Modular Theory, Non-Commutative Geometry and Quantum Gravity

This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.Comment: Special Issue "Noncommutative Spaces and Fields

A Remark on Gelfand Duality for Spectral Triples

We present a duality between the category of compact Riemannian spin manifolds (equipped with a given spin bundle and charge conjugation) with isometries as morphisms and a suitable "metric" category of spectral triples over commutative pre-C*-algebras. We also construct an embedding of a "quotient" of the category of spectral triples introduced in arXiv:math/0502583v1 into the latter metric category. Finally we discuss a further related duality in the case of orientation and spin-preserving maps between manifolds of fixed dimension.Comment: 15 pages, AMS-LaTeX2e, results unchanged, several improvements in the exposition, appendix adde

Enriched Fell Bundles and Spaceoids

We propose a definition of involutive categorical bundle (Fell bundle) enriched in an involutive monoidal category and we argue that such a structure is a possible suitable environment for the formalization of different equivalent versions of spectral data for commutative C*-categories.Comment: 12 pages, AMS-LaTeX2e, to be published in "Proceedings of 2010 RIMS Thematic Year on Perspectives in Deformation Quantization and Noncommutative Geometry

A Category of Spectral Triples and Discrete Groups with Length Function

In the context of A. Connes' spectral triples, a suitable notion of morphism is introduced. Discrete groups with length function provide a natural example for our definitions. A. Connes' construction of spectral triples for group algebras is a covariant functor from the category of discrete groups with length functions to that of spectral triples. Several interesting lines for future study of the categorical properties of spectral triples and their variants are suggested.Comment: 23 pages, AMS-LaTeX2

Covariant Sectors with Infinite Dimension and Positivity of the Energy

We consider a Moebius covariant sector, possibly with infinite dimension, of a local conformal net of von Neumann algebras on the circle. If the sector has finite index, it has automatically positive energy. In the infinite index case, we show the spectrum of the energy always to contain the positive real line, but, as seen by an example, it may contain negative values. We then consider nets with Haag duality on the real line, or equivalently sectors with non-solitonic extension to the dual net; we give a criterion for irreducible sectors to have positive energy, namely this is the case iff there exists an unbounded Moebius covariant left inverse. As a consequence the class of sectors with positive energy is stable under composition, conjugation and direct integral decomposition.Comment: 25 pages, Latex2

an analytical method to simulate the dynamic performances of truncated cone helix ground heat exchangers

Abstract This paper proposes a dynamic analytical method to simulate the thermal performances of truncated cone helix ground heat exchangers (i.e., the so-called "energy baskets"). These ground-coupled devices are attractive solutions to reduce the initial cost of ground-coupled heat pump systems, as they require lower cost to be drilled and installed with respect to traditional boreholes. However, both design methodologies and performance assessment models are still not well developed, producing substantial uncertainties on final operative performances. This work presents a plain evaluation method based on the heat exchangers theory and the analytical solution of the truncated cone helix heat source in a semi-infinite medium. It can be advantageously used to simulate the thermal performance of truncated cone helix ground heat exchangers as a function of helix geometries and operative conditions evolution (e.g., inlet temperature, fluid flow rate, ground temperature…). Specifically, in this paper, we perform a sensitivity analysis of the thermal performances of a case study by varying the main geometrical parameters. Besides, we compare the heat transfer of the reference configuration with an equivalent cylindrical arrangment. The truncated coil configuration is more effective than cylindrical one as the cone aperture reduces the short-circuits between helix pitch and the equivalent thermal resistance with the ground surface. However, obtained results are notably affected by the assumption of an isothermal surface temperature, which leads to a shallow/plain helix/spiral as the best configuration: different conclusions are expected when a time dependent or adiabatic boundary condition will be accounted in the model