Noncommutative Geometry

Abstract

We propose a geometrized Higgs mechanism based on the gravitational sector in the Connes-Lott formulation of the standard model, which has been constructed by Chamseddine, Fröhlic and Grandjean. The point of our idea is that Higgs-like couplings depend on the local coordinates of the four-dimensional continuum, M 4 . The localized couplings can be calculated by the Wilson loops of the U (1) EM gauge field and the connection, which is defined on Z 2 × M 4 .

Quantum Gravity: Unification of Principles and Interactions, and Promises of Spectral Geometry

“…The Closing round table of the International Congress of Mathematicians (Madrid, August [22][23][24][25][26][27][28][29][30]2006) was devoted to the topic Are pure and applied mathematics drifting apart? As panelist, Manin subdivided the mathematization, i.e., the way mathematics can tell us something about the external world, into three modes of functioning (similarly Bohle, Booß and Jensen 1983, [10], see also [13]):…”

Functoriality of groupoid quantales. I

“…In order to see this it suffices to notice that Gelfand duality yields a contravariant functor from compact Hausdorff spaces to C*-algebras, whereas the universal C*-algebra of a discrete group defines a covariant functor. An interpretation of this discrepancy is that a groupoid C*-algebra can be regarded as a description of the space of orbits (in a generalized sense) of the groupoid and that groupoid functors fail to account for this [9]. In addition, for two such spaces to be considered "the same" one usually requires the algebras to be only Morita equivalent rather than isomorphic.…”

Projective Systems of Noncommutative Lattices

“…As it is evident in those Figures, the values of a function at points which cannot be separated by the topology differ by a compact operator. This is an illustration of the fact that compact operators play the rôle of 'infinitesimals' as is discussed at length in [11]. Furthermore, while in Figs.…”

“…In fact [11], the correct way of thinking of any noncommutative C * -algebra A is as the module of sections of the 'rank one trivial vector bundle' over the associated noncommutative space. For the kind of noncommutative lattices we are interested in, it is possible to explicitly construct the bundle over the lattice.…”

“…Forms ω for which this happens are called junk forms. They generate a Z-graded ideal in Ω * A and have to be quotiented out [11,22]. Then the noncommutative differential algebra is represented by the quotient space…”