Beyond the Standard Model with noncommutative geometry, strolling towards quantum gravity

Noncommutative geometry, in its many incarnations, appears at the crossroad of various researches in theoretical and mathematical physics: from models of quantum space-time (with or without breaking of Lorentz symmetry) to loop gravity and string theory, from early considerations on UV-divergencies in quantum field theory to recent models of gauge theories on noncommutative spacetime, from Connes description of the standard model of elementary particles to recent Pati-Salam like extensions. We list several of these applications, emphasizing also the original point of view brought by noncommutative geometry on the nature of time. This text serves as an introduction to the volume of proceedings of the parallel session "Noncommutative geometry and quantum gravity", as a part of the conference "Conceptual and technical challenges in quantum gravity" organized at the University of Rome "La Sapienza" in September 2014

Twisted spectral geometry for the standard model

The Higgs field is a connection one-form as the other bosonic fields, provided one describes space no more as a manifold M but as a slightly non-commutative generalization of it. This is well encoded within the theory of spectral triples: all the bosonic fields of the standard model - including the Higgs - are obtained on the same footing, as fluctuations of a generalized Dirac operator by a matrix-value algebra of functions on M. In the commutative case, fluctuations of the usual free Dirac operator by the complex-value algebra A of smooth functions on M vanish, and so do not generate any bosonic field. We show that imposing a twist in the sense of Connes-Moscovici forces to double the algebra A, but does not require to modify the space of spinors on which it acts. This opens the way to twisted fluctuations of the free Dirac operator, that yield a perturbation of the spin connection. Applied to the standard model, a similar twist yields in addition the extra scalar field needed to stabilize the electroweak vacuum, and to make the computation of the Higgs mass in noncommutative geometry compatible with its experimental value.Comment: Proceedings of the seventh international workshop DICE 2014 "Spacetime, matter, quantum mechanics", Castiglioncello september 201

Smoother than a circle, or How non commutative geometry provides the torus with an egocentred metric

We give an overview on the metric aspect of noncommutative geometry, especially the metric interpretation of gauge fields via the process of "fluctuation of the metric". Connes' distance formula associates to a gauge field on a bundle P equipped with a connection H a metric. When the holonomy is trivial, this distance coincides with the horizontal distance defined by the connection. When the holonomy is non trivial, the noncommutative distance has rather surprising properties. Specifically we exhibit an elementary example on a 2-torus in which the noncommutative metric d is somehow more interesting than the horizontal one since d preserves the S^1-structure of the fiber and also guarantees the smoothness of the length function at the cut-locus. In this sense the fiber appears as an object "smoother than a circle". As a consequence, from a intrinsic metric point of view developed here, any observer whatever his position on the fiber can equally pretend to be "the center of the world".Comment: Short and non technical version of hep-th/0506147. Proceedings of the international conference on "differential geometry and its application", Deva, October 2005. Cluj university press (Romania

Wave-Function renormalization and the Hopf algebra of Connes and Kreimer

In this talk, we show how the Connes-Kreimer Hopf algebra morphism can be extended when taking into account the wave-function renormalization. This leads us to a semi-direct product of invertible power series by formal diffeomorphisms.Comment: 5 pages, no figure, talk presented in the conference "Brane New World and Noncommutative Geometry", Torino, Villa Gualino,(Italy) Octobe

Deformations of the Canonical Commutation Relations and Metric Structures

Using Connes distance formula in noncommutative geometry, it is possible to retrieve the Euclidean distance from the canonical commutation relations of quantum mechanics. In this note, we study modifications of the distance induced by a deformation of the position-momentum commutation relations. We first consider the deformation coming from a cut-off in momentum space, then the one obtained by replacing the usual derivative on the real line with the h- and q-derivatives, respectively. In these various examples, some points turn out to be at infinite distance. We then show (on both the real line and the circle) how to approximate points by extended distributions that remain at finite distance. On the circle, this provides an explicit example of computation of the Wasserstein distance

Emergence of time in quantum gravity: is time necessarily flowing ?

We discuss the emergence of time in quantum gravity, and ask whether time is always "something that flows"'. We first recall that this is indeed the case in both relativity and quantum mechanics, although in very different manners: time flows geometrically in relativity (i.e. as a flow of proper time in the four dimensional space-time), time flows abstractly in quantum mechanics (i.e. as a flow in the space of observables of the system). We then ask the same question in quantum gravity, in the light of the thermal time hypothesis of Connes and Rovelli. The latter proposes to answer the question of time in quantum gravity (or at least one of its many aspects), by postulating that time is a state dependent notion. This means that one is able to make a notion of time-as-an-abstract-flow - that we call the thermal time - emerge from the knowledge of both: 1) the algebra of observables of the physical system under investigation, 2) a state of thermal equilibrium of this system. Formally, this thermal time is similar to the abstract flow of time in quantum mechanics, but we show in various examples that it may have a concrete implementation either as a geometrical flow, or as a geometrical flow combined with a non-geometric action. This indicates that in quantum gravity, time may well still be "something that flows" at some abstract algebraic level, but this does not necessarily imply that time is always and only "something that flows" at the geometric level.Comment: Contribution to the Workshop "Temps et Emergence", Ecole Normale Sup\'erieure, Paris 14-15 october 2011. To be published in Kronoscope. Intended for a non-specialist audienc