Gandalf Lechner
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Gandalf Lechner has a total of 41 co-authors. Here are the top co-authors by number of shared publications:
- Henning Bostelmann
Fachbereich Ingenieur- und Naturwissenschaften, Hochschule Merseburg, Merseburg, Germany
5 shared publications - Ricardo Correa da Silva
Department Mathematik, FAU Erlangen-Nürnberg, Erlangen, Germany
5 shared publications - Charley Scotford
School of Mathematics, Cardiff University, Cardiff, UK
3 shared publications Harald Grosse
University of Vienna 1 Department of Physics 1090 Vienna, Austria
3 shared publications- Christian Schützenhofer
Vienna University
2 shared publications - Daniela Cadamuro
Zentrum Mathematik M5
2 shared publications
Latest Publications
- Open Access
Causal quantum-mechanical localization observables in lattices of real projections
Quantum-mechanical observables for spatial and spacetime localization are considered from a lattice-theoretic perspective. It is shown that when replacing the lattice of all complex orthogonal projections underlying the Born rule by the lattice of real linear projections with symplectic complementation, the well-known No-Go theorems of Hegerfeldt and Malament no longer apply: Causal and Poincaré covariant localization observables exist. In this setting, several features of quantum field theory, such as Lorentz symmetry and modular localization, emerge automatically. In the case of a particle described by a massive positive energy representation of the Poincaré group, the Brunetti-Guido-Longo map defines a spacetime localization observable that is unique under some natural further assumptions. Regarding possible probabilistic interpretations of such a structure, a Gleason theorem and a cluster theorem for symplectic complements are established. These imply that evaluating such localization observables in states yields a fuzzy probability measure that fails to be a measure because it is not additive. However, for separation scales that are large in comparison to the Compton wavelength, the emerging modular localization picture is essentially additive and approximates the one of Newton-Wigner.
Open AccessCausal quantum-mechanical localization observables in lattices of real projections
Quantum-mechanical observables for spatial and spacetime localization are considered from a lattice-theoretic perspective. It is shown that when replacing the lattice of all complex orthogonal projections underlying the Born rule by the lattice of real linear projections with symplectic complementation, the well-known No-Go theorems of Hegerfeldt and Malament no longer apply: Causal and Poincaré covariant localization observables exist. In this setting, several features of quantum field theory, such as Lorentz symmetry and modular localization, emerge automatically. In the case of a particle described by a massive positive energy representation of the Poincaré group, the Brunetti-Guido-Longo map defines a spacetime localization observable that is unique under some natural further assumptions. Regarding possible probabilistic interpretations of such a structure, a Gleason theorem and a cluster theorem for symplectic complements are established. These imply that evaluating such localization observables in states yields a fuzzy probability measure that fails to be a measure because it is not additive. However, for separation scales that are large in comparison to the Compton wavelength, the emerging modular localization picture is essentially additive and approximates the one of Newton-Wigner.
- Open Access
- 3 October 2025
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 406 (11) , 1-34
Inclusions of Standard Subspaces
Standard subspaces are closed real subspaces of a complex Hilbert space that appear naturally in Tomita–Takesaki modular theory and its applications to quantum field theory. In this article, inclusions of standard subspaces are studied independently of von Neumann algebras. Several new methods for their investigation are developed, related to polarizers, Gelfand triples defined by modular data, and extensions of modular operators. A particular class of examples that arises from the fundamental irreducible building block of a conformal field theory on the line is analyzed in detail.
- 3 October 2025
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 406 (11) , 1-34
Open AccessInclusions of Standard Subspaces
Standard subspaces are closed real subspaces of a complex Hilbert space that appear naturally in Tomita–Takesaki modular theory and its applications to quantum field theory. In this article, inclusions of standard subspaces are studied independently of von Neumann algebras. Several new methods for their investigation are developed, related to polarizers, Gelfand triples defined by modular data, and extensions of modular operators. A particular class of examples that arises from the fundamental irreducible building block of a conformal field theory on the line is analyzed in detail.
- Open Access
- 15 December 2024
- Published by Springer Nature in Annales Henri Poincare
- Vol. 26 (11) , 4109-4139
KMS States on ${\mathbb {Z}}_2$-Crossed Products and Twisted KMS FunctionalsKMS states on $${\mathbb {Z}}_2$$ -crossed products of unital $$C^*$$ -algebras $${\mathcal {A}}$$ are characterized in terms of KMS states and twisted KMS functionals of $${\mathcal {A}}$$ . These functionals are shown to describe the extensions of KMS states $$\omega $$ on $${\mathcal {A}}$$ to the crossed product $${\mathcal {A}} \rtimes {\mathbb {Z}}_2$$ and can also be characterized by the twisted center of the von Neumann algebra generated by the GNS representation corresponding to $$\omega $$ . As a particular class of examples, KMS states on $${\mathbb {Z}}_2$$ -crossed products of CAR algebras with dynamics and grading given by Bogoliubov automorphisms are analyzed 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 $${\mathbb {Z}}_2$$ -crossed product of a CAR algebra which has a unique KMS state.- 15 December 2024
- Published by Springer Nature in Annales Henri Poincare
- Vol. 26 (11) , 4109-4139
Open AccessKMS States on ${\mathbb {Z}}_2$-Crossed Products and Twisted KMS FunctionalsKMS states on $${\mathbb {Z}}_2$$ -crossed products of unital $$C^*$$ -algebras $${\mathcal {A}}$$ are characterized in terms of KMS states and twisted KMS functionals of $${\mathcal {A}}$$ . These functionals are shown to describe the extensions of KMS states $$\omega $$ on $${\mathcal {A}}$$ to the crossed product $${\mathcal {A}} \rtimes {\mathbb {Z}}_2$$ and can also be characterized by the twisted center of the von Neumann algebra generated by the GNS representation corresponding to $$\omega $$ . As a particular class of examples, KMS states on $${\mathbb {Z}}_2$$ -crossed products of CAR algebras with dynamics and grading given by Bogoliubov automorphisms are analyzed 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 $${\mathbb {Z}}_2$$ -crossed product of a CAR algebra which has a unique KMS state. - 9 November 2024
- Published by World Scientific Pub Co Pte Ltd in Reviews in Mathematical Physics
- Vol. 38 (08)
Crossing symmetry and the crossing map
We introduce and study the crossing map, a closed linear map acting on operators on the tensor square of a given Hilbert space that is inspired by the crossing property of quantum field theory. This map turns out to be closely connected to Tomita–Takesaki modular theory. In particular, crossing symmetric operators, namely those operators that are mapped to their adjoints by the crossing map, define endomorphisms of standard subspaces. Conversely, such endomorphisms can be integrated to crossing symmetric operators. We also investigate the relation between crossing symmetry and natural compatibility conditions with respect to unitary representations of certain symmetry groups, and furthermore introduce a generalized crossing map defined by a real object in an abstract [ ]-tensor category, not necessarily consisting of Hilbert spaces and linear maps. This latter crossing map turns out to be closely related to the (unshaded, finite-index) subfactor theoretical Fourier transform. Lastly, we provide families of solutions of the crossing symmetry equation, solving in addition the categorical Yang–Baxter equation, associated with an arbitrary Q-system.
- 9 November 2024
- Published by World Scientific Pub Co Pte Ltd in Reviews in Mathematical Physics
- Vol. 38 (08)
Crossing symmetry and the crossing map
We introduce and study the crossing map, a closed linear map acting on operators on the tensor square of a given Hilbert space that is inspired by the crossing property of quantum field theory. This map turns out to be closely connected to Tomita–Takesaki modular theory. In particular, crossing symmetric operators, namely those operators that are mapped to their adjoints by the crossing map, define endomorphisms of standard subspaces. Conversely, such endomorphisms can be integrated to crossing symmetric operators. We also investigate the relation between crossing symmetry and natural compatibility conditions with respect to unitary representations of certain symmetry groups, and furthermore introduce a generalized crossing map defined by a real object in an abstract [ ]-tensor category, not necessarily consisting of Hilbert spaces and linear maps. This latter crossing map turns out to be closely related to the (unshaded, finite-index) subfactor theoretical Fourier transform. Lastly, we provide families of solutions of the crossing symmetry equation, solving in addition the categorical Yang–Baxter equation, associated with an arbitrary Q-system.
- 27 May 2024
- Published by Springer Nature
- p. 127-144
Twisted Araki-Woods Algebras, the Yang–Baxter Equation, and Quantum Field Theory
- 27 May 2024
- Published by Springer Nature
- p. 127-144
Twisted Araki-Woods Algebras, the Yang–Baxter Equation, and Quantum Field Theory
- Open Access
Twisted Araki-Woods Algebras, the Yang-Baxter Equation, and quantum field theory
This article reviews recent work with Correa da Silva on twisted Araki-Woods algebras, including an introduction to twisted Fock spaces and standard subspaces. We discuss a new family of examples of that framework, coming from the set-theoretic Yang-Baxter equation, and explain the relevance of twisted Araki-Woods algebras in the construction of quantum field theoretic models.
Open AccessTwisted Araki-Woods Algebras, the Yang-Baxter Equation, and quantum field theory
This article reviews recent work with Correa da Silva on twisted Araki-Woods algebras, including an introduction to twisted Fock spaces and standard subspaces. We discuss a new family of examples of that framework, coming from the set-theoretic Yang-Baxter equation, and explain the relevance of twisted Araki-Woods algebras in the construction of quantum field theoretic models.
- Open Access
Modular Structure and Inclusions of Twisted Araki-Woods Algebras
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 $\mathcal{L}_{T}(H)$ 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 $\mathcal{L}_{T}(H)$ 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 $\mathcal{L}_{T}(K)\subset\mathcal{L}_{T}(H)$ 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 $K\subset H$ satisfies an $L^2$-nuclearity condition, $\mathcal{L}_{T}(K)\subset\mathcal{L}_{T}(H)$ has type III relative commutant for suitable twists $T$. Applications of these results to localization of observables in algebraic quantum field theory are discussed.Open AccessModular Structure and Inclusions of Twisted Araki-Woods Algebras
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 $\mathcal{L}_{T}(H)$ 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 $\mathcal{L}_{T}(H)$ 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 $\mathcal{L}_{T}(K)\subset\mathcal{L}_{T}(H)$ 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 $K\subset H$ satisfies an $L^2$-nuclearity condition, $\mathcal{L}_{T}(K)\subset\mathcal{L}_{T}(H)$ has type III relative commutant for suitable twists $T$. Applications of these results to localization of observables in algebraic quantum field theory are discussed. - Open Access
- 22 June 2023
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 402 (3) , 2339-2386
Modular Structure and Inclusions of Twisted Araki-Woods Algebras
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 $$\mathcal {L}_{T}(H)$$ depend on the twist operatorTand a standard subspaceHin the one-particle space. Under a compatibility assumption onTandH, it is proven that the Fock vacuum is cyclic and separating for $$\mathcal {L}_{T}(H)$$ if and only ifTsatisfies 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 $$\mathcal {L}_{T}(K)\subset \mathcal {L}_{T}(H)$$ 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 $$K\subset H$$ satisfies an $$L^2$$ -nuclearity condition, $$\mathcal {L}_{T}(K)\subset \mathcal {L}_{T}(H)$$ has type III relative commutant for suitable twistsT. Applications of these results to localization of observables in algebraic quantum field theory are discussed.- 22 June 2023
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 402 (3) , 2339-2386
Open AccessModular Structure and Inclusions of Twisted Araki-Woods Algebras
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 $$\mathcal {L}_{T}(H)$$ depend on the twist operatorTand a standard subspaceHin the one-particle space. Under a compatibility assumption onTandH, it is proven that the Fock vacuum is cyclic and separating for $$\mathcal {L}_{T}(H)$$ if and only ifTsatisfies 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 $$\mathcal {L}_{T}(K)\subset \mathcal {L}_{T}(H)$$ 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 $$K\subset H$$ satisfies an $$L^2$$ -nuclearity condition, $$\mathcal {L}_{T}(K)\subset \mathcal {L}_{T}(H)$$ has type III relative commutant for suitable twistsT. Applications of these results to localization of observables in algebraic quantum field theory are discussed. - Open Access
- 18 February 2022
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 391 (1) , 269-291
Deformations of Half-Sided Modular Inclusions and Non-local Chiral Field Theories
We construct explicit examples of half-sided modular inclusions $$\mathcal {N}\subset \mathcal {M}$$ 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 $$\mathcal {N}\subset \mathcal {M}$$ with large relative commutant and construct a one-parameter family $$\mathcal {N}_\kappa \subset \mathcal {M}_\kappa $$ , $$\kappa \ge 0$$ , of half-sided inclusions such that $$\mathcal {N}_0=\mathcal {N}$$ , $$\mathcal {M}_0=\mathcal {M}$$ and $$\mathcal {N}_\kappa '\cap \mathcal {M}_\kappa =\mathbb {C}1$$ for $$\kappa >0$$ . 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.- 18 February 2022
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 391 (1) , 269-291
Open AccessDeformations of Half-Sided Modular Inclusions and Non-local Chiral Field Theories
We construct explicit examples of half-sided modular inclusions $$\mathcal {N}\subset \mathcal {M}$$ 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 $$\mathcal {N}\subset \mathcal {M}$$ with large relative commutant and construct a one-parameter family $$\mathcal {N}_\kappa \subset \mathcal {M}_\kappa $$ , $$\kappa \ge 0$$ , of half-sided inclusions such that $$\mathcal {N}_0=\mathcal {N}$$ , $$\mathcal {M}_0=\mathcal {M}$$ and $$\mathcal {N}_\kappa '\cap \mathcal {M}_\kappa =\mathbb {C}1$$ for $$\kappa >0$$ . 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. - 15 June 2021
- Published by Theta Foundation in Journal of Operator Theory
- Vol. 86 (1) , 61-91
High energy bounds on wave operators
The wave operators W±(H1,H0) of two selfadjoint operators H0 and H1 are analyzed at asymptotic spectral values. Sufficient conditions for ∥(W±(H1,H0)−Pac1Pac0)f(H0)∥<∞ are given, where Pacj 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 Schr\"odinger operators with matrix-valued potentials. We discuss an application to the problem of quantum backflow.
- 15 June 2021
- Published by Theta Foundation in Journal of Operator Theory
- Vol. 86 (1) , 61-91
High energy bounds on wave operators
The wave operators W±(H1,H0) of two selfadjoint operators H0 and H1 are analyzed at asymptotic spectral values. Sufficient conditions for ∥(W±(H1,H0)−Pac1Pac0)f(H0)∥<∞ are given, where Pacj 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 Schr\"odinger operators with matrix-valued potentials. We discuss an application to the problem of quantum backflow.