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Localized endomorphisms of the chiral Ising model

Jens Böckenhauer

Published Jan 1, 1996• 29 citations
OpenAlex

Abstract

Based on the treatment of the chiral Ising model by Mack and Schomerus, we present examples of localized endomorphisms ̺ loc 1 and ̺ loc 1/2. It is shown that they lead to the same superselection sectors as the global ones in the sense that unitary equivalence π0 ◦ ̺ loc 1 ∼ = π1 and π0 ◦ ̺ loc 1/2 ∼ = π1/2 holds. Araki’s formalism of the selfdual CAR algebra is used for the proof. We prove local normality and extend representations and localized endomorphisms to a global algebra of observables which is generated by local von Neumann algebras on the punctured circle. In this framework, we manifestly prove fusion rules and derive statistics operators. 1

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CHIRAL DECOMPOSITION AS A SOURCE OF QUANTUM SYMMETRY IN THE ISING MODEL

Kornél Szlachányi

The superselection sectors of three closely related models are studied and compared. I. The full Ising model with observable algebra [Formula: see text], the universal algebra of local even CAR algebras on the circle. [Formula: see text] has four sectors with Z(2) × Z(2) symmetry. II. The chiral Ising model with observable algebra [Formula: see text] (c = L or R), which is the universal algebra of local even Majorana algebras, has three sectors with quantum symmetry Gc ≅ M1⊕M1⊕M2. The Mack-Schomerus endomorphism creating the non-Abelian sector of [Formula: see text], respectively of [Formula: see text] is shown to coincide with the restriction [Formula: see text], respectively [Formula: see text] of a Z (2) charge creating automorphism ρ of [Formula: see text]. III. As an intermediate step the superselection sectors of the algebra [Formula: see text] — which can be interpreted as the observable algebra of the conformal Ising model — are found to have ordinary group symmetry described by the dihedral group D4. The relation between the sectors of [Formula: see text] and [Formula: see text] is explained in terms of a strange 'symmetry breaking': symmetry enhancement in the Neveu-Schwarz sector and symmetry breaking in the Ramond. Covariant charged fields are constructed in all three cases and the truncation in Gc is shown to arise from the failure of the Cuntz algebra relations for the chiral charged fields.

Reviews in Mathematical Physics•Aug 1, 1994•7 citations

Sector Theory

Huzihiro Araki

Abstract In the previous sections we treated the irreducible representation space associated with the vacuum state. In this chapter we treat other representation spaces necessary to describe particle systems with the vacuum as a background. Such representation spaces are characterized as transportable localized excitations on the vacuum background. If we assume Haag’s duality of the vacuum representation, such a localized excitation is described by a localized endomorphism of the observables and its equivalence classes correspond to different representations called sectors(eigensubspaces of the superselection rules in each of which unrestricted superposition principle holds). From the discussion about composition and conjugation of localized endomorphisms, we define the statistics (of a sector) which represents the behaviour of more than two equivalent localized excitations under permutation, and we derive the connection of spin and statistics. We also introduce the lattice of local field algebras as operators intertwining different sectors, and the gauge group acting on them. Then the local observable algebras are the gauge invariant part of the local field algebras and sectors are inequivalent representations of the gauge group. The local field algebras can be chosen to satisfy the standard commutation or anticommutation relations between space-like separated regions and we can derive the connection of spin and statistics of sectors. In this way we obtain a deep understanding of the reason why Fermi fields in spatially separated regions anticommute.

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Charged Quantum Fields Associated with Endomorphisms of CAR and CCR Algebras

Carsten Binnenhei

The appearance of the Cuntz algebras Od is a generic feature of local quantum field theory. This fact has been discovered by S. Doplicher and J. E. Roberts within the algebraic theory of superselection sectors. Generators of the Cuntz algebras arise as charged field operators which implement localized endomorphisms of the observable algebra. Whereas the existence of such operators can be derived from first principles, little is known about the actual construction of these fields in concrete models. In view of this apparent discrepancy, we develop a comprehensive theory of quasi-free endomorphisms of the CAR and CCR algebras which give rise to representations of the Cuntz algebras Od on Fock space. The number d is the statistics dimension of the endomorphism. It can be any power of 2 (including 1 and infinite) in the CAR case, but takes only the values 1 and infinite in the CCR case. We obtain necessary and sufficient conditions for implementability of quasi- free endomorphisms. By studying extensions of partial Fock states, we find that the semigroup of implementable quasi-free endomorphisms has a simple structure: It can be written as a product of a group of automorphisms which are close to the identity, and the semigroup of endomorphisms which leave the given Fock state invariant. We describe the construction of the implementers of a quasi-free endomorphism in terms of annihilation and creation operators in is a generic feature of local quantum field theory. This fact has been discovered by S. Doplicher and J. E. Roberts within the algebraic theory of superselection sectors. Generators of the Cuntz algebras arise as charged field operators which implement localized endomorphisms of the observable algebra. Whereas the existence of such operators can be derived from first principles, little is known about the actual construction of these fields in concrete models. We compare the structure of the semigroup of (gauge invariant) quasi-free endomorphisms with the generic superselection structure of quantum field theory.

•Jan 1, 1998•0 citations

NETS OF SUBFACTORS

R. LONGO, Karl-Henning Rehren

A subtheory of a quantum field theory specifies von Neumann subalgebras [Formula: see text] (the ‘observables’ in the space-time region [Formula: see text]) of the von Neumann algebras [Formula: see text] (the 'field' localized in [Formula: see text]). Every local algebra being a (type III1) factor, the inclusion [Formula: see text] is a subfactor. The assignment of these local subfactors to the space-time regions is called a ‘net of subfactors’. The theory of subfactors is applied to such nets. In order to characterize the ‘relative position’ of the subtheory, and in particular to control the restriction and induction of superselection sectors, the canonical endomorphism is studied. The crucial observation is this: the canonical endomorphism of a single local subfactor extends to an endomorphism of the field net, which in turn restricts to a localized endomorphism of the observable net. The method allows one to characterize, and reconstruct, local extensions ℬ of a given theory [Formula: see text] in terms of the observables. Various non-trivial examples are given. Several results go beyond the quantum field theoretical application.

arXiv:hep-th/9411077•Reviews in Mathematical Physics•May 1, 1995•265 citations

NETS OF SUBFACTORS

Roberto Longo, Karl-Henning Rehren

A subtheory of a quantum field theory specifies von Neumann subalgebras A(phi) (the 'observables' in the space-time region phi) of the von Neumann algebras B(phi) (the 'fields' localized in phi). Every local algebra being a (type III1) factor, the inclusion A(phi) subset of B(phi) is a subfactor. The assignment of these local subfactors to the space-time regions is called a 'net of subfactors'. The theory of subfactors is applied to such nets. In order to characterize the 'relative position' of the subtheory, and in particular to control the restriction and induction of superselection sectors, the canonical endomorphism is studied. The crucial observation is this: the canonical endomorphism of a single local subfactor extends to an endomorphism of the field net, which in turn restricts to a localized endomorphism of the observable net. The method allows one to characterize, and reconstruct, local extensions B of a given theory A in terms of the observables. Various non-trivial examples are given. Several results go beyond the quantum field theoretical application.

•Jan 1, 1995•0 citations

Superselection theory for 2D braided quantum spin systems via Connes fusion

Gregory Faurot, C.H. Li, David Penneys +1 more

The article arXiv:2410.21454 constructed a braided $\mathrm{W^*}$-tensor category of superselection sectors associated to a net of von Neumann algebras on a suitably geometric poset using localized and transportable endomorphisms. In this article, we construct an equivalent tensor product and braiding using the Connes fusion tensor product without localizing, using ideas from arXiv:1812.04470 for conformal nets. As an application, we prove that the category of superselection sectors associated to the 2D braided quantum spin system constructed from a unitary braided fusion category $\mathcal{B}$ in arXiv:2506.19969 is equivalent to $\mathsf{Hilb}(\mathcal{B})^{\rm op}$, the opposite of the unitary ind-completion of $\mathcal{B}$, as a braided $\mathrm{W^*}$-tensor category.

arXiv:2609.20725•Sep 17, 2026•0 citations
Based on the treatment of the chiral Ising model by Mack and Schomerus, we present examples of localized endomorphisms ̺ loc 1 and ̺ loc 1/2. It is shown that they lead to the same superselection sectors as the global ones in the sense that unitary equivalence π0 ◦ ̺ loc 1 ∼ = π1 and π0 ◦ ̺ loc 1/2 ∼ = π1/2 holds. Araki’s formalism of the selfdual CAR algebra is used for the proof. We prove local normality and extend representations and localized endomorphisms to a global algebra of observables which is generated by local von Neumann algebras on the punctured circle. In this framework, we manifestly prove fusion rules and derive statistics operators. 1
The superselection sectors of three closely related models are studied and compared. I. The full Ising model with observable algebra [Formula: see text], the universal algebra of local even CAR algebras on the circle. [Formula: see text] has four sectors with Z(2) × Z(2) symmetry. II. The chiral Ising model with observable algebra [Formula: see text] (c = L or R), which is the universal algebra of local even Majorana algebras, has three sectors with quantum symmetry Gc ≅ M1⊕M1⊕M2. The Mack-Schomerus endomorphism creating the non-Abelian sector of [Formula: see text], respectively of [Formula: see text] is shown to coincide with the restriction [Formula: see text], respectively [Formula: see text] of a Z (2) charge creating automorphism ρ of [Formula: see text]. III. As an intermediate step the superselection sectors of the algebra [Formula: see text] — which can be interpreted as the observable algebra of the conformal Ising model — are found to have ordinary group symmetry described by the dihedral group D4. The relation between the sectors of [Formula: see text] and [Formula: see text] is explained in terms of a strange 'symmetry breaking': symmetry enhancement in the Neveu-Schwarz sector and symmetry breaking in the Ramond. Covariant charged fields are constructed in all three cases and the truncation in Gc is shown to arise from the failure of the Cuntz algebra relations for the chiral charged fields.
Abstract In the previous sections we treated the irreducible representation space associated with the vacuum state. In this chapter we treat other representation spaces necessary to describe particle systems with the vacuum as a background. Such representation spaces are characterized as transportable localized excitations on the vacuum background. If we assume Haag’s duality of the vacuum representation, such a localized excitation is described by a localized endomorphism of the observables and its equivalence classes correspond to different representations called sectors(eigensubspaces of the superselection rules in each of which unrestricted superposition principle holds). From the discussion about composition and conjugation of localized endomorphisms, we define the statistics (of a sector) which represents the behaviour of more than two equivalent localized excitations under permutation, and we derive the connection of spin and statistics. We also introduce the lattice of local field algebras as operators intertwining different sectors, and the gauge group acting on them. Then the local observable algebras are the gauge invariant part of the local field algebras and sectors are inequivalent representations of the gauge group. The local field algebras can be chosen to satisfy the standard commutation or anticommutation relations between space-like separated regions and we can derive the connection of spin and statistics of sectors. In this way we obtain a deep understanding of the reason why Fermi fields in spatially separated regions anticommute.
The appearance of the Cuntz algebras Od is a generic feature of local quantum field theory. This fact has been discovered by S. Doplicher and J. E. Roberts within the algebraic theory of superselection sectors. Generators of the Cuntz algebras arise as charged field operators which implement localized endomorphisms of the observable algebra. Whereas the existence of such operators can be derived from first principles, little is known about the actual construction of these fields in concrete models. In view of this apparent discrepancy, we develop a comprehensive theory of quasi-free endomorphisms of the CAR and CCR algebras which give rise to representations of the Cuntz algebras Od on Fock space. The number d is the statistics dimension of the endomorphism. It can be any power of 2 (including 1 and infinite) in the CAR case, but takes only the values 1 and infinite in the CCR case. We obtain necessary and sufficient conditions for implementability of quasi- free endomorphisms. By studying extensions of partial Fock states, we find that the semigroup of implementable quasi-free endomorphisms has a simple structure: It can be written as a product of a group of automorphisms which are close to the identity, and the semigroup of endomorphisms which leave the given Fock state invariant. We describe the construction of the implementers of a quasi-free endomorphism in terms of annihilation and creation operators in is a generic feature of local quantum field theory. This fact has been discovered by S. Doplicher and J. E. Roberts within the algebraic theory of superselection sectors. Generators of the Cuntz algebras arise as charged field operators which implement localized endomorphisms of the observable algebra. Whereas the existence of such operators can be derived from first principles, little is known about the actual construction of these fields in concrete models. We compare the structure of the semigroup of (gauge invariant) quasi-free endomorphisms with the generic superselection structure of quantum field theory.
A subtheory of a quantum field theory specifies von Neumann subalgebras [Formula: see text] (the ‘observables’ in the space-time region [Formula: see text]) of the von Neumann algebras [Formula: see text] (the 'field' localized in [Formula: see text]). Every local algebra being a (type III1) factor, the inclusion [Formula: see text] is a subfactor. The assignment of these local subfactors to the space-time regions is called a ‘net of subfactors’. The theory of subfactors is applied to such nets. In order to characterize the ‘relative position’ of the subtheory, and in particular to control the restriction and induction of superselection sectors, the canonical endomorphism is studied. The crucial observation is this: the canonical endomorphism of a single local subfactor extends to an endomorphism of the field net, which in turn restricts to a localized endomorphism of the observable net. The method allows one to characterize, and reconstruct, local extensions ℬ of a given theory [Formula: see text] in terms of the observables. Various non-trivial examples are given. Several results go beyond the quantum field theoretical application.
A subtheory of a quantum field theory specifies von Neumann subalgebras A(phi) (the 'observables' in the space-time region phi) of the von Neumann algebras B(phi) (the 'fields' localized in phi). Every local algebra being a (type III1) factor, the inclusion A(phi) subset of B(phi) is a subfactor. The assignment of these local subfactors to the space-time regions is called a 'net of subfactors'. The theory of subfactors is applied to such nets. In order to characterize the 'relative position' of the subtheory, and in particular to control the restriction and induction of superselection sectors, the canonical endomorphism is studied. The crucial observation is this: the canonical endomorphism of a single local subfactor extends to an endomorphism of the field net, which in turn restricts to a localized endomorphism of the observable net. The method allows one to characterize, and reconstruct, local extensions B of a given theory A in terms of the observables. Various non-trivial examples are given. Several results go beyond the quantum field theoretical application.
The article arXiv:2410.21454 constructed a braided W∗\mathrm{W^*}W∗-tensor category of superselection sectors associated to a net of von Neumann algebras on a suitably geometric poset using localized and transportable endomorphisms. In this article, we construct an equivalent tensor product and braiding using the Connes fusion tensor product without localizing, using ideas from arXiv:1812.04470 for conformal nets. As an application, we prove that the category of superselection sectors associated to the 2D braided quantum spin system constructed from a unitary braided fusion category B\mathcal{B}B in arXiv:2506.19969 is equivalent to Hilb(B)op\mathsf{Hilb}(\mathcal{B})^{\rm op}Hilb(B)op, the opposite of the unitary ind-completion of B\mathcal{B}B, as a braided W∗\mathrm{W^*}W∗-tensor category.