88 research outputs found
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
From Monge to Higgs: a survey of distance computations in noncommutative geometry
This is a review of explicit computations of Connes distance in
noncommutative geometry, covering finite dimensional spectral triples,
almost-commutative geometries, and spectral triples on the algebra of compact
operators. Several applications to physics are covered, like the metric
interpretation of the Higgs field, and the comparison of Connes distance with
the minimal length that emerges in various models of quantum spacetime. Links
with other areas of mathematics are studied, in particular the horizontal
distance in sub-Riemannian geometry. The interpretation of Connes distance as a
noncommutative version of the Monge-Kantorovich metric in optimal transport is
also discussed.Comment: Proceedings of the workshop "Noncommutative Geometry and Optimal
Transport", Besan\c{c}on november 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
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
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
Twisted spectral triple for the Standard Model and spontaneous breaking of the Grand Symmetry
Grand symmetry models in noncommutative geometry have been introduced to
explain how to generate minimally (i.e. without adding new fermions) an extra
scalar field beyond the standard model, which both stabilizes the electroweak
vacuum and makes the computation of the mass of the Higgs compatible with its
experimental value. In this paper, we use Connes-Moscovici twisted spectral
triples to cure a technical problem of the grand symmetry, that is the
appearance together with the extra scalar field of unbounded vectorial terms.
The twist makes these terms bounded and - thanks to a twisted version of the
first-order condition that we introduce here - also permits to understand the
breaking to the standard model as a dynamical process induced by the spectral
action. This is a spontaneous breaking from a pre-geometric Pati-Salam model to
the almost-commutative geometry of the standard model, with two Higgs-like
fields: scalar and vector.Comment: References updated, misprint corrected. One paragraph added at the
end of the paper to discuss results in the literature since the first version
of the paper. 39 pages in Mathematical Physics, Analysis and Geometry (2016
On twisting real spectral triples by algebra automorphisms
We systematically investigate ways to twist a real spectral triple via an
algebra automorphism and in particular, we naturally define a twisted partner
for any real graded spectral triple. Among other things we investigate
consequences of the twisting on the fluctuations of the metric and possible
applications to the spectral approach to the standard model of particle
physics.Comment: References updated, minor corrections, result on the unicity of the
minimal twist for manifolds strengthene
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