Algebraic constructive quantum field theory: Integrable models and deformation techniques - Lechner, Gandalf (Advances in Algebraic Quantum Field Theory, 2015)
Several related operator-algebraic constructions for quantum field theory models on Minkowski spacetime are reviewed. The common theme of these constructions is that of a Borchers triple, capturing the structure of observables localized in a Rindler wedge. After reviewing the abstract setting, we discuss in this framework (i) the construction of free field theories from standard pairs, (ii) the inverse scattering construction of integrable QFT models on two-dimensional Minkowski space, and (iii) the warped convolution deformation of QFT models in arbitrary dimension, inspired from non-commutative Minkowski space.
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Approximation numbers of weighted composition operators - Lechner, Gandalf; Li, D.; Queffelec, H.; Rodriguez-Piazza, Luis (Journal of Functional Analysis, 2018)
We study the approximation numbers of weighted composition operators f↦w⋅(f∘φ) on the Hardy space H2 on the unit disc. For general classes of such operators, upper and lower bounds on their approximation numbers are derived. For the special class of weighted lens map composition operators with specific weights, we show how much the weight w can improve the decay rate of the approximation numbers, and give sharp upper and lower bounds. These examples are motivated from applications to the analysis of relative commutants of special inclusions of von Neumann algebras appearing in quantum field theory (Borchers triples).
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Keywords: Approximation numbers; Composition operators; von Neumann algebras

Braid groups, the Yang–Baxter equation, and subfactors - Lechner, Gandalf (Snapshots of Modern Mathematics from Oberwolfach, 2021)
The Yang–Baxter equation is a fascinating equation that appears in many areas of physics and mathematics and is best represented diagramatically. This snapshot connects the mathematics of braiding hair to the Yang–Baxter equation and relates it to current research about systems of infinite dimensional algebras called "subfactors".<br

Classification of unitary R-matrices by braid group characters - Deadman, Tasarla; Lechner, Gandalf (Oberwolfach Workshop 2309a "Skew Braces and the Yang-Baxter Equation", 2023)

Construction of quantum field theories with factorizing S-matrices - Lechner, Gandalf (Communications in Mathematical Physics, 2008)
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.

Crossing symmetry and the crossing map - Correa da Silva, Ricardo; Giorgetti, Luca; Lechner, Gandalf (Reviews in Mathematical Physics, 2024)
We introduce and study the crossing map, a closed linear map of order four acting on operators on the tensor square of a given Hilbert space inspired by the crossing property of quantum field theory. This map turns out to be closely connected to Tomita-Takesaki modular theory; in particular its fixed points define endomorphisms of standard subspaces. We also explain how the crossing property is related to finite index subfactor planar algebras/Q-systems. In the latter case, the crossing map turns out to be a special case of the (unshaded) subfactor theoretical Fourier transform.<br

Deformations of half-sided modular inclusions and non-local chiral field theories - Lechner, Gandalf; Scotford, Charley (Communications in Mathematical Physics, 2022)
We construct explicit examples of half-sided modular inclusions ⊂ of von Neumann algebras with trivial relative commutants. After stating a general criterion for triviality of the relative commutant in terms of an algebra localized at infinity, we consider a second quantization inclusion ⊂ with large relative commutant and construct a one-parameter family κ⊂κ, κ≥0, of half-sided inclusions such that 0=, 0= and ′κ∩κ=ℂ1 for κ>0. The technique we use is an explicit deformation procedure (warped convolution), and we explain the relation of this result to the construction of chiral conformal quantum field theories on the real line and on the circle.<br

Deformations of operator algebras and the construction of quantum field theories - Lechner, Gandalf (XVIth International Congress on Mathematical Physics, 2010)
This note outlines a novel approach to the construction of quantum field theories on four-dimensional Minkowski space, which is based on operator-algebraic techniques. It is explained how Rieffel deformations can be used to deform quantum field theories in a manner which is compatible with Poincaré covariance and locality.
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Keywords: Deformation quantization; Deformations of algebras; Noncommutative geometry; Rieffel deformation

Deformations of Quantum Field Theories and Integrable Models - Lechner, Gandalf (Communications in Mathematical Physics, 2012)
Deformations of quantum field theories which preserve Poincaré covariance and localization in wedges are a novel tool in the analysis and construction of model theories. Here a general scenario for such deformations is discussed, and an infinite class of explicit examples is constructed on the Borchers-Uhlmann algebra underlying Wightman quantum field theory. These deformations exist independently of the space-time dimension, and contain the recently studied warped convolution deformation as a special case. In the special case of two-dimensional Minkowski space, they can be used to deform free field theories to integrable models with non-trivial S-matrix. © 2011 Springer-Verlag.
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Deformations of Quantum Field Theories on Spacetimes with Killing Vector Fields - Dappiaggi, Claudio; Lechner, Gandalf; Morfa-Morales, Eric (Communications in Mathematical Physics, 2011)
The recent construction and analysis of deformations of quantum field theories by warped convolutions is extended to a class of curved spacetimes. These spacetimes carry a family of wedge-like regions which share the essential causal properties of the Poincaré transforms of the Rindler wedge in Minkowski space. In the setting of deformed quantum field theories, they play the role of typical localization regions of quantum fields and observables. As a concrete example of such a procedure, the deformation of the free Dirac field is studied. © Springer-Verlag 2011.
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Fock representations of ZF algebras and R-matrices - Lechner, Gandalf; Scotford, Charley (Letters in Mathematical Physics, 2020)
A variation of the Zamolodchikov–Faddeev algebra over a finite-dimensional Hilbert space H and an involutive unitary R-Matrix S is studied. This algebra carries a natural vacuum state, and the corresponding Fock representation spaces FS(H) are shown to satisfy FS⊞R(H⊕ K) ≅ FS(H) ⊗ FR(K) , where S⊞ R is the box-sum of S (on H⊗ H) and R (on K⊗ K). This analysis generalises the well-known structure of Bose/Fermi Fock spaces and a recent result of Pennig. These representations are motivated from quantum field theory (short-distance scaling limits of integrable models).
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Keywords: Fock space; GNS construction; R-Matrices; Representation theory; Yang-Baxter; Zamolodchikov-Faddeev algebra

High Energy Bounds On Wave Operators - Bostelmann, Henning; Cadamuro, Daniela; Lechner, Gandalf (Journal of Operator Theory, 2021)
The wave operators W±(H1, H0) of two selfadjoint operators H0 and H1 are analyzed at asymptotic spectral values. Sufficient conditions for (Formula Presented) are given, where Pac projects onto the subspace of absolutely continuous spectrum of Hj and f is an unbounded function (f-boundedness), both in the case of trace-class perturbations and in terms of the high-energy behaviour of the boundary values of the resolvent of H0 (smooth method). Examples include f -boundedness for the perturbed polyharmonic operator and for Schrödinger operators with matrix-valued potentials. We discuss an application to the problem of quantum backflow.
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Keywords: high energy behaviour; Scattering theory; Schrödinger operator; wave operator

Inclusions of Standard Subspaces - Lechner, Gandalf; Correa da Silva, Ricardo (Mini-Workshop on Standard Subspaces in Quantum Field Theory and Representation Theory, 2023)

Inverse Scattering and Local Observable Algebras in Integrable Quantum Field Theories - Alazzawi, Sabina; Lechner, Gandalf (Communications in Mathematical Physics, 2017)
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 operator algebras and complex analysis, we identify sufficient criteria on S that imply the solution of the inverse scattering problem. These conditions are shown to be satisfied in particular by so-called diagonal S-matrices, but presumably also in other cases such as the O(N)-invariant nonlinear σ-models.
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KMS states on Z2-crossed products and twisted KMS functionals - Correa da Silva, Ricardo; Große, Johannes; Lechner, Gandalf (Annales Henri Poincaré, 2024)
KMS states on Z2-crossed products of unital C∗-algebras A are characterized in terms of KMS states and graded KMS functionals of A. These functionals are shown to describe the extensions of KMS states ω on A to the crossed product A⋊Z2 and can also be characterized by the twisted center of the von Neumann algebra generated by the GNS representation corresponding to ω. As a particular class of examples, KMS states on Z2-crossed products of CAR algebras with dynamics and grading given by Bogoliubov automorphisms are analysed in detail. In this case, one or two extremal KMS states are found depending on a Gibbs type condition involving the odd part of the absolute value of the Hamiltonian. As an application in mathematical physics, the extended field algebra of the Ising QFT is shown to be a Z2-crossed product of a CAR algebra which has a unique KMS stat

Linear hyperbolic PDEs with noncommutative time - Lechner, Gandalf; Verch, Rainer (Journal of Noncommutative Geometry, 2015)
Motivated by wave or Dirac equations on noncommutative deformations of Minkowski space, linear integro-differential equations of the form (D C λW)f = 0 are studied, where = is a normal or prenormal hyperbolic differential operator on Rn, λ ⊂ C is a coupling constant, and W is a regular integral operator with compactly supported kernel. In particular, W can be non-local in time, so that a Hamiltonian formulation is not possible. It is shown that for sufficiently small λ, the hyperbolic character of = is essentially preserved. Unique advanced/retarded fundamental solutions are constructed by means of a convergent expansion in λ, and the solution spaces are analyzed. It is shown that the acausal behavior of the solutions is well-controlled, but the Cauchy problem is ill-posed in general. Nonetheless, a scattering operator can be calculated which describes the effect of W on the space of solutions to Df = 0. It is also described how these structures occur in the context of noncommutative Minkowski space, and how the results obtained here can be used for the analysis of classical and quantum field theories on such spaces.
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Keywords: Linear hyperbolic PDEs; Noncommutative quantum field theory; Noncommutative time; Quantization

Localization in Nets of Standard Spaces - Lechner, Gandalf; Longo, Roberto (Communications in Mathematical Physics, 2015)
Starting from a real standard subspace of a Hilbert space and a representation of the translation group with natural properties, we construct and analyze for each endomorphism of this pair a local, translationally covariant net of standard subspaces, on the lightray and on two-dimensional Minkowski space. These nets share many features with low-dimensional quantum field theory, described by corresponding nets of von Neumann algebras.Generalizing a result of Longo and Witten to two dimensions and massive multiplicity free representations, we characterize these endomorphisms in terms of specific analytic functions. Such a characterization then allows us to analyze the corresponding nets of standard spaces, and in particular to compute their minimal localization length. The analogies and differences to the von Neumann algebraic situation are discussed.
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Modular nuclearity: A generally covariant perspective - Lechner, Gandalf; Sanders, Ko (Axioms, 2016)
A quantum field theory in its algebraic description may admit many irregular states. So far, selection criteria to distinguish physically reasonable states have been restricted to free fields (Hadamard condition) or to flat spacetimes (e.g., Buchholz-Wichmann nuclearity). We propose instead to use a modular ℓp-condition, which is an extension of a strengthened modular nuclearity condition to generally covariant theories. The modular nuclearity condition was previously introduced in Minkowski space, where it played an important role in constructive two dimensional algebraic QFT's. We show that our generally covariant extension of this condition makes sense for a vast range of theories, and that it behaves well under causal propagation and taking mixtures. In addition we show that our modular ℓp-condition holds for every quasi-free Hadamard state of a free scalar quantum field (regardless of mass or scalar curvature coupling). However, our condition is not equivalent to the Hadamard condition.
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Keywords: General covariance; Modular theory; Nuclear maps; Quantum field theory

Modular nuclearity and localization - Buchholz, Detlev; Lechner, Gandalf (Annales Henri Poincaré, 2004)
Within the algebraic setting of quantum field theory, a condition is given which implies that the intersection of algebras generated by field operators localized in wedge-shaped regions of the two-dimensional Minkowski space is non-trivial; in particular, there exist compactly localized operators in such theories which can be interpreted as local observables. The condition is based on spectral (nuclearity) properties of the modular operators affiliated with wedge algebras and the vacuum state and is of interest in the algebraic approach to the formfactor program, initiated by Schroer. It is illustrated here in a simple class of examples.
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Modular Structure and Inclusions of Twisted Araki-Woods Algebras - Correa da Silva, Ricardo; Lechner, Gandalf (Communications in Mathematical Physics, 2023)
In the general setting of twisted second quantization (including Bose/Fermi second quantization, S-symmetric Fock spaces, and full Fock spaces from free probability as special cases), von Neumann algebras on twisted Fock spaces are analyzed. These twisted Araki-Woods algebras L_T(H) depend on the twist operator T and a standard subspace H in the one-particle space. Under a compatibility assumption on T and H, it is proven that the Fock vacuum is cyclic and separating for L_T(H) if and only if T satisfies a standard subspace version of crossing symmetry and the Yang-Baxter equation (braid equation). In this case, the Tomita-Takesaki modular data are explicitly determined. Inclusions L_T(K)L_T(H) of twisted Araki-Woods algebras are analyzed in two cases: If the inclusion is half-sided modular and the twist satisfies a norm bound, it is shown to be singular. If the inclusion of underlying standard subspaces KH satisfies an L^2-nuclearity condition, L_T(K)L_T(H) has type III relative commutant for suitable twists T. Applications of these results to localization of observables in algebraic quantum field theory are discussed. <br

Noncommutative deformations of Wightman quantum field theories - Grosse, Harald; Lechner, Gandalf (Journal of High Energy Physics, 2008)
Quantum field theories on noncommutative Minkowski space are studied in a model-independent setting by treating the noncommutativity as a deformation of quantum field theories on commutative space. Starting from an arbitrary Wightman theory, we consider special vacuum representations of its Weyl-Wigner deformed counterpart. In such representations, the effect of the noncommutativity on the basic structures of Wightman theory, in particular the covariance, locality and regularity properties of the fields, the structure of the Wightman functions, and the commutative limit, is analyzed. Despite the nonlocal structure introduced by the noncommutativity, the deformed quantum fields can still be localized in certain wedge-shaped regions, and may therefore be used to compute noncommutative corrections to two-particle S-matrix elements.
Keywords: non-commutative geometry; field theories in higher dimensions; nonperturbative effects

On the Equivalence of Two Deformation Schemes in Quantum Field Theory - Lechner, Gandalf; Schlemmer, Jan; Tanimoto, Yoh (Letters in Mathematical Physics, 2013)
Two recent deformation schemes for quantum field theories on two-dimensional Minkowski space, making use of deformed field operators and Longo-Witten endomorphisms, respectively, are shown to be equivalent. © 2012 The Author(s).
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Keywords: deformations of quantum field theories; modular theory; two-dimensional models

On the existence of local observables in theories with a factorizing S-matrix - Lechner, Gandalf (Journal of Physics A: Mathematical and General, 2005)
The scaling limit of the two-dimensional Ising model above the critical temperature is considered as an example for relativistic quantum theories on two-dimensional Minkowski space exhibiting a factorizing S-matrix. In this model, a recently proposed criterion for the existence of local quantum field theories with a prescribed factorizing scattering matrix is verified, thereby establishing a new constructive approach to two-dimensional quantum field theory in a particular example. The existence proof is accomplished by analysing the nuclearity properties of certain specific subsets of fermionic Fock spaces, and yields as a byproduct also a verification of the energy nuclearity condition of Buchholz and Wichmann in models of free fermions in four spacetime dimensions.
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Operator-Algebraic Constructions of QFT Models in Two Dimensions - Cadamuro, Daniela; Lechner, Gandalf (Encyclopedia of Mathematical Physics (Second Edition), 2024)
This contribution to the Encyclopedia of Mathematical Physics provides an overview of some quantum field theory models on two-dimensional Minkowski space that have been constructed in the framework of algebraic quantum field theory. A variety of different construction techniques is explained, and the role of the two-dimensionality of the considered spacetime in these constructions is highlighted.<br

Polarization-free quantum fields and interaction - Lechner, Gandalf (Letters in Mathematical Physics, 2003)
A new approach to the inverse scattering problem proposed by Schroer, is applied to two-dimensional integrable quantum field theories. For any two-particle S-matrix S{_2 which is analytic in the physical sheet, quantum fields are constructed which are localizable in wedge-shaped regions of Minkowski space and whose two-particle scattering is described by the given §{_2. These fields are polarization-free in the sense that they create one-particle states from the vacuum without polarization clouds. Thus they provide examples of temperate polarization-free generators in the presence of non-trivial interaction.

Quantum backflow and scattering - Bostelmann, Henning; Cadamuro, Daniela; Lechner, Gandalf (Physical Review A, 2017)
Backflow is the phenomenon that the probability current of a quantum particle on the line can flow in the direction opposite to its momentum. In this paper, previous investigations of backflow, pertaining to interaction-free dynamics or purely kinematical aspects, are extended to scattering situations in short-range potentials. It is shown that backflow is a universal quantum effect which exists in any such potential, and is always of bounded spatial extent in a specific sense. The effects of reflection and transmission processes on backflow are investigated, both analytically for general potentials and numerically in various concrete examples.
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Recent Mathematical Developments in Quantum Field Theory - Abdesselam, Abdelmalek; Hollands, Stefan; Kopper, Christoph; Lechner, Gandalf (Oberwolfach reports, 2016)
This workshop has focused on three areas in mathematical quantum field theory and their interrelations: 1) conformal field theory, 2) constructions of interacting models of quantum field theory by various methods, and 3) several approaches studying the interplay of quantum field theory and gravity.

Scaling limits of integrable quantum field theories - Bostelmann, Henning; Lechner, Gandalf; Morsella, Gerardo (Reviews in Mathematical Physics, 2011)
Short distance scaling limits of a class of integrable models on two-dimensional Minkowski space are considered in the algebraic framework of quantum field theory. Making use of the wedge-local quantum fields generating these models, it is shown that massless scaling limit theories exist, and decompose into (twisted) tensor products of chiral, translation-dilation covariant field theories. On the subspace which is generated from the vacuum by the observables localized in finite light ray intervals, this symmetry can be extended to the Möbius group. The structure of the interval-localized algebras in the chiral models is discussed in two explicit examples. © 2011 World Scientific Publishing Company.
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Keywords: conformal symmetry; operator algebras; Quantum field theory; scaling limits

SO(d, 1) -Invariant Yang–Baxter Operators and the dS/CFT Correspondence - Hollands, Stefan; Lechner, Gandalf (Communications in Mathematical Physics, 2018)
We propose a model for the dS/CFT correspondence. The model is constructed in terms of a “Yang–Baxter operator” R for unitary representations of the de Sitter group SO(d, 1). This R-operator is shown to satisfy the Yang–Baxter equation, unitarity, as well as certain analyticity relations, including in particular a crossing symmetry. With the aid of this operator we construct: (a) a chiral (light-ray) conformal quantum field theory whose internal degrees of freedom transform under the given unitary representation of SO(d, 1). By analogy with the O(N) non-linear sigma model, this chiral CFT can be viewed as propagating in a de Sitter spacetime. (b) A (non-unitary) Euclidean conformal quantum field theory on Rd - 1, where SO(d, 1) now acts by conformal transformations in (Euclidean) spacetime. These two theories can be viewed as dual to each other if we interpret Rd - 1 as conformal infinity of de Sitter spacetime. Our constructions use semi-local generator fields defined in terms of R and abstract methods from operator algebras.
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Standard Subspaces in Quantum Field Theory and Representation Theory - Adamo, Maria Stella; Lechner, Gandalf; Longo, Roberto; Neeb, Karl Hermann (2024)
Real standard subspaces of complex Hilbert spaces are long known to provide the right language for Tomita-Takesaki modular theory of von Neumann algebras. In recent years they have also become an object of prominent interest in mathematical quantum field theory (QFT) and unitary representation theory of Lie groups. This workshop brought together mathematicians and physicists working with standard subspaces, particularly in QFT (construction of QFT models, characterization of entropy, information-theoretic aspects), nets of standard subspaces on causal homogeneous spaces and aspects of reflection positivity and euclidean models related to standard subspaces and modular theory.<br

Strict deformation quantization of locally convex algebras and modules - Lechner, Gandalf; Waldmann, Stefan (Journal of Geometry and Physics, 2016)
In this work various symbol spaces with values in a sequentially complete locally convex vector space are introduced and discussed. They are used to define vector-valued oscillatory integrals which allow to extend Rieffel's strict deformation quantization to the framework of sequentially complete locally convex algebras and modules with separately continuous products and module structures, making use of polynomially bounded actions of Rn. Several well-known integral formulas for star products are shown to fit into this general setting, and a new class of examples involving compactly supported Rn-actions on Rn is constructed.
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Keywords: Deformation quantization; Locally convex algebra; Rieffel construction; Star product

Thermal equilibrium states for quantum fields on non-commutative spacetimes - Lechner, Gandalf; Schlemmer, Jan (Quantum Mathematical Physics, 2016)
Fully Poincaré covariant quantum field theories on non-commutative Moyal Minkowski spacetime so far have been considered in their vacuum representations, i.e. at zero temperature. Here we report on work in progress regarding their thermal representations, corresponding to physical states at non-zero temperature, which turn out to be markedly different from both, thermal representations of quantum field theory on commutative Minkowski spacetime, and such representations of non-covariant quantum field theory on Moyal Minkowski space with a fixed deformation matrix.
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Keywords: Quantum field theory on non-commutative spaces; Thermal equilibrium (KMS) states

Towards an Operator-Algebraic Construction of Integrable Global Gauge Theories - Lechner, Gandalf; Schuetzenhofer, Christian (Annales Henri Poincaré, 2014)
The recent construction of integrable quantum field theories on two-dimensional Minkowski space by operator-algebraic methods is extended to models with a richer particle spectrum, including finitely many massive particle species transforming under a global gauge group. Starting from a two-particle S-matrix satisfying the usual requirements (unitarity, Yang-Baxter equation, Poincaré and gauge invariance, crossing symmetry,...), a pair of relatively wedge-local quantum fields is constructed which determines the field net of the model. Although the verification of the modular nuclearity condition as a criterion for the existence of local fields is not carried out in this paper, arguments are presented that suggest it holds in typical examples such as non-linear O(N) σ-models. It is also shown that for all models complying with this condition, the presented construction solves the inverse scattering problem by recovering the S-matrix from the model via Haag-Ruelle scattering theory, and a proof of asymptotic completeness is given. © 2013 Springer Basel.
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Towards the construction of quantum field theories from a factorizing S-matrix - Lechner, Gandalf (Rigorous Quantum Field Theory, 2007)
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.
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Twisted Araki-Woods Algebras, the Yang-Baxter Equation, and quantum field theory - Lechner, Gandalf (Geometric Methods in Physics XL, 2024)

Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories - Buchholz, Detlev; Lechner, Gandalf; Summers, Stephen J. (Communications in Mathematical Physics, 2011)
Warped convolutions of operators were recently introduced in the algebraic framework of quantum physics as a new constructive tool. It is shown here that these convolutions provide isometric representations of Rieffel's strict deformations of C*-dynamical systems with automorphic actions of ℝn, whenever the latter are presented in a covariant representation. Moreover, the device can be used for the deformation of relativistic quantum field theories by adjusting the convolutions to the geometry of Minkowski space. The resulting deformed theories still comply with pertinent physical principles and their Tomita-Takesaki modular data coincide with those of the undeformed theory; but they are in general inequivalent to the undeformed theory and exhibit different physical interpretations. © 2010 The Author(s).
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Wedge-local quantum fields and noncommutative Minkowski space - Grosse, Harald; Lechner, Gandalf (Journal of High Energy Physics, 2007)
Within the setting of a recently proposed model of quantum fields on noncommutative Minkowski space, the consequences of the consistent application of the proper, untwisted Poincare group as the symmetry group are investigated. The emergent model contains an infinite family of fields which are labelled by different noncommutativity parameters, and related to each other by Lorentz transformations. The relative localization properties of these fields are investigated, and it is shown that to each field one can assign a wedge-shaped localization region in Minkowski space. This assignment is consistent with the principles of covariance and locality, i.e. fields localized in spacelike separated wedges commute.Regarding the model as a non-local, but wedge-local, quantum field theory on ordinary (commutative) Minkowski spacetime, it is possible to determine two-particle S-matrix elements, which turn out to be non-trivial. Some partial negative results concerning the existence of observables with sharper localization properties are also obtained.
Keywords: non-commutative geometry; space-time symmetries; field theories in higher dimensions; integrable field theories

Wick rotation for quantum field theories on degenerate Moyal space(-time) - Grosse, Harald; Lechner, Gandalf; Ludwig, Thomas; Verch, Rainer (Journal of Mathematical Physics, 2013)
In this paper the connection between quantum field theories on flat noncommutative space(-times) in Euclidean and Lorentzian signature is studied for the case that time is still commutative. By making use of the algebraic framework of quantum field theory and an analytic continuation of the symmetry groups which are compatible with the structure of Moyal space, a general correspondence between field theories on Euclidean space satisfying a time zero condition and quantum field theories on Moyal Minkowski space is presented ("Wick rotation"). It is then shown that field theories transferred to Moyal space(-time) by Rieffel deformation and warped convolution fit into this framework, and that the processes of Wick rotation and deformation commute. © 2013 American Institute of Physics.
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Yang–Baxter endomorphisms - Conti, Roberto; Lechner, Gandalf (Journal of the London Mathematical Society-Second Series, 2021)
Every unitary solution of the Yang–Baxter equation (R-matrix) in dimension (Formula presented.) can be viewed as a unitary element of the Cuntz algebra (Formula presented.) and as such defines an endomorphism of (Formula presented.). These Yang–Baxter endomorphisms restrict and extend to several other (Formula presented.) - and von Neumann algebras, and furthermore define a II (Formula presented.) factor associated with an extremal character of the infinite braid group. This paper is devoted to a detailed study of such Yang–Baxter endomorphisms. We discuss the relative commutants of the subfactors induced by Yang–Baxter endomorphisms, a new perspective on algebraic operations on R-matrices such as tensor products and cabling powers, the characters of the infinite braid group defined by R-matrices, and ergodicity properties. This also yields new concrete information on partial traces and spectra of R-matrices.
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Keywords: 16T25; 16W20 (primary); 46L10; 47L90 (secondary)

Yang-Baxter representations of the infinite symmetric group - Lechner, Gandalf; Pennig, Ulrich; Wood, Simon (Advances in Mathematics, 2019)
Every unitary involutive solution of the quantum Yang-Baxter equation (R-matrix) defines an extremal character and a representation of the infinite symmetric group S∞. We give a complete classification of all such Yang-Baxter characters and determine which extremal characters of S∞ are of Yang-Baxter form. Calling two involutive R-matrices equivalent if they have the same character and the same dimension, we show that equivalence classes can be parameterized by pairs of Young diagrams, and construct an explicit normal form R-matrix for each class. Using operator-algebraic techniques (subfactors), we prove that two R-matrices are equivalent if and only if they have similar partial traces. Furthermore, we describe the algebraic structure of the set of equivalence classes of all involutive R-matrices, and discuss several families of examples. These include unitary Yang-Baxter representations of the Temperley-Lieb algebra at loop parameter δ=2, which can be completely classified in terms of their trace and dimension.
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Keywords: Representations of infinite symmetric groups; Subfactors; Yang-Baxter equations