Daniele Guido

The conformal spin and statistics theorem

We prove the equality between the statistics phase and the conformal univalence for a superselection sector with finite index in Conformal Quantum Field Theory onS1. A relevant point is the description of the PCT symmetry and the construction of the global conjugate charge.

MODULAR STRUCTURE AND DUALITY IN CONFORMAL QUANTUM FIELD THEORY

Making use of a recent result of Borchers, an algebraic version of the Bisognano-Wichmann theorem is given for conformal quantum field theories, i.e. the Tomita-Takesaki modular group associated with the von Neumann algebra of a wedge region and the vacuum vector coincides with the evolution given by the rescaled pure Lorentz transformations preserving the wedge. A similar geometric description is valid for the algebras associated with double cones. Moreover essential duality holds on the Minkowski spaceM, and Haag duality for double cones holds provided the net of local algebras is extended to a pre-cosheaf on the superworld≈M, i.e. the universal covering of the Dirac-Weyl compactification ofM. As a consequence a PCT symmetry exists for any algebraic conformal field theory in even spacetime dimension. Analogous results hold for a Poincaré covariant theory provided the modular groups corresponding to wedge algebras have the expected geometrical meaning and the split property is satisfied. In particular the Poincaré representation is unique in this case.

Modular localization and wigner particles

We propose a framework for the free field construction of algebras of local observables which uses as an input the Bisognano–Wichmann relations and a representation of the Poincare group on the one-particle Hilbert space. The abstract real Hilbert subspace version of the Tomita–Takesaki theory enables us to bypass some limitations of the Wigner formalism by introducing an intrinsic spacetime localization. Our approach works also for continuous spin representations to which we associate a net of von Neumann algebras on spacelike cones with the Reeh–Schlieder property. The positivity of the energy in the representation turns out to be equivalent to the isotony of the net, in the spirit of Borchers theorem. Our procedure extends to other spacetimes homogeneous under a group of geometric transformations as in the case of conformal symmetries and of de Sitter spacetime.

Relativistic invariance and charge conjugation in quantum field theory

We prove that superselection sectors with finite statistics in the sense of Doplicher, Haag, and Roberts are automatically Poincaré covariant under natural conditions (e.g. split property for space-like cones and duality for contractible causally complete regions). The same holds for topological charges, namely sectors localized in space-like cones, providing a converse to a theorem of Buchholz and Fredenhagen. We introduce the notion of weak conjugate sector that turns out to be equivalent to the DHR conjugate in finite statistics. The weak conjugate sector is given by an explicit formula that relates it to the PCT symmetry in a Wightman theory. Every Euclidean convariant sector (possibly with infinite statistics) has a weak conjugate sector and the converse is true under the above natural conditions. On the same basis, translation covariance is equivalent to the property that sectors are sheaf maps modulo inner automorphisms, for a certain sheaf structure given by the local algebras. The construction of the weak conjugate, sector also applies to the case of local algebras onS1 in conformal theories. Our main tools are the Bisognano-Wichmann description of the modular structure of the von Neumann algebras associated with wedge regions in the vacuum sector and the relation between Jones index theory for subfactors and the statistics of superselection sectors.

An algebraic spin and statistics theorem

A spin-statistics theorem and a PCT theorem are obtained in the context of the superselection sectors in Quantum Field Theory on a 4-dimensional spacetime. Our main assumption is the requirement that the modular groups of the von Neumann algebras of local observables associated with wedge regions act geometrically as pure Lorentz transformations. Such a property, satisfied by the local algebras generated by Wightman fields because of the Bisognano-Wichmann theorem, is regarded as a natural primitive assumption.

Extensions of Conformal Nets and Superselection Structures

Abstract:Starting with a conformal Quantum Field Theory on the real line, we show that the dual net is still conformal with respect to a new representation of the Möbius group. We infer from this that every conformal net is normal and conormal, namely the local von Neumann algebra associated with an interval coincides with its double relative commutant inside the local von Neumann algebra associated with any larger interval. The net and the dual net give together rise to an infinite dimensional symmetry group, of which we study a class of positive energy irreducible representations. We mention how superselection sectors extend to the dual net and we illustrate by examples how, in general, this process generates solitonic sectors. We describe the free theories associated with the lowest weight n representations of , showing that they violate 3-regularity for

Dimensions and singular traces for spectral triples, with applications to fractals

n > 2. When n≥ 2, we obtain examples of non Möbius-covariant sectors of a 3-regular (non 4-regular) net.

Singular Traces and Compact Operators

Given a spectral triple (A,H,D), the functionals on A of the form a↦τω(a|D|−α) are studied, where τω is a singular trace, and ω is a generalised limit. When τω is the Dixmier trace, the unique exponent d giving rise possibly to a non-trivial functional is called Hausdorff dimension, and the corresponding functional the (d-dimensional) Hausdorff functional. It is shown that the Hausdorff dimension d coincides with the abscissa of convergence of the zeta function of |D|−1, and that the set of αs for which there exists a singular trace τω giving rise to a non trivial functional is an interval containing d. Moreover, the endpoints of such traceability interval have a dimensional interpretation. The functionals corresponding to points in the traceability interval are called Hausdorff–Besicovitch functionals. These definitions are tested on fractals in R, by computing the mentioned quantities and showing in many cases their correspondence with classical objects. In particular, for self-similar fractals the traceability interval consists only of the Hausdorff dimension, and the corresponding Hausdorff–Besicovitch functional gives rise to the Hausdorff measure. More generally, for any limit fractal, the described functionals do not depend on the generalized limit ω.

Group cohomology, modular theory and space-time symmetries

Abstract We give a necessary and sufficient condition on a positive compact operatorTfor the existence of a singular trace (i.e. a trace vanishing on the finite rank operators) which takes a finite non-zero value onT. This generalizes previous results by Dixmier and Varga. We also give an explicit description of these traces and associated ergodic states onl∞( N ) using tools of non standard analysis in an essential way.

A trace on fractal graphs and the Ihara zeta function

The Bisognano-Wichmann property on the geometric behavior of the modular group of the von Neumann algebras of local observables associated to wedge regions in Quantum Field Theory is shown to provide an intrinsic sufficient criterion for the existence of a covariant action of the (universal covering of) the Poincare group. In particular this gives, together with our previous results, an intrinsic characterization of positive-energy conformal pre-cosheaves of von Neumann algebras. To this end we adapt to our use Moore theory of central extensions of locally compact groups by polish groups, selecting and making an analysis of a wider class of extensions with natural measurable properties and showing henceforth that the universal covering of the Poincare group has only trivial central extensions (vanishing of the first and second order cohomology) within our class.