Local Von Neumann Algebras Research Articles

A simple proof of duality for local algebras in free quantum field theory

New and simple proofs of duality for local von Neumann algebras in free-scalar field models associated with a general class of regions in Minkowski space are presented. The proofs are given for both the massive and massless cases and an abstract result of Araki [H. Araki, J. Math. Phys. 4, 1343 (1963)] is assumed. The properties of the local algebras are analyzed using the associated real linear manifolds. Duality is proved in the massive models using elementary properties of Sobolev spaces and in the massless model using dilatation covariance. A proof of the factor property and the cyclicity and separability of the vacuum for these local algebras is also given.

On the connection between quantum fields and von Neumann algebras of local operators

The relationship between a standard local quantum field and a net of local von Neumann algebras is discussed. Two natural possibilities for such an association are identified, and conditions for these to obtain are found. It is shown that the local net can naturally be so chosen that it satisfies the Special Condition of Duality. The notion of an intrinsically local field operator is introduced, and it is shown that such an operator defines a local net with which the field is locally associated. A regularity condition on the field is formulated, and it is shown that if this condition holds, then there exists a unique local net with which the field is locally associated if and only if the field algebra contains at least one intrinsically local operator. Conditions under which a field and other fields in its Borchers class are associated with the same local net are found, in terms of the regularity condition mentioned.

An application of Tomita's theory of modular Hilbert algebras: Duality for free Bose fields

Tomita's theory of modular Hilbert algebras is applied to prove the duality theorem for local von Neumann algebras in a free field theory.

Algebras of observables with continuous representations of symmetry groups

Concrete C*-algebras, interpreted physically as algebras of observables, are defined for quantum mechanics and local quantum field theory. Aquantum mechanical system is characterized formally by a continuous unitary representation up to a factorU g of a symmetry group $$\mathfrak{G}$$ in Hilbert space ℌ and a von Neumann algebra ℜ on ℌ invariant with respect toU g . The set $$\mathfrak{A}$$ of all operatorsX∈ℜ such thatU g X U −1 , as a function ofg∈ $$\mathfrak{G}$$ , is continuous with respect to the uniform operator topology, is aC*-algebra called thealgebra of observables. The algebra ℜ is shown to be the weak (or strong) closure of $$\mathfrak{A}$$ . Infield theory, a unitary representation up to a factorU(a, Λ) of the proper inhomogeneous Lorentz group $$\mathfrak{G}$$ and local von Neumann algebras ℜC for finite open space-time regionsC are assumed, with the usual transformation properties of $$\mathfrak{G}$$ underU(a, Λ). The collection of allX∈ℜC giving uniformly continuous functionsU (a, Λ)X U −1 (a, Λ) on $$\mathfrak{G}$$ is then a localC*-algebra $$\mathfrak{A}_C $$ , called thealgebra of local observables. The algebra $$\mathfrak{A}_C $$ is again weakly (or strongly) dense in ℜ c . The norm-closed union $$\mathfrak{A}$$ of the $$\mathfrak{A}_C $$ for allC is calledalgebra of quasilocal observables (or quasilocal algebra). In either case, the group $$\mathfrak{G}$$ is represented by automorphisms V g resp. V(a, Λ) — with V g X=U g X U −1 — of theC*-algebra $$\mathfrak{A}$$ , and this is astrongly continuous representation of $$\mathfrak{G}$$ on the Banach space $$\mathfrak{A}$$ . Conditions for V (a, Λ) can then be formulated which correspond to the usualspectrum condition forU (a, Λ) in field theory.