Connes-Lott model building on the two-sphere

In this work we examine generalized Connes-Lott models on the two-sphere. The Hilbert space of the continuum spectral triple is taken as the space of sections of a twisted spinor bundle, allowing for nontrivial topological structure (magnetic monopoles). The finitely generated projective module over the full algebra is also taken as topologically non-trivial, which is possible over $S^2$. We also construct a real spectral triple enlarging this Hilbert space to include "particle" and "anti-particle" fields.Comment: 57 pages, LATE

The Connes-Lott program on the sphere

We describe the classical Schwinger model as a study of the projective modules over the algebra of complex-valued functions on the sphere. On these modules, classified by $\pi_2(S^2)$, we construct hermitian connections with values in the universal differential envelope which leads us to the Schwinger model on the sphere. The Connes-Lott program is then applied using the Hilbert space of complexified inhomogeneous forms with its Atiyah-Kaehler structure. It splits in two minimal left ideals of the Clifford algebra preserved by the Dirac-Kaehler operator D=i(d-delta). The induced representation of the universal differential envelope, in order to recover its differential structure, is divided by the unwanted differential ideal and the obtained quotient is the usual complexified de Rham exterior algebra over the sphere with Clifford action on the "spinors" of the Hilbert space. The subsequent steps of the Connes-Lott program allow to define a matter action, and the field action is obtained using the Dixmier trace which reduces to the integral of the curvature squared.Comment: 34 pages, Latex, submitted for publicatio

Non-commutative Quantum Mechanics in Three Dimensions and Rotational Symmetry

We generalize the formulation of non-commutative quantum mechanics to three dimensional non-commutative space. Particular attention is paid to the identification of the quantum Hilbert space in which the physical states of the system are to be represented, the construction of the representation of the rotation group on this space, the deformation of the Leibnitz rule accompanying this representation and the implied necessity of deforming the co-product to restore the rotation symmetry automorphism. This also implies the breaking of rotational invariance on the level of the Schroedinger action and equation as well as the Hamiltonian, even for rotational invariant potentials. For rotational invariant potentials the symmetry breaking results purely from the deformation in the sense that the commutator of the Hamiltonian and angular momentum is proportional to the deformation.Comment: 21 page

Efficient variational contraction of two dimensional tensor networks with a non trivial unit cell

Tensor network states provide an efficient class of states that faithfully capture strongly correlated quantum models and systems in classical statistical mechanics. While tensor networks can now be seen as becoming standard tools in the description of such complex many-body systems, close to optimal variational principles based on such states are less obvious to come by. In this work, we generalize a recently proposed variational uniform matrix product state algorithm for capturing one-dimensional quantum lattices in the thermodynamic limit, to the study of regular two-dimensional tensor networks with a non-trivial unit cell. A key property of the algorithm is a computational effort that scales linearly rather than exponentially in the size of the unit cell. We demonstrate the performance of our approach on the computation of the classical partition functions of the antiferromagnetic Ising model and interacting dimers on the square lattice, as well as of a quantum doped resonating valence bond state.Comment: 23 pages, 8 Figure

On Pythagoras' theorem for products of spectral triples

We discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes' distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non pure states and arbitrary spectral triples. We show that Pythagoras theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and provide non-unital counter-examples inspired by K-homology.Comment: Paper slightly shortened to match the published version; Lett. Math. Phys. 201

A Comparative Study of Different In Vitro Lung Cell Culture Systems to Assess the Most Beneficial Tool for Screening the Potential Adverse Effects of Carbon Nanotubes

To determine the potential inhalatory risk posed by carbon nanotubes (CNTs), a tier-based approach beginning with an in vitro assessment must be adopted. The purpose of this study therefore was to compare 4 commonly used in vitro systems of the human lung (human blood monocyte-derived macrophages [MDM] and monocyte-derived dendritic cells [MDDC], 16HBE14o- epithelial cells, and a sophisticated triple cell co-culture model [TCC-C]) via assessment of the biological impact of different CNTs (single-walled CNTs [SWCNTs] and multiwalled CNTs [MWCNTs]) over 24h. No significant cytotoxicity was observed with any of the cell types tested, although a significant (p < .05), dose-dependent increase in tumor necrosis factor (TNF)-α following SWCNT and MWCNT exposure at concentrations up to 0.02mg/ml to MDM, MDDC, and the TCC-C was found. The concentration of TNF-α released by the MDM and MDDC was significantly higher (p < .05) than the TCC-C. Significant increases (p < .05) in interleukin (IL)-8 were also found for both 16HBE14o- epithelial cells and the TCC-C after SWCNTs and MWCNTs exposure up to 0.02mg/ml. The TCC-C, however, elicited a significantly (p < .05) higher IL-8 release than the epithelial cells. The oxidative potential of both SWCNTs and MWCNTs (0.005-0.02mg/ml) measured by reduced glutathione (GSH) content showed a significant difference (p < .05) between each monoculture and the TCC-C. It was concluded that because only the co-culture system could assess each endpoint adequately, that, in comparison with monoculture systems, multicellular systems that take into consideration important cell type-to-cell type interactions could be used as predictive in vitro screening tools for determining the potential deleterious effects associated with CNT