annuaire partie 4 pdf - Cours et travaux - Collège - Collège de France

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RÉSUMÉS DES COURS ET CONFÉRENCES
particle interactions. In contrast, general relativity appears to be incompatible
with quantum mechanics. This is related to that the gauge symmetry group that
governs the standard model has a different status than the diffeomorphism invariance
group of the Einstein action. This indicates that the quest to find a consistent
quantum theory of gravity is intimately related to the problem of unification
of all fundamental interactions. At low energies gravity is a classical theory, and
it is only at very high energies of the order of Planck mass that gravity would
manifest its quantum nature. It will then be essential to quantize gravity with
the other three forces at the Planck scale. The search for a unified theory then
necessarily involves finding a common origin for both types of gauge and diffeomorphism
symmetry, and possibly a new geometry. At such high energies there
are many indications that the geometry of space-time gets modified and is no
longer described by Riemannian geometry. The most natural and elegant choice
is to use the noncommutative geometry as formulated by Alain Connes. To
discover the true geometry of space-time we look at the physical particles at
low-energy comprising of quarks and leptons as the spectrum of some Dirac
operator. Indeed spaces in noncommutative geometry are defined by specifying
the spectral triple (A, H, D), where A is an involutive algebra of operators in
Hilbert space H and of a self-adjoint operator D in H. The phenomenological
approach to determine the structure of space-time is then to construct the geometry
from the spectrum of the low-energy sector defined by the Hilbert space of
quarks and leptons and the Dirac operator acting on them. I show that this
strategy can be applied successfully to show that our space-time has a hidden
noncommutative structure given by the product of a continuous four-dimensional
Riemannian manifold times a discrete finite internal noncommutative space,
based on matrix algebras. Although the finite internal space is not uniquely
defined, it satisfies severe consistency conditions from noncommutative geometry.
By applying the spectral action principle which states that the physical action
depends only on the spectrum, I show that the resultant theory is a unification
of general relativity with the standard model of particle interactions. The spectral
action reproduces all the interactions of the standard model, but with a prediction
of the Higgs particle mass. There is no fine tuning in the spectral action and all
terms in the Lagrangian come out naturally with the right signs and orders. This
gives a very strong support to the idea that our space-time is noncommutative
and that the Higgs field, in analogy with the Kaluza-Klein ideas corresponds to
the components of the gauge fields along discrete directions. Further support to
this idea comes by allowing the Dirac operator to depend on some scale through
a dilaton field. I show that the resulting standard model is almost scale invariant
and reproduces the Randal-Sundrum mechanism to generate the low-energy scale
from the Planck scale. These results are very encouraging to take the noncommutative
approach seriously and to use it in an attempt to discover the nature of
space-time and particle interactions at very high energies.
These lectures are intended to give hands on approach and derive in detail the
above mentioned results. Most of the material to be presented is based on joint
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