Daniel Kastler

The dirac operator and gravitation

We give a brute-force proof of the fact, announced by Alain Connes, that the Wodzicki residue of the inverse square of the Dirac operator is proportional to the Einstein-Hilbert action of general relativity. We show that this also holds for twisted (e. g. by electrodynamics) Dirac operators, and more generally, for Dirac operators pertaining to Clifford connections of general Clifford bundles.

On the universal Chamseddine–Connes action. I. Details of the action computation

We give a detailed computation of the bosonic action of the Chamseddine–Connes model which we performed using different techniques.

Lie-Cartan pairs

Abstract The algorithms common to exterior derivation, exterior covariant derivation and vector valued cohomology of Lie-Algebras are presented within a unified frame.

The standard model à la Connes-Lott

Abstract The relations among coupling constants and masses in the standard model a la Connes-Lott with general scalar product are computed in detail. We find a relation between the top and the Higgs masses. For mt = 174 ± 22 GeV it yields mH = 277 ± 40 GeV. The Connes-Lott theory privileges the masses mt = 160.4 GeV and 251.8 GeV.

Relaxing the clustering condition in the derivation of the KMS property

We consider as in [1] an infinite dynamical system idealized as aC*-algebra acted upon by time-translation automorphisms. We show that a stationary state of such a system which is stable for local perturbations of the dynamics and is clustering in time, either gives rise to a one-sided energy spectrum or is a KMS state. The clustering property assumed here is weaker than the one assumed in [1]. The new proof makes explicit use of spectral properties of clustering states.

Fuzzy Mass Relations for the Higgs

The noncommutative approach of the standard model produces a relation between the top and the Higgs masses. We show that, for a given top mass, the Higgs mass is constrained to lie in an interval. The length of this interval is of the order of m2τ/mt.

Noncommutative Geometry and Basic Physics

Alain Connes’ noncommutative geometry, started in 1982 [0], widely develo- ped in 1994 as expounded in his book at this date [0] (it has grown meanwhile) is a systematic quantization of mathematics parallel to the quantization of physics effected in the twenties.This theory widens the scope of mathematics in a manner congenial to physics, reorganizes the existing (“classical”) mathematics of which it produces an hitherto unsuspected unification, and provides basic physics (the synthesis of elementary particles and gravitation) with a programme of renewal which has thus far achieved a clarification of the classical (tree-level) aspects of a new synthesis of the (Euclidean) standard model with gravitation [32],[33]: this is the subject of the present lectures— with the inherent tentative prediction of the Higgs mass.

Noncommutative geometry and fundamental physical interactions: The Lagrangian level—Historical sketch and description of the present situation

These notes comprise (i) a descriptive account of the history of the subject showing how physics and mathematics interwove to develop a mathematical concept of quantum manifold relevant to elementary particle theory; (ii) a detailed technical description, from scratch, of the spectral action formalism and computation.

Central decomposition of invariant states applications to the groups of time translations and of Euclidean transformations in algebraic field theory

AbstractWith