scholarly journals Conformal quantum field theory and half-sided modular inclusions of von-Neumann-Algebras

1993 ◽  
Vol 158 (3) ◽  
pp. 537-543 ◽  
Author(s):  
Hans-Werner Wiesbrock
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Von Neumann Algebras ◽  
Quantum Field ◽  
Von Neumann

scholarly journals Universal bounds for large determinants from non-commutative Hölder inequalities in fermionic constructive quantum field theory

2017 ◽  
Vol 27 (10) ◽  
pp. 1963-1992 ◽  
Author(s):  
J.-B. Bru ◽  
W. de Siqueira Pedra
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Von Neumann Algebras ◽  
Perturbation Expansion ◽  
Perturbation Series ◽  
Interparticle Interaction ◽  
Quantum Field ◽  
Von Neumann ◽  
Constructive Quantum Field Theory ◽  
Strength Formula

Efficiently bounding large determinants is an essential step in non-relativistic constructive quantum field theory to prove the absolute convergence of the perturbation expansion of correlation functions in terms of powers of the strength [Formula: see text] of the interparticle interaction. We provide, for large determinants of fermionic covariances, sharp bounds which hold for all (bounded and unbounded, the latter not being limited to semibounded) one-particle Hamiltonians. We find the smallest universal determinant bound to be exactly [Formula: see text]. In particular, the convergence of perturbation series at [Formula: see text] of any fermionic quantum field theory is ensured if the matrix entries (with respect to some fixed orthonormal basis) of the covariance and the interparticle interaction decay sufficiently fast. Our proofs use Hölder inequalities for general non-commutative [Formula: see text]-spaces derived by Araki and Masuda [Positive cones and [Formula: see text]-spaces for von Neumann algebras, Publ. RIMS[Formula: see text] Kyoto Univ. 18 (1982) 339–411].

scholarly journals NETS OF SUBFACTORS

1995 ◽  
Vol 07 (04) ◽  
pp. 567-597 ◽  
Author(s):  
R. LONGO ◽  
K.-H. REHREN
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Relative Position ◽  
Von Neumann Algebras ◽  
Space Time ◽  
Quantum Field ◽  
Von Neumann ◽  
Time Region ◽  
Theoretical Application ◽  
Theory Formula

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.

TRANSITIVITY OF LOCALITY AND DUALITY IN QUANTUM FIELD THEORY

1994 ◽  
Vol 06 (04) ◽  
pp. 597-619 ◽  
Author(s):  
H. J. BORCHERS ◽  
JAKOB YNGVASON
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Von Neumann Algebras ◽  
The Other ◽  
Quantum Field ◽  
Bounded Operators ◽  
Unbounded Operators ◽  
Von Neumann ◽  
Lorentz Covariance ◽  
Wightman Fields

Duality conditions for Wightman fields are formulated in terms of the Tomita conjugations S associated with algebras of unbounded operators. It is shown that two fields which are relatively local to an irreducible field fulfilling a condition of this type are relatively local to each other. Moreover, a local net of von Neumann algebras associated with such a field satisfies (essential) duality. These results do not rely on Lorentz covariance but follow from the observation that two algebras of (un)bounded operators with the same Tomita conjugation have the same (un)bounded weak commutant if one algebra is contained in the other.

Modular structure of algebras of unbounded operators

1992 ◽  
Vol 111 (2) ◽  
pp. 369-386 ◽  
Author(s):  
Atsushi Inoue
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Von Neumann Algebras ◽  
Modular Structure ◽  
Quantum Field ◽  
Unbounded Operators ◽  
Systematic Analysis ◽  
Von Neumann ◽  
Modular Systems

AbstractA systematic analysis of standard systems and modular systems for which one can develop TomitaTakesaki theory for algebras of unbounded operators is presented. Such systems arise in the Wightman quantum field theory. The connection between such systems and the theory of local nets of von Neumann algebras initiated by Araki and HaagKastler is discussed.

scholarly journals Local Nets of Von Neumann Algebras in the Sine–Gordon Model

2021 ◽  
Author(s):  
Dorothea Bahns ◽  
Klaus Fredenhagen ◽  
Kasia Rejzner
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Von Neumann Algebras ◽  
Mass Term ◽  
Quantum Field ◽  
Functional Formalism ◽  
Von Neumann ◽  
Algebraic Quantum ◽  
Lorentzian Signature ◽  
Sine Gordon Model

AbstractThe Haag–Kastler net of local von Neumann algebras is constructed in the ultraviolet finite regime of the Sine–Gordon model, and its equivalence with the massive Thirring model is proved. In contrast to other authors, we do not add an auxiliary mass term, and we work completely in Lorentzian signature. The construction is based on the functional formalism for perturbative Algebraic Quantum Field Theory together with estimates originally derived within Constructive Quantum Field Theory and adapted to Lorentzian signature. The paper extends previous work by two of us.

Sign in / Sign up

Export Citation Format

Share Document