Remarks on Alain Connes' approach to the standard model

Our 1992 remarks about Alain Connes' interpretation of the standard model within his theory of non-commutative riemannian spin manifolds.Comment: 9 pages TeX, dedicated to the memory of E. M. Polivano

Spectral noncommutative geometry and quantization: a simple example

We explore the relation between noncommutative geometry, in the spectral triple formulation, and quantum mechanics. To this aim, we consider a dynamical theory of a noncommutative geometry defined by a spectral triple, and study its quantization. In particular, we consider a simple model based on a finite dimensional spectral triple (A, H, D), which mimics certain aspects of the spectral formulation of general relativity. We find the physical phase space, which is the space of the onshell Dirac operators compatible with A and H. We define a natural symplectic structure over this phase space and construct the corresponding quantum theory using a covariant canonical quantization approach. We show that the Connes distance between certain two states over the algebra A (two ``spacetime points''), which is an arbitrary positive number in the classical noncommutative geometry, turns out to be discrete in the quantum theory, and we compute its spectrum. The quantum states of the noncommutative geometry form a Hilbert space K. D is promoted to an operator *D on the direct product *H of H and K. The triple (A, *H, *D) can be viewed as the quantization of the family of the triples (A, H, D).Comment: 7 pages, no figure

The uses of Connes and Kreimer's algebraic formulation of renormalization theory

We show how, modulo the distinction between the antipode and the "twisted" or "renormalized" antipode, Connes and Kreimer's algebraic paradigm trivializes the proofs of equivalence of the (corrected) Dyson-Salam, Bogoliubov-Parasiuk-Hepp and Zimmermann procedures for renormalizing Feynman amplitudes. We discuss the outlook for a parallel simplification of computations in quantum field theory, stemming from the same algebraic approach.Comment: 15 pages, Latex. Minor changes, typos fixed, 2 references adde

Cartan Pairs

A new notion of Cartan pairs as a substitute of notion of vector fields in noncommutative geometry is proposed. The correspondence between Cartan pairs and differential calculi is established.Comment: 7 pages in LaTeX, to be published in Czechoslovak Journal of Physics, presented at the 5th Colloquium on Quantum Groups and Integrable Systems, Prague, June 199

Quantum electrodynamics of relativistic bound states with cutoffs

We consider an Hamiltonian with ultraviolet and infrared cutoffs, describing the interaction of relativistic electrons and positrons in the Coulomb potential with photons in Coulomb gauge. The interaction includes both interaction of the current density with transversal photons and the Coulomb interaction of charge density with itself. We prove that the Hamiltonian is self-adjoint and has a ground state for sufficiently small coupling constants.Comment: To appear in "Journal of Hyperbolic Differential Equation

Fluctuation Operators and Spontaneous Symmetry Breaking

We develop an alternative approach to this field, which was to a large extent developed by Verbeure et al. It is meant to complement their approach, which is largely based on a non-commutative central limit theorem and coordinate space estimates. In contrast to that we deal directly with the limits of $l$-point truncated correlation functions and show that they typically vanish for $l\geq 3$ provided that the respective scaling exponents of the fluctuation observables are appropriately chosen. This direct approach is greatly simplified by the introduction of a smooth version of spatial averaging, which has a much nicer scaling behavior and the systematic developement of Fourier space and energy-momentum spectral methods. We both analyze the regime of normal fluctuations, the various regimes of poor clustering and the case of spontaneous symmetry breaking or Goldstone phenomenon.Comment: 30 pages, Latex, a more detailed discussion in section 7 as to possible scaling behavior of l-point function

COMMENTS ABOUT HIGGS FIELDS, NONCOMMUTATIVE GEOMETRY AND THE STANDARD MODEL

We make a short review of the formalism that describes Higgs and Yang Mills fields as two particular cases of an appropriate generalization of the notion of connection. We also comment about the several variants of this formalism, their interest, the relations with noncommutative geometry, the existence (or lack of existence) of phenomenological predictions, the relation with Lie super-algebras etc.Comment: pp 20, LaTeX file, no figures, also available via anonymous ftp at ftp://cpt.univ-mrs.fr/ or via gopher gopher://cpt.univ-mrs.fr