Modular Covariance, PCT, Spin and Statistics

The notion of modular covariance is reviewed and the reconstruction of the Poincar\'e group extended to the low-dimensional case. The relations with the PCT symmetry and the Spin and Statistics theorem are described.Comment: 15 pages, plain TeX, talk presented to the Colloquium "New Problems in the General Theory of Fields and Particles", Paris 1994. To appear in the special issue of the Ann. Inst. H. Poincar\'e devoted to the Colloquiu

Noncommutative Riemann integration and and Novikov-Shubin invariants for open manifolds

Given a C*-algebra A with a semicontinuous semifinite trace tau acting on the Hilbert space H, we define the family R of bounded Riemann measurable elements w.r.t. tau as a suitable closure, a la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions, and show that R is a C*-algebra, and tau extends to a semicontinuous semifinite trace on R. Then, unbounded Riemann measurable operators are defined as the closed operators on H which are affiliated to A'' and can be approximated in measure by operators in R, in analogy with unbounded Riemann integration. Unbounded Riemann measurable operators form a tau-a.e. bimodule on R, denoted by R^, and such bimodule contains the functional calculi of selfadjoint elements of R under unbounded Riemann measurable functions. Besides, tau extends to a bimodule trace on R^. Type II_1 singular traces for C*-algebras can be defined on the bimodule of unbounded Riemann-measurable operators. Noncommutative Riemann integration, and singular traces for C*-algebras, are then used to define Novikov-Shubin numbers for amenable open manifolds, show their invariance under quasi-isometries, and prove that they are (noncommutative) asymptotic dimensions.Comment: 34 pages, LaTeX, a new section has been added, concerning an application to Novikov-Shubin invariants, the title changed accordingl

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

Given a spectral triple (A,D,H), the functionals on A of the form a -> tau_omega(a|D|^(-t)) are studied, where tau_omega is a singular trace, and omega is a generalised limit. When tau_omega 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 t's for which there exists a singular trace tau_omega giving rise to a non-trivial functional is an interval containing d. Moreover, the endpoints of such traceability interval have a dimensional interpretation. The corresponding functionals 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 omega.Comment: latex, 36 pages, no figures, to appear on Journ. Funct. Analysi

A Converse Hawking-Unruh Effect and dS^2/CFT Correspondance

Given a local quantum field theory net A on the de Sitter spacetime dS^d, where geodesic observers are thermalized at Gibbons-Hawking temperature, we look for observers that feel to be in a ground state, i.e. particle evolutions with positive generator, providing a sort of converse to the Hawking-Unruh effect. Such positive energy evolutions always exist as noncommutative flows, but have only a partial geometric meaning, yet they map localized observables into localized observables. We characterize the local conformal nets on dS^d. Only in this case our positive energy evolutions have a complete geometrical meaning. We show that each net has a unique maximal expected conformal subnet, where our evolutions are thus geometrical. In the two-dimensional case, we construct a holographic one-to-one correspondence between local nets A on dS^2 and local conformal non-isotonic families (pseudonets) B on S^1. The pseudonet B gives rise to two local conformal nets B(+/-) on S^1, that correspond to the H(+/-)-horizon components of A, and to the chiral components of the maximal conformal subnet of A. In particular, A is holographically reconstructed by a single horizon component, namely the pseudonet is a net, iff the translations on H(+/-) have positive energy and the translations on H(-/+) are trivial. This is the case iff the one-parameter unitary group implementing rotations on dS^2 has positive/negative generator.Comment: The title has changed. 38 pages, figures. To appear on Annales H. Poincare

Zeta functions for infinite graphs and functional equations

The definitions and main properties of the Ihara and Bartholdi zeta functions for infinite graphs are reviewed. The general question of the validity of a functional equation is discussed, and various possible solutions are proposed.Comment: 23 pages, 3 figures. Accepted for publication in "Fractals in Applied Mathematics", Contemporary Mathematics, Editors Carfi, Lapidus, Pearse, van Frankenhuijse

Inclusions of second quantization algebras

In this note we study inclusions of second quantization algebras, namely inclusions of von Neumann algebras on the Fock space of a separable complex Hilbert space H, generated by the Weyl unitaries with test functions in closed, real linear subspaces of H. We show that the class of irreducible inclusions of standard second quantization algebras is non empty, and that they are depth two inclusions, namely the third relative commutant of the Jones' tower is a factor. When the smaller vector space has codimension n into the bigger, we prove that the corresponding inclusion of second quantization algebras is given by a cross product with R^n. This shows in particular that the inlcusions studied in hep-th/9703129, namely the inclusion of the observable algebra corresponding to a bounded interval for the (n+p)-th derivative of the current algebra on the real line into the observable algebra for the same interval and the n-th derivative theory is given by a cross product with R^p. On the contrary, when the codimension is infinite, we show that the inclusion may be non regular (cf. M. Enock, R. Nest, J. Funct. Anal. 137 (1996), 466-543), hence do not correspond to a cross product with a locally compact group.Comment: LaTex, 9 pages, requires cmsams-l.cl

Spectral Properties of Wedge Problems

This paper presents our recent results on the study of the scattering and diffraction of an incident plane wave by wedge structures. A review about the impenetrable wedge problem at skew incidence and about the penetrable wedge at normal incidence is discussed. In particular we focus the attention on the spectral properties of the solution in the angular domain. These studies seem to provide a new tool to enhance the fast computation of the solution in terms of fields via a quasi-heuristic approac