Topos Quantum Logic and Mixed States

The topos approach to the formulation of physical theories includes a new form of quantum logic. We present this topos quantum logic, including some new results, and compare it to standard quantum logic, all with an eye to conceptual issues. In particular, we show that topos quantum logic is distributive, multi-valued, contextual and intuitionistic. It incorporates superposition without being based on linear structures, has a built-in form of coarse-graining which automatically avoids interpretational problems usually associated with the conjunction of propositions about incompatible physical quantities, and provides a material implication that is lacking from standard quantum logic. Importantly, topos quantum logic comes with a clear geometrical underpinning. The representation of pure states and truth-value assignments are discussed. It is briefly shown how mixed states fit into this approach.Comment: 25 pages; to appear in Electronic Notes in Theoretical Computer Science (6th Workshop on Quantum Physics and Logic, QPL VI, Oxford, 8.--9. April 2009), eds. B. Coecke, P. Panangaden, P. Selinger (2010

Combinatorial Hopf algebraic description of the multiscale renormalization in quantum field theory

We define in this paper several Hopf algebras describing the combinatorics of the so-called multi-scale renormalization in quantum field theory. After a brief recall of the main mathematical features of multi-scale renormalization, we define assigned graphs, that are graphs with appropriate decorations for the multi-scale framework. We then define Hopf algebras on these assigned graphs and on the Gallavotti-Nicol\`o trees, particular class of trees encoding the supplementary informations of the assigned graphs. Several morphisms between these combinatorial Hopf algebras and the Connes-Kreimer algebra are given. Finally, scale dependent couplings are analyzed via this combinatorial algebraic setting.Comment: 26 pages, 3 figures; the presentation of the results has been reorganized. Several details of various proofs are given and some references have been adde