Axioms for nonunital spectral triples, extending those introduced in the unital case by Connes, are proposed. As a guide, and for the sake of their importance in noncommutative quantum field theory, the spaces R 2N endowed with Moyal products are intensively investigated. Some physical applications, such as the construction of noncommutative Wick monomials and the computation of the Connes-Lott functional action, are given for these noncommutative hyperplanes.
In this paper we prove that the one dimensional Schrόdinger operator on 1 2 (Z) with potential given by:has a Cantor spectrum of zero Lebesgue measure for any irrational α and any λ>0. We can thus extend the Kotani result on the absence of absolutely continuous spectrum for this model, to all xeT.
We classify all irreducible, almost commutative geometries whose spectral action is dynamically non-degenerate. Heavy use is made of Krajewski's diagrammatic language. The motivation for our definition of dynamical non-degeneracy stems from particle physics where the fermion masses are non-degenerate.
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