of the model are obtained.
We extend the previously introduced constructive modular method to nonperturbative QFT. In particular the relevance of the concept of ``quantum localization'' (via intersection of algebras) versus classical locality (via support properties of test functions) is explained in detail, the wedge algebras are constructed rigorously and the formal aspects of double cone algebras for d=1+1 factorizing theories are determined. The well-known on-shell crossing symmetry of the S-Matrix and of formfactors (cyclicity relation) in such theories is intimately related to the KMS properties of new quantum-loc…
Many Lagrangians of physical theories can be expressed as eigenvalues of certain, relatively simple, matrices involving Dirac gamma matrices. We give concrete examples for Lagrangian corresponding to a point particle coupled to electromagnetic field, electrodynamics, nonabelian gauge theories, extended objects and gravity. We also discuss (in case of a point particle) what are the implications of the least action principle applied to matrix Lagrangians.
We construct the Drinfeld twists (factorizing $F$-matrices) of the $gl(m|n)$-invariant fermion model. Completely symmetric representation of the pseudo-particle creation operators of the model are obtained in the basis provided by the $F$-matrix (the $F$-basis). We resolve the hierarchy of the nested Bethe vectors in the $F$-basis for the $gl(m|n)$ supersymmetric model.
We investigate the question which communication tasks can be accomplished within a given operational theory. The concrete task is to find out which communication matrices have a prepare-and-measure implementation with states and measurement from a given theory. To set a general framework for this question we develop the ultraweak matrix majorization in the set of communication matrices. This preorder gives us means to determine when one communication task is more difficult than another. Furthermore, we introduce several monotones which can be used to compare and characterize the communication …
We explain how Gaussian integrals over ensemble of complex matrices with source matrices generate Hurwitz numbers of the most general type, namely, Hurwitz numbers with arbitrary orientable or non-orientable base surface and arbitrary profiles at branch points. We use the Feynman diagram approach. The connections with topological theories and also with certain classical and quantum integrable theories in particular with Witten's description of two-dimensional gauge theory are shown.
We construct the Drinfeld twists (or factorizing $F$-matrices) of the supersymmetric model associated with quantum superalgebra $U_q(gl(m|n))$, and obtain the completely symmetric representations of the creation operators of the model in the $F$-basis provided by the $F$-matrix. As an application of our general results, we present the explicit expressions of the Bethe vectors in the $F$-basis for the $U_q(gl(2|1))$-model (the quantum t-J model).
Factorisation of symmetric matrices and applications in gravitational theories
Center for Mathematical Analysis, Geometry and Dynamical Systems, Department of Mathematics, Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等技术学院数学系数学分析、几何与动力系统中心）
Various features of the even order poles appearing in the S-matrices of simply-laced affine Toda field theories are analysed in some detail. In particular, the coefficients of first- and second-order singularities appearing in the Laurent expansion of the S-matrix around a general $2N^{\rm th}$ order pole are derived in a universal way using perturbation theory at one loop. We show how to cut loop diagrams contributing to the pole into particular products of tree-level graphs that depend on the on-shell geometry of the loop; in this way, we recover the coefficients of the Laurent expansion aro…
We derive the gravity duals of noncommutative gauge theories from the Yang-Baxter sigma model description of the AdS_5xS^5 superstring with classical r-matrices. The corresponding classical r-matrices are 1) solutions of the classical Yang-Baxter equation (CYBE), 2) skew-symmetric, 3) nilpotent and 4) abelian. Hence these should be called abelian Jordanian deformations. As a result, the gravity duals are shown to be integrable deformations of AdS_5xS^5. Then, abelian twists of AdS_5 are also investigated. These results provide a support for the gravity/CYBE correspondence proposed in arXiv:140…
In the following paper, which is based on the authors PhD thesis submitted to Imperial College London, we explore the applicability of Yangian symmetry to various integrable models, in particular, in relation with S-matrices. One of the main themes in this work is that, after a careful study of the mathematics of the symmetry algebras one finds that in an integrable model, one can directly reconstruct S-matrices just from the algebra. It has been known for a long time that S-matrices in integrable models are fixed by symmetry. However, Lie algebra symmetry, the Yang-Baxter equation, crossing a…
We present a general formalism to investigate the integrable properties of a large class of non-ultralocal models which in principle allows the construction of the corresponding lattice versions. Our main motivation comes from the su(1|1) subsector of the string theory on AdS_5 x S^5 in the uniform gauge, where such type of non-ultralocality appears in the resulting Alday-Arutyunov-Frolov (AAF) model. We first show how to account for the second derivative of the delta function in the Lax algebra of the AAF model by modifying Maillet's r- and s-matrices formalism, and derive a well-defined alge…
S-matrix is one of the fundamental observables of the quantum theory of relativistic particles. There have been attempts to understand the quantum dynamics of relativistic particles abstractly in terms of S-matrix bypassing a Lagrangian formulation of quantum field theory. Equivalently, the space of possible S-matrices defines an abstract theory space. This thesis examines how to constrain the spectrum of physical theories in the theory space using the basic physical requirements of Poincare invariance, quantum unitarity, and causality. The thesis discusses two distinct but related ways of suc…
Starting from the fusion rules for the algebra $SO(5)_2$ we construct one-dimensional lattice models of interacting anyons with commuting transfer matrices of `interactions round the face' (IRF) type. The conserved topological charges of the anyon chain are recovered from the transfer matrices in the limit of large spectral parameter. The properties of the models in the thermodynamic limit and the low energy excitations are studied using Bethe ansatz methods. Two of the anyon models are critical at zero temperature. From the analysis of the finite size spectrum we find that they are effectivel…
We study solutions of the Thermodynamic Bethe Ansatz equations for relativistic theories defined by the factorizable $S$-matrix of an integrable QFT deformed by CDD factors. Such $S$-matrices appear under generalized TTbar deformations of integrable QFT by special irrelevant operators. The TBA equations, of course, determine the ground state energy $E(R)$ of the finite-size system, with the spatial coordinate compactified on a circle of circumference $R$. We limit attention to theories involving just one kind of stable particles, and consider deformations of the trivial (free fermion or boson)…
We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categories are obtained. We also show that the irreducible representations of orbifolds of rank one lattice vertex operator algebras give rise to unitary modular categories and determine the corresponding modular matrices, which has been conjectured for some time.
