Claudio D'Antoni

The universal structure of local algebras

It is shown that a few physically significant conditions fix the global structure of the local algebras appearing in quantum field theory: it is isomorphic to that of ℜ where ℜ is the unique hyperfinite factor of typeIII1 and the center of the respective algebra. The argument is based on results in [1, 2] relating to the type of the local algebras and an improvement of an argument in [3] concerning the “split property.”

Interpolation by Type I Factors and the Flip Automorphism

Abstract Given two von Neumann algebras A ⊂ B we study the relation between the existence of an interpolating type I factor F , namely A ⊂ F ⊂ B , the implementability of the flip automorphism of A ⊗ A by a unitary in B ⊗ B , and the statistical independence of A and B ′ ( A and B ′ generate a W ∗-tensor product). As an application in Q.F.T. we derive in a natural way a structure theorem of Buchholz for the von Neumann algebras of local observables associated to free fields.

Nuclear maps and modular structures. II. Applications to quantum field theory

A correspondence between spectral properties of modular operators appearing in quantum field theory and the Hamiltonian is established. It allows to prove the “distal” split property for a wide class of models. Conversely, any model having this property is shown to satisfy the Haag-Swieca compactness criterion. The results lead to a new type of nuclearity condition which can be applied to quantum field theories on arbitrary space-time manifolds.

Convergence of local charges and continuity properties ofW*-inclusions

The local generators of symmetry transformations which have recently been constructed from a quantum field theoretical version of Noethers theorem are shown to converge to the global ones as the volume tends to the whole space. The proof relies on the continuous volume dependence of the universal localizing maps which are associated to the local splitW*-inclusions.

Nuclear maps and modular structures. I. General properties

Abstract We establish a correspondence between the split property of inclusions A ⊂ B of von Neumann algebras and nuclearity properties of the natural embeddings φp: L∞(A) → Lp(B), p = 1, 2, given by the modular theory.

Implementation of Conformal Covariance by Diffeomorphism Symmetry

Every locally normal representation of a local chiral conformal quantum theory is covariant with respect to global conformal transformations, if this theory is diffeomorphism covariant in its vacuum representation. The unitary, strongly continuous representation implementing conformal symmetry is constructed; it consists of operators which are inner in a global sense for the representation of the quantum theory. The construction is independent of positivity of energy and applies to all locally normal representations irrespective of their statistical dimensions (index)

Charges in spacelike cones

Starting from a conserved current, operators are defined which measure the charge in certain unbounded stringlike regions which are possible localization regions of charged fields in gauge theories.

Scaling algebras and pointlike fields: A Nonperturbative approach to renormalization

We present a method of short-distance analysis in quantum field theory that does not require choosing a renormalization prescription a priori. We set out from a local net of algebras with associated pointlike quantum fields. The net has a naturally defined scaling limit in the sense of Buchholz and Verch; we investigate the effect of this limit on the pointlike fields. Both for the fields and their operator product expansions, a well-defined limit procedure can be established. This can always be interpreted in the usual sense of multiplicative renormalization, where the renormalization factors are determined by our analysis. We also consider the limits of symmetry actions. In particular, for suitable limit states, the group of scaling transformations induces a dilation symmetry in the limit theory.