On the type of local algebras in quantum field theory

Driessler, W.

doi:10.1007/bf01609853

Cited by 36 publications

(36 citation statements)

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“…(2.10) (Taking the intersection separately for both factors of the tensor product is valid in this situation as can easily be proved using the lattice property of von Neumann algebras M ∧ N = (M ′ ∨ N ′ ) ′ and the commutation theorem for tensor products (M ⊗ N ) ′ = M ′ ⊗ N ′ .) We thus see that in conjunction with the well known fact [29] that the algebras associated with wedge regions are factors of type III 1 the split property for wedges implies that the algebras of double cones are type III 1 factors, too.…”

Section: Proposition 21 the Representation Of The Fixpoint Net A Fulmentioning

confidence: 52%

Quantum Double Actions on Operator Algebras and Orbifold Quantum Field Theories

Müger

1998

Comm Math Phys

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Starting from a local quantum field theory with an unbroken compact symmetry group G in 1 + 1-dimensional spacetime we construct disorder fields implementing gauge transformations on the fields (order variables) localized in a wedge region. Enlarging the local algebras by these disorder fields we obtain a nonlocal field theory, the fixpoint algebras of which under the appropriately extended action of the group G are shown to satisfy Haag duality in every simple sector. The specifically 1+1 dimensional phenomenon of violation of Haag duality of fixpoint nets is thereby clarified. In the case of a finite group G the extended theory is acted upon in a completely canonical way by the quantum double D(G) and satisfies R-matrix commutation relations as well as a Verlinde algebra. Furthermore, our methods are suitable for a concise and transparent approach to bosonization. The main technical ingredient is a strengthened version of the split property which is expected to hold in all reasonable massive theories. In the appendices (part of) the results are extended to arbitrary locally compact groups and our methods are adapted to chiral theories on the circle.

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Section: Proposition 21 the Representation Of The Fixpoint Net A Fulmentioning

confidence: 52%

Quantum Double Actions on Operator Algebras and Orbifold Quantum Field Theories

Müger

1998

Comm Math Phys

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“…After using Lemma 6.1, the proof proceeds along the same lines as those of the analogous results in [18,35]. Indeed, let F , G, H be the central projections of M corresponding to the finite, infinite semifinite, and purely infinite part of M, respectively.…”

Section: The Type Of Von Neumann Algebras Associated To Graded Asymptmentioning

confidence: 88%

“…the strong operator topology. The proof for the cases which include Fermionic systems is more involved than the original one in [18,35]. As usual M will be a Z 2 -graded von Neumann algebra whose grading is generated by an automorphism σ ∈ Aut(M) with σ 2 = id.…”

Section: The Type Of Von Neumann Algebras Associated To Graded Asymptmentioning

confidence: 99%

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Disordered Fermions on Lattices and Their Spectral Properties

Barreto¹,

Fidaleo

2011

J Stat Phys

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Abstract. We study Fermionic systems on a lattice with random interactions through their dynamics and the associated KMS states. These systems require a more complex approach compared with the standard spin systems on a lattice, on account of the difference in commutation rules for the local algebras for disjoint regions, between these two systems. It is for this reason that some of the known formulations and proofs in the case of the spin lattice systems with random interactions do not automatically go over to the case of disordered Fermion lattice systems. We extend to the disordered CAR algebra, some standard results concerning the spectral properties exhibited by temperature states for disordered quantum spin systems. We discuss the Arveson spectrum and its connection with the Connes and Borchers Γ-invariants for such W * -dynamical systems. In the case of KMS states exhibiting a natural property of invariance with respect to the spatial translations, some interesting properties, associated with standard spinglass-like behaviour, emerge naturally. It covers infinite-volume limits of finite-volume Gibbs states, that is the quenched disorder for Fermions living on a standard lattice Z d . In particular, we show that a temperature state of the systems under consideration can generate only a type III von Neumann algebra (with the type III 0 component excluded). Moreover, in the case of the pure thermodynamic phase, the associated von Neumann is of type III λ for some λ ∈ (0, 1], independent of the disorder. Such a result is in accordance with the principle of self-averaging which affirms that the physically relevant quantities do not depend on the disorder. The present approach can be viewed as a further step towards fully understanding the very complicated structure of the set of temperature states of quantum spin glasses, and its connection with the breakdown of the symmetry for the replicas.Mathematics Subject Classification: 46L55, 82B44, 46L35.

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“…In particular it is interesting to ask for the Connes , [5], [8], [6], [4] or [2]). In particular it is interesting to ask for the Connes , [5], [8], [6], [4] or [2]).…”

Section: Introductionmentioning

confidence: 99%