On the relation between a conformal structure in spacetime and nets of local algebras of observables

In this Letter, we study the question of how to reconstruct the conformal structure (causality relation) of spacetime from a net of local algebras of observables on the underlying manifold.Partly supported by DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

Fock spaces with generalized statistics

Fock representations for quantum fields which obey generalized statistics are explicitly constructed. The main features of these representations are investigated.     

On the intrinsic definition of local observables

In a theory of local field algebras satisfying the split property, we introduce an algebra generated by the local energy-momentum operators obtained from the canonical local implementation of translations, and possibly by the local charges operators. We discuss the relations of this algebra to the given algebra of local observables and take some steps toward the characterization of theories where they coincide. In the presence of spontaneously broken symmetries, we present a no-go theorem.     

On the topological theory of self-dual connections over noncompact Riemann surfaces

A topological action for self-dual connections over noncompact Riemann surfaces is proposed. TheJ formulation and the associated linear system are obtained. A new connection is constructed, depending on a Kac-Moody parameter such that its flatness condition is theJ-equation associated to the self-dual problem. The algebra of infinitesimal Bäcklund transformations depending on this Kac-Moody parameter is constructed.     

On Bell's inequalities and algebraic invariants

Some algebraic invariants associated with Bell's inequalities are defined for inclusions of von Neumann algebras and studied within the context of general algebraic quantum theory. More special results are proven for quantum field theory which establish that these invariants take infinitely many values. Sharp short-distance bounds on the Bell correlations are also demonstrated in the context of relativistic quantum field theory.     

On modular inclusion and spectrum condition

For a family of von Neumann algebras, we investigate the relations between spectrum condition, half-sided modular inclusion, and reflection symmetry. It is shown that the geometric interpretation of modular actions puts strong restrictions on the family.     

Modular inclusion,the Hawking temperature,and quantum field theory in curved spacetime

A recent result by Borchers connecting geometric modular action, modular inclusion and spectrum condition, is applied in quantum field theory on spacetimes with a bifurcate Killing horizon (these are generalizations of black-hole spacetimes, comprising the familiar black-hole spacetime models). Within this framework, we give sufficient, model-independent conditions ensuring that the temperature of thermal equilibrium quantum states is the Hawking temperature.     

Nuclearity,split property,and duality for the Klein-Gordon field in curved spacetime

Nuclearity, split property and duality are established for the nets of von Neumann algebras associated with the representations of distinguished states of the massive Klein-Gordon field propagating in particular classes of curved spacetimes.Supported by the DFG.     

Generalized nuclearity conditions and the split property in quantum field theory

Generalized nuclearity conditions that are applicable in arbitrary superselection sectors of a quantum field theory and to theories with a maximal temperature are discussed. They are shown to imply the (distal) split property and to impose specific restrictions on the spectral properties of modular operators associated with local algebras and vectors of compact energy support.     

Endomorphism semigroups and lightlike translations

Borchers and Wiesbrock have studied the one-parameter semigroups of endomorphisms of von Neumann algebras that appear as lightlike translations in the theory of algebras of local observables, showing that they automatically transform under the appropriate modular automorphisms as under velocity transformations. Here, these results are abstracted and analyzed as essentially operator-theoretic. Criteria are then established for a spatial derivation of a von Neumann algebra to generate a one-parameter semigroup of endomorphisms, and all of this is combined to establish a von Neumann-algebraic converse to the Borchers and Wiesbrock results. This sort of analysis is then applied to questions of isotony and covariance for local algebras, to show that Poincaré covariance together with a domain condition for the translations can imply isotony.This research was partly supported by a fellowship from the Consiglio Nazionale delle Ricerche.     

Operator ideals and the statistical independence in quantum field theory

We consider analytic properties of a class of dynamical systems which are defined by the action of certain homomorphism groups on von Neumann algebras if restricted to subalgebras. In particular, the analyticity of nuclear maps in the nuclear norm is shown. Furthermore, the statistical independence will be derived from nuclearity conditions. These results give new insight in the statistical independence of commuting algebras.This paper is a result of a collaboration with H. J. Borchers.     

Symmetries and half-sided modular inclusions of von Neumann algebras

LetN, be a von Neumann algebras on a Hilbert space , a common cyclic and separating vector. Assume to be cyclic and also separating forN . Denote by , _N , _N the modular operators to (, ), (N, ), resp (N , ). Assume now ^-it N ^it N for allt 0. (Such type of inclusions ((N U, ) , ) are called half-sided modular.) Then the modular groups ^it , _N ^ir , _N ^is ,t, r, s generate a unitary representation of the group S1(2, )/Z ₂ of positive energy.Another result is related to two half-sided modular inclusions (₁ , ) and (₂ , ). Under proper conditions the three modular groups ^it , ₁ ^ir , ₂ ^is ,t, r, s generate the three-dimensional subgroup of O(2, 1) of two commuting translations and the Lorentz transformation.Partly supported by the DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

A comment on a recent work of Borchers

We establish a converse of a recent result of BorchersPartly supported by the DFG, SFB 286 Differentialgeometric und Quantenphysik.     

A note on strongly additive conformal field theory and half-sided modular conormal standard inclusions

Given a half-sided modular standard inclusion (N , ) of von-Neumann algebras which is conormal, we construct a chiral conformal quantum field theory on the circle. Conversely, a conformal quantum field theory on the circle which fulfills, in addition, the property of strong additivity, yields such data in a natural way.Partly supported by the DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

Theory of superselection sectors for generalized Ising models

We apply the theory of superselection sectors in the same way as done by G. Mack and V. Schomerus for the Ising model to generalizations of this model described by J. Fröhlich and T. Kerler. The sector generating morphisms are found.Supported by the DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

About matrix elements of spatial translations in local field theories

Some results of Fredenhagen concerning the asymptotic behavior of matrix elements of translation operators in local quantum field theories are discussed. An alternative derivation and generalizations are presented.     

Causal Construction of the Massless Vertex Diagram

The massless one-loop vertex diagram is constructed by exploiting the causal structure of the diagram in configuration space, which can be translated directly into dispersive relations in momentum space.     

Dispersive Calculation of the Massless Multi-loop Sunrise Diagram

The massless sunrise diagram with an arbitrary number of loops is calculated in a simple but formal manner. The result is then verified by rigorous mathematical treatment. Pitfalls in the calculation with distributions are highlighted and explained. The result displays the high energy behaviour of the massive sunrise diagrams, whose calculation is involved already for the two-loop case.     

The construction of pointlike localized charged fields from conformal Haag—Kastler nets

Starting from a chiral conformal Haag-Kastler net on two-dimensional Minkowski space we construct associated charged pointlike localized fields which intertwine between arbitrary superselection sectors with finite statistics of the theory. This amounts to a proof of the Spin-Statistics theorem, the PCT theorem and the existence of operator product expansions.This Letter generalizes similar results of a recently published paper by Fredenhagen and the author from the neutral vacuum sector to the the full theory with arbitrary charge and finite statistics.