Noncommutative geometry

Slides (PDF, 0.65 MB)

Abstract

Who ordered the Higgs?
Noncommutative geometry.
Who is she?
The geometry of curved spaces equipped with an uncertainty relation.
What has she to do in physics?
She pretends to derive the electromagnetic, weak and strong forces as
pseudo-forces of gravity.

In detail:

By definition, a pseudo-force can be reduced to a simpler force by a
transformation. For centrifugal and Coriolis forces the
transformation is a rotation, the simpler force is zero. For some
magnetic forces the transformation is a Loretz-transformation, the
simpler force is electro-static. The equivalence principle states
that locally gravitational forces are pseudo-forces. Its
transformations are accelerations, i.e. general coordinate
transformations.

The above transformations form groups, in the three examples even
automorphism groups of geometries. The three geometries are the
Euclidean one of Newtonian mechanics, Minkowskian geometry of special
relativity and Riemannian geometry of general relativity.

These lectures are about a fourth example: the pseudo-force is a very
special Yang-Mills-Higgs force, as the electro-magnetic, weak and
strong force. Its transformations are gauge transformations, the
simple force is gravity. The underlying geometry is noncommutative
geometry.

Noncommutative geometry was discovered by Alain Connes in the 80ies
and allows to equip curved Riemannian geometries with an uncertainty
relation. According to Connes, a noncommutative geometry is defined
by a spectral triple consisting of a (noncommutative) algebra, a
representation and a Dirac operator.

Historically the first example of a noncommutative geometry was
quantum mechanics, however without curvature. The algebra is the
algebra of observables and its representation is on the Hilbert space
of wave functions.

The algebra of a finite dimensional spectral triple is a matrix
algebra. Its automophisms form the gauge group. The representation is
defined on the Hilbert space of fermions. The Dirac operator induces
the Higgs representation, spontaneous symmetry breaking and the
fermion masses.

Thomas Schücker

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